Some further sensitivity theorems inactive equivalent - network theory

J. Dunning-Davies, Faith D. Faulkner and J.I. Sewell

Indexing terms: Active networks, Sensitivity analysis

Abstract: The relevance of various sensitivity measures in the study of active equivalent networks is inves-tigated. It is known that the summed differential sensitivity of the passive components in an active RCnetwork is invariant under the transformation; it is now shown that the similar sums of large-change sensitivitiesand the large-change multiparameter sensitivity are both invariant for component changes within specifiedbounds. Even more practical sensitivity measures derive from integral definitions over a specified frequencyinterval. It is demonstrated that the sums of integral sensitivities are noninvariant under transformationbut the integral of the multiparameter sensitivity function again remains invariant. These results are ofsignificance in any attempted optimisation of the sensitivities of active-filter structures.

1 Introduction

The theory of equivalent networks1'2 has been extendedto the case of networks containing active devices such asoperational amplifiers together with passive elements,3

and some interesting results have followed. The activedevices are regarded as constraints on the passive networkN with a nodal admittance matrix Y from which a con-strained matrix YR is obtained by deleting appropriaterows and columns. The characteristic network functionsare available from YR by the usual ratio of cofactors anddeterminants. In particular, the voltage transfer functionTv, which is a function of the complex frequency s andthe passive elements, is

ND (1)

Whole series of equivalent networks, maintaining thisfunction constant, but which are optimised with respectto component size and spread,3 and component differ-ential sensitivity4 have been generated. Some basic theo-rems on differential sensitivity have been noted.4

In this paper expressions are obtained for both thelarge-change and multiparameter large-change sensitivities,which obviously have considerable practical significance.5

In each case it is shown that the summed sensitivity forcomponent changes within certain practical bounds isinvariant under a generalised scaled Howitt transformation.The next Section investigates the question of invariance ofthe integral definition of the differential, large-change,and multiparameter large-change sensitivities.

2 Large-change sensitivity

Two networks N, N', consisting of active devices (in thesame positions in both networks) together with passiveelements, are scaled equivalent3 with respect to somevoltage transfer function Tv iff, T'V=HTV; H^S?. De-fining the entry sensitivities E^ and E%t of the numeratorand denominator of Tv (eqn. 1) with respect to the entry

ykl of YR as

bN

bD

the complete relationships between entry and elementsensitivities are outlined elsewhere.4

Now Tv = Tv(s, e) where e is the passive element ofinterest between nodes p, q and sensitivity calculationsassume the bilinear properties of network functions.5

Hence

ATV = Tv(e + Ae)-Tv(e)

_ N(e + Ae) N(e)D(e + Ae) D(e)

D(e)[N(e) + Ae

I be J

Tv(e)ATV = k i

(2)

k i

where

rpND _ '•W{e) 1 dD(e)

N bykl D bykl

Now large-change sensitivity S^j, is defined to be

Tv _ e_ ATV

A* ~ T ~Ae~

From eqn. 2

Paper T258 E, received 20th June 1978Dr. Dunning-Davies is with the Department of Applied Mathematics,University of Hull, Hull HU6 7RX, England. Ms. Faulkner and Dr.Sewell are with the Department of Electronic Engineering,University of Hull, Hull HU6 7RX, England

ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6

*̂ A 0 —k I

ND

be(3)

193

0308-6984/78/060193 + 06 $01-50/0

IbN/be bD/be]

\ N D I

Ae

as Ae -* 0,,

bP/beP

-> 5Ju (differential sensitivity). Now

(4)

k I D be< 1

and for changes in e of up to 10%, eqn. 3 can be expandedto give a bounded large-change sensitivity function

k i be be

The computation of such a sensitivity measure duringeach cycle of optimisation would obviously be expensive.However it is possible to update eqn. 5 without furtherevaluation of the complete function. Direct applicationof the updating theorem4 to eqn. 5 gives the currentsensitivity function of some element e', during a generalisedscaled equivalence transformation, in terms of the originalentry sensitivities as

sir -k I

7be

foT]2[*"DURr]b \

, v v v y1 — Ae L L L L

r Tl -1 1 r r D , rfcTi -1[V J ha £ lh J ab [£ J bl

k i

, 1

P ab bl be(6)

Theorem 1: The sum of the bounded large-change sensi-tivities of all passive components of one type in an RC net-work with embedded operational amplifiers is invariantunder a generalised scaled Howitt transformation.

