THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING THE DESIGN OF EFFICIEB B. F. Barton This report was originally distributed as Technical Report No. 44 by the Electronic Defense Group, Department of Electrical Engineering, the University of Michigan, under Engineering Research Institute Project 2262, on Contract No. DA-36-039 sc-63203, Signal Corps, Department of the Army. October, 1955 IP-136

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ACKNOWLEDGEMENT We wish to express our appreciation to the Engineering Research Institute, the University of Michigan for permission to distribute this report under the Industry Program of the College of Engineering.

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT v ACKNOWLEDGEMENTS vi I. INTRODUCTION 1 II. DEFINITION OF THE OPTIMUM TCHEBYCHEFF NETWORKS 7 III. DERIVATION OF THE OPTIMUM MAXIMALLY FLAT NETWORKS 13 APPENDIX I 30 APPENDIX II 32 APPENDIX III 38 APPENDIX IV 40 REFERENCES 42 DISTRIBUTION LIST 43 ii

LIST OF ILLUSTRATIONS Fig. No. Page 1 A General Matching Problem 1 2 A General Matching Problem with a Lossless Coupling Network 2 3 The Problem of Matching to a Parallel RC Load 4 4 The Tchebycheff Power Transfer Characteristic 5 5 Lowpass Ladder Structures with All Transfer Zeroes at Infinity 5 6 The Maximally Flat Power Transfer Characteristics 6 7 Loci of Poles of p(p)p(-p) for n = 4 9 tanh nx 8 Sketch of h x vs. X 10 cosh x 9 Dual Realization of Design Curve Networks 12 10 The Maximally-Flat Power Transfer Characteristic 13 11 The Rectangular Lowpass Characteristic 15 12 Loci of Poles and Zeroes of Maximally Flat Functions of Interest 16 13 Possible Arrays for Zeroes of Maximally Flat 3-Pole Reflection Coefficient 20 14 Maximum Loss and Ripple of 2, 3 and 4 Pole Tchebycheff Networks with Optimized Power Characteristics 22 15 Design Curves for 2-Pole Tchebycheff Network with Optimized Power Characteristic 23 16 Design Curves for 3-Pole Tchebycheff Network with Optimized Power Characteristics 24 17 Design Curves for 4-Pole Tchebycheff Network with Optimized Power Characteristics 25 18 Maximum Loss of 2, 3, and 4-Pole Maximally Flat Networks with Optimized Power Characteristics 26 19 Design Curves for 2-Pole Maximally Flat Network with Optimized Power Characteristic 27 20 Design Curves for 3-Pole Maximally Flat Network with Optimized Power Characteristic 28 iii

LIST OF ILLUSTRATIONS (Cont'd) Fig. No. 21 Design Curves for 4-Pole Maximally Flat Network with Optimized Power Characteristics 22 A General Matching Problem with a Lossless Coupling Network 23 Antenna Approximating Circuit 24 Comparison of Vertical Monopole Impedance of Approximating Circuit 25 Lowpass Equivalents of Antenna with Negligible Base Capacitance 26 4-Pole Tchebycheff Matching Network 27 Composite Antenna Network 28 Identities Useful in Modifying Source-to-Load Impedance Ratio 29 Experimental t|2 vs. Frequency 30 Three Pole Network from Figure 20 31 Modified Networks from Figure 30(b) 32 A Three Pole Network with Maximally Flat Behavior Page 29 30 32 33 34 34 35 36 37 40 40 41 iv

Abstract Some introductory comments on the problem of broadband matching of arbitrary impedances are presented. The analysis of Fano (Ref. 2), leading to minimum loss networks when matching to an RC load with a prescribed %cRC, is summarized. Design curves for the Tchebycheff networks defined in the analysis are presented for the 2, 3, and 4 pole cases. It is hoped that the curves will lead to a wider application of Fano's results. The optimum networks (under the same criterion) with maximally flat behavior of the power transfer characteristic are defined. Design curves for the resulting 2, 3,and 4 pole maximally flat networks are presented. Although these networks are less efficient than the Tchebycheff networks, they are preferable in certain applications. A sample calculation illustrating the principles of this report is presented in Appendix III. V

Acknowledgements The author wishes to express his appreciation to Professor A. B. Macnee, who suggested several technical clarifications incorporated in the final draft of this report, and reached some of the results of Section III independently. Mrs. Jean Childs made the calculations from which the Tchebycheff design curves were plotted. vi

I ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN THE DESIGN OF EFFICIENT COUPLING NETWORKS I, INTRODUCTION The transfer of power is a fundamental problem of electrical engineering. The optimum coupling networks under a particular criterion are defined in this report. Consider the matching problem suggested by Fig. 1, where power is to be delivered from a generator with a resistive internal impedance R to an arbitrary load impedance ZL. Darlington (Ref 1) has shown that any driving point XZ N' N EI FIG. I. A GENERAL MATCHING PROBLEM. impedance can be realized in the form of a lossless network terminated in one ohm. If the coupling network N' is also required to be lossless, the circuit of Fig. 1 can be replaced by Fig. 2. The matching problem is thus reduced to the specification of the network N for a prescribed power transfer vs. frequency characteristic, when a number of elements of N are determined by the load impedance ZL. Note that the Darlington network N" frequently contains an ideal I -

