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Application of graph theory to the analysis of active and mutually coupled networks
APPLICATION OF GRAPH THEORY TO THE ANALYSIS OF ACTIVE A N D MUTUALLY COUPLED N E T W O R K S * BY WAN HEE KIM i SUMMARY
This paper presents a unified graphical approach to the analysis of an electrical network which contains linear multi-terminal devices such as vacuum tubes, transistors, transformers, gyrators, circulators, etc. The concept of the mathematical equivalent circuit of an active device or a passive but mutually coupled device which was introduced by Percival, is related to the indefinite admittance matrix of a multipole defined by Zadeh and Shekel. Various graphical formulas for the evaluation of network functions of a multipole are shown, including the node-condensation techniques. A number of examples are presented. INTRODUCTION
The method of topological analysis (application of the theory of linear graphs) of a network comprising linear, passive and bilateral twoterminal devices such as resistors, inductors and capacitors, has been proved very useful (1, 2). 5 It is particularly suitable for digital computers (3). However, the method is not directly applicable to a network containing linear active devices, such as vacuum tubes and transistors, or mutually coupled passive devices, such as transformers and gyrators. The difficulty is the graphical representation of mutually dependent node-pairs due to a dependent source or the mutual coupling effect imbedded in a device (4). In order to characterize a dependent node-pair in a tube and transformer by graphical representation, Percival (5, 6) recently introduced artificial two-terminal elements, so-called "current and voltage elements," and hence made it possible for the theory of linear graphs to be extended to the analysis of active and mutually coupled networks (7, a). This paper presents first an expository review and extension of the concept and use of "current and voltage elements" in graphical characterization of dependent node-pairs in a network. Then, through investigation of the topological aspects of the techniques of multipole analysis proposed by Zadeh (9, 10) and Shekel (11, 12), a unified approach to the analysis of an electrical network containing multiterminal active and passive devices is pursued. * This work was supported by National Science Foundation Grants G-6020 and G-10354. Preparation of this paper was partially supported by the Marcellus Hartley Fund. 1 D e p a r t m e n t of Electrical Engineering, Columbia University, New York, N. Y. 2 The boldface numbers in parentheses refer to the references appended to this paper. 200
Mar., 1961.]
GRAPtt THEORY IN NETWORK ANALYSIS
2OI
I. GRAPHICAL CHARACTERIZATION OF MUTUALLY DEPENDENT NODE-PAIRS
Let us consider node-pairs (pq) and (ran) in the linear network N shown in Fig. 1. If the current ipq(t) flowing from node p to node q depends not only on the voltage-difference between nodes p and q,
N P
--
Ipq
--
+
+
Vpq
Vmn
q
om
On
FIG. 1. A linear network with mutually dependent node-pairs.
v~ (t), b u t also on the voltage difference between nodes m and n, vm~(t), then, since the network is linear, one m a y write y~q Vpq + y~.m~ V,~,~ = I~q
(1)
where the y's are proportional factors and I -- ~ei(t), V = .ev(t). In Eq. 1, since yp~ relates the voltage and current for the same node pair, it is called the "self-admittance" or simply " a d m i t t a n c e " of the element (or elements) connected between nodes p and q. On the other hand, y~q,m~relates the voltage and current for different node-pairs, and hence is called the " m u t u a l admittance." If the network N consists entirely of passive devices, then Eq. 1 is divided into two isolated parts, t h a t is, y,.V~
or
= I~
(2)
It is of course clear t h a t if N contains only the ordinary two-terminal devices (R, L and C elements), then ypq -- yq~ for all node-pairs in the network, which is said to be "passive and reciprocal." However, if
W A N ]:tEE KIN[
202
[J. F. I.
transformers or gyrators are contained in N, then for some node-pairs in the network ypq.,,,,, ~ ym,,.~,q, and N is called "passive and non-reciprocal." Therefore, Eq. 1 holds for some node-pairs in N if it contains a linear active device or devices. In Eq. 1, Ipq, which depends on both Vpq and V~,, can be decomposed into two components: the one due to Vpq is denoted by I~q' and the other, due to Vm~, is denoted by Ip~". T h u s
a. b.
y,,~V~,q = I,,~' y~,~,,~, V,~, = Ip~"
(4)
c. Ip~' -q- Ivq" = Ip,,. The voltage-current relationship of Eq. 4a represents an ordinary twoterminal element. Equation 4b shows the dependent relationship of node-pairs (pq) and (ran). A voltage-difference between nodes m and n, with m positive, causes a current flowing from nodes p to q.
IJ
b
bd~
UtU'bd/~_~
ULU*bd~/
Z
W
+
N FIG. 2. Use of voltage and current elements to represent dependent node-pairs.
In order to apply the concept of graphs, it is therefore desirable to introduce artificial two-terminal elements into node-pairs (ran) and (pq) to characterize their d e p e n d e n t relationship. The element inserted between nodes m and n, called a "voltage element," is to sense a voltagedifference in the node-pair. The element introduced in node-pair (pq), called a "current element," produces a current of magnitude I " flowing from node p to node q due to the voltage-difference V in node-pair (ran). T h e voltage and current elements are thus related by m u t u a l a d m i t t a n c e ypq .... Note t h a t voltage and current elements always occur in a pair
Mar., r96I.]
GRAPH
"]'HEORY IN N E T W O R K
ANALYSIS
203
and between different node-pairs. If they occur in the same node-pair, that is, p = m and q = n, the mutual admittance ypq,mn is reduced to the self-admittance ypq. Note also that the voltage and current elements introduced cannot be reduced to zero in the evaluation of a network function although all independent voltage and current generators are always reduced to zero. The graphical representation of voltage and current elements for dependent node-pairs (pq) and (ran) is shown in Fig. 2. Since the current and voltage elements are two-terminal elements they can be represented by the edges 3 of a linear graph as current and voltage edges, respectively. The weight of the edges is the mutual admittance. The orientation of a current edge is determined by the direction of the current-flow in the corresponding current element, the orientation of a voltage edge by the polarity of the corresponding voltage element. The graphical representation of the network N (Fig. 2) is shown in Fig. 3, where G denotes the graph corresponding to the net-
G ) rl]
P
Ypq Ypq ,mn
V Ypq,mn
q Fro. 3.
Graphical representation of dependent node-palrs.
work N. The current edge is indicated by double arrows, and the voltage edge by a triangular-shaped arrow. The following examples illustrate the concept and use of current and voltage elements in characterizing a network containing linear active devices as well as mutually coupled nmltitermlnal passive elements. Example 1. A triode of Fig. 4 is characterized as a linear active device by I e = g,~VoK + g~,VeK (4) 3 The term " b r a n c h " may be substituted occasionally for "edge," but the term "element" will only be used in the sense of "element of a network" or "element of a matrix."
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WAN HEE KIM
IJ. I;. I.
P "rp
Vp~
G
K FIG. 4. A triode. where gm is the transconductance and gp the plate-conductance. An equivalent circuit of the triode characterized by Eq. 4 in terms of voltage and current elements is known as the mathematical equivalent circuit (5, 6) and is shown in Fig. 5a. Figure 5b shows the conventional equivalent circuit of a triode for comparison. T h e graphical representation of Fig. 5a is given in Fig. 6.
P
G
gm "~///
P
~
G0
gP
gmTpK
~
VGK
K
K
(a) MATHEMATICAL EQUIVALENT CIRCUIT IN TERMS OF VOLTAGE AND CURRENT ELEMENTS
(b) CONVENTIONAL EQUIVALENT CIRCUIT
FIG. 5. Equivalent circuits of a triode.
Mar., I96I.]
GRAPH THEORY IN NETWORK ANALYSIS
205
P
\ gp
c3m
K FIG. 6. Graphical representation of Fig. 5a. E x a m p l e 2. A two-channel gyrator, shown in Fig. 7, may be characterized by the following set of equations (12):
(5)
I1 = otl V2 12 = -
a2V1
O<2 Jo
~
~
o5
v,
VI
2 o
04 F.;. 7. A gyrator.
where al = as = a is the gyrator admittance. The mathematical equivalent circuit of the gyrator represented by the terminal characterization of Eq. 5 is shown in Fig. 8a and its graphical representation in Fig. 8b. Now a graph representing a device or a network can be decomposed into two subgraphs (7,8). One is the "current graph" containing the edges corresponding to the ordinary elements and current elements. The other is the "voltage graph," with the edges corresponding to the
206
WAN HEE KIM 5
I
[J. F. I 1
5
2
4
~C
I
2
4
(a) AN EQUIVALENT CIRCUIT OF A GYRATOR FIG. 8.