Proof: Consider all the conductances in the network. Byinduction (see Appendix 7) it may be shown that

allconductances

diagonal

1_ - I YlEND{AYG?ED

D diagonal

where YG is the conductance component of YR the con-strained admittance matrix.Let S = YlEND then

S Y%END = [S]ppdiagonal P

and for a scaled equivalent network

Zdiagonal

YtEND'

diagonal

= I .diagonal

IP <* b

= the Kronecker delta)

= Z = Idiagonal

ND

?D

(8)

then

diagonal

Now

z

p

& _ y

?D\'

diagonal

= Zdiagonal

- Idiagonal

= Zdiagonal

Similarly, the third term gives

^T" (9)

diagonalT r?ND

G E

E

— V2-diagonal u

p(AYG)

u

(10)

Hence

194

d S a ldiagonal

r-D

Zallconductances

allconductances(11)

The same proof will hold when considering capacitances.(7)ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6

Theorem 2: The bounded large-change multiparameter sensi-tivity of all the passive components of one type in an RCnetwork with embedded operational amplifiers is invariantunder a generalised scaled Howitt transformation.

Proof: In reality Tv = Tv{s,ex, en) hence

Nie, + Ae,,. . .en + Aen) N(elt...en)

Die, + Ae,,...en + Aen) Die,,...en)

Using the same methods of proof as for eqns. 8-10, itfollows that

~TV' _ ~TV / 1 7 \JAej ~~ ^Ae,- v '

Hence theorem 2, and of course this also implies the in-variance of the differential multiparameter sensitivity of thesame networks.

37V b ND i e , , . . . e n ) [ N i e , , . . . e n ) + A e , — ( e , , . . . <?„) + . . . + A e n — ( e , , . . . e n ) ] - N i e , , . . . e n ) [ D ( . e , , . . . e n )

b e , 3 e n

3DAe, ( < ? , , . ..en)

be,.. + Ae

bD

bey,

bD bDDie,,. . . en)[Die,,. . .en) + Ae, — ( e , , . . . en) + . . . + Aen — ( < ? , , . . . cn)

be, ben

Tvie ^*)'kl

Z Z Zi k I

bet

For convenience define the large-change multiparametersensitivity as:

STV =*Jv ( n )

For

i k I uci

i ' k i D

< i

NDkl

(14)

(12)

3 Integral sensitivity

The previous definitions of sensitivity and the resultanttheorems do have both theoretical and practical implic-ations and although proved in general, their meaningfulinterpretations are restricted to a point-by-point frequencybasis. It has been rightly pointed out6 that a more signif-icant sensitivity measure over a relevant frequency bandwill involve the integral of the appropriate functions.

The relative sensitivity is commonly defined as

rv _ e_ dT\, _ [ dN/de dD/de

Tn be N D(18)

Now N = ansn + an-.xs

n~y + . . . + a0 where at = at

(ex,. . . ep) where p is the number of network elements.Hence

y y^ki_ dykl\k i D det ) dan/de

sn +(dajde) (dajde)

(15) Nwhich is a definition for bounded large-change multi-parameter sensitivity.Using an induction proof similar to that given in Appendix Let W have n distinct roots st=at+jfa t=l,...n and

7, it may be shown that

diagonal

diagonal

Zdiagonal

et e isii/de = al. Then

N an

where

— sr

(19)

diagonal

diagonal

a'n an

D and similarly for

I (AYo)r S_

^ diagonal £/ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6

(16) D bm r=i(20)

