Applications
The “classical” duality of electric network theory was introduced in Section 4.1. There we used the word “inverse” rather than the more usual word “dual”. Our reason to do this was that we wish to use “dual” for a concept obtained from matroidal duality.
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Applications
Section 10.1 Duality in electric network theory II: Inverse, dual and adjoint multiports; the full symmetry The "classical" duality of electric network theory was introduced in Section 4.1. There we used the word "inverse" rather than the more usual word "dual". Our reason to do this was that we wish to use "dual" for a concept obtained from matroidal duality. Hence we shall also emphasize that there are several symmetries in electric network theory. Recall that n-ports, as introduced in Section 8.1, were described as Au + Bi = =0 with r(AIB) = n. We mentioned already then that one and the same multiport can have several matrix descriptions. If the pairs A, B and A', B' determine two multiports Nand N' respectively, then Nand N' will be considered equivalent (denoted by N ~ N') if, for any pair u, i of vectors, either both Au + Bi = 0 and A'u + B'i = 0 hold or none of them. (In other words, they describe the same voltage-current relations, although by different pairs of matrices.)
Statement 10.1.1 N
S(AIB)
= (A'IB')
~
N' holds if and only if there exist matrices S, T so that
and T(A'IB')
= (AlB).
Proof'. 'frivial since equivalence means that Au + Bi = 0 and A'u + B'i = 0 determine the same (n-dimensional) subspace of the 2n-dimensional space, and then Sand T are just transformations of the systems of coordinates.
o
Of course, in order to have equivalence, r(AIB)
= r(A'IB') must hold and
if this common rank is, at the same time, the number of rows in A, B, A', B' as wel~ then S, T are nonsingular square matrices. If N ~ N' then their matroids MN and M N , are identical. (But this can happen for nonequivalent multiports as well, see Exercise 10.1.3.) In Section 4.1 a correspondence among network elements was obtained by the voltage-current symmetry. Let us extend this to multiports at first. If Au + Bi = 0 describes an n-port N then the inverse of N (denoted by inv N) is the n-port N' satisfying Bu + Ai = 0 . This is simply the interchange of the roles of voltage and current. Obviously if N' =inv N then their matroids MN and MN' are isomorphic; the one-one correspondence 4> simply gives 4>(ui) = ii and 4>(ii) = ui for every i = 1,2, ... ,no A. Recski, Matroid Theory and its Applications in Electric Network Theory and in Statics © Springer-Verlag Berlin Heidelberg 1989
10.1 Duality in electric network theory II
207
Once again, observe that N' =inv N implies MN' = ~(MN) but not viee tlersa (see Exercise 10.1.3).
°
Matroid theory suggests an entirely different duality concept among electric network elements. H Au + Bi = describes an n-port N then the dual of N (denoted by dual N) is an n-port N' with matrices A', B' if, for every pair u,i satisfying Au + Bi = and for every pair u', i' satisfying A'u' + B'i' = 0, the relation uTu' + iTi' = 0 holds. Since r(AIB) = r(A'IB') = n, this condition means that (AlB) and (A'IB') are orthogonal complements. Hence if N and N' are dual n-ports then their matroids MN and M N , are also dual to each other (and this implication is also not true vice vers
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