Network Analysis and Feedback Amplifier Design HENDRIK W. BODE, Ph.D., Research Mathematician, BELL TELEPHONE LABOP. ATOKIE$ INc. TI/FELFTH PRINTING D. VAN NOSTRAND COMPANY, INC. PRINCETON, NEW JERSEY TORONTO LONDON NEWYORK ----------------------------------------------------------- D. VAN NOSTRAND COMPANY, iNC. 120 Alexander St., Princeton, New Jersey 257 Fourth Avenue, New York 10, New York 25 Hollinger Rd., Toronto 16, Canada .41I correspondewe sho,ld be addressed to the pri,c[pal olce of the compazy at Princetoy, N.J. CovyntcmT, 1945 BY D. VAN NOSTRAND COMPAXY, All rights in this book are 'eserved. I. Vitho,t written a,thorixatiol from D. Yau Nostra*d Compaly, [lC., 120 AIexa,der Street, Priceton, New Jersey, it llay not be reprod,ced i, any form il whole or il part (except for q,otatio in critical articles or reviews), 1lOt -may it be used for dramatic, motira1-, talki,g-pictlv'e, radio. televisio1 or al 3. other similar purpose. Firsl Fublished Sep:ember 1945 Reprbted Jamt::r! 196, Febrtary 1947 October 197, April 1949, October I950 September 195I, July 1952, December 1953 Jamtary 1955, April I956, July 1957 Produced by 'TECHNICAL COMPOI5ITION CO. BOSTON, MASS. PRINTED IN THE UNITED STATES OF AMERICA ----------------------------------------------------------- PREFACE This book was originally written as an informal mimeographed text for one of the so-called "Out-of-Hour" courses at Bell Telephone Labora- tories. The bulk of the material was prepared in 1938 and 1939 and was given in course form to my colleagues there in the winters of 1939-40 and 1940-41. During the war, however, the text has also been supplied as a reference work to a considerable number of other laboratories en- gaged in war research. The demand for the text on this basis was un- expectedly heavy and quickly exhausted the original supply of mimeo- graphed copies. It has consequently been decided to make the text more widely available through regular channels of publication. In revising the material for publication, the original theoretical dis- cussion has been supplemented by footnote references to other books and papers appearing both before and after the text was first written. In addition, an effort has been made to simplify the theoretical treatment in Chapter IV, and minor editorial changes have been made at a number of points elsewhere. Otherwise, however, the text is as it was originally written. The book was first planned as a text exclusively on the design of feed- back amplifiers. It shortly became apparent, however, that an ex:ensive preliminary development of electrical network theory would be necessary before the feedback problem could be discussed satisfactorily. With the addition of other logically related chapters, this has made the book pri- marily a treatise on general network theory. The feedback problem is still conspicuous, but the book also contains material on the design of non-feedback as well as feedback amplifiers, particularly those of wide band type, and on miscellaneous transmission problems arising in wide band systems generally. Much of this is material which has not hitherto appeared in previous texts on network theory. On the other hand, trans- mission line and filter theory, which are the primary concerns of most earlier network texts, are omitted. Two further explanatory remarks may be helpful in understanding the book. The first is the fact that, although the feedback amplifiers en- visaged in most of the discussion are of the conventional single loop, absolutely stable type, the original plan for the text called for two final chapters on design methods appropriate for multiple loop and condition- ally stable circuits. Invincible fatigue set in before these chapters could iii ----------------------------------------------------------- iv PREFACE be written. In anticipation of these chapters, however, the preliminary nalysis in the early portions of the book was carried forward in more general terms than would otherwise have been necessary. In Chapters IV-VI, particularly, this appreciably complicates the discussion, and the reader interested only in conventional feedback amplifiers can afford to omit the more difficult portions of these chapters. The second general remark concerns the apparently unnecessary re- finemerit to which the design methods described in the book are sometimes carried. This is explained by the fact that the amplifiers of particular interest to the class for which the notes were originally prepared were those used as repeaters in long distance telephone systems. Since a long system may include many repeater points, the cumulative effect of even quite small imperfections in individual amplifiers may be serious. Thus, the amplifier design requires more care than might be justified in an ordi- nary engineering application. Under the circumstances in which the text was originally prepared, it naturally benefited by suggestions from many sources. I am indebted for such help to too many of my colleagues to enumerate individually. Special mention should, however, be made of Mrs. S. P. Mead for her assistance in the final preparation of the material for publication. It is a particular pleasure also to express my thanks to Dr. Thornton C. Fry, without whose support and encouragement the book could scarcely have been written. H. W. BODE Bell Telephone Laboratories, Inc. New York City April 1945 ----------------------------------------------------------- CONTENTS CHAPTER PAGE I. MESH AND NODAL EQUATIONS FOR AN ACTIVE CIRCUIT .... II. THE COMVLEX FREQUENCY PLANE .......... 18 IlL FœEDBACC .................. 31 IV. MATHEMATICAL DEFINITION OF FEEDBACK ........ V. GgnXL Togus roe FEEDBACK CIRCUITS  A ..... 66 VI. Gu Tos vo FgDC CICUXTs--B ..... 83 VII. STXSITV AnD PisIcxn RgxuIZXnIITr ......... 103 VIII. Coxoua ITEGKATON AND NYqUIST'S CTEO FOK STABILITY . 137 IX. PHYSICAL REPaESENTATIO OF DgIVING PoI IMPEDANCE FUNC- TIONS .................. 170 X. ToPIcs i THE DESIGN OF IMPEDANCg FUNIONS ...... 1 XI. PHYSICAL REPRESENTATION OF TRANSFEK IMPEDANCE FUNIONS 226 'XII. Tocs  x DEsiGn or Equuzs ......... 249 XIII. GENERAL RESTRICTIONS ON PHYSICAL NETWORK CHAEKISTICS AT Rxn FREQUENCIES .............. 276 XIV. RLATIONS TW REAL AND IxGIaKY COMPONENTS OF NET- WO FuNexIOS ............... 303 exg Co.rodenTS or Ngxwo Fueo,s ....... 337 I, APnICATION Or GENERAL THEOREMS TO INPUT AND OUTVUT CIRCUIT DsIan .................. 360 XVII. Avvuxcxo or Ggegxu Toms To INTERSTAGE NETWORK Dsoe .................. 403 III. DsIaN oF SINOL Loov AaSOUUTy STXSU Avungs 451 XIX. IuusgxIW Dsss o Ssu Looe Fo.c vuvgzs . 489 IEx ................... 531 ----------------------------------------------------------- LIST OF SYMBOLS Norm: Asterisks indicate relatively unimportant symbols or subsidiary meanings restricted to one section of the book. d real component of 0 230, 278 *d, B, C, .. ß nodes 1 //, maximum obtainable feedback, under different assumptions 466, 477, 484 At circuit loss 477 do loop feedback in the useful band 455 B imaginary component of 0 '230, 278 *B number of branches in a network 3 C generic symbol for capacity 1 D generic symbol for stiffness or reciprocal capacity 2 *Da, Do, De,... branch stiffnesses 2 Dii mutual stiffness, i  j 4 Dis self-stiffness, i = j 4 E generic symbol for voltage 2, 12 *Ex, EB, Ec, ß ß ß node voltages 2 ER output voltage 32 Ei impressed voltage in mesh i, i = 1, 2,..-, n 4 Ei response voltage on node i, i = 1, 2, ..., n 10 E0 input voltage 22, 32 *El output voltage, of tube 404 *El "returned" voltage 32 *E 0 fi-circuit output voltage 385 *E /-circuit input voltage 385 F return difference or feedback 47 F(k) return difference of an element/3, when ' /7/= k 50 F/(/) return difference of/47 for reference k 47 vii ----------------------------------------------------------- viii LIST OF SYMBOLS F(0) *F *F(a) or F *F(b) or Fb G Gij' I 'I, Ib, I,,. ß ß *lib *œ1 L *L,, Lb, L,, ' ß ß LO' *N *N *p return difference of/V, when a given pair of terminals in the network is short- circuited return difference of///, when the given pair of terminals is open-circuited dissipation function value of dissipation function obtained by usihg real components of complex cur- rent coefficients as instantaneous physical currents value of dissipation function obtained by using imaginary components of complex current coefficients as instan- taneous physical currents average value of dissipation function with respect to time generic symbol for conductance transconductance, equal to /Ro mutual conductance generic symbol for current output or receiver current instantaneous branch currents response current in mesh i, i = 1, 2, ...,r impressed current on node i, i = 1, steady-state mesh current in mesh j; coefficient of exp (icoO real component of Ij imaginary component of œj input current plate current IS-circuit input current /-circuit output current generic symbol for inductance branch inductances mutual inductance, i  j self-inductance, i = j number of nodes in a network number of zeros within a contour number of poles within a contour teacrance-resistance ratio Page 67 67 127 133, 171 133, 171 130 13 14 13 19 385 2 11 8 129 129 22 404 385 385 1 2 4 4 3 148 148 219, 367 ----------------------------------------------------------- LIST OF SYMBOLS ix R Rv *Ra, Rb, Re, ß ß ß R v and Rg RO Ro Ro *R0 R and R2 S S' *Sra T *T *T(a) or T *T(b) or Tb *V(a) or V *?(b) or k% generic symbol for resistance transfer resistance, real component of Zv branch resistances plate and grid resistances, respectively mutual resistance, i  j self-resistance, i = j image impedance of a constant-R net- work internal tube resistance terminating resistance terminating resistances sensitivity relative sensitivity transconductance return ratio stored magnetic energy function value of stored magnetic energy function obtained by using real components of complex current coefficients as instan- taneous physical currents value of stored magnetic energy function obtained by using imaginary compo- nents of complex current coefficients as instantaneous physical currents average value of stored magnetic energy function with respect to time stored electric energy function value of stored electric energy function obtained by using real components of complex current coefficients as instan- taneous physical currents value of stored electric energy function obtained by using imaginary compo- nents of complex current coefficients as instantaneous physical currents average value of stored electric energy function with respect to time generic symbol for immittance; symbol for a selected unilateral or bilateral element in the network output immittance 1 433 2 404 4 4 241 14 385 227 52 63 156 48 127 133, 171 133, 171 130 127 133, 171 133, 171 133, 171 15, 48, 51 53 ----------------------------------------------------------- x LIST OF SYMBOLS X Y Yv Z zI Zo Z and Z v Zo f A andfb fc andfa A i, k k *k and k *k2 and k transfer immittance reference value of an element/4 /4 / -- /4/0, the effective value of generic symbol for teacrance generic symbol for admittance transfer admittance mutual admittance, j / k self-admittance, j = k generic symbol for impedance image impedance transfer impedance grid impedance arbitrary impedance added to the nth mesh branch impedances of the lattice mutual impedance, j 5 k self-impedance, j = k circuit impedance, when a prescribed ele- ment vanishes branches zeros of Z:,, i = 1, 2,'.' , n poles ofZT, i= 1,2,'.',n voltage ZoI across the grid instantaneous voltage in an element frequency in cycles per second horizontal step frequencies in loop cut- off characteristic horizontal step frequencies with gain and phase margins frequency at edge of useful band frequency at which excess phase for feed- back loop is equal to 2n/,r radians frequency of equal resistance and react- ance in a capacity-resistance network figure of merit frequency instantaneous current filter type arbitrary reference value of/4 / volume performance characteristics contributions of input and output cir- cuits to loop transmission character- istic 25 61 62 206 15 15 11 11 8 231 9 6 68 231 4 4 67 1 23O 230 6 126 22 465 467 455 484 511 477 126 373 49 385 385 ----------------------------------------------------------- LIST OF SYMBOLS xi *ka and k m p and lip P *p. and p q, q, Y :got *q/I, "t'2, 'Y3, o 'o *OA O0 *X transmission characteristics between  circuit and input and output lines, respectively parameter in filter theory slope of asymptote in units of 6 db per octave differential and integral operators, re- spectively zeros and poles of immittance, respec- tively charge on condenser time gain margin in db phase margin, expressed as a fraction of rr radians transmission characteristic of subsidiary feedback path transmission characteristic of backward or feedback path or the backward cir- cuit itself transmission characteristics of tube cir- cuit with tube dead generic symbol for complex quantity representing transmission, i.e. loss (or gain) and phase shift. By extension, symbol for a general complex network function (= z/+ lB) transfer loss and phase; image transfer constant (of a constant-resistance structure) phase angle of driving-point impedance fractionareal gain,/F = 0 fractionated gain,/F = /4z0 gain from point z/to point B direct transmission gain ratio of number of stages to the optimum number of stages in a feedback ampli- tier amplification ratio of a vacuum tube transmission characteristic of forward circuit or the forward circuit itself 385 373 459 2 22 25 126 8 453 453 158 31 78 45, 53, 278 230, 231 171 81 87 100 53 478 14 32 ----------------------------------------------------------- xii LIST OF SYMBOLS p *Fg./ A A  A o A  Aij, Aijb etc. *'o L œo reflection coefftclent frequency variable, sin -t co/coo frequency variable, sin -t coo/co angular velocity, 2rf reciprocal inductance determinant of the mesh equations value of A when/V = k value of A when H/= 0 value of A when/V' = 0 or determinant of the nodal equations cofactors of A analytic function, A + lB, representing some network characteristic value of,I, when dissipation is neglected integration in the complex plane, over a contour C integration in the complex plane, around a circle integration around a semicircle in the right half-plane, centered at the origin and terminating on the imaginary axis integration around the closed path con- sisting of the semicircle above and the included portion of the imaginary axis (with indentations if necessary to ex- clude singularities on the imaginary axis) Page 364 415 419 8 13 8 50 48 62 15 8-10 218 219 140 141 144 144 ----------------------------------------------------------- CHAPTER I MESH AND NODAL ](UATIONS FOR AN ACTIVE CIRCUIT 1.1 2rntroduction TE networks to be considered consist of ordinary lumped inductances, resistances, and capacities, together with vacuum tubes. The accessible terminals of the vacuum tubes will be taken as the grid, plate, and cathode. Auxiliary electrodes, such as a suppressor or screen grid, are thus ignored, and the analysis assumes, in effect, that they are grounded to the cathode at signal frequencies. For purposes of discussion the tubes will be replaced by equivalent structures consisting of ordinary circuit elements connected between the accessible terminals, together with a source of current or volt- age to represent the amplification of the tube. This ignores such effects as transit time and distributed inductance in the wires inside the tube envelope, which may appear in physical tubes at sufficiently high frequencies. ' It will be assumed throughout that all the elements are linear. This chapter is intended principally as  recapitulation of the conventional theory for networks including vacuum tubes in a form which can be used as a foundation for the chapters to follow.* 1.2. Branch Equations for a Passive Circuit It is simplest to begin by ignoring the active elements in the circuit. The network can then be regarded as an arrangement of individual branches, which may include any combination of the elements R, C, and L in series, connected together at various junctions or nodes. An example is shown by Fig. 1.1. The circuit contains six branches, as indicated by the subscripts a ß ß -f, and four nodes represented by the points At. ß ß D. Generators to furnish the driving forces on the circuit are shown in three of the branches. ß A good general reference to the mesh analysis of passive networks is Guillemin "Communication Networks," Vol. I. See also Shea "Transmission Networks and Wave Filters" for a brief discussion emphasizing the stock theorems, such as the superposition theorem, reciprocity theorem, and Thvenin's theorem, which follow readily from the mesh analysis. The theorem on the use of an equivalent plate generator to represent the amplification of a vacuum tube, on which the extension of the mesh analysis to active circuits depends, is described in most books on radio engineering. See, e.g., Terman "Radio Engineering" or "Radio Engineer's Hand- book," or Everitt" Communication Engineering." ----------------------------------------------------------- 2 NETWORK ANALYSIS c,^P. t The condensers are specified in units of stiffness, or reciprocal capacity, D = I/C, in order to simplify later equations. Each branch has been shown as including all three types of elements but in an actual network many of the elements might, of course, be omitted. Fundamental expressions for the analysis of such a network can be set up by equating the instantaneous voltage drops in each branch of the net- work to the voltage applied to that branch. 'For example, if Is represents t FIG. l. [ the instantaneous current in the first branch of Fig. 1.1, the voltages across the individual elements of that branch are RIa, pLaI, and (1/p)Df, where p and lip represent respectively differentiation and integration with respect to time. The sum of the voltage drops through these three ele- ments must be equal to the voltage of the generator E plus the difference between the voltages at the nodes//and D at which the branch terminates. If we let E:t and Ez) represent the node voltages, we therefore have There will be one equation similar to (1-1) for each branch of the net~ work, or B equations in all if B represents the number of branches. In addition to these equations, however, further equations follow from the fact that, since no electrical charge can accumulate at any node, the sum of the instantaneous currents leaving each node must be equal to the sum of the currents entering it. In Fig. 1.1, for example, this leads to the condition Ia = 16 + It. There is one such equation for each node. One of the ----------------------------------------------------------- MESH AND NODAL EQUATIONS 3 equations, hovever, is superfluous, since if the law of conservation of charge is satisfied at all but one of the nodes, it will automatically be satis- fied at the last one also.* If the number of nodes is represented by N, there will then be N - 1 current equations. The original branch equations included, in addition to the branch currents, the N nodal voltages. One of these voltages, however, can be chosen arbitrarily, since the branch equations involve only voltage differences. There are thus B + N- 1 unknowns to be determined, and the N- 1 current equations together with the original B branch equations are just sufficient to permit a solution. The N- 1 conditions at the nodes allow us to express N - 1 of the branch currents in terms of the others so that a corresponding number of the branch voltage equations similar to (1-1) cau be eliminated. This reduction becomes particularly easy if we follow the familiar device of regarding the remaining branch currents as flowing through complete closed loops in the network. The assumption of closed loops or meshes has two advantages. In the first place it evidently leads to automatic satisfaction of the condition of conservation of charge at each node, since in ,ach mesh as much current flows away from any node as flows into it. In the second place, it eliminates the differences in node voltages which appeared in the original branch equations, since the sum of all such voltage differences around a complete loop must be zero. We may also notice that, since there were originally/3 branch currents and N - 1 of them have been eliminated, the number of remaining currents or meshes is given by the Theorem: In any conductively united network the number of inde- pendent closed meshes or loops is one greater than the difference between the number of branches and the number of nodes. An illustration of the reduction from branch to mesh currents is fur- nished by Fig. 1.2, which shows a choice of mesh currents which is appropri- ate for the circuit of Fig. 1.1. The independent branch currents in terms of which the other currents are expressed are those flowing through branches a, d, and f, each of which is included in only one mesh. There are three meshes since the circuit contains six branches and four nodes. It is apparent that in general the meshes can be chosen in a variety of ways. Thus in Fig. 1.2 the independent branch currents might be chosen as those flowing through, for example, a, d, and e, or a, b, and c. These * This analysis neglects mutual inductance couplings as a matter of simplicity. If the network consists of a number of isolated fragments connected only by mutual inductance, there is evidently one superfluous condition of this sort for each con- ductively separate fragment of the network. ----------------------------------------------------------- 4 NETWORK ANALYSIS Fla. 1.2 possibilities are useful since they allow us to assign branches in which we may have particular interest, such as the generator or receiver impedances, to individual meshes. In a given physical circuit such assignments cannot be made with unlimited freedom. In Figs. 1.1 and 1.2, for example, it is not pos- sible to assign branches a, b, and d to three separate meshes because the corresponding branch currents are related by the condition at node .4 and are not independent variables. For purposes of future an- alysis, however, it will be assumed that there are no restrictions on the choice of meshes, since an adequate mesh system can always be obtained by the addition of ideal transformers or other elements of vanishing physical importance. 1.3. Mesh Equations for a _Passive Circuit It is evident that each mesh equation can be obtained by adding together the branch voltage equations around the complete loop and at the same time eliminating the superfluous branch currents by means of the nodal current conditions. Since this introduces only linear combinations of the coefficients in the original branch equations, the resulting system of equa- tions must be in the general form Zll-/i - Z1212 -Jr- ß ß ß + Zlnln --- E1 Z2I + Z2212 + ." + Z2d = E2 (1-2) where the Z's in the left-hand side are of the form Zq = PLi5 + Rq q-  Dq and p still represents d/dr. The mesh currents are indicated by numbered subscripts to distinguish them from the branch currents. The coefficients Zlb Z==, etc., will be called the self-impedances of the various meshes and the coefficients Z12, Za, Z2a, etc., the mutual or coupling impedances between meshes. The mesh equations are expressions of voltage equilibrium. They express, in other words, the fact that the sum of the driving voltages around ----------------------------------------------------------- MESH AND NODAL EQUATIONS a closed loop must be equal to the sum of the voltage drops in the loop. This makes it easy to evaluate the E's and Z's in the equations. In the first mesh equation, for example, let it be supposed that we set I2 = Ia ..... I, = 0. This can be done without disturbing the first mesh by inserting sufficiently high impedances in each of the other meshes. The first mesh equation then reduces to (p 1D,) I = E. (1-3) L + R +  Since there are no other currents flowing in the structure, the left-hand side of this expression evidently represents the voltage drop due to the flow of the current I through all of the elements in the first mesh. The coefficients L, Rn, and Dli thus represent respectively the sum of the inductances, resistances and stiffnesses in the first mesh. Correspondingly, E on the right-hand side represents the sum of the generator voltages in this mesh. Now, if we allow I2 to flow, an additional voltage drop appears in the first mesh. This must evidently be due to the flow of through the elements which are shared by the first and second meshes. Similarly, Za represents the elements which are common to the first and third mesh, etc. The coefficients in the equations for the other meshes can be determined in analogous fashion. In the purely passive circuits now under considera- tion, the coefficients representing a coupling between two meshes must be the same in each mesh equation. In other words, Zi.i in the ith equation must be the same as Z5i in thejth equation, since either quantity merely represents the elements which are common to the two meshes. The determination of the coefficients in the mesh equations can be illus- trated by reference to the structure of Figs. 1.1 and 1.2. The self-impedance Zn of the first mesh is equal to the sum of the impedances around that mesh. We thus have Ln =Lsq-L6+Lc, Rn = Rsq-Rb+Rc, and Dn = Ds q- Db q- D. Similarly, the voltage E is equal to the total voltage Es + Eb q- E of all the generators in this mesh. The impedances Z2 and Za represent the elements which the first mesh shares respectively with the second and third. As Fig. 1.1 is drawn, however, the positive direction of the first mesh current opposes that of the second and third mesh currents in each common branch. The coupling elements must there- fore be taken negatively to account for the fact that the voltage drops across them due to the flow of the second and third mesh currents are opposite to those produced by the flow of the first mesh current. We thus have L2 = --Lb, R2 = --Rb, etc. The terms appearing in the other mesh equations can be determined in a similar fashion. ----------------------------------------------------------- 6 NETWORK ANALYSIS C.^P. I 1.4. Mesh Equations for an dctioe Circuit To generalize equation (1-2) to fit a circuit containing vacuum tubes, we may suppose that only one of the E's on the right-hand side of (1-2) is an actual driving voltage and that the remaining E's are apparent plate generators representing the amplifications of the tubes. For exampl. e, in one particular tube, let us suppose that the jth mesh current flows from grid to cathode and the kth mesh current from cathode to plate as shown by Fro. 1.3 Fro. 1.4 Fig. 1.3. Following the usual assumptions, the amplification of the tube can then be represented by inserting an equivalent generator -ue in series with the plate impedance R0, where e is the grid voltage, as shown by Fig. 1.4. The passive impedances of tl':e tube can be incorporated as part of the passive circuit and play no part in this analysis. Since e = ZeI  in Fig. 1.4, the equivalent plate generator voltage can also be written as -uZoI.. The kth of equations (1-2) can therefore be written as ZkI + ... + ZI + ... + Zk,I = -ZoI or ZI + .. ß + (Z + Z)I + ß .. + ZI = 0 (1-4) where Zi is the passive coupling between the two meshes. It is obvious that the equation is still in the same form as the original kth equation of (1-2) provided we redefine Z i to include the added quantity/Z o. This is the familiar result that the amplifications of the tubes can be represented by modifications in the various coupling terms in the mesh equations. So far as the general form of the equations goes, the only distinction between active and passive structures is the fact that we can no longer assume in general that the principle of reciprocity holds. In other words, we can no longer assume that Zii = Zii. The quantity uZ o will be called the mutual impedance or transimpedance of the tube, after the analogy with trans- conductance in the following discussion. In order to prevent future confusion with signs, it is important to notice here the convention adopted in Fig. 1.3 for the positive direction of grid ----------------------------------------------------------- MESH AND NODAL EQUATIONS 7 and plate currents. It has been so chosen that the transimpedances in the left sides of the mesh equations will be positive when the u's are positive, as they are in normal tubes, and also so that a uniform convention of sign can be adopted for a number of tubes in tandem coupled by ordinary interstage networks. With this choice, however, the equivalent plate generator volt- age is negative, so that successive tubes in an amplifying circuit give suc- cessioe phase reversals, in addition to any phase shifts which may be ascribed to the purely passive ele- ments of the circuit. Similar re- marks apply to the nodal analysis given later. As an example of the processes in- dicated by (1-4) we may consider the mesh equations for the circuit of Fig. 1.5. The structure represents Fro. 1.5 broadly one stage of an amplifier with grid plate coupling. The coupling is indicated by the impedance Z4 and the preceding and following interstages by the impedances Zx and Za. Z2 is the grid cathode capacity of the tube and Za represents its plate impedance. The circuit has three meshes. They are chosen in the form shown by Fig. 1.5 in order to assign the generator impedance, the grid impedance and the plate impedance each to only one mesh. If we assume for the moment that the tube has no amplification the mesh equations are readily set up in the form (Z + Z + Z)I - (Z4 + Zs)[2 + Z513 = E - (Z4 + Z,)I + (Z2 + Z4 + Z)I2 - ZIa = 0 (1-5) ZsI - ZsI + (Za + Z,)la = O. Since the voltage across the grid is +I2Zu when the currents are taken in the directions shown in Fig. 1.5, the equivalent generator in the plate circuit is -ZI. This appears as an effective voltage in the third mesh equation. When this term is transposed to the left side of the equation in the manner described previously, the third equation thus becomes ZsI1 q- (sZ: - Zs)I2 + (Za + Zs)Is = 0 (1-6) the other mesh equations remaining unaffected. 1.5. Steady State Solution for the Mesh Equations As the mesh equations have been developed thus far, they have always represented differential equations for the circuit. Thus, for example, in ----------------------------------------------------------- 8 NETWORK ANALYSIS CHap. (1-2) the E's and I's represent instantaneous values of voltages and currents and p represents differentiation with respect to time. In order to find the response of the circuit when one of the E's is a voltage varying sinusoidally with time, therefore, we should, strictly speaking, substitute sin cot or cos 0t for the appropriate E and attempt to find expressions for the I's as sums of sine and cosine terms in a form which would satisfy the set of differential equations. In accordance with the usual practice, this procedure can be much simpli- fied if we represent a physical sinusoid by the exponential eiøt. * The currents and voltages in the system are then written in the form Iie 'øt and Eje i't, where the œ's and E's are now merely constants instead of being quantities varying with time as they were in (1-2). The advantage of this substitution results from the fact that differentiating or integrating e with respect to time merely multiplies or divides the exponential by iw. Thus, any quantities of the form pe i't or  (1/p)e 'ø which result when the currents Ie 'ø are substituted for the original currents in (1-2) become simply icoe iøt and (1/ico)e øt when the differentiation and integration symbolized by p and 1/p are carried out. Each p on the left-hand side of (1-2) is then replaced by iw. The time factors e 4'øt in the current and volt- age expressions are unchanged, and can be divided out of the final equations. 1.6. Driving Point and Transfer Impedance It follows from the considerations just advanced that the differential equations (1-2) can also be regarded as a solution for the steady state response of the network to sinusoidal voltages of frequency w/2r provided p is replaced by iw and that we regard the I's and E's as representing merely the constant coefficients in the general current and voltage expressions Ie t and Ee it. With this understanding, the determination of any particular current flowing in response to a particular voltage is equivalent to the solution of a set of ordinary linear equations. As an example, the current Ie it in the first mesh flowing in response to the voltage Exe also in that mesh is given by ie,  = zXn. EeO  (1-7) where ,x is the determinant of the coefficients in the left-hand side of (1-2) and ,xn is the determinant obtained when the first row and the first column of A are omitted. The driving point impedance Z in the first mesh is by definition the ratio * A discussion of the physical meaning of this substitution is avoided here, since the subject is taken up again in the next chapter. ----------------------------------------------------------- MESH AND NODAL EQUATIONS' 9 of the voltage to the current in equation (1-7). It is given in other words by Z ..... 0-8) In a similar fashion the equations can be solved to determine the current in any other mesh in response to this same voltage. For example, the current in the second mesh is given by 2e = ' (1-9) where /x. is the determinant of the coefficients in the left-hand side of (1-2) after the elements in the first row and second column have been omitted.* The ratio between the voltage E1 and the current I2 will be called the transfer impedance, Zv, from the first to the second mesh. It is given by E A ..... (1-O) 1.7. Z and Zv as Functions of a Single Element In future discussion, we will have frequent occasion to study the depend- ence oœ the driving point and transfer impedance upon a single element in the network. Let it be supposed, for example, that we are interested in the variation of Z with respect to a bilateral impedance z in the jth meshdr This can be investigated by examining the way in which z enters the deter- rainants A and Xt of (1-8). In general, any determinant can be regarded as the sum, with appropriate signs, of all possible products formed by multiplying together elements of the determinant, when each product includes just one element from each row and column of the determinant. Since z is in thejth row and column of A, it must therefore be multiplied by all possible products of elements taken from every row and column of  except the jt h. These, however, evidently form the minor /i of the original determinant. Similarly, in *Strictly speaking, the symbols A, A, etc., represent cofactors here. In other words, they are the determinants as defined in the text multiplied by +1 or -I in accordance with the usual rules of determinant theory. In particular, /x Js negative. This may be ignored for theoretical analysis, however, since it is only necessary to treat the symbols as cofactors consistently. 'It is assumed here, in other words, that z is found i thejth mesh and in none of the others so that it is a constituent of only the self-impedance Zii in (1-2). ----------------------------------------------------------- 10 NETWORK ANALYSIS c.^,. 1 forming An the terms by which z s multiplied must be the minor Alljj obtained by omitting both the first and jth rows and columns. If we let ),o and Aøn represent, respectively, A and Ai when z = 0, therefore, we have Aø + zaJi ß (1-11) Z = ao n + Since Aii and Allii are evidently independent of z they can equally well be written as/x. and 0 Anii. This will occasionally be done in later analysis in order to facilitate further transformations. The relation between Zz, and z can be found in similar fashion. It is given by Aø + zAJi (1-12) If z represents a unilateral coupling term, instead of a bilateral element, the expansion is essentially the same. Thus, if we suppose that z is a part of Zij in the original determinant, we readily find A ø + zAq (1-13) and A ø + zAq (1-14) ZT = A012 q_ YgA12i j ' 1.8. Nodal Equations for a Passive Circuit* In the mesh equation formulation, the driving sources are regarded as voltages. The dependent variables, whose determination constitutes the solution of the structure, are the currents in the several closed loops or meshes. There is one equation for each mesh and each equation represents the fact that it is physically necessary for all the meshes to be in voltage equilibrium. As we might expect, it is also possible to set up a system of equations in reciprocal form with the activating forces taken as currents and their responses as voltages. In this case, the nodes replace the closed loops in the mesh equation analysis. Figure 1.6 shows the forln which such an analysis may take. The driving sources are the currents I1-.. I, impressed on the nodes i ß ß ß n from some outside sources. The responses are the voltages E1 ß ß ß E for the individual nodes. Each voltage is sup- * The writer is indebted to Prof. R. M. Foster, of the Polytechnic Institute of Brooklyn, for pointhg out the superiority of the'nodal analysis. ----------------------------------------------------------- MESH AND NODAL EQUATIONS 11 posed to be measured with reference to some particular node which is chosen as ground. The fundamental equations in the nodal system are expressions of current equilibrium. They represent, in other words, the fact that the driving current flowing into any node from the outside must be equal to the total current flowing away from that node into the rest of the network, iust as Ftc. 1.6 ß i the mesh equations represent an equilibrium between driving voltages and voltage drops in any mesh. In Fig. 1.6, for example, the current flowing into the first node from the outside is I. The current flowing from that node directly to ground must be.YE. The current flowing from that node to the second node must be Y2(Ei - E2), etc. The complete equa- tion is therefore YiE q- Y2(E - Eo,) + ... q- Y(E - E) = l (1-15) which can evidently be written as YE --F2E - YaEa ..... YE, = I (1-16) where Y]I = Y1 q- Y12 4. Ya 4. '" 4- Y. (1-17) In equation (1-17) Yll is obviously the total admittance between the first node and all the others when the others are shorted together. It will be called the self-admittance of the node and is evidently analogous to the self-impedance of a mesh, which can be defined as the impedance of the mesh when all other meshes are opened. Similarly, the terms Yt5 are mutual admittances corresponding to the mutual impedances appearing in a set of mesh equationsß Since an equation analogous to (1-16) can be written for each node, the ----------------------------------------------------------- 12 NETWORK ANALYSIS C,AP. t complete system of equations becomes YnE - Y12E ..... Y,E, = I1 -YiE + Y22E ..... Y2E, = I (1-18) --YnlE1 - Y,2E.... + Y,,E, = It is not necessary to write a separate equation for the last or" ground" node. Since as much current must leave the network as a whole as enters it, the condition of current continuity will automatically be satisfied for this node if it is satisfied for each of the others. We thus have the In any conductively united network the number of inde- pendent nodal equations is one less than the total number of nodes. At first sight, it might appear that the cases in which we can regard the energizing sources as constant current generators or, in other words, as generators with infinite internal impedances would be rather rare. In the mesh equation analysis, however, we seldom deal with generators having zero internal impedance and it is customary to allow for this by adding the .o i œ/g0 t o i Z o Infinite < Z o Impedance Source < oj oj œ/Z ø Fro. 1.7 Fro. 1.8 internal impedance of the generator to the impedance of the mesh in which it appears. When consideration is given to this fact the two methods stand on an absolute parity. To show this, let us suppose that the actual driving source is a generator of internal emf E and internal impedance Z0 connected between terminals i andj as shown by Fig. 1.7. It is easy to see that this must be equivalent to the circuit shown in Fig. 1.8 for any connections between i and j. In other words, the source shown in Fig. 1.7 can be represented in the nodal admittance analysis merely by choosing the energizing currents li and I i ----------------------------------------------------------- MESH AND NODAL EQUATIONS' 13 as E/Zo and -E/Zo, respectively, and adding the admittance l/Z0 across terminals i and j. In this discussion we are concerned with the use of current rather than voltage sources only to establish the broad possibility of writing network equations in the general form given by (1-18). It is interesting to note, however, that the formal symmetry between the current and voltage methods of analysis can also be extended to the individual terms in these equations. This follows from the fact that the current and voltage rela- tions for a resistance or conductance can be written as E = RI and 1  = GE, while the corresponding expressions for a capacity or inductance are E = Lpœ and œ -- CpE, where p may be either i0 or d/dt. It is obvious from the symmetry of these expressions that we can erect a set of nodal equations formally identical with a given set of mesh equa- tions by interchanging R and G and L and C wherever they appear. In other words, the general term Zo = pLij + Ri + Di/p in (1-2) is re- placed by Yii '' p'ij' q-Gi5 + Pii/P in (1-18), where F stands for a reciprocal inductance, just as D represents a reciprocal capacity. The two sets of equations will evidently be equal, term for term, provided we set Lii = Ci5, lij = Gij, and Di5 = Iij ß The recognition of these general possibilities constitutes the so-called principle of duality in network theory.* If the mesh equations for one network correspond, term for term, with the nodal equations for another, the two networks are called inverse structures. It is not always possible to obtain the exact inverse of a given structure. There are diculties, for example, with networks including mutual inductance coupling, since the capacitance dual of a coupling between coils does not exist. The inverse may also fail because the inverse set of equations does not corre- spond to any conceivable arrangement of impedance branches. In most of these instances, however, it is possible to obtain a network which will behave like the desired inverse so far as external connections are con- cerned, though it may have a different internal structure. The detailed discussion of these possibilities is beyond the scope of this chapter. The subject is resumed in Chapter X. 1.9. Nodal Equations for an Aletire Network The modifications which are necessary in order to include vacuum tubes in a nodal admittance analysis are essentially similar to those we have already made in the mesh analysis. Suppose, for example, that the grid, ß Good general discussions are given in Guillemin "Communication Networks," Vol. II, and Gardner and Barnes "Transients in Linear Systems," Vol. I. The latter reference may also be cited for its detailed description of the method of setting up a system of nodal equations, especially in circuits containing mutual inductance. ----------------------------------------------------------- 14 NETWORK ANALYSIS C.A,. t plate, and cathode of a given vacuum tube are respectively nodes./', k, and m of the complete network. The voltage between grid and cathode is then Fro. 1.9 E s -E,, and in accordance with our preceding discussion the effect o(the am- plification of the tube can be represented by introducing an equivalent generator -u(E s - E) in the plate circuit. It follows from Figs. 1.7 and 1.8, however, that this equivalent generator can in turn be replaced by two current sources of strengths -(E s -E,)/Ro and tz(E i - E,,)/Ro applied to the plate and cathode, respectively, where R0 is the internal resistance of the tube, pro- vided the admittance 1/Ro between plate and cathode is incorporated as part of the network. With the application of these two current sources, the kth and ruth nodal equations become Ro Ro The terms on the right-hand side can now be transposed and incorporated as part of the mutual admittance terms appearing in the left-hand side. In most cases, the ruth or cathode node will be at ground. If we make this assumption, which corresponds to the assumption made in connection with Fig. 1.3, that the grid and plate circuits are in separate meshes, the second of equations (1-19) can be ignored. The first equation then becomes --Y,Ei - Y.:E.o ..... (YS - G)Es ..... Y,E = 0 (1-20) where G = u/Ro and is the quantity usually described as the transcon- ductance of the tube. As in the mesh analysis, the effect of adding vacuum tubes is not to change the form of the equations but merely to destroy the reciprocity condition Yii = Yii. As an illustration of these processes, nodal equations will be developed for the circuit shown in Fig. 1.9. This is the same network as the one previously shown by Fig. 1.5, redrawn to suit the nodal analysis. Since the bottom or cathode node can be taken as ground, there are two equations. If we suppose initially that the apparent current generator -GmE in the ----------------------------------------------------------- MESH AND NODAL EQUATIONS 15 plate circuit is zero, the equations are readily found to be (Yr q- Y2 q- Y4)Et - Y4Es= I (1-21) The introduction of the plate generator is equivalent to adding -GreEt to the right-hand side of the second of these equations. After this term is transposed to the left-hand side, this equation becomes --(Y4 -- G,n)E q- (Y3 q- Y4 q- Ya)E2 = 0 (1-22) the first of equations (1-21) remaining unchanged. A solution of the nodal equations to find the steady state voltages corre~ sponding to any given set of sinusoidal driving currents can evidently be obtained by the processes already used for mesh equations. For example, the driving point admittance Y between the first node and ground will be defined as the ratio between the driving current entering that node and the resulting voltage at the node. It is evidently given by Y- E - a (1-23) where the primes are used to indicate that the determinants refer to the system of equations given by (1-18). Similarly, the transfer admittance Yv between the first and second node will be defined as the ratio of current .applied at the first node to the resulting voltage at the second node. It can be written as ¾v = E--} = ' (1-24) In view of the obvious analogy between the mesh and nodal methods of analyzing a circuit, the two methods will be used indifferently in most of the following discussion. The primes, which were used in equations (1-23) and (1-24) to distinguish the nodal determinants from those obtained from the mesh equations, will ordinarily be omitted. The determinant A will thus be used to refer to either system unless there is some particular reason for distinguishing between them. The symbol/3, which may perhaps be called an" adpedance" or" immittance," will be used to refer to an element in either system. 1.10. Choice between Mesh and Nodal Alnalysis The above discussion has emphasized the fact that mesh and nodal equations can be used symmetrically in a general theoretical analysis. ----------------------------------------------------------- 16 NETWORK ANALYSIS Caxp. I The reader is cautioned, however, against concluding from this that the choice between the two systems is a matter of indifference when one is dealing with a definite physical circuit. In most circumstances the nodal analysis will be found appreciably more convenient. The advantages of the nodal analysis may be traced to several causes. The most obvious is, of course, the fact that many circuits contain screen grid tubes having a very high plate resistance. Since such tubes are very nearly constant current devices, circuits containing them can evidently be analyzed more conveniently on the nodal than on the mesh basis. Another advantage of the nodal formulation results from the fact that the equations can be more directly correlated with the physical structure of the network than is possible with the mesh formulation. The nodal equations can be written down directly, but to use the mesh analysis it is at least necessary to begin by selecting a suitable system of closed loops. In a complicated circuit, this may not be as easy a problem as it appears. The difference becomes particularly conspicuous in the inverse situation, when one has been given a set of equations and wishes to determine a correspond- ing physical structure. It is evident that the corresponding structure can be written down directly if we use nodal equations. If we begin with mesh equations, on the other hand, the process may be quite difficult. In fact, it is theoretically possible to write down a plausible looking set of "mesh equations" for which no corresponding circuit configuration exists. The final consideration is the fact that, although either mesh or nodal equations can be used in analyzing any given circuit, it is not necessarily true that the two formulations will require the same number of equations. The preceding discussion gives the required number of equations as B - (N- 1) for the mesh system and as N- 1 for the nodal system. In order to compare these expressions, suppose that the network is originally very simple and is built up to its final form by the addition of one node at a time. Obviously, each new node must be connected with the original circuit with at least two new branches if the node is to be an operative part of the structure. We may expect therefore that B.will be at least twice as great as N - 1, so that in general the number of mesh equations will not he less than the number of nodal equations and may be much greater if the circuit is complicated.* For example, it required three mesh equations and only two nodal equations to analyze the structure shown by Figs. 1.5 * These conclusions are true only "in general" because of the possibility of simul- taneously creating two new nodes by means of a cross-connection between them, so that one branch serves for both. An example is furnished by a balanced ladder line, the cross-connections being the shunt branches. These, however, are excep- tional cases which are not representative of ordinary physical circuits. ----------------------------------------------------------- MESH AND NODAL EQUATIONS 17 and 1.9. In general, the nodal analysis appears to be particularly adapted to complicated high frequency circuits where we must consider many capacities to ground. Evidently, ground capacities from any of the exist- ing nodes will not greatly complicate the nodal equations, but they may considerably increase the number of meshes in the circuit. ----------------------------------------------------------- CHAFFER II THœ COMVLEX FRECttJrC¾ PIANE 2.1. Introduction IN actual engineering applications we are concerned with the response of a circuit only to currents and voltages at real frequencies, that is, to ordinary sinusolds. For purposes of analysis, however, it is often neces- sary to give attention also to the response of the circuit to driving forces whose frequencies are complex. This chapter will consider the physical meaning which may be assigned to the term "complex frequency" and some of the elementary ways in which the conception of complex frequencies may be used in describing circuit characteristics. 2.2. The Single Resonant Circuit It will be recalled that the general circuit equations in the last chapter were first developed in differential form, and that integrated or "steady- FIC. 2.1 state" solutions for sinusoidal driving forces were obtained by supposing that the exponen- tial e it could be substituted for a physical sinu- soid. The meaning of a complex frequency can be understood most easily if we return for a moment to this last step. It will be sufficient to examine the solution for the single resonant circuit consisting of resist- ance, inductance, and stiffness in series, as shown by Fig. 2.1. Let the sinusoidal driving voltage be written as E0 cos tot. If q repre- sents the charge on the condenser, so that the current I = dq/dt, the differ- ential equation of the circuit is dq dq L  q- R  q- Dq = Eo cos tot. (2-1) We may assume that the solution of this equation can be written in the general form q = .,/cos ,.,t + B sin cot (2-2) or I - dq _ rico sin tot + Bco cos tot dt where .'/and B are constants still to be determined. (2-3) ----------------------------------------------------------- THE COMPLEX FREQUENCY PLANE 19 The substitution of the assumed form (2-2) for q in (2-1) gives -dLco 2 cos cot - BL02 sin cot - .4Rco sin cot q- BRco cos cot q- .rid cos cot q- BD sin cot = E0 cos cot. (24) This equation must hold for all values of t. In particular, it must hold for values of t at which sin cot is zero and also at values of t for which cos cot is zero. But when the sine terms are zero (24) becomes -.4b.o 2 + BRco q- .4D = Eo (2-5) and when the cosine terms are zero it becomes -Blxo 2 - .4Rco + BD = 0. (2-6) These equations can be solved simultaneously for ,4 and B. This gives (D - L0)E0 ,4 = (Rco) 2 + (D - L=)  (2-7) B = (Rco)Eo (Rco) 2 + (D - Lco) 2 (2-8) from which the assumed solution for q becomes q = E0 (Rco?- Z Zoo) cos cot + (Rco)= + (W - sin cot (2-9) or [ = E0 [R 2 R Lo - D/co + (L:o - D/co)  cos ot + R2 + (L(o --- 3/co)' sm 0t ß (2-10) The fact that these are correct solutions is easily established by direct sub- stitution in equation (2-1). The coefficients in equation (2-10) are, of course, the familiar expressions for the in-phase and quadrature components of the total current. 2.3. Exponential Representation of Physical Sinusolds* The expression given by (2-10) is evidently the true physical current which would flow in response to the assumed sinusoidal driving voltage. ß The use of the exponential solution in electric circuit theory goes back at least as far as Heavlside, "Electromagnetic Theory." For later discussions see G. A. Campbell, "Cisoidal Oscillations," Trans. A.I.E.E., April, 1911; J. R. Carson," Elec- tric Circuit Theory and Operational Calculus," 1926 (Bibliography); T. C. Fry, "Elementary Differential Equations," 1929. The last reference gives a particu- larly complete discussion. ----------------------------------------------------------- 20 NETWORK ANALYSIS CI'AP. 2 The method required to derive (2-10), however, is cumbersome and labori- ous and these objections would appear still more forcefully if we had dealt with a multi-mesh system. The use of the exponential e t to represent the actual physical sinusoid provides a way of analyzing the circuit much more expeditiously. The justification for the use of e 4t in place of a physical sinusold depends upon the principle of superposition. It depends, in other words, upon the fact that in a linear system such as (2-1) the current flowing in response to two driving forces acting together is the sum of the currents which would flow in response to the two separately. Thus, in (2-1), if q(t) is the response of the network to E (t) so that d2 q  dq L 5- -I- tr - q- Dq = E (t) (2-11) and q2(t) is the response to E2 (t) so that d292 dq2 L- + R  + Dq2 = œ2 (t) (2-12) then L d2(ql q- q2) dt  d(q d- q2) q- R q- D(q q- q2) = E(t) q- E2(t) (2-13) dt follows obviously from simple addition of equations (2-11) and (2-12). This principle is usually applied to find the response to E(t) + Es(t) from the responses to El(t) and E2(t) separately. In this application, however, the principle is made to work backward to give the responses to E(t) and E2(t) separately when the response to E(t) q- E2(t)is known. Obviously, it is not always possible to do this, since the knowledge merely of the sum q(t) q- q2(t) does not necessarily tell us how much is ql(t) and how much is q2(t). The decomposition can, however, be effected without ambiguity if El (t) is real while E2(t) is a pure imaginary quantity, since it follows from the fact that the coecients of (2-1) are real quantities that the corresponding q(t) and q2(t) must then be real and pure imaginary, respectively. In this special case, therefore, we can work backward from equation (2-13) to equations (2-11) and (2-12) merely by picking out the real and imaginary components of the q which is a solution of (2-13). In the present application, we have e it = cos cot q- i sin cot. The real and imaginary components of the q which corresponds to the driving voltage e it must therefore be the q's which would correspond respectively to the voltages cos cot and i sin cot. For example, let q and iq2 be the solutions which would correspond to the voltages E0 cos cot and leo sin cot in (2-1), ----------------------------------------------------------- THE COMPLEX FREQUENCY PLANE 21 and letq= q +iq=. We then have d2q dq L- + R  q- Dq = Eo cos cot L d2(iq2) + R d(iq2) dt   + D(iq.) = iEo sin cot. (2-14) (2-15) Adding (2-14) and (2-15) together gives us d2q  Eo e'øt L  q- R q- Dq = = Eod 't (2-i6) where p has been written for ico. By the previous argument, the real com- ponent of the q which satisfies this equation will be the q which satisfies equation (2-14). Upon assuming that q = qoe vt xve find readily qo(pL + pR + D)d 't = Eod'L (2-17) It follows that E0 qo p2L q- pR + D (2-18) or I = pEøeVt (2-19) pL q- pR q- D Upon substituting ico for p in (2-19) we secure I = Eo(cos cot + i sin cot). (2-20) R q- i coL -- D The real component of (2-20) should be the current flowing in response to the voltage E0 cos o#. It turns out to be (2-21) which agrees with equation (2-10). The method also gives as a by-product the current which will flow in response to the voltage E0 sin wt. We have merely to take the imaginary component of (2-20), discarding the i. ----------------------------------------------------------- 22 NETWORK ANALYSIS C.^.. 2 This gives t"'O ß This process can evidently be extended directly to multi-mesh circuits. If we begin with a driving voltage Ee vt the solution of the circuit equations for any one of the currents will appear in the general form Ievt, and if the real component of Ee vt is taken as the true physical voltage the real com- ponent of Ievt will be the physical current. It will be convenient to summarize this discussion in a form in which it appears as a set of definitions of the meanings we shall ascribe to the terms "frequency" and "impedance." Thus (I) A voltage of frequencyf will be written as Eoe  where p = Physically, we shall interpret such an expression by taking only its real component. E0, which was taken as a real quantity in the previous example, may in general be complex. The use of a complex value of E0 amounts simply to a shift in the phase of the physical voltage, as we can readily see by taking the real component of (E0 + iEo)e vt. () We shall take as the current in any mesh the quantity which satisfies the differential equations of the circuit with the voltage of (i) as the driving force. It will appear in the form Ioe v where I0 is another complex constant. The actual physical current corresponding to the acal physi- cal voltage will be the real component of this expression. For brevity, the constants E0 and l0 alone will sometimes be spoken of as "voltage" and "current." (3) The self- or transfer impedance, depending upon whether the current and voltage are in the same or different meshes, will be defined as the ratio E0: lo of the constants in the voltage and current expressions of (1) and (2). (4) The impedance is obtained as an algebraic quantity from the solu- tion of the set of linear equations which result when the differential opera- tor d/dr is replaced by p = i in the differential equations of the circuit. 