Topological formulae for sensitivity coefficients of network functions

TOPOLOGIC ~ L

fORmUL~E

fOR SENSITI VITY COEffICIENTS Of NETW ORK f UNCTIONS mirKo m. milic

Electrical Engineering

f a culty, Belgrade, Yugosl a via

I nt rodu c ti on Sensitivit y coefficients pldY an important role in the analysis of dynamic a ccuracy of networks /1/.

They are related to network func-

tions determined under open-circuit or short-circuit conditions at some termin d l

pdirs.

Thus the

kno~ledge

of network o pen-cir c uit or s hort-

-circuit functi o ns is essential for the evaluation of sensitivity coeffi ci ents. Although the functions of a linear multi terminal-pair network may be determined by well-known algebraic procedures, the computational difficulties rapidly increa s e with the complexity of networks owing to the nece s sity of computing high order determinants. the interest in topological network analYSiS, was laid by Kirchhoff

On the other hand,

the groundwork for which

/2/, is growing steadily.

The topological approach

has made possible the establishment of rules for the direct evaluation of determindnts and cofactors occuring in network functions by inspection of the network graph, i.e., without their actual

expansion /3,4/.

This method is particularly suit a ble for programming on a digital

/5/.

computer

The purpose of this paper is to show the application of graph theory to the evaluation of sensitivity coefficients of network functi 0 ns.

The paper first deals with the derivation of sensitivity coef-

ficients for the functions of a 3-terminal-pair network with respect to variations of one imittance, in terms of open-circuit impedance functions.

The sensitivity coefficients dre then expressed in terms

of the determinant and cofactors of the original network from which the 3-termin a l-p a ir network was darived. cofactors are expressed topologicdlly,

finally,

the determinant and

first for ordinary-element net-

works, and then for networks contdining both ordinary and multlterminal elements.

• 220-

Sensitivi ty Coef fi c i e nt s For t h e p urpo s e of ge n e r a l

a n a l ys i s o f d ynam i c d c c uracy /1 / , th e

3-ter min d l-p a ir n e t work s h ow n in Fi g . 1 is co nsi d ere d .

b

+

N d

This network may be derived fr o m a n origin a l

network cont a ini ng no less

than two indepe ndent node pairs b y ma kin g three points of a ccess fr o m three termin a l

p a irs denot e d b y (a , b ) ,

( c,d ) a nd (e , f) .

Th e so urce is

connected between termin a ls a a nd b wh ile the el em ent of int e r e st wh ic h may change its param e ter v a lue is connected between t e rmin d ls e a n d f. Terminal p a ir (c, possible a lso a t

d ) is a point of observati o n a lt ho u gh obs e rvation i s termin a l

p a irs

(a ,

f) .

b ) a n d ( e,

The 3-termina l-pair network may be ch a r a ct e riz e d b y o pen-circuit impedance functions wh ich are the elements of the c oeffici e nt ma trix in the matrix equation

(1)

If the net wo r k is termin a ted in an impedance (e,

Z

at the termin a l

p a ir

f ) wh ile the terminal pair (c, d ) is left open e q uations ( 1 ) read

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(2)

From equations (2)

we readily derive the driving point and the tran-

sfer functions

zll

z13 z 31 z33 + Z

z21

z23 z31 z33 + Z

El Zll

=~=

Z21

=~=

Z31

=~=

E2

E3

z31 Z + Z

z33

12

Y21 =

""El""

= 0

-

13 Y31 =(1

zll Z

+

E2 v T21 =~

z21 Z z11 l

+

z11 Z

.. zll z33

i

T21

v

T31

+

z31 z11 z 33

- z13 z31

z21 z33 - z23 z 31 z11 z33 - z13 z31

12

=~= 0

E3 ,.--,. El

13 i T31 =-r; ,.

z31 Z

-

z31

z33

+

-

z13 z31

(3)

Z

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Taking the first derivatives of the above network functions with respect to

Z

and

Y, we obtain the following sensitivity coef-

fici ants

ca

Zll

~)-

"

(;~2l)"