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN N' N" N R^ J> R LOSSLESS ZL LOSSLESS 1 ZI LOSSLESS Z2 1 FIG. 2. A GENERAL MATCHING PROBLEM WITH A LOSSLESS COUPLING NETWORK. transformer, and ZL actually may contain a number of resistors. In the representation of Fig. 2 all the power delivered to these resistors is absorbed by the single one ohm resistor. If ZL contains only a single resistor, a simple impedance level transformation produces a normalized one ohm load.l In this case the transformation ratio (i.e., the ratio of load-to-source resistance) is the significant parameter. A generator of internal resistance R and open circuit voltage E can deliver only a finite power IE2 to an external load. This maximum or "available' 4R power is delivered under the matched condition. Efficiency of power transfer is commonly measured in terms of this power; i.e., by the transmission coefficient t which is defined by It2 Power delivered to load Power available from source If the network N is lossless, the transmission coefficient tl for the circuit of Fig. 2, is related to the reflection coefficient p1 at the input to network N by the equations IPl 1 ItlI2 = l R2 1-1 T - Ir 1 In Appendix 1 it is shown that the efficiency of power transfer is unaffected by a variation of impedance level. 2

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN where Z1 is the impedance seen by the generator. It is also worth noting that if N is lossless, the same efficiency of power transfer is obtained when the network is driven from the one ohm end; that is, Ipll2 = - It 12 - It = ip212 Z 12 The rather general matching problem outlined above is very difficult of solution. Fortunately, the solution of far simpler problems are very useful. Fano has shown that if a fixed shunt capacitance (or series inductance) is associated with the load resistance, the maximum loss over a desired band is minimized by approaching the rectangular characteristic (Ref. 2). A further simplified problem is concerned with obtaining circuits whose transfer function approximates the lowpass rectangular characteristic (See Fig. 11). These networks are useful in themselves, and by the well known lowpass-to-bandpass transformation yield circuits giving approximations to the bandpass rectangular characteristic. The following discussion is concerned with networks whose transfer functions approximate the lowpass rectangular characteristic. The |t|2 of a finite lumped network between a resistive generator and a finite lumped load is an even rational function. Rational functions can at best only approximate the rectangular characteristic. Realizable transfer functions can be specified, giving, among others, maximally flat and Tchebycheff (equal ripple) approximations to the rectangular lowpass characteristic. The type of approximation used in a particular application depends on the requirements to be met. Consider the particular case of Fig. 3a where the load R2 1 is to be shunted by a capacitance Cout. Figure 3(b)resulting from an impedance level transformation indicates that the important parameter is the product R2Cout. When an optimum match over a band co is desired, a gain bandwidth factor for the 1 In the dual case the terminal resistance is associated with a series inductance. The results of this report are directly applicable to both cases. 3

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Rl 7 RI < N R2 R2 ' < N2 OCo { OUT 2C2OUT2 (a) (b) FIG. 3. THE PROBLEM OF MATCHING TO A PARALLEL RC LOAD. load cR2Cout is specified. Sharpe (Ref. 3) has shown that where it is desired to approximate a constant over a band, the optimum function at a fixed order of approximation on a gain bandwidth basis is the Tchebycheff rational function with all its zeroes at infinity. Now it is obvious that with a fixed finite number of elements, there is a minimum "maximum loss" which can be achieved. It follows that there is a Tchebycheff network withall transfer zeroes at infinity producing this minimum realizable loss for a given number of coupling elements and oc R2Cot of the load. It can be shown that the Tchebycheff rational function with all its zeroes at infinity is simply the reciprocal of a Tchebycheff polynomial. One is led to consider a transmission coefficient in the form t2 kI t|2 =,k (1) 1 + E Tn2(c) where k'< 1 and e are positive constants and Tn is the Tchebycheff polynomial of degree n. This function behaves as sketched in Fig. 4, with n equal amplitude ripples over the symmetrical lowpass band. 4 r i i 4

L - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Ii-_L_, -wc Wc FIG. 4. THE TCHEBYCHEFF POWER TRANSFER CHARACTERISTIC. The transmission coefficient of Eq 1 can be realized by either of the network forms shown in Fig. 5, which are exact duals. These n-pole coupling networks consist of n elements in a ladder structure of series inductances and shunt L3 LI L, L R { C4 TC2 I t C3 TCI (a) (b) FIG. 5. LOWPASS LADDER STRUCTURES WITH ALL TRANSFER ZEROES AT INFINITY. condensers. The subset of these networks which minimizes the maximum loss for a prescribed n and load product McR2COut are defined in a later section. Design curves are included (See Figs. 14, 15, 16 and 17) for the 2, 3, and 4 pole networks as well as a sample calculation. I 5

I I ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - The networks of Fig. 5 can also be designed so that the transmission coefficient is given by |t|2 k' (2) coZ)db ) This is a maximally flat function with all transfer zeroes at infinity, and behaves as sketched in Fig. 6. In a later section, the optimum maximally flat networks as a function of the product %cR2Cout are determined. Design curves are included (See Figs. 18, 19, 20 and 21) for the 2, 3, and 4 pole networks. -~kI kl/2 WC3db FIG. 6. THE MAXIMALLY FLAT POWER TRANSFER CHARACTERISTICS. In general, the maximally flat networks are found to be inferior to the Tchebycheff networks on a gain-bandwidth basis. However, in some cases the maximally flat networks may be preferable. For example, it is well known that where transient response is important, steep skirts are undesirable. In general, Tchebycheff networks are inferior on this basis. In addition, the compounding of ripple which occurs when Tchebycheff interstages are cascaded may be a problem. 6