(b) GRAPHICAL REPRESENTATION OF A GYRATOR Representationof a gyrator.
ordinary elements and voltage elements. One may ilnmediately see t h a t if a network consists entirely of ordinary elements, the current and voltage graphs of the network are identical ; t h a t is, a graph corresponding to an ordinary network is the current graph of the network and at the same time the voltage graph. The orientation of each edge of a graph of an ordinary network is arbitrarily assigned. When a network contains active devices or mutually coupled devices, the current and voltage graphs of the network m a y not be identical. This is illustrated in Example 3.
Gz
3
0 ....
3
5
~
z
Gz
oC I
4
4
Gl
gm
(a)
(b) MATHEMATICAL EQUIVALENT OF
Fro. 9.
FIG. 9(o)
Mar., 1961.]
20 7
GRAPH THEORY IN NETWORK ANALYSIS
Example 3. Given the network of Fig. 9a. Then the mathematical equivalent circuit of the network is shown in Fig. 9b. The current and voltage graphs of the network are shown in Figs. 10a and 10b, I
5
5
G2
Gz
o2
oC 2
4~5gp
G,~~~///
5
gp
6
6 (b) VOLTAGE GRAPH
(a) OURRENT GRAPH
FIG. 10. Current and voltage graphs of Fig. 9b.
respectively, where the orientation of the ordinary edges is not indicated since it can be arbitrarily assigned. Next, let us consider the three-terminal transistor shown in Fig. 11, where e, c and b represent the emitter, collector and base terminals,
,v.
f ~
-
.
.
.
r b
b FIG. 11. A three-terminal transistor.
0 C
208
WAN HEE KI~
[J. F. I.
It may be characterized as a linear active device as
respectively. follows:
a. b. c. d.
L = g.(v~ - v.) I¢ = - ag~(v, - V , ) + g~( V¢ I~ = g b ( V , - - V,) I.+I~q-Ib =0
V,)
(6)
where g, = emitter conductance, gc = collector conductance, g~ = base conductance, and a = current amplification factor. If one characterizes the first term of the right-hand expression of Eq. 6b by current and voltage elements with mutual admittance ag,, then the mathematical equivalent circuit of a three-terminal transistor and its graphical representation are as given in Fig. 12. It may, however, be desirable that node n in the equivalent circuit of a threeterminal transistor (Fig. 12) should be suppressed, since we are in-
ge
n
gc
ge
n
g¢
age
ag e
i gb b (a) EQUIVALENT CIRCUIT
gb ~.,.'~ b (b) GRAPHICAL REPRESENTATION
FIG. 12. A linear equivalent circuit of a three-terminal transistor.
terested in the voltage-current characteristics of the transistor at terminals e, c and b. Topological fornmlas for condensation of a node of a network are discussed in the next section. 2, MATRIX REPRESENTATION~ AND TOPOLOGICAL FORMULAS FOR MULTIPOLE ANALYSIS
~lultipole A n a l y s i s
Techniques proposed by Zadeh (9, 10) and Shekel (11, 12) for the analysis of multipoles are briefly reviewed before topological formulas are derived. A multipole is another name for a multiterminal device or network. Let us consider a linear network N which is a collection of multi-
Mar., 1 9 6 1 . ]
GRAPH THEORY IN NETWORK ANALYSIS
20 9
poles, as shown in Fig. 13. In order to find the voltage-current characteristics of the network N as a five-terminal device, one must first set up the indefinite admittance matrices* of multipoles M~ and Mb, and then 5 I
o 5
4 N FIG. 13. Network N comprising multipoles Ma and Mb.
add the elements of the indefinite admittance matrices corresponding to the nodes which are common to both multipoles. Let us denote the indefinite admittance matrices of Ma, Mb and N by Y~, Yb and Y, respectively. Then Yis found to be 1
2
3
4
5
1
y.,,
yal,
Yol,
ya,,
0
2
Y.,~
Ya.
Y.,,
Yo2,
0
yo,,
ya,,
yo,, + Yb,,
yo,, + yb,,
yb,,
4
yo,,
y~,,
yo,, + yb,,
y~,, + yb,,
yb,,
5
0
0
yb.,
yb,,
Yb,,
Y=3
(7)
where Y~ll
Y~ =
Y~, ya,,
ybla
yba4 yb~
yb.
y~.
Yb Y..
Y~,.
)'b,.
Yo.
and Y~o -- I i / V i if all other terminal voltages in M. are reduced to zero. Note t h a t in Eq. 7 y,j = y.~j + yb~
for
i, j = 3, 4.
(8)
4 The H-matrix defined by Percival in the reference cited is actually identical to the indefinite admittance matrix proposed by Zadeh and Shekel.
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WAN HEE KIM
[J. F. I.
Generalizing Eq. 8 for a n u m b e r of multipoles, Ma, Mb, • •., M~, interconnected at terminals 1, 2, - . . , k, the a d m i t t a n c e coefficient at each c o m m o n node is found to be Y~ = E Y~v
for
u, v = 1, 2, . . . , k
and
n = a, b, . . . , n
(9)
n
If it is desired to suppress a node, say node 3 in Fig. 13, t h a t is, if one wishes to characterize N as a four-terminal device (or network) the element in (3, 3)-position in Y can be condensed by applying the pivotal condensation m e t h o d to the d e t e r m i n a n t of Y. As an illustration, let us consider the three-terminal transistor shown in Fig. 11. F r o m Eq. 6, the following set of nodal equations can be found :
[ g,
If' I Ib
0
-- a D
c
0
0
-go
go
gb
0
-gb
Vb
0
g~
--gb
.--g,(1--a)
--g~
a g o - - g~
Vc
gb+g~+g,(1--a)
V~
(10)
To characterize the transistor as a three-terminal device, the element in (4, 4)-position in the coefficient m a t r i x of (10) is condensed. T h u s one gets
Ie Ib
go(gb+g~) 1
=b
--gbg~
--gbg~(1--a)
L
- go(go+ag
)
gbg~(1--a)+gbg¢ -
( g o - ago)
--g~g~
VO
--gbgo
Vb
go(g
+go)
(11)
Vo
where D = g~ + g~ + go(1 - a). In setting up the indefinite a d m i t t a n c e matrix from the m a t h e matical equivalent circuit with the c u r r e n t and voltage elements of a multipole, a n element of the indefinite a d m i t t a n c e matrix of the multipole, Y [Yo], is defined as (5, 6) =
Ylj
:
e~
adnfittances of the o r d i n a r y edges connected between nodes i and j sign X (mutual a d m i t t a n c e of a corresponding active edge-pair such t h a t the c u r r e n t edge t e r m i n a t e s at node i and the v o l t a g e edge t e r m i n a t e s at node j)
(12)
wheree = lifi = jande = - lifi# j. T h e sign of the second t e r m is positive if both active edges in a pair are directed a w a y or toward the nodes i and j, respectively, and negative otherwise. Let us apply the rules stated in Eq. 12 to the equivalent circuit
Mar., 1961.]
GRAPH THEORY IN NETWORK ANALYSIS
21I
shown in Fig. 9b. Then, the indefinite admittance matrix Y of the network of Fig. 9a is found to be
y =
1
2
3
4
5
6
l
0
()
al
0
--aj
0
2
0
(;,2 - G 2
0
0
0
3
--012
a~.
0
0
G1
a1
-G2
G2
4
0
0
-al
5
o~2
0
0
6
0
0
0
g,~--ce2 -gin-G1
g~ -gp
(13)
-G1 --g,,~--gp gm+gp+G1
Topological Formulas
Let us denote the incidence matrices of a pair of current graphs G~ and voltage graph Gv of a multipole by A v and A,, respectively. Y, denotes the diagonal admittance matrix in which each diagonal element is the a d m i t t a n c e of an edge in the graphs. Then, it can be shown t h a t (7, 8) the indefinite admittance matrix of the multipole, Y, is Y = A I Y M v'
(14)
where A ' is the transpose of A, and the incidence matrix of an oriented graph, A = [a~j], is defined for all nodes and edges in the graph by (13) aii
=
aij
~
alj
=
1
--
1 0
if edge j is incident at nodes i and the orientation of the edge is away from node i. if edge is incident at nodes i and the orientation of the edge is toward node i. if edge is not incident at node i.
