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N-port synthesis of N-order positive entry resistance matrices
N-Port Synthesis of N-Order Positive Entry Reshtance Matrices b y DAVID P. BROWN Department of Electrical Engineering University of Wisconsin, Madison, Wisconsin
ABST~CT: New relationships between n-order positive entry resistance matrices of graphs of nullity n and network topology are established. A method of decomposition of positive entry resistance matrices into a sum of simple resistance matrices is given, and a number of necessary conditions for synthesis as a transformerless n-port are proven. A procedure to synthesize symmetric, positive entry matrices of order n as resistance matrices of waphs of nullity n is developed and illustrated. Introduction
In recent years a number of results related to the solution of the problem of synthesis of n-order matrices as conductance matrices of transformerless n-port networks have been obtained (1). However, little has appeared in the literature on the problem of synthesis of n-order matrices as resistance matrices of transformerless n-port networks. Conditions for synthesis of a very special class of resistance matrices which have been developed and used to show paramountcy is not a sufficient condition for synthesis (2). A procedure is known (3) for the synthesis of matrices as resistance matrices of star-port structure networks. Recently some results concerning the synthesis of matrices as resistance matrices of planar n-ports have appeared (4). In this paper we establish some new results on the problem of synthesis of matrices as resistance matrices of transformerless n-ports. In particular, relationships between network topology and the entries in corresponding resistance matrices are determined for positive entry resistance matrices. These relationships are used in developing a method of decomposition of positive entry resistance matrices as a sum of simple resistance matrices. A number of necessary conditions for synthesis of n-order positive entry matrices as resistance matrices of graphs of nullity n are proven. In order to synthesize a specified positive entry matrix, a procedure is given to form a tree, including element values, from the simple matrices in the decomposition of the specified matrix. The synthesis process is completed by the addition of chords to this tree. Positive E n t r y R e s i s t a n c e M a t r i c e s a n d N e t w o r k T o p o l o g y
We establish a method of classifying element orientations useful in relating circuit orientations and network topology.
26
N-Port Synthesis of N-Order Matrices Definition 1. If some f-cut set S of a graph contains elements cs and cj such that the orientation of both cs and ci is directed from one vertex set of S to the other, then cs and ci are similarly oriented in S. A resistance matrix of some graph G is BIReB s' = ['R~], where BI is an f-circuit matrix of G, B~ is the transpose of BI and Re is diagonal with nonnegative entries. T h e lemma and theorem given next relate the topology, chord orientation pattern and signs of resistance matrices. The proofs are omitted since they appear elsewhere (S). ][,emma 1. Consider chords cs, i = 1, 2, 3 of a graph G where each pair cs, ci define f-circuits with common elements P~j. If c~ and ci are similarly oriented in the f-cut set defined by some element of P~s, for i, j = 1, 2, 3, i ~ j, then (1) all the elements of at least one of Ply, P13, P23 are in the other two, and (2) the subgraph of G consisting of P12, P13, P~3 is a path.
Theorem 1. If and only if a resistance matrix BIReB/ = ER~i]~ of some graph G of nullity n contains a principal submatrix of order r with all positive entries, then r chords c;, i = 1, 2, - - - , r of G define f-circuits with common elements Psi and all cs are similarly oriented in the f-cut set defined b y some element of Psi for i, j -- 1, 2, - - . , r, i ~ j. I t is next shown that the set of elements of G common to r, r >_ 2, f-circuits is a subgraph of G which is common to some two of the r f-circuits. [ , e m m a 2. Consider r, r >__ 2, chords cs, i -- 1, 2, -- -, r of a graph G which are similarly oriented in some f-cut set of G. If P is the set of elements of G common to the f-circuits defined by all cs, then P is some, not necessarily one, Psi, i, j = 1, 2, . . . , r, i ~ j, where P~i is the set of elements common to the f-circuits defined b y chords cs and cj.
Proof: Since the hypothesis and conclusion are the same for r -- 2, consider r -- 3. B y L e m m a 1-(1), all elements of at least one of PI~, Pla, P~3, say, Pn, are in the other two: Since the f-cut set defined b y any elements of P contains c~, c2 and ca, each element of P is in P12. Thus P contains all and only elements of Pn. This establishes the conclusion for r -- 3. Suppose the theorem is true for r = k - 1 and consider/c chords satisfying the hypothesis. Let Pk-1 and P be the sets of elements common to the f-circuits defined by chords cs, i = 1, 2, . . . , / c - 1 and i = 1, 2, - . . , k, respectively. B y induction hypothesis Pk-1 = Ps0i0 where i _< /0, jo ~ k - 1,/0 ~ j0. Since each element of P is in all P~, P is a subset of the elements of Ps0i0. If P = Ps0~0 the conclusion follows. Suppose at least one element e of Ps0i0 is not in P. The element e is not in P~k, P~k, " " , Pk-lk. B y L e m m a 1-(1), all elements of at least one of P~0i0, Ps0k, P~ok are in the remaining two. Therefore all elements of P~0k(Pi0k) are in Ps0~0 and PJok(P~0k). This implies P = Ps0k(P~0k) and completes the proof. The minimum entry in a principal submatrix of a resistance matrix of G is related to the elements of G common to the corresponding f-circuits in what follows.
Vol. 284, No, 1, July 1967
27
David P. Brown Theorem 2. I f a resistance matrix BzRoBs' = ER~iJn of some graph G of nullity n contains a principal submatrix of order r, r >__ 2, with all positive entries, then the elements of G corresponding to the (one of the) m i n i m u m entry (entries) R~i, i, j = 1, 2, . . . , r, i ~ j, j > i, are common to the f-circuits defined b y all chords c~, i = 1, 2, . . . , r. Proof: B y Theorem 1 and L e m m a 2, the elements of G corresponding to some entry R~0i0 in the principal submatrix R1 of hypothesis is the set of elements common to the f-circuits defined b y all the corresponding chords c~, i = 1, 2, . - . , r. Therefore, the elements of G corresponding to R~0i0 are a subset of the elements corresponding to any other entry R~ljl of R1. This requires t h a t Ri0j0 _< Riljl which proves the theorem. Graphs which contain a vertex common to all f-circuits for some tree are of interest since, b y T h e o r e m 1, the graph corresponding to a n y resistance matrix containing all positive entries m u s t contain at least two such vertices. L e m m a 3. Let G be any graph, T any tree of G and suppose G contains one and only one vertex v common to all f-circuits of T. (1) I f P is the set of elements common to two (or more) f-circuits of G for tree T, then v is some vertex of P. (2) Suppose each P~, i = 1, 2, . - - , t, t _> 2, is the set of elements common to some r f-circuits defined b y chords c~k, k = 1, 2, . . . , r of G for tree T. If some b u t not all of the chords c~_~k, k = 1, 2, . . . , r, are also chords c~k, k = 1, 2, . . . , r for i = 2, 3, . . . , t, and G contains no elements common to more t h a n r f-circuits of T, then the sets of common elements P~ form a star where the common vertex v is an end vertex of each P~. (3) Suppose each P~, i = 1, 2, . . . , t, t _> 2, is the set of elements common to some r~ f-circuits defined b y chords c~k, k = 1, 2, . . . , r~ of G for tree T. If all of the chords c~-ik, k = 1, 2, - . . , ri-1 axe also chords c~k, k = 1, 2, - - - , r~ for i = 2, 3, . . . , t, then the sets of common elements P~ form a p a t h such t h a t P~ is a subgraph of P~-I. I n addition, if v is an end vertex of P1, v is an end vertex of all P~. Proof of (1): Since P is the set of elements common to the f-circuits C~, i _> 2, each vertex of P is common to all C~ and no other vertex of any C~ is common to all C~, for otherwise there is a circuit in tree. Since G contains a vertex in all f-circuits of T, this vertex is in all C~ and therefore it is a vertex of P. Proof of (2) : Consider i = 2 and let C be some f-circuit defined b y a chord of cik, k = 1, 2, - - . , r which is also a chord of c2k, k = 1, 2, . - . , r. Since each chord of clk spans a p a t h in tree containing all the elements of Px and each Chord of c~k spans a p a t h in tree containing all the elements of P2 the chord defining C spans a p a t h in tree containing elements of both P~ and P2. By (1) of conclusion, some vertex of P1 and some vertex of P~ is v. If v is not an end vertex of both P1 and P2, then Pl and P2 h a v e at least one common element e. Thus e defines an f-cut set containing at least r q- 1 chords of T. Therefore, e is in at least r q- 1 f-circuits of T. Since this contradicts the hypothesis, v must be an end vertex of both P~ and P~ and the conclusion follows for i = 2. Suppose the
28
Journal of The Franklin Institute
N-Port Synthesis of N-Order Matrices conclusion is true for i = k and consider k A- 1 sets of common elements satisfying the hypothesis. Application of the argument used for the case i = 2 to Pk-1 and Pk establishes the conclusion for i = k A- 1.