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN II. DEFINITION OF THE OPTIMUM TCHEBYCHEFF NETWORKS Fano (Ref. 2) has considered the networks for which the transmission coefficient is in the form of Eq 1, repeated here for convenience. 2 kI iti2 = l 1 T (-) (1) The results are summarized briefly below. The reader is referred to the original work for a more thorough treatment. The transmission coefficient can be considered more generally as a function of the complex variable p = a + j>. Fano found it profitable to consider the Taylor Series expansion in p of n 1 around infinity. In P particular, if the transmission coefficient has n zeroes at infinity, the function Ti- approaches unity at a definite rate near infinity. Under these conditions, a number- of the coefficients of the even powers of the expansion of In - around P infinity are zero. Then In - can be written in the form P lnl = jP + A p + A3 p +....A23 (2n-3) A(21)+.... - 3 A2n-3 p (23)+ A2n-1_ p where X is 0 or n depending on the sign of p. It can be shown that A = 1 2k+1 2k+l \ A2k+l P2k+l o, -Z (3 1k+1 - \ P i iPpi (3) where poi and Ppi are the zeroes and poles, respectively, of the reflection coefficient. Note that the A2k+l are functions only of the poles and zeroes of '00 the reflection coefficient. It can also be shown that the A2k+l can be expressed 2k+l C1 n be expre1 in terms of the k elements closest to the load for the structure of Fig. 5a. The first few of these relations are: i The elements of Fig. 5a are normalized to a one ohm termination. 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Define 2 A3 1 a == 2 ~3 " (Al)3 3 00 a5 = 2 (Azl)5 5 co 26 A7 1 a7 = 2 (Al~)7 7 Then Li = 2 (4) A1 L C - 2 a 3 a3 L1 L3 = 1 + a3 - (a/a3) + a3 - (a/a3)] L1 a [1 + a3 - (a/a3) + (ala3)- (a7/a3)] As stated above, the A can be expressed in terms of the loci of the poles 2k +1 and zeroes of the reflection coefficient, which from Eq 1 is given by 2 It2 1 - k' + e Tn2(/Oc) IPI = 1- It = 1 + e Tn ()/oW) The poles are located on an ellipse specified by a parameter "a" as indicated in Fig. 7. The zeroes are similarly located and specified by a parameter "b". Fano tabulated the first few relations between the A2"+1 and the parameters a and b as shown below. A = (cD (sinh a - sinh b sin n/2n 8

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A0 0 -2 3 (sinh 3a-sinh 3b sinh a-sinh b =n -= 2 1_'3 + 3j - -c \ 3 sin 3A/2n sin t/2n / 5 -4 5 sinh 5a-sinh 5b A5 W2 (D\ 5 sin 5/2n + sinh 3a-sinh 3b + 2 sinh a-sinh b sin 3n/2n sin i/2n 00 A? 2-6 7 ( sinh 7a-sinh 7b + sinh 5a-sinh 5b + 3 sinh3a - sinh 3b + = " c 7 sin 7v/2n sin 5x/2n sin 3n/2n sinh a-sinh b ) sin Jt/2n / (5) I jw x ' L x x sinh a p PLANE l ----v 0o -L JL x x. -.. 1 I / r cosh a I x FIG. 7. LOCI OF POLES OFp(p) p (-p) FOR n=4. In evaluating these a's, the left half plane zeroes were chosen for the reflection coefficient since it is known that this leads to better networks on a gain bandwidth basis. The poles must be chosen in the left half plane for physical realizability. It can further be shown that the maximum loss is given by cosh nb max cosh na (6) The maximum loss can be minimized for a fixed wccLl using Eq 6 and 9

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A = snh a sinha - sinhb ) 2 is found that a and b should ben so that It is found that a and b should be chosen so that tanh na = tanh nb cosh a cosh b (7) Since tah nx varies as sketched in Fig. 8, it is always possible to pick an "a" and "b" satisfying Eq 7 for given values of 0c, Ll, and n in Eq 8. tanh nx cosh x b x FIG. 8. SKETCH OF nx VS. x. cosh x The procedure used in obtaining the design curves is summarized below, From the sets of equations 4 and 5, it is found that 2 sinh a -sinh b cL sin i/2n (8) For a fixed n and product wcL1 an "a" and a 4'b" can be found satisfying Eq 8 subject to the optimizing Eq 7. The sets of equations (4) and (5) are then used to determine a number n-l of ratios,........ In addition,the transformation ratio R:l between the generator resistances is easily determined from the zero frequency loss. For n even, the zero frequency loss is determined by 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN IPImax. Note that for the circuit of Fig. 5 k' It2 4R =' t 200 = (R + 1)2 Then for n even 1 - It(jo)12 = Ipl2 1 - 4R n even P max (R + 1)2 Solving R - 1 + |pI max n even 1- Ip max Similarly, for n odd R = 1 + PI min n odd 1- IPI min This can also be calculated since 1| sinh nb 'l'min = sinh na The maximum loss in db ( = - 20 loglo tlmin) is also determined from a, b, and n; using Eq 6. The ripple is given by ripple (db) = -20 [log tlmin - log Itl| max] The design curves resulting from the above approach are given in Figs. 14, 15, 16 and 17 for the two, three, and four pole networks (n = 2, 3, and 4). Note that the network complexity n and two other parameters [for example, bandwidth mc and the terminal element L1 in the one ohm network as well as impedance level may be chosen independently. It should be noted that an equally valid interpretation of the curves is obtained if L, L3.... are regarded as shunt capacitances; C2, C4.... are regarded as series inductances; and R is regarded as a conductance. The resulting networks are the exact duals of the networks 11