195

where D has m distinct roots s = yr + jer, r = 1 , . . . m. Sub-stituting eqns. 19 and 20 into eqn. 18 yields

N D

F(st)

an bm t=l

^ Z

sTv

+ I(co - $t) bm r=i yr + / Or -

_„„ = eU + eV I= JW

eWG(sr) (21)

where

Hence, from eqn. 21,

nzt=l

\F{st)\

\G(sr)\

l°e ls = >i

+ \eW\ £

and for the case of a lowpass filter the integral of sensitivityover the passband interval is

IV"(.,.e)\tmjWdcj<\\eU\cj+\eV\ £ \F(st)\

sinh"1 - — - - j e H / jr * "" J w=o

(22)Theorem 3: The sum of the integrals of the differentialmodulus sensitivities of all the passive components of onetype in an RC network with embedded operational ampli-fiers is noninvariant under a generalised scaled Howitttransformation.

Proof: Since the summations concerned are finite

f \ \S?.v\du= \ l\Se]v

»=1 ^0 ° ' = 1

(23)

Under a generalised scaled Howitt transformation it has al-ready been established that

i.e. invariance.

Proof: From eqns. 4 and 5

SA- =e*N~D -Ae-+ | A e - (26)

The first term corresponds with that of the differentialsensitivity already studied. Now consider the second term

= -eAeWG(sr)

r S—Sr

t s — st s~sr

Integration over the passband interval co = [0, 1] gives

• U - l Pt UJ\ ITT

sinh — \U —

\U\t*-\V\ £ \F(st)\t

\F(st)\\G(sr)\sinh"1 »

- | H / |

er2 j ^V7r , 7 r 2 ;

+ sinh

tan"

- l

7r-w

u;=0

A similar expression may be obtained for the third term.Direct application of theorem 3 establishes the non-

invariance of the sum of the integrals of the moduli ofexpressions of the type given in eqn. 26. Hence thecorollary.

Theorem 4: The integral of the modulus of the boundedlarge-change multiparameter sensitivity of the passive com-ponents of an RC network with embedded operationalamplifiers is invariant under a generalised scaled Howitttransformation.

Proof: With reference to eqns. 15—17 the bounded large-change multiparameter sensitivity function is invariantunder transformation, and hence the integral of the mod-ulus is also.

However

and so

(24)

I f |5^|£/W^Z f \<Se?)'\du (25)

Hence the theorem.

Corollary: The sum of the integrals of the bounded large-change modulus sensitivities of all the passive componentsof one type in an RC network with embedded operationalamplifiers is noninvariant under a generaliseJ scaled Howitttransformation.

4 Examples and conclusions

Fig. 1 shows an initial circuit for a 3rd-order Butterworthfunction and Fig. 2 an equivalent network produced bytransformation but having a reduced value for the sumsquared sensitivity function evaluated at a typical valueof s=jl. For the first circuit this numerical value takenover all components is 7-7016 and for the second circuit6-8393, a moderate reduction. Examining the summedsensitivities according to the definitions given, over allconductances say, yields for the differential case values of2-9154 and 2-9008. For component changes of 10% thebounded large-change sensitivity sums are 2-9030 and2-8795 and for the bounded large-change multiparameter

196 ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6

sensitivity 0-2416 and 0-2416 for the respective circuitsat s=j]. So considering the limits of accuracy in thecalculations, these results confirm the invariance theorems.

For the integral definitions the following values are ob-tained, over all conductances, for the first circuit:

Jo- 1-9999

and for the transformed circuit

r1

Jn\du<-7-3133

and bounded large change integral sensitivity evaluationsgive

I £ i*2and

|</co<- 2-7486

- 6-7217

The results demonstrate again the care required in speci-fying sensitivity measures. The sums of the integrals of themoduli of differential and large-change sensitivities arenoninvariant and can be grouped with the other non-invariant measures such as the sum of sensitivity squares ormoduli.