P r o o f o f (3): Since the set of elements P , is common to all f-circuits defined by chords c,k, k = 1, 2, .. -, r~, P~ is in any subset of these f-circuits. In particular, P , is in the f-circuits defined b y chords c,-ik, k = 1, 2, . . . , r,_~ since all of the chords c~-lk are also chords of c;k. Therefore P , is a subgraph of P,_~. Since P~ is a path, it follows that the sets of common elements P,, i = 1, 2, - . . , t, t > 2 form a path. Suppose v is an end vertex of PI. Then by (1) of conclusion, v is an end vertex of all P,. Necessary C o n d i t i o n s : f o r S y n t h e s i s Using the properties established in the foregoing section, we present a simple method of decomposition of positive entry resistance matrices. Based on this decomposition a number of necessary conditions are established for synthesis of n-order positive entry matrices as resistance matrices of graphs of nullity n. Suppose a resistance matrix B R ~ ' = I-R~s-]n of some graph G and tree T contains all positive entries. Let r0 be equal to the minimum value of all Ro" and define Ro = [ro]n, i.e., an n X n matrix which contains r0 in every position. B y Theorem 2, the elements of G corresponding to an off-diagonal entry of B R , B ' equal to r0 are common to all f-circuits for tree T. Let b0 be a matrix consisting of a set of identical columns each of which contains + 1 in each row and the number of columns is equal to the number of elements of G common to all n f-circuits for tree T. Suppose Do is a diagonal matrix with diagonal entries d~ each of which is equal to the resistance of one of the elements common to all n f-circuits of T such that ~'~ d~ = r0. I t therefore follows that Ro = boD&o' and B R , B ' - Ro (1) is a resistance matrix of the graph Go (or any graph 2 - isomorphic to Go) obtained from G by coalescing the vertices of each element common to all n fcircuits of T. Suppose the set of numbers I11 ---- {il l, i21, ". "it t } is a subset of the numbers l, 2, . . . , n such t h a t all entries of the matrix Eq. 1 in positions (ia I, ibl), a, b = 1, 2, . . . , r are non-zero and no other subset of the numbers 1, 2, . . . , n containing more than r terms satisfies this property. Let rl be equal to the minimum value of the entries of the matrix Eq. 1 in the positions (i~1, ib1) and define where
r~=l
rl
for
(u,v) = (i~l, ib1)
/o
for
(u, v) ~ (ia 1, ib1)
Since the entries in positions (ia 1, ibl), a, b = 1, 2, . . . r comprise a principal submatrix of some resistance matrix of Go, the elements of Go corresponding to an entry of the matrix Eq. 1 equal to rl are common to r f-circuits of Go for tree To where To is the subgraph of T in Go containing no self loops.
Vol. 284, No. 1, July 1967
29
David P. Brown
Let bl be a matrix consisting of a set of identical columns each of which contains + 1 in rows il ~, i21, . - . , iJ and the number of columns is equal to the number of elements of Go common to the aforementioned r f-circuits of To. Suppose D~ is diagonal with diagonal entries d~1 each of which is equal to the resistance of one of the elements common to the r f-circuits such that ~ d~1 = rl. It therefore follows that R1 = b~Dlb~' and BReB' -
(Ro+R1)
(2)
is a resistance matrix of a graph GI (or any graph 2 - isomorphic to G1) obtained from Go by coalescing the vertices of each element common to all r fcircuits of To. If the matrix Eq. 2 contains non-zero entries in positions (in ~, ib2), a, b = 1, 2, . . . , r where the set of numbers 112 ---- {il 2, i2 2, "" ", it s} is a subset of the numbers 1, 2, . . . , n, repeat the above decomposition procedure. If there are t sets of numbers I11, I1~, . . . , I1t, the decomposition results in a matrix BR~B' -
(Ro + R1 + "'" + Rt)
(3)
which is a resistance matrix of a graph Gt (or any graph 2 - isomorphic to Gt) obtained from G by coalescing the vertices of each element in r or more ( < n ) f-circuits of T. This decomposition is next applied to the matrix Eq. 3 and the procedure repeated until a diagonal matrix with nonnegative entries results. A diagonal matrix must result since a matrix is determined which is a resistance matrix of a graph containing no elements common to two or more f-circuits. L e m m a 4. The graphs G~, i = O, 1, . . . , s, just defined contain one and only one vertex common to all self loops and f-circuits for tree T~ where T~ is the subgraph of T in G~ containing no self loops. P r o o f : The lemma conclusion follows directly from the method of forming the graphs G~ from G. L e m m a 5. Let G~ be the graph obtained from G b y coalescing the vertices of each element common to more than r f-circuits of G for tree T and suppose Ra = [Ru,]n is a resistance matrix corresponding to G~ for tree T~ as just defined. Then for each u, at most 2(r - 1) entries Ru, are non-zero where u # v, v = 1, 2, • .., n. P r o o f : To prove this lemma it is only necessary to show that each f-circuit of G~ for tree T~ contains elements in at most 2(r - 1) other f-circuits of Ti. Let C be any f-circuit of Gi for tree Ti and suppose P~ is the set of elements of C in some other f-circuit C~ of Ti. B y Lemma 4 some vertex v of G~ is common to all f-circuits of Gi for tree T~. I t follows by Lemma 3-(1) that v is a vertex of Pi for all i. Since all Pi are in C, the set of all P~ form a path where v is in all Pi. B y hypothesis any clement of C is in at most r f-circuits of Ti. Therefore, any element of some P~ is in at most r -- 1 other P~'s. Let v be an end vertex of
30
Journal of The Franklin Institute
N-Port Synthesis of N-Order Matrices
all P~ such that the elements of P~. are in Ps+~ f o r j = 1, 2, - - . , s -- 2 a n d j = s, s -[- 1, . . - , 2s - 1. For this arrangement, the elements of C are in 2(s - 1) other f-circuits of T~. If v is not an end vertex of some P~, less than 2(s -- 1)P~ are incident to v and therefore the elements of C are in less than 2(s - 1) other f-circuits of T;. This completes the proof. Consider any stage of the resistance matrix decomposition, B R ~ B ' -- ( Ro -4- R1 A- . . . A- R.)
(4)
where each of the matrices R0, R1, - - ' , R, contains non-zero entries in at least r -{- I rows. Suppose there exists t and only t sets of numbers I~" = {ilJ, ij, . . . , ir j} where each set is some subset of the numbers 1, 2, - . - , n such that the entries in the matrix Eq. 4 in all positions (ia 3, ibi)a, b = 1, 2, "" ", r are not zero and no other subset of the numbers 1, 2, . . *, n containing more than r numbers satisfies this property. By interchanging row i and j and column i and j of the matrix Eq. 4 a new resistance matrix of G8 results. Therefore, b y interchanging rows and corresponding columns of the matrix Eq. 4 any set of entries (ia~ /b~'), a, b = 1, 2, - . - , r can be located as a leading principal submatrix of a resistance matrix of G,. In terms of this notation, we establish a number of properties of the resistance matrix decomposition. The first property presents a restriction on common numbers of index sets. P r o p e r t y I. Any number ~, 1 < a ~_ n can be common to at most two index sets I i, j = 1, 2, - . . , t, where each index set contains r numbers. P r o o f : Contrary to the conclusion, suppose a, 1 < a < n is in three (or more) index sets I ~, 12 and 13. Let P1, P~ and P3 be the sets of elements of G, common to all f-circuits defined b y the chords corresponding to the numbers in I ~, 12 and 13, respectively. Such sets of elements exist b y Theorem 1. B y Lemma 3 - ( 2 ) , P1, P2 and P3 form a star and the common vertex v is an end vertex of each P~, i = 1, 2, 3. Since a is in all three index sets, P1, P2 and P3 must be in the f-circuit defined b y chord c~. Thus, this f-circuit contains a vertex of degree 3 (or greater). Since this is impossible, the conclusion follows. A property of the magnitude and location of the minimum entry (one of the minimum entries) in principal submatrices of resistance matrices is proven next. P r o p e r t y 2. Suppose (i.~, i~J) is the location of the minimum entry (or some one of the minimum entries) of all positions (ij, ibJ)a, b = 1, 2, . . . , r in the matrix Eq. 4. The (1) entry in location (i~ j, i~~') -4- entry in location (i, k, i~k) < entry in location ( i j k, i J k) where ( i J "k, ibjk) is any location common to the sets of locations (i~i, i~J) and (ia ~, ibm)a, b = 1, 2, . . . , r, a ~ b , j ~ k. (2) (i. j, i~s) # (ia k, ibk) for a l l a , b = 1, 2, - ' - , r, a # b arid all k = 1, 2, . . . , t , k C j . P r o o f o f (1): Let P~ and Pk be the sets of elements of G~ common to all
Vol. 284, No. I, July 1967
31
David P. Brown f-circuits defined by chords corresponding to the numbers in P' and I k, respectively. By Theorem 2 the entry in location (ia~ iBj) and the entry in location (in k, ia k) is equal to the resistance of the elements of Pi and Pk, respectively Let P,'k be the set of elements of G, corresponding to the entry in location ( i j ~, ibik). Since the elements of Pik are common to the f-circuits defined by chords ca and Cb and a and b are both in I~ and both in I ~, all the elements of Pj and all the elements of Pk are in Pjk. By Lemma 3-(2), P~. and Pk are disjoint. Since the entry in location (inck, ib~'~) is equal to the resistance of the elements of Pik, (1) of the conclusion follows.