(

-

z23 z3l

)_ - -

z3l z33

0

=~

(

z33

~Z3l

'JY

0

-

z13 z 3l

(zll z33 -

:~3l1

z13 z 3l)

2

(zll z33 -

z13 z 3l)

z3l (zll z23 -

2

'CY

-

(VT~l)_ VY -

2

" (

z13 z 3l

VT~l }_-

V,Y

0

~ vr;l)'

z3l

(VTL) __ 'QY -

~=-2--

o

z33

=0

dnd

z3 1 zll

z3l(zllz23 - z13 z 21) 2 zll

z3l(zllz33 - z13 z 3l) 2 zll

z3l

(4)

0

The subscript Z

z13 z 3l 2 zll

0

z3l z 11 z33 -

(~) -

z13 z 21)

(zllz33- z13 Z 3l)

!;~l)o

'(JY (~)-

"

z3l zll

(:~~l)o

z13 z 3l

CJY (~)-

ll

(

-

"

z23 z3l =--2z33

1 t:: 1 (;~3l

'QY (~)-

z13 z3l 2 z33

Y

0

indicates

th ~t

the derivatives have been evaluated for

O.

Topological Formulae ~s

mentioned earlier, the network under consideration may be

derived from an ori ginal network containing no less than two inue pe naent

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node pairs.

To express the sensitivity coefficients topologically,

we

have first to express the open-circuit impedance functions in e4uations (4)

in terms of the determinant and cofactors of the original

network.

This can be accomplished by solving the node e4uations of the original network for

the terminal voltages va ' expressing the terminal-pair voltages

termindl voltages.

o (a+b)

Thus equation (1)

Vb'

vc'

El'

E2 ,

Cl

(C+d)

I

(e+f) (e+f) where

0

3

and v f' and in terms of .he

becomes

(a+b)

o (e+ f)

vd' v e' and t::

11

El

12

E2

13

E3

(5)

is the determinant of the node-admittance matrix of the

original network,

and

D(p+q) (r+s)

is the dbbrevidtion for the cofactor

sum Dpr - Dps - Dqr + Dqs In fact D(P+4) (r+s) represents the first cofactor of the element in the (p,r) - position of the determinant 0 in which row

q

and column

and column r + column s,

s

first addition cofactor of be defined in an analogous cofactor rows

way.

s

have been replaced by

respectively, and columns column t

+

q

and

column u respectively.

This cofactor is called the

~ddition cofactors

0 /6/.

D(m+n) (p+q), (r+s) (t+u) and r and columns p

m

nand

have been replaced by row p + row q

respectively.

For instance,

of any order may

the second addition

is the second cofactor of and

t

have been deleted,

row m + row n u

by

and

0

in which

while rows

row r + row s

column p + column

q

and

It can be seen that this addition

cofactor stands for the sum of 16 ordinary co factors of the type Dpq,rs making use of addition cofactors,

we may obtain very compact expressions

for network functions and sensitivity coefficients which lend themselves to and easy topological interpretation. open-circuit impedances from equation (5)

To show this, in equations

we substitute the (4).

simplifications we find the sensitivity coefficients to be

~!:ll)o =

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After some

(~) -3Z o

D(a + b) ( e+ f)D(e + f) (c +d ) D2

-~) \ () z

D ( a+ b) ( 9+f ) D( 9+f) ( e+f)

o

( '3Z 31 ) _ _ '(IY ,

-

0

UY \"'(~) o

D(a+b) (e+f) D(a+b) (a+b)

t~ T~l)• - 'i>Y

-

D (a+b) (e+f) D(a+b) (a+b), (9+f) (c+d)

D2

(a+b) (a+b)

(1)T~l ~

_ \1)Z J.o

D(a+b) (a+b), (a+F) (B+f)

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,_ ~ ~T;l -ay L-o

(

~ T~l)

\ 3 Z

o

O(a+b) (e.f) O~a.b) (a.b), (e.f) (e.f) O(a.b) (a.b)

DD~a.b) (e.f) "

O(s.f) (e.f)

(9)

We can now give topological interpretation to all addition cofactor. in for.ulae (9).