The sensitivity measures having real practical signific-ance, the bounded large-change multiparameter sensitivityand the integral of the modulus of this function exhibit acomplete invariance, when the network is subject to a fairlygeneral class of equivalence transformations, and this con-firms an underlying practical feature.

Fig. 1 Initial realisation of a 3rd-order Butterworth function

Elements in Siemens and farads

Fig. 2 Reduced sum-squared-sensitivity realisation of a 3rd-orderButterworth function

5 Acknowledgment

The work reported in this paper was supported by the UKScience Research Council.

6 References

1 HOWITT, N.: 'Group theory and the electric circuit', Phys. Rev.,1931, 37, pp. 1583-1596

2 SCHOEFFLER, J.D.: 'The synthesis of minimum sensitivitynetworks', IEEE Trans., 1964, CT-11, pp. 271-276

3 GRANT, L.G., and SEWELL, J.I.: 'A theory of equivalent activenetworks', ibid., 1976, CAS-23, pp. 350-354

4 GRANT, L.G., and SEWELL, J.I.: 'Sensitivity minimisation andpartitioned transformations in active equivalent networks', IEEJ. Electron. Circuits & Syst., 1978, 2, pp. 33-38

5 FIDLER, J.K.: 'Network sensitivity calculation', IEEE Trans.,1976, CAS-23, pp. 567-571

6 ROSENBAUM, A.L., and GHAUSI, M.S.: 'Multiparametersensitivity in active RC networks', ibid., 1971, CT-18, pp.592-599

7 Appendix

Consider the second terms in eqns. 5 and 6. It is necessaryto show that

all conductances de IEk I

D be

diagonal

(27)

(28)

where A YG is the matrix of entry increments.For a network with n nodes and m operational ampli-

fiers, for simplicity consider single input/output type,where Ix is the input at the node / and Ot is the output atnode /. Let f=n—2m, the number of nodes notconnected to an amplifier. Also ei;- = eyj- and e^Ae^ = 0,

( ^ ^ 0 . 27gives:

all conductancess" = —

i f

— yD a=l = o + i l « i « = i

-Eg -

ND (A Y\TFD —

Equating to eqn. 28

y^ A g

D diagonal

For a network of n + 1 nodes, m amplifiers and/ ' = / + 1free nodes

+

ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6 197

] r f' f f'~ - - \y e Ae pNDFD + y y A

r» L LaaL*taaI-J aa &aa ~ L L eab LieabLf a = l a=l b=o+l

(fND i n-ND pND t?ND\(h aa -T t bb —tab —tba)

\Paa+ tbb h ab ~ & ba)

f m

+ I Z eao^eao^-E^a = l l=i

+ Z I eai^iE™ -E™)(Eil-E?ia)a=l 1=1

+ £ Z «no.AW-*S8.)(-*&>.>*n + 1, m

Similarly for a network with n nodes, m + 1 amplifiers,/'=/-2

f f m + lm + l

e,to^e,lOs(-E^s)(-E?lOs)

= P,

Finally, for a network with n + 1 nodes, m + 1 amplifiers,/ ' ="/- 1

1 r f' f m + l m + l

J Li L, Li eaa^eaa^ aa & aaa = l b = a + l 1=1 s = l

„ / crND i L- ND _ 17iVD _ cJV£hLab V-*3 aa ^ bb •• ~tba)

(Eaa + Ebb ~Eab ~ & ba)

— p1n + l , m + 1

For a specific case of n = 3, m = 1 these expressions hold,hence the second term can be written

bb ~ *• ab ~ b ba)

-E?ia)

Iall

conductances

= ~ Z-^ diagonal

The same process can be used to identify the other terms ineqn. 6.

198 ELECTRONIC CIRCUITS AND SYSTEMS, NOVEMBER 1978, Vol. 2, No. 6