Copyright © 1964, by the author(s).

All rights reserved.

Permission to make digital or hard copies of all or part of this work for personal or

classroom use is granted without fee provided that copies are not made or distributed

for profit or commercial advantage and that copies bear this notice and the full citation

on the first page. To copy otherwise, to republish, to post on servers or to redistribute to

lists, requires prior specific permission.

Internal Technical Memorandum M - 61

COMPUTER GENERATION OF

EQUIVALENT NETWORKS

by

D. A. Calahan

March 3, 1964

COMPUTER GENERATION OF EQUIVALENT NETWORKS

D. A. Calahan

Electronics Research LaboratoryUniversity of California

Berkeley, Californiaon leave from

University of IllinoisUrbana, Illinois

Summary

The theory of continuously equivalent networks is extended to the state equations and toinclude a scaling option. The theory is thenapplied to the classical problem of the design oflow- and band-pass filters with loss in inductorsonly. The alignment problem in active filters isalso approached from an equivalence viewpoint.

Introduction

Although the subject of equivalent networkshas long been recognized to be of academicinterest, it is only recently that a consciousattempt has been made to utilize the theory inpractical design problems. This paper extendssome recent work in this subject and then considers some examples of use of the computer ingeneration of equivalent networks.

The Approach

A differential approach to the equivalent network problem was first proposed by Schoeffler^and has been investigated by other authors. 2» 3It will be found useful to work with the A matrix

rather than the nodal or branch admittance ma

trices, so that the development of Schoefflermust be altered.

Assume that the state equations of the circuit are of the form (with no mutual inductance,capacitor loops, etc.)

Jl ..L

'1 .

d

~3T

rsi 0

. 0

•-

k v

0

Vc

m 0

(1)

- M W = W M - H vm <2>th

Assume vjn appears only in the r equation asshown and further assume that one of the state

variables (the g**1) is the output voltage. Both ofthese conditions can be satisfied by placing aninductor of zero-value in series with the source

and a capacitor of zero value across any pair ofnodes regarded as the output. Further, assumethe equations are arranged so that r = 1,g = p = m + n.

Taking the Laplace Transform of both sidesyields (for zero initial conditions)

H vm = (sM + HI M (3)where

[Y] =[Y,a>] = V. = V. (s) (4)in in* ' * '

Now consider the problem of adjustment of thenetwork elements while maintaining constant theratio of the output voltage to the input voltage.Letting each element be a function of some dummy variable x, then one may postulate a differential change in the element values that producethe new state variables

[y(x +Ax)l = {[V] + Ax [a] } [y(x)]1 + Ax a

(5)

where

M-721 °22

p-1,10 0

PP

Ip

, [Y(x)]=[Yks)]

pp (6)

so that yp(x + Ax) = yp(x) and equivalence ismaintained. Eq. (3) then becomes

[b] Vin =\s [M(x +Ax)| +fA (x +Ax)] UY(x +Axtt

={a [M(x +AxJ +[a(x-1-AxJ}{[u| +Ax[a|} [y(x)J1 + Ax a

PP (8)

Taking note that

1) (B(x)] has only one nonzero entry2) V. (x + Ax) = V. (x) since V. is a'in1 ' inx ' in

source.

then

[B(x+Axl V.n(x+Ax) ={fu]+Ax[b]}[B(xjvin{X)1 + Ax b.

if

W

bll b12' IP

0 b22* . b72p

0•

•

0 P2 . bPP

11(9)

(10)

Multiplying both sides of Eq. (8) by{[u] + Ax [bj} /(l +Ax bn) gives

ISK

{HO + Ax[b]| {s[M(x +AxJ +Aftx +Ax)]}

Thus

(1 + Ax a ) (1 + Ax b„)pp' x 11'

{[U] +Ax [a]} [Y(x)](11)

={s[M(x| + [A(x)]} [Y(x)] (12)

L[M(xjJ +[A(xj| ={[u] +Ax[b]}{s[M(x +Ax)] +[A(x +Axj)}{lu]+ Ax [j)

(1 + Ax a ) (1 + Ax bn)pp 11

Separating powers of s yields

[M(x +Ax)j - [m(x)]Ax

(13)

