ENGINEERING RESEARCH INSTITUTE THE UNIVERSITY OF MICHIGAN ANN ARBOR SYNTHESIS OF LOSSLESS NETWORKS FOR PRESCRIBED TRANSFER IMPEDANCES BETWEEN SEVERAL CURRENT SOURCES AND A SINGLE RESISTIVE LOAD Technical Report No. 76 Electronic Defense Group Department of Electrical Engineering By: A. B. Macnee Project 2262 TASK ORDER NO. EDG-7 CONTRACT NO. DA-36-039 sc-63203 SIGNAL CORPS, DEPARTMENT OF THE ARMY DEPARTMENT OF ARMY PROJECT NO. 3-99-04-042 SIGNAL CORPS PROJECT 194B November 1957

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iii ABSTRACT iv ACKNOWLEDGMENT v 1. INTRODUCTION 1 2. THE SYNTHESIS METHOD 1 3. EXAMPLES OF THE METHOD 6 4. PRACTICAL CONSIDERATIONS AND CONCLUSIONS 11 REFERENCES 16 DISTRIBUTION LIST 17 ii

LIST OF ILLUSTRATIONS Page Figure la Parallel Lossless Networks with a Single Resistive Load 3 Figure lb Parallel Lossless Networks Driven by a Resistive Source 3 Figure 2 The Situation for One Typical Lossless Network in Fig. la 4 Figure 3 Example of a Low-Pass, Band-Pass Frequency Multiplex Network 9 Figure 4 Example of a Two-Channel Bandpass Network Realization 12 Figure 5 Experimental Set-Up of a Two-Channel LowPass Partitioning Network 14 iii

ACKNOWLEDGMENT The writer is happy to acknowledge discussions with Dr. B. F. Barton which contributed to the ideas presented here. iv

ABSTRACT A technique is presented for the synthesis of lossless networks open-circuited at one end and paralleled across a single resistance at the other end. The synthesis is for prescribed transfer impedances between the open-circuited terminals and the resistive termination. Such networks can be applied to a variety of frequency multiplexing problems, including the design of multi-channel amplifiers. Examples are included, and some practical limitations of such networks are considered.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN SYNTHESIS OF LOSSLESS NETWORKS FOR PRESCRIBED TRANSFER IMPEDANCES BETWEEN SEVERAL CURRENT SOURCES AND A SINGLE RESISTIVE LOAD I. INTRODUCTION The synthesis of a lossless network for a prescribed transfer impedance when terminated in a resistive load was developed by Cauer (Reference 1) and has been described by Guillemin and others (References 2 and 3). The purpose of this note is to present a generalization of that technique to the case of multiple current sources. The resulting networks are potentially useful in the synthesis of wideband multi-channel amplifiers, and in frequency multiplexing systems of all types (References 4 and 5). 2. THE SYNTHESIS METHOD The circuit considered is illustrated in Figure 1 a and b. In 1 2 n Figure la n lossless networks driven by current sources I I....I1 are paralleled at their outputs across a single resistive load. This situation is of interest at the output of a multi-channel amplifier employing pentode tubes or in the multiplexing of a number of carrier channels to a single transmission facility. In Figure lb a single resistive generator drives a number of lossless networks in parallel to produce output voltages E....E This situation might be encountered at the input of a channelized wide-open receiver connected to a single antenna or in the input of a wideband multi-channel amplifier. In the

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Figure la the output voltage E2 is given by 1 1 2 2 2 12 1 12 I1 12 ( where z is the transfer impedance from a current source I at the input to the 12 1 J network to the voltage across the load resistance R when all networks are paralleled across R. As long as the lossless networks are also bi-laterial, the output voltages in Figure lb are given by 1 1 E =Z I E2 Z12 I1 2 2 =E2 2 Z12 1 En n I (2) 2 12 1 by reciprocity, the transfer impedances Z12 are the same as those in Eq. 1. A technique for synthesizing the lossless networks of Figure 1 for prescribed 21 Z2 Zn will now be presented. Z12 Z12 12 Since the inputs of the lossless network of Figure la and the outputs of the networks of lb are open-circuited, one is naturally led to consider the open-circuit impedance parameters of these networks. These parameters are defined by the equations = z I +z I (3) 1 11 1 12 2 - z Ij + z (4) 2 21 1 22 2 for the positive voltage and current directions shown in Figure 2. With a load impedance ZL connected at the output terminals, and a bi-lateral network, the transfer impedance for this network becomes A 2 12 L (5) 12 z + Z 22

