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 va