[b][M(x +Ax)] +[m(x+Ax)] [a] - (bn +app)[M(x+Axj+ 0(Ax,

[A(x •>• AxQ - fA(x)1Ax

(14)

b][A(x +Ax)] +[A(x +Ax)][a] - (bn +app) [a(x +Ax)]+ 0{Ax)

which in the limit become

(15)

• (bn+ v H (16)

tl =[b][A]+ [A][a]- (bn+ aPP> W • <n>

d x

d x

pp

The form of this equation is somewhat moregeneral than Schoeffler's. 1

If the right-hand side of Eq. (9) is notdivided by 1 + Ax b... then Eqs. (16)and (17)become

W--WW ♦[*][.].. w «.9,

(18)

Further, Eq. (9) may be solved in part to yield

V. (x + Ax) = (1 + Ax b„) V. (x) (20)in 11 in

so that taking the limit as before gives

Vin(x) =Vin(0) exp fbllX) (21)

Hence eliminating the term 1 + Ax b.. resultsfunction by

t tion willfollowing example

nCHUC CJ.XIlli.IlCkli.tlg CllC ICIIIl X T UA u«« xc

in a scaling of the voltage transfer functioan amount expf-bi^x}. The scaling opti<be found useful in the following examples:

Application 1: Nonuniformly Lossy Ladders

The object of the following examples will beto use the previous theory to generate lossyresistively terminated networks from losslessones. Thus, the state equations are a naturalapproach to the problem since the lossless elements are contained in [Mj , whereas the losscomponents are contained in [a].

Example 1: Singly-terminated Ladders:Geffe* has given exact formulae for design ofsingly-terminated ladders with lossy inductors.A simple example is the network of Fig. 1. Asan exercise to test the mettle of this equivalenceviewpoint, one could attempt to obtain similardesign formulae for the simple case. Thematrices of interest are

[M>

[*

Cl/Gl

"11

0

0

12

'22

'32

&

13

'23

'33

1 Rj 0 1

0

0

(22)

•1 R2 1 B =

0>

11 a12 **13

21 a22 a23

0 a33

(23)

where G, = 0. Inserting the above in Eqs. (18)and (19) results (in part) in trivial constraintswhich require fbl and fa] to be of the form

A similar statement applies if 1 + A x a isPPeliminated in Eq. (5).

w-

bll -al2<Cl/L2Gl> -a13<Cl/C2Gl>

'11-a23(L2/C2)

"a22

H- 0

0

l12

l22

«24)

13

23<

Further, since the voltage transfer function cannot be sealed by growing a resistor in series withLr bu =0 .

The remaining constraints generate thedifferential equations

= all Cl/Gl

Ll =(a22 " *Yl> Ll

C2 = (-a22 C2)

(25)

(26)

(27)

R,' =i =an R2 +a12 +a13 (C1/C2 Gl> +a22 Rl

R2 =(all " a22) R2 " *12

(28)

(29)

where a12 =-au L^Gj/Cj) a13 =-au . (30)

These equations are readily solved to yield

anx

Cl/Gl =C?/Gl e

° „(a22 " all)x

22 x

L1 = L1 e

C2 = C2 e

(31)

(32)

(33)

R2 =(L° G°/C°).exp(a22 - a^x [l - expCfe^x]

+ R2 exp (a22 - an) (34)

Rx =-R2 +[r° +R^ +(C°/G° C°)(l -exp ^ xj]

exp|a22 xjr (35)

thus achieving exact design formulae for anychoice of a., and a.

It has not been found possible to extend theabove analysis to include Geffe's general case;however, the correct matrix manipulation mayyield a closed form result.

Example 2; Doubly-terminated Laddero:

An attempt has been made to find a closedform solution for the simplest doubly-terminatedladders (Fig. 2a and 2b). The matrices for thenetwork of Fig. 2a are identical to those ofExample 1, except that now G3 £ 0, b^ 4 0,C[ =0, and G3= 0. If there is a closed-formsolution, it is not apparent from the differentialequations and so a computer solution was necessary. It may readily be shown that there arefour remaining linear constraints on the five unknowns of Eq. (24), leaving one unknown to beselected arbitrarily. Selecting b^ to be thisconstant and letting bn =1,**then the voltagetransfer function will be scaled down, as indeedit must if resistance is to be inserted in serieswith the inductors. Further, the flat loss introduced is readily shown to be (20x/2. 3) db. Theflow chart is shown in Fig. 3.