If we denote a submatrix of an incidence matrix A by A 1, which is obtained from A by deleting a row corresponding to the reference node of the network, the node-determinant zXof the network N is given by
z~ = IA,x Y,A lv'l.
(15)
Example 4. Given an ideal three-channel circulator as shown in Fig. 14. The circulator is characterized by I, = a~(V~ -
V~)
12 = G2(V1 -
V3) -7[-y2V2
Ia = a~(V2 -
V,) + yaV3
(16)
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WAN
H E E KI]Vl
[J. F. I.
~2 Tz "11
TI
IO
>
Z,+
V[
[3
_2_
05
?'3 FIG. 14. A three-channel circulator.
where al = as = aa = a is the circulator a d m i t t a n c e . T h e m a t h e matical equivalent circuit and its c u r r e n t and voltage graphs are given in Fig. 15, where node n is the reference node and the orientation of the I
e~
2
I
2
%~ 5
I
°(3
°~~ °¢1
Yz
3
(a) EQUIVALENT CIRCUIT
2
5 (c) VOLTAGE GRAPH Gv
(b) CURRENT GRAPH Gz
Fro. 15. Graphical representation of tile circulator of Fig. 14.
ordinary edges is arbitrarily assigned. circulator, 4, is given as
The n o d e - d e t e r m i n a n t of the
1 1 A =
]A~IY,Air'
I = 2 3
0
a2 -aa
2 --0/1
y2 aa
3 O~1 --al
ya
Mar., [96r.]
GRAPH THEORY IN NETWORK ANAI,YSIS
2I 3
and 1
=2 3
al
if2
if3
y2
ya
1
0
0
0
0
0
1
0
1
0
0
0
1
0
1
Y. =
al
Alv =
a2
aa
y2
ya
0
1
--1
0
0
--1
0
1
1
0
1
-- 1
0
0
1
ai
0
0
0
0
0
a.~ 0
0
0
0
0
aa
0
0
0
0
{)
y~
0
0
0
0
()
ya
(18)
F r o m the B i n e t - C a u c h y t h e o r e m and the p r o p e r t y of the incidence matrix of a connected graph (7, 8, 13) it is well-known t h a t zX = sum of the p r o d u c t of the corresponding majors of (A~, Y,) and
Av~ = ~ el X (complete tree-product tl of current graph Gx and i
voltage graph Gv of n e t w o r k N)
(19)
where a complete tree is a tree which is c o m m o n to both c u r r e n t and voltage graphs, and a tree-product is the product of a d m i t t a n c e s of a set of edges constituting a tree. ~¢ is the sign of the complete treep r o d u c t t¢ taking the value of 1 or - 1 . W h e n a n e t w o r k contains only the ordinary edges, the c u r r e n t and voltage graphs of the n e t w o r k are identical, t h a t is, A~ = A v. Therefore the sign of a tree-product is always positive (14). However, when a n e t w o r k includes d e p e n d e n t node-pairs, its c u r r e n t and voltage graphs are different because of active edges in the graphs. Hence, a treeproduct which is c o m m o n to both c u r r e n t and voltage graphs m a y not always be positive. In order to propose a m e t h o d of sign-determination of a complete tree-product, it m a y first be necessary to introduce a n u m b e r of concepts dealing with active edges. Let us consider a complete tree consisting of edges (el, e2, • •., en-i) of a n e t w o r k of n nodes. T h e tree constituted by the same set of edges in a c u r r e n t graph is called a " c u r r e n t tree" and the tree in the corresponding voltage graph is called a "voltage tree." A pair of active edges (current and voltage edges) included in the corresponding c u r r e n t and voltage trees is called an "active edge-pair" of the complete tree. We shall label the two end-points of c u r r e n t and voltage edges in a pair (i, j) and (m, n). The nodes are chosen such t h a t node i is located
2[ 4
WAN HEE KtM
[J. F. I.
on the side farther than node j from the reference node in the current graph, and node m on the side farther than node n from the reference node of the corresponding voltage graph. Nodes i and m are called "principal nodes" (7, a) and nodes j and n are called "minor nodes" of the current and voltage edges, respectively. (This is always possible since a tree contains no loops but is connected.) Then, the sign of an active edge-pair is defined by
Sign of active edge-pair yk =
if both current and voltage edges in the pair are directed away or toward their minor nodes otherwise
(20)
Finally, the sign of h, the complete tree product, e~, is determined by ~i = ( - 1)~ I I (sign of active edge-pair y~ in h)
(21)
k
where k is the number of active edges contained in the complete treeproduct t~, and ~/ is the number of interchanges needed to give all active edge-pairs in the complete tree the same principal nodes. The proof for this method of sign-determination is somewhat lengthy and will be found elsewhere (7, 15).
Example 5. Referring to the network shown in Example 3, the node-determinant of the network, with node 6 as the reference node, is the sum of the complete tree-products of the current and voltage graphs shown in Figs. 10a and b, respectively. It is clear that there exists only one tree which is common to both the current and voltage graphs. This is shown in Fig. 16. I
5
G2 2
I
6
3
6
(a) CURRENT TREE
(b) VOLTAGE TREE FIG. 16.
Complete tree of Fig. 10.
Gz 2
Mar., I96T. ]
GRAPH THEORY IN NETWORK ANALYSIS
215
In order to determine the sign associated with the complete treeproduct, let us first find the sign of all active edge-pairs in the complete tree. There are two active edge-pairs, a~ and a2 as shown in Fig. 17. 3
0¢ I
4
(a)
o¢ 2
5
ACTIVE EDGE PAIR (x:j
I
o~ 2
5
4
(b) A C T I V E EDGE PAIR (x: 2
FIG. 17. Active edge-pairs in Fig. 16.
Applying the rules stated in (20) for each active edge-pair shown in Fig. 17, one finds t h a t the sign of active edge-pair al = 1 and the sign of active edge-pair a2 = - 1 One also nodes of nodes as product,
sees t h a t one interchange is necessary to give the principal the current and voltage edges in the pair a2 with the same in the pair al. Therefore, the sign of the complete treee, is found as e = ( - 1)~(1)( - 1) = 1.
Hence, the node-determinant, A, of the network should be evaluated as A --- ala2g~G1G2.
Generalizing (19) into a cofa~tor of (p, q)-position, Apq, of the nodedeterminant, A, of a network, one obtains Ap. = E ~ ' X (a complete tree-product u~ of current graph G/~') and the corresponding voltage graph G v (qr)
(22)
where Gz (p') is the subgraph of current graph Gr of the network derived from G with node p and the reference node r made coincident, and
216
WAN HEE KI51
IJ. F. I.
Gv (~') is the subgraph of the corresponding voltage graph Gv with node p and the reference node made coincident. The sign for a complete tree u~, e/, is determined by the rules stated in (21) for the corresponding current and voltage trees in GI ~p'~ and G v ~ L 2
I,n 'I~ ~3 ~
I,n Y3
Y2
~a
3
3
(In) (a) G1:
(~n) (b) Gv FIG. 18.
Subgraphs of Figs. 15b and c.
Example 6. Find the driving-point impedance ZH at channel 2 of the circulator shown in Fig. 14 for y2 = y~ = a. The node-determinant of the circulators, 4, is evaluated either from (17) or from (19), (20) and (21), as zX
=
ala3y2
2
+
alOt2y3
=
2
n
2
n
(:X3~
I3 Oc:~
Y3
3 CURRENT TREE
VOLTAGE TREE
(o) COMPLETE TREE UI FIG. 19.
(23)
2a ~.
GURRENT TREE
2
n Y3 3 VOLTAGE TREE
(b) COMPLETE TREE
Complete trees of GI (in) and Gv (t').
U2
Mar., I96I.]
GRAPlt THEORY IN" NETWORK ANALYSIS
217
In order to evaluate the cofactor of the node-determinant, All, by the topological m e t h o d proposed in (22), we must first obtain the subgraphs of current and voltage graphs of the circulator (Figs. 15b and c), Gz ('') and G v (~). T h e y are shown in Fig. 18. Inspecting the current and voltage subgraphs of Fig. 18, there exist two complete trees, u~ and us, as shown in Fig. 19. Since u,, consists of the ordinary edges, the sign of tree products is positive. For u,, active edge-pairs a2 and a3 (shown in Fig. 20) will have the following signs : edge-pair a2 = -- 1 edge-pair a.~ =
2
3
3
0<2
°(2 n
(o)
1. 2
~3 t n
i1
E D G E PAIR o( 2
(b)
C~3 n
E D G E PAIR o¢. 3
FIG. 20. Active edge-pairs in Fig. 19a.