Proof of (2) : If the location (i~, i~') = (in ~, ibk) for some k # j, then from (1) of the conclusion (in k, iak) <_ O. Since this is impossible, the conclusion follows. In the next two properties the pattern of some of the zero entries of the matrix Eq. 4 is given. Property 3. If the number a, 1 _~ a _~ n, is common to two and only two index sets 11 and I ~, then the entries of the matrix Eq. 4 in all locations (a, d) are zero where d is any number not equal to the numbers in I x and I ~. Proof: Suppose some entry in location (a, d), as noted in the hypothesis, is not zero and let Pad be the set of elements of G, corresponding to this entry. Let P1 and P2 be the sets of elements of G, common to all f-circuits defined by chords corresponding to the numbers in 11 and 12, respectively. B y LeInma 4 and Lemma 3-(1), some vertex of Pad is V, the common end vertex of P1 and P2. Since P1, P2 and Pad are in the f-circuit defined by the chord ca, some element e of P1 or P, is in P,d. Therefore, the element e is common to r + 1 fcircuits of G,+t, all those corresponding to the numbers in P (or 12) plus d. Since this is impossible, the conclusion follows. Property 4. Suppose there exists t index sets I i, j = 1, 2, - . . , t ordered such t h a t at at least one any number all locations
least one number in I x is in 12, at least one number in I * is in P, . . . , number in I t-1 is in I*. If i i is any number in I~ not in I k and i k is in I k not in I~ where k > j, then the entries of the matrix Eq. 4 in (i~ i k) are zero for all j + k equal to an odd integer.
Proof: Let Pi be the set of elements of G, common to all f-circuits defined by chords corresponding to the numbers in U, j = 1, 2, . . . , t. By Lemma 3-(2), the sets of common elements P~ form a star and the common vertex v is an end vertex of each P~.. Suppose some entry in location (i ~', i k) as indicated in the hypothesis is not zero and let Pik be the elements of G, corresponding to this entry. No elements of P~ or P , can be in P~'k for if e is such an element, it is common to at least r + 1 f-circuits: All those defined by chords corresponding to the numbers in IJ (or I *) plus at least one from I k (or I J). By Lemma 4 and Lemma 3-(1) some vertex of Pik is v. If v is not an end vertex of Pjk some fcircuit of G, has a vertex of degree three or larger. Since this is impossible v is an end vertex of P~k. Let ci be a chord spanning elements Pj and Pj+~ for j = 1, 2, . - . , t - 1. Such chords exist by hypothesis. Suppose for some fixed j, c~ is
32
Journal of The Franklin Institute
N-Port Synthesis of N-Order Matrices oriented so that the f-circuit orientation of the elements of P i is directed toward (away from) v. Since all entries in the matrix Eq. 4 are non-negative, the fcircuit orientations on elements common to two or more f-circuits must coincide. Therefore, all the f-circuit orientations on P~+, must be directed toward (away from) v for a even and directed away from (toward) v for a odd. Hence, all t h e f-circuit orientations on Pk, k > j, are directed away from (toward) v for j W k equal to an odd integer less than 2t since j ~- k = 23" W a and a is an odd integer. This requires the f-circuit orientations on Pj'k defined by ci and ck not to coincide. Since this is impossible, the entry in location (i~, i ~) of the matrix Eq. 4 is zero. Three additional structural properties of principal submatrices corresponding to index sets I~' are given next. Property 5 follows from Property 1 and 4 and Property 6 is proven by the same procedure used in the proof of Property 4.
Property 5. If a and t~ are two numbers in the index set Iq+l where a is also in the index set IqJ and not in Iq k and ~ is also in the index set Iq k and not in I ~ then IqJ and I~ k are disjoint. Property 6. If there exists t index sets I j, j = 1, 2, . . . , t ordered such t h a t at least one number in I 1 is in 12, at least one number in 12 is in P, . - . , at least one number in I H is in I t and at least one number in I t is in I ~, then t is an even integer. Property 7. Suppose some number of I~ ~ is in I~2 and some number of I~ ~ is in I , 2 where u > v. If some number of I~ ~ is in I~ 1 and some number of I~2 is in I , 2, then any number in I , 1 and I , 2 must be in I , 2 and any number in I , ~ and I J must be in I , ~.
Proof: Let P,1, P,2, Pul and P ~ be the sets of elements of G common to all f-circuits defined b y chords corresponding to numbers in I 1, i 2 , i 1 and I~~, respectively. B y Lemma 3 - ( 2 ) , P,1 and P,2 form a star with common vertex v an end vertex of both P , I and P ~ . The same is true for P ~ and P~2. Since some number in I~ 1 is in I , 1, P,~ is in P ~ and since some number in I~~ is in I~2, P ~ is in P~2- If a is any number common to I , 1 and I~2, then P,~ and Pu~ are in the circuit defined by the corresponding chord and so is P,~. Hence, any number common to I~ ~ and I~2 is in I , 2. B y a similar argument it follows that any number common to I , 2 and I~~ is in I , ~. This completes the proof. A number of properties of n-order resistance matrices containing all positive entries are established. These results are summarized as follows: Any symmetric matrix M = [-m~i] of order n with positive entries is realizable as a resistance matrix of a graph of nullity n only if M can be decomposed as, M = M0 ~MI ~ . . . -F Mj W . . . M r where (1) M0 = Em0], and m0 is equal to the minimum value of all m~j, (2) for the set of numbers Ii = {il j,/~J, . . - , iri}, r ~ 2, which is a subset of
Vol. 284, No. 1, July 1967
33
David P. Brown the numbers 1, 2, . - . , n such that the entries of
(5)
M - (M0 + M1 + - " + Mj-1)
in all positions (ij, ibm), a, b = 1, 2, . . . , r are non-zero and no other subset of the numbers containing more than r terms satisfies this property, j = 1, 2, ---, t - 1, and
m u v = l mi
for
(u,v) = (ij, i j )
/o
for
(u, v) ~ (i j , ib~')
Mi = [-m~o],
where m~ is equal to the minimum value of the entries in the matrix Eq. 5 in all positions ( i ~ ib0, a, b = 1, 2, . - - , r, (3) Mt is diagonal with non-negative entries, (4) The sets of numbers Ii and numerical pattern of the entries of M must be such that Lemma 5 and Properties 1-7 are satisfied.
Synthesis of Resistance Matrices Suppose M is decomposed, as discussed in the previous section, and the index sets of the decomposition are, I; I11, I12, . . . , Iltl; I~1, I22, .-.I2t2; . . . , where index sets with the same subscript contain the same number of terms. A method of forming a tree T of a graph corresponding to M is discussed next. Classify the index sets I1j into two groups AI 1 and A~~ such that A~I contains all I~~"which are disjoint and AI 2 contains all other IlJ. Each index set in A~~, if not null, has a number in common with some one index set in A~L Since a number can be in at most two index sets, the index sets in A12 are disjoint. Let the 2
"°÷I •
%
A..
,,, /
1
/
\\f/ "..J
•
m?+=: i • Jv
'"'/ /~a+t,
/
m;'-,
//] A: a ÷ t , / " ' J v/
=/ ~
i
"
I
_?
"'zo
:
\.-"
"
\m',
\'
\
m~o \
20
~',+= 2
~,
~ ""~
[
=
.
man_,+ ' 'a + "~m u-, '
rn I " FIG. ].