2.4. Tae Complex Frequency Plane The definitions of frequency and impedance which have just been given were developed on the assumption that the driving force would be a simple sine wave. The frequencyf is then a real quantity and the new variable p is a pure imaginary. ite evidently however, the definitions can be extended formally to situations in which bothf and p are complex. The ----------------------------------------------------------- THE COMPLEX FREQUENCY PLANE 23 physical meaning of such an assumption is easily determined. Suppose, for example, that we are dealing with the driving voltage Eoe vt. Let Eo and p be, respectively, E0t q- leo2 and Pt q- ip2. The volt,age can then be written (Eo q- iEo2)e (w+iw) t = (E01 cos pt -- Eo2 sin p2t)e wt q- i(Eoi sin pt q- Eocos p)e wt. (2-23) By the definitions just established the physical voltage is the real com- ponent of this expression or, in other words, (E0t cos pt - Eosin p.t)e wt. It is obviously a sinusoidal oscillation with positive or negative damping depending upon whether px is negative or positive. The physical current F. 2.2 corresponding to this voltage is obtained by dividing the complex voltage by the impedance and taking the real component of the result. It will evidently be a damped sinusold with the same frequency and damping as the driving voltage. We will hereafter consider that frequency is in general a complex quan- tity. It can conveniently be represented on a plane such as that shown by Fig. 2.2. As the figure is drawn, the horizontal axis represents real values of p, and the vertical axis imaginary values ofp or real values of fre- quency. Real frequencies are therefore obtained by reading up the vertical scale. This arrangement is normally the most convenient one in theoretical ----------------------------------------------------------- NETWORK ANALYSIS analysis, sincepis a more convenient variat)le t}anf. If we prefer, how- ever, the diagram can be given a quarter turn i a clockwise direction, so that real values of frequecy are found on a scale reading from left to right in the normal fnshion. In this event, complex freqtencies are found ab(ve and below the real frequency axis. T}e other axis, corresponding to real valtles cf p or ptlre imaginar? values of frequenc?, represents the liniting case in which tle driving voltage and responses are exponentially increasing or decreasing withut oscillation. I will be noticed that the diagram represezt ne=ative as well as posi- tive values of frequency. The lower half of the plane, in which negative frequencies are found, is seldom of much actual concern in network analysis. In any physical circuit, the real component of the impedance {s an even function of fruqueicy, and the imaginary component is an odd function. In other words, the real component of the impedance at a negative quenc>-is equal to its value at the correspoiding positive frequency, while the imaginary component at a negative frequency is the negati'e of the imaginary component at the corresponding positive 0aquency. Simple relaticns of symmetry, thercfore connect the upper and lower halves of the p'/,mO'//p0r/a,re. This arises from the fact that on one of these halves, the driving voltage and response corespond to fmctions which decrease ex- ponentially with tim% hile on the other half they represent exponentially increasing functions. As our later discussion will show, there is a close connection between the steady state response characteristics of the net- wor, and its transient characteristics. Since a network whose transients increase as time goes on is unstable, or, in other words, non-physical, the characteristics of physical networks in the half of the plane corresponding to exponentialls' increasin fnctions are severely limited. The functions whose behavior ot the cmplex plane will be of chief interest are the driving-point and transfer impedances Z and Zr, and the crresponding admittances }'and t'. Each of these can be expressed terms of determinants whose elements are relatively simple functions of frequency. In the mesh system, for example, the general impedance coecient can be written as Z,¾ = (?2L:_i + ;oR.i  D;: p. Since any of the determinants 2, 2, : used in the definitions of Z and Zz can be expressed as tIqe sum of products f quantities of this type, it is clear that they must all be polynomials in p divided bv some power of p. The same result, of course, holds for determinants taken from the nodal system. ----------------------------------------------------------- THE COMPLEX FREQUENCY PLANE The individual functions, Z, Z:, Y, and Y,, are each expressible as the ratio of two determinants, from equations (1-8), (1-10), (1-23) and (1-24). Evidently, therefore, they must each appear, in general, as the ratio of two polynomials, as shown by `4,,pm q_ `4__p- q_ ... q- `4p q- .40 IF(v) = Bp q- B_p_  q- . . . q_ Bp q_ Bo (2-24) Such an expression is called a rational function of p. In studying the behavior of such a function as (2-24) on the complex frequency plane, it is convenient to give special attention to its zeros and poles, which are respectively the points at which the function becomes zero and infinite. This is easily expressed by rewriting both numerator and de- nominator of (2-24) as a product of factors, so that the equation becomes - p)(p - ... (p - IF() -- B,(p - pl)(.p - . . . (p - ' (2-25) Evidently Pt ß ' 'Pm are the zeros, and p - ß ß p, are the poles. Ordinarily the p's and p% will all be different, so that the zeros and poles are all of the first order, or "simple." In special cases, however, two or more zeros or poles may coincide to give a multiple zero or pole. The zeros and poles are obviously the analogues, for general networks, of the resonances and anti-resonances which are familiar in purely reactive structures. The prin- cipal difference is the fact that the "resonances" and "anti-resonances" in a general network may occur at complex frequencies. The consideration of the zeros and poles is important for two reasons. The first is the fact that except for the constant multiplier `4/B, they evidently specify (2-25) completely. Assuming, then, that IF represents a driving-point impedance or admittance, we can conclude that two driving-point impedances or admittances having the same zeros and poles can differ only by an ideal transformer. Similarly, if IF is a transfer impedance or admittance, we can say that two transfer impedances or admittances having the same zeros and poles can differ only by a constant gain or loss. The other reason for paying particular attention to the zeros and poles will appear more clearly in later chapters. It depends broadly upon the fact that the location of the zeros and poles in the frequency plane furnishes our best index in classifying networks. Thus, unless the zeros and poles meet certain restrictions, the impedance functions which they specify can- not be furnished by a physical network. Assuming that these restrictions are met, further study of the zeros and poles permits the function to be assigned to one of several general categories. ----------------------------------------------------------- 26 NETWORK ANALYSIS C.... 2 2.6. Zeros and Poles of a Resonant Circuit Impedance As an illustration of this discussion we may return to the resonant circuit which was analyzed earlier in the chapter. The impedance of this circuit, as given by (2-19), can be written as Z = L (p - p) (p - p) (2-26) P where The quantities p and P2 are evidently the zeros of the impedance. Their location depends upon the two quantities, R/L and D;'L. If we multiply R 'L by any quantity, and D/L by the square of that quantity, however, p and P2 will merely be multiplied by the same quantity. It is, therefore, sufficient to study the possible locations of p and P2 when R,/L is varied +ca Fro. 2.3 while D/L is held fixed. If R/L is small compared to D, L, which corre- sponds to a resonant circuit with small damping, the quantities under the square root signs will be negative, andp and p2 will therefore be conjugate complex numbers with negative real parts. Typical locations for p and p2 are represented by the circles in Fig. 2.3. The cross at the origin repre- sents the pole of impedance which is found when p = 0. It is customary to consider that there is another pole at p = , since the impedance is also infinite there. It is easily shown that, as R/L varies, p and p.o move along the circular paths indicated by Fig. 2.4. At the extreme points .4 and .4', for which R vanishes, p and p2 tie on the real frequency axis. This corresponds to the ordinary resonance of a non-dissipative resonant circuit, in which the impedance vanishes at a real frequency. The points B and B' represent the ----------------------------------------------------------- THE COMPLEX FREQUENCY PLANE 27 zeros when the circuit contains a moderate amount of dissipation. This is similar to the case previously illustrated by Fig. 2.3. At C, on the other hand, (R/2L) 2 = D/L and the two zeros are equal. In other words, the impedance has a double zero at this point. This is the critically damped case. Since C is found on the real p axis the corresponding physical volt- age and current are non-oscillatory exponentially decreasing functions. If R/L is still larger, p and P2 are found respectively to the right and left of C on the real p axis as illus- trated byD andD . It will be no- ticed that, although the zeros can be t t Ftc.. 2.4 p plane assigned a great variety of positions by varying the relations among R, L, and D, they are always found in the left half of the p plane. 2.7. /inalytic Functions The introduction of complex values of frequency is equivalent in mathe- matical terms to studying such quantities as the driving-point and transfer impedance by the methods of function theory. In this field, one of the most important tools available to the mathematician is the conception of an analytic function. Definition: A function is said to be anaJytic at a given point in the plane of the independent variable provided it has a finite derivative, independent of direction, at that point. The function is analytic over a given region provided it is analytic at every point in that region. Points for which it is not analytic are called singular points or singularities. The restriction that the derivative be independent of direction is rela-. tively unimportant for engineering purposes. It is effective only ia elimL nating suclx functions as the real component of Z, or the absolute value of Z. For example, I zJ cannot be an analytic function ofp at any point because dJ Z J must be a real quantity, and the phase angle of the derivative dJ Z J/alp must therefore change as we change the phase angle, or direction in which dp is taken. As long as we restrict ourselves to functions which are in general complex, such as Z or log Z, however, the fact that the deriva.- 'tire will be independent of direction can be taken for granted. The essen- -tial feature of the definition, then, is the fact that if the function is to be analytic the derivative must befinite. ----------------------------------------------------------- 28 NETWORK ANALYSIS c.^P. 2 The points at which the derivative of a rationat function, such as (2-25), becomes infinite are readily determined. If, for example, we let N and D represent respectively the polynomials in the numerator and denominator of (2-25), the ordinary rules for differentiation give dN NdD di//(v ) D pp -- pp - (2-28) dp D 2 Since N and D are ordinary polynomials, neither they nor their derivatives can become infinite for any finite value of p. We can thus conclude that (2-28) will become infinite only at the points at which D vanishes, or in other words only at the poles of the original function. The singular poins of animpedance or admittance function are therefore its poles, and the function will be analytic in any part of the p plane which contains no poles. It will be seen that the analyticity of the impedance or admittance func- tion /'//is not dependent upon the location of its zeros. If k//is a trans- fer impedance or admittance, however, it is usually convenient to specify it in terms of attenuation and phase shift. This is equivalent to dealing with the function log /4/, rather than with /V itself. The expression corresponding to (2-28) for the derivative of log///is D dN N dD d log ti/(v) dp dp -- (2-29) dp ND This is evidently infinite whenever either N or D vanishes. The sinular points* of the logarithm of an impedance or admittance are therefore the zeros and poles of the originl function. Log/4/will be analytic only in regions which contain no zeros or poles of The properties of analytic functions furnish the most direct method of establishing Nyquist's criterion for stability. The first application of this material will be made in Chapter VIII, where Nyquist's criterion is discussed. 2.8. Physical I/alidity of Complex Frequencies The conception of a complex frequency can be looked upon in several ways. If we like, we can think of complex frequencies as having real *The singular points are "logarithmic singularities" and not poles. For the point P0 to be a pole the function must approach infinity near P0 as 1/(p -- p0)% where nis an integer. Although log/4/approaches infinity at the zeros and poles of P?, the approach is at a much slower rate. For example, it is shown in ordinary calculus that, although log x = -- log (i/x) approaches - oo as x vanishes, it in- creases so slowly that the limit of x log x is zero. ----------------------------------------------------------- THE COMPLEX FREQUENCY PLANE 29 physical existence. The definitions of complex frequency and impedance have been so drawn that an analysis stated in terms of complex frequencies can be submitted to physical verification. There is no difficulty in suppos- ing that a generator can be constructed to give a driving voltage varying as an exponentially increasing or decreasing sinusold for a reasonable period of time. By energizing a network with such a generator, the response charac- teristics of the structure can be obtained by direct physical measurement. The conception of complex frequency can thus be checked in the laboratory by a direct comparison of measurement and computation. Although this physical possibility is present, another point of view is more illuminating. We are finally interested in the response of the network only at real frequencies. It is only this characteristic which is specified in ordinary design problems. Moreover, the Fourier integral analysis tells us that if we know the responses of the network to driving voltages repre- sented by pure sinusolds, we can find its response to any other driving voltage. The real frequency characteristic, therefore, tells the whole story. So far as the purely theoretical relations are concerned, we might start with the response at real frequencies and compute the response to the exponentially increasing and decreasing sinusoids corresponding to complex frequencies by Fourier integral methods. Although the complex frequency conception is thus not essential, its introduction is of great value in facilitating the mathematical treatment of the theory. From a purely mathematical point of view, it is simpler to study the impedance function on the complex frequency plane than it is to consider only real frequencies. We have already noticed an analogous situation in the discussion of the response of the resonant circuit to a sinu- soidal driving voltage. The addition of an imaginary component to the voltage, although it is later discarded, makes the mathematical expressions so much more symmetrical that the algebra is actually much simplified. Somewhat the same advantages are obtained when we generalize the con- ception of frequency to include complex as well as real values. In this book we will use the idea of a complex frequency chiefly as a tool to specify what kinds of network characteristics are physically realizable. The same conclusions theoretically should be obtainable by the use of Fourier meth- ods on the real frequency characteristic, but the mathematics required with that treatment is much more difficult. A curious and interesting qualification of this discussion of the relation between the complex and real frequency response arises when we consider the physical significance of a complex frequency in more detail. The characteristics we are examining are, of course, those which correspond to the steady state response of the network. Since we never have a network which has been acted upon by a given voltage for an infinite length of time, ----------------------------------------------------------- 30 NETWORK ANALYSIS C,).p. 2 the steady state is never realized exactly in any experimental situation. We are accustomed to supposing, however, that a physical measurement of the steady state response can be obtained with sufficient exactness svith a suddenly applied voltage if we delay the measurement until the transients have had time to decay sufficiently. There is evidently no diculty about doing this when the driving voltage is a pure sinusoid. It is also possible if the driving voltage lies on the right side of the ? plane, since then the steady state characteristic will emerge as an increasitg exponential, while the transient terms are dying out. If the driving voltage is su- ciently far to the left of the 2 plane, on the other hand, the" steady state" response will diminish with time even more rapidly than do the transients. Evidently for frequencies in this part of the plane no physical measurement can be made which xvill lead to a response which is chiefly determined by the steady state characteristic of the network. Since the physical response can always be computed from the real freqtency characteristic by the Four- ier integral method, this suggests strongly that the connection between the steady state characteristics in the extreme left of the p plane and the characteristics at real frequencies is somewhat tenuous. It should be possible to manipulate the characteristics at the extreme left of the p plane with considerable freedom without affecting the characteristics at real frequencies appreciably, if at all. These possibilities have been exploited in some branches of network theory. A description of these nethods, hov- ever is beyond the scope of this book. ----------------------------------------------------------- CHAPTER III FEEDBACK 3.1. Introduction THus and the following three chapters are devoted to a general analysis of feedback circuits and a discussion of the meaning of feedback. The princi- pal object of the analysis is the development of a general feedback theory in terms of the mesh or nodal equations of the amplifier as a whole without distinction between u and  circuits. This is attempted parfly because the mesh or nodal formulation is the most satisfactory one for analytical work, and partly because without such a general foundation it is difficult to pro- vide a satisfactory theory for the multiple loop circuits which appear with increasing frequency in current design practice. As an introduction to this discussion, however, the present chapter gives a summary of the familiar theory of feedback amplifiers in terms of t circuits and  circuits and also a description of some of the commonest feedback arrangements. This part of the discussion is given only in outline form since a general acquaintance with feedback circuits is assumed in this book.* 3.2. Elementary Theory  Feedback Circuits In its simplest form, a feedback amplifier can be regarded as a combina- tion of an ordinary amplifier, or u circuit, and a passive network, or B cir- cuit, by means of which a portion of the output of the/ circuit can be Line P1 '____/P2 Line FIG. 3.1 returned to its input. Such a combination is shown by Fig. 3.1. Both the  and  circuits are, of course, actually four-terminal structures. The circuits are represented by single lines in Fig. 3.1 for simplicity. When a portion of the output voltage is returned to the input, the circuit * See H. S. Black "Stabilized Feedback Amplifiers," B.&TJ., or "Electrical Engineering" for Jan. 1934, also U.S. Patent No. :2,102,671. Good textbook refer- ences are Terman "Radio Engineer's Handbook," or "Applied Electronics" by the Electrical Engineering Staff of M.I.T. 31 ----------------------------------------------------------- 32 NETWORK ANALYSIS CHAP. 3 may, in fact, break into spontaneous oscillation. In this event the circuit is normally inoperative as an amplifier. If we suppose for the moment that oscillations are avoided, however, the characteristics of the structure can be obtained without difficulty. It is merely necessary to recognize the fact that the operation of the u and/ circuits separately is fully defined by the voltages appearing across their terminals, without regard to the fact that they are parts of the feedback loop. For example, let/E0 and E represent, respectively, the signal voltage applied to the input and the final voltage delivered to the output, as is shown in Fig. 3.1, and let E represent any additional voltage supplied at the input by the return of a part of the output voltage through the fi circuit. Then the u circuit, operating as an ordi- nary amplifier, must satisfy the equation ER = (E0 + E). (3-I) Similarly, if we let/3 represent the transmission characteristic of the circuit, the voltage which it supplies at the input terminals must be given by & = ER. (3-2) Upon eliminating El between these two equations, we find ER = Eo + uE, (3-3) or in other words  Eo. (3-4) E - 1 - Without the / circuit, the output voltage would be given by E_ =/E0. We therefore have the Theorem: Feedback reduces the gain of an amplifier by the factor 1 - .* The quantity u can be called thefeed3ackfactor. It evidently repre~ * All the theorems in this chapter are to be taken as approximate, in the sense that they will be superseded by the more general propositions given in Chapters V and VI. We may also notice that in many statements of this theorem the factor by which the gain is reduced is written as 1 + #8. The choice of the sign of depends upon the way in which the phase shifts of the tubes are counted. Ordinary vacuum tubes give a phase reversal of the signal, in addition to any phase shifts contributed by the interstage impedances. In the standard/ circuit containing an odd number of tubes, therefore, there will be one net phase reversal. If this is in- cxuded as part of/ the factor appears as 1 -/13. If the phase reversal is counted separately, on the other hand, the proper expression is 1 +  Cf. Terman, loc. cit. p. 395. The term "feedback" will he used in the follow- ing chapters for a quantity analogous te 1 -- //. ----------------------------------------------------------- FEEDBACK 33 sents the transmission around the complete loop from the input of the amplifier back to the input again. In ordinary practice, ta/S is very much larger than unity. Under these circumstances, equation (3--4) is con- veniently rewritten as E0 eø - 1 - u// (3-5) and since the first factor on the right-hand side of (3-5) must be substan- tially unity in absolute value when u is large, we can conclude that the gain of the amplifier varies approximately inversely with the transmission through the  circuit or, in other words, is approximately proportional to the  circuit loss. The error in this conclusion due to the departure of I /(1 - u) [ from unity will be called the  ect or the u eor in subsequent discussion. Equation (3-5) evidently implies that the gain of the amplifier may be much affected by slight variations in the  circuit but that it is almost inde- pendent of variations in . In order to show this more clearly. we may differentiate (3-4), keeping  constant, to give dE n 1 d -  ß (3-6) In this equation, the quantities dEn/En and d/ evidently represent corre- sponding changes in the amplifier gain and in the gain of the u circuit when both gains are expressed in logarithmic units, such as nepers or deci- bels. We therefore have the Theorem: The variation in the final gain characteristic in rib, per db change in the gain of the  circuit, is reduced by feedback in the ratio (1 - g): 1. The final property of feedback of fundamental engineering importance is the fact that it reduces the effects of extraneous noise or nonAinear dis- tortion in the u circuit. In a broad physicM sense, extraneous noise and non-linear distortion in any element can be regarded as "variations" in that element, and the sensitiveness of the circuit to such variations is always correlated with its sensitiveness to normal variations in the value of the element.* Fundamentally, therefore, this property is merely a reflection of the theorem just established. In order to demonstrate it independently, however, let it be supposed that a generator D0 is inserted somewhere in the interior of the  circuit as shown by Fig. 3.2. D0 may represent either an extraneous noise voltage, such as would be produced, for ß This is shown generally in Chapter V. ----------------------------------------------------------- 34 NETWORK ANALYSIS CH,v. 3 example, by a bad contact or by hum in the power supply, or it may be taken to represent the voltages of the modulation products arising from non-linear distortion in the u circuit. Let Ed represent the actual output Line Line Fro. 3.2 voltage which appears on the line in consequence of this noise generator and let D represent the additional voltage which appears between /q and by transmission around the u loop. Since the total voltage at this junc- tion is Do q- D and the gain between this junction and the output line is 2, we must have E = 2(D0 + D). (3-7) The voltage D which is returned to the junction by transmission through the  circuit and through  is evidently given by D = t3E,. (3-8) Upon eliminating D we therefore have /d -- (3-9) 1 -  where u has been written for the total gain 2. Since the noise which would appear in the output in the absence of feedback is g2D0, this result is equivalent to the T/eorem: The noise level in the output of a feedback amplifier is reduced by feedback in the ratio (1 - ): 1. We cannot conclude from this that the signal-to-noise ratio is reduced by this factor, because feedback may also change the effective signal level in the z circuit. An accurate statement can, however, be easily obtained by comparing the structure with a non-feedback amplifier which has the same final gain /(1 - /) and the same input and output voltages 150 and Ea. The comparison is made most easily if we suppose that the complete circuit is broken up into/t and/2 portions, as in Fig. 3.2, having respec- tively the gains 1 - / and g/(1 - ). Then since both /2 and the comparison non-feedback amplifier have the same gain and deliver the same output voltage E, they will have the same signal levels throughout, and we can conclude that feedback is fully effective in improving the signal- to-noise ratio for any noises originating in this part of the circuit. In , ----------------------------------------------------------- FEEDBACK 35 on the other hand, the signal level is less than it is in any portion of the comparison amplifier and the improvement in the signal-to-noise ratio for noises originating in this portion of the  circuit is consequently only partial. At the input terminals of the first tube, where the signal is also reduced by the factor 1/(1 - ufi), feedback has no effect on the signal-to- noise ratio. Feedback is thus a useful tool in combating troubles due to modulation and perhaps power supply hum, in the case of tubes with directly heated cathodes, which are characteristic of output stages. It is of little value, however, in dealing with noises due to thermal agitation, shot effects, etc., which may be expected to be troublesome in the input stages. The engineering importance of feedback circuits results from the possi- bilities they present of diminishing markedly the effects of noises or varia- tions in gain in the  circuit. The decrease in the external gain which follows from the use of feedback is unfortunate and makes it necessary in general to use a more complicated  circuit to obtain adequate final gain. This, however, is an easy sacrifice to make to secure the improvements which are available in other directions. As an example, we may consider an amplifier having 40 db external gain and 40 db feedback. The u circuit is then required to furnish 80 db gain, so that it represents an increase of 2 to 1 over the gain which would be required of a non-feedback amplifier. For this 2 to 1 increase in the complexity of the tz circuit, however, we secure an improvement of 100 to 1 in its effective linearity and gain stability. 3.3. Types of Feedback Circuits The principal circuit configurations useful in feedback circuits can be classified most easily in terms of the way in which the  and fi circuits are connected to each other and to the line at the ends of the amplifier. The ß ,. O 1 ,.,r,e o'1 ..... L.2 ....... 1'3 f ' ' ' ' Fro. 3.3 varieties of connections which may be made do not appear very clearly from a single line drawing such as that of Fig. 3.1. Physically, however, the/ circuit, the/ circuit, and the line must all be two-wire circuits. The actual situation is therefore that shown broadly by Fig. 3.3 in which the three circuits are connected together by means of a six-terminal network. The classification of feedback circuits thus depends upon the forms which these six-terminal connecting networks assume. ----------------------------------------------------------- 36 NETWVORK ANALYSIS C,AP. 3 There may, of course, be an unlimited variety of six-terminal arrange- ments to select from. The simplest ones, and the ones which appear to be most useful arc, however, shown by Figs. 3.4 to 3.8. In each structure, the terminals are labeled iu accordance with the notation used in Fig. 3.3. Figure 3.4, for example, shows a series type of feedback circuit. The t circuit is taken as a conventional three-stage amplifier, the internrage imped- 3.4 ances being indicated by lrl and 12. The IS circuit is represented for con- cretehess as the w of branches ., B, and C, but it may, of course, reduce to a single branch or it may assume a still more elaborate form. Tbe effective line terminals of and e'-f' are indicated at the high sides of the trans- FI. 3.5 formers since the line and transformer characteristics evidently add di- rectly.* The characteristic feature of this amplifier is the fact that the g and 19 circuits, as seen from the line, are in series at each end of the amplifier. Figure 3.5 shows a shunt type feedback system. The fl circuit is here represented as a T, but, as in Fig. 3.4, it may in general be taken as any * It is also possible to feed back on the low sides of the teamformers. In thla case the transfomxers become part of the  circtfit. ----------------------------------------------------------- FEEDBACK 37 four-terminal structure.* The characteristic feature of this type of feed- hack is the fact that the t circuit,/ circuit, and line are all in parallel at each end of the amplifier. Series and shunt feedhack circuits are the simplest and probably the most convenient arrangements for most applications. In ordinary circumstances they are also the circuits which give a maximum amount of feedhack. They suffer, however, from two major disadvantages. The first, which is Fro. 3.6 discussed in more detail in Chapter V, is the fact that in these circuits feedback changes the impedance of the amplifier as seen from the line to either a very high or a very low value. They are thus not convenient arrangements to use with amplifiers which must have a good reflection coecient against the line. The second is the fact that the line impedances form a part of the  loop. Variations in the line impedance may therefore affect the u characteristic and in some cases the effect may be great enough to cause instability. These difficulties are overcome by the use of a bridge type feedback circuit, such as that shown by Fig. 3.6. This circuit includes three new branches, represented by Z, Z, and Z4 in Fig. 3.6, at each end of the amplifier. A fourth branch, which is represented by Z, is also included to permit control of the input and output impedances of the  circuit if neces- sary. The three new branches, together with the impedances of the  circuit, the g circuit, and the line, give a network having a total of six * It should he noticed, however, that if the  circuit in Fig. 3.4 were chosen as a T, or that in Fig. 3.5 as a , the extreme branches could in either case be assimilated as part of the line impedances. Since the insertion of unnecsary impedanc in the line is likely to waste power, it is clear that thee are unacceptable configura- tions unless the contributions of the extreme branches are so small as to be almost meaninglis. The configurations actually shown in Figs. 3.4 and 3.5 are thus repre- sentative of those which would be appropriate in practical cases. These considera- 'tions are discussed in more detail in subsequent chapters. '  See the discussion of the effect of omitting Zx given later in Chapter V. ----------------------------------------------------------- 38 NETWORK ANALYSIS caAv. 3 branches. If anyone of the six is taken as a generator impedance, the remain- ing five can be arranged as the four arms of a bridge plus a gMxranometer arm. For example, if the generator impedance is taken as the line the galvanometer arm becomes the 3 circuit impedance. When the bridge is balanced in this arrangement the / loop is independent of the lit*e imped- ance. The conjugacy between the line and the 3 circuit also destroys the effect of feedback on the amplifier impedance so that it becomes compara- œ Fro. 3.7 tlvely easy to secnre a moderate impedance which can be adjusted to match a given line by controlling the elements in the bridge. The bridge type circuit suffers from the general disadvantages that it may require extreme impedance levels and that a portion of the output power may he consumed by the branches added to secure a bridge bance. 'These difFtculties can be ameliorated By replacing the bridge by a three- winding transformer or hybrid coil. In view of the several known equiva- lences between a bridge and a three-winding transformer there are several way in which this substitution may be effected. Figure 3.7, for example shows a" high side" hybrid coil feedback. In this case Z, represents the "balancing" impedance. Figure 3.8 shows a" Iow4ide" feedback. In the preceding figures, the same circuit connections have been shown at each end of the amplifier as a matter of simplicity. The number of available configurations, however, is much increased by the possibility of ----------------------------------------------------------- FEEDBACK 39 combining different connections at input and output. For example, Fig. 3.9 shows a series connection at the input terminals in combination a I d I e I Fro. 3.9 Fro. 3.10 with a shunt connection at the output. Figure 3.10 shows a combination of series input and hybrid coil output. 3.4. Cathode Feedback Circuits In addition to these general arrangements, a wide variety of other feed. back circuits may be used in practice. A particularly important example, for practical purposes, is furnished by the so-called "cathode" feedbacks. These may exist in two forms, depending upon the number of stages in the 3t circuit. In either case, the arrange- ment is essentially a modification of a series feedback amplifier. Figure 3.11.4, for example, shows a series feedback for two stages in compari- son with the corresponding cathode feedback shown by Fig. &liB. The  circuit is represented by the single branch Z. In this instance, the cathode connection is used to secure aphase reversal. As the discussion in Chapter I pointed out, the successive Fie. 3.11 ----------------------------------------------------------- 40 NETWORK ANALYSIS Cu.. 3 tubes in the 3t circuit produce successive phase reversals. With an odd number of tubes it turns out that the net resulting phase is of a sign suitable for feedback without instability. If there are an even number of stages as shown by Fig. 3.11.4, however, the current delivered by the  circuit has the wrong sign for direct return to the input. This is avoided in Fig. 3.1lB by crossing the terminals in the g circuit to secure an additional phase reversal. The circuit is called a "cathode" feedback because the cathode of the first tube is off ground.* Fo. 3.12 The use of a cathode feedback circuit to replace a corresponding series feedback circuit when the  circuit contains an odd number of stages is shown by Fig. 3.12. Here the cathode feedback is introduced principally to minimize distributed capacities to ground. As Fig. 3.12.4 shows, the conventional series feedback circuit is grounded at the cathode junction, P1. The junction P2, to which the transformers are connected, is off ground and their capacities to ground fall effectively across the  circuit. No improvement is obtained by transferring the ground terminal from _P to P2 because this leaves the ground capacity of the  circuit, which is at least equally large, to be accounted for. The total capacity can, however, be minimized by grounding most of the forward circuit in the manner shown by Fig. 3.12B. Since the cathodes of both input and output tubes are off grotnd there is no net phase reversal. A special feature of the cathode circuits is the fact that some feedback may exist for the tubes whose cathodes are off ground even when the remaining tubes are dead. Thus in Fig. &lib the plate current for the * We can evidently cross terminals without a change in ground by including a transformer in the loop. In ordinary situations, however, the inclusion of a trans- former so restricts the available feedback, as determined by the methods described later, that Fig. 3.11 represents a preferable solution. ----------------------------------------------------------- FEEDBACK 41 first tube can return to its cathode only by flowing through the S circuit impedance, so that some voltage would be returned to the first grid even if the second tube were removed. In Fig. 3.12B a similar situation holds for both the first and third tubes. Speaking rather roughly, we can suppose that the S circuit impedance operates independently in producing this residual feedback and in produc- ing the principal feedback. For example, Fig. 3.13 gives the approximate equivalent of Fig. 3.12B under this method of treatment. It is obtained Fro. 3.13 from the original series feedback amplifier of Fig. 3.12A by inserting new impedances equal to the  circuit impedance in the cathode leads of the first and third tubes. The first and third tubes can evidently be regarded by themselves as miniature feedback amplifiers of the series type. These tubes thus have more total feedback than would appear if we considered only the transmission around the principal loop. On the other hand, since the local feedback reduces their gain, the transmission around the principal loop will be decreased unless some compensating change is made. 3.5. Multiple Loop Feedback Amplifiers The circuits of Figs. 3.1lB and 3.12B are examples of multiple loop amplifiers, or in other words of amplifiers in which voltage can be returned to some of the grids by more than one path, so that the effective feedbacks on the various tubes are different. In these particular structures the subsidiary paths are accidental results of the type of feedback connections adopted. In current amplifier development, however, there appears to be an increasing tendency to turn to multiple loop circuits deliberately in order to obtain results not available from single loop structures. One simple type of multiple loop structure is shown by Fig. 3.14. The circuit is a series feedback amplifier with additional feedback on the last tube through the insertion of an impedance in its cathode lead. The structure is thus similar to the" equivalent "amplifier previously shown by Fig. 3.13, except that since the local feedback is now produced by the impedance Zt2 , which is independent of the principal feedback impedance Z, it can be chosen arbitrarily. We can look upon the circuit as a device ----------------------------------------------------------- 42 NETWORK ANALYSIS C.^.. 3 for securing more reduction in the non-linear distortion in the last tube than can be obtained, according to the principles laid down later, by feedback around the main loop alone. P'w,. 3.14 Figure 3.1S shows a second type of multiple loop structure. It is similar to that shown by Fig. 3.14 except that thelocal path represents shunt rather than series feedback. The subsidiary path may be regarded either as a F. 3.15 branch deliberately added to improve the characteristics of the output tube, as in Fig. 3.14, or as a representation of a large parasitic grid-plate capacity, such as is found, for example, in the power triodes used for radio broadcasting. Fc,. 3.16 Still a third example is shown by Fig. 3.16. Here local series feedback is applied around the first two stages of the complete u circuit. We may imagine the local feedback to be regenerative, so that it provides a higher  gain around the complete loop than would otherwise be obtainable. In addition to the particular structures shown by Figs. 3.14 to 3.16, many other multiple loop amplifiers can evidently be secured either by combining ----------------------------------------------------------- FEEDBACK 43 two or three of the local feedback paths shown by these figures in a single amplifier or by providing still more paths. 3.6. Other Feedback Circuits The preceding sections have been intended as a brief sketch of the types of physical configurations directly envisaged in this book. They are com- posed characteristically of linear vacuum tubes and passive elements. Feedback circuits may, however, also be designed to include non-linear or non-electrical elements. Many of these are sufficiently similar in funda- mentals to a linear electrical circuit to be treated by the same methods, provided the proper precautions are taken. The diversity of applications will be indicated by two illustrations. The first consists of a feedback circuit including frequency translating devices. Figure 3.17, for example, shows a radio transmitter in which a portion of the output is de- modulated and returned to the signal input as voice fre- quency or "envelope" feed- back. If the modulator and demodulator are nearly ideal and the carrier frequency is Fro. 3.17 much higher than the voice band this can be analyzed essentially as a lin- ear circuit. It is merely necessary to consider the transmission of an equivalent voice frequency around the complete loop. If the modulator outputs include a variety of products which can be transmitted around the loop, however, or if the carrier frequency is within a few octaves of the top of the voice band, the situation is more complicated and will not be considered here. The second general example is furnished by regulator circuits for such purposes as speed, voltage, or frequency control. Here the fact that the control circuit acts as a valve, producing a large change in output for the comparatively slight expenditure of energy required to change the control, gives an equivalent of vacuum tube amplification. The use of a portion of the output to adjust the control circuit is, of course, feedback. There is no definite useful bat,d, in the sense in which this term is ordinarily understood in communication circuits, but an approximate effective band can ordinarily be assigned the circuit from a consideration of the rapidity with which the controls should operate. The essential problem of course, is to avoid hunting, which is the equivalent of instability in a feedback amplifier. ----------------------------------------------------------- CHAPTER IV MATHEMATICAL DEFIrIT0I OF FEEDBACK 4.1. ]ntroduction TH conception of a feedback amplifier developed in the preceding chapter can be summarized in the following words: The amplifier consists of a forward or  circuit and a backward or  circuit. The feedback can then be determined from the product /, which represents the transmission around the complete loop formed by the .and y circuits together. The circuit has the fundamental physical property that the effects of variations in the  circuit, whether they are taken as changes in the normal/ gain or as departures from strict linearity or from freedom from extraneous noise, are reduced by the factor 1 -  in comparison with the effects which would be observed in a non-feedback amplifier. This set of conceptions is almost indispensable in describing a feedback amplifier or in reasoning generally about the functions of the various parts. They will be retained here for this general purpose. For future analytical work, however, they are extended in this chapter to provide a purely mathe- matical definition of feedback. The mathematical definition is framed in terms of the general mesh or nodal equations introduced in'the first chapter. The system of equations is taken with reference to the complete amplifier, without distinction between t and  circuits, so that these conceptions disappear from the formal analysis. This change is made for two reasons. The more obvious one is the fact that the mesh or nodal analysis furnishes a convenient foundation for further theoretical work. It is especially appropriate in discussing the relationship between feedback and stability. The second reason for developing a general definition of feedback in terms of the equations of the circuit as a whole is that it allows us to avoid the ambiguities and uncertainties which appear if we rely exclusively upon an analysis in terms of separate/ and  circuits. The/ and t analysis supposes that these circuits are clearly distinguishable entities to which can be ascribed definite properties independently of one another. This was suggested, for example, in the generalized sketch shown by Fig. 3.1 of the preceding chapter. In fact, however, the actual physical configurations shown by the figures which appeared later in the chapter do not permit such a clear-cut separation letween the two circuits so that what we are to ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 45 call/ and what fi remains somewhat vague. Since the properties of gain stabilization and distortion reduction hold only for the  circuit, and the eventual gain is determined by the/S circuit, this is a matter of considerable importance. The simplest example of the diftSculty of distinguishing sharply between tz and  is furnished by the computation of gain from the familiar equation œ0 __ 1 - The computation requires a knowledge of  and u. The product , representing the transmission around the loop, is itself well defined. The  which must be used in order to make the equation an accurate expression for the amplifier gain is, however, not so apparent. h depends in part upon the way in which the current divides in the six-terminal connecting net- works shown at the ends of the amplifier in Fig. 3.3 of the preceding chapter. In evaluating u we must therefore make some allowance for the  circuit impedance, instead of removing  entire17, since otherwise the division of current in these networks will, in general, be changed. For particular cir- Fro. 4. cuits this can be examined by setting up detailed circuit equations, but without further theoretical study it is dicuh to see, in general, just what branches of the  circuit should be included in making the allowance, and in ally event it is clear that the problem of designing a  circuit to give a specified external gain characteristic may be confused by the fact that any elements we put in affect both  and The diculty of separating the amplifier into  and fi parts may become much greater in a multiple loop structure containing several feedback paths. A particularly extreme example is furnished by the cathode feed- back circuit shown by Fig. 3.12B in the preceding chapter. As drawn there, the circuit includes only the elements which would be supplied in the design process. In a physical embodiment, however, it would be necessary to consider also the parasitic capacities between grid and cathode and between plate and cathode in each tube. When these are added the circuit appears in the form shown by Fig. 4.1. For design purposes it is possible to divide the elements of the circuit into a group which is most important ----------------------------------------------------------- 46 NETWORK ANALYSIS C.AP. 4 in determining forward gain and another which is chiefly effective for feed- back. It is clear, however, that no sharp division into  and fi circuits can be made. Every element in the structure enters to some extent into both forward and backward transmission. 4.2. Return Ioltage and Reduction in Effect of Tube Fariations The consideration of multiple loop structures leads to another reason for developing 'a general mathematical definition of feedback, which may be less obvious than those previously discussed. In a single loop structure the fundamental quantity appears to be the loop transmission/z/. This is the same as the return voltage which would appear by transmission around the complete loop if we applied a unit voltage to any grid and opened the circuit just behind it. In such a structure we know that the factor measuring the reduction in the effect of tube variations is 1 - #/, so that it is always closely correlated with the return voltage. In a multiple loop structure voltages may be returned to the grids of the tubes by various paths which differ from tube to tube. For any particular Fro. 4.2 tube, however, the total return voltage can be obtained, at least on paper, by adding together the contributions from all available paths through the network. This is illustrated by Fig. 4.2. N represents the complete circuit exclusive of the tube in question and Pt and P2, connected to- gether, the grid terminal for normal operation. The return voltage can then be defined as the voltage which would appear between P and G in response to a unit voltage between P2 and G when the connection between -Pt and/2 is broken. The grid-plate and grid-cathode capacities C and Cs are shown as going to P to indicate the fact that opening the loop should not disturb the admittances seen from the end point P. Given any individual tube, it is also possible to determine the ratio between a prescribed small variation in its gain and the resulting change in the transmission characteristic of the complete circuit. It is natural to suppose that the correlation between this ratio and the return voltage on the tube will be the same for a general circuit as it is for a single loop structure. Thi is substantially true in the simplest and most common circuits. In exceptional circuits, however, the actual effect of individual tube variations on the final transmission characteristic may be much greater or much less than would be predicted from the return voltage. One of the objects in setting up a general mathematical definition of feedback is therefore to determine when the return voltage computation is a reliable ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 47 index of the effect of tube variations and what corrections must he applied when it fails. One other aspect of the general situation deserves attention. Since the vacuum tubes are ordinarily the most variable and non-linear constituents of a complete amplifier, feedback is of engineering importance chiefly in correcting for their characteristics. An incidental result of the application of feedback, however, is the fact that it also reduces the effect of variations in some of the bilateral elements of the circuit. The effects of variations in the elements of an interstage impedance, for example, are reduced by feedback to the same extent as are those of variations in the transconduct- ances of the associated vacuum tubes. In any discussion of the relation between feedback and the effects of element variations, it is therefore legitimate to extend consideration to bilateral as well as unilateral elements. The analytical treatment of feedback developed in this chapter applies, in fact, equally well to elements of either type. In order to simplify exposi- tion, however, each step in the development is introduced as though uni- lateral elements only were in question, the extension of the analysis to bilateral elements being described subsequently. 4.3. Return Ratio, Return Difference, and Sensitivity The preceding section has indicated that the usual conception of feedback includes two distinct ideas. The first is that of a loop transmission or return of voltage, and the second that of a reduction in the effects of varia- tions in the tube characteristics. In normal circuits these two are related by simple mathematical laws so that the term "feedback" can refer generically to both. In exceptional circuits, when the correlation between the two breaks down, the first idea is evidently the one which most nearly agrees with the usual physical conception of feedback. It will therefore be taken as the basis for the definition of feedback in the general case. To prevent any possible confusion, this idea will also be described by the new name' return difference. It is still worthwhile, however, to retain the general idea of a reduction in the effects of tube variations. This will be referred to by the name sensitivity. The return difference, or feedback, and the sensitivity will be repre- sented by the symbols F and S, respectively. They are to be regarded as the analogues, in general, of the quantity 1 -  in a single loop structure. Thus, "return difference" is an abbreviation for "return voltage differ- ence," meaning by this the voltage difference existing between P1 and P= in Fig. 4.2 under the conditions of measurement indicated there. The quantity I - ta, rather than  itself, is chosen as the fundamental unit, because it turns out to lead to simpler and more compact formulae in most ----------------------------------------------------------- 48 NETWORK ANALYSIS situations. In order to have a symbol corresponding to the loop trans- mission # itself, however, we will also write F = 1 + T. Tires, T = -#8 in an ordinary amplifier.* T will be called the return ratio. To complete the nomenclature, we might similarly introduce a symbol for the quantity $ - 1, but the number of occasions when such a symbol would be useful is too small to make this step worthwhile. 4.4. Definitions of Return Ratio and Return Didference In order to secure more precise definitions of the quantities described in the preceding section, let the input of the general circuit be taken as the first mesh or node, and the output as the second mesh or node. We will also suppose that the grid and plate terminals of the tube under examina- tion are labeled respectively 3 and 4, and tha. t its transconductance or mutual impedance is represented by/4 7. //? is thus a constituent of Z4a or Y4a in the general system of mesh or nodal equations. In later sections the definitions of re'turn ratio and return difference will be extended to bilateral elements. The form of these statements remains the same when IF is a bi- lateral element, except that it is taken as a constituent of the self-impedanc= or admittance Zaa or Yaa, rather than of the coupling term Za or The loop transmission or return voltage in Fig. 4.:2 can be obtained by multiplying the transimmittance,/V, of the tube itself by the backward transmission fi'om the plate to P. In making the latter calculation, the open circuit which appears between Pt and Pa can evidently be represented by supposing that P and P are connected together as in normal opera- tion, but that the tube is dead. If we let A ø represent the circuit deter- minant when/47 = 0, therefore, equations (1-10) and (1-24) of Chapter I give the backward transmission as Aa//xø. Since the negative sign intro- duced by the phase reversal in the tube is canceled by the fact that T is analogous to -, we therefore have a4a (4--1) F= 1 '4'- T= 1 -q-k// Ao'' But it follows from the discussion in connection with equations (1-11) to (1-14) that 0 q_/f&x4a is the value which the circuit determinant assumes when..;the tube transimmittance has its normal value////. If we represent the normal circuit determinant by the usual symbol /, therefore, equa- * The introduction of the minus sign may be explained by t]e fact that an ordinary feedback amplifier contains an odd number of tubes, which contribute an odd number of phase reversals to the loop. Thus T, as defined, is equal to the loop transmission without these phase reversals, and will ordinarily be a positive quantity except for the effect of possible phase shifts in the interstage or feedback networks. The sign chosen for T is also more convenient in dealing with bilateral elements.. ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 49 tion (4-1) can also be written as F = a-  ß (4-2) In order to emphasize the importance of this last formula, and to pave the way for the treatment of bilateral elements in a subsequent section, the relation embodied in (4-2) will be restated as the Definition: The return difference, or feedback, for any element in a complete circuit is equal to the ratio of the values assumed by the circuit determinant when the specified element has its normal value and when the specified element vanishes. Equation (4-2) probably represents the most convenient working for- mula for the analytic treatment of feedback. A number of examples of its use in feedback circuit analysis will be given in the next chapter. The fact that the equation expresses F in terms of the determinant of the system is particularly convenient in studying the relation Between feedback and stability since, as we shall see later, the roots of the determinant tell whether or not a system is stable. The formula is also especially useful in studying multiple loop systems, since if we once know the determinant we can readily evaluate the individual feedbacks without making a complete sepa- rate calculation for each tube. 4.5. Return Difference for a General Reference It is convenient to introduce here a generalization of the conception of return difference whose meaning will probably not be fully apparent until a considerably later point. In developing equation (4-1), we based the calculation, in a sense, upon the reference condition of the circuit obtained . by setting/F = 0. Thus the backward transmission from plate to grid was obtained for this condition of the circuit, and the forward transmission/4 , by which the backward transmission was multiplied to produce the com- plete loop gain, may be thought of as/3' - 0, or the surplus of the actual tube transimmittance over this reference value. We can evidently perform a similar computation for any reference con- dition/? -- k. The "loop gain," then, becomes the effective transimmit- rance,/F - k, multiplied by the backward transmission from plate to grid evaluated for the condition/g = k. Since the tube is no longer completely dead, this backward transmission must include the effects of a certain amount of physical feedback, but this is a practical rather than a theoretical complication. The reference k can be anything we like. For example, it might be the value of transimmittance at which the tube would be dis- carded in favor of a new one, or it might be the transimmittance which ----------------------------------------------------------- 50 NETWORK ANALYSIS c,Ap. 4 would lead to a certain specified gain through the over-all circuit. The latter condition is the one which will be used in future applications of this concept. The return ratio and return difference resulting from this computation will be spoken of as the return ratio and return difference of kk' for the reference k. If F represents this return difference, we evidently have witere A k is the value assumed by A when  k. But since h  = xø q- k4a and A = zX ø q- kFA.3, where fi0 is, as betore, the value of 3. when W = 0, equation (4-3) can be rewritten as F = . (44) This equation is obviously analogous to (4-9) and, like (4-2), will be regarded as a definition in future discussion. Equation (44) leads to an easy method of cmnputing the return differ- ence for the reference k from the return difference for zero reference. Thus, if we multiply and divide the right side of (4=t) by aø: we have A .0 (4-5) F() Stated in words, this result is the Theorem: The return difference of/4 / for any reference is equal to the ratio of the retutu differences, with zero reference, which vould be obtained if F/7 assumed, first, its normal valu% and, second, the chosen reference value. The conception of a return difference for a reference other than zero will be utilized at the end of this chapter. Meanwhile, it cau be assumed that the term" return difference" applies only to the zero reference ease. 4.6. Return Difference for a Bilateral Element In setting up equation (4-2) as a definition of return difference we evi denfly extended the analysls formally to bilateral as well as unilateral ele, ments, since A and A ø are meaningful quantities for elements of either type. The physical significance of the return difference of a bilateral element, on ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 5I the other hand, is most easily studied if we replace (4-2), for a bilateral element, by an equation more nearly in the form of (4-1). This is readily done. Thus, if IF is a constituent of Y3a or Zaa, we evidently have zX = zX ø +/FzX s in the bilateral case. Substitution of this relation in (4-2) gives F=I+T= I+IF-, (4-6) which is like (4-1) except that A43 is replaced by/xsa. The meaning of the return difference for a bilateral element is easily understood from an examination of the terms in (4-6). Let it be sup- posed, for example, that IF represents an impedance. Then Xø//xaa repre- sents the impedance which would be seen by a generator in the mesh con- taining IF if/4/were zero. In other words, it is the impedance which IF faces. The return ratio T = IFAaa/X ø is therefore equal to the ratio of the impedance IF to the impedance presented to IF by the rest of the circuit. The return difference F is equal to the ratio of the complete imped- ance, including IF, to the impedance of the external circuit. Similarly, if IF represents an admittance, the return ratio T and the return difference F are, respectively, equal to the ratio of the admittance IF to the admittance of the rest of the circuit, and the ratio of the admittance of the complete circuit, including IF, to the admittance of the rest of the circuit.* Viewed in this light, the conception of return difference for a bilateral element appears as an expression of the fact that a generator with internal impedance cannot be fully effective in driving an external circuit. The internal voltage drop is the" returned" voltage. It is "returned" to the source in the sense that it is unavailable to drive the external circuit. Thus, suppose that IF is the impedance Z and that the impedance of the external circuit is represented by Z0. In the absence of Z a unit generator would produce a current 1/Zo in the circuit. The insertion of Z into a cir- cuit carrying this current is equivalent to adding or" returning" the volt- age -Z/Zo to the source. The current strength is not supposed to be changed when Z is added since this is the logical equivalent of opening the loop in the unilateral case to prevent the return voltage itself from produc- ing a response. The return difference is then the difference between the * These relations hold, of course, for both active and passive circuits. If the circuit does in fact contain vacuum tubes, however, it is important to notice that the imped- ance assigned to the external circuit must be the active impedance obtained when the tubes are lit. This may be quite different from the impedance which would be obtained from the passive elements alone. Methods of computing the active imped- ance from the passive impedance are described in the next chapter. ----------------------------------------------------------- 52 NETWORK ANALYSIS Ct,. 4 original and the returned voltage and measures the net voltage available to drive the external circuit. 4.7. Definition of Sensitivity We turn now to the second leadJug conception of the present chapter that of sensitivity. This concep6on can be illustrated by reference to equation (3-6) of the preceding chapter, which appeared as dE  l .... (3-6) Evidently, the equation states in effect that I - ttB is the factor relating any given percentage variation in the t circuit to tire resulting percentage variation in the output voltage. In other words, i - t0 is a measure of the sensitiveness of the over-aI1 circuit to small variations in #. Equation (3-6) is, of course, [mited to the g elements in an ordinary feedback circuit. In order to generalize appropriately to any circuit, let the gain through the complete system be represented by & We then have the Definition: The sensitivity, S, for an element .r is given by l s = (4-7) c log The definltlon is intended to apply to both unilateral and bilateral elements. The relation between (4-7) and (3 6) may be made more apparent if we express 0 in terms of the logarithm of the output voltage En, and replace the partial derivative by ordinary differentiation, on the assumption that /F is the only element in the circuit which varies. This allows (4-7) to be written as dE 1 E S I' (4-8) Thus, S is the ratio between a given percentage change in .F, in the general case, and the resolting percentage change in tlre delivered voltage Ee, just as 1 - / expresses the corresponding ratio between changes in in the special case of the single loop amplifier. In an average situation, we may expect S to be of the order of magnitude of unity. In an ordinary non-feedback amplifier, for example, the over-all gain varies by I db for each db change in the gain of any one of the tubes, and S for any tube is evidently 1 exactly. On the otlrer hand, $ may be much greater than unity. Thus, ignoring phase angles, if the final gain ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 53 varies by 0.01 db for 1 db variation in///7, the sensitivity S is 100. This is the result we would expect for the elements in the forward circuit of an amplifier with 40 db feedback. We might also secure such a res'ult, how- ever, even in a purely passive circuit, if//7 were an impedance element having comparatively little to do with the over-all transmission. It is also possible for S to be much smaller than unity. This might occur, for example, in a regenerative amplifier at the point of singing or in an ordinary circuit which depends on a critical bridge balance or on sharply tuned teaclance branches. It is to be noticed that in the discussion of the return difference we labeled the input and output terminals of the system, but the input and output terminals did not actually enter into the analysis. Since the sensitivity, on the other hand, depends upon the transmission through the circuit, it must in general depend upon the nodes or meshes which we choose to regard as the terminals of the system, as well as upon the chosen element //7 itself. 