II 0 E2 Z 12, 22 FIG. I (a). PARALLEL LOSSLESS NETWORKS WITH A SINGLE RESISTIVE LOAD. 0i~~st no 0 tE2 Iz, 22 FIG. I (b). PARALLEL LOSSLESS NETWORKS DRIVENTH A SINGLBY A RESISTIVE SOURCE. n n z2 22

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN FIG. 2. THE SITUATION FOR ONE TYPICAL LOSSLESS NETWORK IN FIG. I (a). For the j network in Figure la the load impedance is zT 1 (6) L 1 1 I 1 +sy +..+ + s +.. ~R 1.~i-1 j+l n z22 Z 222 22 Dividing through by z22 ZL Eq 5 can be written z 22 112 = 1 1 1 (7) + + + — R 1~n z22 z22 It is clear from this result that all the transfer impedances in Eqs 1 and 2 will have common poles. Thus Eq 1 can be written1 E = (8) R+ The poles of Zl2 must lie in the left half of the complex p-plane. Given E NjIj E2 (9) A + pB 0 0 this means A + pB is a Hurwitz polynomial. To identify Eq 9 with Eq 8 two 0 0 possibilities exist: (1) split the odd part of the denominator pB into n parts 1. This concise formulation was pointed out to the writer by B. F. Barton.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN and'write nE (10) A o / parts giving B j=l A ~ where Z pBs pB or (2) split the even part of the denominator A into n j1 0 12 BJ 2 ( 14) parts giving casem n ~, Nj (ij) 22 po ~~~~~and~~ j=1 \A0J n where Z Aj = A ~ In case 1 one identifies 12 Nj 12(12) -- - (A2) 22 0 and pB-2. (13) so that zj = Nj (14) 12 PBJ Similarly in case 2 (15) pB0j z22 = (16) A0 and z12 i A o~~~~~~

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Examination of Eqs 14 and 17 points up a limitation of restricting ourselves to lossless networks and serves as a basis for choosing between case 1 and 2. Since z12 must be purely imaginary for p = jcu, we see that the numerator polynomials N. must all be even or odd for this identification to be realizable. If the Nj are even polynomials, case 1 is applicable; if they are odd, case 2 J j would be chosen. In either case z12 and z22 have the same denominator and hence, in general, common poles. The only other requirement necessary to assure the success of this identification is to split A = A1 + A +n 0 0 0 0 or 1 2 n PBo pBo + pB +.. + pBo so as to make z22 a positive real function. A division that is always realizable is Ai = Ao/n or pBj = PBo/n. In general, one has a considerable range of freedom in making this division which can be used to simplify some of the networks to be synthesized or to control other parameters of interest (such as the ratio of capacities at the inputs to the reactive networks). Once a satisfactory division of Ao or pBo has been chosen each lossless network is synthesized by expanding zJ22 for the prescribed zJ (Ref. 2). This can 22 12 always be achieved to within a constant multiplier. The importance of this multiplier will depend upon the specific application of the multiplexing network. 3. EXAMPLES OF THE METHOD To illustrate the ideas of the previous section consider a two channel example such that1 1. For this function the components of |E2| due to I1 and I1 alone add up to a Tschebycheff lowpass filter characteristic with a maximum 1 db attenuation in the passband from zero to one radian per second. 6 "

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 1 22 I +2 p I 1 p11 (18) 1 + 3.200 p + 3.497 p2+ 2.941 p3 Since the numerator polynomials N1 = 1 and N2 = 2p2 are both even functions of p, we must use case 1. The most general split for pB is pB = kop + kp = ko(+- P) (19) o 3 pB2 = (3.200-ko) p + (2.941 - kl) p3 (20) while A = + 3.497p. (21) 0 Therefore, according to Eqs 12 and 13, we identify 1 1 + 3.497 P (22) 22 (22) koP (1 + kOO P2 1 1 (23) z12 (23) 21 2 2 _ + 3.497 p2 -z~[ ro ](24) 2 2p2 1/ (25) (3.200-k )p ( + 1 2 o [1+ y3.200-k P The conditions imposed by the desired positive real character of Z2 are: k > O 0 and k< 3.497 * (26) 0