We consider first the cass whsn ths nstwork con-

aiste of 2-terminal elements without mutual coupling.

The node deter-

.inant and its cofactor. are then expressed in terms of tree admittance producte and k-tree admittance products network without product. in

Z

and let

/3,4/.

Let

N

be the given

V(y) denote the sum of tree admittance

N, where a tree admittance product is the product of admit-

tanc •• of all tree branches.

Then

o

(10)

= V(y)

Th3 symmetrical addition cofactor O( .b) (a+b) may be interpreted as the node determinant of the network Nta=b) obtained from N by identifying nodes a and b. Hence, D(a+b) (a+b) is given by the sum V(a=b) (y) of tree admittance products in N(a=b), or by the sum of 2-tree admittance products Wa,b(y) in N, where all 2-trees in N have the nodes a and b in two different connected parts. Therefore we have

(11) and Similarly (12 )

The asymmetrical addition cofactors D(a+b) (e+f) be put in tha form D(a+b) (e.f) = D(e+f) (a+b)

iIIae,bfo(Y) -

and

O(e+f) (c+d)

Uiaf,beo(Y)

Wbe,afo(Y) + Wbf,aeo(Y)

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may

(13 )

D(a+l') (c+d) " D(C+d) ('HI') " IIIca,dl'o(Y) - IIICI',deO(Y) (14)

- Wda,cI'O(Y) + IIIdl',cao(Y)

~hara 0 is tha raferanca n~de. Here Waa,bl'o(Y) is tha su~ of 2-traa admittanca products ovar all 2-traes 01' N having nodes a and e in ona connactad part, and b, 1', and 0 in the other. I I' one 01' tha nodae is required to ba in dil'l'arant connacted parts, a.g. a=o (~hich

is aquivalent to saying that node a is takan for ral'aranca),Woe,bI'O(Y)= = 0 by dafinition. These I'ormulaa may ba obtained by axpressing the addition cofactors in terms 01' ordinary col'actors, tha lattar baing expressed in terms of 2-tree admittance products, and then aliminating tha 2-tree admittance products wnich cancal on substraction. In this ~ay there are no suparfluous terms in formulaa (13) and (14). Addition cofactors in ~hich two rows and columna have baan deletad may be traatad in a similar ~ay. Lat N(:~~) ba tha network darivad from the original ona .. ith node pairs

a,b

and

a, f

mads

coincidant respectively. Then D(a+b) (a+b), (a+f) (a+l') is the node (a- b) determinant of N e=f. Hanca it .. ay ba put in one 01' the four aquival ent forms

(a - b) () D(a+b) (a+b), (e+f) (e+f) = V e=f (y) = III ::~ (y)

()

= III

::~

(y) (15 )

Here: ( a- b)

V e~ I' (y)

sum of tree admittanca products of

( a- b)

N e= f

I

W(::~) (y) sum of 2-tree admittance products of N(a=b) over all 2-trees having the nodes e and f in two different connected parts, and similarly for

w(:::)(y);

Ua,e,bf(Y) = sum of 3-tree admittance products over all 3-trees of N having the node groups (a) , (e), and (b, f) in three different connected parts, and similarly for the other terms in expression (15). Finally, D(a+b) (a+b), (e+f) (c+d) is of the same form as D(a+f) (c+d) , the only difference being that it refers to the network N(a=b). Hence

- 227-

(,,=b) () U/(a=b) () U/ ec, fdo Y ed, fco Y -

(a= b) () U/ (a= b) () fc,edo Y + U/ fd,eco Y •

(16)

The formulae so far derived a re applicable to networks consisting only of ordinary elements.

If the network under consideration contains

both 2-terminal and multi terminal elements such as coupled coils, vacuum tubes, transistors, etc., then it must first be repl a ced by its topological model. This model is a graph composed of elements of a single kind, each element consisting of a "current branch" and a "voltage branch" both with the same weight /3,4/.

The graph is then decomposed into a "cur-

rent graph" consisting only of current branches and a "voltage graph" consisting only of volt"ge branches.

