Sensitivity minimisation and partitioned transformations in active equivalent networks

Sensitivity minimisation and partitionedtransformations in active equivalent

networksL.G. Grant and J.I. Sewell

Indexing terms: Active networks, Minimisation, Sensitivity

Abstract: Following the extension of equivalence transformations to a general class of electrical networkscontaining active devices such as operational amplifiers, the theory is developed to accommodate the mini-misation of the sensitivity of the network response function to component variations. Theorems for passive-component sensitivity invariance and for updating routines applicable to the optimisation step are proved.Previous limitations imposed upon the optimisation routine by equality realisability constraints are com-pletely eliminated by a partitioned approach. This leads to a new form of transformation. The partitionedtransformation can be implemented in both discrete and continuous forms, and applied to the optimisationof component values and sensitivities in active networks.

1 Introduction

Equivalent-network theory is well established in the passive-filter field,1'2 and the application to sensitivity studies andcomponent value optimisation is well documented. Foractive networks it has been shown recently3 that

Y'R = nM 0)is a scaled equivalence transformation with respect to thevoltage transfer function, where YR is the constrainedadmittance matrix of a passive network with embeddedoperational amplifiers and x\, § are suitably constrainedgeneralised Howitt transformation matrices. The solution ofproblems involving the minimisation of component spreadand size has been successfully attempted using this trans-formation. The implications of this theory upon sensitivityminimisation of active RC networks obviously requireinvestigation. The initial implementation of the transform-ation3 resulted in large arrays of equality and inequalityconstraints that must be satisfied to maintain realisability.In particular, the equality constraints can prove to be quitea nuisance in the constrained optimisation routine. A newtransformation is developed from a partitioned approach,and this completely eliminates all equality conditions.

2 Active-network sensitivities

Assuming that the active elements are ideal operationalamplifiers, whose effects upon the network can be rep-resented as ideal constraints, then only the sensitivities ofthe passive components need be considered.

The relative sensitivity Sp% of the voltage transfer func-tion Tv to some passive element epq between the nodes pand q is defined as

bTv

Tn dePQ

where

^T,, -

(2)

(3)CPQ

is termed the semi-relative sensitivity. Noting that

= det [YR]U = mv det [YR]a D(s)

Paper T138 E, first received 28th October 1977 and in revised form6th January 1978Dr. Grant and Dr. Sewell are with the Department of ElectronicEngineering, University of Hull, Hull HU6 7RX, England

ELECTRONIC CIRCUITS AND SYSTEMS, MARCH 1978, Vol. 2, No. 2

(4)

we define as follows the entry sensitivities Eî, Eî of thenumerator and the denominator to the entry ykl of YR:

JV _ bN[YR]U' U' kl (5)

dD 3(det[F,i]H)= det [YR]iit kl (6)

Hencerv - __L y y d^p d-Vfei

N D de(7)

PQ

Sensitivity polynomials are considered, since this enablessensitivity optimisation over a range of frequencies to beundertaken without further development.

We define the variable

Akplq = 8kl(8k

dePQ

•p

kl

j + 8kP8kQ) — (8kp8lq + 8kq8lp) (8)

j Q is a conductance

kl WsApq if epq is a capacitor

Therefore, once the matrix of entry sensitivities isobtained, the relative sensitivity of any element may befound using eqns. 9, 7 and 2.

During the generation of equivalent active networks, it isadvantageous to be able to calculate the current-sensitivityvalues (in terms of the current transformation matrices) andthe entry-sensitivity values of the original network withoutfurther sensitivity analyses. The following theorem demon-strates how the current entry sensitivity matrix may beevaluated at any point of the transformation:Theorem 1

N' = Mr T7

33

0308-6984/78/138E-0033 $1-50/0

where EN is the current entry numerator sensitivitymatrix, EN the original entry numerator sensitivity matrix,and i\, § the current transformation matrices.

Proof

\EN'

\3(det [YR]U)

JPQ N' dy'PQ det[Y'R)u by'pq

= det[YR]UiPq

det [YR]V

Now

det [YR]U = det [ n K^] l 7

= det 11 det [YR]udet£

for a scaled equivalence transformation.3 Form^ such that

dety = det [YR]V

=> det [YR]ij = det r\ det y det §

= det &

det

therefore

N' dyL

= det

= Z Z det [r)]pa deta 6

det

Theorem 2The sum of the relative sensitivities of all passive compo-nents of one type in an RC network with embeddedoperational amplifiers is invariant under a generalised scaledHowitt transformation.