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN defined on the design curves; the alternative form is illustrated below for n = 4. Optimization is for a fixed cocCl for the dual networks of Fig. 5b. For the sake of clarity, the character of these networks is restated: "Let the number of lossless coupling elements, and the gain bandwidth product ( = ccL1 or wcCl for Fig. 5b) of a parallel RC or series RL termination, be specified. Then, the networks specified by the design curves of Figs. 14, 15, 16 and 17 are those which result in an absolute minimization of the maximum passband loss." 04 HENRIES C2 HENRIES R OHMS I L, L FARADS FARADS FIG. 9. DUAL REALIZATION OF DESIGN CURVE NETWORKS. Bode (Ref. 4) has shown that where power is to be delivered over a band wcD to a load R2 shunted by a capacitance Cout (i.e., for a fixed cocCl in Fig. 5b) the limiting value of the transmission in nepers is a = ln -. 2 \2 RC ) This curve is plotted in Fig. 14. It can be shown that this limit is approached as the number of elements in the networks defined above becomes large ( n-oo). More significant, however, is the rapidity with which the limit is approached even for a small number of elements (see for example n = 4 in the loss curves). The loss curves for the maximally flat networks which are optimized in the same way in Section III are given in Fig. 18. It can be shown that as the number of ------- - ~~~~~~12

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN elements is increased, these loss curves approach the same limit. The curves indicate,however, that the limit is approached much more slowly for these networks as n is increased. The use of the design curves is illustrated by a sample calculation in Appendix II. III. DERIVATION OF THE OPTIMUM MAXIMALLY FLAT, NETWORKS Maximally flat behavior of |t|2 with all the transfer zeroes at infinity requires that ItI2 kIl 1 +, 2n 3+[j~ \, / Referring to Fig. 10; at zero frequency |t|2 = k' while at the frequency coc the It2 is defined as |t|. Of course, at = W3db |t| = k2, 2 I crmtcrin = 3db' It12 it- - l2 Iti MIN. kw ~CA Wc 4db FIG. 10 THE MAXIMALLY FLAT POWER TRANSFER CHARACTERISTIC. 13

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN From Eq 9 t in (10) c 2n 3db Consider the product clt|2min which by Eq 10 is It I2 = - (n) c m in 2n 1 + W aCl A \3db If k is specified, the zero frequency point on the curve of Fig. 10 is known. Further, if W3db is specified (at which I|t = k/2) the complete curve can be drawn for a given n. The choice of k specifies the generator-to-load-resistance transformation ratio. This specifies the network of given complexity except for a scale on the frequency coordinate, which is specified when W3db is chosen. Thus, if k, w3db, and n are held constant, a particular curve and a particular network are being considered. Now, it is reasonable to ask whether under these conditions it is possible to pick wc in such a way that the areawc t2min is maximized. This is determined by setting [cltl min] 0 ac When this is done it is found that one should choose c = 3db 2n (12) As n becomes very large, this equatin indicates c should approach 03db. Referring to Fig. 11, this is reasonable, since for a rectangular characteristic one should choose Wc = (3db for gain bandwidth efficiency. [~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~11 J

I - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN i 2 ki k/2 --- --- O0 ' CWc = s3db co FIG. II. THE RECTANGULAR LOWPASS CHARACTERISTIC Substituting Eq 12 in Eq 10 k1 2n lmin 2n- 1 This equation determines the resulting variation in transmission over the band. 2 2 The function I p associated with the It| of Eq 1 is Ip12 l -kl + ((J/odb) = j /W'dbL2n 1 + (W/AD3db) Considered as a function of p/u3db, the function has 2n ed on 1 the unit circle, and 2n zeroes equally spaced on the circle of radius (1-) 2. It is known that for best gain-bandwidth characteristics, one should choose as the.zeroes of the reflection coefficient the n zeroes of Ip12 in the left half plane. The poles must, of course, be the n poles in the left half plane, if the network is to be realizable. These poles and zeroes are located as indicated in Fig. 12. [ 15

I i ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN / 2n n EQUALLY SPACED POLES AND ZEROS. II FIG. 12. LOCI OF POLES AND ZEROES OF MAXIMALLY FLAT FUNCTIONS OF INTEREST. A number of alternative procedures may be used in obtaining a network in a particular application. If oc and t|2 are specified, for example, Eqs 10 and 12 may be used to determine the required kI and o3db' The arbitrary constants of Eq 9 are then specified, and Ip|2 is determined by Ip12 = 1- Itl2 The function p(p) p(-p) is obtained by replacing a2 by -p2. The reflection coefficient is constructed from the left half plane poles and zeroes of p(p)p(-p). Then p(p) is given by p(p) D Z+ D Z + 1 P where Np and D are Hurwitz polynomials, and Z is the driving point impedance seen from the one ohm termination.l From the above equation, One choice of sign in this equation leads to a network with a shunt capacitance across the one ohm termination. The other choice of sign leads to a series inductance at the one ohm end. The cases are easily distinguished from the behavior of p at infinity. I J L,

I ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN z = (DP ) (DpFN ) The ladder structure is readily obtained from a continued fraction expansion of Z. Alternatively,the technique previously employed f6r the Tchebycheff case can be applied here to give equations from which the element values for the 2, 3, and 4 pole networks can be calculated directly. Following the procedure used in obtaining the Tchebycheff networks, the functions A2k+ (See Eq 3) are evaluated. Note in Fig. 12 that the sum of the imaginary parts of the pole positions is zero, since the complex poles occur in conjugate pairs. A single real pole at p = -1 is present for n odd. Consider the term of the coefficient Al: Ppi This is ~ Pi =[2{cos(g/2+n/2n) + cos(g/2+3t/2n) +.... + cos(n/2+ t 2/ )} -6] db 2n 3= 3db2 sinx/2n+sin 3tr/2n +... + sin (2[ +1 8 where [n/2] is the largest integer in n/2, 8 = 0 for n even and 6 =1 for n odd. Similarly Poi = - 3db(2 2n /2n+sin3/2n+ + sin3 n +.. 1) 2n Then A = {l - (1-) l/2 }~ sin/2n + sin 3n/2n +... + sin (2[n/2-) +~ ~ }l~~ 3db l \ ~2n + 5} 3db - -- 17