4.8. General Formula for Sensitivity The definition of sensitivity given by equation (4-7) can be made more concrete by an examination of the functional relationship between 0 and If we retain the notation used in the preceding sections and represent the output impedance or admittance by///R, the gain through the circuit can be written in the general form e0 = a k/-"R. (4-9) The discussion of Chapter I shows, hoxvever, that both zX2 and/x must be linear functions of P/". If we let Xø2 and /x ø represent the values of these determinants when/F = 0, we can therefore write equation (4-9) as e ø = Aø= + /4VA1248 //7 R. (4-10) A 0 q-- /P'A4a This equation of course holds for any value of p/7. For purposes of future discussion, it will be convenient to pay particular attention to the case when/f7 is zero. The gain under these conditions constitutes the so-called direct transmission gain.* If we let O0 represent this gain, we evidently have e0 0 = mR. (4-11) * So-called because it represents a current transmitted directly to the output, without the intervention of the element/. ----------------------------------------------------------- 54 NETWORK ANALYSIS Casv. Returning to the general formula (410), if we apply the definition of given hy (4-7) to it directly, the result, after some mmipulation, appears as 1 (A=  'a)(A a + a) (12) This can  almplifi by means of a general idmxtlty in determinant theory, wch is of frequent application in network anysis. The identity where  is aoy determinant, a and c a any two rows of , d b and d any two columns of . If we let 0 of (4-12) be the gener deteiuant which appears in this equation and makc the proper identifications of s=bscrlpts, this allows (4-12) to be written as 8 = g/Aa (14) lfwe assume tat IV is a bilater dement in a or Yaa, rather than a nui- lateral element in Za or Ya, 1 the steps from (9) to {14) can reated exactly, except that each subscript 4 is replied by a subscrlpt 4.9. Retrn Derence nd Sensitivity in Simple Cases Th genial formula (14) in the prdlng section was developed lely as a mtter of completenms. In actual pratice, it is ordinmily easier to evaluat the scns{tivlty indictly from the tam dience. In general, te sensitivity and th mn dierence for  gen ement a not equal so that if we a to calculate R fm  it is rst nsary establish the raton Setween them. This will  te subjt of te next sever stions. For the moment, however it is convenient to dispose of the especially simple ce when thg two rc, in fct, u. This occurs when the dit transmission term (11) vanish. If we assume, that A2 is ze, the analysis of file pcding section becomes very much simpler. Thus, if we sutitute this ntion dictly in uation (12), we dily find A ø + Aa 8 ao A0 (4-15) This, however, is exactly the same formula as the one which was developed for the return ffergnce in equation ( 2). We therefo have the Tor: The sensitivity and return difference are equ for any ele- ment whose vanishing lda to  u'ansmlssion through the cidnit as a whole. * See f example, Stt and Mae They of ternlnas, p. . ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 55 The most familiar examples of elements meeting this condition are probably the tubes in the forward circuit of an ordinary feedback amplifier. We can assume, for practical purposes, that the transmission through the structure will be zero if any one of the tubes fails. In strict accuracy, this is seldom exactly true. Some current will ordinarily* trickle through the / circuit into the load, even when/ = 0. This trickle, however, is usually so much smaller than the normal output current that it can be neglected, so that the forward circuit can be regarded as falling within the scope of the theorem for practical purposes. In this case, of course, the theorem express- es nothing new. Since the theorem requires no assumption except that of negligible direct transmission, however, its application can evidently be extended to circuits which differ fairly substantially from the conventional single loop configuration. In the field of bilateral elements, simple examples of the theorem are obtained from series-shunt or ladder networks. We can obtain zero trans- mission when kF = 0 in circuits of this type by . adopting an impedance analysis if///represents    an element in shunt, or an admittance analysis if /3' is an element in series. A specific example is furnished by the circuit of Fig. 4.3. The transmission is supposed to take place from Z1 to Z, while Z represents the Fro. 4.3 variable /7. The return difference is an impedance ratio which can be written down by inspection as F = Z + Z2 _ ZZ2 + Z(Z + Z) . (4-16) ZZ ZZ2 Z, + Z2 On the other hand, the current flowing in Z2 in response to a unit generator in Z1 is given by Z e ø = ß (4-17) ZZ + Z(Z + Z) Hence, Z1Z2 dZ dO = ZtZ2 + Z(Zt + Z2) ' (4-18) Since the coefficient on the right-hand side of (4-18) is I/S, by (4-8), the theorem is verified for this case. * That is, in the absence of a balanced bridge at either input or output. ----------------------------------------------------------- 56 NETWORK ANALYSIS cvxp. 4 A second example, this time for a bridge circuit, is furnished by Fig. 4.4. The transmission is from Zt to Zs and the variable element is taken as 4.4 Z, the remaining impedances being so chosen that the bridge is balanced when Z vanishes. For simplicity, let every impedance but Z be taken as 1. This makes Z and Z4 con- jugate so that Z, can be removed in deter- mining the impedance wlfich Z faces. Wid the help of this simplification, we readily find that Z faces the impedance 2. We therefore have 2 dZ  z z (4.49) This can be verified by direct consideration of the transmission equations for the bridge, but the algebra is too lengthy to be included here. 4.10. Circtdts vith lppredable Direct Transmission , We turn now to situations in which the assumption of negligible direct transmission is no longer valid. Instances of elements giving a substantial direct translnission term are readily found even in conventional single loop amplifiers. For example, the B circuit elements belong gentreally to this class, as do many of the elements in customary input and output circuits. In the field of passive circuits, the elements of bridge type networks are usually of this type.* More difficult situations involing a substantial mount of direct trans- mission may be found if  is the transimmittance of a tube in a multiple loop circuit. An example is shown by Fig. 4.5. The structure is drawn as a single stage feedback amplifier but it may also be taken as the last stage in the double loop feedback structure shown by Fig. 3.14 of the preceding chapter. The iaapedances Z and Z.5 can be regarded as the terminating impedances in the single stage cse. Z represents the feedback branch and Z and Z, are, of course, parasitic grid4:athode and plate-cathode irapeal- antes. When the gain of the tube vanishes, the circuit reduces to the form shown by Fig. 4.6 and in the slng[e stage case the transraission through this net- work evidently represents the quantity eaa defined in equation (4-11). By proper adjustment of the elements Ze, g3 and Z, the transmission thxugh this path can be made anything we like in comparison with that through * That is, in the absence of special situations like that of Fig. 4.4, whexe the bridge is supposed to balance when the variable element is zero. ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 57 the tube. For example, if Za is very small while Z2 and Z4 are quite large: the direct transmission becomes insignificant. If we make Z2 and Z4 small enough, however, and Za very large, it may be much more important than the transmission through the tube. By proper adjustment of the impedances, we can also secure an intermediate case in which the two paths exactly cancel, so that the net output under operating conditions is zero. In ordinary physical cases, Za will, of course, be small, while Z2 and Z4 will be quite large so that we can regard the directly transmitted current as being much smaller than that flowing through the tube. 4.5 Fm. 4.6 When the circuit represents a complete amplifier, this means that the directly transmitted current can be neglected in any ordinary situation. If the circuit is the last stage of a multiple loop structure, on the other hand, the rest of the structure must also be considered in determining the direct transmission to the final output impedance. In this case, even a slight trickle of current directly through the passive elements of Fig. 4.6 may be important in some circumstances. The reason for making this distinction will appear in a later section. 4.11. General Relation between Sensitivity and Return Difference When the direct transmission is substantial, it is simplest to use it as a reference from which the remainder of the actual output voltage or current is calculated. We are then concerned explicitly only with the difference between the normal output and the directly transmitted term. Thus, from (4-10) and (4-11) we can write = o+a -  (AøA124a -- A2A4a ) This can be simplified with the help of the general relation (13). The result is eo _ eoo = Aø(A ø + A4a) ' (21) ----------------------------------------------------------- 58 NETWORK ANALYSIS C,Ap. 4 Let us now consider the" sensitivity" of the quantity e ø - e øø, using the term in our customary fashion to mean the ratio between a given percent- age variation in///and the corresponding percentage variation in e ø - e øø. As a function of/¾/, the right-hand side of (4-21) is very like (4-10) in the special case zXø; = 0. The only difference is the fact that z-Xls4a in (4-10) is replaced by - AaA/A ø in (4-21). But when we calculated the sensitivity from (4-10) for the special case ZXs = 0, we were led to (4-15), which does not depend upon Asa4. We may therefore draw the following conclusion: Theorem: The sensitivity of the difference, e ø- e øø, between the normal output and the direct transmission for any element k?is equal to the return difference for/¾/. This result, of course, includes our earlier theorem on circuits with zero direct transmission as a special case. If we begin with that earlier theorem, the present result is an obvious one for a circuit composed of two inde- pendent parallel paths, one of which contains/(/and has no direct trans- mission, and the other of which furnishes the over-all direct transmission and is independent of/3'. This is a situation which is very unlikely to occur physically, since there would almost always be interaction between the two paths at input and output terminals, if nowhere else, but the theo- rem states in effect that asy circuit can be thought of in these terms even when the physical separation into two independent paths cannot be achieved. The theorem just established can also be stated in an analytic form which is somewhat more convenient for purposes of calculation. It is obvious that if the output voltage of the system varies by a given amount, the per- centage change which the given variation represents will be inversely pro- portional to the output we are considering. Thus, the percentage changes in e ø and e ø - e øø, corresponding to a given variation in the element/¾/, will be in the same ratio as the quantities e ø - e øø and e ø. Since sensitivity is an inverse measure of percentage change, from (4-8), the result expressed by the theorem can therefore be transformed immediately to the relation F e 0 -- e 0ø e 0ø s- e ø (4-22) where S is, as before, the sensitivity for the complete output e ø. This result can also be established by direct calculation from equations (4-2), (4-10), (4-11), and (4-14). It holds for any circuit and for either uni- lateral or bilateral elements. Equation (4-22) is of particular interest as a means of estimating quickly whether the return difference is a reliable measure of sensitivity or whether a more elaborate calculation hould bg made. Since we are ordinarily ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 59 interested in the sensitivity only to within several db, we can say, in general, that the return difference will be a conservative measure of sensitivity as long as the absolute value of e øø is not greater than that of e ø. It will, however, be a very pessimistic estimate if the two quantities happen to be nearly equal in phase angle as well as magnitude. On the other hand, the sensitivity is much poorer than the return difference in circuits for which the absolute value of e øø is much greater than that of e ø. The use of equation (4-22) will be illustrated in more detail by a con- sideration of three different situations. As a first example, let it be sup- posed that///is the transconductance of one of the tubes in a normal feed- back amplifier. We m, ay suppose for concreteness that the normal gain e ø is 40 db. The transmission e øø which is obtained vhen /3' vanishes will depend somewhat upon the type of circuit which has been chosen. If either the input or the output is a balanced bridge, so that the fi circuit and the line are conjugate, for example, this quantity is zero. In other circum- stances it will not be precisely zero but we can estimate its value as -40 db from the general rule that the external gain is equal to the fi circuit loss. Thus, the ratio eøø/e ø is of the order of magnitude of -80 db and the dis- tinction between return difference and sensitivity is entirely negligible. As a second example, let it be supposed that/is in the fi circuit. It may be taken to represent a shunt impedance, a series admittance, or the trans- conductance of the final tube in the circuit shown later by Fig. 4.9. In any of these cases setting  = 0 opens the feedback so that e øø is much greater than e ø. Variations in//z are thus much more important in affecting the final transmission characteristic than a calculation of the return voltage would indicate. This is, of course, to be expected for elements in the fi circuit. The third situation is represented by the circuit shown previously by Fig. 4.5. If this structure is taken as a complete feedback amplifier, the situation is essentially the same as that first described. The only difference results from the fact that, since the circuit contains only a single tube, 0o and 0 would probably be numerically smaller than was assumed there. We might suppose, for example, that the ratio eøø/e ø is -30 db. This would still give a negligible distinction between return difference and sensitivity for most applications. An entirely different situation, on the other hand, may be obtained if the circuit is the last stage of a double loop amplifier. In these circumstances 0o and 0 refer to the transmission characteristics of the complete amplifier and in virtue of the feedback around the principal loop this may not be much affected even by a considerable change in the transmission of the last tube. For example, if the normal feedback around the principal loop is 40 db, the assumed decrease of 30 db in the gain of the circuit of Fig. 4.5 when/3  vanishes will still leave a net feedback of 10 db ----------------------------------------------------------- 60 NETWORK ANALYSIS cuav. 4 around the principal loop. The difference between e eø and e e is thus only that due to the change in the /effect in the principal loop caused by the reduction fram 40 to 10 db. It is clear therefore that F will be much smaller than S in (4-20), so that the actual stabilization of the circuit against variations in the last tube is much greater than would be indicated by a computation of the return voltage on that tube.* 4.12. Reference Value for k' The method of computing sensitivity which we have thus far considered consists essentially in separating out the directly transmitted component m. 4.7 of the total output current, so that in effect it becomes the origin from which the net output current is computed. This is illustrated for an ordinary single loop amplifier by Fig. 4.7. The actaal bilateral/ circuit in the amplifier is represented symbolically as the sum of the two rmilateral * A physical interpretation of this appm'ealfly surprising result can be obtained by noticing that in the multiple loop structure voltage can be returned from the plate of the last tube to its grid by two di -flea'cat paths. The first passes through the prit cipal / drcuit and the first stages of the fot-ard circuit while the second passes direcdy through the local feedback elements. These two paths together can be legarded as forming a feedback amplifier, the  circuit of which is the first path, ahile the  oh-cult is reprented by the second. Cinder the conditions which have been assumed, there is a net gain around the complete feedback loop of this amplifier and the insertion of the feedback path must therefore diminish its gain. The insert'on of the local feedback elements ia the final structare, in other word% redtrees the  eturn voltage on the last tube Speaking approximately, the difference between F and $ is an indication that this effect should be neglected. The return voltage which most nearly mpreqents the effective stabilization of the circuit against variations in k/z is that which would be obtained if the local feedback network were omitted. To a first approximation, the insertion of the local feedback circuit does not affect the fetalback on the last tub% but it does of course affect the feedback on the remaining tabes by ehanglng the trans- mission characteristic around the principal loop. This is discus.ed in more detail ia a later chapter. ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 61 components fi and fi2. If we suppose that the variable element k/" is here identified with the whole / circuit, the component /52will provide the directly transmitted term. The use of this term as a reference is equivalent to saying that the contribution of fis to the final output is to be considered separately from the contribution of the ideal feedback amplifier represented by the combination of t and fi enclosed by the broken lines. As an alternative to this procedure, we may also take account of the direct transmission term by changing the origin from which the variable element/3' is measured. In the circuit of Fig. 4.7, for example, we might Fro. 4.8 begin by lumping ta and fi2 together, as shown by Fig. 4.8. The structure thus becomes an ideal single loop amplifier, without direct transmission, in which the effective forward gain is /z t =  -3- fi. This is equivalent to computing  from the origin -fi rather than from zero. The use of an offset reference point for the variable element in this manner is merely an unnecessary complication in most elementary situations, where the methods we have already developed are adequate to deal with the problem. It is worth some attention, however, since in certain circuits it leads eventually to a simplified analysis. This will appear more clearly in Chapter VI. For the general case the new origin for/4 z will be called the reference value of/(/'. It will be symbolized by/fro and is specified by the Definition: The reference value of any element is that value which gives zero transmission through the circuit as a whole when all other elements of the circuit have their normal values. It was indicated earlier in the chapter that return difference computations could in general be based upon any arbitrary reference value for/4/. From this point of view, P?0 is only a special case which is called the reference in recognition of the unique output current to which it leads. The reference condition is evidently somewhat like a bridge balance and expressing/F in terms of its departure from 10 is similar to expressing the impedance of one arm of a bridge in terms of its departure from the impedance which would ----------------------------------------------------------- 62 NETWORK ANALYSIS C.,. 4 give an exact balance, rather than in terms of its actual value. It will be recalled that this is the device which was used to simplify the analysis of the circui in Fig. 4.4. It is apparent from such an expression as (4-10) that  is gluten, for unilateral and bilateral elements respectively, by 'o = - ' (4-23) A1243 and /o = -- A2a ' (4-24) If we let P/ represent the departure, PP' -- /o, froru the reference value, such an equation as (4-10) therefore becomes . (4--2s) A o __ A2 A43 - ,tA43 A1243 This expression has the same form as a function of Y  as the original equation (4-10) had as a function of f/when we assumed A2 = 0. Thus we can apply to it the procedures we used previously to establish equation (4-15) for the sensitivity in the case of zero direct transmission. Since a given percentage change in /x/s will not be equal to the same percentage change in  /V, unless/V and If-' happen to be equal, however, the" sensitiv- ity" computed fi'om (4-25) will not in generM be equal to the sensitivity defined in (4-7) or (4-8). To prevent confusion, therefore, the result of the present coruputation will be called the relative sedtisi'y syrubolized by S . With this understanding, we can evidently write $s = I = I + lg- ' , = A (4-26) o0 0 log II zs where the symbol A t is given by A' = A ø -- A2 as, (4-27) and evidently represents the value assumed by A when/'P' -- 0. If (4-27) is simplified by means of (4-13), the expression for S t can also be written as AtaAo AaA42 ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 63 It is evident that there is a complete formal parallelism between this analysis and that of an ordinary circuit with zero direct transmission. For example, (4-26) is exactly like (4-15) except for the substitution of for k? and zX 2 for zX ø. These, however, are exactly the modifications which are made in convering a return difference for zero reference into a return difference for the reference/4z0o We therefore have the Theorem: The relative sensitivity for any element/4 z is equal to the return difference of/P' for the reference 0. There remain the problems of determining S' from more immediately measurable quantities and of relating S  to the actual sensitivity S. Of a variety of equations which can be used to determine S', perhaps the simplest is $,_ F() --, (4-29) F(0) where F(//?) is, as usual, the return difference for/$7 when kF has its normal value, and F(/4z0) is the return difference for/P', calculated for/4/= This result follows immediately from (4-5). Another simple formula, useful in special circumstances, is d 0 o 82 - , (4-30) d 0  -- d 0 where e øø' stands for (Al:43/A43)/''R and is, from (4-10), the transmission through the system when the variable element  is infinite. If  repre- sents a tube, this condition is, of course, an unrealizable one. It is also possible to determine S  from measurements made when  = 0, by modify- ing the circuit in certain special ways. The development of these methods, however, is postponed until Chapter VI. The most straightforward relation between S  and S is probably This equation can be established immediately if we recall that the distinc- tion between S and S  is due only to the fact that a given actual change in the physical network will produce different percentage changes in  and  when these two quantities are unequal. Other useful formulae for the relation between S and S  are F S = F (8 - 1), (32) ----------------------------------------------------------- 64 NETWORK ANALYSIS CHP. 4 and 00 = . (4-33) S S t + e0 _ e0 ø They are both readily established from the preceding general equations for F S, and S  and the identity (4-13). The various situations which may arise in these equations for different relations betveen e øø and e ø can be illustrated again by the examples used in the discussion of equation (4-22). 4.13. Reference Falue of [IF as an Index of Location in the l Loop If we exclude the special problems presented by multiple loop amplifiers, the introduction of the reference value///o into computations of sensitivity is, in a broad sense, the analytical counterpart of the physical fact that the properties of feedback circuits vary with the location of the element in the loop. It corresponds in other words to the fact that the stabilizing and dis- tortion reducing properties of feedback hold only for elements in the t circuit. Since we cannot, at best, decide what part of the complete loop is Fro. 4.9 t and what part is  until we have chosen the input and output terminals, these properties thus depend not so much upon the fact that a feedback loop exists as they do upon the location of the element in question with respect to the transmission path which is eventually of interest. The refer- ence value/F0, since it depends upon the particular choice of input and out- put terminals, takes this factor into account. As the preceding examples have shown, the reference value for an element in the t circuit is ordinarily quite small so that with a large return voltage the effective sensitivity is also large. When the element is in the B circuit, on the other hand, the value of k//which will produce zero trans- mission in the complete system is in general large and variations in /F when computed against this extreme reference correspond to relatively little stabilization of the final amplifier transmission characteristics. The way in which the reference value appears as an index of location can be illustrated concretely by the circuit of Fig. 4.9. The structure is a normal single loop feedback amplifier with the output impedance taken as R, with the exception that the second interstage includes a transformer- ----------------------------------------------------------- MATHEMATICAL DEFINITION OF FEEDBACK 65 resistance combination instead of some more conventional configuration. Letting/4 z represent the transconductance of the output tube and assum- ing that the reference value for W is negligibly small, we readily find that dER X ø dl4 z 1 dlV .... , (4-34) ER A/F I+T where T = (4/ ø) and can be identified with the negative of the trans- mission characteristic around the complete loop. This expresses the Gmiliar result that feedback reduces the effect of variations in the tube gain by the factor 1 - u. Let it be supposed now that the output impedance is taken as R but that R is retained as an ordinary circuit element. The feedback loop, regarded as a complete loop, is exactly the same as it was before. The change in the choice of output impedance has, however, transferred the last robe to the 3 circuit so that we may expect that the stabilizing properties of feedback have disappeared for variations in the gain of this tube. The situation can be analyzed by using the formula for relative sensitivity given by equation (4-26). If we set  = - Wa, this formula can be written as dE  - W' 4a dW E   (35) We now determine the o which will lead to zero transmission through the complete amplifier. In the present instance o must obviously be infinite since zero transmission can be obtained only with an infinite g circuit gain. If o is infinite, however, W must also be infinite and (35) therefore reduces to dE - (36) Upon multiplying and dividing the right-hand side of (36) by  and rio and comparing with (34), this becomes dE o a d - T d (37) where T still represents -u3 for normal operation. We can readily verify that this is the correct formula by direct differentiation of the ordinary equation for the gain of a feedback amplifier as a function of B. ----------------------------------------------------------- CHAPTER V GENERAL THEOREMS FOR FEEDBACK CIRCUITS- A 5.l. Introduction T-IlS chapter and the one which follows will continue the general dis- cussion of feedback circuits begun in the preceding chapter in terms of the definitions of return difference and sensitivity which were established there. They have for their principal object the development of general theorems on the relation between these quantities and impedance, gain, non-linear distortion, etc. The theorems of the present chapter axe developed from simple mathematical identities which remain valid whatever the reference values for the elements may be. They are thus stated in terms of the return difference for a general reference, including the relative sensitivity and the return difference for zero reference as special cases. 5.2. Impedance of an 4ctioe Circuit* The first general theorem relates to the effect which feedback may have upon the impedance measured between any two points of the circuit. I n addition to its general interest the theorem is of particular application with respect to the calculation of the return difference for bilateral element% since it was shown in the preceding chapter that that depended upon the imped- ance of the circuit to which the element was connected. In developing the theorem it is supposed that the impedance which would be obtained in the absence of active clements is first determined by ordinary circuit methods. The theorem then is concerned with the modification produced in tbls impedance by the addition of the active elements. This is, of course, the heart of the problem. The fact that the active elements must in general produce sone effect is easily seen if we consider, for example, the input impedance of an ordinary feedback amplifier. By definition this impedance must be the ratio of the input voltage to the current which flows through the llne into the anplifier. The net current which flows past the input terminals, howeveG is a com- posite of the current which would flow if we considered only the passive elements and of the current which is returned to the source through the feedback circuit. Tlxe presence of this feedback current may obviously * The material of this section is a modifmd version of results originally due to R, B, Blackman (B.&T.],, Octobeh 1943). 66 ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- A 67 make the impedance of the amplifier quite different from the impedance which we would measure if the tubes were not operating. Although the impedance of an active circuit may be quite different from that of the passive structure the relation between the two is easily built up. Let it be supposed, for example, that we are interested in the active imped- ance Z which would be measured from the terminals of a resistanceless generator inserted in the nth mesh of the circuit. This is obviously z = a (5-1) Now suppose that we choose any element,/4/, within the network. It is convenient to assume that /4 / represents some mutual impedance Zij, although the final results are the same whether /4 / is a unilateral or a bilateral element. We can rewrite (5-1) as A A ø A Z- a,-  a ø ' (5-2) where, as in the preceding chapter, ao and ,, represent a and ann when In equation (5-2), a0/ is evidently the impedance which would be measured if  = 0. Assuming that  or Zi5 is the mutual impedance of one of the vacuum tubes, then, we can call this the passive impedance Z0, or the impedance which would be measured if this tube were dead. More- over, /o is the return difference for  with the circuit in its normal condition, that is, with the terminals between which Z is measured shorted together. In addition, A,, and a],, are the coefficients of Z,, in  and 0 respectively. The ratio &,/,, is therefore the limit approached by a/a ø as Z becomes indefinitely great. It consequently follows that &,,,/a] represents the return difference for  when the self-impedance of the nth mesh is made infinite, or in other words when the terminals between which Z is measured are left open. We can therefore write equation (5-2) as _F(0) z = zo , (5-3) where F(0) and F(m) are the return differences for  when the terminals between which Z is measured are respectively short-circuited and open- circuited. If we base the analysis on admittances instead of impedances the result is the same and we can write F(0) Y = v0 , (34) ----------------------------------------------------------- 68 NETWORK ANALYSIS easy. 5 where F(0) and F(o ) now represent the return differences with respect to PF' when zero and infinite admittance, respectively, axe added across the terminals between which Y is determined. Equations (5-3) and (54) describe the impedance or admittance at any part of a feedback circuit in terms of the impedance or axlmittanee which would be obtained vith any arbitrary element vanishing, and the return difference for that cleanrut. If the arbitrary element kF' is the mutual impedance or transconductance of a vacuum tube, therefore, we cm dis- count the effect of this active element in the circuit. In ordinary feedback amplifier zero ffain in any one tube will interrupt the feedback circuit so that the actual impedance or admittance can be computed directly from (5-3)or(5-4)bychoosiuganyoneofthetubesasP, r/. In raore complicated cases a single dead tube may not reduce the calculation of impedances to the completely passive case. Evidently, however, by starting with all the tubes as dead and applying (5-3) and (5-) repeatedly as each tube in turn is assigned its normal gain we can cover all cin_-ait. The analysis used in developing (5-3) and (5-q) has been based upon the assumption that the refeaence for / is zero. Since (5-2) is merely an identical form of (5-1), however, the zero value for/ is a matter of indif- ference and we can choose any reference we like as long as we choose the same reference for both 's- The general result can therefore be stated in the following words. Tricorein: The ratio of the impedances seen at any point of a network when a given element P//is assigned two different values is equal to the ratio of the return differences for IP' when the terminals between which the impedance is measured axu first short-circuited and then open-circuited, if the return differences are computed by letting the first value of PP' be the operating value and the second the reference. The relation between feedback and impedance can also be stated in another way. Let it be supposed that an arbitrary impedance Z is added in series with the nth mesh, and let fit and ,t represent A and Z a, respec- tively, after the inuoduction of Z. The return difference for any/4/after g is added can be written as h  A + F - $, - zx  + ZA,' (5-5) Now let ., be so chosen that - = 0. Upon comparing the result with (5-1) we reaxtily establish the ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- A 69 Theorem: The impedance seen in any mesh is the negative of the impedance whose insertion in that mesh would give zero return difference for an arbitrarily chosen element in the circuit. This is an obvious theorem in the light of the discussion of stability given in a later chapter since it will appear that either a zero return difference or zero impedance at any frequency corresponds to the possibility of a natural oscillation in the circuit at that frequency. Fo. 5.1 5.3. Examples of `4ctive Impedances To exemplify these relations we will consider the series feedback amplifier shown by Fig. 5.1. Let the Z of equation (5-3) be the impedance which would be measured in series with any one of the series connected branches such as Z2, Z6 or either of the high side transformer windings. In other words it is the impedance which would be measured between any such pairs of terminals as `4`4', CC', or DD' in Fig. 5.1. It will also be assumed that the k//of equations (5-1) and (5-2) is the transconductance of any one of the tubes. With terminals ,4d', CC', or DD t shorted together the return difference with respect to /4/ is F(0) = 1 - fl, where fl is the transmission around the loop computed i'n the normal fashion. With the terminals opened, on the other hand, the return difference with respect to kk'is unity. Equation (5-3) consequently gives Z = Z0(1 - ufl), (5-6) where Z0 is the impedance which would be measured with one of the tubes dead and is evidently the ordinary passive impedance. The impedance measured in any series line is thus much larger than the passive impedance. For the impedance between ,4 and ,4 t for example we find Z = (1 - ,#)(Zl + Z + Za), (5-7) ----------------------------------------------------------- 70 NETWORK ANALYSIS CaAv. upou the assumption that the input and output impedances of the tubes are very large in comparison with the impedances in the/$ circuit. Next consider the apparent impedance which would be measured between any such points as z/or C and grinrod. We now find that the normal return ratio will be obtained when the impedance connected between .'/or C' and ground is infinite and that the return ratio vanishes 5vhen the termi- nals are short-clreulted. In other words F(0) = 1 and F() Equation (5-3) thus giv zo z = --. (54) [he impedmce measured acm.s the path of the feedback loop is therefore reduced by feedback. For the impedance between / and ground, for exampl% we have 1 Z(Z + Z = -- (5-9) As a more complicated example we may consider the impedance meas- ured across the terminals E, E t in Fig. 5.1. Here we have zz(g + Zo) Z0 Zs + Z + Za + Zo ' (5-10) while F(0) = i - , 1 z z+z0 +' the factor multiplying  in e second equation being obtained by calculat- ing the change produced in the transmission characteristic of the intentage when E, E  is openmlreuited. Tbe substitution of thee values in equa- tion (5) then giv the impdance taught for. If in partil we assume that  is very gret the result becomes Z = Z + Z. (i2) This sult, of course, might have been forespn fi'om (5). If we sider that  and Za mgeth present a seri impance it follows from this equation that the impedance of the cirit m which they are connect must  very high if the febback is lot . Only Z 7 and Zs tberefom ne be consided in determining the impance at terminals E, These ca]ofiations have been based upon the first of the two thems Wen in the prodding section. The stone resulm foow from the second ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- A 71 theorem. As an example, we may return to the discussion of the effect of feedback on a series impedance, as expressed by (5-6). Let the imped- ance whose insertion in the series arm would reduce the return difference to zero be represented by Z'. Its insertion in series with Z0 will produce the loss Zo/(Z' 4- Zo) in the transmission around the loop. For the return difference to vanish, however, the loop transmission, u/g, must be reduced to unity. We therefore have - , (5-13) Z' + Zo or z'= (fi - 1)z0, (5-14) which is the negative of the active impedance given by (5-6). 5.4. Feedback for Bilateral Elements A knowledge of the active impedances of the circuit makes it a simple matter to compute the return differences and sensitivities of its bilateral elements in accordance with the methods of the preceding chapter. As an example we may choose the impedance Zo of Fig. 5.1. By the previous analysis, the return ratio for this element is equal to the ratio of its imped- ance or admittance to the impedance or admittance of the circuit which it faces, the return difference is equal to the return ratio increased by unity, and the sensitivity is equal to the return difference suitably modified to take account of direct transmission. If we exclude the slight trickle of current directly through the fi circuit, zero output current is obtained when the branch Z6 is an open circuit. It is obviously convenient, therefore, to use an admittance analysis, in which case the direct transmission term is zero and the sensitivity can be taken equal to the return difference. It follows from (5-3) that the impedance seen at terminals C, C  of Fig. 5.1 is (1 - ufi)(Z4 + gs + Z6) and the admittance which Z6 faces is therefore the reciprocal of (1 -//5) (Z4 q- Zs + Z0) - Zo. Upon divid- ing the admittance of Z0 by this admittance the return ratio and the return difference or sensitivity for the element Z0 are obtained in the form T = Y[(1 - /3)(Z + Z + Z6) - and Z4 + Z + Zo F = $ = (1 -u/) (5-15) Z6 The factor (1 --//5) in this expression is self-explanatory. The remain- ing factor (Z 4- Z q- Zo)/Za reflects the fact that the/z circuit gain does ----------------------------------------------------------- 72 NETYVORK ANALYSIS not vary in strict proportion to Z6 because of the presence of the other impedances. If Z6 were very small, for example, its impedance might vary considerably in per cent without greatly affecting/ and a corresponding term in the sensitivity expression must therefore be included in virtue of the fundamental definition given by equation (4-8) of the preceding chapter. If we consider a shunt impedance such as Z4 the procedure is essentially the same. In this case, the reference condition is a short circuit and it is convenient to use impedances rather than admittances in the analysis. Since the impedance which Z4 faces, however, is now reduced by feed- back the ratio between Z4 and the impedance of the rest of the circuit is correspondingly increased. The essential result is of the same general type as equation (5-15). As a third example we may consider the impedance Z2 in the fi circuit of Fig. 5.1. So far as the calculation of return ratio and return difference is concerned, the situation with respect to this element is exactly the same as it was for Z6, and we can make use of (5-15) again, with appropriate substitution of Z, Z, and Za for Z4, Z.5, and Z0. The presence of a large direct transmission term, however, complicates the computation of sensitiv- ity. It is simplest to begin by determining the relative sensitivity S'. We can evidently secure zero transmission from the amplifier as a whole by assigning the fi circuit a large gain equal to that of the  circuit and a phase which will cancel the  circuit output. The reference value for Z2 must therefore be very nearly - (Z q- Za) or, in other words, very nearly the negative of the passive impedance which it faces. The effective impedance, /4/, can therefore be taken as Z - Z q- Za. The impedance which P//' faces must be the difference, Ifi(Zt q- Z2 + Za), between/P' itself and the total inqpedance calculated in equation (5-7). The relative and absolute sensitivity are readily found from these facts, plus the relation $/S = /4z'//F,, to be and 1 -/45 Z q-Z2q-Za 8 - , (5-16) and are obviously small in normal situations. The result is easily checked by direct differentiation of the gain equation for the amplifier in accordance with the fundamental definition of Chapter IV. It is interesting to notice that the difference between the very large sensitivity represented by equation (5-15) and the low value obtained in equation (5-16) is the result ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- A 73 entirely of the difference in the two reference conditions. The situations otherwise are exactly the same. 5.5. Effect of Feedback on Input and Output rmpedances of ,4mphfiers The distinction between the active and passive impedances of a feedback circuit is particularly important in considering the effect of feedback on the impedance which an amplifier presents to the line. The principal results, for the basic connections described in Chapter III, can be listed as follows. 1. The active impedance of a series feedback amplifier is (1 -/atS) times its passive impedance. Since the input and output impedances of tubes are normall 7 high anyway, the active impedance is, in general, almost infinite. A similar statement can be made for a cathode feed- back circuit. 2. The active impedance of a shunt feedback amplifier is 1/(1 - u/) times its passive impedance. It is thus relatively low. 3. The active impedance of a balanced bridge amplifier is the same as its passive impedance. Tiffs connection is therefore intermediate be- tween the series and shunt connections. 4. If the balance of the bridge in the circuit of the preceding paragraph is disturbed by a change in the final tube impedance, the reflection coefficient between the active impedance so obtained and the active impedance before the change is 1/(1- /5) times the reflection coefficient which would be obtained if the circuit were passive, where ta/ represents the loop transmission after the change is made. The first three of these statements can be dismissed briefly. The line impedance in a series or shunt feedback amplifier is merely a special case of a general series or shunt impedance, the results for which have already been given by equations (5-.6) and (5-8). In the balanced bridge circuit, the bridge balance produces conjugacy between the line and the  circuit. It follows from this that the loop transmission is independent of the line impedance.* We therefore have F(0): F() in (5-3), so that feedback does not affect the impedance. The fourth statement may require amplification. In the theoretical balanced bridge connection the tube impedance is one of the arms through which the balance is obtained. Since tube impedances are ordinarily quite variable, the balance which can be relied upon in practice is imperfect. Moreover, it may be necessary to shunt the tube with a dissipative branch * This follows readily from the principle of reciprocity. See, for example, the dis- cussion in the next chapter under the heading "Reference Feedback as a Balanced Bridge." ----------------------------------------------------------- 74' NETWORK ANALYSIS cas. 5 in order to secure an impedance whose phase angle and magnitude axe appropriate to produce a balance with permissible impedances in the other arms of the bridge. This is particularly unfortunate in an output bridge because of the wastage of output level to which it leads. The final state- ment says in effect that if the feedback is large the departures produced in the impedance which the amplifier presents to the line will be extremely small even when no effort is To/ CircuH' Fzo. 5.2 made to control the impedance of the tube. Naturally, how- Z d ever, the other property of a bridge circuit, that the loop transmission is independent of the line impedance, will no longer hold. This effect of feedback is easily demonstrated by using (5-3) in two different ways. Let Z in Fig. 5.:2 represent the impedance whose removal produces the disturbance under consideration. It will be supposed that with Z present the bridge is perfectly balanced. Let Za represent the llne impedance. Let Z,x and Z repesent respectively the passive imtdances of the circuit to the right of Z when Za has its norm-d value and when Za is replaced by a short circuit. Finally, let Z,, and Z represent respec-. tively the active impcdanees looking into the amplifier when Z is present and when Z is removed by opening the teninals P, The first step is the computation of the active impedance Z looking int the terminals P, P_ when the neighboring impedance is taken as Let this be the 3g of (5-3) and let the F's of this equation refer to the last ttxbe. Let the loop transmission with Z, absent be represented by so that F(o) = 1 -- . From ordinary circuit considerations, the intro- dmction of Z changes the loop transmisslon to [Za/(Za + Zb)]t4. More- overs tile passive impedance Za is Z + Jbl- Wig therefore have 1--- Z + Z Z (Z + Zt)  _  (47) [Now consider the impedance Z corresponding to Z,. The passive impedance becomes g q- Z. F(0) is the same as it was in developing ($-17), since with g piesent the bridge is balanced and a change in the line impedance does not affect the feedback loop. The mtlo between the loop transmissions with Zo present and g absent is ga/(Z q- g2). We there- ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- A 75 fore ave Z. + Z2 = (Z + Zb2) Z + Zb  (5-18) 1 Z +  In this computation the F's of (5-3) have referred to the last tubeß We now apply (5-3) again with the F's taken with respect to Za. It foliows from (5-3) that the ratio of the return differences for Za with Za present and absent must be the same as the ratio of Z1 to Z2. We can therefore write Z + Za = --, (549) z + Za Z o Z,i(Zc2 + Za) Z - ß (5-20) If we set Za = Zx the left-h'and side of (5-20) is the reflection coefficient between the active impedances of the network before and after the removal of Za. On the right-hand side all the quantities except the factor 1 - u represent the network in its passive state. The original statement is therefore proved. 5.6. Use of Impedance Measurements to Determine Feedback The theorems in the first section were developed as a means of computing the active impedance of a circuit when the return differences are known. In practice, however, they are perhaps more frequently applied as a means of determining the return difference from impedance measurements. This is often a more convenient method of obtaining the return difference than a direct transmission measurement would be, since it does not require open- ing the feedback loop. The method can be applied even to unstable struc- tures by including in the measurement a known impedance of a magnitude which will stabilize the circuit. 5.7. Relation between Feedback for Two Elements The process used to develop the formula for active impedances can also be applied to obtain a theorem relating the return differences for two tie- ----------------------------------------------------------- 76 NETWORK ANALYSIS ments iu the circuit under actual operating conditions to the return differ- cnces which would be found for each element if the other vanished. Let the two elements be represented by ff/ and kF=. To express the fact that the determinant of the system will depend upon both .? and $?=, we may write it, in general, as /x(/K,/4/2). Then .(0 ,/4/2) represents the determinant when/F is zero, A(/F, 0 ) the determinant when ?r'2 is zero, and A (0,0) the determinant when both/ and/472 are zero. The return difference for either element can be expressed as the ratio of the complete determinant to the determinant obtained when that ele- ment vanishes. Letting F and F2 be the return differences for//z and Fz, respectively, these rdations in our present notsrio% are F a(0 ,k/' 0 ' (W,0 )  = a(/,o) = x(/,o) a(0,0) F, a(O,'2) x(0,0) a(0,/=) (5-1) (5-22) = F(w= = 0). F2 (/, = o) Equation (5-22) is evidently unaffected if the /K's are assigned any reference values, as long as the reference values are taken as the same on both sides of the equation. We can therefore state the Theorem: The ratio between the actual return differences for any two elements for any reference conditions, is the same as the ratio which would be obtained if the retrn difference for each element were computed with the other element at its reference value. As an example, we may take ,r-F' as Y in Fig. ,g.1 and /= as the transconductance of one of the tubes. We see by inspection that F(/x - 0) = i and ]'(k?2 = O) = (7, + Zn q- ZO/Ze. The theorem states that the ratio between these two F's will be preserved for any values of the/4/'s. This is, of corn'st, verified by equation (5-15). 5.8. Thgeeni's Theorem in lctiee Circuits The general formula for return difference also can be used to develop another type of identity which is even simpler than those described previ- ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- A 77 ously. Let it be supposed, for example, that Wrepresents the transimmit- rance of the tube whose grid and plate are labeled respectively i and j. / must be a constituent of Zji or Yii in the general determinant. The return difference for/can be written, from (4-2), Chapter IV, as F- A0 - zXki X ø' (5-23) where/o represents A when F/= 0 and k is any other node or mesh in the circuit. In equation (5-23) the determinant Aki can equally well be written as xi since it contains no terms from the ith column of the original determi- nant and is therefore independent of V. The ratios Ai// and Ai/ø are thus the transmissions* from k to i when/V has its normal value and when /V vanishes. Moreover, the identity evidently holds equally well if we use any arbitrary value instead of zero as a reference for/. We can there- fore draw the following conclusion: Theorem: The ratio between the transmissions from any point of the network to the grid of a given tube for an arbitrarily chosen reference condition and for the normal operating condition is equal to the return difference of the tube for the chosen reference. A simple example is furnished by the transmission from the input line to the/ circuit of an ordinary amplifier. The effective signal level on the grid of any tube is 1/(1 - #fi) times the level which would exist if that tube were dead. We can also write the return difference equation as F = 6 --- aj. X ø (5-24) The quantities Ajk/A and Aik/A ø evidently represent transmissions from the plate to k under normal and reference conditions. We therefore have the Theorem: The ratio between the transmissions from the plate of a given tube to any point of the network for an arbitrarily chosen reference condition and for the normal operating condition is equal to the return difference of the tube for the chosen reference. This is best exemplified by the discussion of the following sections. * "Transmission "is used here as an abbreviation for transfer admittance in a mesh analysis or transfer impedance in a nodal analysis. ----------------------------------------------------------- 78 NETWORK ANALYSIS If IF is a bilateral element the situation is essentially the same except that no distinction need be made between the "grid" and "plate" ends of/4. We therefore have the Theorem: The ratio between the transmissions from a given bilateral element to any point in the network, or vice versa, for an arbitrarily chosen reference condition and for the normal operating condition is equal to the return difference of the given element for the chosen reference. The last theorem gives a clue to the characterization of the three theo- rems as a whole. If/'/' is a bilateral element the return difference for IF corresponding to any given reference is the ratio of the total immittances seen from tF when/4 / has its normal and reference values. But the state- ment that this is the same as the ratio of the transmissions from k to IF under the two conditions is merely another way of expressing Thvenin's theorem.* On this account the group of three theorems on the relation between return difference and transmission will be described as the general ized Thdvenin's theorem, applicable to unilateral as well as bilateral elements. In other words the return difference for a unilateral element plays the same role in determining the final response that the impedance relations at generator or receiver terminals would play in an ordinary transmission calculation. 5.9. Computation of I4'o As an example of these theorems we will consider the determination of the reference//70 for one of the tubes in the circuit. It will be recalled that IF0 is the value which IF must assume in order to provide zero trans- mission through the complete structure. An equation for IF0 has already been given by (4-23) of the previous chapter but the zX's which appear in it are not easily recognized as quantities which could be determined by physical measurement. With the help of the generalized Thvenin's theorem of the preceding section it is possible to develop an alternative formula for IF0 involving quantities of more direct physical significance. Let the input and output of the circuit as a whole and the grid and plate of the tube IF be labeled respectively 1, 2, 3, and 4. The quantities ' = A?2/ ø, 3'2 = ZXa/A ø, ?a = A42/zxø, and 3'4 = A4a/A ø represent respectively the transmissions from input to output, from input to grid, from plate to output, and from plate to grid, all evaluated on the assump- * Thvenin's theorem is discussed in most books on communication circuits. See, e.g., Shea, "Transmission Networks and Wave Filters," p. 55, or Terman, "Radio Engineer's Handbook," p. 198. ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- A 79 tlon that the tube is dead. It will be supposed that all these transmissions are known. If we begin with the tube dead, the excitation on the grid for a unit source in the input will be 2. The fact that the tube has the residual gain in the reference condition can therefore be represented by inserting an equivalent generator -P//0'2 in the plate circuit.* If this generator were actually an independent source of current or voltage it would evidently pro- duce the response -0,a in the output. The reference condition could then be established by finding what value of P//o would lead to exact cancel- tation between this response and the direct transmission . But the introduction of the equivalent generator coincides with a change in tube gain from zero to/4/0. In accordance with the theorems of the preceding section, this must reduce the transmission from plate to output by a factor equal to the return difference of the tube when PV = /F0. This last quantity can be found from a knowledge of the transmission T4 from plate to grid. The correct relation is thus easily seen to be 1 +/Fo4 ' (5-25) or o = , (5-26) in which all the quantities can be measured directly. The fact that this is actually the same as the original formula for/4/o can be established by means of equation (4-13) of the preceding chapter. 5.10. Reductionof Distortion by Feedback One of the principal practical advantages of feedback is the fact that its use reduces the flow of modulation currents in the load due to the non- linear distortion of the elements in the y circuit. In order to investigate this, let it be assumed that the nonAinear distortion is represented by the addition of a separate "distortion generator" in the plate circuit of the distorting tube, while the circuit itself remains linear. This supposes that the level of the fundamental components of the signal has been established in advance, so that the amount of nonAinear distortion can be calculated, and also that the distortion is a small part of the signal, so that second order effects representing "distortion of the distortion" can be ignored. The distortion generator may also be used to represent a source of extraneous noise rather than a source of modulation products. * The negative sign is due to the phase reversal in the tube. ----------------------------------------------------------- 80 NETWORK ANALYSIS An appropriate relation can be developed immediately from the generali- zation of Thvenin's theorem described previously. It is merely necessary to choose the point k to represent the output circuit. The second of the preceding three theorems can then be restated as the Theorem: The noise or distortion current in the output produced by a prescribed distortion generator in one of the elements of the circuit is equal to the current which would be found with the element in an arbitrarily chosen reference condition divided by the return difference of the element for the chosen reference. But, if we deal only with the portion of the output current which flows because the given element is activated, the return difference is also a measure of the sensitivity of the circuit to variations in the linear properties of the given element. It thus appears that the contributions of the given element to the distortion and to the fundamental frequency currents in the output are governed by the same laws. This is not surprising if it is recalled that a slight change in the linear properties of a circuit can be represented by the introduction of a small generator at the disturbed point.* The circuit must naturally have the same properties whether the generator represents distortion or a change in the linear characteristics of the circuit. 5.11. Exact Formula for External Gain with Feedback The relation between feedback and external gain is customarily expressed by the statement that the gain is reduced by the amount of feedback. Equation (34) of Chapter III, for example, gives this result for the simple analysis in terms of independent/ and/3 circuits. If we wish to make very precise gain calculations, this statement suffers from two objections. The first is that the meaning of gain in the absence of feedback is somewhat uncertain, on account of the interaction between the impedances of the  and/3 circuits at the ends of the amplifier. It is not perfectly clear whether we should simply remove the feedback circuit entirely in making the calculation of gain before feedback, or whether we should make some allowance for the energy absorption of the /3 circuit elements at input and output, and if so, what that allowance should be. The second difficulty is the fact that the relation between gain and feedback was developed only for the conventional single loop amplifier. It is not clear how the relation should be applied to other situations, and in particu- lar to situations in which there is an appreciable direct transmission term. As a final example of the methods established in Chapter IV, therefore, we * See the "Compensation Theorem," in Shea, p. 56, or Terman, loc. cit., p. 198. ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- A 81 will develop an exact expression for the external gain in the presence of feedback. It is convenient to begin with equation (4-21) of Chapter IV. If we multiply and divide by X ø, this equation can be written as A 0 --/A 13 A42 . eø -- eøø - a ø +/Fa3 a- ø ' (5-27) The quantity/xø/(/x ø q-/-P'/t4a) in this expression will be recognized as the reciprocal of the return difference F. If we replace the remaining terms by e ør, the expression as a whole becomes e ø - e øø = -eø, (5-28) while if we make use of (4-22), Chapter IV, the equation can also be written as eø = 2 eø' (5-29) The quantity e øv will be called thefractionated gain. It may be regarded as an exact statement of what is meant by "gain before feedback." We notice that it is essentially the product of three factors. Two of them, &la/& ø and (/42//ø)/d/a, represent, respectively, the transmission from the input to the grid and from the plate to the output with the tube dead. They thus include the input and output impedances of the/ circuit just as it stands. The third is the gain/-F of the tube itself. In a single loop struc- ture the fractionated gain is then the gain which would be realized if it were possible to open the fi circuit without affecting its impedance at either end. An example is furnished by the circuit of Fig. 4.5 in the preceding chapter. If this structure is taken as a complete amplifier, the fractionareal gain is readily computed to be eO v = _ Z2 Z4 ZtF , Za(Z4 + Z) (Z + Z.)Za Z + Z + + Z + Z Za+Z4+Z Z+Z+Zs (5-30) where Z on the right-hand side is identified with/4"a in the general expres- sion (5-27) and the two preceding factors will be recognized as the input- grid and plate-output transmissions,/la//ø and 32//0, for this particular structure. Equations (5-28) and (5-29) offer alternative ways of treating the gain reduction due to feedback in systems with appreciable direct transmission. ----------------------------------------------------------- 82 NETWORK ANALYSIS C,^P. 5 In equation (5-28), the gain reduction is applied only to the surplus of the total output over the direct transmission term. This is the most natural relation if we continue to think of the system as made up of two non- interacting paths in parallel, one of which is simply a fixed structure fur- nishing the over-all direct transmission, while the other contains the vari- able P/ and exhibits the essential phenomena of feedback. Equation (5-29) shows, however, that it is also permissible to apply the gain reduction due to feedback to the complete output provided we take" feed- back" to be F2/S. Equation (5-27) can be regarded as a relation which is appropriate if we wish to give special attention to the reference condition k/ = 0. The quantities e øø and /0 evidently apply to this state. Just as with most of the other equations in this chapter, however, an analogous expression can be developed for any reference. The use of the reference k/0 is of particular interest, since it leads to an alternative "gain before feedback" expression Based upon measurements made with an interrupted feedback path. This is discussed in the next chapter. ----------------------------------------------------------- CHAPTER VI GENERAL THEOREMS FOR FEEDBACK CIRCUITS- B 6.1. Introduction Tins chapter will continue the development of general feedback theorems begun in the preceding chapter. The center of attention in the present chapter, however, is the relative sensitivity, S', and its use in expediting feedback and gain calculations. A large part of the discussion is concerned with multiple loop circuits, where the conception of relative sensitivity is most useful. The chapter can be omitted by readers interested only i simple feedback circuits. 6.2. Reference Feedback as a Balanced Bridge In ordinary circuit calculations we frequently encounter a condition of bridge balance between two branches by means of which transmission calculations can be considerably simplified even when the transmission is not taken di- rectly between the two branches in question. As an example we may consider the calcula- tion of the current which would flow in branchF of Fig. 6.1 as a consequence of a generator in branch//under the assumption that branches B and F are conjugate. Such a problem might be encountered, for example, in connection with the design of a constant R Fa. 6.1 equalizer structure. Since//and B are not conjugate and current must flow in B as a result of the generator in z/, it might appear at first sight that the conjugacy condition allows no simplification in computing transmission from z/to F. It follows from the principle of reciprocity,* however, that the current flowing in F as a result of the generator in ,4 must be equal to the current which would flow in //when the generator is inserted in F. When the generator is inserted in F, however, no current can flow in B and we can consequently choose any value we like for this impedance without affecting the result. Obviously convenient values of B are zero and infinity, since with either one the circuit is reduced to a simple series-shunt * See Shea or Terman, loc. cit., pp. 52 and 198, respectively, or Guillemin "Com- munication Networks," Vol. I, p. 152. 83 ----------------------------------------------------------- 84 NETWORK ANALYSIS configuration which is readily computed. A third convenient value for B is that one which balances the bridge composed of branches B, C, E, and F. This allows us to omit D, if we assume that the generator is in//, so that we can again reduce the structure to a simple series-shunt network. In a broad sense computations on a feedback circuit in its reference con- dition present an analogous situation. Evidently, the reference, since it demands zero output current for any input generator, is somewhat similar to a bridge balance between input and output. Since the principle of reciprocity breaks down in circuits containing unilateral elements, we can- not use as simple a device as was suggested in connection with Fig. 6.1 in exploiting this possibility. This complicates the analysis without essen- tially affecting the results, however. We will find that in a number of sub- sequent theorems computations in the reference condition can be made with arbitrary choices of the impedances in the input and output circuits. The choice of an impedance which will simplify the calculation then becomes principally a matter of ingenuity. 6.3. Return Difference and Relative Sensitivity The simplest illustrations of these possibilities are furnished by a set of relations between the return difference, the sensitivity, and the trans- mission from input to grid and output to plate terminals of the tube in question. As in Chapter IV, let 1, 2, 3, and 4 denote, respectively, the input, output, grid and plate. we can write Then from (4-2) and (4-26) of Chapter IV S  A A ø A ø Aa A  (64) where, as before, the superscripts 0 and  indicate that the determinants to which they are attached are to be evaluated with //7 = 0 and /(/' = 0 respectively. We observe that the determinant a in (6-1) is independent of/47 and might equally well be written as zXøa or zX' Thus the factor /xa/Aø in (6-1) is the transmission from input to grid with the tube dead, while the factor zX'//ha is the reciprocal of the transmission between the same points when the tube is in its reference condition. If we begin by multiplying and dividing F/S' by z42, instead of/Xa, we can also obtain an analogous expression involving the transmissions from plate to output for these two values of The principal difficulty with these expressions as they stand is the fact that the input to grid or plate to output transmission in the reference state cannot be calculated without allowing for the residual feedback which ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS--B 85 exists because the residual transimmittance kF0 remains in the tube For most circuits, however, the idea of bridge balance between input and out- put in the reference condition allows the problem to be much simplified. Since the balance cannot depend upon the input and output impedances, we can study the input to grid transmission for an arbitrary value of the impedance connected to the output terminals, or the plate to output trans- mission for an arbitrary value of the input impedance. By choosing the proper values in each case it is generally* possible to interrupt the residual feedback path. These possibilities are reasonably obvious physically, but it will simplify later analysis if we also verify them mathematically. To represent the effect of a change in the output line upon the input to grid transmission in the reference condition, then, we can rewrite (6-1) as F l a  + "zX2 where  is an arbitrary immittance added at the output terminals when the tube is in the reference condition. But we can also write from the general identity (4-13), Chapter IV, if we recall that 2 = 0, since there is zero transmission from input to output in the reference state. It follows from (6-3) that (6-2) is independent of , so that we can choose any value we like for this quantity without vitiating the original relationship between S  and F given by (6-1). In particular, then, we may give  a value which will interrupt the return path from plate to grid, or in other words will make fi4a = 0. With this choice the second factor of (6-2) becomes independent of 0, so that we are at liberty to suppose that the tube is dead rather than that it is in its reference condition. We can therefore state the following Theorem: The ratio between the return difference and the relative sensitivity for any tube is equal to the ratio between the transmission from the input circuit to the grid of the tube when the output impedance has its normal value and the transmission between the same two points when the output impedance is assigned the value which interrupts the return path from the plate to the grid of the tube, if the robe itself is dead in both cases. * That is, in the absence of some such special situation as that represented by the bridge-type feedback amplifiers described in Chapter III, in which the loop trans- mission is independent of the input and output line impedance. ----------------------------------------------------------- 86 NETWORK ANALYSIS C.,P. 6 If the transmission path is taken from plate to output the analysis is precisely similar and we have the Theorem: The ratio between the return difference and the relative sensitivity for any tube is equal to the ratio between the transmission from its plate to the output circuit when the input circuit has its normal value and the transmission between the same points when the input circuit is assigned the value which interrupts the returu path from the plate to the grid of the tube, if the tube itself is dead in both cases. Simple illustrations of these theorems are furnished by ordinary single loop amplifiers. If we apply the first theorem to a series feedback amplifier, for example, the interruption of the return path is accomplished by open- circuiting the output line. This evidently produces a slight change in the input impedance of the g circuit, which would otherwise be terminated by the output line impedance in series with the output impedance of the circuit. Since the input line, the input of the  circuit, and the input impedance of the/ clvcuit are all in series at the input ternfinals, there is a corresponding slight change in the transmission from the input line to the  circuit. Ina shunt feedback strocture the situation is similar except that the interruption in the return pth is produced by short-circuiting the out- put terminals. In either instance, of course, the change in transmission is small in any ordinary application. A more specific example can be obtained by returning to the structure sixown by Fig. 4.5, in Chapter IV. ill we use the first theorem the interrup- tion of the return path is accomplished by open-circuiting Z. For either the open-clmult or the normal value of Zs, however, the transmission from a generator in series with Z to the grid is inversely proportional to the impedance seen from the generator terminals. We can therefore write by inspection F Z + 2 + Za (64) " gs (Z4 + Zs) Z + g + Za + Z + Z.5 6.4. External Gain with Feedback It was suggested at the end of the last chapter that gain expressions analogous to the ones given there could be developed by starting with any reference for the variable element/4 / . If we begin, in particular, with the reference/Ko, we are led to formulae involving considerations very similar to those we have just discussed. The appropriate gain equation for calculations based on the reference ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS--B 87 is easily written from analogy with (5-27) of the preceding chapter. It is (6--5) or = , (6-6) if e ø is written in place of the last group of factors in (6-5). The validity of (6-5) can be verified by direct calculation from equations (4-25) and (4-28) of Chapter IV, if we make use of the condition which follows from an argument similar to that used for (6-3) in the present chapter. In view of the various relations among S, S t, and F which were developed in the last section and in Chapter IV it is also possi- ble to write (6-5) and (6-6) in a variety of other obvious ways. If we confine our attention to equations (6-5) and (6-6) as they stand, we are concerned principally with the quantity e ø) This is evidently a fractionareal gain expression very similar to the original fractionareal gain e øv which appeared in Chapter V, except that each of the three transmission factors of which it is composed is calculated with respect to the condition P//= /4/o rather than with respect to the condition///= 0. As in the pre- ceding section, the input and output-transmission factors ZXs/ t and Aa2/A t can be calculated with an arbitrary value for the line impedance not directly involved in the transmission path. If we choose in particular the values which interrupt the return path, the calculations can be made with the tube dead. Thus the difference between these factors and those appearing in e øv is that at each end they include the/5 circuit impedance as it would appear with the feedback loop interrupted at the other end, rather than as it would appear for the circuit connections as they stand. A simple example is furnished by the series feedback amplifier shown by Fig. 4.5 of Chapter IV, which we used previously to illustrate the calcula- tion of fractionsted gain in the zero reference case. Evidently, the trans- mission from Z to the grid in this structure is most easily evaluated if we suppose that Z5 is infinite and the transmission from plate to Z5 if we assume Z to be infinite. The fractionsted gain for the reference/(/0 can, there- for% be written down as Z Z4 e¾' = Zt + Z2 + Za Zs + Z4 + Z ZP?t' (6-7) This may be compared with equation (5-30) of the preceding chapter. ----------------------------------------------------------- 88 NETWORK ANALYSIS CuaP. 6 6.5. Simplified Computation of tf7o The material of the previous sections has been chosen principally to provide the simplest possible illustrations of the use of the bridge balance condition when the analysis, as a whole, is based upon the reference//70. It is somewhat misleading, however, in the sense that we are, in fact, likely to choose/470 rather than zero for the reference only if a relatively elaborate computation is to be attempted. The reason is apparent if we notice that the analysis in terms of/0 depends essentially upon the variables//7' and S', which are obviously more difficult to evaluate than are the correspond- ing variables//7 and F in the zero reference analysis. Thus, the use of the reference/0 calls for an initial investment in labor not required with the other procedure. On the other hand, it leads in general to simpler rela- tions. For example, (6-7) is simpler than its zero reference counterpart, and the simplification is enhanced if we include the fact that (6-7) can be applied directly to find the final output, while with the zero reference method it is still necessary to compute the direct transmission. We also need to know the direct transmission to find the absolute sensitivity in the zero reference case, whereas equation (4--3 1) of Chapter IV gives S directly if we begin with F?' and S'. In general, it appears that these advantages should outweigh the extra difficulty of determining/47' and S' initially if the circuit is complicated or if a long series of results is to be obtained, but the zero reference analysis is probably more advantageous in elementary situations. Since the computation in terms of/3'o hinges primarily upon/s,7' and S', it is of considerable interest to consider how these variables can best be evaluated. /3", of course, depends directly upon /470. S' can be deter- mined indirectly from F by the methods described earlier in this chapter and in Chapter IV. This, however, involves the intermediate step of computing F. If we wish to determine S' directly, we are concerned, in general, with the backward transmission from plate to grid in the reference condition, since it was shown earlier that S' is equal to the return difference for the reference/47o. Fortunately, the computation both of//70 and of the backward trans- mission in the reference state can be simplified by means of the bridge balance condition we have already discussed. The situation is particularly favorable if the circuit belongs broadly to any one of the general types illustrated by Figs. 6.2, 6.3, 6.4, and 6.5. In each figure the networks Ns and N= are arbitrary, but it will be seen that the relations between either the source and the grid or the plate and the load are particularly simple. For example, in Fig. 6.2, the plate and the load are" effectively in parallel" in the sense that if the plate-cathode impedance is a short circuit, there can be 'no transmission between either the input or the grid and the load. Similarly, Fig. 6.3 represents a series arrangement for tle plate and load, ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- B 89 while Figs. 6.4 and 6.5 give analogous relations between the input and the grid. The circuit need belong only" broadly" to one of these classes since minor departures will not seriously affect the results. For example, there may be other paths between input and output in addition to those shown by the figures, provided the transmission through these paths by them- selves is relatively small, since Chapter IV shows that the distinction between S  and S or F depends only upon the ratio of e øø to e ø. Fro. 6.2 Fm. 6.4 Fro. 6.5 This section will deal only with the computation of//F0. If we consider in particular the circuit of Fig. 6.2, we notice that since no voltages can exist in the output in the reference condition, no voltage difference can exist across terminals z/_,/ either. We can therefore determine the refer- ence condition equally well if we begin by short-circuiting these terminals, provided we define the reference condition as that one which gives zero current through the short-circuit. This evidently demands cancellation between the current which would be supplied to the short-circuit by the rest of the ntwork with the tube dead and the current supplied directly by the tube. In evaluating the latter, however, we need make no allowance for residual "feedback" since the short-circuit destroys the return path. The reference transconductance of the tube for the circuit of Fig. 6.2 is therefore equal to the ratio of the current flowing between z/and z/ to the voltage between grid and cathode, both quantities being evaluated with z/z/ short-circuited and the tube dead. It will be noticed that this requires a knowledge of only two transmissions, in comparison with the four appear- ing in (5-26) of the previous chapter. A simple example is furnished by the structure of Fig. 6.6. Obviously a voltage E o between grid and cathode will deliver a current YaE to a short- circuit between plate and cathode when the tube is inactive. We therefore ----------------------------------------------------------- 90 NETWORK ANALYSIS have at once = Ya. (6-8) A structure belonging to the general class of Fig. 6.3 can be analyzed in a similar fashion if we replace the short-circuit between d and zff by an open-circuit between B and B'. The reference transimpedance is equal to Fro. 6.6 Fro. 6.7 the ratio between the voltage across BB  and the current in the grid circuit.. both quantities being evaluated with BB  open-circuited and the tube dead. For example, in the structure of Fig. 6.7 we have /4"0 = Za. (6-9) We may algo continue to specify the reference condition in Fig. 6.7 in terms of admittances. Thus if we begin with any voltage between grid and cathode in that figure and compute directly the transconductance which will give a balance between the voltages across Za and Z4, with Z5 open, we readily find that P/"0, as a transconductance, is given by Z 8 . tlzø = Z2Z--' (6-10) In a circuit belonging to the general class shown by Fig. 6.4 the interrup- tion of the residual feedback path can be accomplished by supposing that a voltage generator, of zero internal impedance, is applied between grid and cathode, while in Fig. 6.5 we may assume that the circuit includes a current generator, of infinite impedance, in series with the grid lead. The reference transimmittance is equal to the ratio between a current or voltage source in the plate circuit and this voltage or current source in the grid circuit, when the plate and grid sources are adjusted to produce the same response in the output with the tube dead. These relations can be exemplified by using the structures of Figs. 6.6 and 6.7 again, and lead to the results we have already found in (6-8) and (6-9). Although these results are physically obvious it will simplify the dis- cussion in the next section to show how they can be demonstrated mathe- ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS--B 91 matically. We will consider in particular the structure of Fig. 6.2. It is convenient in this structure to use a nodal analysis, with the cathode of the tube on ground. In agreement with our earlier conventions, the input, output, grid, and plate will be taken as, respectively, the first, second, third, and fourth nodes. The short-circuit between _d and .d' will be repre- sented by adding the arbitrarily large quantity Y4 to the self-admittance of the fourth node. In terms of this notation, the voltage on the grid and plate corresponding to a unit source applied to the input with the tube dead can be written as Aløa q- Y4Ala,4 Ea = A øq-Y4A44 ' (6-11) and 3'14 E4 - A0 + Y, A4 (6-12) where A ø represents the system with the tube dead and Y4 = 0. The current in Y is E4Y4. The statement to be established is that the reference transconductance of the tube is equal to the ratio of this current to the grid voltage Ea when Y4 becomes infinite. A general formula for the reference is, however, given by (4-23) of Chapter IV. Upon inspecting (6-11) and (6-12) to find the current-voltage ratio when Y4,becomes infin- ite we therefore obtain the required relation in the form -- ,, (6-13) To prove this equation, let the voltage on the output node be written as E2 = 5o + Y4544 (14) When Y4 becomes infinite, however, the confiration in Fig. 6.2 is such that Es vanishes. -We must therefore have A244 = 0. Upon identifying 1244 with a,.a in (4-13), Chapter IV, this gives AA44 = 14&42 ß (6-15) The result (6-13) follows readily &ore (6-15) if we use (13) of Chap- ter IV again to replace A1344 and A1243 by their values in terms of first order minors. 6.6. Simpled Computation Transmission,from P&te to id The fact that the input and output must be conjugate in the reference condition, which we have just used to simplify the computation of the ----------------------------------------------------------- 92 NETWORK ANALYSIS C,Ap. 6 reference/V0 itself, can also be applied to the computation of the plate-grid transmission when/4/=/4/0. This can be illustrated by an examination of Fig. 6.2. For example, it follows from the conjugacy condition that the impedance looking to the left from terminals z/z/' in Fig. 6.2 must be inde- pendent of the input circuit when /4/=///0. Otherwise, if we were to vary the input circuit, we would expect to find a varying impedance across /ff for a prescribed plate generator and consequently a varying current in the output circuit. Since a variation in the input impedance can be represented by keeping the input impedance constant and adding a suit- able generator in series with it, this is impossible by the conjugacy condi- tion. Similarly, once the current gets over to the input impedance and the associated elements in N, the way in which it divides in the various meshes of N must be independent of the output impedance. We can therefore divide the total transmission between plate and grid in the reference con- dition into two factors, one of which depends broadly upon the load imped- ance and upon the elements of N2 but is independent of the input imped- ance, and another which depends upon the input impedance and the ele- ments of N, but is independent of the output. These relations may be expressed by the following If the structure is in any one of the forms shown by l:i'igs. 6.2, 6.3, 6.4, or 6.5 the actual circuit used in computing the transmission .between plate and grid in the reference con- dition can be replaced by an equivalent circuit in which the output impedance is assigned an arbitrary value, provided the strength of the energizing source in the equivalent cir- cuit is so chosen with respect to the source in the original circuit that they give the same voltages on the input side of the tube for any one arbitrarily chosen value for the imped- ance of the amplifier input circuit. The equivalent source may be associated either with the plate circuit or with the load and the comparison of voltages may be made either at the grid itself or at the input circuit terminals. In the application of the theorem, of course, one would attempt to choose the output impedance in a way to facilitate the final computation of feedback, while the input impedance would be chosen to facilitate the intermediate step of comparing the voltages. The notation of the preceding section will be retained in the proof of the theorem. The fact that the output circuit is arbitrary in the equivalent structure will be represented by adding the arbitrary quantity Y2 to the self-admittance ¾2'2, while the arbitrary input mesh assumed in the voltage ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS--B 93 comparison will similarly be represented by the addition of Let -/4 represent the actual plate source and I2 the equivalent source, while E is the voltage which each produces across the input. When the voltage comparison is made, we have A4 E1 = I4 At _3_ Y1A I , (6-16) and E = I2 a, + y1/x + Y2ai2 + , , (6-17) where /t is the determinant of the actual circuit when Y = Y2 = 0 and  = 0- In accordance with the conditions of the theorem Iu must be so chosen with respect to 14 that the E's determined by the two equations are equal. On the other hand, when the input circuit is assigned its actual admit- tance value, the equivalent source I2 will produce a voltage between grid and cathode given by Ea = I , + Y22 (6-18) If we replace I by its value in terms of ]i as determined from equa- tions (6-16) and (6-17) this can also be written as It follows &ore (13), Chapter IV, however, that     (6-20) if we recall that, since there can be no transmission &ore input to output in the reference condition, we can set Ae= 0. With the help of (6-20), it is readily seen that the second factor of (6-19) must be equal to unity. This equation therefore reduces to A4123 (21) * With corresponding changes in wording, if we use an impedance rather than an admittance analysis. As in the preceding section, it is assumed as a matter of simplic- ity that the input, output, and cathode are all grounded so that changes, for exmple, in the input and output affect only a self-admittance tm. ----------------------------------------------------------- 94 NETWORK ANALYSIS cAp. 6 But the transmission from plate to grid for the actual circuit is given by A4a Ea = 14 -' (6-22) The theorem is therefore demonstrated provided we can assume that -- 1. (6-23) The final step is to establish the fact that (6-23) holds for any structure of the general type illustrated by Figs. 6.2 to 5.5. It is suflqcient to examine Fig. 6.2. From an argument similar to that used to establish equation (6-15) it is clear that fi244 = /x234 = 0 for this structure. Correspond- ing to (6-15) itself we must therefore have /x,/x4 = /x44, (6-24) and A23A44 = 4A43, (6-25) from which (6-23) follows by direct division. The proof of (6-23) for the other configurations can be made by the same methods. We may also notice that although (6-23) was established on the assumption that the equivalent source was associated with the output and that the voltage comparison was made at the input, it would also have been obtained if we had introduced the equivalent source in the plate and com- pared the two voltages at the grid, so that the theorem holds for this con- dition also. As a simple example of the theorem, we may consider the structure previously shown by Fig. 6.7. Z in this figure will be taken to represent the input circuit and Zs to represent the load. For the equivalent source, it is convenient to suppose that Zs = oo, since this removes all the plate side elements from the computation. In making the voltage comparison, on the other hand, it will be supposed that Z = oo since this allows us to ignore the grid elements. If the original plate current source is I., the voltage across Za (or across Zs) for the comparison condition is given by I4[ZaZ4/(Za + Z4 + Za)]. The equivalent source must of course be adjusted to give this same voltage across Za. The equivalent source, how- ever, corresponds to an open plate circuit. When we restore the input impedance to make the actual measurement, therefore, we find that a fraction Z2(ZI + Z2+ Z) of the voltage which it would produce across Za in the comparison condition must appear between grid and cathode. If we include also the factor/4/ to give complete loop transmission, therefore, ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- B 95 the return ratio for the reference/4" 0 can be written as T' = Z2 Z ZakFt. (6-26) Zt + Z2 + Za Za + Z + Z Equation (6-26) is evidently the expression for the return voltage which would be deduced by inspection upon the assumption that Zs is so small compared to the other impedances that there is no interaction between the two ends of the network. The choice of the reference value/4'o is equiva- Fro. 6.8 lent in effect to destroying the interaction between input and output, so that in terms of this reference value the equation becomes an exact expres- sion for T t even when Za is not small. In other words, in the reference con- dition the two forward couplings represented by Za and the transconduct- ance of the tube cancel one another. The transmission backward from plate to grid is therefore unilateral and the two ends of the network are independent of one another in exactly the same way that the plate circuit and grid circuit of an ideal vacuum tube are independent. 6.7. /Implifter with Local Feedback- Computation of tFo These various theorems will be exemplified by means of the structure shown in Fig. 6.8. The circuit is a multiple loop amplifier of the general type illustrated by Figs. 3.14 and 3.15 of Chapter III. The main feedback is provided by the branch Y8. The last tube is provided with additional local feedback by means of branches Ya and Yd. This stage is evidently similar to the structures which we have already analyzed, as complete amplifiers rather than as constituents of a multiple loop circuit, in connec- tion with Figs. 6.6 and 6.7 of the present chapter. Although the analysis does not depend upon any particular assumption concerning the elements, we may conveniently suppose that Y0 is a para- sitic grid plate capacity and that Ya is a physical element deliberately ,added to enhance the total feedback on the tube. Yz and Y are intro- ----------------------------------------------------------- 96 NETWORK ANALYSIS C,AP. 6 duced to represent the fact that in a physlcal tube a portlon of the total grid and plate admittances must be considered as going directly to the cathode and this portion must be distinguished from the portion which goes to ground when the cathode is off ground, as it is in this case. Y and ¾ represent normal parasitic capacities and design elements connected to ground while Y7 is used to represent the total output admittance. The presence of both Ya and Y6 does not appreciably complicate the structure in theory, but it leads to considerably more complicated circuit equations, principally because the circuit with both elements present is essentially a bridge rather than a series-shunt configuration. In order to simplify the discussion, therefore, each stage of the analysis will be begun on the assumption that only one of these two ele- ments is present and the complete equation will be .4 B supplied only as a final step. f Since the properties of the circuit for the first and  second tubes are similar to those which would be found in a single loop amplifier, we can turn im- mediately to the output stage. The first step is to determine the reference value/'F0 for the transcon- ductance. Since no current can flow in the output circuit for the reference condition, we can sup- Fro. 6.9 pose that Y? is removed and the fundamental con- dition then becomes that the sum of the voltages across ¾8 and Y must vanish. The voltage across Ys, however, is obviously very small and will be neglected also. The circuit is thus reduced to the form shown by Fig. 6.9 and the problem becomes that of determining a transconductance/'70 such that there is zero transmission from _d to B. It follows from the discussion early in this chapter that/4-"0 must be inde- pendent of Y and Y, so that any convenient values for these admittances can be assumed in making the computation. If one of the branches 3 or 6 is missing the structure reduces to one of the types shown by Figs. 6.6 and 6.7, for which the reference transconductance has already been calculated by equations (6-8) and (6-10). With suitable changes in notation to agree with Fig. 6.9 the results may be reproduced here as Y2Y4 /4'0 - , (6-27) if ¾6 vanishes, and w0 = Y6, (6-28) if Ya is infinite. In the general case, when neither Ya nor Y0 can be ignored, we can con- ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS--B 97 tinue to determine/3'0 from a transmission computation, using arbitrary values of Y1 and Ys. A convenient choice is now Y5 = 0 and Y = This choice interrupts the return path from plate to grid, so that the net output voltage, which must be set equal to zero, can be calculated by simple superposition of the voltage due to the original source and the voltage due to the flow of plate current. With the tube dead, and these values for and Y, it is easy to calculate that a current source ,rx applied to node in Fig. 6.9 will produce the voltage i`4 E&- ya(y4+ yo) (YaY6+Y2Y6+ Y2Y+ YaY6) (6-29) from node B to ground, that is, across Y. The grid-cathode voltage pro- duced by the same energizing current is -I,/Ya. Allowing for the phase reversal in the tube, the corresponding plate current in the reference con- dition is lxtfo/Ya. When this current source is applied to the network, again with the tube dead and the chosen values inserted for Y and Ys, the resulting voltage drop across Y is ,x/4Zo 1 (1 _ Y6) (6-30) E&- Ya Y4+ Y6 But the sum of the two voltages in (6-29) and (6-30) must be zero. The correct value of/3'o is consequently Y4Y6 q- YY6 q- Y2Y4 q- YsY6 kFo = , (6-31) Ya- Y6 from which (6-27) and (6-28) follow as special cases. It is also possible, on the other hand, to determine H/0 directly from the nodal equations without using any special devices. Since this procedure is perfectly general, it is worth illustration. For the circuit of Fig. 6.9, the nodal equations appear as Ea (Y + Y= + Yo) - EoYo - EeY= = Ix, -Ex(Y6 - F/l) + Eo(Y4 + Y + Y) - Ec(Y +/F) = 0, (6-32) if we assume that the circuit is energized by the current 1.4 flowing into node z/. When PF = /470 we must have zero transmission from `4 to B. This corresponds to A`4B = 0 so that k?0 is the solution of - (¾6 - W0) - (Y + 0) (Y= + P//o) (Y2 q- Ys q- Y q- 70) = 0. (6-33) When the determinant is expanded, we obtain again the formula for//7 o already found in (6-31). ----------------------------------------------------------- 98 NETWORK ANALYSIS CAp. 6 6.8. lmplifier with Local Feedback -- Computation of Local Feedback We will assume that the final object of the analysis of the circuit of Fig. 6.8 is the determination of the relative sensitivity for the last tube. The absolute sensitivity for this tube can, of course, be determined immedi- ately from the relative sensitivity and the ratio kF/kF , which is fixed by the known value of W0. It is convenient to base the computation of S  for the last tube upon the theorem following equation (5-22) of Chapter V. We will take/F1 to represent the transconductance of the output tube and /F2 that of one of the preceding tubes. The reference values which appear in the statement of the theorem will be chosen as/40 and zero, respectively. The return difference of the output tube for the reference $F0 is, of course, the same as S . Moreover, when W assumes its reference value the return difference for W is unity, since the main loop is opened. Similarly, with W at reference the return difference of 1 for the reference W0 is merely that which would be obtained from a consideration of the "local" structure of Fig. 6.9, including the associated line and B circuit impedances. It follows from the theorem, therefore, that the actual relative sensitivity for x is the product of the return difference for e and the "local" relative sensitivity for W. This section will be concerned only with the computation of the local sensitivity. If Yo = 0, the local circuit is identical with that previously shown by Fig. 6.7 except that Z7 + Z8 has been added in parallel with Z. The local sensitivity can, therefore, be immediately written down from equation (6-26) in the form S[ = 1 + Z Z4 Z, (6-34) Z1 + Z2 + Z3 Z3 + Z4 + Z9 where Z has been written for brevity to represent the complete impedance composed of Z7 and Zs in parallel with Zs. If Za vanishes, on the other hand, the circuir is of the type represented by Fig. 6.2. The theorem on the computation of the feedback by the use of an equivalent source is, therefore, still valid. In this instance it is con- venient to suppose that the equivalent source is defined by Yo = m and that the comparison of grid responses is made for the condition Y = m. With Y = m, a current Ii in the plate circuit will evidently produce a voltage Ii/(Y4 + Yo + Yo between B and C. With Y normal, on the other hand, a generator of unit voltage and zero internal impedance applied across B and C will produce a voltage Ya/(Ys + Ys + Yo) between  and C. The local sensitivity is, rherefore, S  (6-35) = ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- B 99 If neither of the branches 3 or 6 can be ignored the analysis becomes considerably more complicated. Since the circuit no longer falls in any one of the classes represented by Figs. 6.2 to 6.5, it is not possible to use the theorem on equivalent sources to compute the feedback. We can, how- ever, develop a suitable expression directly from the expansion of the system determinant. As an alternative which requires substantially the same algebraic work, although it may seem simpler, it is also possible to derive the sensitivity from the return difference. By ordinary circuit analysis the return difference for the local circuit can be found as where YYg + Y6(¾ + t'o) + F = 1 -{- /¾/, (6-36) a + bYa + cY6 + dYaY o a = Yy'¾o +++ , b = (¾ + Y)(Y4 + Yo), (6-37) c = (Y, + Y)(Y2 + Y), d=Yi+Y=+Y+Yo. We know, however, that S[ is equal to the F of (6-36) divided by the value which F would assume if we set  = W0. From the known value of 0 this gives (YY= + YaY)(YY + YaY0+ Y( + 3Ya+ dYaY)' (6-38) 6.9. /Implifter with Local Feedback -- Final Properties In accordance with our preceding discussion the actual S' for the third tube in Fig. 6.8 can be obtained by multiplying (6-38) by the return differ- ence for one of the other tubes. The return ratio for either the first or second tube, hovever, is simply the transmission around the main loop. This in turn can be broken up into two components, one representing the transmission from .4 in Fig. 6.9 to some point such as B, say, and the second representing the transmission around the rest of the loop. The second will be symbolized by K and will be assumed to be known since it presents no special problem. Since we already know S[ for the last tube, equations (6-5) and (6-6) allow us to compute the transmission from//to B as soon as the fraction- ated gain of this tube for the reference condition is determined. It will be ----------------------------------------------------------- i00 NETWORK ANALYSIS C.,P. 6 recalled that the grid transmission term in this gain can be calculated for an arbitrary choice of the load impedance and the plate transmission term for an arbitrary choice of the input impedance. Let it be supposed, first, that Y6 = 0. It is then convenient to choose the arbitrary impedance as an open circuit in each computation. This has already been examined in connection with (6-7), For the present circuit the resulting transmission from /to B can be written as* 1 Z1Z2 ZZo œOAB -- 8[ Z I '-{- Z 2 '-Jr- Z 3 Z 3 '- Z 4 .- Z 9 t/fart' (6-39) where, corresponding to the fact that we have assumed Y6 = 0, S[ must be determined from (6-34). If we assume Za--0, on the other hand, it is most convenient to determine the grid transmission for the condition Z0 = 0 and the plate transmission for the condition Z = 0. With these two assump- tions the two transmissions are readily seen to be 1/(Y + Yg + Y6) and 1/(Yt + Yo + Yo). The gain from a/to B consequently becomes 1 1 1 eøAB = S- Y + Y + Y6 Y4 -- Y6 -- Y0 //rt, (6-40) where S[ is determined from (6-35). If neither of the branches 3 or 6 can be neglected the analysis is naturally somewhat more complicated but it can be made by the same general methods. For example, in computing the transmission to the grid, we can conveniently assume that Y0 = -Y[(Y1 + Ya)/(Y1 -3- Y)]. This is the value of Y0 which gives zero transmission from Y4 to Y2 so that the flow of current in Y4 due either to transmission in the passive parts of the net- work or to transmission through the residual transconductance/3'0 will not affect the voltage across Y2. The computation can thus be made for any assumed value, such as a short circuit, for Y4- Similarly in computing the transmission from plate to load we can assume Y = -Y6[(Ya -1- Y4)/(Y -3- Y0)] which allows us to short-circuit * The numerator of (6-39) includes the factors Zt and Z, for which no correspond- ing terms exist in (6-7). These factors are introduced to express the result in nodal rather than mesh terms. Thus in (6-7), where an impedance analysis was used, the driving force was taken as a unit generator in series with Z and the response was stated in terms of the current through the load. The introduction of the factor Z in effect expresses the driving force as a unit current applied to Z, while the intro- duction of Z0 is equivalent to expressing the response as the voltage across the load. A nodal analysis is chosen here for consistency with the other equations of this section. ----------------------------------------------------------- THEOREMS FOR FEEDBACK CIRCUITS -- B 101 The expression for the transmission from ,4 to B is accordingly ,OAB  S[ YY= + YYa + Y=Ya + YaY6 x Y6 (641) YaY4 q- YaYo + YaYo + Y4Yo where the first and second expressions involving the s are, respectively, the transmission from the source to the grid and from the plate to the load. Upon multiplying the appropriate one of these expressions by K, which represents the transmission from B around the rest of the loop, including the transconductance of the second tube, we secure the complete fi charac- teristic. This then is -T for either te first or the second robe. In accordance with the theorem on the relation between two return differ- ences, the actual relative sensitivity for the third tube can be obtained by multiplying the corresponding F for the first or second tube by the S[ for the third tube, as expressed by equations (6-34), (6-35), or (038). For example, if we assume Ya = m and write S  for the total relative sensitivity of the third tube, the result from (6-35) and (60) is ( 1 1 1 'K) 1 1 =  + (Y + K)'. As the final step in the analysis we may compute the distortion which would appear in the load as the result of a prescribed distortion generator in the plate circuit of the third tube. The theorems of Chapter V show that this is equal to the distortion whick would flow in the load when the third tube is in the reference condition divided by S  for that tube. We have, however, already computed the ratio between a given plate current and the voltage between B and ground for the reference condition. If we let k represent the ratio between the voltage at B and the resulting voltage across the final load impedance with the amplifier input circuit open, there- fore, the results can be immediately written down as for Y6 = 0 for Za=0 YaYo + (643) in general, ----------------------------------------------------------- 102 NETWORK ANALYSIS o,^p. 6 where Ij is the prescribed distortion generator and S  in each case is the appropriate relative sensitivity for the third tube. It will be recalled that a double loop feedback circuit essentially similar to the one under discussion here was used in Chapter IV to' :illustrate the fact that the scnsitlvity of a tube in the t circuit of a multiple loop structure was not necessarily equal to its return difference. The ilhmtration can he made somewhat more specific with the help of the presetat equations. For example, suppose we set ,r'F t = -/'Fo in (6,2). This is cqulvalcnt to setting/'P' = 0, so that the corresponding return difference will be unity. It is clear, however, that the ratio of relative sensitivity to return difference is independent of/F, so that it will he the same for actual operating con- ditions as it is fro' this special choice. Upon thtroducing/3 o = Ys, from (6-28), for the case represented by ((z,-42) we therefore have 8' ¾n(Y + K) if- = I -- (y + _o + Y6)(4 + Y6 + Y)' (644) It is evident from (6-44) that if we can make K large enough the sensi- tivity* can be made much greatcr than the retrain difference. On the other hand, by choosing special values for K and the various Y's we can also secure a sensitivity which is much smaller than the return difference. The values of these quantities which would appear natural[y in normal design practice are probably not such as to make either extreme very likely. The fact that the sensitivity and return difference are not necessarily identi- cal is of considerable theoretical interest however since the limitations on available "feedback" developed in the following chaptcrs are actually limitations only on the return difference. * No distinction between $ and S' need be made here, since we can readily choose a k/r0 small entmgh to make the two approximately equal, without affecting the rest of the argument. ----------------------------------------------------------- CHAPTER VII STABILITY AND PHYSICAL REALIZABILITY 7.1. Introduction THE preceding chapters have been devoted largely to the problem oœ active network analysis. It has been assumed, in other words, that the structure under consideration was given, and that we were interested in finding out what it would do. To this end, the mesh and nodal equations were first introduced. The succeeding chapters consist principally of applications of these equations to various situations, with particular atten- tion to what they could tell us about the relation between a single given element and the,characteristics of the complete network within which it appears. Beginning with the present chapter, attention will be turned broadly œrom problems of analysis to those of synthesis or design. It will be assumed in other words that our primary interest is in working backward from a prescribed type of response characteristic to a network which might exhibit it. This chapter will serve only to introduce the subject. It is devoted principally to a consideration of the requirements which a net- work must meet if it is to be stable and of the limitations which this imposes on the network characteristics which are available ['or design. 7.2. Design Methods and the Problem of Physical Realizability The development of final design methods for feedback amplifiers is approached here by way of a lengthy and perhaps indirect introduction. Before beginning the discussion it may consequently be desirable to say a few words concerning the point of view which motivates this approach, It must be recognized to begin with that the processes of synthesis or design are in some respects essentially different from those of analysis. If a net- work is given, only one response to any prescribed force is possible, and that response can, in theory, be obtained by a mechanical computation, so that the whole operation is reduced to a routine level. The design process can- not be described so exactly. In a broad sense it consists in the construction of a larger unit by the establishment of a pattern of relationships among a number of smaller and more easily controlled units. In a feedback ampli- tier, for example, we are concerned in the first instance with the provision of suitable characteristics for the amplifier as a whole by the establishment 103 ----------------------------------------------------------- 104 NETWORK ANALYSIS CP. 7 of an appropriate pattern of relationships among the separate units, such as tubes, input and output circuits, feedback and interstage networks, etc., of which it is composed. Beyond this point we may be concerned with the relation between any one of these circuits individually and the various elements from which it is built. In almost all design situations several or many patterns of relationships may yield a satisfactory result. For example, we may obtain a given for- ward gain for a feedback amplifier from various combinations of input and output circuits, tubes, and interstage networks. On a smaller scale, a given interstage characteristic can usually be represented, within tolerable limits, by structures of several different physical configurations. The choice between the possible solutions may depend upon ulterior considera- tions, such as economy, reliability, power consumption, the speed with which parts can be secured, etc., which are not readily taken into account, at least in detail, in a theoretical discussion. Or it may be purely arbitrary. In any event the establishment of any one pattern involves essentially an effort of imagination on the part of the designer. As such it is a creative operation, on a more or less difficult plane, and defies exact analysis. In a group of structures which are very much alike, such as a set of amplifiers meeting similar requirements in about the same frequency range, a general type of pattern may become so well established that much of the work is reduced to a routine level. As the diversity of application increases, how- ever, the essentially creative nature of the design process becomes more apparent. It follows from this discussion that design methods suitable for a variety of applications can never be reduced entirely to a set of rules. They are best when they leave the final synthesis in the hands of the designer but stress the development of conceptions and processes which make the establishment of any particular set of relationships as simple and easy a matter as possible. This can be done in part by pointing out types of relationships which are plausible but either cannot be carried out or lead to unsatisfactory results. It is futile, for example, to plan a feedback ampli- fier about an assumed input transformer whose gain is greater than can be obtained with the existing parasitic capacities. On the positive side, design can be expedited by the construction of general patterns of relations which can be extended to a variety of situations by the choice of numerical values for a few parameters, and by the discovery of simple methods of specifying the subsidiary units which make up a complete structure. An excellent example here is furnished by conventional filter theory. The general pattern is the composite filter with matched image impedances. The subsidiary units are the discrete sections. They are particularly easy ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 105 to deal with since an individual section is specified, in essentials, by a single parameter, and in their significant properties the sections are directly additive. The choice of any particular combination of sections to meet a particular set of requirements, however, is left in the hands of the designer. It is evident from this background that what we need. most of all in developing design methods for feedback structures is a characterization of the available units which may enter the complete structure in terms which are as easy as possible to handle in planning the over-all design. This is, of course, necessary if we are to avoid blind alleys of the type described previously. It is also required in planning any general design patterns which are likely to be of practical value and it is necessary again in fixing the proportions of any specific pattern. As a matter of actual experience, it appears that if the characteristics of the units of the amplifier can be properly specified in broad terms the road to a final detailed design is relatively straight. In network synthesis, a characteristic is "available" in the broadest sense if it can be furnished from some combination of physically obtainable elements. The restriction to physical elements is one which does not appear in network analysis. It makes no difference in the routine of deter- mining the response of a given structure whether the elements are positive or negative, to say nothing of whether or not they are accompanied by parasitic effects of the types which might occur in practice. In network design, however, the restriction is fundamental and will be the next object of investigation. It is unfortunately a dicult topic and will require several chapters. The quantities which appeared most conspicuously in the preceding analysis were the driving point and transfer immittances, the return differ- ence, and the sensitivity. They may be lumped together under the general name, network functions. They are all defined as ratios of determinants so that they are all rational functions ofp. It will be recalled from Chapter I that any rational function can be specified, except for a constant multiplier, by its zeros and poles. In the next few chapters the condition of physical realizability will be discussed in terms of the restrictions it imposes upon the location of the zeros and poles of the various network functions on the complex p plane. Following this discussion, the restrictions on the zeros and poles will be converted into equivalent restrictions on the behavior of the functions on the real frequency axis. This background is necessary in order to provide a specification in useful form of what is available in design- ing a feedback structure. With it as a foundation we will at length be able to approach the actual design problem directly. ----------------------------------------------------------- 106 NETWORK ANALYSIS CH^P. 7 7.3. Criteria of Physical Reatizability Before we can study the restrictions which the condition of physical realizability places upon available network functions, it is evidently neces- sary to find some formulation of what we mean by physical realizability which can be used as a basis for deduction. Perhaps the most obvious for- mulation is expressed by the statement that a physically realizable network is a combination of vacuum tubes and positive inductances, capacities, and resistances. This, however, is both awkward and misleading. Except in the very simplest configurations a study of the relationship between the signs of the elements and the resulting network characteristic entails intolerable algebraic complexities. Moreover, it can readily be shown* that any negative element can be simulated, at least in the ideal case, by a suitable combination of tubes and positive elements. The distinction between positive and negative elements thus cannot be the heart of the problem. Although the sign of the elements cannot be used as a basis for analysis, some importance can be attached to the fact that the elements must at least be real It follows immediately from this that if the frequency vari- able is taken as p, the coefficients in the mesh and nodal equations, and therefore the coefficients in the network functions, must also be real. If we replace i0 by its conjugate in any term of a network function, conse- quently, that term must assume the conjugate of its original value. Since conjugate values everywhere in the function must lead to a conjugate result, this establishes the Theorem: A physically realizable network function assumes conjugate complex values at conjugate complex points on the p plane. For most applications this theorem can be expressed more conveniently by means of the following two corollaries: 1. Any zeros and poles of a physical network function which are not real must occur in conjugate complex pairs. 2. The real and imaginary components of a physical network function on the real frequency axis have respectively even and odd symmetry about the origin. The first of these evidently follows from the fact that zero and infinite values of a network function are their own conjugates, while the second is estab- lished if it is noticed that symmetrical positive and negative real frequencies are a special case of conjugate p's. We may also observe that since the zeros and poles specify the network completely except for a constant * See, for example, the circuits described near the end of Chapter IX. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 107 multiplier, and the multiplier must be real if the second corollary is to hold, the two corollaries together are equi'alent to the original theorem. The theorem on real element values is sufficient to restrict the range of available characteristics only very generally. The field can be narrowed much further from a consideration of the stability of the network. It is a familiar fact that many hopefully designed feedback structures "sing," or break into spontaneous oscillation, when the circuit is closed. This is customarily explained by regarding the free oscillation as a manifestation of one of the natural transients of the system. It is assumed, in other words, that the system has been exposed to some small shock which pro- duces a normal transient response. In most systems transients are expon- entially decreasing functions of time and quickly die out. If the system sings, however, it is supposed that one of the transients is negatively damped, and so increases with time. In this case it will eventually become very large, no matter how small the initial shock may have been. Since random small shocks, on the level of thermal vibrations at least, are una- voidable, the phenomenon must occur if the system has any possible tran- sient response which increases with time. In a physical situation the amplitude of the oscillation may become large enough to burn out part of the system. Otherwise, it is limited by the inability of the system to maintain a linear response characteristic for amplitudes beyond a certain range. This is true, for example, in an ordi- nary oscillator, where the amplitude is limited normally by the physical possibilities of the output tube. Since the analysis in this book is con- fined to linear circuits either eventuality removes the structure from our purview.* It may seem at first sight that although the possibility that the network may break into free oscillation may be important, it should be considered separately from our immediate problem, which is the investigation of the steady state characteristics of the netvork. A connection between the two problems, however, appears from the welLknown fact that the tran- sient response of a network can be predicted from its steady state charac- teristics. The analysis given in later chapters shows that this connection is so close that the steady state characteristics which may be obtained from stable structures are radically limited in comparison with the characteris- tics obtainable from mathematical functions chosen at random. Since there is no point in discussing the hypothetical" steady-state "character- * In some modern oscillator circuits the amplitude of the oscillation is limited by a thermally controlled element. These are essentially linear circuits, since the change in the thermistor over one cycle is negligible, and it is not intended to exclude them here. After the thermistor reaches its steady value they can be regarded as stable structures, but with a root on the real frequency axis as described later. ----------------------------------------------------------- 108 NETWORK ANALYSIS CHAP. 7 istic of a structure which will in fact sing when it is constructed, there is economy of thought in combining the two ideas to begin with. The essential statement of what we will mean by physical realizability can therefore be expressed by the following Definition: A network function will be said to be physically realiz- able if it corresponds to a network of real elements having no modes of free vibration whose amplitudes increase indefinitely with time. This will also be regarded as a definition of what is meant by a stable* circuit. The relationship between the modes of free vibration and the steady state network functions is described in the following sections and, more generally, in later chapters. The definition just given is the foundation upon which the analysis of general circuits, including both vacuum tubes and passive elements, will be based. A structure composed exclusively of passive elements, on the other hand, cannot give as wide a variety of characteristics as would be admissible from this definition alone. Since many of the units of which a typical feedback amplifier is composed, such as the interstage networks and the feedback circuit itself, are purely passive, it is of interest to determine what these additional restrictions on passive circuits may be. An analysis of this problem is given at the end of this chapter. Pending this analysis, the following results will be assumed: 1. A passive circuit is always stable. 2. The real component of a passive immittance is never negative at real frequencies. 3. If a passive network is driven by a single real frequency generator the power delivered to the network as a whole is always at least as great as the power consumed by any one resistance in the structure. The second and third of these conditions are evidently merely consequences of the principle of conservation of energy, in combination with the fact that a passive network cannot contain a source of power. The justification for the first may not be quite so obvious, but the proof given later estab- lishes it on the same general grounds, using the methods of classical dynamics. * It is to he noticed that stability as defined here includes, as a limiting case, the possibility of purely sinusoidal transients which neither increase nor decrease with time, such as characterize purely reactive structures. This limiting case is discussed in more detail in a later section. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILIT¾ 109 7.4. Stability and the Roots of zX Our first object will be the development of some analytic tool for investi- gating the relation between the steady state characteristics of the network and its stability. The stability of the circuit depends upon its possible transient responses and is therefore best determined from a study of the differential equations representing it. This is facilitated by the fact that the general mesh and nodal equations of Chapter I were first developed in differential form. Equations (1-2) of that chapter, for example, give the differential mesh equations and can be rewritten here as di f L - q- Ri + Dl idt q- ß ß ß f + Ll - + Ri + D idt = 0 L  + Ri + D idt + ß ß . L di f +  + Ri + D2 idt = 0 ..................................................... (74) di f L  + Ri + D idt + ß . ß f + L  + Rd + D idt  O. These are essentially the same as the original expressions, but the instan- taneous currents have been represented by small rather than capital letters and differentiation and integration with respect to time have been written out explicitly in order to avoid confusion with later notation. The driving voltages on the right-hand side of the equations have' also been omitted, since we are interested only in the free response of the system. Let it be supposed that the possible transients are exponentials of the general form e vt. The individual currents i, i, ß ß., i can be written as lie , IVt, ß .., Ie vt, where the ]'s are constants whose magnitudes will depend upon the original disturbance. In general, the p's representing possible transients may exist either as real quantities or as conjugate complex pairs. If p is complex the "currents" Je vt, Ie *, etc., must also be complex. As in Chapter II, however, the real components of these "currents" satisfy the differential equations by themselves and may be taken to represent the actual physical transients. Upon substituting ]e , ] vt, etc., in (7-1) and dividing out the com- mon time factor, e v, the result appears as ----------------------------------------------------------- 110 NETWORK ANALYSIS C..P. 7 pl pl -½(pLln- Rln P/ p/ p/ + (pL2, + R2 + D I = O ..................................................... (7-2) P/ Pl +(pL+R+DI=O. P/ It is evident that I = 12 ..... In = 0 is always a solution of equations (7-2). Since there are n equations and n J's we may expect, in general, that the I's will be uniquely determined, so that this is the only solution. If the transient is to exist physically, however, some, at least, of the I's must be different from zero. This will be possible provided p is so chosen that the n equations represent fewer than n independent conditions on the I's. We might find, for example, that with a special choice of'p one of the equations was equal to the sum of two others. It can be shown* that the general condition for the n equations to represent fewer than n independent relations is that the determinant of the coefficients in the equations shculd vanish. The expression which fixes the values of p which may represent transients is therefore ,x = 0 (7-3) where zX is, of course, identical with the/x we have previously used and is a polynomial in p divided by some power of p. If one of the p's which satisfies (7-3) lies in the left half-plane, it follows from the discussion in connection with Fig. 2.2 that the correspond- ing physical transient will be a damped sinusoid of the general form e --t cos/t. If p lies in the right half-plane, on the other hand, the transient will be of the form e t cos/t, where a is positive in either case. A sinusold with exponentially increasing amplitude, such as e t cos /t, is, however, a * See, for example, Dickson's Modern dlgdrraic Teories, p. 55, or B6cher's Intro- duction to Higher Algebra, p. 47. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 111 runaway transient of the type barred out by our preceding definition of physical realizability. We can therefore state the following Theorem: None of the zeros of the, principal determinant of a physical network can lie in the right half of the p plane. An example of a permissible dis- tribution of zeros is given by Fig. 7.1. As the figure shows, some of the zeros are taken as real and others as conjugate complex pairs. Most o p plane of the zeros are found in the interior of the left half-plane, but there is in ---o. o , addition one zero at the origin and o a pair of conjugate zeros on the real frequency axis. Zeros of this type o correspond to transients whose am- plitudes are maintained with time but do not increase. There is thus Fro. 7.1 no physical reason for barring them out on grounds of instability, but they represent the extreme limit which can be attained in a stable structure. A more detailed example of permissible zeros can be obtained by return- ing to the damped resonant circuit which was used as an illustration in Chapter II. The zeros were given by equation (2-27) of that chapter as P = + - -Z; pz= -  - (7-4) They were described there as the zeros of impedance but since Z = zX//x they are evidently the same as the zeros of zX. When R = 0 the two p's lie on the real frequency axis. With moderate damping they occupy con- jugate points in the left side of the p plane, while when the damping is extreme they are found on the negative real axis. This is illustrated by Fig. 2.4 of Chapter II. It is interesting to notice that in this simple case the stability requiremen.t corresponds almost exactly to the requirement that all the elements be positive. If we change the sign of any one or any two of the elements at least one of the zeros will be found in the right half-plane. The only possibility is the obviously symmetrical situation obtained by making all three elements negative. 7.5. Zeros of  on the Real Frequency 4xis The possibility of securing zeros of zX at real frequencies, which was exemplified by Fig. 7.1, merits further discussion. In passive structures ----------------------------------------------------------- 112 NETWORK ANALYSIS CH,P. 7 zeros must be found at real frequencies if the network is composed only of pure reactances.* In the resonant circuit just described, for example, real frequency zeros were obtained by setting R = 0. They may also be obtained, even with dissipation, in active circuits containing sources of power which just balance the dissipative losses. As an example of this condition we may imagine a feedback amplifier which is normally stable but can le made unstable by an appropriate change in some continuously variable control. Zeros would be found on the real frequency axis in this circuit if we could set the controlling element on the exact point dividing the regions of stability and instability. The probability of securing such exact balances or such ideally dissi- pationless elements in a physical structure is evidently infinitesimal. We are thus entitled to assume, if we wish, that all the zeros in physical circuits are somewhat to the left of the real frequency axis. This possL bility will not be utilized in dealing with ordinary reactive resonances in passive circuits. The assumption of zero dissipation is frequently a con4 yenlent idealization, especially in dealing with driving-point immittances. On paper, it may also arise in transfer immittance problems, as it would, for example, if we were computing the transmission through a dissipationless filter which is either open- or short-circuited at both ends. For practical purposes, however, the consideration of four-terminal problems will be restricted to circuits in which the terminations, at ]east, are dissipative. The other possibilities of securing real frequency zeros arise in circuits containing active elements. Here it will be convenient to suppose that the zeros lie, in fact, slightly to the left of the real frequency axis. Aside from the question of convenience, there are special physical reasons for making this assumption. At a real frequency zero a driving force of corresponding frequency inserted in any part of the circuit will produce an infinite response everywhere else in the circuit. For example, if the input and output of an amplifier are represented respectively by 1 and 2 the output current in response to a unit input voltage is /x2//x, so that there should be infinite gain to a driving force whose frequency coincided with one of the zeros of In a physical situation, of course, we would expect the active elements to become overloaded and excessively non-linear as soon as this frequency was approached. Since the exact location of the zero would be immaterial in any case if we were interested only in driving forces at more remote fre- quencies, there is thus a special justification for the assumption on grounds of linearity. A convenient example is furnished by the thermistor controlled oscillator described in a previous footnote. If the amplitude of the oscillation is * See, for example, the discussio qiven near the end of the chapter. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 113 small and there is reasonable selectivity in the thermistor circuit this should be a linear network for signal voltages at frequencies remote from that of the natural oscillation. In these ranges, however, the steady state charac- teristics are negligibly affected if we move the zero of/x slightly away from the real frequency axis. If driving forces in the neighborhood of the zero are applied the circuit must become nonAinear, since we can no longer assume high discrimination against signal currents in the thermistor circuit, and the thermistor temperature will be affected by the heat generated due to the passage of signal currents through it. If zeros of zX are assumed to occur at real frequencies they are subject to one restriction which has not appeared heretofore. It was tacitly assumed in discussing (7-3) that all the zeros were separate. In special cases, however, multiple zeros may occur. It is known that in such circumstances the form of the transient solution may be changed. Instead of consisting solely of exponentials it may also include exponentials multiplied by powers oft. For example, ifp is a double zero of (7-3) the corresponding transient appears in the general form Ae vt + Bte vt. If p is in the interior of either the right or left half-plane the extra factor t in the second term is of no significance in determining whether the transient will increase or diminish with time, since it is overwhelmed by the exponential. In the special case when p is on the real frequency axis, however, it makes an increasing tran- sient of one which would otherwise be merely persistent. Since transients which increase with time are inadmissible we can therefore state the Theorem: Zeros of  on the real frequency axis must be simple.* 7.6. Zeros of Other Determinants In addition to zX itself, the network formulae which have been developed involve other determinants derivable from zX in various ways. One group of these includes /x ø and what may be called the "symmetrical" minors Xii, zXii, zXiij5, etc. Each of these quantities can be regarded as the form to which /x reduces when some prescribed change is made in the network * This theorem is not rigorously true in degenerate circuits. Suppose, for example, that the system consists of two identical but entirely independent units. A single set of mesh equations may be used to describe both units. The determinant of the system will be the product of the determinants for the two units separately, and must have a double zero at any real frequency at which the determinants of the separate units have simple zeros. The slightest coupling between the _units, however, will destroy this relation. In any event, such an exception does not destroy the physical consequences of the theorem, since we are eventually interested in the zeros, not of A itself, but of the ratio of A to one of its principal minors. If A has a multiple zero because of such a degeneracy, the minor will have a zero one order lower, so that the zero of the ratio is still simple. ----------------------------------------------------------- 114 NETWORK ANALYSIS CaA. 7 and their zeros can consequently be limited in the same way as those of A itself if the network is known to be stable after the change is made. Thus A ø is the form to which ,x reduces when some given element/47 vanishes and can have no zeros in the right half-plane, and only simple zeros at real frequencies, if the circuit is stable with N / absent. This would certainly be true, for example, if  represents one of the tubes in an ordinary single loop amplifier, since when/f vanishes the loop is opened. Similarly, such quantities as A or AS are the cofactors of///ii or/fi in A. They are thus the forms to which A reduces* when N7ii or Nzji becomes infinite. This is equivalent to open-circuiting the ith or jth mesh, if we are using a mesh analysis, or short-circuiting the ith orjth node to ground, if we are using a nodal analysis. In the same way, Aiiij gives the result when the open or short circuit is applied both at i and atj. The zeros of any of these quantities are restricted in the same way as those of A itself if the network is stable after the open or short circuit is applied. This is true, for example, if we are dealing with a series impedance or shunt admittance in a single loop amplifier, since an open-circuited series branch or a short-circuited shunt branch will break the feedback loop. All these relations become particularly simple in passive networks. Obviously, a passive network is still passive after any of these various operations is performed upon it. The proposition stated previously, that a passive network is always stable, therefore allows us to establish the Theorem: In a passive circuit none of the zeros either of A ø or of any of the symmetrical minors of A can lie in the right half of the p plane, and any zeros on the real frequency axis must be simple. The remaining determinants which appear in the network formulae are "unsymmetrical "minors of the general types Aij, Aijkk, 'Sijkl, etc. These can evidently be regarded as the forms to which A reduces when indefinitely large unilateral couplings are added to the circuit. For example, since Aii is the cofactor of/(7ij, it is the limit which would be approached by A if we introduced into the circuit an ideal vacuum tube of extremely high gain with grid terminals at j and plate at i. Unfortunately, there is ordinarily no simple method of determining whether the circuit will be stable after this modification is made, so that such a physical interpreta- tion is of no great value. For circuits of general physical configuration it appears that the zeros of these unsymmetrical cofactors may appear in any part of the plane, even when the structure is made up entirely of passive * Obviously, the limits approached by A in the two cases are actually//'i/Ail and 14"iiAii. Since we are concerned only with the location of the zeros of A, however, the multipliers/i and I4'ii can be disregarded. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 115 elements. They are restricted only by the conjugac¾ condition. Restric- tions on these zeros can sometimes be imposed when the network is known to have one of certain special physical configurations, but this is more con- veniently discussed in a later chapter. 7.7. Zeros in an Illustrative Circuit These principles will be exemplified by means of the circuit shown by Fig. 7.2. The structure will be taken ini- tially as the Bridged-T of purely passive elements. The broken lines shown going to _d and B are connections made R 2 Fro. 7.2 to the vacuum tube at a later stage to show how an active element affects the stability of the circuit. It will be assumed for concreteness that all the passive elements are of unit magnitude. If the meshes are chosen as shown in the figure, with the tube deleted, the mesh equations in the absence of any driving force are P -pI + (2p q- 1) 12 - p/ = 0 (7-5) 1 11 -- pI2 q- q- 1 q- Ia O. P The equation corresponding to (7-3) is consequently or 1 1 p+l+- -p -- p p -p 2p+l -p 1 1 -p p+l+- p p = 0 (7-6) 1 A = - (3p a q- 4p 2 q- 72 q- 2) -- 0. (7-7) p The roots of (7-7) are p = -« and p = -«(1 q- ix7). They are indicated by the circles in Fig. 7.3. They are all on the left half of the p plane, as of course they should be, since the network, being passive, is necessarily stable, ----------------------------------------------------------- 116 NETWORK ANALYSIS C.^p. 7 Fro. 7.3 We may next proceed to verify that the zeros of zX ø and of the symmetrical minors of fi are also confined to the left half-plane, for the passive structure. Let it be supposed that zX ø represents the system when L2 = 0. The disap- pearance of L2 is equivalent to replac- ing the Z22, Z3, Za2, and Zaa terms in (7-6) respectively by (p + 1), 0, 0, and (1 + l/p). We readily find that the equation corresponding to (7-7) appears as A0 = _2 (i + p)2. (7-8) P This has a double root at p = -1, which is, of course, in the left half-plane. The double root is represented by the crosses in Fig: 7.3. Similar results hold if rio represents the system after any other element has vanished. As an example of a symmetrical minor we will take fi. This quantity is given from (7-6) as A22 --- 1 1 p+l+- -- p p 1 1 -- p+l+- p p 1 (p3 q_ 2p q_ 3p q_ 2). (7-9) p The roots are in the left half-plane at the points -1 and -«(1 4- ix/q). They are indicated by the squares in Fig. 7.3.* The second order symmet- rical minors are still simpler since they are the same as the self-impedances * The fact that many of the roots of these various expressions happen to coincide is due to the specially simple and symmetrical form of the network, and would not be true in general. For example, Rt, Ra L, and 14 constitute a balanced bridge as seen from R2. A generator in series with R= can therefore produce no current in D, so that the driving point impedance measured in the second mesh must be much simpler than the structural complexity of the circuit would indicate. This is reflected by the fact that A and A have common roots, which cancel out in the ratio z/A2, repre- senting this impedance. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 117 of the several meshes. For example, we have Allaa = 2p + 1, with a root 1 at p = -. The properties of the unsymmetrical minors will be illustrated by means of Xal. We find from (7-6) that -p A31  2p+l 1 P -p _ P (pa + 2p + 1). (7-10) The roots occur at p = -0.4.53 and atp = +0.227 4- il.47, as shown by the triangles in Fig. 7.3. They are thus found in both halves of the plane. This must be anticipated, in general, whenever we are dealing with unre- stricted circuits. By choosing special configurations or special element values, on the other hand, the roots of an unsymmetrical minor may be con- fined to the left half-plane, just as are those of the determinants previously considered. As an example, suppose that a resistance R is added in series with Lz of Fig. 7.2. We may suppose that R is simultaneously subtracted from R and R2, so that the change affects Zx9 and Zm but not the self- impedances Zz and Z29.. It is readily shown that (7-10) becomes a =  (pa + Rp + 2p + 1). (7-11) All the roots lie in the left half-plane when R > «. For example, if R = -theyare found atp = -0.52 and atp = -0.074 4-il.38. These locations are shown by the primed triangles in Fig. 7.3. The reason for paying particular attention to networks for which the roots of at least certain specified unsymmetrical minors can be confined to the left half- plane is that this leads to the "minimum phase" condition, which is of considerable importance in amplifier theory. "Minimum phase" net- works are mentioned again in one of the following sections, but a detailed discussion of their properties is reserved for a later chapter. In order to exemplify the changes which may be produced in these results by the presence of an active element, we may suppose that the vacuum tube is added to the network by closing the connections indicated by the broken lines in Fig. 7.2. We can take R and Ra to represent the grid and plate impedances of the tube. Since R = 1, the transimpedance of the tube, which is equal in general to t times its grid impedance, becomes simply/, and the incorporation of the tube is equivalent to adding t to Za in the mesh equations. The new determinant of the system is readily ----------------------------------------------------------- 118 NETWORK ANALYSIS C}Ap. 7 found from this as 1 1 pq. 1 q.- P P -p 2p+l 1 1 /-- -p p = _1 [3pa + 4P 2+ 7p + 2 + (p3 + 2p + 1)1. (7-12) p When/ is very small the zeros of (7-12) will evidently be very close to those originally determined from (7-7). As I is made larger and larger, however, some of them will eventually appear in the right half-plane, so that the network will become unstable. This can be studied most easily by observing that since the zeros must vary continuously with t they can go from one half-plane to the other only by crossing the real frequency axis. If we assign a pure imaginary value to p in (7-12), however, the real and imaginary components of the expression can be separated and equated to zero individually. This gives (3 q- u)p a q- (7 q. 2)p = 0 (7-13) and 4p 2 + (2 + ) = 0. (7-14) If we eliminate,between (7-13) and (7-14) the result is 4p  q- 7p  - 3p = 0, (7-15) which is satisfied by p = 0, ]1) 2 = -2.1, and p2 = q'0.35. The last of these can be disregarded, since it evidently does not correspond to a point on the real frequency axis. It represents an accidental solution of (7-13) and (7-14) in another part of the plane. The first two, however, are valid solutions and correspond respectively to  = -2 and  = q-6.4. We can therefore conclude that the network will be stable for 6.4 >/ > -2, and will sing when t is taken beyond these limits. For example when/ = 9 the zeros are p = -0.43 and p -- q.0.05 4- il.45, while when / = -2.5 they are p = q-0.18, p = -0.74 and p = -7.5.* These relations are illustrated by Fig. 7.4. As the gain of the tube is changed from / = 0 to t = q-o the zeros move along the approximate * The negative 3t assumed here could not, of course, be obtained from an ordinary tube, but it might be secured by using one of the" negative transconductance" tube3 vhich have been developed experimentally. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 119 paths shown by the solid lines in the drawing, following the directions indi- cated by the arrows. The paths for the range =0 to -- -oo are shown by the broken lines. The crosses and squares correspond to the roots just determined for the special values  = 9 and  = -2.5. The circles give the original positions of the zeros when  = 0 and the triangles their final positions when  = 0o. As a comparison of (7-10) and (7-12) indicates, the final positions are the same as those of the roots of Fro. 7.4 Ast. Since some of the roots of zXa are found in the right half-plane in this illustration, it is evident without further analysis that the circuit must sing if  is made sufficiently large in either direction. This may serve to explain to some extent the reason why so much stress was laid in our previ- ous discussion on the possibility of confining the roots of this determinant to the left half-plane. The modifications produced in the other determinants of the network by the addition of the active element are of a similar type. The chief point to notice is that the zeros of zX ø and of the symmetrical minors necessarily occur in the left side of the plane only when the circuit is passive. After the addition of the tube they will, in general, appear in the right side for 's beyond a certain range. As a final example we may consider the effect of the tube on zX22. When 3* is included this determinant becomes 2 = -l(p a + 2p  + 3p - 2 -1- t). (7-16) P We readily find with the help of methods similir to those used in connection with (7-12) that all the zeros of this expression lie in the left halLplane ----------------------------------------------------------- 120 NETWORK ANALYSIS Ca.P. 7 for 4 > a > - 2, but that some of them occur in the right half-plane for 's outside this range. It is to be observed that the range of stability for X22 is not identical with that for/. For example, if we were to choose  = 5 the network as it stands would be stable, since all the zeros of /x are still in the left half-plane, but the structure would sing if R2 were open-circuited, since with this value of a some of the zeros of 22 have crossed the real frequency axis. 7.8. Summabv of Requirements on Network Functions It is convenient to pause here to summarize the implications of the preced- ing discussion for the ½'arious network. functions. The network functions can be listed as the driving point immittance/'F = /x/zij , the transfer immit- rance/dz, = zX//xo, the return difference F = /x//x ø, the absolute sensitivity ' = --AA12/J//ZAlaA42, and the relative sensitivity S  = --AA1248/A18A42. The two sensitivities and the return difference are included here largely for the sake of completeness. Design methods to give direct control of sensitivity, in cases where it departs materially from the return difference, have not yet been developed. The return difference is under better design control, but in ordinary circumstances it is most easily treated in terms of the return ratio, which, since it is a loop transmission characteristic, has properties essentially similar to those of the transfer immittance. We will begin by listing the requirements which must be met by network functions corresponding to any stable physical circuit, and continue with additional requirements which are satisfied by special classes of struc- tures of particular interest, though not by all structures. The most obvious requirements arise from the fact that the driving point and trans- fer immittances, the return difference, and the two sensitivities are all rational functions of p with real coefficients. They must therefore meet the following conditions: 1. Zeros and poles are either real or occur in conjugate complex pairs. 2. The real and imaginary components are respectively even and odd functions of frequency on the real frequency axis. These are the only requirements which can be placed upon the sensitiv- ities, in general. We cannot even restrict the location of their zeros, since the numerators of the two expressions include, respectively, the unsym- metrical minors zX12 and zX24a, whose roots may lie anywhere. These functions will therefore not be considered further here. The numerators of the remaining three functions consist of zX alone. For these functions, consequently, we can state the following additional requirements: ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZAB1LITY 121 3. None of the zeros can be found in the right half-plane, 4. Zeros on the real frequency axis must be simple.* These four requirements are the only ones which can be stated for the driving point and transfer immittances and the return difference in the general case. For example, in terms of the notation adopted earlier in this section the poles of these several functions are respectively the roots of the determinants zXj.,/xii , and zx ø. It will be recalled from our earlier discus- sion that nothing in general could be said about the roots of /xis. The roots of/xys and A ø were interpreted as the natural modes of vibration of the network after it was modified in certain special ways, and therefore could not appear in the right half-plane if the modifications did not make the circuit unstable. In general, however, there is no necessary connection between the stability of the modified and unmodified structures. For example the illustrative circuit described in the preceding section was stable in its normal condition when the gain of the tube lay in an intermediate range near st = 5, but became unstable in this range for the condition repre- sented by making A22 the criterion. We must therefore conclude that in the most general case the poles of the return difference and of the driving point and transfer immittance functions may lie anywhere in the plane. Nevertheless, the special conditions for which they are confined to the left half-plane are of particular interest. They may be listed as follows: 5a. None of the poles of the return difference can lie in the right half- plane, and poles on the real frequency axis must be simple, if the circuit remains stable when the specified element/3' vanishes. This requirement is always met by a passive network. 5b. None of the poles of a driving point immittance can lie in the right half-plane, and poles on the real frequency axis must be simple, if the circuit remains stable when an infinite immittance is added between the driving terminals. This requirement is always met by a passive network. 5c. Poles of the transfer immittance may occasionally be found in the right half-plane, even for passive networks. Transfer immittances having no poles in the right half-plane, however, have the special property of being "minimum phase shift" functions. The reason for adopting this name, and the significance of the minimum phase relation, will be discussed in later chapters. It makes no difference for the minimum phase property whether poles on the real frequency axis are simple or multiple. These five requirements complete the list of conditions of special interest * With the restriction that zeros exactly on this axis may sometimes be regarded as inadmissible from the considerations discussed previously. ----------------------------------------------------------- 122 NETWORK ANALYSIS CHAP. 7 for the return difference, but it is desirable to carry the consideration of driving point and transfer immittances one or two steps further. The principal remaining point is the fact that either of these immittance func- tions can satisfy all the preceding requirements and still not correspond to a passive network. The additional requirements which must be satisfied by passive structures are, however, readily derived from the conditions described earlier and can be written as follows: 6a. The real component of the driving point immittance of a passive circuit cannot be negative at real frequencies. 6b. If a transfer immittance function corresponds to a passive net- work, the response which it specifies in the final branch, representing the load, must not be so great that the power consumed in the load at any real frequency would exceed the power which would be delivered by the generator if it were separated from the network and connected to a load equal to the conjugate of its own internal immittance. Condition 6a is evidently only a restatement of the second of the three conditions given previously for passive structures. The fact that it is not a consequence of the first five conditions is easily seen if we notice that they would be satisfied equally well by the negative of a passive immittance. It is also possible to satisfy them with an immittance function whose real component is positive in some frequency ranges and negative in others, as is shown by the examples given in the next section. Condition 6b can be understood if it is recalled that the maximum power obtainable from a generator with a prescribed internal immittance is secured when the load is equal to the conjugate of the internal immittance.* This maximum must evidently be at least as great as the power which would flow from the generator into the actual network, and therefore, from the last of the three power conditions, at least as great as the power consumed by the actual load. It is important to notice that 6a and 6b, although they both refer to passive circuits, are in other respects quite dissimilar and cannot be inter- changed. For example, another way of expressing 6a is to say that the . phase angle of a driving point immittance cannot exceed 4-90 ø. This would be an entirely irrational limitation in most transfer immittance problems, where the phase shifts may, in general, be made as large as we please. Similarly, in dealing with 6b we may notice that the transfer immittance, since it is a rational function of frequency, is completely deter- mined by its zeros and poles together with a multiplying constant. The * See, for example, K. S. Johnson, Transmission Circuits for Telephonic Communica- tion, p. 14. ----------------------------------------------------------- 124 NETWORK ANALYSIS c,,^,. 7 Z's at real frequencies, and are determined by substituting p = ico and rationalizing in the ordinary manner. Each impedance is obtained from Fo. 7.5 the preceding one by combining it in parallel with a resistance of value - 1.* This is illustra- ted by Fig. 7.5, where the internal impedance of the generator is assumed to be zero. The short- circuit generator impedance is important, since the impedance zeros which determine stability are those of the complete network, including the generator. If the generator impedance is not zero the addition of the successive negative resistances may evidently affect the zeros of the com- plete impedance, and therefore the stability. Turning first to Z in (7-17), we notice that it is a rational function ofp with real coefficients whose zeros and poles are all in the left half-plane. It thus meets requirements 1, 2, 3, 4, and 5b of the preceding list. In the corresponding R, the denominator, being a sum of squares, is always positive at real frequencies. The numerator is also always positive, since it can change sign only by passing through zero, and it is readily seen that it has no zeros for real values of co. Zt therefore meets requirement 6a also and represents a passive impedance. Since it has no zeros or poles on the real frequency axis, it also meets, incidentally, the" minimum teacrance" and" minimum susceptance" conditions as given by requirement 7. As we add negative conductance gradually in parallel with Z, there is at first no change in the character of the function.'f The resistance component however diminishes and may at length become negative. The boundary condition is represented by Z2. R2 is still positive everywhere, but touches zero at co = 4-1. With further increments of negative conduct- ance, condition 6a is no longer satisfied, although the remaining conditions, including 5b, may still be valid. For example, in Za the resistance com- ponent changes sign at o = 4-V'5/3-. The poles are still in the left half- plane, although one of them is on the boundary at p = 0. Finally, Z4 represents an impedance satisfying only the first four conditions. The addition of more and more negative conductance in the circuit of Fig. 7.5 will evidently not make the circuit unstable, so that beyond Z4 the first four conditions are always satisfied. An example of an unstable * The introduction of the negative resistance is adopted merely to provide a system- atic way of going from one impedance expression to the next, and is not intended to raise any questions concerning the physical construction or characteristics of such a device. The purposes of the present section are served if we take the impedance expressions one at a time without regard to any physical relation between them.  That is, it still meets the passive requirements and could be represented by some network including only positive passive elements. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 125 circuit can, however, be obtained by adding an appropriate negative resistance in series with the early Z's, or either a negative or a positive resistance in series with Z4. Thus if we add +2 to Z4 the result is +/' - 2 Z = 2p  - 2 (7-18) This has zeros at p = -0.74 and p -- -I-0.54 and therefore represents an unstable structure. The same set of rational expressions can also be used to exemplify the other network functions. For example, if we regard the various Z's as representing transfer rather than driving point impedances, we can imme- diately classify Z, Z2, and Za as physically realizable expressions of the minimum phase type. Z4 is physically realizable but non-minimum phase, since it includes a pole in the right halLplane, while Z, is non-physical. The chief differences occur in the application of the passive network con- ditions. For the transfer impedance case, the vanishing of the real com- ponent on the real frequency axis, as exemplified by Z2, is no longer a matter of particular significance. We are interested, on the contrary, in the minimum absolute values of the various functions on the real frequency axis. For example, it is readily shown that the minimum absolute value of Z2at real frequencies is 0.19. This means that the maximum current flow- ing in the load in response to a unit generator in the source will be 1/0.19 = 5.29, so that if we represent the load resistance by R i the corre- sponding power is 28.0Rj. The maximum power obtainable from a unit generator is however 1/4Ri, where Ri is the internal resistance of the generator. The passive network condition therefore demands that 1 28.0Ri  4Ri (7-19) Z as it stands will therefore represent a passive function if Ri and Rj are sufficiently small. In other cases it can be made into a passive function by multiplying it by a suitable constant. If the rational functions are taken as return differences, the first four Z's represent physically realizable expressions, although Z4 corresponds to a network which would be unstable if the prescribed ///vanished. If the expressions represent sensitivities the situation is still simpler, since there is no limitation even on the zeros of this function, and all five expressions can be regarded as physically realizable. 7.10. Energy Relations in a Passive Network As the final step in this discussion, we will turn to the consideration of the three special conditions on passive networks which were postulated, with- ----------------------------------------------------------- 126 NETWORK ANALYSIS Cp. 7 out proof, near the beginning of the chapter. Since the distinctive feature of a passive network is the fact that it does not contain a source of power, an obvious point of departure in establishing these conditions is found in a study of the power and energy relations in the circuit.* The instantaneous power dissipated in any one resistance R in the struc- ture is i2R, where i is the instantaneous current flowing through the element. Similarly, the instantaneous energy stored in the magnetic field of an inductance is «i2L, while the instantaneous energy storage in a condenser is «q2D, where q = fi dt is the charge on the condenser. Each of these quantities must be positive if the corresponding R, L, or D is positive and in a network containing only positive elements the total stored energy or dissipated power must therefore be positive for any choices of the instan- taneous i's and q's. This is the fundamental condition upon which the analysis is based. The expression of the total stored energy or dissipated power directly in terms of the individual elements of the network is not very useful, princi- pally because none of our other formulae are stated in these terms. It is a comparatively simple matter, however, to construct alternative power and energy formulae in terms of the coefficients in the mesh or nodal equations. We may begin, for example, with a set of differential mesh equations similar to (7-1), except that the equations will be referred to the steady state condition by introducing the instantaneous voltages e, ß ß., e, on their right- hand sides. Let it be supposed that the first equation is multiplied by i, the second by i, etc., and that the equations are then added. The result is ES-;. Rrsiri d- ZZ Lri  + EE Drdq --- Z ed. (7-20) r,s= l r,s= l r,s= l r=l where the first summation, for example, represents a series of terms of the form Rtti + R2ii= + ß ß ß + Rxii + R2i2ix + R2i + ß ß ß + R,,i, and q, in the third summation has been written, for brevity, in place of f i, dt. On the right-hand side, each term of the form eri is evidently equal to the instantaneous power fed into the circuit by the rth generator, so that the summation gives the total instantaneous power supplied to the circuit ß The method given here is a paraphrase of the standard dynamical treatment 'dr small oscillations. See, for example, Webster's Dynamics, Chapter V, or Whittaker /lnalytical Dynamics, Chapters II and VII. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 127 from the outside. The first summation on the left-hand side must repre- sent the instantaneous power dissipated by the resistances, since it is the only term which would be present in a purely resistive network. It can be written as twice* the "dissipation function" F, where F is defined by F = « Z E Ri,i. (7-21) r=l s =1 The remaining terms on the left-hand side represent the rates of change of the stored energies associated with the coils and condensers of the net- work. For example, if r = s in the second summation we can write the corresponding term as Lrrir(dir/dt) = (d/dt) Lrrr. Ifr  s we may make use of the fact that since this is a passive circuit we must have L, = Lr.' The sum of the corresponding rs and sr terms can therefore be written as ( dis dish= L d dl L ir- + i dt /  dt (iri) =  (Lriri + Lii). Evidently, the complete second summation becomes dT/dt, where T is the stored magnetic energy and can be written as T = « Z Z Lii. (7-22) In the third summation it is convenient to set i5 -- dqs/dt. Following the procedure just used, this allows us to write the summation as df//dt, where //represents the stored energy in the condensers and is given by I/= « ]] Z Dqrq. (7-23) r=l 8=1 The essential result of this discussion has been the development of the expressions for the quadratic forms F, T, and f/, as given by (7-21), (7-22), and (7-23). It follows from our previous discussion that in a net- work composed only of positive elements, F, T, and ;Zmust all be positive. Moreover, we can regard the individual i's and q's as assuming arbitrary values in making this statement, since we began with arbitrary generators in each mesh. F, T, and/z must therefore remain positive if we assign the * The factor two is introduced arbitrarily to secure symmetry with the functions considered later. The use of the symbols F, T, and k' for the energy functions follows standard dynamical usage. There should be no confusion with the other meanings, of return difference, return ratio, etc., assigned to the same symbols, since the energy function discussion is not continued beyond the present chapter. ' The use of the reciprocity condition here and in later sections should be noticed particularly, since it explains why this type of analysis is restricted to passive networks. ----------------------------------------------------------- 128 NETWORK ANALYSIS cH,p. 7 i's and q's any other values whatever, positive or negative, and can vanish only if all the i's and q's vanish.* In mathematical language the three functions are positive definite. The positive definite conditions can best be understood as a set of restrictions on the values which the mesh coefficients Rr, Lr, and Dr can assume if the system of mesh equations as a whole is to correspond to a passive network. Suppose, for example, that all the i's except i vere 1 .2 chosen equal to zero. F would reduce to Rxx. The positive definite condition evidently requires that Rlx > 0. Similarly, all the other coeffi- cients of the type Rji , Lij , or Djj must be positive. If the energy functions included only "self" coefficients of this type, as they would if they repre- sented sums of powers and energies for the individual physical elements, this would be the whole story. Account must, however, also be taken of coupling terms such as Rdri, where r  s. Whatever the sign of Rr may be, this term may evidently be made negative by proper choice of the signs of ir and it. The positive definite condition therefore requires that the absolute values of such mutual coefficients as R, be not too great in comparison with the self coefficients. An example is furnished by the function F i q- kili + i 2 (7-24) If ] k I < 2, this expression is positive for all real values of il and i2, as we can see most easily by setting the expression equal to zero and noticing that the roots, in terms of ii/i2, must be complex. For other values of k, how- ever, the expression may be made to cross zero and become negative by varying i/iv, appropriately. For I k ] > 2, therefore, the expression is no longer positive definite. With more than two i's the situation is more complicated but the essential pattern of relationships is preserved. In general, the self and mutual coefficients obey the same laws as the self and mutual inductances in a set of coils with physically realizable coefficients of coupling, as we might expect, since both conditions reflect a positive energy requirement. 7.11. Impedance and Energy Relations at Real Frequencies With the development of expressions for the functions F, T, and k' we are prepared to prove the three special conditions on passive networks postulated near the beginning of the chapter. The present section will consider only the second and third of these conditions. * The last part of this statement is intended as a characterization of networks in general, and may have exceptions in special cases. For example, if the first mesh includes no inductance the stored magnetic energy will evidently be zero for any choice of i, as long as the other i's vanish. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 129 Since the second and third conditions are stated in terms of steady state characteristics, it is natural to begin with the ordinary steady state mesh equations for the circuit. The analysis will be based upon a set of energy expressions built up by multiplying each mesh equation by a corresponding current and then summing all the equations, much as was done in obtain- ing (7-20). One modification, however, must be made to take account of the fact that since (7-20) was developed from the differential equations of the circuit, its energy functions were expressed in terms of the true instan- taneous currents and voltages in the structure. The I's and E's which appear in a set of ordinary steady state mesh equations, on the other hand, are merely complex quantities which are brought into the analysis when the true currents and voltages are replaced by fictitious expressions of the type Iie vt and Eie vt, in accordance with the conventions described in Chapter II. This, however, still allows us to secure a meaningful result if, instead of multiplying each mesh equation by the corresponding I, we multiply it by the conjugate of that I. If the equations are then added, the result appears as i E LriI8 + E RL + -- E.. DiI8 = Ei (7-25) r,el r,s=l 160 r,s=l where p has been replaced by ico since we are interested only in real fre- quency characteristics, and i$ represents the conjugate of I s. Only the single generator E is included, in order to state the eventual result in terms of the impedance seen in the first mesh. Since the I's in (7-25) are not functions of time the three summations cannot represent the actual instantaneous physical energy functions. The summations can, however, be identified term-by-term with multiples of the time aoerages of the corresponding terms in the true energy expressions. To show this, we may begin by considering a" self"-inductance term such as LlI. Upon replacing I and i1 by I q-ill and I- iI, respectively, this term becomes Ll(I + I). The corresponding term in the expression for the true electromagnetic energy of the circuit is i 1'9. L 1 , where I represents the instantaneous physical value of the first mesh current. It follows from the definition of/i, however, that I' = Real component of (I q- iI)(cos ot + i sin cot) (7-26) = Ii cos cot - [l sin cot. This term in the true electromagnetic energy can therefore be written as r r* «Ln (I2 cos 2 cot - 2ItJ, sin cot cos cot q- I sin ' cot). (7-27) If we average this expression over a long period of time, the sin ot cos cot ----------------------------------------------------------- 130 NETWORK ANALYSIS cp. 7 term disappears while the cos 2 cot term and the sin  cot term each becomes «. The average value of the electromagnetic energy due to the flow of I in Ln is therefore  .2 = x L tI2 (zLnIx ),r  n  + I) (7--28) which is just  of the value found for the term LIi in the first summation of (7-25). Similarly, when we examine a pair of "mutual" inductance terms of (7-25), such as L212ix + L2Ji2, we readily find, with the help of the relation L2 = L2, that they may be written as 2Lg(II2 + I). The corresponding term in the expression for the true electromagnetic energy is r /*r* L2[Ii.I cos  wt -- (I,Z,a + I,,ta) sin t cos t + Id2a sin  wt] (7-29) whose average value is + Id). This is again just  of the amount given by the summation of (7-25). We conclude, therefore, that when all the terms of this summation are evaluated they will represent four times the average value of the total electromagnetic energy T, taken over a long period of time. Obviously the second summation of (7-25) is in a precisely similar Gshion equal to four times the average value of the dissipation function F. The third summation of (7-25) cannot be identified directly with a multiple of the average value of the final energy Gnction F since it depends upon productsof currents, and if it were to represent F, the quantities should, on the contrary, be charges. We may notice, however, that since a current is the derivative of a charge, the effect of introducing a current in place of a corresponding sinusoidally varying charge is to produce a shift in phase, which is of no importance for averages taken over a long period of time, and to multiply the expression by . Since there are two I's in the third sum- mation, the introduction of currents for charges thereGre increases its value by a hctor ofw 2, and we can conclude that the summation is equal to 4w"times the average value of the stored energy of the condensers. Using the values just Gund Gr the three summations replaces (7-25) by 4ir + 4F - 4ig, = E. (740) If we assume that the driving voltage E is of unit amplitude, the current I will be equal to the admittance of the network. i is, of course, the same as I except Gr a change in the sign of the imaginary part. Equa- tion (7-30) can thereGre be used to furnish a relation between the energy functions of the network and its input admittance. We find Y = + - r.)l ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 131 where Fv, ?v, and Tv, are to be evaluated under the assumption that the network is energized by a voltage of unit amplitude.* The second of the three conditions on passive networks postulated at the beginning of this chapter is to the effect that the real component of a driving point immittance is never negative at real frequencies. Since F must be positive this is established immediately by (7-31). 'IChe third condition states that the total power fed into the network hy an outside generator at real frequencies must be at least as great as the power con- sumed.by any one resistance. This can be shown by investigating the way in which any individual element enters the expression for F. A simpler method, however, is to suppose the given resistance removed and replaced by a generator having a voltage equal to the drop across the resistance produced by the prescribed external generator. This will leave the dis- tribution of currents in the rest of the network unaltered. If we repeat the analysis which led to (7-31) for the modified network driven by both generators, however, the real component of the right-hand side of the result- L C = L/R z Fro. 7.6 Fro. 7.7 ing expression will still represent the average F for the modified structure and must be positive or zero. Evidently, therefore, the power consumed by the rest of the network cannot be negative, so that the total power con- sumed by the complete structure must be at least as great as that consumed by the prescribed resistance. As examples of (7-31) we may take the networks of Figs. 7.6 and 7.7. The impedance of the network of Fig. 7.6 is a pure resistance when the inductance and capacity resonate. The average values of the energies stored in the inductance and capacity must therefore be equal at this frequency. The network of Fig. 7.7 is equivalent to a pure resistance at all frequencies. By the same reasoning, therefore, the average values of its T and F must be the same at all frequencies. Both of these conclusions can, of course, be readily checked by calculation. * "Unit amplitude" here means that the maximum value of the sinusoidal wave representing the voltage is unity. Since we are dealing with energy, it is perhaps more natural to use the rms or "effective" emf, whiclx is 1/%/' times the maximum value. If we use a unit effective voltage, therefore, the constant 4 in the right-hand side of (7-31) should be replaced by 2. ----------------------------------------------------------- 132 NETWORK ANALYSIS CHAP. 7' 7.12. Stability of Passive Networks The first of the three special conditions on passive circuits mentioned near the beginning of the chapter states that a passive circuit is always stable. As the final step in this analysis we will prove that this is a conse- quence of the fact that the three energy functions F, ?; and/z, of a passive network are positive definite. The relation between stability and energy arises, of course, from the fact that it takes energy to set a circuit into motion and that, generally speaking, the greater the disturbance the greater the energy. It might appear at first sight from this argument that stability will be assured from the positivehess of F alone, since if F is positive the circuit as a whole will lose energy continuously, whatever the sign of T and ? may be. In fact, however, the positivehess of T and //is equally important. If one of these functions may be negative the circuit may lose energy through IR losses continuously and still remain very far from its position of equilibrium provided more and more negative energy* is stored. Insolvency is no bar to a spendthrift life as long as one's credit at the bank is good. The relation between stability and the energy functions can be developed most easily if we return to the set of equations given by (7-2) of the present chapter. These were identical with the ordinary mesh equations except that the driving voltages were set equal to zero and p was supposed to assume one of the special values corresponding to a transient oscillation in the network. Upon treating the equations by the processes used in developing (7-:25) the result is readily seen to be p EE LrsirI + EE RrirT + - EE DriL = 0 (7-32) which is the same as (7-25) except that the right-hand side has been set equal to zero and ic0 has been replaced by p since transient oscillations are not necessarily restricted to real frequencies. In the previous discussion we identified each of the summations of (7-32) with the average value of one of the energy functions of the network. We cannot make the same identification here, since if the frequency is * If a "negative energy" is difficult to visualize, we may suppose that the circuit under consideration is a passive structure except for the inclusion of the equivalent of a negative inductance, provided by means of one of the vacuum tube circuits described later. As long as the negative inductance is taken as an entity the complete circuit can still be analyzed by the methods used for passive structures, since the prin- ciple of reciprocity is maintained. The" negative energy" stored in the inductance, however, can be regarded physically as positive energy drawn from the vacuum tubes and transmitted to the rest of the circuit. ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 133 complex the physical currents in the network will be increasing or decreas- ing and there is no good way of taking a time average. Fortunately, how- ever, no such precise physical interpretation is necessary. It will be re- called that when I in (7-25) was replaced by its value in terms of Ira and I1b the LI term of the first summation became L (Ia q- Ib). Similarly, the sum of the L and L2 terms became (L2 + L) Both the "self" and" mutual" inductance terms therefore broke up into the sum of two terms, one involving products of the [a's and the other, products of the Ib's. Corresponding results, of course, held for the R and D terms. Even without the help of a physical interpretation of the sum- mations, therefore, we can rewrite (7-32) in the same way as p   Lr, (IraI,, + IrI,b) +   Rrs (Ir,I,a + Ird, b) q- - 5. Drs([r[a + IrJ, b) = 0. (7-33) The summation  LrIrI,, is obviously twice the energy function T when for each instantaneous physical current we use the corresponding quantity Ira. We can, therefore, represent this portion of equation (7-33) by 2T(a). Similarly, the summation 5- LrIr[b represents twice the T function when each physical current is replaced by the corresponding I. It can therefore be written as 2T(b). The other parts of equation (7-33) represent in the same fashion twice the F and k' functions when we substi- tute the Ijs and I's for the corresponding physical currents and charges. The complete expression can therefore be written as 1 p[T(a) + r(b)] + F(a) + F(b) +  [V(a) + ?(b)] = 0. (7-34) The T's, F's, and F's which appear in (7-34) do not necessarily corre- spond to any physical energies actually present in the circuit. They are merely certain mathematical expressions secured by replacing the instan- taneous currents in the true energy expressions by the i s and Ia.b s. In general, they may be expected to have different values as we go from one possible transient to another, since the distribution of currents in the net- work, and therefore the i s and ]i s, will depend upon the transient fre- quency. However, we at least know that the original energy expressions were positive for all possible values of the instantaneous currents. It fol107(s that the new T's, F's, and k"s must be positive in all cases. Any p corresponding to a possible transient must therefore satisfy a quadratic equation like (7-34) in which all the coefficients are positive. From the ----------------------------------------------------------- I34 NETWORK ANALYSIS Cu.,. 7 usual formula for the roots of a quadratic we can readily deduce that per- missible p's, or zeros of A, must satisfy the following conditions: 1. The zeros will be found at negative real values ofp if either T or//is identically zero. In other words, impedances corresponding to networks containing only capacities and resistances, or only inductances and resist- ances, must have zeros on the negative real axis ofp. 2. The zeros will be found on the negative real axis even if both T and// are present provided F is sufficiently great. This means that very highly dissipative networks will have negative real zeros even when both kinds of reactive elements are present. 3. If F is identically zero, the zeros will occur on the imaginary axis. In other words, the impedance of a non-dissipative network vanishes only at real frequencies. 4. If none of these conditions is met, the zeros ordinarily occur in con- jugate complex pairs. The real parts of the zeros are always negative. These propositions are best exemplified by the discussion given in a later chapter. They evidently contain much more detailed information than is provided by the bare statement that a passive network must be stable. It is clear, however, that among them they at least confirm that statement. 7.13. Comparison of Criteria of Physical Realizability The preceding discussion has developed the properties of physically realizable structures from a variety of criteria. In dealing with passive structures, for example, we began with the statement that the circuit could contain only positive elements and later replaced it by the statement that its energy functions must be positive definite. In dealing with active structures, on the other hand, we relied chiefly upon the postulate that a physical circuit must be stable. As they stand, these criteria are not readily compared directly, chiefly because the formulae for the energy functions were developed on the assumption that the circuit met the-reci- procity condition Zij = zi, so that they are not easily extended-to the general case. To put the criteria on the same footing, it will be assumed that the active elements appear only as parts of- negative- impedance devices, so that if these devices are taken as entities the circuit can be regarded as made up exclusively of bilateral elements.