ProofLet YG be the conductance component of YR. Then, where'sum of diagonal entries' is represented by s.d.e.,

Zall

conductances

£ENDST" = s.d.e. of Y£E

Putting

= YgEND\PP

Z ST'v = s.d.e. of [Y^]TEND

allconductances

= s.d.e. of ?YgEND(STrl

= ZZZp a b

= zzza b P

-1bp

R]ap

Now

(12)

det T| (

x I I

= 1 1a b

ab det

3(detti)

1 3(det§)

det?-1

dyab

Hence the theorem is proved.Using theorem 1 and its equivalent for the entry

denominator sensitivity matrix, the current relative sensi-tivity of some element e'pq during a generalised scaledequivalence transformation may be found in terms of theoriginal entry sensitivities using the relation

i - i,T I cJVDV"̂ T"^ ^*^ T~i

LLLL \\r\ \E §k I a b \ \ \ka\ \ab\ \bl

-1

where

\FN ED

END = £_ _ £ _

TT'U (I")

(11)

Z [flaptClpb — &aP

=> Z sT'v = z Za b

'aball

conductances

= zZ Qa//

conductancesHence the theorem is proved, and an alternative proof is

given of sensitivity invariance of active networks generatedby the scaled equivalence transformation (eqn. 1).

The theory presented here has been used to formulateobjective functions designed to minimise sum squaredsensitivity, using the computer program described in Refer-ence 3 and utilising a s.u.m.t. optimisation technique.4

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

Conductances in Siemens (S): capacitances in farads (F)

34 ELECTRONIC CIRCUITS AND SYSTEMS, MARCH 1978, Vol. 2, No. 2

Fig. 1 shows an initial circuit for a 3rd-order Butterworthfunction; the value of sum squared sensitivity function ats = / l is 7-7016. The circuit of Fig. 2 results from sensitivityminimisation and has a comparative sensitivity value of6-8393. This solution is a local as opposed to a global mini-mum. The increase in number of elements is apparent, butthis is not unexpected as the limiting solution to theseproblems is a fully connected network with equal elementsensitivity values.

The results shown here were obtained using one-sidedtransformations only, because of the problems reported inReference 3 and the difficulties arising from the nonlinearform of the equality constraints generated by the quadratictransform.

Fig. 2 Reduced sensitivity realisation of a 3rd-order Butterworthfunction

Conductances in Siemens (S): capacitances in farads (F)

3 Partitioned transformation

The equality constraints, which are necessary to keep sym-metric the passive part of the admittance matrix duringoptimisation, have prevented, up to now, the formulationof a continuous quadratic transformation for active RCnetworks like that proposed for passive networks.5 Essen-tially, continuous techniques depend on the transformationbeing small and the ability to make linear approximationsto quadratic constraints. If the constraints are inequalities,good linear approximations may be made; however, it isalmost impossible to make efficient linear approximationsto quadratic equality constraints. The elimination ofequality constraints in the discrete transformation willreduce the complexity of the optimisation problem.

Here, a transformation is constructed which eliminatesthe equality constraints by means of topological transform-ations of the constrained nodal admittance matrix, andwhich utilises the redundancy of the row of the constrainednodal admittance matrix corresponding to the input nodeto simplify the application of the transformation.

Consider an «-node RC network N with m embeddedoperational amplifiers. The constrained admittance matrixYR, formed by deleting rows corresponding to amplifieroutput nodes and columns corresponding to amplifier inputnodes, has dimensions (n — m)x (n —m) and is symmetri-cal in (n — 2m — \){n — 2m)/2 pairs of entries. From eqn.4,

71 _

v -det det [Yr]j _ N(s)

(13)det [YR]U det [Yr]j D(s)

where Yr is formed by removing the ith row of YR. Inspec-

tion reveals that Yr is symmetrical in (n — 2m — 2)(n —2m — l)/2 pairs of entries. Now consider Yr re-ordered soas to place these symmetrical entries in an (w — 2m — l ) x(« — 2m — 1) symmetrical submatrix in the top right handcorner. That is, form

Y* =

/= n-2m-\

(14)

where Ys is a n / x /symmetric matrix.The following well-known matrix properties are used:(i) the interchange of columns i and / of a matrix A is

obtained by forming the product ATt +,-, and(ii) the interchange of rows / and / of a matrix A is

obtained by forming the product 7} +}-Awhere the topological transformation matrix Tt+j is the unitmatrix with its ith and/th rows interchanged.

Let the unloaded nodes (those not connected to anamplifier, and not the input node /) be px, p2,..., Pf; letthe corresponding rows and columns be ritr2,. • • ,rf;c1,c2,... ,cf. Then the topological transformations fromYr -• Y* take the form

—> V Tr, -»1 MrMCf-*n-

T•••if

(15)

where the 7^,- are (« — m — 1) x (n — m — 1) and theTc.+k are («—m)x (n—m). Hence, taking inverse trans-formations, and noting 7}.»;- is self inverse,

(16)

In practical terms, this amounts to simply re-ordering thenodes.