I I ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN =- ( l-kln') /2n I2sin 2 2]/ 2n+ sint/2n db by a well known identity. Similarly A" = /(1-k) -l{2[rcos[I + 2n+ cos[3 + 323Xl+ L. 3 /JLL=2 2nj.2 2J + Cos + (2[] - 1)2n )] -8}3db By.l 3 si 2+ln be 3ed rs tm 1/3{ l(-k)3/n i2 iJ+ a 3db sin }2! 2n ssinally, the A general expressd ion riss o t 2k+1 1 2n I2k+l si2n s n 2k+l A0 (1 J d 25",2(-Ir ----- 4 52(-1) k< ) AFrom l__k_ thi e n plots o sim 2k+l)s n 3db2k+l 2ks+ln mn n (2k+l)n. L 2n Using the Eq 4, which axpressed again applicablerms of thEq 14 with k ore useful parameters wc and It|J2 Thus 2c {(1 l2n 2n 2)k+l k si (22-i[2 + A2k+l 2 k n l- (1- 2-1 Itl in) 2- 2n (24) n~ and Eq 14] were used to obtain a set of design curves (Figs. 19,/2n0, ) for the Ltl -0 - -2 2n + 8 (2n-1) C L1 2n-1 min sin i From this equation, plots of maximum loss in db (= -10 log |t| ) versus 2 10 min W Li were obtained for n= 2, 3, and 4. See Fig. 18. The two sets of equations Eq 4 and Eq 14] were used to obtain a set of design curves (Figs. 19, 20, 21) for the maximally flat networks (n = 2, 3 and 4) similar to the curves previously discussed for the Tchebycheff networks. The equations are tabulated in Appendix III. 18 I

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The resistance transformation ratio R is again determined by the zero frequency (minimum) loss; i.e., k' = 4R (R + 1)2 The minimum loss is given by min loss (db) = max loss(db) -10 log10 2n2n - 1 For n = 2, 3 and 4; 10 log0 2n is 1.25 db,.79db, and.58db respectively. L 2n-l Fano notes that where the reflection coefficient p1 is written P() K Po(PPl)( - Po2).........Pon) Pl(P)= K (P -Ppl)(-p2 )...... (P Ppn) the reflection coefficient P2 is given by P2(p) = K(1-)n+l (P + Pol) +Po2 ) +- (+Pon) (P - Pol)(P-p2) (p P pn) Thus the zeroes of P2 are the negatives of the zeroes of pi- In the previous discussions the n zeroes of pi were arbitrarily chosen as the n zeroes of |p|2 in the left half plane. For these networks the n zeroes of P2 are, therefore, the n zeroes of Ip|2 in the right half plane. A new set of A2k determined from 2k+l p can be used with the set of Eqs 4 to determine the elements starting furthest 2 00 from the one ohm end. For example, the function (A')1 associated with P2 is (A')l {1 +(1-k)2n} 2 sin2[n/2] /2n + } ( l) 1 lsin it For n odd, (A')l -2 and thus 1 co Al - l-(l-k )2j (A') L1 +-k) 19

L --- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN For the optimum networks L 1 1-(1- 1 Itlmin) 1 1 +(1- 2n 2 2n-1 min It is easily shown that as k-0 (and R-aoo) the ratio L approaches 1. The ~L! ~ f n limiting networks as the transformation ratio becomes large are discussed briefly in Appendix IV. Choosing the zeroes of the reflection coefficient p1 in the left half plane led to one of two alternative networks for the maximally flat networks with n= 3 or 4. For example, if the zeroes of p1 are chosen in the left half plane, the zeroes of P2 occur in the right half plane. Alternatively, the zeroes of p1 0 0 (a) J (b) OPTIMUM ARRAY FOR ZEROES OF p\ FOR N= 3 RESULTING ZEROES OFP2 FOR N= 3 0 0 0 0 (c) (d) ANOTHER ARRAY FOR ZEROES OF Pi FOR N=3 RESULTING ARRAY OF ZEROES OF p1 FOR N: 3 0 0 FIG. 13. POSSIBLE ARRAYS FOR ZEROES OF MAXIMALLY FLAT 3-POLE REFLECTION COEFFICIENT. 20

I - ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN might be chosen as in Fig. 13c, in which case the zeroes of P2 occur as in Fig. 13d. The 3 pole case was arbitrarily chosen for illustration. The four arrays represent the only possible choices leading to realizable networks for the 3 pole case illustrated. There are, similarly, four alternatives for the four-pole case, leading again to two networks, of which the one chosen in the analysis is known to give superior power transfer efficiency. If the zeroes of p1 are chosen as 00 in Fig. 13c, the resulting (At'") is (A)" ' (2sin2[n/2] /2n 1 2sin2 [n/2] /2n 1 L sin i/2n sin i/2n + 8 c3db Further, for the associated p2,(Af) F 2in /2]n +/)2n 8]+(1 - 2si n2[ n/ /2n - 8 L3db 1. sin 7t/2n J L sins' /2n JJ The ratio L - 1 = R L1 J L is then L1 2sin[n/2]i/2n +1 +(l kl)- 2sin2[n/2]/2n +1 sin /2n sinsin /2n I I I s02S n/2 ]1/2n + (n 1 sin n/2n L sin n/2n This ratio is not finite in the limit as k-O. 21