If we interchange the principal nodes of the edges in the pair a3 only once, t h e y will be the same as the principal nodes of the edges in the pair as. Therefore, the sign associated with the complete tree-product of Fig. 19a, ~1', is found to be ~1'= ( -
1)I( - 1)(1) = 1.
Hence, A l l = a]a2 + y2y~ = 2a 2
and ZI1
--
A1, A
1 a"
N o d e Condensation
In order to derive topological formulas for the condensation of a node in a network, the following notation is necessary (14, 16) :
218
WAN HEy; KI~
[J. F. I,
Tr n T v = Sum of all complete tree-products of current and voltage graphs, Gz and G v , of a network N,
with n nodes including the sign of each complete tree-product, e, determined by (20) and (21). ¢:
T I (ii) n T("") = S u m of complete tree-products of Gz.i~ and GvOnn). T1 <"'~> n T v <",~> = Sum of all complete tree-products of G~ <",~> and G v<".,>. Gr <",~> is the subgraph of Gr
with nodes 1, 2, . . . , n, but not node i, made coincident in GI. G v<",~> is the subgraph of G v with nodes 1, 2, • •., n, but not node j made coincident in Gv. Then, f r o m (19) and (22) ,5 = T I n
Tv
(24)
t:
where node r is the reference node of the network.
K
P
Kt
YI'
pi
!
(a) WYE CONNECTED CIRCUIT N
(b) DELTA CONNECTED CIRCUIT N
FIG. 21. Commongrid transformation.
Mar., I96I.]
219
GRAPH THEORY IN NETWORK ANALYSIS
Let us assume that node n is to be condensed in the network N or its graphical representation G. Then, the indefinite admittance matrix of the condensed network, Y* = [Y~,0*], is found to be (16)
(25)
Y.o* = ~" 1/y,,,, I T , <'-I,'~> CI Tv <'-''~>]
where y,,,, is the element in (n, n)-position of the indefinite admittance matrix of the network, and is determined by (12). ~ = 1 if i = j and e= - lifi#j.
Example 7. The wye-delta transformation of tube circuits will be investigated in terms of graphical formulas proposed in (25). Let us consider the so-called "common grid transformation" (3) which is shown in Fig. 21. First the graphical representation of both circuits is
y,
s
gp
K
P
n
p'
KI
13'
(0) GRAPH OF FIG. 21(o)
(b) GRAPH OF FIG. 21(b)
FIG. 22. Graphs of tube circuits of Figs. 21a and b.
found. This is shown in Fig. 22, where g~, g~' are the transconductances and g~ and gp' the place-conductances of the tubes in Figs. 21a and b, respectively. The indefinite admittance matrices of networks N and N*, Y and Y*, are found by (12) or by the method proposed by Zadeh (9, 10) as
220
WAN
HEE
KIM
[J. F. t.
P
K
n
G
g~+y2
-gp-gm
0
--y2+gm
0
--yl --gm
n
0
gp'-[-Yl+gm 0
ya
--y3
G
--Y2
-yl
P Y=K
pP
Yl+y2+ya
--y3
p,
K'
1¢t
g~t Acy 1, +Y3'
--gp'--y/--g,.'
--y3t +gm '
--gp' --y/
g~' +Yl' +Y2' +g,.'
--y2' __gin,
-y~'
--y2 t
Y~' +Y3'
Y* = K' nt
(26)
(27)
By applying (25), Y* is found, in terms of current and voltage graphs and their subgraphs of N, to be e
TI(Kn) N T v (K"),
_ [T/K,O (2] Tv(P.).],
-- [T,(K.) A Tv (eK)]
TI (Pn) N Tv (Pn),
_ [ T ( p . ) f2] Tv(Vm]
-- ['TI(PK)('~ Tv(P.)-],
T (PK) (~ Tv (PK)
1
yl +y~ +Y3
-- ['Tf :P'O I'~ Tv(K")'], -
ETt(V.)~
Tv(K")'],
gT, (yt "-[-y2 -Jr-y3)"+'y2(yl "J-y3"I-gin)
-- g~ (yt q-Y2-l-y3) --Y2 (yl +g~) -- Y3g~
-- gp (Y, q-Y2 +Y3) -- Y2 (yt +g,,,)
ep (Y, -[-Y'~--bye) --[-yl (y2 q--y3) q- e,,, (y~ q-y3)
-- Y2y3
- yly3
1
Y, -t-y2 +Y3
--Y3(Y2--gm) 1
(28)
From (27) and (28) one finds that Ylt
=
y2(yl + g,.)/S
Y2' = yly3/S
gp' = gp(y, + y2 + y3)/S ] g,.' = g.,y3/S
(29)
y~' = y2y3/S w h e r e S = yl + y2 + y3. CONCLUSION
When a network contains linear multiterminal devices, both active and passive, the theory of linear graphs is always applicable as long as there exists the mathematical equivalent circuit in terms of current and voltage elements. This opens a possibility of analyzing an active and mutually coupled network by a digital computer. The concept of the mathematical equivalent circuit of a network having dependent nodepairs is found to be useful also in the analysis of a magnetic amplifier (17).
Mar., I96I.]
GRAPH THEORY IN NETWORK ANALYSIS
22I
REFERENCES
(1) Y. H. Kv, "Resume of Maxwell's and Kirchhoff's Rules for Network Analysis," JOUR. FRANKLIN INST., Vol. 254, pp. 211-224 (1952). (2) W. S. PERCIVAL,"The Solution of Passive Electrical Networks by Means of Mathematical Trees," Proc. l E E [-London'], Vol. 100, pt. IIl, pp. 143-150 (1953). (3) W. H. KIM, D. H. YOUNGER,C. V. FREIMANAND W. MAYEDA,"On Iterative Factorization in Network Analysis by Digital Computer," Proc. 1960 East Joint Computer Conference, December 1960, pp. 241-253. (4) H. W. Hsu, "On Transformations of Linear Active Networks with Applications at UltraHigh Frequencies," Proc. IRE, Vol. 41, pp. 59-67 (1953). (5) W. S. PERCIVAL, "Improved Matrix and Determinant Methods for Solving Networks," Proc. I E E [London], Vol. 101, pt. IV, pp. 258-265 (1954). (6) W. S. PERCIVAL, "The Graphs of Active Networks," Proc. I E E [London], Vol. 102, pt. C, pp. 270-278 (1955). (7) C. L. COATES, "General Topological Formulas for Linear Network Functions," Genera] Electric Res. Lab. Rep. No. 57-RL-1746, Schenectady, N. Y. (8) W. MAYEDA,"Topological Formulas for Active Networks," Interim Tech. Report No. 8, Contract No. DA-11-022-ORD-1983, University of Illinois, Urbana, Ill., January~ 1958. (9) L. A. ZADEH,"A Note on the Analysis of Vacuum Tube and Transistor Circuits," Yroc. I R E , Vol. 41, pp. 989-992 (1953). (10) L. A. ZADEH, "Multipole Analysis of Active Networks," I R E Trans. on Circuit Theory, Vol. CT-4, pp. 97-105 (1957). (11) J. SHEKEL,"Matrix Representation of Transistor Circuits," Proc. I R E , Vol. 40, pp. 14931497 (1952). (12) J. SHEKEL, "The Gyrator as a 3-Terminal Element," Proc. I R E , Vol. 41, pp. 1014-1016 (1953). (13) S. SESHU, "Topological Considerations in the Design of Driving Point Functions," I R E Trans. on Circuit Theory, Vol. CT-2, pp. 356-367 (1955). (14) W. H. KIM, "Topological Evaluation of Network Functions," JOUR. FRANKLIN INST., Vol. 267, pp. 283-293 (1959). (15) I. FRISCI-IAND W. H. KIM, "Properties of 2-semi-isomorphic Graphs and Their Applications" (to be published). (16) W. H. KIM, "Topological Analysis of Linear Multipoles," The Matrix and Tensor Quarterly, London, Vol. 11, No. 2, pp. 33-41 (1960). (17) K. A. PULLEN, JR., "The Use of Network Topology with Active Circuits," Report No. 1096, Ballistic Research Labs., Aberdeen Proving Grounds, Maryland; February 1960.