The following results emerge and are proved wherenecessary.

Theorem 3

= 0, or —l,or 1

Theorem 4

For fixed p, q, i there exists only one / such thatdet [Tpû ± 0

For fixed p,q,j there exists only one / such thatdet [Tp+q]ij * 0

Theorem 5

det [AB...Z]tj

= Z I • • • I det [A)ia det [B]ab . . . deta b z

Proof

See Reference 3.

ELECTRONIC CIRCUITS AND SYSTEMS, MARCH 1978, Vol. 2, No. 2 35

Theorem 6

If A is (/ - 1) x (/ - 1), Y is ( / - 1) x /, B is / x /, then

det [AYB]

Proof

Construct

A' ="l :

0 :0 :0 :0 :

then

A'Y'B =

jt

Is

A

-

A

I» = 1

7

/

Y

Y

det A det

Y'

B

B

b

.

\—

[Y].b

0

Y

1

det [B]bj

/

0 0

J/

Hence, from theorem 5,

det [A'Y'Bhj = Z E det &4']la det [Y']ab det [*]w

Now

det [A']la = 0 for all a =£ 1

=> det [i4'y'^]v = Z d e t M n d e t I^'lift d e t M «

det [>lK»].y =

Theorem 7

det [F].b det [B]bj

If A is (/ - 1) x (/ - 1), Y is (/ - 1) x / and B, C . . . Z are/ x /, then

det [A YBC. . . Z]j = Z Z • • • Z det /I det [F].b6 c z

x det [ 5 ] b c . . . d e t [Z] 2 y

Proof

By extension of theorems 5 and 6, writing

using eqn. 16 and theorem 6, we get

det [Yr)j

= Z Z • • • Z det A det [F*].a det [TC{ _ m + 2 ] a b . . .a b z

. . .det [7; r n_m] zy (17)

and

det [rr].i

= IE---a b

. . . det [7

de t [F*] . a de t [ r C i . m + 2 ] a / ? . . .

Hence, from theorems 3 and 4,

det[yr].y = ±det[r*].z

det[Fr],- = ± d e t [ n . *

(19)

(20)

where /, t are the unique solutions of the equations. Thesigns are ignored, as inverse transforms are taken later. Then

det

det [TCi ^ det [TCj

. . . de t 0

(21)

(22)

Eqns. 21 and 22 are solved easily by a computeralgorithm using the result of theorem 3.

We observe from eqns. 19 and 20 that the problem ofapplying transformations to Yr to retain invariant [det Yr] Jdet [Yr]j is equivalent to that of applying transformationsto Y* to retain invariant det [Y*]Jdet [Y*]t.

We now wish t o form Y*' b y t ransformat ions upon Y*,such tha t

det [Y*'h _ Hdet [Y*}jdet [Y*']t ~ det [Y*].t

where HE *% ( ^denotes real space).Put

Y*' = t]F*§

(23)

(24)

where r| is (n — m — 1) x (n — m — 1) and § is (n — m) x(n — m), and where

/ m

n =m no

m(25)

m + 1 /

Considering eqn. 14, eqn. 24 becomesn —m

Y*' =

F'

n — m — 1

(26)

(27)

where ^ =r)'Ysg. Putting i] 'T = f' gives

YS' = n'rsn'T, (28)

a transformation which retains the symmetry of t h e / x /top-right-hand submatrix. Now

detfr*']., = dethr*5].i

= Z det x\ det [F*].bdetb

(29)


from Theorem 6, and

det[F*']., = Idetndet[F*].adet[$]a,a

Putting the /th row of § = 05,7 and the f th row of(0, co E 3>), then from eqns. 29 and 30 we have

(30)

= CO6J 7

det [Y*']A _ det n det [F*]./0~l det§

det [F*']. t ~ det v\ det [F*].^ co"1 det§

det [Y*h= H

det [Y(32)

where//= o>/0.Hence we have the following theorem:

Theorem 8

Constraining r\, § such that

0) §' = vl7 and § , TJ have the partitioned form ofeqn.25

(ii) §, x\ are nonsingular(iii) the /th row of § = 05 y(iv) the rth row of § = a)5i;-

then the transformation F*' = i]F*§ is a scaled equivalencetransformation with scaling factor to/0, and the necessarysymmetry properties of Y are conserved under the trans-formation.