34
J o u r n a l of
The Franklin
Institute
N-Port Synthesis of N-Order Matrices
non-zero entry in the matrix of the decomposition of M associated with an index set inA~ ~ a n d A 1 2 b e m f f , p = 1,2, . . . , a a n d p = a - ~ 1, a ~ 2 , . . . , a - ~ t ~ respectively. Suppose each number m~" represents the resistance of an element of a tree T where the elements mff, p = 1, 2, . . . , a are all incident to one of the vertices of m0 and the elements raft, p = a ~ 1, a ~- 2, . . . , a ~ tl are incident to the other vertex of too. The n u m b e r m0 is the resistance of the element of T corresponding to the index set I. See Fig. 1 for this subgraph of T. Classify the index sets 12j into two groups A21 and A~2 such t h a t all I~J which are disjoint are in A2 ~ and all other I2j are in A~9-. The index sets in A22 are disjoint. Note t h a t a particular number in any I~j is in at most one index set in A1~ or A~2 since the index sets in A1 ~ and the index sets in A~2 are disjoint. F o r m subgroups of the index sets in A~1, A~I~, A~ 1, . . . , A ~ I such t h a t some number in each index set in A~ .~ is in some one I1j and all index sets in A2~.~ are disjoint for j = 1, 2, . . . , a. Let the sets of indices of A~t not classified be A~ ~. Suppose the nonzero entry in the matrix of the decomposition of M associated with the index sets in Asi ~ are m2'~ -~+~, m2 a~-~+2, . . . , m2"i, j = 1, 2, . . . , a, ao = O. Let each n u m b e r m~( ) represent the resistance of an element of the tree T formed such t h a t the elements rn~p, p = aj_l -t- 1, ai_~ + 2, . . . , a~. are incident to m j as shown in Fig. 1. Suppose the elements of the tree corresponding to the index sets in A201are labeled m20t, m~0~, . - . , m~ "0. To the subgraph of T already formed, these elements are added incident to the vertex of mo closest to the elements m~i (see Fig. 1). The foregoing procedure is next applied to the index sets in A22. I n forming T, the elements should be placed so t h a t for m~ incident to m~i which is incident to m0, a n y number in the index set associated with m0 and m~~ is also in the index set associated with m~~. T h e procedure of forming T is continued until all index sets associated with the decomposition of M are considered. I n general, for the index sets Iu j classified into the two sets A~ 1 and Au 2 just defined, if Iu i has a number in common with I j where u > v and no other number vl exists such t h a t I~J has a number in common with I~l i and u > vl > v, place the element m j associated with I~i incident to the vertex of the element m j associated with I j such t h a t the p a t h in T with end elements m0 and m j contains mJ'. If no index set I j exists, m j is put incident to a vertex of m0, as discussed in the case of I2i. Consider Mr, the diagonal matrix in the decomposition of M. Let the nonzero entries in Mt be mr, and suppose the number i is in the index set Iu j and there is no index set I j which contains i such t h a t v > u. Let the number mr, be the resistance of an element of T which is placed incident to mu i such t h a t the p a t h in T with end elements mr, and rn0 contains mui. The tree T thereby formed is such t h a t the index sets associated with the elements in any p a t h in T containing m0 have one or more common numbers. To form a graph corresponding to M, chords are placed incident to vertices of
VoL 2~, No. ~, July I~7
35
David P. Brown
T such that if P is a path in T containing m0 where the index sets corresponding to the elements of P have a common number i and the index sets associated with the elements incident to end elements of P do not contain i, chord ci is placed so as to span P. The graph obtained by placing n such chords incident to the vertices of T has M as its resistance matrix. Cederbaum (2) showed that the following paramount matrix Eq. 6 is not the resistance (or conductance) matrix of a 4-port network: -3 3
2
1
3 -~a- 6 5 (6)
M1
2
6
6 4
.1
5
4
5.
A less restrictive conclusion is evident from the decomposition technique of the previous section. Consider the partial decomposition of M1, 1
1
0
1 il
ff 1
-1
1
1
1-
1
1
1
1
+
--
0
1
1 0
.0
1
0
1_
"0 0 0
O
033
3
+ 1
1
1
1
.1
1
1
1_
"1
1
1
0-
1
1
1
0
1
1
0
+ 033
3
1
_0 3 3
3.
.0 0
00.
(7) In the first matrix on the righthand side of Eq. 7, row 2 is common to three index sets, {1, 21, {2, 3} and {2, 4}. This contradicts Property 1 and therefore M1 is not the resistance matrix of a graph of nullity 4. Consider the following matrix (6) : -7
3
2
1-
3
18
6
5
2
6
15
4
;1
5
4
12"
M2 =
(8)
Decomposition of M~ indicates that M~ is not the resistance matrix of a graph of nullity 4 since Property 1 is not satisfied. Therefore, consider an enlarged or
36
Journal of The Franklin Institute
N-Port Synthesis of N-Order Matrices augmented matrix M2c, where M2 is augmented by row and column 4, (7)
M~ =
-7~
3~
3
2
2 7
3-~
21½
11
10
3
11
22½ 15
11½
2
10
15
15
_2
10
11½ 15
10
30
(9)
19½J
We show that M2o is realizable as a resistance matrix of a graph of nullity 5 and therefore the same is true of M~. The complete decomposition of Mu is shown as follows:
M2. =
-2 2
2
2
2-
-0 0
2
2
2
2
2
0 8 8 8 8
2
2
2
2
2
2
2
2
2
_2 2 2 -000
2
000
+
+
0 0-
1
1
0
0-
1
1
1
0
0
- 4 - 1 1 1 0 0
2
0 8 8 8 8
0 0 0 0 0
2_
_08888_
_000003 -000
0
0-
0 0 0
0
0
0
0
0
0
~ooo
1½ 1½ 1½ +
00
1½ 1½ 11
0
0
0
0
0 0 0 0
0
0 0 0 0
0
o o o 3½ 3½ [_0 0
0 +
o o 3~ 3½ o o 0 3½ 3½ 0
[00000
1½ 1½ 1½_ LO 0 0 0 0 0 0 0
+
-1
0 8 8 8 8
o o
,'0 0
0
0
3½ 3 !2 -
00~ -3;v
+
0
.0 0 0
0
0
OJ
0
0
o
9~ o
o
o
0
0
61
0
0
0
0
0
111
0
_0
0
0
0
(10)
4½
Using the procedure of tree formation, the tree of Fig. 2 (a) is determined where the numbers in parenthesis are the index set corresponding to the element. Adding the chords to the tree of Fig. 2(a) and coalescing the vertices of chord c4 results in the graph shown in Fig. 2(b). The resistance matrix of the graph of nullity 5 in Fig. 2(b) is M~. Vol. 284, No. I, July 1967
37
David P. Brown
z/IA I 3
eli,
=
9 tls
'/,
{$4}
/
3~s
11451
t
|£348,1
'|llZJ4ff~)
'
(
Ill}
.~4 '/. (a)
9 ~/3
/_
c) /
c,_
\ "/,
6 '/z Fro. 2. (b)
An alternative realization of Ms as an n 4- 2 node n-port is given by Watanabe
(8) ~eferences (1) E. A. Guillemin, et al., "The Realization of n-Port Networks Without Transformers-A Panel Discussion," I R E Trans. on Circuit Theory, ¥ol. CT-9 pp. 202-214, 1962. (2) I. Cederbaum, "ToPological Considerations in the Realization of Resistive n-Port Networks," I R D Trans. on Circuit Theory, Vol. CT-8, pp. 324-329, 1961. (3) T. J. Harrison, "Impedance Matrix Synthesis of a Class of n-Port Networks," Ph.D. Thesis, Stanford Univ., 1964. (4) D. P. Brown, "Synthesis of a Class of Planar n-Ports," I E E E Trans. on Circuit Theory (corres.) Vol. CT-12, pp. 612-613, 1965. (5) D. P. Brown, "Tropical Properties of Resistance Matrices," J. Math. and Phy. (5) P. Slepian and L. Weinberg, "Synthesis Applications of Paramount and Dominant Matrices," Proc. NEC, Vol. 14, pp. 611-630, 1958. (7) D. P. Brown and Y. Tokad, "On the Synthesis of R Networks," I R E Trans. on Circuit Theory, Vol. CT-8, pp. 31-39, 1961. (8) H. Watanabe, "On the Synthesis of Conductive n-Port with n + 2 Nodes," Presented at the Internat. Conf. on Microwaves, Circuit Theory and Info. Theory, Tokyo, Japan, Sept. 1964.