In transforming the network of Fig. 2b,seven equations in eight unknowns are generated,so again selection of bji =1 generates a uniqueset of constraints. It snould be noted that, tokeep,identical dissipation factors in each inductor, a constraint of the form

t22'

must be imposed as the integration proceeds.Element value s for the fourth order.

Butterworth filter are given in Table 1. ' Forr = 1 (equal termination), it was found impossible to generate any equivalent nonuniformlylossy networks.~T*or 0 < r < 1, either of twolossless realizations may be used as initialnetworks in the transformation process; thefirst is found in Ref. 5, the second is the dualand impedance scaled version of the first. Thedual network was found to allow for significantly

*The network could be normalized to unit sourceresistance by using the expression for Rj as animpedance scaling fact or.

Note that the a's and b's may be seal ed arbitrarily, since this is equivalentlo scaling x.+

Typical computing, time was two minutes .on anIBM 7090.

more loss, except for r = 0 when no dual exists.The maximum inductor loss was found in everycase to strongly depend on the termination ratio,and the largest d is approximately twice thatallowable for uniformly-lossy networks.

The nonuniformly lossy band-pass ladder (Fig.4) may, of course, also be approached from theequivalence viewpoint. In this case, true equivalence cannot be maintained since any inductorloss will shift a transmission zero from theorigin to the negative real axis. However, if oneconsiders the output to be the inductor current ofL? rather than the capacitor (output) voltage C^,then v (s)

IL(s) = R2 + s L2

the bothersome transmission zero disappears, andonly a small error is made in keeping the actualfrequency response of the filter invariant (ifd =R2/L2 is small). * It now happens that thedual network allows less inductor loss (Table 2).Further the maximum value of d allowed forany value of Q was approximately twice thenormalized real part of the pole closest to theimaginary axis and so is again twice that expectedfrom uniform predistortion arguments. Finally,calculation shows the filters with the largest inductor loss are overall somewhat less sensitivethan the lossless variety. The price paid is thelarger element value spread in the former.

Application 2: Active-RC Networks

One of the difficulties in the design of active-RC filters is the sensitivity problem. Classically,this has been approached from an a priori viewpoint, i. e. , how to minimize the effects ofchanges in the active element on the network response. Viewing a posteriori, the problemchanges to that of alignment or "how to restore theoriginal transfer function" or, formally, the equivalent network problem. We may therefore consider the application of the computer in establishingan alignment procedure for such networks.

Example 1; Consider the low-pass filter ofFig. 5, b but normalized to a source resistanceof 1R. The matrices of interest are

M = A =

YA + YB + 1

Y - YL.M B

Jll-a12(Cl/C2)

all + a12b =

U22-J 22

(39)

The flat loss is now (20x/2. 3) - 20 log^Q L-,(x)/L2(0); for the dual band-pass network, it happensbil must now be normalized to -1 to grow positiveresistors.

There are six elements and five a's andb's to be chosen. In solving for the elementalderivatives as in Eqs. (25)-(29), it happens thatajj and a22 appear only as the differenceall " a22* 3° tnat only four a's and b's are distinct, xhus, constraining three elements toremain unchanged (Yc, Ci, and C? being themost different to change in lumpedcircuits),bn can be chosen to be the only arbitrary constant. Interpreting Yy^, Yg, etc. in terms oftransistor parameters, Re(x) and Rf(x) maybe obtained to maintain equivalence for a givenchange in p* ((J(x) Fig. 6 ). It is noteworthythat

1. a minimum value of p" exists belowwhich the transfer function cannot be maintained

(without a change in Yc, C^, or C2).2. a region (3 < x < 7) exists over which

j3(x) is relatively constant. If the circuitshould operate in the region then, a smallchange in p* would have to be compensated bya large change in both Rf and Re— this inspite of the low sensitivity in this region.There does exist a simple relationship betweenthe sensitivity functions and the curves of Fig.7: Taking the derivative of the transfer function T(s), one has

dT

d x

_ 8T dp + 8TdR

e

8 p dx 3Re dx

8T ^f8Rr dx

(40)

so that the sensitivities (8T/8p, 8T/8Re,8T/8Rf) and the slopes of the curves of Fig. 7(dp/dx, dRe/dx, dRf/dx) are orthogonal. Eq.(40), however, .gives no hint to the relativemagnitude of each set.