This paper presents a unified graphical approach to the analysis of an electrical network which contains linear multi-terminal devices such as vacuum tubes, transistors, transformers, gyrators, circulators, etc. The concept of the mathematical equivalent circuit of an active device or a passive but mutually coupled device which was introduced by Percival, is related to the indefinite admittance matrix of a multipole defined by Zadeh and Shekel. Various graphical formulas for the evaluation of network functions of a multipole are shown, including the node-condensation techniques. A number of examples are presented. INTRODUCTION
The method of topological analysis (application of the theory of linear graphs) of a network comprising linear, passive and bilateral twoterminal devices such as resistors, inductors and capacitors, has been proved very useful (1, 2). 5 It is particularly suitable for digital computers (3). However, the method is not directly applicable to a network containing linear active devices, such as vacuum tubes and transistors, or mutually coupled passive devices, such as transformers and gyrators. The difficulty is the graphical representation of mutually dependent node-pairs due to a dependent source or the mutual coupling effect imbedded in a device (4). In order to characterize a dependent node-pair in a tube and transformer by graphical representation, Percival (5, 6) recently introduced artificial two-terminal elements, so-called "current and voltage elements," and hence made it possible for the theory of linear graphs to be extended to the analysis of active and mutually coupled networks (7, a). This paper presents first an expository review and extension of the concept and use of "current and voltage elements" in graphical characterization of dependent node-pairs in a network. Then, through investigation of the topological aspects of the techniques of multipole analysis proposed by Zadeh (9, 10) and Shekel (11, 12), a unified approach to the analysis of an electrical network containing multiterminal active and passive devices is pursued. * This work was supported by National Science Foundation Grants G-6020 and G-10354. Preparation of this paper was partially supported by the Marcellus Hartley Fund. 1 D e p a r t m e n t of Electrical Engineering, Columbia University, New York, N. Y. 2 The boldface numbers in parentheses refer to the references appended to this paper. 200
Mar., 1961.]
GRAPtt THEORY IN NETWORK ANALYSIS
2OI
I. GRAPHICAL CHARACTERIZATION OF MUTUALLY DEPENDENT NODE-PAIRS
Let us consider node-pairs (pq) and (ran) in the linear network N shown in Fig. 1. If the current ipq(t) flowing from node p to node q depends not only on the voltage-difference between nodes p and q,
N P
--
Ipq
--
+
+
Vpq
Vmn
q
om
On
FIG. 1. A linear network with mutually dependent node-pairs.
v~ (t), b u t also on the voltage difference between nodes m and n, vm~(t), then, since the network is linear, one m a y write y~q Vpq + y~.m~ V,~,~ = I~q
(1)
where the y's are proportional factors and I -- ~ei(t), V = .ev(t). In Eq. 1, since yp~ relates the voltage and current for the same node pair, it is called the "self-admittance" or simply " a d m i t t a n c e " of the element (or elements) connected between nodes p and q. On the other hand, y~q,m~relates the voltage and current for different node-pairs, and hence is called the " m u t u a l admittance." If the network N consists entirely of passive devices, then Eq. 1 is divided into two isolated parts, t h a t is, y,.V~
or
= I~
(2)
It is of course clear t h a t if N contains only the ordinary two-terminal devices (R, L and C elements), then ypq -- yq~ for all node-pairs in the network, which is said to be "passive and reciprocal." However, if
W A N ]:tEE KIN[
202
[J. F. I.
transformers or gyrators are contained in N, then for some node-pairs in the network ypq.,,,,, ~ ym,,.~,q, and N is called "passive and non-reciprocal." Therefore, Eq. 1 holds for some node-pairs in N if it contains a linear active device or devices. In Eq. 1, Ipq, which depends on both Vpq and V~,, can be decomposed into two components: the one due to Vpq is denoted by I~q' and the other, due to Vm~, is denoted by Ip~". T h u s
a. b.
y,,~V~,q = I,,~' y~,~,,~, V,~, = Ip~"
(4)
c. Ip~' -q- Ivq" = Ip,,. The voltage-current relationship of Eq. 4a represents an ordinary twoterminal element. Equation 4b shows the dependent relationship of node-pairs (pq) and (ran). A voltage-difference between nodes m and n, with m positive, causes a current flowing from nodes p to q.
IJ
b
bd~
UtU'bd/~_~
ULU*bd~/
Z
W
+
N FIG. 2. Use of voltage and current elements to represent dependent node-pairs.
In order to apply the concept of graphs, it is therefore desirable to introduce artificial two-terminal elements into node-pairs (ran) and (pq) to characterize their d e p e n d e n t relationship. The element inserted between nodes m and n, called a "voltage element," is to sense a voltagedifference in the node-pair. The element introduced in node-pair (pq), called a "current element," produces a current of magnitude I " flowing from node p to node q due to the voltage-difference V in node-pair (ran). T h e voltage and current elements are thus related by m u t u a l a d m i t t a n c e ypq .... Note t h a t voltage and current elements always occur in a pair
Mar., r96I.]
GRAPH
"]'HEORY IN N E T W O R K
ANALYSIS
203
and between different node-pairs. If they occur in the same node-pair, that is, p = m and q = n, the mutual admittance ypq,mn is reduced to the self-admittance ypq. Note also that the voltage and current elements introduced cannot be reduced to zero in the evaluation of a network function although all independent voltage and current generators are always reduced to zero. The graphical representation of voltage and current elements for dependent node-pairs (pq) and (ran) is shown in Fig. 2. Since the current and voltage elements are two-terminal elements they can be represented by the edges 3 of a linear graph as current and voltage edges, respectively. The weight of the edges is the mutual admittance. The orientation of a current edge is determined by the direction of the current-flow in the corresponding current element, the orientation of a voltage edge by the polarity of the corresponding voltage element. The graphical representation of the network N (Fig. 2) is shown in Fig. 3, where G denotes the graph corresponding to the net-
G ) rl]
P
Ypq Ypq ,mn
V Ypq,mn
q Fro. 3.
Graphical representation of dependent node-palrs.
work N. The current edge is indicated by double arrows, and the voltage edge by a triangular-shaped arrow. The following examples illustrate the concept and use of current and voltage elements in characterizing a network containing linear active devices as well as mutually coupled nmltitermlnal passive elements. Example 1. A triode of Fig. 4 is characterized as a linear active device by I e = g,~VoK + g~,VeK (4) 3 The term " b r a n c h " may be substituted occasionally for "edge," but the term "element" will only be used in the sense of "element of a network" or "element of a matrix."
204
WAN HEE KIM
IJ. I;. I.
P "rp
Vp~
G
K FIG. 4. A triode. where gm is the transconductance and gp the plate-conductance. An equivalent circuit of the triode characterized by Eq. 4 in terms of voltage and current elements is known as the mathematical equivalent circuit (5, 6) and is shown in Fig. 5a. Figure 5b shows the conventional equivalent circuit of a triode for comparison. T h e graphical representation of Fig. 5a is given in Fig. 6.
P
G
gm "~///
P
~
G0
gP
gmTpK
~
VGK
K
K
(a) MATHEMATICAL EQUIVALENT CIRCUIT IN TERMS OF VOLTAGE AND CURRENT ELEMENTS
(b) CONVENTIONAL EQUIVALENT CIRCUIT
FIG. 5. Equivalent circuits of a triode.
Mar., I96I.]
GRAPH THEORY IN NETWORK ANALYSIS
205
P
\ gp
c3m
K FIG. 6. Graphical representation of Fig. 5a. E x a m p l e 2. A two-channel gyrator, shown in Fig. 7, may be characterized by the following set of equations (12):
(5)
I1 = otl V2 12 = -
a2V1
O<2 Jo
~
~
o5
v,
VI
2 o
04 F.;. 7. A gyrator.
where al = as = a is the gyrator admittance. The mathematical equivalent circuit of the gyrator represented by the terminal characterization of Eq. 5 is shown in Fig. 8a and its graphical representation in Fig. 8b. Now a graph representing a device or a network can be decomposed into two subgraphs (7,8). One is the "current graph" containing the edges corresponding to the ordinary elements and current elements. The other is the "voltage graph," with the edges corresponding to the
206
WAN HEE KIM 5
I
[J. F. I 1
5
2
4
~C
I
2
4
(a) AN EQUIVALENT CIRCUIT OF A GYRATOR FIG. 8.