38
Journal of The Franklin Institute
ABST~CT: New relationships between n-order positive entry resistance matrices of graphs of nullity n and network topology are established. A method of decomposition of positive entry resistance matrices into a sum of simple resistance matrices is given, and a number of necessary conditions for synthesis as a transformerless n-port are proven. A procedure to synthesize symmetric, positive entry matrices of order n as resistance matrices of waphs of nullity n is developed and illustrated. Introduction
In recent years a number of results related to the solution of the problem of synthesis of n-order matrices as conductance matrices of transformerless n-port networks have been obtained (1). However, little has appeared in the literature on the problem of synthesis of n-order matrices as resistance matrices of transformerless n-port networks. Conditions for synthesis of a very special class of resistance matrices which have been developed and used to show paramountcy is not a sufficient condition for synthesis (2). A procedure is known (3) for the synthesis of matrices as resistance matrices of star-port structure networks. Recently some results concerning the synthesis of matrices as resistance matrices of planar n-ports have appeared (4). In this paper we establish some new results on the problem of synthesis of matrices as resistance matrices of transformerless n-ports. In particular, relationships between network topology and the entries in corresponding resistance matrices are determined for positive entry resistance matrices. These relationships are used in developing a method of decomposition of positive entry resistance matrices as a sum of simple resistance matrices. A number of necessary conditions for synthesis of n-order positive entry matrices as resistance matrices of graphs of nullity n are proven. In order to synthesize a specified positive entry matrix, a procedure is given to form a tree, including element values, from the simple matrices in the decomposition of the specified matrix. The synthesis process is completed by the addition of chords to this tree. Positive E n t r y R e s i s t a n c e M a t r i c e s a n d N e t w o r k T o p o l o g y
We establish a method of classifying element orientations useful in relating circuit orientations and network topology.
26
N-Port Synthesis of N-Order Matrices Definition 1. If some f-cut set S of a graph contains elements cs and cj such that the orientation of both cs and ci is directed from one vertex set of S to the other, then cs and ci are similarly oriented in S. A resistance matrix of some graph G is BIReB s' = ['R~], where BI is an f-circuit matrix of G, B~ is the transpose of BI and Re is diagonal with nonnegative entries. T h e lemma and theorem given next relate the topology, chord orientation pattern and signs of resistance matrices. The proofs are omitted since they appear elsewhere (S). ][,emma 1. Consider chords cs, i = 1, 2, 3 of a graph G where each pair cs, ci define f-circuits with common elements P~j. If c~ and ci are similarly oriented in the f-cut set defined by some element of P~s, for i, j = 1, 2, 3, i ~ j, then (1) all the elements of at least one of Ply, P13, P23 are in the other two, and (2) the subgraph of G consisting of P12, P13, P~3 is a path.
Theorem 1. If and only if a resistance matrix BIReB/ = ER~i]~ of some graph G of nullity n contains a principal submatrix of order r with all positive entries, then r chords c;, i = 1, 2, - - - , r of G define f-circuits with common elements Psi and all cs are similarly oriented in the f-cut set defined b y some element of Psi for i, j -- 1, 2, - - . , r, i ~ j. I t is next shown that the set of elements of G common to r, r >_ 2, f-circuits is a subgraph of G which is common to some two of the r f-circuits. [ , e m m a 2. Consider r, r >__ 2, chords cs, i -- 1, 2, -- -, r of a graph G which are similarly oriented in some f-cut set of G. If P is the set of elements of G common to the f-circuits defined by all cs, then P is some, not necessarily one, Psi, i, j = 1, 2, . . . , r, i ~ j, where P~i is the set of elements common to the f-circuits defined b y chords cs and cj.
Proof: Since the hypothesis and conclusion are the same for r -- 2, consider r -- 3. B y L e m m a 1-(1), all elements of at least one of PI~, Pla, P~3, say, Pn, are in the other two: Since the f-cut set defined b y any elements of P contains c~, c2 and ca, each element of P is in P12. Thus P contains all and only elements of Pn. This establishes the conclusion for r -- 3. Suppose the theorem is true for r = k - 1 and consider/c chords satisfying the hypothesis. Let Pk-1 and P be the sets of elements common to the f-circuits defined by chords cs, i = 1, 2, . . . , / c - 1 and i = 1, 2, - . . , k, respectively. B y induction hypothesis Pk-1 = Ps0i0 where i _< /0, jo ~ k - 1,/0 ~ j0. Since each element of P is in all P~, P is a subset of the elements of Ps0i0. If P = Ps0~0 the conclusion follows. Suppose at least one element e of Ps0i0 is not in P. The element e is not in P~k, P~k, " " , Pk-lk. B y L e m m a 1-(1), all elements of at least one of P~0i0, Ps0k, P~ok are in the remaining two. Therefore all elements of P~0k(Pi0k) are in Ps0~0 and PJok(P~0k). This implies P = Ps0k(P~0k) and completes the proof. The minimum entry in a principal submatrix of a resistance matrix of G is related to the elements of G common to the corresponding f-circuits in what follows.
Vol. 284, No, 1, July 1967
27
David P. Brown Theorem 2. I f a resistance matrix BzRoBs' = ER~iJn of some graph G of nullity n contains a principal submatrix of order r, r >__ 2, with all positive entries, then the elements of G corresponding to the (one of the) m i n i m u m entry (entries) R~i, i, j = 1, 2, . . . , r, i ~ j, j > i, are common to the f-circuits defined b y all chords c~, i = 1, 2, . . . , r. Proof: B y Theorem 1 and L e m m a 2, the elements of G corresponding to some entry R~0i0 in the principal submatrix R1 of hypothesis is the set of elements common to the f-circuits defined b y all the corresponding chords c~, i = 1, 2, . - . , r. Therefore, the elements of G corresponding to R~0i0 are a subset of the elements corresponding to any other entry R~ljl of R1. This requires t h a t Ri0j0 _< Riljl which proves the theorem. Graphs which contain a vertex common to all f-circuits for some tree are of interest since, b y T h e o r e m 1, the graph corresponding to a n y resistance matrix containing all positive entries m u s t contain at least two such vertices. L e m m a 3. Let G be any graph, T any tree of G and suppose G contains one and only one vertex v common to all f-circuits of T. (1) I f P is the set of elements common to two (or more) f-circuits of G for tree T, then v is some vertex of P. (2) Suppose each P~, i = 1, 2, . - - , t, t _> 2, is the set of elements common to some r f-circuits defined b y chords c~k, k = 1, 2, . . . , r of G for tree T. If some b u t not all of the chords c~_~k, k = 1, 2, . . . , r, are also chords c~k, k = 1, 2, . . . , r for i = 2, 3, . . . , t, and G contains no elements common to more t h a n r f-circuits of T, then the sets of common elements P~ form a star where the common vertex v is an end vertex of each P~. (3) Suppose each P~, i = 1, 2, . . . , t, t _> 2, is the set of elements common to some r~ f-circuits defined b y chords c~k, k = 1, 2, . . . , r~ of G for tree T. If all of the chords c~-ik, k = 1, 2, - . . , ri-1 axe also chords c~k, k = 1, 2, - - - , r~ for i = 2, 3, . . . , t, then the sets of common elements P~ form a p a t h such t h a t P~ is a subgraph of P~-I. I n addition, if v is an end vertex of P1, v is an end vertex of all P~. Proof of (1): Since P is the set of elements common to the f-circuits C~, i _> 2, each vertex of P is common to all C~ and no other vertex of any C~ is common to all C~, for otherwise there is a circuit in tree. Since G contains a vertex in all f-circuits of T, this vertex is in all C~ and therefore it is a vertex of P. Proof of (2) : Consider i = 2 and let C be some f-circuit defined b y a chord of cik, k = 1, 2, - - . , r which is also a chord of c2k, k = 1, 2, . - . , r. Since each chord of clk spans a p a t h in tree containing all the elements of Px and each Chord of c~k spans a p a t h in tree containing all the elements of P2 the chord defining C spans a p a t h in tree containing elements of both P~ and P2. By (1) of conclusion, some vertex of P1 and some vertex of P~ is v. If v is not an end vertex of both P1 and P2, then Pl and P2 h a v e at least one common element e. Thus e defines an f-cut set containing at least r q- 1 chords of T. Therefore, e is in at least r q- 1 f-circuits of T. Since this contradicts the hypothesis, v must be an end vertex of both P~ and P~ and the conclusion follows for i = 2. Suppose the
28
Journal of The Franklin Institute
N-Port Synthesis of N-Order Matrices conclusion is true for i = k and consider k A- 1 sets of common elements satisfying the hypothesis. Application of the argument used for the case i = 2 to Pk-1 and Pk establishes the conclusion for i = k A- 1.