Conclusions

There are several means of extending theresults given for nonuniformly lossy filters:

1. Generate tables of element values for alllow-order Butterworth, Tchebycheff andmaximally-flat time delay networks.

2. Explore the possibility of using the degrees of freedom available in the interactionprocess to obtain selected element values. Forexample, if the constraint of Eq. (36) is notimposed, one of the a's may be selected arbitrarily and the problem of using this freedomproperly is posed.

3. Search for exact formulae for each ele

ment value, either using the classical approachof Takahasi? or, using Takahasi's results asinitial conditions, generate, by means of matrix manipulation in Eqs. (18)-(19), elementalvalues as functions of x as in Eqs. (31)-(35).

Concerning active filter design, it would beof interest to

1. Develop design graphs for active filters(particularly band-pass) which permits a choicein the characteristics of the active element

(similar to Fig. 7).2. Investigate more thoroughly the possible

degeneracies which arise in the constraint equations, so that one could predict if possible, thenumber of "alignment" elements he must beprepared to vary to accommodate for an uncontrolled variation in some other element. Withthe advent of integrated circuits, which can bedesigned with resistive elements that vary in acontrolled fashion, such graphs could also beused to determine the variation required tomaintain a response characteristic over wideenvironmental changes.

Acknowledgment

The use of the facilities of the ComputingCenter, University of California (Berkeley), isgratefully acknowledged. The research reported herein was supported in part by AFOSRGrant AF-177-63_, University of Illinois.

Bibliography

1. J. D. Schoeffler, "Continuously equivalentnetworks and their application, " Tech. Rpt.No. 5, Case Inst, of Tech. ; December 2, 1962.

2. A. D. Waren, "Equivalence of nonrecipro-cal active and passive networks," Tech. Rpt.No. 4, Case Inst, of Tech. ; April 20, 1962.

3. D. A. Calahan, Modern Network Synthesis:N-ports, Active Networks and Related Topics,"Vol. II, Chap. 5, Hayden Book Co. ; 1964.

4. P. R. Geffe, "A note on predistortion, "IRE Trans. , Vol. CT-6, No. 9, p. 395;December 19 59.

5. L. Weinberg, Network Analysis andSynthesis , McGraw-Hill; 1962.

6. I. M. Horowitz, "Optimum design of singlestage gyrator-RC filters with prescribed sensitivity, " Tran^IRE, Vol. CT-8, pp. 88-99;June 1961.

7. H. Takahasi, "On the ladder-type filter network with Tchebycheff response, " J. Inst.Elec. Comm. Engrs. , Japan, Vol. 34, No. 2,February. 1951. (In Japanese.)

*1 JEv_

•vwv^

L,

Fig. 1, Singly-terminated ladder.

-WWII Ll-WW—^

r "1 *1

o—VW\—VWv—r>r

(a)

(b)

R2 L2

Fig. 2. Doubly-terminated ladder.

Element Values for Fourth Order Nonuniformly Lossy Butterworth Lowpass Filter (Figure 2)(d =R^L.)

r = 0 (not dual) * = lA (dual)

0.00

0.050.1

0.150.2

0.250.30.350.40.450.50.550.60.650.680.70

.0.710.72

0.730.7360.740.7450.7480.750.7530.7550.757

1.5311.641

1.7571.8772.002

2.1282.2542.3752.486

2.5772.6332.6292.5252.250

1.9551.6791.5H1.310I.092

0.9450.83730.70000.60740.5449O.44760.38070.3124

0.00

0.030.060.090.12

0.150.180.21

0.240.270.300.330.360.390.420.450.48

0.510.540.570.600.610.620.6244

1.5711.6321.6951.7591.8251.8911.9592.0272.0942.1612.2272.2892.3482.3992.4422.4722.4842.4702.4192.3112.104