Once we have Y*' in symbolic form, inverse transform-ations are applied to produce Fr'; thus

*r ~ *r, ->1 • • • *rf*f* (33)

The matrix Y' is then rebuilt from Y'R by inserting suitablesymbolic rows and columns corresponding to the con-strained nodes, to form the symmetric nodal admittancematrix of the passive part of an equivalent active network.The symmetry of this matrix has been ensured entirely bythe form of the transformation and not by equality con-straints imposed on the optimiser in a previous implemen-tation.3

4 The continuous partitioned transformation

The partitioned transformation may be implemented in acontinuous form. Let

= If+m (34)

where If+m, If+m + i are identity matrices,?], § are of theform described in theorem 8 and JC is a small real scalar.Then rj, § clearly obey the conditions of theorem 8 and

Y*' = fjF*| (35)

is a generalised scaled equivalence transformation. Substitut-ing eqn. 34 into eqn. 35 gives

= (If+m +T]x)Y*(If+m + l

= F* + (nF* + Y*S)x + (36)

If the transformation parameter* is small, a linear approxi-mation can be made to eqn. 36:

Y*' = Y* + (nF* + Y*$)x (37)

The consequence of eqn. 37 is that, for a small x, a trans-formation may be applied producing an equivalent network

which is close in some metric sense to the original. Therealisability constraints obtained from eqn. 37 are linear,and if a linear approximation to the objective function ismade then a linear programming problem may be solved fora sequence of step sizes. Using eqn. 36 as a check of realis-ability, and cutting the step size x if realisability iscontravened, a series of networks which tend towards someoptimum may be obtained.

From eqn. 10 it can be seen that the matrices of entrysemirelative sensitivities for an active network underequivalence transformation are related as follows:

but

= If+m+r)X => i\T =

~T\-l _) - lf+m — r\ x + T r_2xz + ...and

(38)

(39)

(40)

therefore

= (If+m

- rfx + • • • )

= END - ENDf)x (41)

The sensitivity minimisation techniques of Section 1 cannow be implemented using linear programming methods;this avoids matrix inversion each time the semirelativesensitivities are updated.

5 Conclusion

A method for the minimisation of the sensitivity of activenetworks using equivalent-network transformations hasbeen produced, and appropriate sensitivity updatingroutines derived. A new transformation is given whicheliminates equality constraints and enhances the applicationof the quadratic transformation, both in the minimisationof element spread and in the sensitivity of active networks.It also enables a continuous form of the transformation tobe used. The choice between discrete and continuous formsis somewhat subjective, as some assessment of the effect ofthe truncation steps in the continuous form must beweighed against the avoidance of matrix inversion presentin the discrete implementation. The equivalent networksproduced by the two forms can differ quite considerably.Both methods will produce new networks containing moreelements than the original, an effect observed particularlywhen the optimisation of total sum squared sensitivity isattempted.

6 Acknowledgments

The efforts of D.G. Spridgeon in writing and running someof the programs used in this work are particularly appreci-ated. The work was supported by the Science ResearchCouncil.

ELECTRONIC CIRCUITS AND SYSTEMS, MARCH 1978, Vol. 2, No. 2 37

References

HOWITT, N.: 'Group theory and the electric circuit', Phys. Rev.,1931,37, pp. 1583-1596SCHOEFFLER, J.D.: 'The synthesis of minimum sensitivity net-works', IEEE Trans., 1964, CT-11, pp. 271-276GRANT, L.G., and SEWELL, J.I.: 'A theory of equivalent activenetworks', ibid., 1976, CAS-23, pp. 350-354

FIACCO, A.V., and McCORMICK, G.P.: 'Nonlinear program-ming: sequential unconstrained minimisation techniques' (Wiley,1968)CHEETHAM, B.M.G.: 'A new theory of continuously equivalentcircuits', IEEE Trans., 1974, CAS-21, pp. 17-20

L.G. Grant was born in South Shields,Tyne and Wear, England on 8thSeptember 1952. In 1973 he receivedthe B.Sc. degree in mathematics fromthe University of Hull. From 1973 to1976 he was with the Departmentof Electronic Engineering at theUniversity of Hull, undertaking re-search leading towards a Ph.D. In 1976he joined Ferranti Ltd. in Edinburgh,Scotland, where he is currently with

the Inertial Systems group working on high-accuracymeasurement systems.

John I. Sewell was born in Cumbria,England, on 13th May 1942. Hereceived the B.Sc. degree in electricalengineering from the University ofDurham in 1963, and the Ph.D. degreefrom the University of Newcastle-upon-Tyne in 1966. From 1966 to1968 he was a Senior ResearchAssociate at the University ofNewcastle-upon-Tyne, where he wasengaged in the study of computational

methods in the design of active networks. He joined theUniversity of Hull in 1968 as a Lecturer in the Departmentof Electronic Engineering; he is now Senior Lecturer.


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Sensitivity minimisation and partitioned transformations in active equivalent networks