(b) GRAPHICAL REPRESENTATION OF A GYRATOR Representationof a gyrator.
ordinary elements and voltage elements. One may ilnmediately see t h a t if a network consists entirely of ordinary elements, the current and voltage graphs of the network are identical ; t h a t is, a graph corresponding to an ordinary network is the current graph of the network and at the same time the voltage graph. The orientation of each edge of a graph of an ordinary network is arbitrarily assigned. When a network contains active devices or mutually coupled devices, the current and voltage graphs of the network m a y not be identical. This is illustrated in Example 3.
Gz
3
0 ....
3
5
~
z
Gz
oC I
4
4
Gl
gm
(a)
(b) MATHEMATICAL EQUIVALENT OF
Fro. 9.
FIG. 9(o)
Mar., 1961.]
20 7
GRAPH THEORY IN NETWORK ANALYSIS
Example 3. Given the network of Fig. 9a. Then the mathematical equivalent circuit of the network is shown in Fig. 9b. The current and voltage graphs of the network are shown in Figs. 10a and 10b, I
5
5
G2
Gz
o2
oC 2
4~5gp
G,~~~///
5
gp
6
6 (b) VOLTAGE GRAPH
(a) OURRENT GRAPH
FIG. 10. Current and voltage graphs of Fig. 9b.
respectively, where the orientation of the ordinary edges is not indicated since it can be arbitrarily assigned. Next, let us consider the three-terminal transistor shown in Fig. 11, where e, c and b represent the emitter, collector and base terminals,
,v.
f ~
-
.
.
.
r b
b FIG. 11. A three-terminal transistor.
0 C
208
WAN HEE KI~
[J. F. I.
It may be characterized as a linear active device as
respectively. follows:
a. b. c. d.
L = g.(v~ - v.) I¢ = - ag~(v, - V , ) + g~( V¢ I~ = g b ( V , - - V,) I.+I~q-Ib =0
V,)
(6)
where g, = emitter conductance, gc = collector conductance, g~ = base conductance, and a = current amplification factor. If one characterizes the first term of the right-hand expression of Eq. 6b by current and voltage elements with mutual admittance ag,, then the mathematical equivalent circuit of a three-terminal transistor and its graphical representation are as given in Fig. 12. It may, however, be desirable that node n in the equivalent circuit of a threeterminal transistor (Fig. 12) should be suppressed, since we are in-
ge
n
gc
ge
n
g¢
age
ag e
i gb b (a) EQUIVALENT CIRCUIT
gb ~.,.'~ b (b) GRAPHICAL REPRESENTATION
FIG. 12. A linear equivalent circuit of a three-terminal transistor.
terested in the voltage-current characteristics of the transistor at terminals e, c and b. Topological fornmlas for condensation of a node of a network are discussed in the next section. 2, MATRIX REPRESENTATION~ AND TOPOLOGICAL FORMULAS FOR MULTIPOLE ANALYSIS
~lultipole A n a l y s i s
Techniques proposed by Zadeh (9, 10) and Shekel (11, 12) for the analysis of multipoles are briefly reviewed before topological formulas are derived. A multipole is another name for a multiterminal device or network. Let us consider a linear network N which is a collection of multi-
Mar., 1 9 6 1 . ]
GRAPH THEORY IN NETWORK ANALYSIS
20 9
poles, as shown in Fig. 13. In order to find the voltage-current characteristics of the network N as a five-terminal device, one must first set up the indefinite admittance matrices* of multipoles M~ and Mb, and then 5 I
o 5
4 N FIG. 13. Network N comprising multipoles Ma and Mb.
add the elements of the indefinite admittance matrices corresponding to the nodes which are common to both multipoles. Let us denote the indefinite admittance matrices of Ma, Mb and N by Y~, Yb and Y, respectively. Then Yis found to be 1
2
3
4
5
1
y.,,
yal,
Yol,
ya,,
0
2
Y.,~
Ya.
Y.,,
Yo2,
0
yo,,
ya,,
yo,, + Yb,,
yo,, + yb,,
yb,,
4
yo,,
y~,,
yo,, + yb,,
y~,, + yb,,
yb,,
5
0
0
yb.,
yb,,
Yb,,
Y=3
(7)
where Y~ll
Y~ =
Y~, ya,,
ybla
yba4 yb~
yb.
y~.
Yb Y..
Y~,.
)'b,.
Yo.
and Y~o -- I i / V i if all other terminal voltages in M. are reduced to zero. Note t h a t in Eq. 7 y,j = y.~j + yb~
for
i, j = 3, 4.
(8)
4 The H-matrix defined by Percival in the reference cited is actually identical to the indefinite admittance matrix proposed by Zadeh and Shekel.
210
WAN HEE KIM
[J. F. I.
Generalizing Eq. 8 for a n u m b e r of multipoles, Ma, Mb, • •., M~, interconnected at terminals 1, 2, - . . , k, the a d m i t t a n c e coefficient at each c o m m o n node is found to be Y~ = E Y~v
for
u, v = 1, 2, . . . , k
and
n = a, b, . . . , n
(9)
n
If it is desired to suppress a node, say node 3 in Fig. 13, t h a t is, if one wishes to characterize N as a four-terminal device (or network) the element in (3, 3)-position in Y can be condensed by applying the pivotal condensation m e t h o d to the d e t e r m i n a n t of Y. As an illustration, let us consider the three-terminal transistor shown in Fig. 11. F r o m Eq. 6, the following set of nodal equations can be found :
[ g,
If' I Ib
0
-- a D
c
0
0
-go
go
gb
0
-gb
Vb
0
g~
--gb
.--g,(1--a)
--g~
a g o - - g~
Vc
gb+g~+g,(1--a)
V~
(10)
To characterize the transistor as a three-terminal device, the element in (4, 4)-position in the coefficient m a t r i x of (10) is condensed. T h u s one gets
Ie Ib
go(gb+g~) 1
=b
--gbg~
--gbg~(1--a)
L
- go(go+ag
)
gbg~(1--a)+gbg¢ -
( g o - ago)
--g~g~
VO
--gbgo
Vb
go(g
+go)
(11)
Vo
where D = g~ + g~ + go(1 - a). In setting up the indefinite a d m i t t a n c e matrix from the m a t h e matical equivalent circuit with the c u r r e n t and voltage elements of a multipole, a n element of the indefinite a d m i t t a n c e matrix of the multipole, Y [Yo], is defined as (5, 6) =
Ylj
:
e~
adnfittances of the o r d i n a r y edges connected between nodes i and j sign X (mutual a d m i t t a n c e of a corresponding active edge-pair such t h a t the c u r r e n t edge t e r m i n a t e s at node i and the v o l t a g e edge t e r m i n a t e s at node j)
(12)
wheree = lifi = jande = - lifi# j. T h e sign of the second t e r m is positive if both active edges in a pair are directed a w a y or toward the nodes i and j, respectively, and negative otherwise. Let us apply the rules stated in Eq. 12 to the equivalent circuit
Mar., 1961.]
GRAPH THEORY IN NETWORK ANALYSIS
21I
shown in Fig. 9b. Then, the indefinite admittance matrix Y of the network of Fig. 9a is found to be
y =
1
2
3
4
5
6
l
0
()
al
0
--aj
0
2
0
(;,2 - G 2
0
0
0
3
--012
a~.
0
0
G1
a1
-G2
G2
4
0
0
-al
5
o~2
0
0
6
0
0
0
g,~--ce2 -gin-G1
g~ -gp
(13)
-G1 --g,,~--gp gm+gp+G1
Topological Formulas
Let us denote the incidence matrices of a pair of current graphs G~ and voltage graph Gv of a multipole by A v and A,, respectively. Y, denotes the diagonal admittance matrix in which each diagonal element is the a d m i t t a n c e of an edge in the graphs. Then, it can be shown t h a t (7, 8) the indefinite admittance matrix of the multipole, Y, is Y = A I Y M v'
(14)
where A ' is the transpose of A, and the incidence matrix of an oriented graph, A = [a~j], is defined for all nodes and edges in the graph by (13) aii
=
aij
~
alj
=
1
--
1 0
if edge j is incident at nodes i and the orientation of the edge is away from node i. if edge is incident at nodes i and the orientation of the edge is toward node i. if edge is not incident at node i.