P r o o f o f (3): Since the set of elements P , is common to all f-circuits defined by chords c,k, k = 1, 2, .. -, r~, P~ is in any subset of these f-circuits. In particular, P , is in the f-circuits defined b y chords c,-ik, k = 1, 2, . . . , r,_~ since all of the chords c~-lk are also chords of c;k. Therefore P , is a subgraph of P,_~. Since P~ is a path, it follows that the sets of common elements P,, i = 1, 2, - . . , t, t > 2 form a path. Suppose v is an end vertex of PI. Then by (1) of conclusion, v is an end vertex of all P,. Necessary C o n d i t i o n s : f o r S y n t h e s i s Using the properties established in the foregoing section, we present a simple method of decomposition of positive entry resistance matrices. Based on this decomposition a number of necessary conditions are established for synthesis of n-order positive entry matrices as resistance matrices of graphs of nullity n. Suppose a resistance matrix B R ~ ' = I-R~s-]n of some graph G and tree T contains all positive entries. Let r0 be equal to the minimum value of all Ro" and define Ro = [ro]n, i.e., an n X n matrix which contains r0 in every position. B y Theorem 2, the elements of G corresponding to an off-diagonal entry of B R , B ' equal to r0 are common to all f-circuits for tree T. Let b0 be a matrix consisting of a set of identical columns each of which contains + 1 in each row and the number of columns is equal to the number of elements of G common to all n f-circuits for tree T. Suppose Do is a diagonal matrix with diagonal entries d~ each of which is equal to the resistance of one of the elements common to all n f-circuits of T such that ~'~ d~ = r0. I t therefore follows that Ro = boD&o' and B R , B ' - Ro (1) is a resistance matrix of the graph Go (or any graph 2 - isomorphic to Go) obtained from G by coalescing the vertices of each element common to all n fcircuits of T. Suppose the set of numbers I11 ---- {il l, i21, ". "it t } is a subset of the numbers l, 2, . . . , n such t h a t all entries of the matrix Eq. 1 in positions (ia I, ibl), a, b = 1, 2, . . . , r are non-zero and no other subset of the numbers 1, 2, . . . , n containing more than r terms satisfies this property. Let rl be equal to the minimum value of the entries of the matrix Eq. 1 in the positions (i~1, ib1) and define where
r~=l
rl
for
(u,v) = (i~l, ib1)
/o
for
(u, v) ~ (ia 1, ib1)
Since the entries in positions (ia 1, ibl), a, b = 1, 2, . . . r comprise a principal submatrix of some resistance matrix of Go, the elements of Go corresponding to an entry of the matrix Eq. 1 equal to rl are common to r f-circuits of Go for tree To where To is the subgraph of T in Go containing no self loops.
Vol. 284, No. 1, July 1967
29
David P. Brown
Let bl be a matrix consisting of a set of identical columns each of which contains + 1 in rows il ~, i21, . - . , iJ and the number of columns is equal to the number of elements of Go common to the aforementioned r f-circuits of To. Suppose D~ is diagonal with diagonal entries d~1 each of which is equal to the resistance of one of the elements common to the r f-circuits such that ~ d~1 = rl. It therefore follows that R1 = b~Dlb~' and BReB' -
(Ro+R1)
(2)
is a resistance matrix of a graph GI (or any graph 2 - isomorphic to G1) obtained from Go by coalescing the vertices of each element common to all r fcircuits of To. If the matrix Eq. 2 contains non-zero entries in positions (in ~, ib2), a, b = 1, 2, . . . , r where the set of numbers 112 ---- {il 2, i2 2, "" ", it s} is a subset of the numbers 1, 2, . . . , n, repeat the above decomposition procedure. If there are t sets of numbers I11, I1~, . . . , I1t, the decomposition results in a matrix BR~B' -
(Ro + R1 + "'" + Rt)
(3)
which is a resistance matrix of a graph Gt (or any graph 2 - isomorphic to Gt) obtained from G by coalescing the vertices of each element in r or more ( < n ) f-circuits of T. This decomposition is next applied to the matrix Eq. 3 and the procedure repeated until a diagonal matrix with nonnegative entries results. A diagonal matrix must result since a matrix is determined which is a resistance matrix of a graph containing no elements common to two or more f-circuits. L e m m a 4. The graphs G~, i = O, 1, . . . , s, just defined contain one and only one vertex common to all self loops and f-circuits for tree T~ where T~ is the subgraph of T in G~ containing no self loops. P r o o f : The lemma conclusion follows directly from the method of forming the graphs G~ from G. L e m m a 5. Let G~ be the graph obtained from G b y coalescing the vertices of each element common to more than r f-circuits of G for tree T and suppose Ra = [Ru,]n is a resistance matrix corresponding to G~ for tree T~ as just defined. Then for each u, at most 2(r - 1) entries Ru, are non-zero where u # v, v = 1, 2, • .., n. P r o o f : To prove this lemma it is only necessary to show that each f-circuit of G~ for tree T~ contains elements in at most 2(r - 1) other f-circuits of Ti. Let C be any f-circuit of Gi for tree Ti and suppose P~ is the set of elements of C in some other f-circuit C~ of Ti. B y Lemma 4 some vertex v of G~ is common to all f-circuits of Gi for tree T~. I t follows by Lemma 3-(1) that v is a vertex of Pi for all i. Since all Pi are in C, the set of all P~ form a path where v is in all Pi. B y hypothesis any clement of C is in at most r f-circuits of Ti. Therefore, any element of some P~ is in at most r -- 1 other P~'s. Let v be an end vertex of
30
Journal of The Franklin Institute
N-Port Synthesis of N-Order Matrices
all P~ such that the elements of P~. are in Ps+~ f o r j = 1, 2, - - . , s -- 2 a n d j = s, s -[- 1, . . - , 2s - 1. For this arrangement, the elements of C are in 2(s - 1) other f-circuits of T~. If v is not an end vertex of some P~, less than 2(s -- 1)P~ are incident to v and therefore the elements of C are in less than 2(s - 1) other f-circuits of T;. This completes the proof. Consider any stage of the resistance matrix decomposition, B R ~ B ' -- ( Ro -4- R1 A- . . . A- R.)
(4)
where each of the matrices R0, R1, - - ' , R, contains non-zero entries in at least r -{- I rows. Suppose there exists t and only t sets of numbers I~" = {ilJ, ij, . . . , ir j} where each set is some subset of the numbers 1, 2, - . - , n such that the entries in the matrix Eq. 4 in all positions (ia 3, ibi)a, b = 1, 2, "" ", r are not zero and no other subset of the numbers 1, 2, . . *, n containing more than r numbers satisfies this property. By interchanging row i and j and column i and j of the matrix Eq. 4 a new resistance matrix of G8 results. Therefore, b y interchanging rows and corresponding columns of the matrix Eq. 4 any set of entries (ia~ /b~'), a, b = 1, 2, - . - , r can be located as a leading principal submatrix of a resistance matrix of G,. In terms of this notation, we establish a number of properties of the resistance matrix decomposition. The first property presents a restriction on common numbers of index sets. P r o p e r t y I. Any number ~, 1 < a ~_ n can be common to at most two index sets I i, j = 1, 2, - . . , t, where each index set contains r numbers. P r o o f : Contrary to the conclusion, suppose a, 1 < a < n is in three (or more) index sets I ~, 12 and 13. Let P1, P~ and P3 be the sets of elements of G, common to all f-circuits defined b y the chords corresponding to the numbers in I ~, 12 and 13, respectively. Such sets of elements exist b y Theorem 1. B y Lemma 3 - ( 2 ) , P1, P2 and P3 form a star and the common vertex v is an end vertex of each P~, i = 1, 2, 3. Since a is in all three index sets, P1, P2 and P3 must be in the f-circuit defined b y chord c~. Thus, this f-circuit contains a vertex of degree 3 (or greater). Since this is impossible, the conclusion follows. A property of the magnitude and location of the minimum entry (one of the minimum entries) in principal submatrices of resistance matrices is proven next. P r o p e r t y 2. Suppose (i.~, i~J) is the location of the minimum entry (or some one of the minimum entries) of all positions (ij, ibJ)a, b = 1, 2, . . . , r in the matrix Eq. 4. The (1) entry in location (i~ j, i~~') -4- entry in location (i, k, i~k) < entry in location ( i j k, i J k) where ( i J "k, ibjk) is any location common to the sets of locations (i~i, i~J) and (ia ~, ibm)a, b = 1, 2, . . . , r, a ~ b , j ~ k. (2) (i. j, i~s) # (ia k, ibk) for a l l a , b = 1, 2, - ' - , r, a # b arid all k = 1, 2, . . . , t , k C j . P r o o f o f (1): Let P~ and Pk be the sets of elements of G~ common to all
Vol. 284, No. I, July 1967
31
David P. Brown f-circuits defined by chords corresponding to the numbers in P' and I k, respectively. By Theorem 2 the entry in location (ia~ iBj) and the entry in location (in k, ia k) is equal to the resistance of the elements of Pi and Pk, respectively Let P,'k be the set of elements of G, corresponding to the entry in location ( i j ~, ibik). Since the elements of Pik are common to the f-circuits defined by chords ca and Cb and a and b are both in I~ and both in I ~, all the elements of Pj and all the elements of Pk are in Pjk. By Lemma 3-(2), P~. and Pk are disjoint. Since the entry in location (inck, ib~'~) is equal to the resistance of the elements of Pik, (1) of the conclusion follows.