* It is then readily seen that the various criteria are not logically equivalent. As the list:was * Cf. the discussion in an earlier footnote. In accordance with the ass-fiMt3qn made here, the word" element "in the present section will be taken to mean a bilateral element. ' ': : ----------------------------------------------------------- STABILITY AND PHYSICAL REALIZABILITY 135 given, the criteria appear in the order of diminishing severity. In Other words, a network all of whose elements are positive always has positive definite energy functions and a network with positive definite energy func- tions is always stable. The converses of these propositions are, however, not true. A network which is stable does not necessarily possess positive definite energy functions, and a network with positive definite energy functions is not necessarily composed exclusively of positive elements. The fact that the positive energy condition is not equivalent to the posi- tive element condition is easily seen in trivial cases. For example, the energy condition will evidently be maintained if we add a negative resist- ance either in series with an actual network containing an equal or larger positive series resistance, or in series with an equivalent of such a network. A more elaborate example is suggested by the equivalent 5/' of a two-winding transformer. It will be recalled that the central branch of the T con- tains a negative inductance representing the mutual coupling. The energy stored in the inductances as a whole, however, is always positive. Evi- dently the energy would still be positive if we replaced the transformer by a corresponding arrangement of three separate positive and negative induct- ances. The energy conditions will also be fulfilled if, instead of using inductances, we insert positive and negative impedances of any description in the same ratio. On the other hand we will be able to show that any impedance function meeting the requirements derived from the energy conditions can always be represented by some network containing only positive elements together with systems of ordinary mutual inductances. In this sense, therefore, the energy conditions and the positive element conditions are equivalent. The relation between the conditions that the network be stable and that its energy functions be positive definite is less easy to understand. If the network includes only two kinds of elements, it can be shown that it will be stable only when both associated energy functions are positive definite. When all three kinds of elements are present, however, the positive definite condition is not necessary. This may be exemplified by means of the illustrative impedance formulae given by (7-17). For example, the last two of these represented impedances which met the stability requirement but had real components which were negative in some portions of the real frequency spectrum. Evidently in such impedances the positive definite condition does not apply to the dissipation function F. We may a!,s.o recall that, in order to restrict the location of the poles of impedance, it ws necessary to suppose that the structure would be stable when its driving point terminals were open-circuited. Since positive definite energy func- tions remain positive definite when the current in the driving mesh is set ----------------------------------------------------------- 136 NETWORK ANALYSIS c.... 7 equal to zero, open circuit stability is assured when the energy conditions are met. As such examples as Z4 of (7-17) show, however, open circuit stability is not a property of all structures which are stable under their nominal operating conditions, so that this represents another way in which a stable structure which does' not meet the energy conditions can be obtained. ----------------------------------------------------------- CHAPTER VIII CONTOUR INTEGRATION AND NY(lUIST'S CRITERION FOR STABILITY 8.1. Introduction 'I'vE analysis of the preceding chapter consists, in essentials, of an investigation of the restrictions which must be placed upon the zeros and poles of the several network functions if the structure is to he stable under various conditions. This is obviously a necessary first step in attacking the general problem of determining the characteristics obtainable from physically realizable structures. Of itself, however, it is of limited utility. Its chief limitation is the fact that the restrictions are stated in terms of the behavior of the function at complex frequencies, while for practical design purposes only the real frequency characteristics are ultimately of interest. As the situation stands, the relations between the two are too indirect to be of much value. For example, it is not very clear from the restrictions on the zeros and poles just what sorts of real frequency characteristics are physically possible. Moreover, if we have a known structure, whose com- puted real frequency characteristics are satisfactory, it is a long and tedious process, in general, to determine whether the roots of a meet the stability requirement. If some of the roots turn out to be in the wrong side of the plane we are still at a loss to know whether we have merely made an un- fortunate choice in some unimportant feature of the design or whether the result is inevitable in any circuit hazing the desired type of behavior. What we evidently need in order to bring the analysis to a useful con- clusion is some mathematical too by means of which the restrictions on the behavior of network functions at complex frequencies can be transformed directly into equivalent restrictions on their behavior at real frequencies. The real frequency axis can be looked upon as the boundary of the right half-plane, which is the region in which special restrictions on the network functions occur, so that the broad mathematical problem is that of relating the behavior of a function inside a given region to its behavior on the boundary of the region. The most useful tool for this purpose is found in Cauchy's theory of analytic functions in terms of integrals around closed contours. This chapter is intended primarily as a sketch of some of the elementary aspects of this theory.* The most extensive applications of * For supplementary reading, reference may be made to any book on the theory of functions of a complex variable. Particularly good accounts are found in Goutsat, Townsend, or Pierpont. 137 ----------------------------------------------------------- 138 NETWORK ANALYSIS CHAP. the material are made several chapters later, after an intermediate discus- sion of the general properties of driving point and transfer immittances. The theory is exemplified in the present chapter by a discussion of the Nyquist diagram method of determining stability. The chapter also includes two specific theorems which are useful in the discussion of driving point and transfer immittances given in the next few chapters. The analy- sis relies upon the general framework of ideas given by Chapter II, and this material should be reread if necessary before the present chapter is under- taken. 8.2. Integration in the Complex Plane In ordinary calculus we are familiar with the conception of an integral as the area under a curve. Figure 8.1, for example, shows the approximation to the area under a given curve by means of a number of thin vertical strips. We say that the integral of the function from x to x2 is equal to the limit Fro. 8.1 Fro. 8.2 approached by the area of the strips when the number of strips becomes indefinitely great and each one is made indefinitely thin. The area of any strip, such as the shaded one in the figure, however, is evidently equal to its height times its breadth or, in other words, to f(xj.)(xj.+- xi). This definition of an integral can therefore be expressed by the equation f(x)dx = lira Y',/(xj)(xj+ -- x). (8-1) Integrals of functions of a complex variable are defined in a precisely similar way. If we suppose, for example, that the functionf (z) is to be integrated along a prescribed curve running from z to z2 in the complex z plane, as shown by Fig. 8.2, we may begin by choosing a number of inter- mediate points, z 5. For any given choice of the intermediate points we can ----------------------------------------------------------- CONTOUR INTEGRATION 139 set up the corresponding sum  f(zj)(zj+ - zs). The integral, then, is defined as the limit* approached by this sum when the number of points of division is made indefinitely great and the successive points are brought indefinitely close together. In other words, the integral may be expressed by ,qf(z)dz = lira Zf(zj)(zj+ - zj). (8-2) It will be seen that this is formally similar to equation (8-1). The only difference lies in the fact that sincef(z.) and (zj.+ - zj) will in general be complex quantities, the final result will ordinarily be complex. ' 8.3. Integrals in Limiting Cases The definition of an integral given by equation (8-2) leads immediately to a simple consequence which we will use repeatedly in later discussion. We observe from (8-2) that the absolute value of the integral cannot be greater than the sum of the absolute values of all the component terms, f(zj)(z5+ --zj). Now suppose that M represents Fro. 8.3 the largest absolute value off(z) over the path con- sidered. The absolute value of each of the component terms can be no greater than the absolute value which would be obtained iff(z) were re- placed by M. We therefore have As an example of this relation, let it be supposed that the path of inte- gra, tion is the semicircle with radius R shown by Fig. 8.3, where it is sup- posed that R can be made indefinitely large. The path length is rR. Iff(z) varies as some positive power of z for very large values of z, xve can evidently say nothing about the integral on the basis of equation (8-3) since both M and the path length will become very large as R becomes large. Equation (8-3) also fails to provide a limit iff(z) approaches a constant value other than zero as z approaches infinity. On the other hand, iff(z) varies as some negative power of z, such as z -s, M must vary as R -2. We see from equation (8-3) that the integral must therefore vanish for a sufficiently large value of R in spite of the fact that the path length is * This discussion, of course, ignores such questions as the demonstration that the limit exists, for an appropriatef (z), and that it is independent of the precise choice of the zi's, which would require consideration in a formal analysis. ----------------------------------------------------------- 140 NETWORK ANALYSIS C,^,. 8 indefinitely great. The same result, of course, holds iff(z) varies as any larger negative power of z. If the semicircle of Fig. 8.3 is supposed to be very small, rather than very large, essentially similar results are secured. The path length now vanishes in the limit, so that it is clear from (8-3) that the integral also vanishes if f(z) either approaches a constant value or behaves as any positive power of z near the origin. Iff(z) behaves as z -2, or as any larger negative power of z, on the other hand, M increases so rapidly as R diminishes that we can say nothing about the integral on the basis of (8-3). In both situations, an intermediate case occurs iff(z) varies as z -. When the semicircle is very large this gives an M which diminishes as the d0 - [riG. 8.4 as a complete semicircle. dz = iRdødO, so that the integral from O to 02 becomes path length increases, while when the semi- circle is small M increases as the path length decreases, the relative rates of increase and decrease being such that in either case the product of the two is a constant. Equation (8-3) thus gives a finite upper limit to the integral, but we are not sure just what its exact value may be. This situation can be treated by specifying z in terms of a polar angie 0, as shown by Fig. 8.4. For the sake of generality the path is shown as an arbitrary arc of a circle, with end-points at 0 = 0h and 0 = 02, rather than If we write z = Rd ø we evidently have f e 7 1o, iReiOdO o2 .... idO = --i(O -- 02), t] o ]eiø  0. (8-4) while if the integral is taken in the other direction the result is evidently the same except for a reversal in sign. We thus have the Theorem: The integral of z - over an arc of a circle centered at the origin is .+i or -i times the central angle of the arc, in radians, accordingly as the integration is taken in a counter- clockwise or in a clockwise direction. The importance of these results lies in their utility in evaluating integrals in many limiting cases. For example, in future discussion we will have fre- quent occasion to consider the integrals, over a very large semicircle, of func- tions which behave near infinity like (z/_/z) -3- (/l-/z 2) + (//-a/z ) -3-" '. Evidently if the semicircle is sufficiently large we can discard all the ----------------------------------------------------------- CONTOUR INTEGRATION 141 terms in this series except the first and evaluate that one by means of the theorem just given. For purposes of future analysis this discussion requires amplification in one particular. We have thus far merely rejected cases, for either the very large or very small semicircle, in whichf (z) varies as such a power of z that the product M X path length becomes indefinitely large as the limiting case is approached. We can draw no conclusions about such situations from (8-3) alone, and nothing significant can be said, in fact, as long as the path of integration is an arbitrary arc of a circle. In the following sections, however, there will be occasion to consider paths of integration extending around a complete circle and back to the starting point. This gives a particularly symmetrical situation for which the integral can be much simplified. To show this, we may repeat the analysis of (8-4), replacing z - by z ' and the upper and lower bounds of integration by -r and r. This gives ,foZ'dz = f, -= [R,,ei,,OliReiOd 0 = iR '*+ [cos (n + 1)0 + i sin (n + 1)O]dO. (8-5) If n is any positive or negative integer except -1 these expressions must vanish since the integral of either a sine or a cosine over a complete cycle is zero. If n = - 1 the result is -2*ri, for the clockwise direction of inte- gration indicated, as we can see either from (8-5) or by the preceding theorem. We therefore have the Theorem: The integral of z  around a complete circle centered at the origin is zero unless n = -1. Ifn = -1 it is -2ri for -integration in a clockwise direction and q-2ri for integra- tion in a counterclockwise direction. 8.4. Relation between the Integral and the Path of Integration In spite of the parallelism which exists between the definitions of real and complex integration as indicated by equations (8-1) and (8-2), one difference exists which has not been previously emphasized. In defining the real integral in equation (8-1) it was sufficient to give the integration limits xt and xz, since it was clear that the points x 5 were necessarily taken on the x axis between these limits. For the complex variable z, on the other hand, it was necessary to specify not only the limits zt and z2 but also the particular curve between those limits on which the points of subdivision z i were supposed to be chosen. The question naturally arises whether the ----------------------------------------------------------- 142 NETWORK ANALYSIS CHAP. S choice of the path between zl and z2 is significant or whether the same result would be secured if we connected z and z2 by some different path, as shown by Fig. 8.5. This question is answered by an important theorem, due to Cauchy, which is sometimes called the" Principal Theorem of Analysis." Cauchy's theorem states that the integral between z 1 and z2 will be the same for either of the two paths provided the function to be integrated is analytic on FIG. 8.5 both paths and in the intermediate region bounded by the paths. In most circum- stances it is convenient to replace the con- ception of an integration from z to z2 along two different paths by the conception of an integration around a complete closed loop and back to the starting point. In Fig. 8.5, for example, we might regard the loop in- tegration as composed of a forward integra- tion from z to z along path `4 and a back- ward integration from z= to z along path B. Clearly, however, the integral from z= to z along B must be the negative of the integral from z to z= along B. It must therefore also be the negative of the integral from z to z2 along `4, if the integrals from z to zs along the two paths are equal. Cauchy's theorem can consequently be stated in the following words: Theorem: If a functionf (z) is analytic within a closed curve and also on the curve itself, the integral off(z) taken around that curve is equal to zero. This theorem will be assumed here without proof. Cauchy's theorem is readily illustrated by our preceding discussion of the integration of powers of z on circular paths. Let it be supposed, for example, that the closed loop is taken as a circle about the origin and that f(z) is chosen as the polynomial .40 + /z + ß .- + `4,z . Then f(z) is analytic on and within the circle so that according to Cauchy's theorem the integral around the complete circle must vanish. This is, of course, verified by the preceding discussion, which showed that the integral of each term in the polynomial vanishes. We may next suppose that the term k/z is added to the polynomial. This additional term produces a pole at the origin so that the function is no longer analytic at all points within the circle and the conditions of Cauchy's theorem are not met. Correspondingly, we find from our preceding dis- cussion that the integral no longer vanishes, but becomes 2rik, if the inte- gration is supposed to take place'in a counterclockwise direction. On the ----------------------------------------------------------- CONTOUR INTEGRATION other hand, if we were to add k/z , rather than k/z, to the polynomial the loop integral would remain zero, although the new function would still not be analytic at the origin. It thus appears that the converse of Cauchy's theorem is not true. In special cases the loop integral may be zero even though the function is not analytic at all points within the contour. In these examples the closed contour has been taken as a circle. This, of course, is a partic- ularly easy path for purposes of computation. It is important to notice, however, that Cauchy's theorem shows that the same results would be secured if the circle were distorted into a path of any other shape. To illustrate this, we will consider the integral of z 2 around the square path shown in Fig. 8.6. The side of the square is taken as 3 units and the ½ B Fro. 8.6 ' corners //, B, C, and D as 1 -i, 1 + i2, -2 + i2, and -2- i, re- spectively. If we write z = x q- iy, z 2 becomes x  - y2 q- 2ixy. On the: side//B we have x = 1 and dz = idy. This portion of the complete loop integration can therefore be written as fB z=dz = [(1 -y2) q- 2iy]idy. (8-6) 1 This can be evaluated by the methods of ordinary calculus and is equal to p plane Fro. 8.7 --3. Similarly, overthesideBCwehave y = 2, dz = dx, and the integral becomes z2dz = [(x  - 4) + 4ix]dx = 9 + 6i. (8-7) The integrals over the remaining two sides can be treated in the same way and are equal re- spectively to - 6 - 9i and 3i. The sum around the complete loop is easily seen to vanish,_thus confirming Cauchy's theorem for this case. In future discussion, Cauchy's theorem will be applied chiefly to closed loops in the p plane of the type shown by Fig. 8.7. The loop consists broadly of a large semicircle in the right half of the p plane closed by a diameter lying on the real fre- quency axis. The small indentations away from the real frequency axis ----------------------------------------------------------- 144 NETWORK ANALYSIS C..v. 8 are supposed to be very small semicircles introduced to avoid any singu- larities which may be found there. The integral around the complete path will be represented by the symbol f and the integral around the large semicircle by the symbol f. This path is chosen because our previous discussion on the location of the zeros and poles of physically realizable network functions can readily be converted into a specification of the analyticity either of the network functions themselves or of certain derived functions in the right half of the p plane. Cauchy's theorem can thus be used in studying the integrals of these expressions around the complete loop. If we suppose that the path is very large, however, the integrals around both the large semicircle and the small indentations can be dismissed easily by means of the methods described in a preceding section. What is left, then, is an integral along the real frequency axis from some very large negative frequency to a corre- spondingly large positive frequency so that Cauchy's theorem allows .us to relate the real frequency characteristics of the structure directly to the conditions of physical realizability. 8.5. The Calculus of Residues Before studying these possibilities in detail, it is desirable to consider briefly what happens to the integral of a given function around a closed path when the function is not analytic inside the path. The results have already. been suggested by the examples given previously. To study the general case, suppose that the function is analytic except for a simple pole at z, so that near za it can be expanded in the form* f(:g) -- '--1 q- d 0 q- /l(g -- Za) q- /o(Z -- Za) 2 q- '' '. (8--8) Now choose the path of integration shown by Fig. 8.8. The function is analytic within this closed path so that the integral around the complete path * The series in (8-8) is introduced here as a convenient way of characterizing the behavior of the function in the neighborhood of z. From the point of view of pure mathematics its use is somewhat illogical, since the justification for such an expansion depends upon an analysis of the type under consideration at present. We may notice, however, that all we really need to know is thatf (z) can be represented as the sum of the first term in (8-8) and a remaining portion which is bounded in the neighborhood of z. This is readily established from the definition of a pole. It follows from (8-5), however, that the integral of the bounded portion around a very small circle near z can be ignored, so that the correct result is secured without using the com- plete expansion. ----------------------------------------------------------- CONTOUR INTEGRATION 1-,5 must be zero. The contributions of the integrations along the path between P and P2 in each direction, however, evidently cancel out. The integral around the outside loop from P back to P again must therefore be the negative of the integral around the small inside circle enclosing the point If we integrate jr(z), as given by (8-8), around this small circle, however, all the terms except the first must drop out, while if we transfer the origin to za the first can be evaluated by the methods of equation (8-4). We therefore find that the integral around the outside loop is given by* f(z)dz = - 2ri-//_t. (8-9) The coefficient .4_ is called the residue of the function at the pole' za. If there are a number of poles 8.8 in the interior of the loop, then by a continuation of the same process we can include them one by one, thus securing in general, an expression of the form f f(z)dz = -2ri[A_ q- B_ + C_ + ... q- N_]. (8-10) Since equation (8-5) shows that only simple poles have integrals different from zero around small circular paths enclosing them, only the coefficients of the first order poles at any point should be considered in building up an expression such as (8-10). Illustrations of the calculus of residues can be obtained by using the same material as was previously employed to illustrate the general Cauchy theorem. Since the fact that the integral of 1/z over a small circle about the origin is equal to 2,ri was used in establishing (8-9) and (8-10), we are perhaps not justified in regarding this result as an illustration. We can, * It is important to notice that the negative sign in the right-hand side of (8-9) appears because of the direction of integration (clockwise around the outer loop) which is chosen in Fig. 8.8. It is convenient to choose this direction here because it leads to a positive direction of integration along the real frequency axis when we eventually apply the result to contours of the type shown by Fig. 8.7. In most treat- ments of the Cauchy integral, however, the loop integration is conventionally taken in the opposite direction, so that the equation corresponding to (8-9) appears without the minus sign. ----------------------------------------------------------- 146 NETWORK ANALYSIS tsar. s This can be evaluated by ordinary calculus _x log 5 « log 2 + i tan -a (2) i tan -x (- 1). that the integral from B to C can be written as however, at least exemplify the fact that the result is independent of the shape of the path by considering the integral of the same function around the square contour of Fig. 8.6. Setting z = x + iy, as before, we have 1/z = [x/(x 2 +y2)]_ [iy/(x 2 +y2)]. The integral from .4 to B is readily written from this as f .4  dz = ' ' z f_tI'l 1 Yy2'3 idy' (8-11) q- y I and gives the result In the same way we find - -log8-«logS-itan - (-1)+itan - («). (8-12) Using similar methods, the results for the other two sides are found to be, respectively, Slog 5 - « log 8 - /tan - (-1) + itan - (3) and x_ log 2 - « log 5 + i tan -1 (1) - i tan - (-2). If theintegrals for all four sides are added together it is readilyseen that the sum of the real components vanishes. The sum of the imaginary compo- nents can be studied most readily by observing that each component sepa- rately is equal to the central angle sub- tended at the origin by the correspond- component i tan -x 2 - i tan -x (-1) ! obtained from the integration along .4 1 t'"''t z/B is equal to the angle, in radians, between straight lines drawn from the origin .to the corners .4 and B. Fro. 8.9 Evidently, :' the total central angle subtended by all four sides is one revolution, or 2r radians. The complete loop integral is therefore 2ri. This agrees with (8-9) if account is taken of the fact that the direction of integration in the present instance is counterclockwise. The fact that the imaginary component produced by integration along each side is equal to the central angle subtended by that side is important, since it indicates why the significant feature of the situation is not the exact shape of the path, but the fact that the pole at the origin is inside the path. Evidently, any slight distortion of the path would still leave the total central angle subtended equal to one revolution, or 2r radians. On the other hand, suppose that the path were translated without distortion to some position such as ,'/'B'CD  in Fig. 8.9 for which it no longer included ----------------------------------------------------------- CONTOUR INTEGRATION 147 the pole at the origin. Then evidently the total central angle subtended by all four sides would be zero, so that the loop integral would vanish. An example of a different sort is furnished by one of the classical theorems in the calculus of residues. Let g(z) be a function which is analytic on and within a given closed contour and let q be any point within the contour. Then g(z)/(z - q) is a function which is analytic in the same region except for a simple pole at z = q. The residue at this pole must beg(q), the value assumed by g when z = q, as we can easily see by expanding g(z) near this point in the Taylor's series g(q) -3- g (q) (z - q) q- (1/2!)g  (q) (z - q)2 +.. ', and noticing that after division by z - q the series takes the same'form as that given forf (z) in (8-8). If we identify g(z)/(z - q) with f(z), therefore, (8-9) allows us to write f g½) & -2ig(q) (8-13) where, as before, the integration is taken in a clockwise direction. This theorem is of interest here because of its bearing on the general problem of relating the values assumed by an analytic function within a given region to its values on the boundary of the region, which was dis- cussed earlier in the chapter. Evidently, if we know g(z) we can perform the integration on the left-hand side of (8-13) and calculate the special value g(q) directly. In order to make this possible, however, we need know g(z) only on the path of integration, that is, only on the boundary. Equa- tion (8-13) thus provides a method of determining an analytic function anywhere inside a given region from a knowledge of its behavior only on the Boundary of the region. The problem with which we are actually con- cerned is that of determining what properties a function must have on the boundary of the region when it is known to have certain properties in the interior. This problem is evidently in many respects the converse of that solved by (8-i3), although it is much more general, since we begin with a specification only of the general properties of the function rather than with a knowledge of its behavior in detail. On this account it is not possible to present an adequate answer in terms of a single compact formula such as (87-13). The range of questions of practical interest requires the develop- ment of a considerable variety of formulae, only a fewof which are given in the present chapter. Except for these qualifications, however, the solu- tion of the converse problem will be found to imply relations between the values of a function on the boundary of a region and in its interior as tightly knit as that given by (8-13). 8.6. Integral of the Logarithmic Derivative For the immediate purposes of the present chapter, the preceding dis- ----------------------------------------------------------- 148 NETWORK ANALYSIS cuap. 8 cussion is valuable chiefly because it permits the development of a theorem which is of direct interest for amplifier design. Let it be supposed that f(z) is some given function which may, in general, have both zeros and poles, but no singularities aside from poles, within some prescribed con- tour. The object of the theorem is to determine, as far as possible, how many zeros and poles lie within the contour from an inspection of the values assumed byf(z) on the contour itself. The theorem is developed from a study of the integral of the derivative of the logarithm of the function. In other words, we let 0 = `4+ iB = logf (z), and write  dz = zz q- i zz dz = d f- dz. (8-14) The integrand in the last expression of (8-14) will evidently be analytic within the contour except possibly for points at whichf (z) is either zero or infinite. If we suppose that z0 represents one such point and that the func- tion has an nth order zero or pole at z0, we can write f(z) = (z - /'(z) = n(z - + (z - (8-15) if'C) n f(z) z - z0 g(z) where n will be positive if z0 is a.zero and negative if z0 is a pole and g(z) is analytic and not zero in the neighborhood of %. We thus see that f'(z)/f(z) has a simple pole of residue  at z = z0. The integral around the complete contour, as expressed by (8-14), must be -2ri times the sum of all these residues, if the integration is taken in a clockwise direction. At the points for whichf (z) is zero, however, n will be positive and the sum of such residues is therefore equal to the total number of zeros within the contour when each zero is counted in proportion to its multiplicity. Similarly, at a pole n will be negative, and the sum of all such residues will therefore be equal to minus the number of poles when multiple poles are weighted according to their multiplicity. The complete equation (8-14) must therefore be f /' f- dz = 2-i(P - N) (8-16) where N and P are respectively the number of zeros and the number of poles, and the integration is supposed to take place in the clockwise direc- tion. On the other hand, the first and second expressions in (8-14) are merely the integrals of the derivative of O, or .4 + iB, and can therefore be inte- ----------------------------------------------------------- CONTOUR INTEGRATION 149 grated directly. The result must be the difference between the initial and final values of 0, or _d d- lB, as we go around the complete loop. Since the right-hand side of (8-16) is pure imaginary, we have only to consider the imaginary term lB. If we let 1 and 2 symbolize the initial and final points, this equation consequently becomes 1 2- I B I = P -- N. (8-17) The relation expressed by (8-17) can be given a simple graphical inter- pretation. If we represent f(z) on a complex plane of its own, the values whichf (z) assumes as z traverses the prescribed contour can be represented as a moving point in that plane. But the left-hand side of (8-17) is 1/2r times the total change in the phase angle off(z) as z itself travels around the complete contour. Since 2r radians is one revolution, this is the same as saying that the left-hand side of (8-17) is equal to the number of times the moving point representingf (z) revolves around the origin in thef (z) plane while z itself moves once around the path of integration. In order to evaluate the left-hand side, therefore, we need merely plot the values of f(z) which correspond to z's on the prescribed contour and count the num- ber of loops of the plot which encircle the origin. The result given by (8-17) can consequently be expressed as the following Theorem: If a function f(z) is'analytic, except for possible poles, within and on a given contour the number of times the plot off(z) encircles the origin of thef (z) plane in the positive direction,* while z itself moves around the prescribed con- tour once in a clockwise direction, is equal to the number of poles off(z) lying within the contour diminished by the number of zeros off(z) within the contour, when each zero and pole is counted in accordance with its multiplicity. As an elementary example of this theorem, let it be supposed that f(z) = z and that the contour of integration in the z plane is chosen either as the unit circle or the square of Fig. 8.6. Evidently in this case the paths traced out by the moving point in thef (z) plane are the same as these con- tours in the z plane. They are shown by I and II in Fig. 8.10. Corre- sponding to the fact that each contour in the z plane includes one zero and no poles, each of these paths is traversed once in the clockwise direc- tion as z moves clockwise once around the associated integration contour. * The positive direction is, of course, that one for which the phase angle off(z) is increasing. In other words, it represents a counterclockwise encirclement of the origin in thef(z plane by. the movin point. ----------------------------------------------------------- 150 NETWORK ANALYSIS C..P. 8 On the other hand, ill(z) = l/z, the paths in thef (z) plane corresponding to the circle and the square in the z plane are respectively the circle and the curvilinear quadrilateral shown by I and II in Fig. 8.11. Each of these paths is traversed once in the counterclockwise direction as z moves clock- wise once around he associated z contour, corresponding to the fact that f (z) plane C B D .'/ plane 8.10 Fro. 8.11 each contour now includes one pole and no zeros off(z). If we choose more complicated expressions forf (z) the paths will, of course, ordinarily become more complicated and may encircle the origin more than once. Such situations, however, can best be illustrated by the examples given in later sections. As the preceding examples may suggest, the theorem on the logarithmic derivative amounts, in simple cases, to a statement of a certain corre- spondence between specified areas in the z andf (z) planes. Thus suppose -plana 8.12 Fro. 8.13 that the z contour is that shown by Fig. 8.12, and that it encloses one zero and no poles off(z), although zeros and poles may be found outside the contour. The associatedf (z) path must encircle the origin in the f(z) plane once, as shown by Fig. 8.13. Then the theorem says, in effect, that ----------------------------------------------------------- CONTOUR INTEGRATION 1 1 in a certain sense the interiors of the two contours correspond to one another. For example, there is, by assumption, one point in the interior of the z contour of Fig. 8.12 at which f(z) = 0. Correspondingly, the pointf (z) = 0, or the origin in thef (z) plane, is found in the interior of the contour in that plane, as shown by Fig. 8.13. Suppose, however, that we were to choose any other point z0 inside the z contour. Then the new functionf(z) -f(z0) would still have a zero but no pole inside this con- tour, so that its plot must enclose the origin in its own plane. The plot of f(z) -f(Zo) can be obtained from that off(z) in Fig. 8.13, however, merely by a translation of amountf (z0), so that this result is only possible iff(z0) lies inside the contour in Fig. 8.13.* Thus every point in the interior of the contour of Fig. 8.12 corresponds to some point in the interior of the contour of Fig. 8.13. If, on the other hand, the contour of Fig. 8.12 includes a pole but no zeros we can show by an argument of the same type that any point in the interior of the z contour must correspond to a point which is outside the contour for f(z). Thus the interior of one contour corresponds to the exterior of the other. This is manifested by the fact that thef(z) contour is traversed in the reverse direction. As the number of zeros and poles in the interior of the z contour is increased, these relations, of course, grow more complicated. We must, in general, think of the interior of the z contour as being broken up into several subregions, some of which correspond to the interior and others to the exterior of thef (z) plot, or to the interior and exterior of specified loops in thef (z) plot if the plot crosses itself several times. 8.7. Nyquist's Criterion for Stability -- Single Loop Case The importance of the theorem just established arises from the fact that it leads immediately to the familiar criterion for stability due to Nyquist. To show this, let the independent variable, which has hitherto been taken as z, be represented by p. Let the integration contour be the path in the p plane shown previously by Fig. 8.7. It will be supposed that this path is made indefinitely large. We will let the function whose logarithmic derivative is integrated around this path be the return difference F = ///x ø for one of the tubes in the circuit. It will be assumed that F has no * The interior loop in Fig. 8.13 has been drawn to illustrate the fact that for special values off(z0) the plot of the new functionf (z) -f(zo) may encircle the origin more than once. Thus there may be more than one point inside the z contour corresponding to a prescribedf (z0). This is evidently only possible when thef (z) contour crosses itself. In other circumstances there is a one-to-one correspondence between the points in the two interiors.  Regeneration Theory, B.S.T.J., Jan. 1932. See also Peterson, Kreer, and Ware, Regeneration Theory and Experiment, Proc. I.R.E., Oct. 1934. ----------------------------------------------------------- 152 . NETWORK ANALYSIS C,,r. 8 singularities on the real frequency axis so that the small indentations shown by Fig. 8.7 can be ignored. The Nyquist diagram for determining the stability of a circuit is in essentials a plot of the values of F corresponding to p's lying on this con- tour, prepared in the manner described in the preceding section. In draw- ing the diagram, however, advantage may be taken of a number of simpli- fying possibilities. In the first place, any physical tube must contain parasitic plate-cathode and grid-cathode capacities which vill short-circuit the transmission path from plate to grid at extremely high frequencies. This is equivalent to saying that the return ratio of the tube will vanish, or its return difference will approximate unity, if p is made indefinitely great. As we make the contour in Fig. 8.7 larger and larger, consequently, the moving point which traces the path of F in the F plane will become more and more nearly stationary as p moves around the semicircular part of the complete contour. In the limit, this part of the contour can be dis- regarded entirely so that the complete diagram becomes a plot of only the real frequency values of F for the complete real frequency axis from --oo to q-. The second simplification arises from the even and odd symmetry, respectively, of the real and imaginary components oœ F on the real fre- quency axis, which was discussed in the preceding chapter. This makes it necessary to compute the path in the F plane only for positive frequencies. The half of the path which corresponds to negative frequencies can be inserted as the mirror image of this part with respect to the real axis of the plane. The third simplification is perhaps more important than either of the first two. The zeros and poles of F are respectively the roots of /x and of zX ø. If we make the path of integration in Fig. 8.7 sufficiently large we can suppose that all the roots in these two quantities which lie in the right halLplane will fall within the contour. We can therefore determine the difference between the number ofrootsof,x and iX ø whichlie within the right halLplane by counting the number of loops of the plot off which encircle the origin. But the stability of the structure depends upon the location of the roots of/x only, so that counting the loops gives only ambiguous infor- mation concerning the stability of the circuit unless we know how many roots of zX ø are included in the total. For the time being this difficulty will be avoided by assuming that the circuit is known to be stable when the prescribed/'F vanishes. This is true, for example, for an ordinary single loop amplifier, which was the case actually considered by Nyquist in his original treatment of the problem, since the failure of any one of the tubes will open the feedback loop. In these circumstances 6 0 can have no roots in the right half-plane so that the stability or instability of the circuit cat', ----------------------------------------------------------- CONTOUR INTEGRATION 153 be determined unambiguously from the Nyquist plot. Evidently the con= dition for stability is that the plot shall not encircle the origin, while any encirclements which do occur must be in the clockwise direction. A number of illustrative plots are shown by Figs. 8.1,, 8.15, and 8.16. In each case the region of maximum F is taken as a band centered about some point coo. As we go from coo to infinite frequency the return difference must, of course, reduce to unity for the reasons mentioned previously. In each drawing F is shown as reducing to unity at zero frequency also, since plate supply coils and blocking condensers will normally interrupt the d-c F lane F plane 8.14 Fro. 8.15 Fro. 8.16 feedback path.* The portions of each plot corresponding to positive and to negative frequencies are shown respectively by the solid and the broken lines. The directions in which the plots are traced as co varies from - oo to q-oo are indicated by the arrows. Evidently, Fig. 8.14 represents a stable structure. Figure 8.16, on the other hand, represents a structure which is unstable, with four roots in the right half-plane, since the plot encircles the origin four times. At first glance, it may appear that the structure of Fig. 8.15 is also unstable. It is easy to see, however, that the net phase rotation of a vector connecting the origin to a moving point on the path, over the complete contour, is zero, so that this structure is stable. The foregoing description of the Nyquist diagram has been based upon the return differenceF as a matter of theoretical simplicity. In practice,however, the diagram is usually plotted in terms of the return ratio T, or the loop * By using special circuits, however, it is possible to provide a d-c return path, so that maximum feedback can be assumed to occur over a band centered about zero frequency. It is also often convenient to assume such a case, ignoring the power supply elements, in analytic work in order to make use of the transformation from symmetrical band-pass to low-pass characteristics described in one of the following chapters. The two examples given later in this section are of this type. Approxi- mate illustrative characteristics can be obtained from Figs. 8.14 to 8.16 by omitting the portions of each plot between --coo and +coo and identifying -4-coo with zero frequency, ----------------------------------------------------------- 154 NETWORK ANALYSIS C.A,'. 8 transmission characteristic u. Since F = 1 + T = 1 - u/J the relations among the three plots are easily ascertained. For example, Fig. 8.17 shows the diagram of Fig. 8.14 plotted in terms of T. It is the same as Fig. 8.14 except for a translation one unit to the left. Figure 8.18 shows the same plot in terms of/, and is the same as Fig. 8.17 rotated through 180 ø . In each figure the negative frequency characteristic has been omitted for simplicity. Evidently a loop around the origin in the F Fro. 8.17 Fro. 8.18 diagram is the same as a loop around -1,0 in the T diagram, or a loop around 1,0 in the t diagram. Calling these three the critical points, therefore, the general result of this discussion can be summed up in the Theorem: If a structure is stable when a given element vanishes, the necessary and sufficient condition for it to remain stable when the element assumes its normal value is that the Nyquist diagram for the return difference, return ratio3 or loop transmission of the element should not encircle the appropriate critical point. The choice between the T and/fi diagrams can conveniently be related to the well-known fact that under normal circumstances a feedback ampli- fier must contain an odd number of stages in its forward circuit. In an ordinary design, for example, the purely passive parts of the feedback loop will give a very small phase shift in the neighborhood of the band center o0, while on each side of 0o they will vary in a manner somewhat similar to that shown by the diagram of Fig. 8.17. It is clear that if the passive circuits furnish the complete tfi characteristic such a diagram will encircle the point 1,0 and produce instability unless the loop transmission is very small. There is, however, a phase reversal associated with each tube. By using an odd number of tubes we secure one net phase reversal. This rotates the/ diagram into the position shown by Fig. 8.18, and permits ----------------------------------------------------------- CONTOUR INTEGRATION 155 the use of a substantial amount of feedback near coo without instability. I'he/a diagram is thus appropriate when we consider the complete phase shift around the feedback loop, including the tubes, while the use of the T diagram, in an amplifier containing an odd number of stages, is equivalent to considering the phase shifts of only the passive parts of the structure. The amplifier can, of course, be built with an even number of stages by using one of the devices described in Chapter III. r plane --4.0 t :6,0 FIo. 8.19 As a quantitative example of a Nyquist diagram we may consider the circuit shown by Fig. 7.2 in the preceding chapter. It is apparent from equation (7-12) of that chapter that the return ratio T for the tube can be written as pa+2p+l T = ta 3p a + 4p" + 7p + 2 (8-18) If we assume, for definiteness, that u -- 5, this yields the values of T given in the following table. co T  T 0 2.5 - i0 1.7 -0.26 + i0.87 0.5 1.73 - il.05 1.8 0 + il.02 1.0 0.5 - il.5 2.0 0.44 q- i1.12 1.2 -0.33 - i1.17 3.0 1.29 + i0.81 1.4 --0.79 - i0.26 4.0 1.48 + i0.59 1.5 --0.73 + i0.23 6.0 1.59 + i0.38 1.6 -0.53 + i0.61 10.0 1.64 + i0.22 The Nyquist diagram obtained by plotting these points is shown by Fig. 8.19. The plot does not enclose the point -1,0 so that the system is ----------------------------------------------------------- 156 NETWORK ANALYSIS CHA,. 8 Fro. 8,20 stable. We may notice, however, that the whole diagram is proportional to  and would enclose - 1,0 if t were multiplied by perhaps 1.25 or 1.3. This agrees with the calculations made in the preceding chapter, where it appeared that some of the zeros of z would be found in the right half-plane for t > 6.4. The stability of the circuit for negative t's can be examined conveniently by using the same diagram with the critical point taken as 1,0. We observe that to make the system stable under these conditions we must multiply  by a factor of about 0.4, which agrees with the limit t = -2 determined in the preceding chapter. We may also notice that the values of 0 at which the Nyquist plot crosses the real axis, which, of course, mark the places at which the plot encounters the two critical points when  is assigned its limiting values, are respectively  1.45 and 0. These agree with the values given in Chapter VII Gr the points at which the vari- ous zeros of  cross from the left to the right side of the plane. Fro. 8.21 A second example is furnished by the circuit of Fig. 8.20. The structure represents a normal three stage amplifier with shunt feedback except that to simplify the computations all the branches have been taken as propor- tional to a given admittance y. Let the transconductances of the three tubes be represented by St, S2, and Sa. If we ignore the phase reversals due to the tubes, the voltage gains from the first grid to the second and from the second to the third are respectively S/y and S2/y, while that from the third grid back to the first is kSa/(1 + 2k)y. The product of these three is the return ratio T for any one of the tubes. We therefore have k 1 T - 1 q- 2k SSøSaji' (8-19) To plot the Nyquist diagram, we will suppase that y = 1 + p + (l/p). This correspo:.ds to a resistance, capacity, and inductance all in parallel. ----------------------------------------------------------- CONTOUR INTEGRATION 157 Such a structure might represent a simple form of amplifier transmitting a band of frequencies in the neighborhood of the resonance of the coil and condenser. If we choose kSS2Sa/(1 + 2k) -= 6, this gives the Nyquist diagram shown by Fig. 8.21. Only the positive half of the diagram is shown, since with the symmetrical characteristics chosen the negative half is an exact duplicate. It will be seen that the circuit is stable and gives a return difference in the center of the band of 17 db. The circuit becomes unstable, however, if the tubes are assumed to have slightly more gain. 8.8. Nyquist's Criterion for Stability -- Multiple Loop Case The discussion in the foregoing section has been based upon the assump~ tion that none of the roots of/x ø can be found in the right half-plane, or in other words, that the circuit is stable when the prescribed /F vanishes. This assumption is, of course, valid if/F represents a tube in any single loop amplifier, and it can also be expected to hold for the majority of multiple loop cases. On the other hand, certain multiple loop circuits may be stable under operating conditions but become unstable when specified tubes fail. This section will consider the application of the Nyquist diagram to such situations. If some of the roots of X ø are found in the right half-plane, it is evident that the circuit will not be stable if the Nyquist diagram fails to encircle the critical point. In accordance with (8-17) such a situation implies that there are as many roots of ;x as there are of zX ø in the right half-plane. To assure stability the Nyquist plot should encircle the critical point in a counterclockwise direction as many times as there are roots of x ø to consider. It is therefore necessary to know the number of these roots. This can be determined from the Nyquist diagrams for the other tubes of the circuit with the original tube dead. To analyze the situation generally, let it be supposed that the tubes are originally all dead and are as- signed their normal gains one by one in some chosen order. As each tube is restored to its operating condition we may compute its return difference for the condition of the other tubes existing at that stage of the process and plot the corresponding Nyquist diagram. It follows from (8-17) that the diagram for thejth tube will encircle the critical point PS - NS times in a counterclockwise direction, if Ps and N i represent respectively the numbers of poles and zeros of the jth return difference which appear in the right halLplane. The total number of encirclements for all plots is (P1 - N) q- (P2 - N2) q- '" q- (P, - N,). But the / which appears in the numerator of any return difference is the same as the /x ø in the denominator of the succeeding return difference. We therefore have Ns = -Pi+I. Moreover, P = 0, since the circuit with all tubes dead must be stable. The final circuit will be stable if N, = 0. We therefore have the ----------------------------------------------------------- 158 NETWORK ANALYSIS CaAP. 8 Theorem: If a circuit is stable when all its tubes have their normal gains, the total number of clockwise and counterclockwise encirclements of the criticalpoint must be equal to each other in the series of Nyquist diagrams for the individual tubes obtained by beginning with all tubes dead and restoring the tubes successively in any order to their normal gains. In applying this theorem, it is important to notice that the gains of the tubes may be restored in a variety of orders. If the amplifier contains n tubes there are, in the general case, n! possible arrangements. Although the final index of stability or instability must be independent of the order in which the tubes are chosen, the diagram for any individual tube may be vastly affected by the point in the series Fro. 8.22 at which its gain is supposed to be restored. An example of this theorem is furnished by the circuit of Fig. 8.22. The structure is the same as that shown previously by Fig. 8.20 except for the addition of a sub- sidiary feedback around the first two stages of the forward circuit.* A two- stage subsidiary loop is a convenient choice here, where we are interested in il- lustrating a circuit which may become un- stable when one of the tubes fails, since, as shown previously, an even number of stages leads to a returned voltage which is broadly of the wrong sign for stability. In the present instance we may therefore expect the circuit to sing when the output tube fails, if the gain of the first two stages is suFfi- ciently great. It will be assumed, for definiteness, that k = 0.001 and k2 = 0.01. We may expect from these numbers that questions of stability will arise as soon as the voltage gain per stage is greater than about 10. The several Nyquist diagrams which are required to determine whether the structure will be stable can, of course, be obtained from loop transmission compu- tations, as was done in connection with Fig. 8.20. For the variety of cases to be considered here, however, it is simpler to base the analysis upon the determinant of the system. Using nodal methods, we readily find that * This general type of circuit was described by F. B. Llewellyn, (U.S. Pat. No. 2,245,598), who called the subsidiary feedback the c circuit, in distinction to the principal, or/0, feedback. The present example is, of course, not intended to illus, trate the contemplated engineering applications of such a circuit. ----------------------------------------------------------- CONTOUR INTEGRATION 159 (1 + kx + k)y 0 -k -kly Sx y 0 0 -k2y S (1 + k)y 0 -ky 0 Sa (1 + kx)y = y4 + kk2Sa _ k2y.SS2 + kySzSSa (8-20) where Sx, S2, and Sa are, as before, the transconductances of the three tubes, and the second expression has been simplified by ignoring the small quantities k and k in comparison with unity. Since the circuit contains three tubes there are 3! = 6 orders in which the gains of the tubes can be restored. The first two tubes, however, can be regarded for analytic purposes as a single tube, since they are directly in tandem and cannot affect the stability of the circuit unless both are opera- tive. This is evidenced by the fact that S and S appear only as the product SS2 in (8-20). We need consider, consequently, only two possi- bilities, one in which the gains are restored in the order Sa;SxS2, and the other in which they are re- stored in the order Sx S=;Sa. The simpler Nyquist diagrams are found if we begin by restoring the gain of Sa. After Sa is re- Fro. 8.23 stored, its return ratio is readily found from (8-20) by means of the formula T = (/x//x ø) -- 1, where, of course, / represents the determinant when Sa has its normal value, A ø the determinant when Sa vanishes, and both quantities are to be evaluated under the assumption that SS2 = 0, since the gains of these tubes have not yet been restored. This gives kkSa 7' = -- (8-21) Y Upon choosing y = 1 + p q- (l/p), as before, this leads to a Nyquist diagram whose positive frequency half is shown by Fig. 8.23. For practical values of $a the path would be very small because of the very small value of kk2, but in any event it is clear that it does not encircle the critical point We next restore the gains of the tubes S and $2 to their normal values. The resulting return ratio for these tubes can be found by the same general method as was used in obtaining (8-21) and appears as k Sa -- ky T = kk2y2S a + ya SS2. (8-22) ----------------------------------------------------------- 160 NETWORK ANALYSIS cH^v. 8 A series of curves for the positive frequency half oft for 0 > 1 is shown by Fig. 8.24.* In each case it has been assumed that S S. = 200. The re- sults for other values of S1Ss can, of course, be obtained merely by expand- ing or shrinking the curves actually given. Assuming that SiS2 = 200, Curve I gives the Nyquist plot when Sa = 40. It will be seen that the plot encircles the critical point - 1,0. In the preceding diagram of Fig. 8.23, on the other hand, the critical point was not encircled. The net number of encirclements in the two plots together is therefore not zero and in accord- ance with the preceding theorem the structure is consequently unstable. Fro. 8.24 Curve II gives the result when Sa is assumed to be 20. The circuit is now on the edge of instability, since the plot passes directly through the critical point. As Sa is diminished below 20 the circuit becomes definitely stable. Curve III, for example, shows the result when Sa = 10. Very low values of Sa, on the other hand, lead once more to instability. For examole, when Sa = 4 the plot is that shown by Curve IV and once more encircles the critical point. Instead of following this arrangement we can also restore the gains in the order S S2;,5'3. The return ratio for S and S. with Sa = 0 can be obtained from (8-20) as - ks 81 T = yS (8-23) * The images of these curves about the real axis correspond to values ofw < 1. ----------------------------------------------------------- CONTOUR INTEGRATION 16l The corresponding Nyquist diagram where SS2 = 200 is shown by Fig. 8.25. It will be seen that the curve in Fig. 8.25 encircles the critical point once in a clockwise direction.* In accordance with the preceding theorem the final plot of the return ratio for Sa must consequently encircle the critical point once in a counterclockwise direction if the complete circuit is to be stable. This can be examined by setting up the return ratio for Sa as T= k kY + SS y(y--5 -__- SS-----2) Sa. (8-24) The Nyquist diagram corresponding to this equation when S = 10 and SS2 = 200 is shown by the solid curve of Fig. 8.26, while the diagrams obtained for the same value of S0, but with S1S2 chosen as 100 and 400, are shown respectively by the broken line Curves I and II. Considering in particular the solid Fro. 8.25 curve, it will be seen that the plot does in fact loop around -1,0 once in a counterclockwise direction, so that the final structure is stable. This is, of course, in agreement with the conclusion previously reached in con- nection with Fig. 8.24, since the assumed S's are the same as those which apply to Curve III of that figure. / I I 8.26 By varying Sa it is also possible to confirm the conclusions reached previ- ously for the conditions represented by the other curves of Fig. 8.24. The changes in So can be represented in Fig. 8.26 by expanding or contracting * Figure 8.25 gives only the positive frequency half of the complete plot. A second loop around the critical point is, of course, provided by the negative half, ----------------------------------------------------------- 162 NETWORK ANALYSIS C.AP. 8 the diagram or, more conveniently, by keeping the plot fixed and moving the critical point. If we retain the choice S$2 --- 200, it will be observed that the circuit remains stable if $a is varied by a small amount in either direction from the original value 10, but that it becomes unstable for larger changes. As an example we may select Sa = 4 which corresponds to Curve IV in Fig. 8.2,. This is equivalent to moving the critical point to the position P in Fig. 8.26. The critical point is thus placed outside the solid curve, which corresponds to instability in this situation. On the other hand, the choice Sa = 40, which corresponds to Curve I in Fig. 8.24, moves the critical point to the position P2. With this change, the point is still encircled by the curve, but the encirclement takes place in the wrong direction. 8.27 Fro. 8.28 8.9. Conditional and Unconditional Stability In a formal mathematical sense, the above criteria of stability, based entirely upon the encirclement of the critical point in the Nyquist diagram, require no qualifications. Any structure that meets them is stable. For practical engineering purposes, however, it is desirable to pay some atten- tion to the general shape of the Nyquist plot in addition to counting the number of times it loops around the critical point. This gives rise to two general classes of stable structures, as illustrated by the return ratio dia- grams for single loop structures shown by Figs. 8.27 and 8.28. Both dia- grams represent stable circuits. The first, however, is absolutely or uncon- ditionally stable, while the second is merely Nyquist* or conditionally stable. The reason for making this distinction appears if it is recalled that for practical purposes we are really interested in the stability of an amplifier * So-called because it was generally assumed before Nyquist's work that it was not possible to obtain a positive real./t8 greater than unity without instability. ----------------------------------------------------------- CONTOUR INTEGRATION I63 over a period of time. Most of the elements of the amplifier can be expected to remain fairly constant. The gains of the tubes, however, are likely to diminish with age and since one of the usual objectives in applying feedback to a circuit is to allow a large variation in the tubes without much effect on the external gain, it must be supposed that this diminution will be substantial. Since the return ratio diagram in a single loop feedback structure swells or shrinks in direct proportion to the gain of the tubes, the effect of aging therefore will be to contract the loop. If a diagram such as that shown in Fig. 8.28, in which the return ratio path goes beyond 180 ø for an interval in which there is a net gain around the loop, is sufficiently decreased the plot will take the form shown by Fig. 8.29 and evidently represents an unstable circuit. If the diagram is of the uncondition- ally stable type shown by Fig. 8.27, on the other hand, it can be de- creased indefinitely without produc- ing instability. Another possibility of securing a change in tube gains with time occurs when the power is first applied to the tubes. Until the cathodes are warm the gain of the tubes will be very small. Fro. 8.29 As power is applied to the circuit, therefore, we must imagine that the return ratio diagram begins by being very small and expands continuously to its final position as the cathode temperatures increase. If the final diagram is of the type shown by Fig. 8.28, there will be an intermediate point in the course of this expansion for which the system is unstable. When this intermediate point is reached natural oscillations begin and build up exponentially. At the same time, of course, the gains of the tubes increase as the cathodes approach their operating temperatures so that there is a tendency for the amplifier to pull itself out of the unstable condition. In most circumstances, however, the sing develops so rapidly that the tubes are overloaded before the gain is sufficient to bring the Nyquist diagram out of the unstable condition. Since overloading usually reduces the effective gains of the tubes, the system is very likely to persist in an unstable condition permanently. These difficulties are not necessarily unanswerable. For example, we may close the feedback loop after the gains of the tubes have reached their normal values, or we may apply "B" battery to the tubes after the cath- ode temperatures are sufficiently high. For practical purposes, however, these devices represent undesirable complications. Moreover, even if ----------------------------------------------------------- 164 NETWORK ANALYSIS CaxP. 8 they are used the amplifier is somewhat unreliable, since it may still sing if the tubes age sufficiently or if the power supply is momentarily interrupted. For these reasons, most of the analysis which follows will assume that the amplifier is to be unconditionally stable. On the other hand, it turns out that under equivalent circumstances a conditionally stable amplifier may exhibit much more feedback than would be obtainable from an uncon- ditionally stable structure. Conditional stability thus represents an important possibility when adequate feedback is hard to secure. A second qualification on this discussion is also pertinent. In describing the characteristics of a conditionally stable circuit it was tacitly assuned that the structure contained only a single feedback loop. Evidently, the same physical considerations affect multiple loop structures also. In the single loop case, however, we can distinguish between conditionally and unconditionally stable situations merely by inspecting the shape of the Nyquist diagram, since changes in tube gains affect only the size of the diagram. In a multiple loop circuit, on the other hand, both the shape and the size of the Nyquist diagram for any one tube may be affected by changes in the gains of the other tubes. Examples are furnished by the preceding Figs. 8.24 and 8.26. This evidently produces a much more complicated situation. The analysis of the problem is too lengthy to be given here and will be presented at a later point. 