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 2 Similarly the conditions for z22 to be realizable are ko < 3.200 kl < 2.941, (27) and 2.941-kl < 3.497 3.200-ko Since z12 is itself realizable, the bandpass network can be directly 2 2 2 synthesized as an L network, with z22 - z12 in the series arm and z 12 as the shunt arm, provided 2 2 = 1 + 1.497 p (8) 22z 12 ( (3.200-ko) P +. p2 o L.200-k is also realizable. This will be true if 2.941-k1 1497 (29) 3 200 kl s 1.497 * (29) 3.200-k 0 A minimum of elements are required when condition 29 is met with the equal sign. For this case the condition 29 becomes 1.497 k - k1 = 1.849. (30) o The network parameters are then 2 2 1 z22 -z = (31) (3.200-ko)p and 2 12 2p (32) (3.200-k0)p (1+1.849 p2) The network is shown in Fig. 3a. 1 1 Since all the zeros of z12 lie at infinity, expanding z22 in Cauer's first form (alternatively removing poles of admittance and impedance at infinity) produces a satisfactory ladder network. This realization of Eq 22 is shown in Fig. 3b. Within the conditions 26, 27, and 30 one still has a considerable

(3.200 - ko) f (a.) 0.7485(3.200-ko)f 3.200- ko}h *- 22 3.497 o-3.497 ke. I T m k, 2.616 h (C,) I3a.a}, k- k,337 f FIG. 3. EXAMPLE OF A LOW-PASS, BAND-PASS FREQUENCY MULTIPLEX NETWORK; (a,) BAND-PASS NETWORK, (b.) LOW-PASS NETWORK, AND (c.) COMPLETE NETWORK WITH CONSTANTS ADJUSTED FOR EQUAL INPUT CAPACITIES.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN freedom in the choice of k1 or k0. Choosing k1 = 0, for example, will reduce the number of reactive elements required. Another choice of interest is to try to make the input capacities of the two networks equal. This will be true provided 1.7485 ko - 0.2859 k1 = 2.395. (33) This condition together with Eq 30 solved simultaneously give the particular values ko = 1.413 and kl = 0.2663. Since none of the other conditions 26 and 27 are violated, this is a useful result. The complete network using these numerical values is given in Fig. 3c. As a second example consider the synthesis of a two-channel bandpass circuit such that - Pi + p3 E 4 (34) 1 + 2p +3p +2p +p4 The numerator polynomials are both odd in this example, so one is lead to case 2. The odd part of the denominator is pB = 2p + 2p3 (35) and the even part is split into two parts A = k + k p + k2p (36) and A = 1 - ko + (3-k1) 2p+ (1-ko)p. (37) Then one identifies k 2p + 2p3 p 12 - k2 2 - 4 (38) k + k1p + k2p la23, z 1 (39) k +k 2

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 2 2p + 2p3 z22 2 4 1-ko + (3-k1)p + (l-k_)p and 2 P p3 12 1 - ko + (3-k1)p2 + (1-k2)P With constants ko, kl, and k2 to be specified the possibilities open to the network designer are numerous. A choice which simplifies the structure of the number one channel considerably is to let ko = k2 = 0. The network realization for this case is shown in Fig. 4a. The transfer impedance of the second channel has three zeros at zero frequency and one at infinity. A ladder structure having the correct poles and this distribution of zeros is obtained by removing one pole of admittance at infinity. The ladder obtained by this development is given in Fig. 4b. Checking, however, one finds that the transfer impedance for this ladder is - _p3 z12 2 4 (42) 1 + (3-k1)p2 + p which for all positive, non-zero kl is less than Eq 41 by the constant factor 2-2k1/2-kl. Normally an overall scale factor times Eq 34 would not be serious, but the relative responses are important. These can be equalized by a tapping down on the output of the first channel. The result of making this adjustment and picking kI = 0.5 is the complete network of Fig. 4c. Alternately one might go back and make another choice for k and k2. 4. PRACTICAL CONSIDERATIONS AND CONCLUSIONS At the out-set it was hypothesized that the synthesis of a number of lossless channels with a single resistive load was a desirable objective. Such a 1. For the particular choices of k = k2 = 0 it is not possible to obtain equal input capacities in this case. 11