9S-81-S W3P b-8d-3 OL61 0 ItItF IIIIlI1111111! 1iz t1t#C 11f:# ittiillI1 I I 1111111! I1S I1 H, I I L _UIII I I III 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I II 0 flil l] ----- X lillt Il1l||I0IF|t; X > - < mW Xt I~~~~~gIIIIIIIIIIIIIIIIIIIIITIITII~~~~~~~III I HIM III IIIIIIllIII I I - 0 ffl — 1 - tmtitte - -0t44+1-4-0 I I I HllI l44+-l --- 11|11|1111t1111 Iliiiiiiil a) WI|| CW~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~C xtm O g0-0tti; ISTWIWTW0+ 01t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t1 gE0 S 2~~~~HllI Z:1 -j-t i_~- t M+i~tL~l4W X41-01< i-FIIIIIIIIIIII:_:tlt~fl~iitli l TI~tI~il~tli ~l-il - IIHIMTT TII III IT w ~ 0t e ^ F " T T E ~~~~~~~~MHL UH I-~ -~ 11111111101 1 11111i1111 F1/ nI It H ill~ LI 1 o z z W 0. I/TIAtil illlllllf jD T1X-ILl g O a < <W X00W X01/1 1 1 ST~~~~~~~~~~~tIVIIIIGIIIIIIIIIIIIIIIIIIIIIII I I I I -NIlIill n IlI I 0 I FEFHll@ fll w - > 1. i1f IL T II1+:.lI g O < s — f -t —t-t — lit ttx fii lll fELQ -1 11 1 1 11 1!I IITIT N;~~~~~~~~~~~~~~I~ ITT II IllII1!II m~ C.)~ Hil Ill1 111 MI m g 0 1 1 201EE1 1L:S S 2 m g 01-g; S 1 1 1 15 11 11;107 Joj{{ 1!1 -H 1!Ii I I g~~ll10 M S | 11 15112/1 1 1 1L/milfi~~~~~~~~il jl~~li 1S1;1 t 1; Xt | {1{ M1(1 1li 1 F H dl1111 IHI W IIIL III I I 1 AtI I 111g111111 W 1111g111111W 11.111111 1 1 t t I M!IIIIIIIIIIIIIIIIIIIIIIIITIIIIIIIIIIIXIIIIIIIfil1.11 1 1 1I ff~ ~ ~~~~~~~~~~~~~M li7il I I 1 lllllg1~iil~ 1 1 1 11111111iiiiiiilllllllllll~IIIIIII 10 | || ei O C111111if1V S X T i111111111 W IQ LLL1IM B t11 1 1 1 1 1 11 1 1110111111!11111111111|1111 111111111111111111 1 19111 11 11111 1 0-1I11 IM I E tM1|1111111121q1E1 1 1- I I I 1 -; It~llII I 1111 S1111 TrlI I I I I Irl~l~l HIIHID~lililililiill1 11111111111111161Y I i1 1 1 1ill1 1 1 1 1 U) W W l Cflllllllllllllll) N m I I -I I I IIIIIIIHV F 11111111111111 111111111 111 1 1 1 1 1 1 1 [ I I 111111111111111111111F111111 111111111111111111111111 1 1 1 1 1 1 17 N.-I c,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~d 1 E fillllllll-l~ll 11 11 1 II 11111111g i 1111111 011;1111-1 I II FII IIF II T1111111r111111111 I 1 IIIIIITT II-IIl III III l C.,l LL IL Fl I 11111 fill[. II4 I AF -11-I Al l Il I I I 0 00 q N 0 00D (0 q M 0 OD 0 0 C ~o5dd ~ ^ (i - od d ~~ ~.0 22

g9-81-S n3P 'l-8d-3 OZ61 10.0 8.0 6.0 4.0 2.0 1.0 0.8 0.6 0.4 0.2 0.10 0.08 0.06 0.04 0.02 0.01 LI R I C2 FIG. 15. DESIGN CURVES FOR 2-POLE TCHEBYCHEFF NETWORK WITH OPTIMIZED POWER CHARACTERISTIC. U.l 0.2 0.4 0.6 OB 1.0 2 (C LI 2.0 4.0 6.0 8.0 10.0 23

*g-81-9 VdI'3 Z-8d-O OL61 10.0, 8.0 6.0 4.0 2.0 1.0 0.8 0.6 0.4 0.2 0.10 0.08 L$ LI 0.06 0.04 FIG, 16. DESIGN CURVES FOR 3-POLE TCHEBYCHEFF NETWORK 0.02_ WITH OPTIMIZED POWER.CHARACTERISTICS. 0 001 o, I 0,2 0.4 0.6 0.8 1, 0 2.0 4.0 6.0 8.0 I0.0 2 c LI 24

r9S-1-L W3r L-Bd-V 329 10.0 8.0 6.0 4.0 2.0 1.0 0.8 0.6 0.4 0.2 0.10 0.08 R...~~~~~~~~~~~~~~~C4.~~~~~~~~~~~~~.....2 0.04~~~~~~~~~~~~~~~~~~~~ --- --- FIG. 17.;_ ~~~~~~DESIGN CURVES FOR 4~-POLE TCHEBYCHEFF NETWORK 0.02 '- 11111 I LLLLLIlU WITH OPTIMIZED POWER CHARACTERISTICS, 0.01 0.1 02 0.4 Q6 0.8 1.0 2.0 4.0 60 8.0 10.0 2 WC Li 25

C) cJ c0J OJ CM <\i q (0 3 O C-D 0o d0 0 o o d 0 0 10 0 s) - In - O o 0 26

10 5 1.0 0.5 0.1 0.05 0.01 0.1 FIG. 19. DESIGN CURVES FOR 2-POLE MAXIMALLY FLAT NETWORK WITH OPTIMIZED POWER CHARACTERISTIC. 0.5 1.0 2.s cLi 5 10 27

99-IZ-1 Mg t?Z-9d-O Z9ZZ 10 5 1.0 0.5 0.1 0.05 0.01 Ql 0.5 1.0 5 10 2 uCLI 28

yi-IC-l No SC-Ud-U 6C9d 10 5 1.0 0.5 0.1 0.05 FIG. 21. DESIGN CURVES FOR 4-POLE MAXIMALLY FLAT NETWORK WITH OPTIMIZED POWER CHARACTERISTIC. 0.01 I l r 0.1 0.5 1.0 5 2 WCLI 29 10