Topological formulae for node

determinant dnd dddition co factors of a multiterminal-element network may be given in terms of complete trees, where a complete tree is a set of brdnch e s which constitute a tree both in the current and in the voltage graph.

D

=

Thus, instead of formula (la) we have

V (y)

L:f I(

where

El(

= + 1

complete tree ~ll

(complete tree Tk admittance product)

(17)

/(

is the value of the sign permutation associated with the Tk /3,4/.

addition co factors may be determined topologically by rec-

ognizing thdt the two indices of a row- addi tion subscript and the two ones of a column-additian subscript denote the nodes which are made coincident in the current graph and in the voltage graph, respectively. In order to find addition cofactors of the type D(a+b) (e+r) we place a current branch of weight Yl between the nodes a and b, and a voltage branch of weight Yl between the nodes toward node band f respectively,

e and f, with orientation and take all complete trees con-

taining Yl Let WJ(ab)~(ef) denote the sum of-complete tree admittance products (including the sign (permutation) of the given network containing

Yl

(13), and (14),

Then it is clear that instead of formulae (11),

(12),

the following formulae are valid

(18)

(19)

- 228-

( 20)

(21)

The addition co factors of the type D(m+n) (r+S), (p+q) (t+u) may be found by placing a current branch of weight Yl between nodes m and

n,

one of weight

branch of weight

Yl

Y2

between nodes

between

nodes

p

rand

and

q, then a volt a ge

s, and one of weight

Y2

between nodes t and u. Let UJ(mn) (pq)V(rs) (tu) (y) denote the sum of complete tree a dmittance products (including the sign permut a tion) containing both

Yl

and

Y2.

Then instead of formulae (15)

a nd (16)

we have 1

D(a+b) (a+b), (e+f) (e+f) = Y1Y2 UJ(ab) (ef)V(ab) (ef) (y) 1

D(a+b) (a+b), (e+f) (c+d) = Y1Y2 UJ(ab) (ef)V(ab) (cd) (y).

(22)

(23)

Conclusion Explicit formulae for sensitivity coefficients of 3-terminal-pair-network functions have been obtained by applying topological concepts. The topolo gical approach leads to

a

useful interpretation of

the system tr a nsmission properties on which the sensitivity to changes in network p a rameters is dependent. pologic a l

moreover, computation based on to-

formulae does not involve any unnecessary steps, such as

cancellation of terms inherent in evaluation of network determin a nts a nd it may therefore be considered as one of maximal efficiency. Making use of this method, multiparameter and higher order sensitivity coefficients /1/ may be derived.

A dual procedure may be car-

ried out on impedance basis /7/ for obtaining sensitivity formulae expressed in terms of co- trells for networks containing both ordinary and multiterminal elements.

- 229-

REFERENCES

1.

M.L. Bykhovsky, "Funaamentals of Dynamic riccuracy of Electrical ctnG Mechanical Networks" (in Russian), I1cad. Sci. USSR, 195 8 .

2. L. Weinberg, "Kirchhoff's Third and F o urth Laws", I RE Trdn s . Circuit Theory, vol. CT- 5 , pp. 0-:50 (March 1958). 3. W. Mayed", "found a tion of Mod ern Network Theory", linois, Urban", Ill., 195 8 .

on

University of Il-

4. S. Seshu and M.B. Reed, "Linedr Graphs "nd Electric a l Hddison-Wesley, Reading, M"ss., 1961.

Networks",

5. W. Mayeda dnd m.E. Van Valkenburg, "Network !\nalysis and Synthesis by Digital Computers", IRE Wescon Conv. Record, pt. 2 , pp. 137-144, 1957. 6.

V.P. Sigorsky, "Electronic Circuit I1nalysis", 2nd ed., Tekh. Lit. Uknlinian SSR, Kiev 1963, (in Russian).

Gos.

IZll.

7. M.M. Mi1it, "Mpp1ication of Graph Theory in the ~ndlysis of Multiterminal-Element Networks", M.Sc. thesis, Electrical Eny. faculty, Belgrade, Yugoslavia, 196:5, (in Serbian).

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