Ls-9Z -8 9 ~I1-IS-v k, f 2 h k,f e —--— ~~Z 01F ~~~~ —' {I --, f 03/4 f (C.) {3/4 3/2 f ~ 4/9 h /82 h FIG. 4. EXAMPLE OF A TWO-CHANNEL BANDPASS NETWORK REALIZATION. 12

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN structure is attractive for power amplifiers employing pentodes, or the input to low-noise amplifiers. On the other hand, nature exacts a considerable price for this freedom from excess dissipation. One limitation, already indicated, is that the numerator polynomials used must always be either even or odd functions of p. This may considerably limit the approximation techniques one can employ. Another practical limitation can be best illustrated by considering the network of Fig. 3c. The load impedance of the lowpass portion of this network is the parallel combination of the one ohm load resistor and the output impedance of the bandpass network, Z22 According to Eq 34 this impedance has a zero at C = + 0.535 radians per second. This zero is not, however, a zero of the overall transfer impedance from the low-pass input, z12, specified by Eq 18. This is 1 2 physically accomplished by having a zero of y11 coincide with the zero of z22l This means that when the output of the lowpass network is short-circuited by the 2 1 zero of z22, the impedance seen by the current source I1 becomes infinite. The resulting infinite current through the 2.616 henry inductance is just sufficient to produce the desired finite voltage across the short-circuited load resistor. It is clear that such a pole-zero cancellation can be expected to require very critical trimming of at least one component in a practical network. Experimental investigations reveal that these lossless networks terminated in a single resistance can indeed by rather critical of adjustment. Figure 5a illustrates the result of a 10% change in the shunt capacity of each channel on the individual responses E2/I1 and E2/I2 for the network of Fig. 3. While the resulting distortion of the responses are severe, it was also found that the shunt inductance of the bandpass channel always could be used to trim out the shape distortion. 13

LG-9Z-8 8 ~1 —1S-Vt IN79 200 C 40 0.331kLfId IN79l OSCILLATOR TO OSCILLOSCOPE ANE C =0./248 Lfd 0.348mh 248 MEG LAND C =0.2738 fd 403020 1C0 7.5 KC (b) __ANGES C,-0,248, fd O.I p. fd, X~~~~~CMR 14,

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN The fact that these networks are critical of adjustment is not surprising; this is a well known drawback of a double-tuned circuit only loaded on one side, see for example Reference 6. The network designer can reduce these effects by employing Barton's insertion loss design method or by introducing some dissipation into the lossless structures through Darlington's predistortion techniques (References 6 and 7). In each case the effect is to move the polezero cancellations into the interior of the complex p-plane where they have less influence on the response along the jw-axis. Another approach is to control the approximation problem so as to move these pole-zero cancellations along the jwaxis to some region of less importance to the desired overall response. 15

REFERENCES 1. W. Cauer, "Ausgaugsseitig Leerlaufeude Filter," Elek. Nachr. Tech.y Vol. 16, No. 6, 1939, pp.161-163. 2. E. A. Guillemin, "A Summary of Modern Methods of Network Synthesis," Advances in Electronics, Vol. III, pp. 276-279, Academic Press, New York, 1951; also "Synthesis of Passive Networks," pp. 445-452, Wiley and Sons,- New York, 1957. 3. J.E. Storer, "Passive Network Synthesis," pp. 181-195, McGraw-Hill, New York, 1957. 4. J. Linvill, "Amplifiers with Prescribed Frequency Characteristics and Arbitrary Bandwidth',"MIT Res. Lab. of Electronics Tech. Report No. 163, 1950. 5. B. F. Barton, "Synthesis of Multi-Channel Amplifiers," Electronic Defense Group Technical Report No. 70, The University' of Michigan Engineering Research Institute, Ann Arbor, Michigan, 6. Wallman and Valley, MIT Rad. Lab. Series, Vol. 18, p. 300. 7. Bode, Net. Analysis and Feedback Amplifier Design, pp. 216-222, Van Nostrand Co., New York, 1945.

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