I -- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX I Consider the network of Fig. 22a. The reflection coefficient p1 is. RI E N LOSSLESS (a) R2 E2,.. I (b) FIG. 22. A GENERAL MATCHING PROBLEM WITH A LOSSLESS COUPLING NETWORK. given by 2 1 ' R12 1 + R1 1Z1+R. (1) The circuit of Fig. 22b is obtained from Fig. 22a by an impedance level transformation producing a one ohm load. For this circuit the reflection coefficient P1 is given by I P 12 |Pzll 2 mp (2) However, Z and Z1 are driving point impedances. It is known that when the 1 1 impedance of the elements of Z1 are lowered by a factor R2, the new driving point impedance Zl is Z1 = (3) R 1I 30

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Substituting Eq 3 in Eq 2, it is found that IP1' 12 = |P12 Since, in general 1p12 = l-|t|2, is follows that 22I2/ 2 2 'B 2 E' _, 2 It112 = IE21R 2Ep 2 4R l I 12 = IE21l 4R1 1E112 4R1 B1 R2 1El12 - E1 2 R2 It is further proved that under these conditions E2' = E2 for a constant generator voltage E1. 31

L ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX II Consider the problem of designing a broadband matching network for a monopole antenna. Hallen (Ref. 5) calculated the input impedance of a lossless uniform monopole over a lossless ground (neglecting base capacitance). He found that if this impedance is plotted against "the antenna length in radians" (=pi = 2n ), the impedance has as a parameter only the length-to-radius ratio I/a. Since no general technique is available for designing matching networks for distributed impedances, a practical approach is to work with an approximate lumped equivalent to the antenna. Assume, then, that the input impedance of a monopole with l/a =60 over a range from Pa = 1.52 radians to Al = 4.71 radiansl is satisfactorily approximated by the input impedance of the circuit of Fig. 23. If the base capacitance of the antenna is neglected, the bandpass to lowpass transformation for which bandwidth is conserved yields the antenna lowpass equivalent of 36 0.00388 ' CBASE (O 50.3 - 0.00278 465 2.675 I FIG. 23. ANTENNA APPROXIMATING CIRCUIT. 1 This corresponds to 3.1'1 frequency coverage. These figures were chosen rather arbitrarily for the example. Note that to make impedance calculations using Fig. 23 one uses the quantity Ai rather than actual frequency. The quality of the approximation of Fig. 23 is indicated in Fig. 24. 32 I

tg-~I1-g W3 1-89-0 OL61 +200 450 FIG. 24. COMPARISON OF VERTICAL MONOPOLE IMPEDANCE WITH IMPEDANCE OF APPROXIMATING CIRCUIT. +150 400 i! I +100 350 +50 300 - 250 I I /0 12 3 AH(ANTENNA LENGTH IN RADIANS) 33 -100 150 - /,' -150 100 -O250 0 -,,.=.. 0 I 2 3,BH(ANTENNA LENGTH IN RADIANS) 33 4 5

- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Fig. 25 (a). By making an impedance level transformation the network of Fig. 25(b) with a one ohm termination is determined. Then the problem of designing a matching network for the antenna is reduced to the problem of specifying a matching 36 0 ---. ~ -- 0.0774 -- 0.00278 465. I (a) (b) FIG. 25. LOWPASS EQUIVALENTS OF ANTENNA BASE CAPACITANCE. WITH NEGLIGIBLE network for Fig. 25(b) over a frequency band from 0 to coc=4.71-1.52=3.19. Using 2 2 =2 — =. —.-)2 9 3=.485, and L1 = 1.293 in Fig. 17 the 4 pole network in the WcLl (3-19)(1.293) dual form of Fig. 9 is as indicated in Fig. 26. 0.159 0.0816 0.0774 0.0885 FIG. 26. 4-POLE TCHEBYCHEFF MATCHING NETWORK. 1 The four pole network rather than the two or three pole was chosen for illustration to allow for downward adjustment of generator impedance level in the final bandpass circuit without ending up with transformers. Note also that although the series inductance of the antenna equivalent in Fig. 25(b) was.0774, no attempt was made to use this in picking the matching network. This is because when )c, L1, and impedance level are chosen, no further freedom in picking the elements is allowed, if optimum networks are used. These points are discussed later. ~~_ _,~... 34

I ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN From Fig. 14, the resulting maximum passband loss is found to be 1.44 db, the ripple.22 db, while the Bode limit is 1.08 db. Note that although the series inductance C2 did not come out to be exactly.0774 (which would allow us to absorb some of the base capacitance in L3) it is fortunate that C2 is larger than.0774 rather than smaller. The matching network can be started out under these conditions with a series circuit. This design is satisfactory where base capacitance has a negligible effect. After making the necessary impedance level and bandpass transformations, the resulting system is shown in Fig. 27 (a). A 41.2 0.0034 1 379 36 0.00388 0.00369 138.9 0.00299s 746.7 50.3 465 (a) 27.96 0.005 37.9 36 0-00388 0.00369 o f ae4F 1 465 (b) | 0029{ I46.7 50.3 278.00 99 0000278 IDEAL 27.96 0.00413 37.9 36 0.00388 I94.4 ~ I- - 71 0.003 465 (69 0.00226 94.4 o,, T ' I A. AL o465 (C) I A (NOTE: VALUES IN OHMS, FARADS, AND HENRIES). FIG. 27. COMPOSITE ANTENNA NETWORK. 35