1.9951.843I.581

1.5771.5181.4581.3901.322

1.251I.1781.1041.030

0.95750.88860.82710.78010.76560.79520.8545O.90820.99701.1371.2721.4031.6321.8462.032

2.5272.818

3.389

1.0821.150

1.2251.3101.4071.5171.6451.7941.9702.181

2.4372.7583.1683.7204.1554.5074.7064.9365.1775.3355.4495.5935.6905.7545.8545.922

5.991

r = 1/8 (dual)

1.6261.5861.5451.5031.4611.4181.3741.3291.2831.2381.1911.145I.0961.0541.010

O.96660.92580.8884O.85650.83410.83270.84270.86640.9340

1.1161.1571.200

1.2471.2971.3511.409I.4721.5411.6161.6991.790I.8902.002

2.1272.2692.4302.6162.8363.1023.4453.5903.7674.020

0.38270.39790.4l44

0.4323O.45190.47330.49680.52280.5516O.58380.62000.6610

0.7077O.76170.79820.8244O.83810.85290.86740.87640.88270.8903O.8952O.89850.9033O.90660.9099

0.39^0.40350.41300.42300.43350.44460.45640.46870.4818

0.49570.5105O.52620.5430O.5609O.58010.6008O.62320.64750.67420.70380.73770.75050.76500.7830

Flat

loss

(db)

0.00

1.132.273.394.525.656.737.818.919.95

11.0

11.912.913.814.314.614.714.814.915.0

15.115.115.115.215.215.215.2

0.00

0.6241.25I.872.503.12

3.754.374.995.616.236.84

7.458.068.669.259.83

10.4

10.911.511.912.1

12.312.4

0.00

0.030.060.090.12

0.150.180.210.240.270.30

0.330.360.390.420.450.480.510.530.540.55

0.000.020.040.060.080.10

0.12

0.140.160.180.20

0.22

0.240.260.280.300.320.340.360.380.39

h1.5961.6521.7101.7681.828I.8871.9502.0072.0662.1232.1762.2252.2662.2982.3152.3112.2742.1832.0641.9681.791

1.5931.6231.6541.6841.7141.7441.7741.803I.8311.857I.8831.9061.9271.9441.9561.963I.9611.9471.9121.8351.743

I.6721.6281.5831.537I.4911.444

1.3971.3491.3021.2541.2061.1581.112

1.0661.023O.98190.94590.91910.91230.91730.9432

1.1511.192

1.2371.2851.3371.3931.4541.5201.5921.6711.7581.854I.9612.0812.2172.3752.5612.7902.9843.1073.284

r » 1/2 (dual)

1.7651.7301.6941.6591.6241.5881.5531.5171.4821.4471.412

1.3771.3431.3091.2761.244

1.2131.1841.1591.1421.144

1.226I.2561.2881.3211.3561.3921.4311.4721.515I.5611.6111.6641.7201.7811.8471.9191.9992.0882.1922.3242.421

0.40720.4l650.42620.43650.4473O.45870.47080.48360.49720.51160.5270

0.54350.56110.58020.60080.62330.64830.67670.69870.7H50.7284

0.43500.44i60.44840.45540.46280.47040.47830.4866O.49520.50420.51360.52350.53390.54490.55650.56890.58220.59670.61310.63280.6469

Flat

loss

(db)

0.00

0.57*»1.151.722.312.883.474.054.625.205.786.356.927.498.058.599.139.649-9510.1

10.2

0.000.3280.6570.9871.321.651.982.322.652.983.323.664.00

4.334.664.995.325.655.966.256.38

(a) main program.

Fig. 3. Flow chart.

Transfer: current

element values

Set-upconstraint equations

in matrix form

Invert

Matrix

Calculate

Derivative

(b) derivative subroutine

-Ecc

r 1o y/W vAAA

RgHVf-

>RL

\AA r. VVv Lci - "C2

o

» e

———o

Fig. 4. Band-pass filter.

o-vW

R1 = R + r + r, (1 - a)e e e b '

Fig. 5. Active RC low pass filter.