Proof of (2) : If the location (i~, i~') = (in ~, ibk) for some k # j, then from (1) of the conclusion (in k, iak) <_ O. Since this is impossible, the conclusion follows. In the next two properties the pattern of some of the zero entries of the matrix Eq. 4 is given. Property 3. If the number a, 1 _~ a _~ n, is common to two and only two index sets 11 and I ~, then the entries of the matrix Eq. 4 in all locations (a, d) are zero where d is any number not equal to the numbers in I x and I ~. Proof: Suppose some entry in location (a, d), as noted in the hypothesis, is not zero and let Pad be the set of elements of G, corresponding to this entry. Let P1 and P2 be the sets of elements of G, common to all f-circuits defined by chords corresponding to the numbers in 11 and 12, respectively. B y LeInma 4 and Lemma 3-(1), some vertex of Pad is V, the common end vertex of P1 and P2. Since P1, P2 and Pad are in the f-circuit defined by the chord ca, some element e of P1 or P, is in P,d. Therefore, the element e is common to r + 1 fcircuits of G,+t, all those corresponding to the numbers in P (or 12) plus d. Since this is impossible, the conclusion follows. Property 4. Suppose there exists t index sets I i, j = 1, 2, - . . , t ordered such t h a t at at least one any number all locations
least one number in I x is in 12, at least one number in I * is in P, . . . , number in I t-1 is in I*. If i i is any number in I~ not in I k and i k is in I k not in I~ where k > j, then the entries of the matrix Eq. 4 in (i~ i k) are zero for all j + k equal to an odd integer.
Proof: Let Pi be the set of elements of G, common to all f-circuits defined by chords corresponding to the numbers in U, j = 1, 2, . . . , t. By Lemma 3-(2), the sets of common elements P~ form a star and the common vertex v is an end vertex of each P~.. Suppose some entry in location (i ~', i k) as indicated in the hypothesis is not zero and let Pik be the elements of G, corresponding to this entry. No elements of P~ or P , can be in P~'k for if e is such an element, it is common to at least r + 1 f-circuits: All those defined by chords corresponding to the numbers in IJ (or I *) plus at least one from I k (or I J). By Lemma 4 and Lemma 3-(1) some vertex of Pik is v. If v is not an end vertex of Pjk some fcircuit of G, has a vertex of degree three or larger. Since this is impossible v is an end vertex of P~k. Let ci be a chord spanning elements Pj and Pj+~ for j = 1, 2, . - . , t - 1. Such chords exist by hypothesis. Suppose for some fixed j, c~ is
32
Journal of The Franklin Institute
N-Port Synthesis of N-Order Matrices oriented so that the f-circuit orientation of the elements of P i is directed toward (away from) v. Since all entries in the matrix Eq. 4 are non-negative, the fcircuit orientations on elements common to two or more f-circuits must coincide. Therefore, all the f-circuit orientations on P~+, must be directed toward (away from) v for a even and directed away from (toward) v for a odd. Hence, all t h e f-circuit orientations on Pk, k > j, are directed away from (toward) v for j W k equal to an odd integer less than 2t since j ~- k = 23" W a and a is an odd integer. This requires the f-circuit orientations on Pj'k defined by ci and ck not to coincide. Since this is impossible, the entry in location (i~, i ~) of the matrix Eq. 4 is zero. Three additional structural properties of principal submatrices corresponding to index sets I~' are given next. Property 5 follows from Property 1 and 4 and Property 6 is proven by the same procedure used in the proof of Property 4.
Property 5. If a and t~ are two numbers in the index set Iq+l where a is also in the index set IqJ and not in Iq k and ~ is also in the index set Iq k and not in I ~ then IqJ and I~ k are disjoint. Property 6. If there exists t index sets I j, j = 1, 2, . . . , t ordered such t h a t at least one number in I 1 is in 12, at least one number in 12 is in P, . - . , at least one number in I H is in I t and at least one number in I t is in I ~, then t is an even integer. Property 7. Suppose some number of I~ ~ is in I~2 and some number of I~ ~ is in I , 2 where u > v. If some number of I~ ~ is in I~ 1 and some number of I~2 is in I , 2, then any number in I , 1 and I , 2 must be in I , 2 and any number in I , ~ and I J must be in I , ~.
Proof: Let P,1, P,2, Pul and P ~ be the sets of elements of G common to all f-circuits defined b y chords corresponding to numbers in I 1, i 2 , i 1 and I~~, respectively. B y Lemma 3 - ( 2 ) , P,1 and P,2 form a star with common vertex v an end vertex of both P , I and P ~ . The same is true for P ~ and P~2. Since some number in I~ 1 is in I , 1, P,~ is in P ~ and since some number in I~~ is in I~2, P ~ is in P~2- If a is any number common to I , 1 and I~2, then P,~ and Pu~ are in the circuit defined by the corresponding chord and so is P,~. Hence, any number common to I~ ~ and I~2 is in I , 2. B y a similar argument it follows that any number common to I , 2 and I~~ is in I , ~. This completes the proof. A number of properties of n-order resistance matrices containing all positive entries are established. These results are summarized as follows: Any symmetric matrix M = [-m~i] of order n with positive entries is realizable as a resistance matrix of a graph of nullity n only if M can be decomposed as, M = M0 ~MI ~ . . . -F Mj W . . . M r where (1) M0 = Em0], and m0 is equal to the minimum value of all m~j, (2) for the set of numbers Ii = {il j,/~J, . . - , iri}, r ~ 2, which is a subset of
Vol. 284, No. 1, July 1967
33
David P. Brown the numbers 1, 2, . - . , n such that the entries of
(5)
M - (M0 + M1 + - " + Mj-1)
in all positions (ij, ibm), a, b = 1, 2, . . . , r are non-zero and no other subset of the numbers containing more than r terms satisfies this property, j = 1, 2, ---, t - 1, and
m u v = l mi
for
(u,v) = (ij, i j )
/o
for
(u, v) ~ (i j , ib~')
Mi = [-m~o],
where m~ is equal to the minimum value of the entries in the matrix Eq. 5 in all positions ( i ~ ib0, a, b = 1, 2, . - - , r, (3) Mt is diagonal with non-negative entries, (4) The sets of numbers Ii and numerical pattern of the entries of M must be such that Lemma 5 and Properties 1-7 are satisfied.
Synthesis of Resistance Matrices Suppose M is decomposed, as discussed in the previous section, and the index sets of the decomposition are, I; I11, I12, . . . , Iltl; I~1, I22, .-.I2t2; . . . , where index sets with the same subscript contain the same number of terms. A method of forming a tree T of a graph corresponding to M is discussed next. Classify the index sets I1j into two groups AI 1 and A~~ such that A~I contains all I~~"which are disjoint and AI 2 contains all other IlJ. Each index set in A~~, if not null, has a number in common with some one index set in A~L Since a number can be in at most two index sets, the index sets in A12 are disjoint. Let the 2
"°÷I •
%
A..
,,, /
1
/
\\f/ "..J
•
m?+=: i • Jv
'"'/ /~a+t,
/
m;'-,
//] A: a ÷ t , / " ' J v/
=/ ~
i
"
I
_?
"'zo
:
\.-"
"
\m',
\'
\
m~o \
20
~',+= 2
~,
~ ""~
[
=
.
man_,+ ' 'a + "~m u-, '
rn I " FIG. ].