If we denote a submatrix of an incidence matrix A by A 1, which is obtained from A by deleting a row corresponding to the reference node of the network, the node-determinant zXof the network N is given by
z~ = IA,x Y,A lv'l.
(15)
Example 4. Given an ideal three-channel circulator as shown in Fig. 14. The circulator is characterized by I, = a~(V~ -
V~)
12 = G2(V1 -
V3) -7[-y2V2
Ia = a~(V2 -
V,) + yaV3
(16)
212
WAN
H E E KI]Vl
[J. F. I.
~2 Tz "11
TI
IO
>
Z,+
V[
[3
_2_
05
?'3 FIG. 14. A three-channel circulator.
where al = as = aa = a is the circulator a d m i t t a n c e . T h e m a t h e matical equivalent circuit and its c u r r e n t and voltage graphs are given in Fig. 15, where node n is the reference node and the orientation of the I
e~
2
I
2
%~ 5
I
°(3
°~~ °¢1
Yz
3
(a) EQUIVALENT CIRCUIT
2
5 (c) VOLTAGE GRAPH Gv
(b) CURRENT GRAPH Gz
Fro. 15. Graphical representation of tile circulator of Fig. 14.
ordinary edges is arbitrarily assigned. circulator, 4, is given as
The n o d e - d e t e r m i n a n t of the
1 1 A =
]A~IY,Air'
I = 2 3
0
a2 -aa
2 --0/1
y2 aa
3 O~1 --al
ya
Mar., [96r.]
GRAPH THEORY IN NETWORK ANAI,YSIS
2I 3
and 1
=2 3
al
if2
if3
y2
ya
1
0
0
0
0
0
1
0
1
0
0
0
1
0
1
Y. =
al
Alv =
a2
aa
y2
ya
0
1
--1
0
0
--1
0
1
1
0
1
-- 1
0
0
1
ai
0
0
0
0
0
a.~ 0
0
0
0
0
aa
0
0
0
0
{)
y~
0
0
0
0
()
ya
(18)
F r o m the B i n e t - C a u c h y t h e o r e m and the p r o p e r t y of the incidence matrix of a connected graph (7, 8, 13) it is well-known t h a t zX = sum of the p r o d u c t of the corresponding majors of (A~, Y,) and
Av~ = ~ el X (complete tree-product tl of current graph Gx and i
voltage graph Gv of n e t w o r k N)
(19)
where a complete tree is a tree which is c o m m o n to both c u r r e n t and voltage graphs, and a tree-product is the product of a d m i t t a n c e s of a set of edges constituting a tree. ~¢ is the sign of the complete treep r o d u c t t¢ taking the value of 1 or - 1 . W h e n a n e t w o r k contains only the ordinary edges, the c u r r e n t and voltage graphs of the n e t w o r k are identical, t h a t is, A~ = A v. Therefore the sign of a tree-product is always positive (14). However, when a n e t w o r k includes d e p e n d e n t node-pairs, its c u r r e n t and voltage graphs are different because of active edges in the graphs. Hence, a treeproduct which is c o m m o n to both c u r r e n t and voltage graphs m a y not always be positive. In order to propose a m e t h o d of sign-determination of a complete tree-product, it m a y first be necessary to introduce a n u m b e r of concepts dealing with active edges. Let us consider a complete tree consisting of edges (el, e2, • •., en-i) of a n e t w o r k of n nodes. T h e tree constituted by the same set of edges in a c u r r e n t graph is called a " c u r r e n t tree" and the tree in the corresponding voltage graph is called a "voltage tree." A pair of active edges (current and voltage edges) included in the corresponding c u r r e n t and voltage trees is called an "active edge-pair" of the complete tree. We shall label the two end-points of c u r r e n t and voltage edges in a pair (i, j) and (m, n). The nodes are chosen such t h a t node i is located
2[ 4
WAN HEE KtM
[J. F. I.
on the side farther than node j from the reference node in the current graph, and node m on the side farther than node n from the reference node of the corresponding voltage graph. Nodes i and m are called "principal nodes" (7, a) and nodes j and n are called "minor nodes" of the current and voltage edges, respectively. (This is always possible since a tree contains no loops but is connected.) Then, the sign of an active edge-pair is defined by
Sign of active edge-pair yk =
if both current and voltage edges in the pair are directed away or toward their minor nodes otherwise
(20)
Finally, the sign of h, the complete tree product, e~, is determined by ~i = ( - 1)~ I I (sign of active edge-pair y~ in h)
(21)
k
where k is the number of active edges contained in the complete treeproduct t~, and ~/ is the number of interchanges needed to give all active edge-pairs in the complete tree the same principal nodes. The proof for this method of sign-determination is somewhat lengthy and will be found elsewhere (7, 15).
Example 5. Referring to the network shown in Example 3, the node-determinant of the network, with node 6 as the reference node, is the sum of the complete tree-products of the current and voltage graphs shown in Figs. 10a and b, respectively. It is clear that there exists only one tree which is common to both the current and voltage graphs. This is shown in Fig. 16. I
5
G2 2
I
6
3
6
(a) CURRENT TREE
(b) VOLTAGE TREE FIG. 16.
Complete tree of Fig. 10.
Gz 2
Mar., I96T. ]
GRAPH THEORY IN NETWORK ANALYSIS
215
In order to determine the sign associated with the complete treeproduct, let us first find the sign of all active edge-pairs in the complete tree. There are two active edge-pairs, a~ and a2 as shown in Fig. 17. 3
0¢ I
4
(a)
o¢ 2
5
ACTIVE EDGE PAIR (x:j
I
o~ 2
5
4
(b) A C T I V E EDGE PAIR (x: 2
FIG. 17. Active edge-pairs in Fig. 16.
Applying the rules stated in (20) for each active edge-pair shown in Fig. 17, one finds t h a t the sign of active edge-pair al = 1 and the sign of active edge-pair a2 = - 1 One also nodes of nodes as product,
sees t h a t one interchange is necessary to give the principal the current and voltage edges in the pair a2 with the same in the pair al. Therefore, the sign of the complete treee, is found as e = ( - 1)~(1)( - 1) = 1.
Hence, the node-determinant, A, of the network should be evaluated as A --- ala2g~G1G2.
Generalizing (19) into a cofa~tor of (p, q)-position, Apq, of the nodedeterminant, A, of a network, one obtains Ap. = E ~ ' X (a complete tree-product u~ of current graph G/~') and the corresponding voltage graph G v (qr)
(22)
where Gz (p') is the subgraph of current graph Gr of the network derived from G with node p and the reference node r made coincident, and
216
WAN HEE KI51
IJ. F. I.
Gv (~') is the subgraph of the corresponding voltage graph Gv with node p and the reference node made coincident. The sign for a complete tree u~, e/, is determined by the rules stated in (21) for the corresponding current and voltage trees in GI ~p'~ and G v ~ L 2
I,n 'I~ ~3 ~
I,n Y3
Y2
~a
3
3
(In) (a) G1:
(~n) (b) Gv FIG. 18.
Subgraphs of Figs. 15b and c.
Example 6. Find the driving-point impedance ZH at channel 2 of the circulator shown in Fig. 14 for y2 = y~ = a. The node-determinant of the circulators, 4, is evaluated either from (17) or from (19), (20) and (21), as zX
=
ala3y2
2
+
alOt2y3
=
2
n
2
n
(:X3~
I3 Oc:~
Y3
3 CURRENT TREE
VOLTAGE TREE
(o) COMPLETE TREE UI FIG. 19.
(23)
2a ~.
GURRENT TREE
2
n Y3 3 VOLTAGE TREE
(b) COMPLETE TREE
Complete trees of GI (in) and Gv (t').
U2
Mar., I96I.]
GRAPlt THEORY IN" NETWORK ANALYSIS
217
In order to evaluate the cofactor of the node-determinant, All, by the topological m e t h o d proposed in (22), we must first obtain the subgraphs of current and voltage graphs of the circulator (Figs. 15b and c), Gz ('') and G v (~). T h e y are shown in Fig. 18. Inspecting the current and voltage subgraphs of Fig. 18, there exist two complete trees, u~ and us, as shown in Fig. 19. Since u,, consists of the ordinary edges, the sign of tree products is positive. For u,, active edge-pairs a2 and a3 (shown in Fig. 20) will have the following signs : edge-pair a2 = -- 1 edge-pair a.~ =
2
3
3
0<2
°(2 n
(o)
1. 2
~3 t n
i1
E D G E PAIR o( 2
(b)
C~3 n
E D G E PAIR o¢. 3
FIG. 20. Active edge-pairs in Fig. 19a.