bS-OZ —I W3P ~gI-1-V Z9ZZ (a0) ZI 1LC L Z2 LC(b- l b LC(I-b)/b2 + L/b (b) Z2 b2 IDEAL Z b2 Z2/b2 (c) (c) C (a.) (b) 2 t C CL/I- a I(- 1 C L/a (a-I) -_C-. — ~ D~ZI I CL IDEAL D^. Z2 o.2z2 [~O2Z (c) (c) FOR REALIZATIONS WITHOUT TRANSFORMERS I b< I + L I <aI + CL LC C ALTERNATIVE FORMS IN WHICH THE IMPEDANCE LEVEL AT END ZI IS MODIFIED ARE OBTAINED BY GENERAL IMPEDANCE LEVEL TRANSFORMATION FIG. 28, IDENTITIES USEFUL IN MODIFYING SOURCE TO LOAD IMPEDANCE RATIO. (a) CIRCUIT FOR WHICH MODIFIED SOURCE TO LOAD IMPEDANCE RATIO IS DESIRED, (b) EQUIVALENT CIRCUIT WITH MODIFIED SOURCE-TO-LOAD IMPEDANCE RATIO USING IDEAL TRANSFORMER. (C) EQUIVALENT CIRCUIT REALIZABLE WITHOUT COUPLED COILS L CL FOR -I< b < I + L I< o.<_+ C LC ' 36

- ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN - Now, assume that it is desired to insert a coaxial line with Zo = 94 ohms at A-A in Fig. 27(a). The generator impedance is adjusted by inserting an ideal transformer, and absorbing it as indicated in Figs. 27(b) and (c).1 The photograph of Fig. 29 is the experimental t|2 vs. frequency curve of a network covering the frequency range 6.24 to 19.35 me based on Fig. 27 (c). The loss in the band is about 1.7 db. i FIG. 29. EXPERIMENTAL Jt|2 VS. FREQUENCY. (CIRCUIT OF FIG. 27c). 1 The circuit of Fig. 27(c) is realized using one of the identities of Fig. 28. These identities are easily proved by considering the open circuit parameters Zll, Z12, and z22 of the coupling circuit. These identities permit limited modification of the source to load resistance ratios without transformers for bandpass circuits. Modification of the source to load resistance ratio of lowpass circuits cannot be accomplished without transformers. 37

- ENGINEERING RESEARCH INSTITUTE APPENDIX Equations of optimum maximally flat networks UNIVERSITY OF MICHIGAN n =2 - OcL1 L1 C2 n= 3 2 wcL L1 C2 C2 = f2 31/ L + = 25/6 1+ 1/A 3 l-( 1 3 Itl min 4 2 3/4 1-(- 4 min) 3/ {1-(1- 3 Itl ) 1f 46 2 \1/6 1 [i(1- tI 2min 16] e 2 11/2.~~ ~ l-l —t in) - I {1-(l- 6 t i)1 } L3 L1 L1/C2 1-(1- 6 |t |2in )5/6 l-(l-ng|t - 1 - L1/C2 + C2/5L1 n=4 - CocLi =.2.7 -(1-/+( 8 1It2 )/8] 0 (sin 22~ + cos 22i ) 1i C2 1 1 + 3 1-(1- 8 t2 )/8 - t min). 0 cos 22- - sin 2212 0 ] {1-(1- - t ) 8}3 7 min 0 0 3 os 22 + in 2 ) (cos 222 + sin 222 ) 38

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN C4 4 + C1 ii 8 11j2 1/8}7 t_ min) where 0, 1 0 A.le r,/' 1 COS 2'2 - >iX L 2 0 o05 (cos 22~ + sin 22 ) o 2 sin 22~ 2 0 (cos 22 + sin 2 39

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN APPENDIX IV 2 Consider the 3 pole maximally flat network specified at-2L =.1 mcL1 obtained by extrapolating the maximally flat 3 pole design curves. From the data 2 )CL.1 L3 L1 1/3 C ='.0027 L1 R a 18 and choosing wc = 1 the networks of Fig.30a and b are determined 20/3 20 0.054 18 (a) (b) FIG. 30. THREE POLE NETWORK FROM FIG. 20. Now by raising the generator impedance level to one ohm, the circuit of Fig.31a is produced. Remembering that U3db = Oc 2n-1 2I a bandwidth transformation 0.972 1.27 18 (a.)c = I (b) )3db = FIG. 31 MODIFIED NETWORKS FROM FIG. 30b. 4o

-- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN is performed to produce the corresponding circuit with unity 3db bandwidth (Fig. 31(b)). The network is approaching the well known network with unit 3db bandwidth producing maximally flat behavior of e2 2 from a current generator (See Fig. 32). il LI - o0 - 4/3 v 3/2 1/2 i e2 FIG. 32. A THREE POLE NETWORK WITH MAXIMALLY FLAT BEHAVIOR OF | This circuit is the limiting case of zero power transfer efficiency, since only finite power is obtained from a generator with infinite available power. 41

REFERENCES 1. S. Darlington, "Synthesis of Reactance 4-Poles", Journal of Mathematics and Physics, XVIII, pp 275-353, Sept. 1939. 2. R. M. Fano, "Theoretical Limitations on the Broadband Matching of Arbitrary Impedances," Journ of Franklin Inst., 1950. 3. C. B. Sharpe, "Tchebycheff RC Filters," Doctoral Thesis on file in the University of Michigan Library, 1953. 4. H. W. Bode, "Network Analysis and Feedback Amplifier Design," Van Nostrand, 1950. 5. E. Hallen, "Admittance Diagrams for Antennas and the Relation Between Antenna Theories," Technical Report No. 46, Cruft Laboratories, Harvard University, June 1948. 42

UNIVERSITY OF MICHIGAN 3 9015 02223 2410