34
J o u r n a l of
The Franklin
Institute
N-Port Synthesis of N-Order Matrices
non-zero entry in the matrix of the decomposition of M associated with an index set inA~ ~ a n d A 1 2 b e m f f , p = 1,2, . . . , a a n d p = a - ~ 1, a ~ 2 , . . . , a - ~ t ~ respectively. Suppose each number m~" represents the resistance of an element of a tree T where the elements mff, p = 1, 2, . . . , a are all incident to one of the vertices of m0 and the elements raft, p = a ~ 1, a ~- 2, . . . , a ~ tl are incident to the other vertex of too. The n u m b e r m0 is the resistance of the element of T corresponding to the index set I. See Fig. 1 for this subgraph of T. Classify the index sets 12j into two groups A21 and A~2 such t h a t all I~J which are disjoint are in A2 ~ and all other I2j are in A~9-. The index sets in A22 are disjoint. Note t h a t a particular number in any I~j is in at most one index set in A1~ or A~2 since the index sets in A1 ~ and the index sets in A~2 are disjoint. F o r m subgroups of the index sets in A~1, A~I~, A~ 1, . . . , A ~ I such t h a t some number in each index set in A~ .~ is in some one I1j and all index sets in A2~.~ are disjoint for j = 1, 2, . . . , a. Let the sets of indices of A~t not classified be A~ ~. Suppose the nonzero entry in the matrix of the decomposition of M associated with the index sets in Asi ~ are m2'~ -~+~, m2 a~-~+2, . . . , m2"i, j = 1, 2, . . . , a, ao = O. Let each n u m b e r m~( ) represent the resistance of an element of the tree T formed such t h a t the elements rn~p, p = aj_l -t- 1, ai_~ + 2, . . . , a~. are incident to m j as shown in Fig. 1. Suppose the elements of the tree corresponding to the index sets in A201are labeled m20t, m~0~, . - . , m~ "0. To the subgraph of T already formed, these elements are added incident to the vertex of mo closest to the elements m~i (see Fig. 1). The foregoing procedure is next applied to the index sets in A22. I n forming T, the elements should be placed so t h a t for m~ incident to m~i which is incident to m0, a n y number in the index set associated with m0 and m~~ is also in the index set associated with m~~. T h e procedure of forming T is continued until all index sets associated with the decomposition of M are considered. I n general, for the index sets Iu j classified into the two sets A~ 1 and Au 2 just defined, if Iu i has a number in common with I j where u > v and no other number vl exists such t h a t I~J has a number in common with I~l i and u > vl > v, place the element m j associated with I~i incident to the vertex of the element m j associated with I j such t h a t the p a t h in T with end elements m0 and m j contains mJ'. If no index set I j exists, m j is put incident to a vertex of m0, as discussed in the case of I2i. Consider Mr, the diagonal matrix in the decomposition of M. Let the nonzero entries in Mt be mr, and suppose the number i is in the index set Iu j and there is no index set I j which contains i such t h a t v > u. Let the number mr, be the resistance of an element of T which is placed incident to mu i such t h a t the p a t h in T with end elements mr, and rn0 contains mui. The tree T thereby formed is such t h a t the index sets associated with the elements in any p a t h in T containing m0 have one or more common numbers. To form a graph corresponding to M, chords are placed incident to vertices of
VoL 2~, No. ~, July I~7
35
David P. Brown
T such that if P is a path in T containing m0 where the index sets corresponding to the elements of P have a common number i and the index sets associated with the elements incident to end elements of P do not contain i, chord ci is placed so as to span P. The graph obtained by placing n such chords incident to the vertices of T has M as its resistance matrix. Cederbaum (2) showed that the following paramount matrix Eq. 6 is not the resistance (or conductance) matrix of a 4-port network: -3 3
2
1
3 -~a- 6 5 (6)
M1
2
6
6 4
.1
5
4
5.
A less restrictive conclusion is evident from the decomposition technique of the previous section. Consider the partial decomposition of M1, 1
1
0
1 il
ff 1
-1
1
1
1-
1
1
1
1
+
--
0
1
1 0
.0
1
0
1_
"0 0 0
O
033
3
+ 1
1
1
1
.1
1
1
1_
"1
1
1
0-
1
1
1
0
1
1
0
+ 033
3
1
_0 3 3
3.
.0 0
00.
(7) In the first matrix on the righthand side of Eq. 7, row 2 is common to three index sets, {1, 21, {2, 3} and {2, 4}. This contradicts Property 1 and therefore M1 is not the resistance matrix of a graph of nullity 4. Consider the following matrix (6) : -7
3
2
1-
3
18
6
5
2
6
15
4
;1
5
4
12"
M2 =
(8)
Decomposition of M~ indicates that M~ is not the resistance matrix of a graph of nullity 4 since Property 1 is not satisfied. Therefore, consider an enlarged or
36
Journal of The Franklin Institute
N-Port Synthesis of N-Order Matrices augmented matrix M2c, where M2 is augmented by row and column 4, (7)
M~ =
-7~
3~
3
2
2 7
3-~
21½
11
10
3
11
22½ 15
11½
2
10
15
15
_2
10
11½ 15
10
30
(9)
19½J
We show that M2o is realizable as a resistance matrix of a graph of nullity 5 and therefore the same is true of M~. The complete decomposition of Mu is shown as follows:
M2. =
-2 2
2
2
2-
-0 0
2
2
2
2
2
0 8 8 8 8
2
2
2
2
2
2
2
2
2
_2 2 2 -000
2
000
+
+
0 0-
1
1
0
0-
1
1
1
0
0
- 4 - 1 1 1 0 0
2
0 8 8 8 8
0 0 0 0 0
2_
_08888_
_000003 -000
0
0-
0 0 0
0
0
0
0
0
0
~ooo
1½ 1½ 1½ +
00
1½ 1½ 11
0
0
0
0
0 0 0 0
0
0 0 0 0
0
o o o 3½ 3½ [_0 0
0 +
o o 3~ 3½ o o 0 3½ 3½ 0
[00000
1½ 1½ 1½_ LO 0 0 0 0 0 0 0
+
-1
0 8 8 8 8
o o
,'0 0
0
0
3½ 3 !2 -
00~ -3;v
+
0
.0 0 0
0
0
OJ
0
0
o
9~ o
o
o
0
0
61
0
0
0
0
0
111
0
_0
0
0
0
(10)
4½
Using the procedure of tree formation, the tree of Fig. 2 (a) is determined where the numbers in parenthesis are the index set corresponding to the element. Adding the chords to the tree of Fig. 2(a) and coalescing the vertices of chord c4 results in the graph shown in Fig. 2(b). The resistance matrix of the graph of nullity 5 in Fig. 2(b) is M~. Vol. 284, No. I, July 1967
37
David P. Brown
z/IA I 3
eli,
=
9 tls
'/,
{$4}
/
3~s
11451
t
|£348,1
'|llZJ4ff~)
'
(
Ill}
.~4 '/. (a)
9 ~/3
/_
c) /
c,_
\ "/,
6 '/z Fro. 2. (b)
An alternative realization of Ms as an n 4- 2 node n-port is given by Watanabe
(8) ~eferences (1) E. A. Guillemin, et al., "The Realization of n-Port Networks Without Transformers-A Panel Discussion," I R E Trans. on Circuit Theory, ¥ol. CT-9 pp. 202-214, 1962. (2) I. Cederbaum, "ToPological Considerations in the Realization of Resistive n-Port Networks," I R D Trans. on Circuit Theory, Vol. CT-8, pp. 324-329, 1961. (3) T. J. Harrison, "Impedance Matrix Synthesis of a Class of n-Port Networks," Ph.D. Thesis, Stanford Univ., 1964. (4) D. P. Brown, "Synthesis of a Class of Planar n-Ports," I E E E Trans. on Circuit Theory (corres.) Vol. CT-12, pp. 612-613, 1965. (5) D. P. Brown, "Tropical Properties of Resistance Matrices," J. Math. and Phy. (5) P. Slepian and L. Weinberg, "Synthesis Applications of Paramount and Dominant Matrices," Proc. NEC, Vol. 14, pp. 611-630, 1958. (7) D. P. Brown and Y. Tokad, "On the Synthesis of R Networks," I R E Trans. on Circuit Theory, Vol. CT-8, pp. 31-39, 1961. (8) H. Watanabe, "On the Synthesis of Conductive n-Port with n + 2 Nodes," Presented at the Internat. Conf. on Microwaves, Circuit Theory and Info. Theory, Tokyo, Japan, Sept. 1964.
38
Journal of The Franklin Institute