If we interchange the principal nodes of the edges in the pair a3 only once, t h e y will be the same as the principal nodes of the edges in the pair as. Therefore, the sign associated with the complete tree-product of Fig. 19a, ~1', is found to be ~1'= ( -
1)I( - 1)(1) = 1.
Hence, A l l = a]a2 + y2y~ = 2a 2
and ZI1
--
A1, A
1 a"
N o d e Condensation
In order to derive topological formulas for the condensation of a node in a network, the following notation is necessary (14, 16) :
218
WAN HEy; KI~
[J. F. I,
Tr n T v = Sum of all complete tree-products of current and voltage graphs, Gz and G v , of a network N,
with n nodes including the sign of each complete tree-product, e, determined by (20) and (21). ¢:
T I (ii) n T("") = S u m of complete tree-products of Gz.i~ and GvOnn). T1 <"'~> n T v <",~> = Sum of all complete tree-products of G~ <",~> and G v<".,>. Gr <",~> is the subgraph of Gr
with nodes 1, 2, . . . , n, but not node i, made coincident in GI. G v<",~> is the subgraph of G v with nodes 1, 2, • •., n, but not node j made coincident in Gv. Then, f r o m (19) and (22) ,5 = T I n
Tv
(24)
t:
where node r is the reference node of the network.
K
P
Kt
YI'
pi
!
(a) WYE CONNECTED CIRCUIT N
(b) DELTA CONNECTED CIRCUIT N
FIG. 21. Commongrid transformation.
Mar., I96I.]
219
GRAPH THEORY IN NETWORK ANALYSIS
Let us assume that node n is to be condensed in the network N or its graphical representation G. Then, the indefinite admittance matrix of the condensed network, Y* = [Y~,0*], is found to be (16)
(25)
Y.o* = ~" 1/y,,,, I T , <'-I,'~> CI Tv <'-''~>]
where y,,,, is the element in (n, n)-position of the indefinite admittance matrix of the network, and is determined by (12). ~ = 1 if i = j and e= - lifi#j.
Example 7. The wye-delta transformation of tube circuits will be investigated in terms of graphical formulas proposed in (25). Let us consider the so-called "common grid transformation" (3) which is shown in Fig. 21. First the graphical representation of both circuits is
y,
s
gp
K
P
n
p'
KI
13'
(0) GRAPH OF FIG. 21(o)
(b) GRAPH OF FIG. 21(b)
FIG. 22. Graphs of tube circuits of Figs. 21a and b.
found. This is shown in Fig. 22, where g~, g~' are the transconductances and g~ and gp' the place-conductances of the tubes in Figs. 21a and b, respectively. The indefinite admittance matrices of networks N and N*, Y and Y*, are found by (12) or by the method proposed by Zadeh (9, 10) as
220
WAN
HEE
KIM
[J. F. t.
P
K
n
G
g~+y2
-gp-gm
0
--y2+gm
0
--yl --gm
n
0
gp'-[-Yl+gm 0
ya
--y3
G
--Y2
-yl
P Y=K
pP
Yl+y2+ya
--y3
p,
K'
1¢t
g~t Acy 1, +Y3'
--gp'--y/--g,.'
--y3t +gm '
--gp' --y/
g~' +Yl' +Y2' +g,.'
--y2' __gin,
-y~'
--y2 t
Y~' +Y3'
Y* = K' nt
(26)
(27)
By applying (25), Y* is found, in terms of current and voltage graphs and their subgraphs of N, to be e
TI(Kn) N T v (K"),
_ [T/K,O (2] Tv(P.).],
-- [T,(K.) A Tv (eK)]
TI (Pn) N Tv (Pn),
_ [ T ( p . ) f2] Tv(Vm]
-- ['TI(PK)('~ Tv(P.)-],
T (PK) (~ Tv (PK)
1
yl +y~ +Y3
-- ['Tf :P'O I'~ Tv(K")'], -
ETt(V.)~
Tv(K")'],
gT, (yt "-[-y2 -Jr-y3)"+'y2(yl "J-y3"I-gin)
-- g~ (yt q-Y2-l-y3) --Y2 (yl +g~) -- Y3g~
-- gp (Y, q-Y2 +Y3) -- Y2 (yt +g,,,)
ep (Y, -[-Y'~--bye) --[-yl (y2 q--y3) q- e,,, (y~ q-y3)
-- Y2y3
- yly3
1
Y, -t-y2 +Y3
--Y3(Y2--gm) 1
(28)
From (27) and (28) one finds that Ylt
=
y2(yl + g,.)/S
Y2' = yly3/S
gp' = gp(y, + y2 + y3)/S ] g,.' = g.,y3/S
(29)
y~' = y2y3/S w h e r e S = yl + y2 + y3. CONCLUSION
When a network contains linear multiterminal devices, both active and passive, the theory of linear graphs is always applicable as long as there exists the mathematical equivalent circuit in terms of current and voltage elements. This opens a possibility of analyzing an active and mutually coupled network by a digital computer. The concept of the mathematical equivalent circuit of a network having dependent nodepairs is found to be useful also in the analysis of a magnetic amplifier (17).
Mar., I96I.]
GRAPH THEORY IN NETWORK ANALYSIS
22I
REFERENCES
(1) Y. H. Kv, "Resume of Maxwell's and Kirchhoff's Rules for Network Analysis," JOUR. FRANKLIN INST., Vol. 254, pp. 211-224 (1952). (2) W. S. PERCIVAL,"The Solution of Passive Electrical Networks by Means of Mathematical Trees," Proc. l E E [-London'], Vol. 100, pt. IIl, pp. 143-150 (1953). (3) W. H. KIM, D. H. YOUNGER,C. V. FREIMANAND W. MAYEDA,"On Iterative Factorization in Network Analysis by Digital Computer," Proc. 1960 East Joint Computer Conference, December 1960, pp. 241-253. (4) H. W. Hsu, "On Transformations of Linear Active Networks with Applications at UltraHigh Frequencies," Proc. IRE, Vol. 41, pp. 59-67 (1953). (5) W. S. PERCIVAL, "Improved Matrix and Determinant Methods for Solving Networks," Proc. I E E [London], Vol. 101, pt. IV, pp. 258-265 (1954). (6) W. S. PERCIVAL, "The Graphs of Active Networks," Proc. I E E [London], Vol. 102, pt. C, pp. 270-278 (1955). (7) C. L. COATES, "General Topological Formulas for Linear Network Functions," Genera] Electric Res. Lab. Rep. No. 57-RL-1746, Schenectady, N. Y. (8) W. MAYEDA,"Topological Formulas for Active Networks," Interim Tech. Report No. 8, Contract No. DA-11-022-ORD-1983, University of Illinois, Urbana, Ill., January~ 1958. (9) L. A. ZADEH,"A Note on the Analysis of Vacuum Tube and Transistor Circuits," Yroc. I R E , Vol. 41, pp. 989-992 (1953). (10) L. A. ZADEH, "Multipole Analysis of Active Networks," I R E Trans. on Circuit Theory, Vol. CT-4, pp. 97-105 (1957). (11) J. SHEKEL,"Matrix Representation of Transistor Circuits," Proc. I R E , Vol. 40, pp. 14931497 (1952). (12) J. SHEKEL, "The Gyrator as a 3-Terminal Element," Proc. I R E , Vol. 41, pp. 1014-1016 (1953). (13) S. SESHU, "Topological Considerations in the Design of Driving Point Functions," I R E Trans. on Circuit Theory, Vol. CT-2, pp. 356-367 (1955). (14) W. H. KIM, "Topological Evaluation of Network Functions," JOUR. FRANKLIN INST., Vol. 267, pp. 283-293 (1959). (15) I. FRISCI-IAND W. H. KIM, "Properties of 2-semi-isomorphic Graphs and Their Applications" (to be published). (16) W. H. KIM, "Topological Analysis of Linear Multipoles," The Matrix and Tensor Quarterly, London, Vol. 11, No. 2, pp. 33-41 (1960). (17) K. A. PULLEN, JR., "The Use of Network Topology with Active Circuits," Report No. 1096, Ballistic Research Labs., Aberdeen Proving Grounds, Maryland; February 1960.