8.10. Extensions of Nyquist's Criterion Thus far we have applied the Nyquist diagram method of determining stability only to the return difference zX/zX ø. Since zX will appear in almost any transmission or impedance expression we care to set up, however, it is clear that the application of the criterion is not necessarily restricted to this one function. Some of the possible extensions are considered here. The discussion is given only in outline, since the essential situation is the same as it is for the return difference function. The chief point to notice is that any transmission or impedance expression will contain zX in combination with some other determinant, just as the return difference includes both zX and d ø. In general, the Nyquist diagram will give only the difference between the zeros and poles of the impedance or transmission function or, in other words, only the difference between the number of zeros of zX and of the determinant with which it is associated. In extending Nyquist's cri- terion, therefore, it is necessary to assume that we have some means of determining the number of zeros in the right-hand half-plane furnished by the other determinant. This is usually equivalent to saying that we must know that the structure is stable for some particular reference condition or if it is not stable what modes of instability it has. ----------------------------------------------------------- CONTOUR INTEGRATION 165 1. Nyquist's Criterion for a Driving Point Immittance. The immittance which will be seen at the terminals of a generator applied to the nth mesh or node of a general network can be written as = (8-23) If we make a Nyquist plot of this expression, it follows from (8-17) that the number of loops encircling the origin will be the difference between the number of zeros of/x and of/x,, in the right-hand half-plane. The quan- tity /x,, however, is the form to which the determinant of the system reduces when an infinite immittance is across the driving terminals or, in other words, when the driving terminals are open-circuited in an impedance analysis or short-circuited in an admittance analysis. If the system is known to be stable under these conditions (8-25) can have no poles in the right half-plane. If it also has no zeros in this region, so that the system is stable under normal conditions, it follows that the Nyquist plot cannot encircle the origin. This result can be generalized. Suppose that instead of adding an infinite immittance between the driving terminals we add only the finite amount//7,. Since zX, must be independent of///,, (8-25) becomes ///"' = /3' + Fr' = (8-26) where/x' represents the new value of zX. Division of (8-25) by (8-26) gives /4/--5 = /4' + k? = a -' (8-27) If the system is known to be stable after the addition of///, (8-27) can have no poles in the right half-plane and the previous argument applies. We therefore have the Theorem: If a system is stable when a prescribed immittance is added between a pair of terminals it will be stable without the given immittance provided the Nyquist plot of the ratio between the total immittances at the terminals in the two cases does not encircle the origin. In particular, it is neces: sary to plot only the normal immittance itself if the system is stable when an infinite immittance is added between the driving terminals. In applying this theorem, it must be borne in mind that the complete Nyquist contour of Fig. 8.7 includes the large semicircle in the right half- ----------------------------------------------------------- 166 NETWORK ANALYSIS C.^P. s plane as well as the real frequency axis. This part of the path-was dis- missed in the consideration of F on the assumption that a physical return differehce always approached unity at infinite frequency. The same simplification obtains here if the quantity which is plotted approaches a constant value at infinity. If it behaves as either a positive or a negative power of frequency near infinity, however, the Nyquist diagram must include an arc of a very large or very small circle to represent the values assumed by the function over thks part of the path. 2. Nyquist's Criterion for a Transfer _[remittance. The transfer immittance from point i to pointj in a general network can be written as /4v = --. (8-28) The Nyquist diagram corresponding to (8-28) will encircle the origin as many times as there are roots of/x in the right half-plane provided there are no roots of/xi i in this region. From the discussion under 5c in the list of general network conditions given in the preceding chapter, the restriction on the roots of/xg i is equivalent to specifying that the transfer immittance must be a minimum phase shift function. We can therefore conclude that if the transfer immittance is known to be of minimum phase type the net- work will be stable provided the Nyquist diagram of the transfer immittance does not encircle the origin. As an example we may consider the familiar expression u/(1 --/) for the gain of an ordinary feedback amplifier. Since a transfer immittance has the physical significance of a loss, this expression can be regarded as the reciprocal of the transfer immittance from input to output. The poles of transfer immittance are consequently either points at which  vanishes or points at which g becomes infinite. None of the latter group of points can be found in the right half-plane, since the/ circuit, being passive, is necessarily stable. None of the former group of points will appear in the right half-plane if the gain u by itself represents a minimum phase shift expression. This will be true for any of the u cir- cuits encountered in ordinary design practice. In any ordinary situation, therefore, the stability of the amplifier can be determined by observing whether or not the Nyquist plot of its external gain encircles the origin.* .This analysis can be generalized by methods similar to those used for the * In making such a plot, however, allowance muse again be made for the face that the complete Nyquist path includes the large semicircle in the right half-plane. In practical situations, the gain of the amplifier must eventually drop off as some nega- tive power of frequency. The final part of the diagram must include an arc of a very small circle to represent the behavior of such a function over the large semicircular portion of the path. ----------------------------------------------------------- CONTOUR INTEGRATION 167 driving point immittance. We observe that aii in (8-28) must be inde- pendent of the self-immittances at i andj. If we make an arbitrary change in either or both self-immittances, therefore, we can write the new transfer immittance as /f,= -- (8-29) where zX  represents the new value of zX. The ratio of (8-28) and (8-29) is (840) and the Nyquist diagram of this function will encircle the origin as many times as there are roots of A in the right half-plane if there are no roots of zX' in this region. We therefore have the Theorem: If a system is stable when prescribed immittances are added at two points in a circuit, it will also be stable without the added immittances if the Nyquist plot of the ratio of the transfer immittances between the two points in the two cases does not encircle the origin. In particular, it is neces- sary to plot only the transfer immittance in the second case if the function is known to be of minimum phase type. 8.11. Two Theorems from Function Theory The discussion of this chapter will be concluded by the demonstration of two standard theorems from function theory. The theorems are developed here for use in the next few chapters. They can conveniently be regarded as by-products of the Nyquist diagram method of treating stability, although they are usually established independently. To develop the first theorem, letf (z) andf2(z) be two functions which are' analytic within and on the boundary of a given region. Bothf (z) and f. (z) may, however, have zeros within the region. It will be assumed that Ifs(z)[ > [f2(z)l at all points on the boundary. Consider the function F(z) defined by P(z) =.riO)+f,.(z). = 1 + f,.(z). (8-31) J (z) (z) In accordance with (8-17), the number of times the origin is encircled by the Nyquist plot of F(z) is equal to the difference between the number of zeros of F(z) and the number of poles of F(z) lying within the region. But the zeros and poles are respectively the roots offt (z) +f(z) and fx(z).. Furthermore, since we have assumed Ifs(z)[ > If2(z)l on the boundary, it is clear from the right-hand side of (8-31) that the Nyquist plot must be ----------------------------------------------------------- 168 NETWORK ANALYSIS CaaP. 8 inside the unit circle in Fig. 8.30. Evidently the plot cannot encircle the origin at all. We therefore have the Theorem: Iff(z) andfz(z) are analytic on and within a given closed contour and If1½)1 > Ifs(z)[ on the contour, the func- tionsf (z) andf (z) +f2 (z) have the same number of roots within the contour. FI. 8.30 The general field of application of this theorem is obviously that of determining rough limits within which changes in a structure can be expected not to affect its stability. As an example, suppose fx(z) represents an impedance looking into some pair of terminals in an ampli- fier. It will be supposed that f(z) is "open-circuit stable" -- so that it has no poles in the right half-plane. Let fz(z) be an ordinary passive impedance added between these terminals. If I.f(z)[ > [fz(z)l at all points on the real frequency axis the addition of the passive impedance cannot affect the stability or instability of the structure. Fro. 8.31 Fro. 8.32 To establish the second theorem, letf(z) be analytic within and on the boundary of some given region. The Nyquist plot off(z) will take one of the forms indicated by Figs. 8.31 and 8.32, depending upon whether or not there is a root off(z) in the region. Let Zo be any point in the region. In accordance with the argument advanced in connection with Fig. 8.13, f(zo) can be represented by some point P lying within the Nyquist plot in Fig. 8.31 or 8.32. Evidently the real component off(z0) cannot be as great as the real component exhibited by f(z) in some parts of the boundary ----------------------------------------------------------- CONTOUR INTEGRATION 169 because in either figure we can find some point P on the plot itself which lies to the right of P. Similarly, the existence of points such as P2 indi- cate that there must be parts of the boundary for which the real component off(z) is less than that off(z0). The points Pa and P4 illustrate similar relations for the imaginary component. In both plots, also, there is a point Ps corresponding to an absolute value off(z) greater than that of f(z0). In Fig. 8.32 we can, in addition, pick out a point P6 corresponding to a smaller absolute value than that off(z0), as well as points P7 and P8 at which the phase angle is greater than and less than that off(z0). But the z0 with which we started was any point in the interior of the region. We have therefore established the Theorem: Ill(z) is analytic within and on a given closed contour the maxima and minima of the real and imaginary components off(z) and the maximum absolute value off(z), for the region composed of the contour itself and the points interior to it, are all found on the contour. Ill(z) has in addition no zeros within the contour the minimum absolute value of f(z) and the maximum and minimum phase angles off(z) are also found on the contour. An example of this theorem is furnished by the common engineering problem of maximizing or minimizing some aspect of the performance of a complete passive network at a prescribed frequency by making the most suitable choice of some branch impedance which is under our control. It will be supposed that the branch impedance may be a reactance, a resist- ance, or some combination of the two. In general, any ordinary passive network characteristic, such as a driving point or transfer impedance, will have neither zeros nor poles considered as a function of one of the branch impedances, as long as the branch impedance has a positive resistance com- ponent.* In other words, both the driving point or transfer impedance and its logarithm will be analytic functions in the right half of the plane repre- senting the branch impedance. It follows that the real and imaginary components of the driving point and transfer impedances, their absolute values, and their phase angles will all assume both larger and smaller values on the imaginary axis than they do anywhere in the right half-plane. Since we cannot assign a negative resistance component to the branch impedance, the maximum and minimum values which are physically obtainable for any of these quantities must therefore be found when the branch impedance is a pure imaginary. It is not necessary to examine dissipative impedances. * An exception to this statement must be made for the transfer impedances for certain types of bridge circuits, in which zero delivered current can be secured by bridge balance. These are the non-minimum phase shift networks described in a later chapter. ----------------------------------------------------------- CHAPTER IX PHYSICAL REPRESENTATION OF DRIVING POINT IMPEDANCE FUNCTIONS 9.1. Introduction THxs and the succeeding chapter are devoted to a general discussion of the properties of driving point impedance and admittance functions on the basis of the requirements laid down in Chapters VII and VIII. The material is not intended to constitute a complete theory. It is presented principally to illustrate the general requirements deduced in preceding chapters by showing some of the more elementary physical consequences to which they lead. For the sake of logical coherence, however, the present chapter will be centered about the general problem of showing that the con- ditions on driving point impedances laid down in Chapter VII are sufficient as well as necessary or, in other words, that any impedance functions meet- ing these conditions can be realized in a physical structure. Miscellaneous additional topics will then be treated in Chapter X. The list of requirements on driving point impedance functions given in Chapter VII includes both general conditions applicable to all networks, active or passive, and additional special conditions applicable only to passive structures. Purely passive impedances, however, are both those for which the greatest experience is available and those of greatest present importance in design. The discussion will consequently be directed princi- pally at impedances of this type. Active impedances are treated by indi- cating the points at which they require formal extensions in the passive analysis. In particular, the present chapter will begin by showing how any impedance function meeting the passive requirements can be realized. The problem of realizing an active impedance expression is then treated by showing that any active impedance can be obtained from a combination of a passive impedance and a negative resistance. 9.:2. Resistance Reduction of Passive Impedances The conditions which must be met by any passive impedance function were given as 1, 2, 3, 4, 5b, and 6a in the list of Chapter VII. Our first object is to show that an actual physical structure can be found which will represent any impedance function meeting these requirements. Methods 170 ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 171 of solving this problem have been invented by Brune* and Darlington. Darlington's structure consists of a four-terminal teacrance network termi- nated in a resistance. He is able to show that by properly proportioning the network the input impedance of the structure can be assigned any functional form which meets these requirements. For the purposes of this book the method developed by Brune is the more useful. Brune's method depends upon two principles. In order to explain the first, let it be supposed for simplicity that the impedance func- tion has no zeros or poles on the imaginary axis. The fact that this assump- tion is immaterial is shown in the next section. Both the impedance and admittance will then be analytic in the right-hand half of the p plane, including the imaginary axis. This is a situation which can be examined by the second of the two theorems developed at the end of the preceding chapter if we regard the right half of the p plane as the region of analyticity and the imaginary axis as the boundary. For our present purposes, we will be particularly interested in the conclusion that the minimum value of the resistance or conductance along the imaginary axis is less than any value of resistance or conductance in the right-hand half-plane. Since the real component of the impedance is positive at all points on the real frequency axis, from 6a of Chapter VII, it consequently follows that it must also be positive throughout the right half of the p plane. Brune described this situation by the statement that a passive impedance is a positioe real function, by which he meant that the real component of Z is always positive when the real component ofp is positive. The same result can also be established by the energy function argument of Chapter VII if we write the right-hand side of equation (7-32) of that chapter as Ei, as in the preceding equation (7-25), instead of zero, so that the equation refers to the steady-state rather than to the transient condition of the network. The term Ei will, of course, be retained in the final expression (7-34) and can be interpreted as the conjugate of the driv- ing point admittance in the same way as was done in connection with (7-30). If we represent the phase angle of the impedance by 0, set p = p + ip2, and write T, F, and F for brevity to represent the sums T + Tb, etc., this allows us to write ( p2 T p2q_p 0 = tan - (9-1) Fq-p Tq-pq_p * Journalof Mathematics and Physics, M.I.T., Vol. X, Oct. 1931, pp. 191-235.  Journal of Mathematics and Physics, M.I.T., Vol. XVIII, No. 4, Sept. 1939 pp. 257-353. ----------------------------------------------------------- 172 NETWORK ANALYSIS The quantities T, F, and k' are, of course, always positive. In the right half-plane, where p is also positive, it is easily seen that the absolute value of 0 is less than, or at most equal to, the absolute value of tan - p2/p. In other words, when p lies in the right half-plane, Z must have a phase angle less than or in the limit equal to that of p itself. Evidently, there- fore, the phase angle of Z cannot reach 4-90 ø, so that the real component of Z must be positive. The fact that the minimum resistance occurs on the real frequency axis may also be used to deduce a second result. Evidently, if we subtract any resistance not greater than this minimum from the impedance function we will still have a positive resistance throughout the right half-plane. The new function can therefore have no zeros in this region. The poles of the function and the various conditions ofconjugacy are, moreover, not affected by the subtraction of a finite real constant. Since an exactly symmetrical situation is obtained if the analysis is expressed in terms of admittances rather than impodances, this allows us to state the Theorem: A passive immittance will continue to meet the conditions of physical realizability in passive networks if it is dimin- ished by any real constant as long as the real component of the resulting expression does not become negative at any real fi'equency. An immittance function will be called a minimum resistance or minimum conductance expression if its real component vanishes at some point on the real frequency axis, so that no further diminution is possible without vio- lating the passive conditions. As an example of these relations we may consider the impedance Z given by the first of equations (7-17) of Chapter VII. The corresponding Rt, also given by these equations, has a minimum at co 2 = 1.63, at which point it is equal to 0.105. The impedance will consequently continue to satisfy the passive conditions if we subtract from it any resistance not greater than 0.105. The limiting, or minimum resistance, expression is given by 0.48p 2 + 0.69p + 1.58 Z = Z - 0.105 = 5p 2 q- 3p q- 4 (9-2) As an alternative to this procedure we may also examine the reciprocal of Z, using an admittance analysis. The real component of 1/Z reaches a minimum value of unity at co 2 = 1. The corresponding minimum con- ductance expression is 1 422 + 2p q- 2 Y =Z 1 - q-pq-2 (9-3) ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 173 This is the same as 1/Z in (7-17) in Chapter VII, as we might expect from the relation between Z and Z2indicated by Fig. 7.5 in that chapter. The principle of resistance or conductance reduction has been introduced here primarily as a step in the development of Brune's method of synthe- sizing networks. It is, however, of occasional value also in actual design problems. As an example, let it be supposed that an interstage network has been designed without regard to plate or grid-leak conductance and that we wish to take account of these quantities. If the interstage design includes a parallel resistance of sufficiently low value there is, of course, no difficulty in making the appropriate changes. The preceding theorem shows, however, that if the minimum conductance of the network is sufficiently large the impedance can always be represented with such a branch, even if the original structure of the network is quite different. In this example, of course, the equivalent circuit, While it may be physically realizable in a theoretical sense, may not be found in a configuration which lends itself readily to actual construction. 9.3. Reactance Reduction of Passive Zmpedances The preceding section has shown that the real component of a passive immittance can be varied by a constant amount, within certain. limits, without affecting the passive character of the complete expression. Similar possibilities may also exist for the imaginary component except that the change, instead of being a constant, is a prescribed function of frequency. These possibilities are associated with the presence of zeros and poles of impedance on the real frequency axis. It will be recalled that zeros and poles of impedance at real frequencies are always simple and occur in plus and minus pairs. Let 4-p0 represent such a pair of zeros or poles. If P0 represents a pole we can write Z = Z/(p - P0), where Z  has no pole at P0 and can consequently be expanded in a Taylor's series about this point. We can therefore write 1 Z - -- [_d o --{- `41(P -- Po) '-[- `4(P - Po) 2 '- '' '] P - P0 `40 - + `4 + `4v.(p - P0) + '" (94) P - Po If P0 represents a zero, we have, similarly, Z = (p - po)[Bo + Bi(P - P0) -1- Bsp - po)  + .. 4. (9-5) When p is very close to P0, the terms do/(p - Po) and Bo(p - Po) in these expressions are much more important than any others. Since P - Po is a positive imaginary for values ofp on the imaginary axis on one side of P0 and a negative imaginary for values on the other side, both .40 ----------------------------------------------------------- 174 NETWORK ANALYSIS CaAp. 9 and B0 must be real quantities if the impedance is not to have a negative resistance component for frequencies sufficiently close to P0 on one side or the other. Both `4o and Bo must also be positive. This is immediately apparent if we make use of (9-I). Unless .40 and B0 are positive the impedance will be approximately a negative resistance, with a phase angle certainly greater than 4-90 ø, for values of p sufficiently close to Po in the right-hand half- plane. The fact that .4o and B0 must be positive can also be shown directly from a Nyquist stability diagram. In applying this method it must be recalled that the integration contour assumed in preparing the Fro. 9.1 diagram may include small indentations away from the real frequency axis, as shown by Fig. 8.7 of Chapter VIII, to avoid singu- larities of the integrand on that axis. Since the integrand in the Nyquist method is the logarithmic derivative of the impedance function such an indentation must be made for each zero and pole of impedance on the real axis. If we consider in particular a pole, the resulting Nyquist diagram may be studied by means of Fig. 9.1. The solid line shows the behavior of the function on the small indentation and adjacent parts of the real frequency path when .4o is supposed to be positive, while the broken line gives a similar plot when .4o is negative. The dotted line indicates the plot corresponding to other parts of the real frequency axis. The exact shape here is unimportant, but this part of the complete plot must of course link up with either the solid or the broken line portions without leaving the right half-plane. It is clear that if we choose the broken line path the complete plot will encircle the origin, so that the stability condition will be violated. From these facts, it is easy to show that a zero or pole at real frequencies can always be represented as an ordinary resonant or anti-resonant network. Corresponding to (9-4), for example, there must be a similar expansion about the conjugate pole at -P0. While the two expansions will not, in general, be identical, it is easy to see that the constant ,40, at least, will be the same in both. The sum of the two terms representing the poles is, however, 2.4op/(p 2 - p). This can be identified with œD/(p  + D/L), which represents the impedance of an anti-resonant network, provided we have 2.40 D = 2.40;. L = p (9-6) ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS' 175 Since z/0 is positive and p0  is negative, both elements must be positive. In the special case when the pole occurs at zero or infinity the anti-resonant network reduces to a condenser or an inductance. In an exactly similar way, of course, we can represent zeros of impedance, or poles of admittance, by series resonant circuits in parallel with the rest of the network. An impedance all of whose real frequency poles have been deleted in this manner will be called a minimum reactance network, while if the zeros have been removed it will be called a minimum susceptance structure. In either case the removed branch is, of course, a pure imaginary on the real frequency axis.* The resistance of the remainder at real frequencies is therefore still positive and we need merely repeat the argument of the preceding section to show that the remainder must consequently meet all the passive conditions. This establishes the Theorem: A passive impedance or admittance will continue to meet the conditions of physical realizability in passive networks if it is diminished by the teacrance or susceptance correspond- ing to its real frequency poles. As an example of this process we may consider the impedance function Z= 2P2 q-P + I (9-7) pa -F p q- p q- l' The exlSression meets all the requirements of physical realizability in a passive network. There ave three poles, one at p = - 1 and the remaining two at p = q-i. The latter pair, since they occur on the real frequency axis, indicate that Z is not a minimum reactance function. In order to extract these poles, it is convenient to begin by noticing that Z/o in (9-4) and (9-6) must satisfy the relation ,40 = lim [(p - p0)Z] = lim r ps ---fiø2Z 1. (9-8) L 2/,0 * In the right half-plane, however, it has a positive real component, as we can see by inspection of the branch immittance expression. This is of interest in connection with the analysis of the preceding section, which was based upon the assumption that the immittance had no singularities on the real frequency axis and the consequent fact that its real component attains smaller values on the axis than it does anywhere in the interior of the right half-plane. It is clear that the argument holds afortiori if we begin with a non-minimum teacrance or susceptance expression. ----------------------------------------------------------- 176 NETWORK ANALYSIS CuAP. 9 In the present instance, where po  = -1, this gives lira[P2+ 1 2p 2+p+ 1 ] - - (9-9) Using this in (9-6), we find that the elements of the corresponding anti- resonant circuit are given by L -- D -- 1. If the impedance of these com- ponents is written separately the complete expression corresponding to (9-7) appears as Z- P 1 + + (9-1o) The first term on the right-hand side represents the anti-resonant network, while the second is the minimum teacrance part  of the complete expression. The second is readily identified with the impedance of a re- sistance in parallel with a condenser so that the Fro. 9.2 complete structure is that shown by Fig. 9.2. In amplifier design, the principle of teacrance or susceptance reduction is chiefly useful as a guide to available interstage configurations. An example is skown by Figs. 9.3 and 9.4. In Fig. 9.3 we observe that, aside Fro. 9.3 Fro. 9.4 from the parasitic capacity, the interstage impedance must have a pole at infinity, since both branches contain series inductances. It is conse- quently possible to represent this portion of the network as a single induc- tance in series with some other physical impedance. This is illustrated by Fig. 9.4, the residual network after the pole at infinity is extracted being represented by the box. The exact configuration of the residual portion will depend somewhat upon the numerical values of the elements in the original structure, but one possibility is indicated by the broken lines. ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 177 Although the two structures are theoretically equivalent the structure of Fig. 9.4 has the practical advantage that it tends to minimize the effects of element capacities to ground. At high frequencies, in Fig. 9.4, we have, in effect, to reckon with the ground capacity of only the single series coil, so that the introduction of interstage elements produces only a slight increase in the total interstage capacity.  As a second example, let it be supposed that the structure of Fig. 9.2, in association with the usual _L  parasitic capacity, represents an interstage impedance. This particular configuration is a convenient one for many design purposes. From a theoretical point of -I- view, however, it is obviously inefficient, since it in- cludes a capacity path through the network at high _ frequencies. This corresponds analytically to a pole -' of admittance at infinity. As the previous discussion shows, the pole can be split out as a separate shunt Fro. 9.5 capacity which can be absorbed as part of the normal parasitic capacity of the interstage, thus allowing the same impedance characteristics to be duplicated at a higher level. The decomposition is effected by writing the admittance corresponding to (9-7) as p 1 p2+p +2 (9-11) Y; 2 + 22p2 + p+1 ' The network corresponding to this expression is shown by Fig. 9.5. The first term in (9-I1) is, of course, represented by the parallel capacity. The method by which a representation of the second term is secured may be less obvious, but it will be explained in a later section. The fact that this part of the network requires mutual inductance for an exact representation is unfortunate, but for most purposes a sufficiently good approximation can be obtained with a network of the same configuration without the mutual coupling. 9.4. Properties of Networks of Pure Reactances* In later chapters it will be shown that minimum resistance and mini- mum teacrance networks have the special property that in each case one of the components of the impedance is fully determined as soon as the other is known. Thus, for example, if a network is of minimum teacrance type its teacrance characteristic can be computed from its resistance characteris- tic. The only possibilities of changing the reactance without affecting the "* The material of this section is based upon the classic paper by R. M. Foster, "A Reactance Theorem," B.S.T.J. April 1924, pp. 259-267. ----------------------------------------------------------- 178 NETWORK ANALYSIS CH,p. 9 resistance lie in the addition of a pure reactance network. Particular interest thus attaches to the properties of purely reactive impedances. As we have already seen, any zeros or poles of impedance on the real frequency axis can be represented by resonant or anti-resonant networks. Conversely, if the network is composed exclusively of pure reactances, this is the only possible location for the zeros and poles. The proof depends merely on the fact that the reactive component of any physical network must always be an odd function of frequency. If the network is composed of pure reactances, therefore, the impedance as a whole must be an odd function. It follows that if the reactive network has a zero or pole at any point P0 in the complex plane, there must be a corresponding zero or 9.6 pole at --P0. Since we can never have zeros or poles in the interior of the right half-plane, how- ever, this means that no zeros or poles can be found in the interior of the left half-plane either. The zeros and poles must consequently be confined to the imaginary axis. They must, of course, then be simple and occur in positive and negative pairs. One more fact will complete the mathematical specification of the impedance of a network of pure reactances. In a general network there is no particular restriction on the relative number or arrangement of the real frequency zeros and poles. In a purely reactive network, on the other hand, the number of zeros must be the same as the number of poles if we include the extreme zeros and poles at zero and infinite frequency, and zeros and poles occur alternately along the real frequency axis. To show why this must be so, let it be supposed, on the contrary, that two zeros were to occur consecutively. The teacrance characteristic in their neighborhood would evidently take some such shape as that indicated by the broken or solid lines of Fig. 9.6. In either case, the derivative of the reactance characteristic is positive at one zero and negative at the other. In equa- tion (9-5), however, B0 can evidently be identified with the derivative at the corresponding zero. If all the B0's are to be positive, therefore, the situation shown in Fig. 9.6 is not possible. A similar argument can be used to show that two poles cannot occur in succession. With this background, we can write a general formula for the imvedance of any reactive network in the following form (p2 _ p)(p2 _ p)..._ (p_____ p}) . (9-12) z -- kp (? (v _ In this expression, the quantity k is a positive real constant, while p, p, etc., are negative real quantities. Each of the factors (p2 _ p) thus represents a pair of zeros or poles at positive and negative real freauencies. ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 179 We can take care of the fact that zeros and poles must alternate by impos- ing the condition that _ > ... >_ > >_ 0. As equation (9-12) is written, the impedance is zero at zero frequency and infinite at infinite frequency, which means that there is an inductive path but no capacitative path through the network. Evidently either a zero or pole must be found at both zero and infinite frequency, but there is no particular reason in general why either point should be one thing rather than the other. We can therefore classify reactive networks'into L-L, L-C, C-L, and C-C forms, depending upon the types of elements which their impedances approximate at these frequencies. For example, equation (9-12) as it stands represents an L-L network. In order to take care of the other cases, we shall suppose that Pt may assume the special value zero iii /// Fro. 9.7 if we wish to represent a network whose reactance is similar to that of a capacity at zero frequency, and that the last factor p2 _ p} may be omitted in order to represent networks which behave like a capcity at infinite fre- quency. A sketch of a typical characteristic corresponding to (9-12) is shown by Fig. 9.7, the modifications necessary to represent other types of networks being indicated roughly by the broken lines. Granted any such general formula as (9-12), a corresponding physical network can be obtained either by representing the poles by anti-resonant networks in series or by representing the zeros by resonant networks in parallel, following the methods already described. The only change arises from the fact that if the structure is composed of pure reactances the repre- sentation of the real frequency zeros or poles gives the complete network. There is no residual "minimum reactance" or "minimum susceptance" network requiring some other form of representation. Thus if we expand in terms of impedance poles the resulting structure takes the general form shown by Fig. 9.8, while if the expansion is taken with respect to the imped- ----------------------------------------------------------- 180 NETWORK ANALYSTS C.AP. 9 ance zeros the result is of the form shown by Fig. 9.9. In Fig. 9.8 the final series inductance and capacity represent poles at infinite and zero frequency, Fro. 9.8 respectively, so that the structure is of the C-L type, in the notation of the previous paragraph. In Fig. 9.9, on the other hand, the parallel inductance Fro. 9.9 work corresponding to the be combined to give cj- prj= p2_dz ß PJ' P The corresponding formula for the elements in Fig. 9.9 is and capacity indicate a structure of the L-C type. In both cases, however, the networks can be modified to suit other conditions by omitting either or both of the odd elements. With either configuration, the element values can be computed by the methods already dis- cussed in connection with (9-4) and (9-6). In the structure of Fig. 9.8, if Lj. and C i rep- resent the elements of the anti-resonant net- poles at -bpi , these equations can conveniently (9-13) (9-14) In most circumstances, the choice between the two configurations depends upon which one leads to more convenient element values. In general, we find that the configuration of Fig. 9.9 is the one which requires the larger inductances and smaller capacities. o T T T T FIe. 9.10 Fa. 9.11 Reactive networks can, of course, be built also in a variety of other configurations. Two fairly obvious possibilities are given by Figs. 9.10 and 9.11. In order to represent any reactance network in the form shown by Fig. 9.10, for example, we may begin by representing the network in the ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 181 form shown by Fig. 9.8, and identifying the first series coil in that structure with the first series coil in Fig. 9.10. The remainder of the reactive net- work can then be converted to the form shown by Fig. 9.9, and the shunt condenser identified with the first shunt condenser of Fig. 9.10. By repeating the process the complete circuit is built up. For general engineering purposes, the most significant aspect of net- works of pure reactances is perhaps the fact that the characteristics which they may exhibit exist in such limited variety. Over the complete positive and negative real frequency axis a simple inductance or capacity sweeps once with positive slope through all values between - o0 and + oo. The most general reactive network characteristic, as illustrated by Fig. 9.7, is merely the same characteristic repeated several times on a distorted fre- quency scale. The distortion of the frequency scale always leads to a reactance charac- teristic whose slope is greater than that of a simple inductance or capacity. This can be shown most easily by returning to the energy analysis given in Chapter VII. Thus in the special case of a purely reactive structure equation (7-31) of that chapter reduces to 1 . = 4co(T- ?). (9-15) For a purely reactive network, however, it is also possible to establish the relation dX - 4Xe(T q- //), (9-16) where, as in (9-15), T and ? are evaluated on the assumption that the network is energized by a voltage of unit maximum amplitude. If the second of these relations is divided by the first the result is [xl , (9-17) dco co T-V- co where the equality sign holds, of course, only if the network consists exclusively of inductances or exclusively of capacities. This establishes the Theorem: The slope of the teacrance characteristic of a general reac- tive network at any frequency is always greater than that of a simple inductance or capacity having the same teacrance at the given frequency. These relations are illustrated by Fig. 9.12 It is to be noticed that (9 15) and (9-16) together determine T and k' from X and dX/dco. This is of some interest in connection with high power ----------------------------------------------------------- 182 NETWORK ANALYSIS CHAP. 9 circuits, such as radio transmitters, where the cost of the elements is largely determined by their kva ratings. It is clear that the total kva rating of the complete network, for any single frequency signal, depends only upon its external characteristics and is independent of its configuration. ,   J   Inducl-ance Fro. 9.12 9.5. Brune's Method of Deweloping a General Passive Impedance The two processes of resistance reduction and teacrance reduction were used by Brune to show that any impedance expression meeting the general passive conditions could actually be represented by a physical network. Brune's method of finding the network is a step-by-step one. The succes- sive branches are found one at a time until the last branch is a pure resist- ance. The process begins by the representation of the impedance poles at real Fro. 9.13 frequencies as a number of anti-resonant net- works in series, in the manner just described. After all the poles of impedance have been removed, the zeros of impedance, or poles of admittance, of the reduced impedance are similarly treated. There will result then as the next few elements of the network a num- ber of resonant circuits in parallel. After the zeros of impedance have been removed we may find new poles which must be taken out, and then again new zeros, and so on. We will thus secure as the first part of the expansion a ladder network of the general type shown by Fig. 9.13. Since each stage in the representation of the zeros and poles decreases the degree of the rational function representing the impedance, it is obvious that the process will either succeed in giving us the complete impedance or else that we must eventually reach a stage at which there are neither zeros nor poles on the real frequency axis. Suppose that Z of Fig. 9.14 repre- sents the impedance after it is no longer possible to subtract purely reactive ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 183 elements from the circuit either in series or in parallel. In order to continue the analysis we artificially introduce a zero along the real frequency axis so that reactive elements in shunt can again be subtracted. The first step in this process is to subtract from the impedance a series resistance (R of Fig. 9.14) equal to the minimum value of the resistance along the axis. This leaves the new impedance Z2, which at some point along the axis is a pure teacrance. The teacrance at this point is eliminated by subtracting a suitable element. An inductance rather than a capacity is chosen for this purpose since we will later require a negative mutual impedance, which can be obtained physically with inductances but not with capacities, to con- struct the network. Suppose first that the required inductance is negative, as shown by -œt on Fig. 9.14. Subtracting it leaves the impedance Za, which must be zero at the frequency at which the resistance component of Z1 was a minimum. Z1.-q, Z2...', Z$.-',iL' Z4  Z 5 Fro. 9.14 We can therefore introduce a corresponding resonant circuit L-D in shunt. This leaves Z4. Now the impedance Z had no pole at infinity, but the introduction of -Li gave us a pole at infinity in Zs and obviously Z must still have such a pole. Let this be removed by the introduction of the element L, leaving the impedance Z5, which again has neither poles nor zeros along the imaginary axis. It is easily shown that if neither Z nor Z, is to have a pole at infinity, the inductances -L, L, and La must represent the equivalent T of a transGrmer having finite inductance and perfect coupling. By using such a transformer, therefore, we can provide the nega- tive inductance -L which is required. If L1 is positive the process is exactly the same except that now La turns out to be negative. It is easily seen that if the original impedance met the requirements of physical realizability, each of the successive new impedances will also meet these requirements.* Zs therefore meets the same conditions as Z1, except that as a rational function it is of somewhat lower degree. By repeating the process, therefore, we will eventually construct the complete network. As an example of this process we may consider the representation of twice * There is a temporary departure from the strict requirements if L is positive. This is amended, however, as soon as La is added. ----------------------------------------------------------- 184 NETWORK ANALYSIS CHAr. 9 the residual admittance (p2 + p + 2)/2(2p2 + p + 1) which appears in (9-11). This expression is already of minimum susceptance and minimum reactance type, so that we can begin immediately with the stage in the expansion represented by Fig. 9.14. Upon identifying the reciprocal of the admittance with the Z1 of Fig. 9.14, we readily find that the corresponding resistance is (1 - 002)  R -= 2 w, _ 3c0 + ,' (9-18) This reaches its minimum value, zero, at o s = 1. In the present instance, therefore, it is not necessary to consider the resistance reduction sym ' O. lO5 -O. IG5 0.338 oT  o--'XAA/V o.3z 9.15 Fro. 9.16 bolized by R in Fig. 9.14. At 00 = 1 we find that Z = i. The inductance represented by -L in Fig. 9.14 is therefore +1. After the subtraction of this inductance, Za is given by 2p 2 + p + 1 (1 -- p)(1 + p2) (9-19) Za- p2q-p+2 P- p2+p+2 ' The factor (1 q-p2) in (9-19) represents the zero corresponding to the resonance of Ls and D in Fig. 9.14. With the help of (9-14) these ele- ments can be evaluated as L2 = D = 1. When their contribution is sub- tracted from Z we obtain, finally, p 1 Z4 = - 5 + ' (9-20) The term -p/2 evidently represents the inductance La in Fig. 9.14. It is, of course, negative, since the first inductance was positive. The term « represents the terminating impedance Zs. In this example it is necessary to carry the process, illustrated by Fig. 9.14 through only one stage in order to reach a terminating impedance which is a constant resistance because the original impedance expression was of only the second degree. The complete structure is shown by Fig. 9.15. Since Z1 in (7-17) of Chapter VII has been used frequently for illustra- tive purposes it is convenient to adopt this expression as a second example of Brune's expansion. The situation is essentially the same as that just examined except for the fact that the present impedance is not initiall 7 in ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 185 minimum resistance form. We have already found in connection with (9-2), however, that the minimum resistance of Z is 0.105. This gives the R of Fig. 9.14 and the remainder of the network follows readily. The complete structure is shown by Fig. 9.16. As an alternative, we may begin with a conductance reduction of the impedance, following the analysis given in connection with (9-3). Since a minimum conductance network is also minimum resistance, this is an equally legitimate method of going from the initial expression to the stage represented by Z2 in Fig. 9.14. In the present instance it yields the struc- ture shown by Fig. 9.17. For practical purposes the chief objections to Bmne's method are the facts that it uses mutual inductance and that a very considerable amount of labor is required to compute the elements one by one. On the other hand, the technique demands a knowledge of the impedance only at real frequencies, so that it has some advantage in the simulation of impedances which are specified only by curves. -o.s i ..,vv . 1 f œ z__ Fro. 9.17 Fro. 9.18 9.6. Negative Resistances The discussion thus far has considered only impedance functions meet~ ing the passive requirements. The corresponding physical structures, of course, then consist of combinations of the three passive elements, resistance, inductance, and capacity. To consider more general cases we need one additional building block. The additional unit can conveniently be taken as a negative resistance, since such an element expresses most distinctively the difference between a passive network and a general circuit, containing a source of power. A negative resistance can be obtained in a variety of rather familiar ways. No attempt will be made to consider this field in any detail here. Broadly, one possibility rests upon the difference between the active and passive impedances of feedback circuits as expressed, for example, By (5-3) or (5-4) of Chapter V. Evidently, a negative resistance can be obtained from any feedback circuit of pure resistances if the circuit is so arranged that the two return differences F(0) and F(oo) in these equations are of opposite sign. An example is shown by Fig. 9.18. If we assume that the ----------------------------------------------------------- 186 NETWORK ANALYSIS Ca.P. 9 vacuum tubes are ideal the passive impedance at the input terminals is Rs(Rs + R4)/(R + Rs + R4). The return difference F(0) reduces to unity, since the return ratio vanishes when the input terminals are short- circuited. The return ratio with the input terminals opened is negative, corresponding to the fact that with the two stages indicated in Fig. 9.18 there is no net phase reversal in the tubes, and is readily evaluated as -GG2RRRa/(R1 q- Ra q- R4), where G, and Gm are the transconduct- ances of the tubes. Substitution in (5-3) of Chapter V therefore gives Rs(Rs + R4) 1 Z= R q- Rs q- R4 G,G,2RRRa l-- R1 q- RS q- R R1 (Rs + R4) = Rt + R3 + R - G,,,,G,2RiRRs (9-21) Z will evidently be a negative resistance if the R's are chosen appropriately. For example, if Rs and R3, which are introduced only to make the idealiza- tion of the circuit appear somewhat less forbidding, are made infinite, Z will always be negative, as evidenced by the expression -1 Z = G,G,,,,R (9-22) Negative resistances can also be secured through a variety of other devices, such as the dynatron or an arc- discharge. An illustrative characteristic for J= the dynatron is shown by the solid line in  c_____ z Fig. 9.19. The ratio eft, representing the resistance to any steady voltage e, is always positive. Near the point C, however, the // ........ slope of the characteristic is negative. If  the impressed e is taken as the sum of a d-c o ½ component and a small superimposed a-c Fro. 9.19 component, as in the analysis of the charac- teristics of a vacuum tube, the effective resist- ance to the a-c component, therefore, will be negative when the operating point is near C. Negative resistances are introduced here merely as convenient devices to explore the purely mathematical implications of the general set of requirements on driving point immittance functions laid down in Chapter VII. For this purpose they will be regarded as idealized elements of exactly the same type as positive resistances. Too much emphasis, how- ever, cannot be laid upon the fact that an actual negative resistance is a ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 187 much more complicated device, subject to many restrictions which are ignored in such an idealization. Depending upon the circuit to which the negative resistance is connected, and perhaps even upon the past history of the circuit, this may lead on occasion to marked departures from the behavior which would be computed from an idealized analysis. For example, Fig. 9.19 represents a characteristic which we might expect to trace physically if the circuit were energized by a battery of controllable voltage and zero internal impedance. Suppose, on the other hand, that the device .is supplied by means of a much higher voltage operating through a high external impedance. Then, in effect, we are controlling the current, rather than the voltage, at the negative resistance terminals. Itis evi- dently possible that the actual characteristic may skip from one branch to +R +R -R Fro. 9.20 Fro. 9.21 the other, as suggested by the broken lines//D and BE, in such a way as to avoid the negative slope part of the nominal characteristic entirely. If the external impedance includes reactive elements the skip may depend upon transient effects or, in other words, upon the past history of the circuit and the rate at which the energizing source is varied. Since some external impedance is required in order to segregate the a-c and d-c components these considerations cannot be avoided entirely in any application. In a negative resistance device which relies upon vacuum tubes compli- cating factors are introduced principally by the unavoidable parasitic capacities of the tubes. These will evidently convert the negative resist- ance into an ordinary passive impedance at sufficiently high frequencies. The change may be unimportant in some applications, but in others it may produce singing. Which type of behavior is actually followed will depend, in general, upon both the external circuit and the type of feedback used to produce the negative resistance. If we do postulate ideal negative resistance elements it follows immedi- ately that negative elements of other types are also available. This can be shown most easily by reference to the well-known circuits shown in Figs. 9.20 and 9.21. A simple computation shows that the input imped- ance Z is given in either case by z =  (9-23) z2 ----------------------------------------------------------- 188 NETWORK ANALYSIS C.^p. 9 Thus any negative impedance, including as special cases a negative capacity and a negative inductance, can be produced by terminating the T in the positive inverse of the required impedance. 9.7. Representation of General Driving Point,]remittance Functions The requirements of the general list in Chap'ter VII which are relevant to driving point immittance functions are 1, 2, 3, 4, 5b, and 6a. Of these, the first four must be satisfied by any immittance function. This suggests that possible functions may be divided into three general classes, depending upon whether they meet the first four requirements alone, the first four and 5b, or all six requirements.* With the addition of two rather obvious sub- classes the scheme is Ia. Functions which have no poles in the right half-plane and whose real components are positive (or zero) at all points of the real frequency axis. lb. lq'unctions which have no poles in the right half-plane and whose real components are negative (or zero) at all points of the real frequency axis. II. Functions which have no poles in the right half-plane and whose real components are positive on some parts of the real frequency axis and nega- tive on others. Ilia. Impedance functions in which some poles occur in the right half- plane. IIIb. Admittance functions in which some poles occur in the right half- plane. The Class Ia is, of course, the class of ordinary passive immittances. The functions in Ib are exactly the negatives of ordinary passive functions. They will be called negative immittances. The more general functions described in the later classes will be called general or active immittances. The conception of a negative immittance is introduced here as a convenient theoretical abstraction. Such a function can evidently be obtained, under idealized circumstances, by the methods suggested by Figs. 9.20 and 9.21. In view of the limitations of physical negative resistance devices, however, it is probable that any actual function would belong to one of the more general Classes II or III. Impedance functions and admittance functions have been written sepa- rately in Class III to emphasize the fact that the driving source for an * The apparent fourth class, consisting of functions which meet the first four requirements and 6a but fail to satisfy 5b, cannot exist. If 5b is not satisfied, so that there are poles in the right half-plane, the Nyquist plot of the function must encircle the origin, which is inconsistent with 6a. The specification of both the sign of the real component and the location of the poles in some of the items of the subsequent list is introduced merely for clarity. ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 189 impedance function is a' voltage generator of zero internal impedance,.while for an admittance function it is a current generator of infinite internal impedance. Networks corresponding to functions of the two types may then be described as short-circuit sta]e and open-circuit stale respectively. If the functions belong to Class III the corresponding networks will not, of course, remain stable if the energizing sources are interchanged. In the other classes, which have zeros and poles confined to the same half-plane, these distinctions are unnecessary. If the active immittance is obtained from a feedback circuit we can frequently de- termine whether it is open-circuit stable or short-circuit stable by inspection. For example, any immittance measured in series with the feedback loop, as at .4.4  or BB  in Fig. 9.22, must be open- circuit stable, while any immittance measured across the loop, as at CC  or Fro. 9.22 DD , must be short-circuit stable, since in either case the introduction of the appropriate generator impedance will interrupt the feedback. For purposes of analytic description, the construction of active imped~ ances is most easily treated by an extension of the processes of resistance and conductance reduction described earlier in the chapter. In discussing passive immittances these processes were limited by the fact that the real component of a passive immittance could not become negative. With the addition of a negative resistance to the normal passive elements this limi- tation is unnecessary and we are led at once to a representation of active immittances by a simple extension of the methods used for passive circuits. To exemplify this process, let it be assumed that the function to be repre- sented is an impedance of Class IIIa. It will also be supposed that none of the zeros of the impedance occur exactly on the real frequency axis.* * If zeros on the real axis do occur, the corresponding residues of 1/Z must be positive real, negative real, or complex. Zeros corresponding to positive real residues can be represented separately by resonant circuits in the manner already described for passive networks. The other possibilities pose a more dicult problem. They may exist theoretically in, for example, a feedback amplifier which is on the point of singing. Consideration of these possib!lities .Will, however, be avoided here on the ground, mentioned in Chapter VII, fhat a physical circuit exhibiting such zeros would be excessively non-linear. Negative real residues can, of course, be represented theoreti- cally by negative reactance elements but the consideration of this possibility is especially unrealistic because, in addition to the question of non-linearity, a structure exhibiting such zeros must necessarily become unstable if it is fed through a generator circuit including the slightest trace of dissipation. ----------------------------------------------------------- 190 NETWORK ANALYSIS CaAP. 9 The reciprocal, Y, of the specified impedance will consequently be-analytic in the right half-plane including its boundary, the real frequency axis. Let G and -G2 represent the maximum and minimum values of the real com- ponent of Y on the axis. In accordance with the preceding theorems these will also be the maximum and minimum values of the real component with respect to the complete right half-plane. If we rewrite the admittance as -G2 + (Y + G2), therefore, the term ¾ + G2 will have a positive real component throughout the right half-plane. In Brune's language it is a "positive real" function and can be represented by an ordinary passive impedance. The first term -G2 represents, of course, a parallel negative E a b a Fro. 9.23 Fro. 9.24 resistance. The combination is shown by Fig. 9.23a. Similarly, if we write Y as G1 -t- (Y -- Gt) the complete impedance appears as a positive resist- ance in parallel with a negative impedance. This is illustrated by Fig. 9.23b. If we begin with a function of Class IIIb the analysis is essen- tially the same, except that we are now led to a series combination of a positive or negative resistance and a negative or positive impedance, as shown by Figs. 9.24a and 9.24b. The results can be summarized as the Theorem: If an active network is stable with an energizing source of zero internal impedance, the impedance facing the source can be represented either by a negative resistance in paral- lel with an ordinary passive network or by a positive resist- ance in parallel with the negative of a passive network. If the network is stable with an energizing source of infinite internal impedance, the network impedance can be repre- sented either by a negative resistance in series with a passive network or by a positive resistance in series with the nega- tive of a passive network. This discussion has been advanced specifically for functions of Class III. It is apparent, however, that it is equally valid for functions of Classes I and II. We need only recognize that functions of these classes are both open-circuit stable and short-circuit stable so that they can be represented in any one of the four ways shown by Figs. 9.23 and 9.24. It may also be ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 191 interesting to notice that the methods of representation can be combined to give still other possible configurations. For example, since the negative impedance in Fig. 9.23b is open-circuit stable as well as short-circuit stable it can itself be represented in the form shown by Fig. 9.24a, leading to a representation of the original expression by a positive impedance and an L of positive and negative resistances. As a matter of emphasis it may be desirable to say once more that the circuits of Figs. 9.23 and 9.24 do not necessarily constitute either a unique way or a physically desirable way of constructing active impedances. They are introduced merely as a convenient method of expressing the physical significance of the conditions on active and passive driving point immit- tances laid down in Chapter VII. It will be seen that the difference between an active and a passive driving point immittance amounts essen- tially to a single negative resistance, appropriately located. There is a close analogy between this result and a result derived later for the distinction between active and passive transfer immittances. 