Element Values for Nonuniformly Lossy Four-Pole Butterworth Bandpass Filters (Figure 4)(Q ="0/BW. o>0 =1, d =R:/L.)

r = 1/4, Q = 5, (not dual) r = 1/2, Q = 5, (dual)

Flat Flat

loss loss

d h cl \ °2 (db) d h cl h C2 (db)

0.00 0.9960 l.oo4 0.03188 31.37 0.00 0.00 8.365 0.1195 0.2231 4.483 0.00

0.01 1.084 0.9208 0.03240 30.92 0.580 0.005 8.243 6.1214 0.2122 4.713 0.320

0.02 1.192 0.8364 0.03325 30.19 1.25 0.01 . 8.092 0.1238 0.2011 4.974 0.6340.03 1.326 0.7502 0.03458 29.10 I.87 0.015 7.905 0.1269 0.1860 5.275 1.080.04 1.501 0.6613 0.03663 27.55 2.42 0.02 7.675 0.1308 0.1776 5.628 1.250.05 1.742 0.5678 0.03988 25.41 3.03 0.025 7.392 0.1361 0.1649 6.054 1-530.06 2.111 0.4654 O.04537 22.50 3.67 0.03 7-039 0.1432 0.1513 6.588 1.850.07 2.742 0.3507 0.05397 19.34 4.62 0.035 6.589 0.1536 0.1359 7.302 2.160.08 3.357 0.2761 0.05245 20.65 7.00 0.04 6.067 0.1686 0.1183 8.304 2.57

0.09 3.963 0.2282 o.o44i8 25.12 10.1 0.044 5.933 0.1753 0.1068 9.042 3-130.10 4.764 0.1867 0.03430 32.89 14.1

0.105 5.290 0.1671 0.02908 39-03 16.4 r = 1/2, Q = 10, (not dual)0.11 5.943 0.1479 0.02374 48.04 19.2

0.115 6.776 0.1292 0.01830 62.56 22.6 0.00 4.483 0.2231 0.02989 33.96 0.00

0.12 7.876 0.1108 0.01279 89.78 27.1 0.005 4.974 0.2009 O.O3092 32.36 0.6390.125 9.395 0.09261 0.00721 159.5 33.6 0.01 5.630 0.1775 0.03264 30.68 1.22

0.13 11.63 0.07468 0.00159 725.5 48.6 0.015 6.597 0.1513 0.03571 28.07 I.870.02 8.540 0.1163 0.04318 23.33 2.51

r = 1/2, Q = 5, (not dual) 0.025 10.31 0.09421 0.04424 23.29 4.10

0.03 11.53 0.08318 0.03973 26.26 6.11

0.00 2.2415 0.4461 0.05977 16.73 0.00 0.035 13.04 0.07296 0.03481 30.23 8.430.01 2.487 0.4014 0.06194 16.18 0.621 0.04 15.00 0.06315 0.02973 35.61 11.0

0.02 2.814 0.3539 0.06550 15.35 1.19 0.045 17.65 0.05331 0.02457 43.30 14.2

0.03 3.294 0.3012 0.07176 14.08 1.86 0.05 21.46 O.04369 0.01934 55.21 18.0

o.o4 4.151 0.2355 0.08457 12.14 2.62 0.055 27.37 0.03416 0.01408 76.04 23.1

0.05 4.879 0.1929 O.O8574 12.44 4.24 0.06 37.78 0.02470 O.OO8781 122.2 29-7

0.06 5.436 0.1691 0.07743 l4.il 6.22 0.065 60.93 0.01530 0.00345 311.3 42.1

0.07 6.100 0.1482 O.06762 16.43 8.490.08 6.495 0.1383 0.06247 17.90 11.2

0.09 8.076 0.1095 0.04645 24.44 14.50.10 9.651 0.09100 0.03546 32.23 18.40.105 10.70 0.08186 O.02989 38.33 20.8

0.110 12.00 0.07280 0.02430 47.26 23.60.115 13.66 0.06380 O.OI867 61.61 27.2

0.120 15.86 0.05487 0.01301 88.47 31.5

A

x \

6 -.4 -.2 6 .2 .4 .6 .8 I.

Fig. 6. Element Values For Alignment of Active Butterworth Filter

Element Values For Alignment Of ActiveButterworth Filter (Figure 5)re=30X1 r$ 5000. rc=a>Rc= 5p00n C,s.3/if x

C2=.0643/if /

Relative Gain,Relative Zero Sensitivity To 0

+