9.8. Combinations of .4ctive [ropedances In dealing with passive circuits we are accustomed to thinking of the individual passive impedances as units which can be combined with one another and associated with a driving generator in any way we like. What- ever arrangement is chosen, the circuit as a whole will remain passive, and therefore stable. In active circuits, on the other hand, no such freedom is possible. Impedances which are stable for one energizing source may become unstable if the source is altered and two impedances which are individually stable for a given source may become unstable when they are added together, even if the source itself is unchanged. In dealing with active circuits, therefore, it is necessary to study the stability of the structure in terms of the complete impedance or admittance facing the. current or voltage source, including the self-impedance or self-admittance of the source itself. This is evidently a grave restriction. It affects both the freedom with which the active network itself can be designed and the freedom with which the energizing source can be chosen. The latter is perhaps particularly important. The analysis thus far has assumed that the self-impedance of the source would be either zero or infinite, whereas most practical sources have a finite, non-zero, self-impedance. The problem of relaxing these restrictions will be attacked here through a consideration of the open-circuit or short-circuit stability of a combination of two immittances in series or parallel, as illustrated by Figs. 9.25 and 9.26. Each of the two immit- tances can be regarded as an active structure if we wish, or one of them can be taken as a representation of the actual self-immittance of a physical generator. ----------------------------------------------------------- 192 NETWORK ANALYSIS C,Ap. 9 The two situations illustrated by Fig. 9.25 can be dismissed easily. If an impedance is to be short-circuit stable, as in Fig. 9.25a, none of its zeros can lie in the right half-plane. But since the zeros of an impedance obtained from a number of branches in parallel are the same as the zeros of the separate branches, each of the individual branch impedances must be Fro. 9.25 Fro. 9.26 similarly restricted. Correspondingly, the zeros of admittance, or poles of impedance, in the structure of Fig. 9.25b are the same as the admittance zeros of the component structures and must be excluded from the right half-plane if the complete structure is to be stable. We therefore have the obvious Theorem: A parallel combination of impedances will be short-circuit stable if and only if,all the individual impedances are short- circuit stable. Similarly, a series combination will be open- circuit stable if and only if all the individual impedances are open-circuit stable. The combinations illustrated by Fig. 9.26 present a more difficult prob- lem. The discussion here will present only a few elementary rules which may be useful in some situations. To give the problem a physical context, we may suppose that Z and Y in Figs. 9.26a and 9.26b are respectively short-circuit stable and open-circuit stable structures and that Z2 and Yv, represent allowances for the self-impedance or admittance of the actual generator. The question which will be attacked is that of estimating under what circumstances Z2 and Y2 can be introduced without upsetting the stability of the circuit. If Z= and Y= are real constants their effect on the stability of structure is most easily determined from an inspection of the Nyquist diagram of the original Z or Y. The addition of a constant Zs or Y2 is equivalent to a lateral translation of the whole diagram. It is clear that the lateral trans- lation will not affect the stability of the circuit as long as it is not large enough to carry any of the points of intersection between the Nyquist path and the horizontal axis from one side of the origin to the other. This leads to the ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 193 Theorem: The series combination of a short-circuk stable impedance and a positive or negative resistance is itself a short-circuit stable impedance if the addition of the resistance leaves the sign of the real component of the impedance unchanged at every point on the real frequency axis at which the imagi- nary component of the impedance vanishes. Similar17, an open-circuit stable structure will remain open-circuit stable when combined in parallel with a positive or negative re- sistance for the same condition on the real and imaginary components of the initial and final admittance. If Z2 or ¾2 are functions of frequency rather than real constants the problem is more difficult, but it is still possible to show that they will not affect the stability of the circuit if they meet certain conditions. The situation can be expressed by the Theorem: The series combination of a short-circuit stable impedance Z and an open-circuit stable impedance Z2will be short- circuit stable if I Z 1 I > ] Z2 I at all points on the real fre- quency axis. Similarly, the parallel combination of an open-circuit stable admittance Y and a short-circuit stable admittance Y2 will be open-circuit stable if] Y } > } Ys } at all real frequencies.* The wording of the theorem is not intended to imply that an immittance which is specified, for example, as short-circuit stable cannot also be open- circuit stable. The stability of the immittances for the non-specified con- ditions is a matter of indifference. The theorem is easily demonstrated by methods similar to those used for the first theorem at the end of the preceding chapter. If we consider in particular the relation between Z and Z2, for example, we can write Z + Z2 = Zl(1 +Z). (9-24) The quantity Zt q- Z can have no zeros in the right half-plane if it is to be short-circuit stable and its poles must be the same as those of Zt since Z2, being open-circuit stable, has no poles in this region. The Nyquist plot of Zt q-,Z2 must therefore encircle the origin the same number of times in the * Throughout this discussion it is assumed for simplicity that none of the zeros and poles of the various immittances occurs exactly on the real frequency axis, including infinity. The theorems are not necessarily invalid even when this assumption is vio- lated, as it might be, for example, in circuits controlled at high frequencies by parasitic capacities, but such situations evidently require careful handling. ----------------------------------------------------------- I94 NETWORK ANALYSIS CuaP. 9 same direction as the plot of Zt alone. It is evident, however, that the number of times the plot of Z1 + Z2 encircles the origin is equal to the sum of the encirclements obtained by plotting the factors Z and 1 q- (Z2/Zi), on the right-hand side of (9-24), separately. Since the plot of 1 q- (Z=/Zi) cannot encircle the origin at all under the assumed conditions, as Fig. 8.30 in the preceding chapter shows, this establishes the theorem. It is evident from the proof of the theorem that the condition ]Z [ > I Z, I or I Y I > I Y2I does not necessarily fix the actual upper limit of values which may be assumed by the added Z2 or Y=. In many circumstances the circuit will remain stable even if the condition is violated over a portion of the frequency spectrum. If we disregard one special case, however, there is a final upper limit beyond which the added immit- tance cannot go without necessarily producing instability. This is shown by the following Theorem: The series combination of a short-circuit stable impedance Zi and an open-circuit stable impedance Z= cannot be short- circuit stable when [Z ] > [Z1 [ at all points on the real frequency axis unless Z is also open-circuit stable and Z= is also short-circuit stable. Similarly, a parallel combina- tion of an open-circuit stable admittance Y and a short- circuit stable admittance Y= can be open-circuit stable when [ Y2 I > I Y I at all real frequencies only if both Y and Y are actually both short-circuit stable and open-circuit stable. The proof of this theorem is essentially similar to that of the preceding theorem. We begin by writing the total impedance as Z 1 q- Z3= Z,(1 +Z). (9-25) Under the assumed conditions the plot of the factor 1 + (Zt/Z=) cannot en- circle the origin. The total number of encirclements by the plot of Z + Z2 must therefore be the same as those by the plot of Z=. They must be in the direction appropriate for zeros since by hypothesis Z2 is open-circuit stable and has no poles in the right half-plane. Just as in the preceding theorem, however, the plot of Z1 + Z= must encircle the origin the same number of times and in the same direction as the plot of Zt if Zi + Z2 is to have no zeros in the right half-plane. This must be in the direction corre- sponding to poles since Z is short-circuit stable. Evidently the require- ments cannot be met unless neither plot actually encircles the origin at all, which is the same as saying that each of the impedances Z1 and Z2 must be both short-circuit stable and open-circuit stable. ----------------------------------------------------------- DRIVING POINT IMPEDANCE FUNCTIONS 195 The preceding theorems cover all combinations of the two impedances except those in which both impedances are open-circuit stable, but not short-circuit stable, or vice versa. A guide to this last situation is fur- nished by the Theorem: A series combination of two impedances cannot be short- circuit stable and a parallel combination of two impedances cannot be open-circuit stable, when both impedances are either short-circuit stable but not open-circuit stable or vice versa, if the absolute magnitude of either impedance is greater than that of the other at all points on the real fre- quency axis. It is assumed that the degenerate case in which the two immittances have strictly coincident poles in the right half-plane can be disregarded. The proof is similar to those of the preceding theorems. If we suppose, for example, that [ Z2 ] > [ Z I and that Z and Z2 are short-circuit stable their sum will be short-circuit stable only if the plot of Z -3- Z2 encircles the origin as many times as there are poles of Zi and Z in the right half- plane. In accordance with (9-25), however, the actual plot will encircle the origin only as many times as there are poles of Z2 alone in this region. The two conditions cannot be reconciled except for the trivial case when Z and Z2 have identical poles in the right half-plane. A curious feature of this result is the conclusion that the stability of a short-circuit stable impedance will not be disturbed by the addition of a small open-circuit stable impedance but it may be entirely upset if the added impedance, even though very small, is also short-circuit stable. Neither of the last two negative theorems applies to combinations of impedances which are both open-circuit stable and short-circuit stable. It is natural to expect that this combination is more likely to give a stable result than any other. If the two impedances are passive, for example, they can be combined in any proportion. In more general cases, however, it is still necessary to pay attention to the possible instability of the final circuit. An example is furnished by a final Theorem: If Zi and Z2 are respectively a positive impedance and a negative impedance it is always possible to find values of the positive constant multiplier X such that the series combina- tion of Z and XZ will not be short-circuit stable and their parallel combination will not be open-circuit stable unless Z1 and Z are exactly proportional to one another. The proof is obvious from a Nyquist plot of (Z + XZ2)/Z. ----------------------------------------------------------- CHAPTER X ToPics iN THE DESIGN OF IMPEDANCE FUNCTIONS 10.1. Introduction THE preceding chapter was essentially an attempt to explore the general physical significance of the list of restrictions on driving point immittance functions given in Chapter VII. The present chapter continues this dis- cussion but in a different way. The material selected consists chiefly of devices and conceptions of direct application in design work. The chapter is thus intended broadly as a resum of design methods, but its scope is limited by the fact that it includes no material not easily related to the analytic framework already established. The discussion is directed prima- rily at driving point immittance functions, but many of the results apply also to network functions of other types. Unless otherwise specified a passive network. will be assumed. Since the chapter does not contribute directly to the theoretical structure of the book as a whole it can be omitted, if necessary, especially if the reader is reasonably familiar with elementary passive network theory. If the omission is made, however, note should at least be taken of the fre- quency transformations described near the end of the chapter, since they will be used in several later discussions. 10.2. Inverse Networks The duality between the impedance and the admittance methods of analyzing a network suggests a conclusion which was mentioned briefly in Chapter I but has not otherwise been dealt with explicitly. This is the proposition that to every network there corresponds an inverse. The result arises, of course, from the fact that the requirements on physical driving point functions are the same whether we consider an impedance or an admittance. If we are dealing with a passive structure, for example, the requirement that the real component of the impedance be positive at real frequencies implies that the real component of the admittance must also be positive. Moreover, the restrictions on the zeros and poles are sym- metrical, so that the interchange of zeros and poles which occurs when an impedance is replaced by its reciprocal does not affect the satisfaction of the conditions of physical realizability. It therefore follows that if a passive impedance is physically realizable, its reciprocal is also realizable. 196 ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 197 In ordinary networks, a suitable structural form for the reciprocal imped- ance can be found by the familiar procedure exemplified by Fig. 10.1.* Each series connection is replaced by a parallel connection, and vice versa. The individual elements are found by replacing resistances by resistances, inductances by capacities, and capacities by inductances, in such a way that the product of corresponding resistances or corresponding inductances and stiffnesses is always constant. In Fig. 10.1 the constant product of corresponding impedances, including the driving point impedances, is taken as R. We can regard the type of inverse network illustrated by Fig. 10.1 as the structural inoerse of the original network. Evidently the procedure which has been suggested for finding the structural inverse is not a general one. For example, since it considers only series and parallel connections, it offers Ro Ro Ro Fro. 10.1 no means of finding the inverse of a Wheatstone bridge. A structural inverse of a bridge network can, however, still be found by an extension of the original process. The extension depends upon the consideration of the network as a geometrical diagram of lines and points by means of which the plane is divided into areas. Physically, the points represent networl junc- tions and the lines the various elements connecting them, while the areas represent closed meshes in the circuit. The process of finding the inverse network consists broadly in an interchange of areas and points. A new point is taken in each area and each such new point is connected with each new point in the neighboring areas by a branch which is the inverse of the branch separating the corresponding areas. The process is illustrated by Fig. 10.2, the new points being .4, B, C, and D. It will be seen that the inverse of the Wheatstone bridge is another bridge. * See, for example, O. J. Zobel, B.S.T.J., Jan., 1923, and July, 1928. A good :extbook reference is Guillemin, "Communication Networlds," Vol. II, p. 203. ----------------------------------------------------------- 198 NETWORK ANALYSIS C.Ap. 10 In spite of this generalization a structural inverse cannot be found for every network. No structural inverse exists, for example, for the Brune network described in the previous chapter since we cannot find the equivalent of a pair of perfectly coupled capacities to represent the reciprocal of the coupled coils in the original network. Moreover, as R. M. Foster has shown,* certain kinds of network configurations may not be representable as configurations of points, lines, and areas on a plane, in the manner assumed by the pre- ceding discussion. No structural inverse exists for such networks even when mutual inductance is ignored. Although a structural inverse is not always obtainable, the analytic argument remains valid. If we disregard the structural relationship, therefore, we can always find some network whose impedance is the recipro- D FIa. 10.2 eal of the impedance of any given network. For example, the inverse of a Brune network is another Brune network. This can be illustrated by the networks shown by Figs. 9.15 and 9.17 of the preceding chapter. The first of these corresponds to the impedance Z = (2p 2 + p + 1)/(p 2 q- p + 2). The second was developed to represent the impedance Zx of (7-17) in Chapter VII. If we remove the parallel resistance at its input, however, it represents the impedance Z2 of the same set of expressions, and satisfies the equation Z2 = «(p + p q- 2)/(2p  + p + 1). Thus if this branch is removed the two networks become inverse structures of impedance product «. This discussion has been directed, for simplicity, at passive networks. There is evidently no difficulty, however, in extending it to negative *" Geometrical Circuits of Electrical Networks," Trans. A.I.E.E., June, 1932. ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 199 impedances or to active impedances belonging to what was described as Class II in the preceding chapter. If we turn to Class III, on the other hand, the restrictions on the zeros and poles of immittance are no longer symmetrical. Only the zeros need be confined to the left half-plane. Nevertheless, the impedance functions of Class Ilia and the admittance functions of Class IIIb are evidently inverse in a certain sense. The differ- ence is merely that in going from Class IIIa to Class IIIb or vice versa, the source as well as the network itself must be reciprocated, while we have thus far assumed that the source itself would remain unaltered. If this change in the source is regarded as permissible, therefore, the general result can be summed up in the Theorem: Corresponding to any physically realizable impedance expression there is an identical physically realizable admit- tance expression, and vice versa. The transformation from one mode of expression to the other need not include the generator if the original impedance or admittance is both open-circuit stable and short-circuit stable. If active impedances are represented by combinations of passive net- works and negative resistances, as was done in the preceding chapter, the previous remarks on the structural inverse of a given network can evidently be carried over to the general case without change. The problem of finding the structural inverse of a circuit containing vacuum tubes explicitly has not been studied. 10.3. Complementary Networks In addition to the inverse of a given immittance function we can also speak of its complement. The complement may be defined by the require- ment that the sum of the original function and its complement must be a real constant. The complement will exist as a passive impedance, pro- vided we meet the requirements of the following Theorem: A passive complement can be found for any immlttance function if the prescribed function has no poles in the right half-plane or on the real frequency axis and if the sum of the prescribed function and its complement is chosen at least as great as the maximum value of the real component of the prescribed function on the real frequency axis. The proof of the theorem is omitted here, since it can readily be obtained by a repetition of the methods used in the previous chapter. If we take as an example a passive impedance the requirement means simply that the impedance must be of minimum reactance type and that the final resist- ----------------------------------------------------------- 200 NETWORK ANALYSIS CHAP. 10 ance must be at least as great as the maximum resistance of the original structure. The familiar constant resistance combinations of ordinary network theory represent simple special cases of the complementary relationship. An example is given by Fig. 10.3. L C R R L=R Fro. 10.3 10.4. Partial Fraction Expansion of a General Impedance We saw in the previous chapter that poles of impedance or admittance on the real frequency axis could be detached from the complete impedance expression and represented separately by reactive networks in series or parallel with the structure as a whole. The same process can be extended, at least formally, to the other poles of impedance or admittance also. The representation of the network impedance which is thus secured is particu- larly valuable for theoretical purposes. Its utility in practical problems is restricted by the fact that in the most important special case, that of passive circuits, it does not invariably lead to a passive network to repre- sent a passive immittance function. Even so, however, it is useful in many situations. It will simplify discussion to restrict our attention to passive circuits and to assume that the prescribed function is an impedance. Let it be supposed, then, that the poles of the impedance are represented by the points p '"Pn. In order to avoid complications in exposition, we will also assume that all the poles are simple. The procedure is essentially similar to that which was followed in connection with equation (9-8) of Chapter IX. Corresponding to any particular pole pj, we can define a quantity C$ by cj = [@ - 00-1) It is easily seen that C i is equivalent to the quantity which was called in the preceding equation (9-8). We can therefore conclude from the discussion of this equation that Ci/(p - p:) affords a representation of the pole pj. In other words, the quantity Z - Cj/(p - Pi) will have no pole at Pi' Let us suppose that all the poles are removed from the original impedance expression by the repeated application of this process. The quantity which remains then has no poles anywhere in the complex plane, and it follows from general function theoretic principles that it must be a ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 201 constant.* We can easily show that the constaPir is a real quantity or, in other words, a resistance.t If we represent it by R0 this is equivalent to saying that the impedance can be represented by the formal expansion cl G Z - -- q-  q- ... q- -- q- R0. (10-2) P -- P P -- P2 P -- P As equation (10-2) stands, it suggests that the impedance can be repre- sented by a number of networks in series, each network corresponding to one term in the expansion. Whether or not such a representation is actually possible with a passive network depends essentially on the con- stant R0. In order to represent any term in the expansion as a simple passive network, it must, of course, meet the condition that its individual resistance characteristic be positive at all real frequencies. If the individ- ual term fails to meet this condition as it stands, it may still be possible to represent it as a passive network if we can add to it a sufficiently high Fro. 10.4 Fro. 10.5 resistance, which must, of course, be subtracted from the R0 term, to satisfy the resistance condition. The essential requirement which must be satis- fied is, therefore, that R0 be large enough to allow all the constituent net- works to furnish a positive resistance at real frequencies without leaving a negative resistance in series with the structure as a whole. There is a close analogy between this result and a proposition in four-terminal network theory. As we will see later, the transmission characteristics of a general four-terminal network can always be represented by a number of simple structures in tandem provided the general level of loss in the original net- work is high enough to allow each of the constituents to furnish a positive loss at all frequencies. In order to illustrate this relationship, let us suppose that P5 is found on the negative real axis. It is then easy to show that the corresponding C s must be a real quantity. If C i is positive, the term C/(p - Pi) can be * Liouville's Theorem -- see any text on function theory. t As (10-2) indicates, the constant is equal to the resistance of the network at infinite frequency. ----------------------------------------------------------- 202 NETWORK ANALYSIS Ca^p. 10 easily identified with the parallel combination of resistance and capacity shown by Fig. 10.4. If C s is negative this representation is non-physical. By adding the resistance Ci/pi, however, the expression becomes CJP/Pi(P - Pi), which corresponds to the inductance-resistance network shown by Fig. 10.5. If the pole is complex, a more elaborate analysis is required. Complex poles, of course, occur in conjugate pairs, and the pairs must be kept to- gether if we are to secure a physical network. Let us suppose that a particular pair of conjugate poles is written as Pa + ipb. It is easily shown that the corresponding C's must also he conjugate quantities. If we repre- sent them as Ca q- iCb we can write the component impedance Zj as Ca + iC Ca - iG Z= + (10-3) Fro. 10.6 Fro. 10.7 Fio. 10.8 can be secured. When the added resistance is the least possible, the struc- ture will take the form of the last stage of a Brune network, as shown by Fig. 10.6. If the resistance component is great enough, other configura- tions are also possible. In general, they will contain one inductance, one capacity and three resistances. The particular configurations which can ß be used, however, depend upon the numerical values of the constants in (10-3). Typical circuits are illustrated by Figs. 10.7 and 10.8. Fro. 10.9 These considerations can evidently be extended to all the poles. If we adopt in particular the Brune representation of the complex poles and regard the structures of Figs. 10.4 and 10.5 as special cases of the Brune network, the complete circuit takes the form shown by Fig. 10.9. The = 2p2Cap -- (Cp= + Cp) ß In many cases, the real component of (10-3) will not be positive for all real frequencies. If we add enough resistance, however, a passive structure ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 203 corresponding parallel combination obtained from an admittance analysis is given by Fig. 10.10. As the figures show, the resistance or conductance which remains after all the poles of immittance have been represented may be either positive or negative. It will, of course, be positive if the original resistance or conductance is large enough. Since the minima of the various component resistance or conduct- ance characteristics will ordinarilyoccur at different - -- frequencies, on the other hand, we may expect o-._-o that the sum of the component characteristics will be substantially greater than zero at all points on the real frequency axis. We may therefore expect that the final branch will be negative if the original immittance approximates the limiting minimum resistance or minimum conductance type. This ' 'vvvv is, of course, a serious practical limitation. In theoretical work, however, the fact that the com- Fro. 10.10 plete immittance is exhibited as the sum of a number of very simple terms may still make the structure quite useful. For these applications, at least, we can therefore formulate the result as the Theorem: A passive immittance having no multiple poles can always be represented as the sum of a number of passive immit- tances, each of which is at most of the second degree, and a positive or negative real constant. The extension of this analysis to active impedances involves only two considerations. In the first place, if the impedance is not both short-circuit stable and open-circuit stable some of the poles either of its impedance or admittance will be found in the right half-plane. In the corresponding expansion the component networks representing these poles can still be built, but they will not be passive structures. The second consideration is the obvious one that if we are dealing with an active circuit anyway the fact that the final resistance or conductance term may be negative should be of no particular consequence. 10.5. Reconstruction of a Passioe Impedance from a Knowledge of Either Component* The discussion in the previous chapter shows that the resistance and reactance characteristics of a passive network can be varied independently * As a general reference to transformations of this sort, see Darlington, "Synthesis of Reactance 4-Poles," Journal of Matlxematics and Physics, Sept., 1939. ----------------------------------------------------------- 204 NETWORK ANALYSIS within certain limits. Thus, we can change the resistance characteristic of a network by a constant amount without changing its reactance, and we can add or subtract a reactance corresponding to poles on the real frequency axis without affecting the resistance. These are, however, the only two ways in which the two components can be varied independently. If we restrict ourselves to minimum resistance and minimum reactance networks, the resistance and reactance are uniquely related. If we know either one, we can determine the other, and therefore the impedance as a whole. Since we are considering the real and imaginary components of the impedance at real frequencies, it is simplest to write Z as a function of co rather than as a function of p. In general, of course, Z can be represented as the ratio of two polynomials in p. On the real frequency axis the even powers in each polynomial will be real quantities and the odd powers pure imaginaries. We can therefore write 4 + icoB Z(co) - C + icoD' (10-4) where .d, B, C, and D are polynomials in 0) 2 with real coefficients. If we rationalize this expression in the usual manner by multiplying the numera- tor and the denominator by C - icoD, the result becomes zlC + w2BD BC - Z(o) - C2 + co2 2+ ico C2 + co2D 2 , (10-5) The resistance is thus an even rational function of co with real coecients, while the reactance is a similar function multiplied by co. Our problem is that of finding the complete expression for Z from a knowledge of either of its components. Let it be supposed that we know the even rational function representing the resistance. We begin by expanding this expression in partial fractions in the manner described in the preceding section. Since the denominator is an even function of co, the poles must occur in positive and negative pairs. To each pole, more- over, must correspond its conjugate since the coefficients in the denomina- tor are all real quantities. The poles thus occur in sets of four symmetri- cally placed about the origin. In the special case in which poles are found on the imaginary frequency axis the sets of four may reduce to pairs. The poles might also reduce to pairs, on the face of the situation, if they occurred on the real frequency axis, but if the assumed resistance characteristic corre- sponds to a physical network, there can be no such poles. The poles of the resistance function which lie below the real frequency axis* were introduced when the numerator and denominator of the original * That is, below the real axis in the frequency plane, or to the right of the imagi- nary axis in the p-plane. Cf. the relations described in connection with Fig. 2.2. ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 205 impedance were multiplied by C- icoD. They must be eliminated in reconstructing the expression for Z if the final result is to correspond to a physical network. Let us suppose that the poles above the axis are repre- sented by col ß ß ß co, and the corresponding residues, i.e., the Ci's of (10-2), by C ß . ß C. The poles below the axis will then be the conjugate points  ß ß ß 5, while it is easily shown that their residues will be the correspond- ing conjugate quantities l ß ' ß ,. If we assign the constant R0 of (10-2) equally to the two groups of poles this allows us to write the complete partial fraction expansion corresponding to (10-2) in the symmetrical form co -- coi co coj The two bracketed expressions in (10-6) evidently represent conjugate quantities on the real frequency axis. Each, therefore, provides half the final resistance characteristic. If we multiply the first by two we secure the required impedance expression in the form Z =  2Ci q- Ro. (10-7) 1 co -- coj The fact that this is actually the sought-for expression for the impedance is easily established. It evidently gives the right resistance characteristic and its poles are in the proper portion of the plane. The fact that the zeros are also in the proper half of the plane follows at once if we remember that the resistance must be positive on the real frequency axis and make use of the general theorem on the location of the maxima and minima of an analytic function. It is easily shown also that (10-7) is the only valid impedance corresponding to the original resistance characteristic if we exclude the possibility of introducing pure reactance networks by the addition of poles on the real frequency axis. If we begin with the reactance characteristic, the procedure is essentially the same. The oniy distinction arises from the fact that because of the presence of the multiplier ico, the residues of the poles below the real fre- quency axis are the negative conjugates of the residues above the real frequency axis. Along the real axis, therefore, the sums of the contribu- tions of the two groups of poles have real components of opposite sign and imaginary components of the same sign. A constant real quantity can therefore be added to one group and its negative to the other without affect- ing the result. This corresponds, of course, to the fact that the reactance component of any network is not changed by the addition of an additive constant to its resistance. The extension of the analysis to active impedances evidently presents, in general, no great difficulty. It is necessary to assume, however, that the ----------------------------------------------------------- 206 NETWORK ANALYSIS C,,P. lO desired type of stability is appropriate for the restrictions on the p61es of the given functions. Thus if we begin with a resistance we can readily construct a corresponding complete impedance function which will be open- circuit stable. It is not so easy, however, to determine an impedance which is short-circuit stable but not open-circuit stable since the poles of the short- circuit stable function may occur in both halves of the plane and there may be several ways of separating the partial fraction expansion of the resistance into two halves. 10.6. Choice of Coecients in Impedance Expressions Thus far in our discussion we have considered the physical restrictions on possible impedance expressions and some of the ways in which a definite circuit corresponding to any particular impedance can be obtained. We have not, however, considered the design problem, which is that of choosing an expression for the impedance to simulate a characteristic which has already been prescribed. There are a number of ways in which this prob- lem can be attacked, especially when the characteristic we have in view is in some analytically simple form. Space does not permit consideration of all these possibilities. For the sake of completeness, however, the simplest and most direct attack is outlined below.* Let it be supposed that the rational function representing the impedance is written as Z = R + iX = ,'1o +/lp + zt2p 2 + ... + ,4,p  ' (10-8) Bo + Bp + B2p 2 + ... + B,p m If we replace p by ic0 on the real frequency axis and multiply through by (Bo + ß ß ß + B,pm), we can equate real and imaginary parts separately to secure the pair of equations (z 0 -- -d2C.O 2 '+' -df40J 4 .... ) -- R(Bo -- B2w 2 + 14 w4 .... ) + Xo(Bt - Bs0 2+ Bo 4 .... ) = 0, (10-9) (dl -- d3 w2 q- /i5W 4 .... ) -- R(B1 - Baw 2 + B5 w4 .... ) x -- -- (BO -- B2co 2 + B4c04 .... ) = 0. Now let co, R, and X in these expressions be assigned particular values chosen from the characteristic we are trying to meet. If we choose a sufficient number of sets of values of these three quantities the result will be a system of simultaneous equations in the//'s and B's whose solution ß The method which is described is essentially a modification of a method due to O. J. Zobel. See "Distortion Correction in Electrical Circuits," B.S.T.J., July, 1928. ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 207 gives a network approximating the desired impedance. Since the equations are all linear the solution is relatively simple. The most straightforward process is obtained if we use both of equations (10-9) at each matching frequency. In some instances, however, a better overall characteristic is found if we choose twice as many matching fre- quencies and apply the equations alternately. Since equation (10-8) will obviously not be affected if the numerator and denominator are divided through by any constant, one of the //'s or B's is arbitrary and can be conveniently set at the value unity. The process evidently carries with it no guarantee that the resulting impedance expression will be physical. Since, as we have already seen, the resistance and reactance characteristics of a physical network can be chosen independently only within narrow limits, this is inherent in the nature of the problem. The same general method can also be applied to the simulation of either component separately. For example, if we begin with a resistance function of the form 0 -[- dl w2 -[- 2/04 q- ''' q- /ra c02m R = /!0 q- Bw2 + Bw4 + . . . + Bo? m , (10-10) we can evidently choose appropriate values of the constants by means of the set of simultaneous linear equations obtained by substituting special values of 0 and R in the equation (/10 q- z{J11.02 q- ''' q- -/fro tO2m) -- R(Bo + Bo? + .-. + B,o? ') -- 0. (10-11) The expression for the complete impedance can then be built up from the formula for R by the method described previously. With this procedure the requirements for physical realizability are much less onerous than they were before. We must still be careful, however, that the rational function which is obtained for R has no poles of any order, and no zeros of odd order, on the real frequency axis. Although this procedure appears to take cognizance of only one component, it may still be appropriate for the simulation of a complete impedance. Since the minimum reactance and the resistance characteristic of a network are always dependent on one another, there is no essential loss of generality in restricting ourselves initially to the resistance characteristic alone. We can always control the reactance characteristic to some extent by the final addition of a series teacrance network. The process of resistance simulation is particularly simple if we make use of the fact that networks whose physical configuration is that of a "con- stant k "high-pass or low-pass filter terminated in a resistance furnish input resistances of the type of (10-10) in which all the terms in the numerator ----------------------------------------------------------- 208 NETWORK ANALYSIS C.AP. 10 except the first or last are zero.* The analytic problem is then that of simulating a prescribed characteristic by a polynomial, and ordinary polynomial or Taylor's series methods are applicable. 10.7. Transformations of the Frequency k'ariable If we turn back to such general equations as (1-2) of Chapter I, we observe that aside from the resistance terms, every quantity is either of the form Lgip or of the form Dii/P. The impedance as a whole, of course, is some function ofp which depends upon the particular values assigned to the L's, D's, and R's. Now suppose that in the given network we replace each inductance by an impedance varying with frequency as some functionf (p) and each capacity by an impedance varying as 1/f(p). We will also sup- pose that the various impedances replacing the original inductances or capacities are in the same proportions as the original inductances or capaci- ties themselves. Evidently this merely replaces p byf(p) in every equa- tion, so that the impedance Z(p) becomes transformed into Z[f(p)]. In other words, if we know the impedance function of a given structure, we can find immediately the impedance function of the structure obtained when each inductance is replaced by a proportional impedance of some other type, and each capacity by the related inverse impedance. It is merely necessary to replace p in the original impedance function by the expression for the impedance which replaces the inductances. While the result has been stated only for driving point impedances, it evidently holds also for the transmission properties of a network. So far as the formal statement of the principle goes, each inductance might be replaced by a dissipative impedance, such as that illustrated by Fig. 10.11. In practical applications, however, the principle is of impor- tance chiefly when each inductance is replaced by a  o network of pure reactances. This can be explained from the fact that if we deal with a network of pure reactances, both the original variable p and the new Fro. 10.11 variable f(p) assume only pure imaginary values at real frequencies. The real frequency characteristics of the transformed network can therefore be obtained from the real frequency characteristics of the original structure merely by correlating corresponding values of p andf (p), whereas if we use a dissipative network the characteristics of the transformed structure must be obtained by computation. Since the most elaborate reactive network can merely run through all teacrance values from -o to +o0 repeatedly, where the original variable p ran through such values only once, the transformed characteristics are, at * Further details are given in "A Method of Impedance Correction," H. W. Bode, B.S.T.J., Oct. 1930. ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 209 most, repeated copies of the original characteristics, with some distortion of the frequency scale and perhaps an inversion in frequency. The best illustrations of this principle are found in filter theory. For example, if we begin with the low-pass structure shown by Fig. 10.12.4, the simplest transformation is effected if we makef (p) = kip where k is a real constant. This replaces each inductance by a capacity and each capacity by an inductance as shown by Fig. 10.12B. The characteristics are the same as those of the original structure except that the frequency scale is inverted and positive and negative frequencies are interchanged. In other words, the new structure is a high-pass filter. The relation between any two corresponding frequencies, such as the cutoffs of the two structures, on the absolute frequency scale, can, of course, be controlled by means of the constant k. The next simplest transformation is J(p) = k(p/o, 4- o)/p). This replaces each inductance by a series resonant circuit, and each capacity by an anti-resonant circuit, as indicated by Fig. 10.12C. The result is easily seen to be a band-pass filter. The resonant frequencies of the networks replac- ing the original coils and condensers corre- spond to zero frequency in the low-pass filter and represent the center of the trans- mitted band. As we go either way from this frequency, we secure a distorted replica of the original low-pass filter characteristics. The frequency at which the center of the band is found is, of course, determined by Fro. 10.12 the constant 0, while the width of band can be controlled by the constant k. These relations are shown in more detail in Fig. 10.13. The solid and broken lines at the top of the figure represent respectively the real and imaginary components of the complete filter characteristic. As the figure is drawn the characteristic itself is regarded as fixed and the changes which occur in going from one type of filter to another are expressed by distorting the frequency scale, as indicated by the horizontal axes at the bottom of the drawing. The topmost axis represents the scale in the low- pass case. Since it may be taken as a reference it has been drawn in the usual arithmetic fashion, without distortion. In order to express the correspondence among the three characteristics completely it is necessary to draw both the positive and negative halves of the real frequency axis in the low-pass case. The fact that the positive ----------------------------------------------------------- 210 NETWORK ANALYSIS C.AP. 10 half is the one of direct design interest is indicated by drawing it very heavily. Since the real and imaginary components of the characteristic must be respectively even and odd functions of frequency in accordance with the general principle outlined in previous chapters, the relations between the two halves are easily determined. The second horizontal axis gives the scale appropriate for the high-pass filter. The constant k has been chosen as unity. The transformation is essentially merely a matter of replacing the frequencies in the low-pass case , I   , , a) Low -3.0 -2.0 -I.O .0 1.0 2.0 i . I 0 I f , [ C. Hirjh Pass 033 05 t. 2.0  -{.0 -0.$ -0.35  I I I I I , r J Band Pass 0.5 0.62 0.78 1,0 1.21 1.5 2.0 r Fro. 10.13 b 7 their reciprocals, but in order to secure an exact correspondence the positive frequency half of the high-pass scale must be identified with the negative frequency half of the low-pass scale. The bottom axis gives the band-pass scale, the constant k being chosen as two. Here the positive and negative halves of the low-pass scale correspond respectively to positive frequencies above and below the center of the band in the band-pass scale. 10.8. Frequency Transformations in Almplifier Design These frequency transformations will be used in later chapters to simplify the discussion of amplifier design methods. Most practical amplifiers are called upon to transmit a band extending from one finite non-zero frequency to another. For purposes of analysis, however, we will take as our point o( departure a structure transmitting from direct current up to some pre- scribed frequency. This will be called the equivalent low-pass amplifier. ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 211 The modification of the characteristics of the equivalent amplifier to suit the actual requirements can be made by either of two methods, depending upon the band width of the actual amplifier on a logarithmic frequency scale. If the band is relatively broad it is simplest to suppose that the characteristics of the equivalent amplifier are the same as those of the actual structure at all high frequencies and to superimpose upon them a set of low-frequency characteristics to take account of the fact that the trans- mission band of the actual amplifier does not extend to zero frequency. The required low-frequency characteristics can be obtained from any suitable high-frequency design by drawing the characteristics on a recipro- cal frequency scale, using the transformation from low-pass to high-pass filters which was described in the preceding section. This is illustrated by Fig. 10.14. The solid line represents the loop gain characteristic in the original equivalent low-pass design and the broken line the modification in the characteristic near the lower edge of the useful band. Flo. 10.14 Fro. 10.15 If the band of the actual amplifier is relatively narrow, on the other hand, it is more desirable to treat the complete characteristic as a single unit, obtaining it from the equivalent low-pass structure by means of the trans- formation relating low-pass and band-pass filters. Fig. 10.15, for example, shows a band-pass characteristic corresponding to the low-pass characteris- tic of Fig. 10.14. Since the lov-pass to band-pass transformation always leads to characteristics which are symmetrical about the center of the band, this leaves amplifiers with dissimilar characteristics at the upper and lower edges of the band to be treated directly. Narrow-band amplifiers with dissymmetrical requirements, however, are very exceptional. These frequency transformations have been introduced here as an ana- lytic simplification. They are, however, frequently convenient also in the preliminary stages of an actual design, since the branches of the equivalent amplifier are usually more easily computed than those of the actual struc- ture. 10.9. Principle of Conservation of Band ?/idth The low-pass to band-pass transformation has one simple property of considerable importance. This is' the fact that the transformation from a ----------------------------------------------------------- 212 NETWORK 'ANALYSIS cusP. lo coll to a resonant circuit or from a condenser to an anti-resonant circuit does not affect the band width, in cycles, over which the impedance or admittance of the branch stays within any prescribed limits if we keep the coil unchanged in the first case and the condenser in the second. For exam- ple, in a low-pass circuit the susceptance of a given capacity C will be less than some fixed value Bo between zero and a point coo for which Bo = cooC. In the band-pass circuit the susceptance of the corresponding branch can be written in general as 1 wB = wC - -' (10-12) L t9 will assume the values 4-Bo at two points on opposite sides of the band. At these points, which may be indicated by w and =, (10-12) becomes 1 L (10-13) 1 -o, B0 = oC - -' L Subtracting the second equation from the first gives (2 2 -- 12)0 = (C0 2 -- O01)J 0 = (C0 2 + l)ooC or w -  = w0. (10-14) The frequency interval between corresponding points in the band-pass characteristic is thus the same as the equivalent interval* in the low-pass characteristic no matter how the mid-band frequency, which depends upon L, is chosen. A similar result evidently follows if we keep the inductance constant in the transformation from a simple coil to a series resonant circuit. The transformation to a single band-pass circuit is the only one of particular engineering interest. As a matter of fact, however, similar relations also hold if we replace an individual coil or condenser by a active network of any arbitrary complexity, subject only to the condi- tion that the network becomes equal to tke element that it replaces at infinite frequency. The impedance or admittance of the branch will * This interval is, of course, only the positive frequency part of the low-pass band. Since the band-pass characteristic was'compared with the sum of the positive and negative frequency characteristics of the low-pass structure in Fig. 10.13 it may appear at first sight that the band-pass interval should be doubled. The apparent discrepancy is explained by the fact that there must be a native frequency band- pass characteristic also. The total interval on the complete real frequency axis in the two cases is the same. ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 213 then lie within specified limits in a number of discrete bands, whose breadth and arrangement depend upon the resonances and anti-resonances chosen for the branch. The sum of all these intervals, however, is equal to the corresponding interval for the original inductance or capacity. The importance of these conclusions will appear more clearly in later chapters. It will be shown that, broadly speaking, most of the characteris- tics of feedback amplifiers are ultimately limited by the parasitic elements in the circuit, which are principally shunt capacities to ground and second- arily series inductances. For example, tube gains are ultimately limited by interstage capacities. Input and output transformers, at least at high frequencies, are restricted principally by leakage inductance and high side capacity. The amount of feedback which can be secured is limited in the same way by the miscellaneous parasitic elements in the feedback loop. Evidently, in all these cases the result just established can be applied to the parasitic elements when they are resonated as part of the transformation from a low-pass to a band-pass system. Since the relative impedance levels of the various branches in the complete circuit are not affected by the trans- formation, however, the teacrance or susceptance of any branch containing a parasitic element is correlated with any* overall response characteristic of the circuit in the same way after the transformation as it is in the low-pass structure. With the understanding that this is what is meant, therefore, the general result can be expressed as the Theorem: The width of the frequency band, in cycles, over which a given response can bemaintained in a circuit of given general configuration containing prescribed series inductances and shunt capacities is independent of the location of the band in the spectrum. 0 C C trequency Fm. 10.16 * That is, any characteristic which can be determined from single frequency imped- ance values of the branches. This would evidently eliminate a delay, for example, since the result here depends both upon the impedances of the branches and upon the rate at which they vary with frequency. It is also assumed, of course, that the fact that the sign of such a response characteristic as a teacrance may be opposite, on one side of the band, to that obtained from a low-pass circuit is immaterial. ----------------------------------------------------------- 214 NETWORK ANALYSIS CaAv. 10 The conservation of band width in the low-pass to band-pass transforma- tion is illustrated by Fig. 10.16. The characteristics are the same as those originally shown in Figs. 10.i4 and 10.15. Typical equal intervals in the two cases are indicated by the horizontal lines//, B, and C. 10.10. Frequency Transformations to Dissipatioe lmpedaces It was suggested in connection with Fig. 10.9 that frequency transfor- mations which replaced the reactive elements in the original structure by dissipative impedances were of comparatively little value. This is gener- ally true. There are, however, two particular cases of such transformations of somewhat special interest. The first occurs when the original structure is composed only of reactances. In this case the transformation method can be used to generalize Foster's results for networks of pure reactance to include networks of any two types of impedance elements whatever. It is not necessary to assume, as was done in the previous discussion, that the impedance elements replacing the coils and condensers, respectively, in the original structure are inverse. The rule for making the transformation can be expressed most easily in terms of equation (9-12) of Chapter IX. The structure to which this equation refers reduces to an inductance at high frequencies. Evidently, the p which multiplies the whole right-hand side of the equation can be thought of as the expression for the impedance of this inductance. Simi- larly, the p2 terms which appear in the various factors of numerator and denominator correspond to resonances between the inductances and capaci- ties of the network and can be thought of as the ratio between the imped- ance of an inductance and that of a capacity. It can be shown by a more detailed analysis that this identification is correct. Evidently, therefore, if we replace the inductances and capacities in a network of pure reactances by proportional impedances of any other two types the new impedance expression can be obtained from the Theorem: The expression for the impedance of a network made up of any two kinds of impedance elements can be obtained from the expression for the impedance of a corresponding net- work of pure reactances by replacing the multiplier p in the pure teacrance expression by the impedance which corre- sponds to a unit inductance and by replacing the p2 terms in the rest of the pure teacrance expression by the ratio of the impedances corresponding to a unit inductance and to a unit capacity. Since the original reactance expression was derived for an L-L configuration it is assumed in the statement of the theorem that a structure of this type ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 215 is in view. Modifications to suit other cases, however, are easily made by the methods described in the preceding chapter. As an example of this theorem we may consider a network of inductances and resistances. The transformation to such a structure from a network of pure reactances leaves a unit inductance as a unit inductance but a unit capacity is replaced by a unit resistance. Thus the multiplier p in the expression for a pure reactance network is unchanged, but each p2 is replaced by p. Substituting in (9-12) of Chapter IX, the new impedance expression becomes (p _ p)(p _ p2) ... (p _ p) (10-15) z = kp ..... ß As a second example, let the network be composed of capacities and resistances. This leaves a unit capacity as a unit capacity while a unit inductance is replaced by a unit resistance. In the impedance formula, the multiplier p is replaced by unity and each p is replaced, as.before, by p. The result is (? - P) - )'" (P - ) z =  (p _ p)(p _ p)...  _ p_) In both (115) and (116) the zeros and poles are found on the negative real p axis and occur alternately. The only distinction between the two expressions is the fact that as we proceed along this axis starting from the origin, the alternation begins with a zero when the network is made up of inductances and resistances and with a pole when the network is made up of capacities and resistances. Fro, 1(}.17 Fro. 10,18 Both types of networks can be represented in partial fraction form. Corresponding to the network of inductances and resistances, for example, we may secure either of the configurations shown by Figs. 10.17 and 10.18. These expansions have already been described in substance, in connection with Figs. 10.3, 10.4 and 10.5 of the present chapter. As the analysis shows, the poles of inductance-resistance and capacity-resistance networks, ----------------------------------------------------------- 216 NETWORK ANALYSIS although they are both found on the negative real axis, correspond to Cs's , or residues, of opposite sign. When we add together corresponding terms from a capacity-resistance and an inductance-resistance network, as in Fig. 10.3, therefore, the poles may cancel out, leaving merely a constant. 10.11. Effects of Parasitic Dissipation The second situation in which frequency transformations which replace reactive elements by dissipative impedances may be useful occurs when we are trying to express the effects of the normal parasitic dissipation of coils and condensers in the formulae for the network. For example, if R is the parasitic resistance associated with a coil L we can write the impedance of the coil as pL q- R = (p q- R/L)L. The effect of dissipation can thus be represented by replacing p by p q- R/L in the impedance of the non- dissipative coil. Similarly, the impedance of a capacity including a para- sitic conductance G can be written as 1/(pC q- G) --- i/(p q- G/C)C, and is the same as the impedance of a non-dissipative capacity with p q- G/C substituted for p. In most networks the ratio R/L is about the same for all coils and the ratio G/C is about the same for all condensers. If, in addition, the two ratios are equal to one another thh network may be spoken of as one having uniform dissipation. It is less often true that this second requirement is satisfied by actual circuits. In ordinary networks, however, the effects of dissipation are much the same whether we regard the dissipation as being concentrated principally in the coils alone or the condensers alone or assume it to be equally divided between elements of the two types.* Under these circumstances we can evidently represent the effects of dissipation by replacing p by p q- « (R/L q- G/C) in the impedance expressions for both coils and condensers. This therefore leads to the * This can be taken as a matter of experience, but it can also be justified, for many networks, theoretically. Thus ifwego back to the energy analysis of Chapter VII it is evident that the effects of parasitic dissipation must be attributed to the power loss in the dissipative elements. The ratio of the power loss, 12R, in a dissipative coil to its stored energy, «IU, however, is simply 2R/L, while in a dissipative condenser the ratio is 2G/C. in a complete network, therefore, the dissipated power must be (2R/L)Tq- (2G/C)I/, which can also be written as (Tq- I/)(R/L q-G/C) q- (T - I/)(R/L - G/C). The first term of this expression evidently represents the average dissipation assumed above while the second term gives the error in this assumption. Since T-- // is proportional to the input susceptance by (7-31) of Chapter VII, the error will be negligible for any network whose impedance is approxi- mately a pure resistance. Even if this condition is not met the second term will be negligible in comparison with the first, as shown by (9-15) to (9-17) of Chapter IX, if the network is a sharply varying two terminal teacrance, or, as shown by later equa- tions of similar type, if the network is any electrically long filterlike structure. ----------------------------------------------------------- THE DESIGN OF IMPEDANCE FUNCTIONS 217 Theorem: If a network can be regarded as uniformly dissipative any of its actual characteristics can be obtained by replacing p by p + «(R/L + G/C) in the equations for the corre- sponding characteristics in the absence of dissipation. In a mathematical sense the theorem states in effect that the changes due to dissipation can b represented by evaluating the function on a line somewhat to the right of the real frequency axis rather than on the real frequency axis itself. This is illustrated by Fig. 10.19. The light solid line represents the new axis when R/L q- G/C is constant with frequency and the broken line the result if R/L q- G/C increases with frequency, which is the usual case in practice. As an alter- native, we can of course say that the computa- tions are still made on the real frequency axis but that the function itself, including its zeros, / / t J \ Fro. 10.19 poles, and other reference points, has been displaced an equivalent dis- rance to the left. ,-x' These relations lead to a simple method of designing networks to give automatic compensation for the effects --x' of parasitic dissipation. The method was first used by :..x' Darlington* in the design of filters which would give o--e flat transmission bands when c