FORMAL RE ALIZ ABILITY THEORY — II 510 



in;i(le hy coimecliiig Ms between Ts and V, , 1 < s < n. This 2/i-polc 

 lias tlie impedance matrix 



Z.{v) = fip)D. 

 Then 



Z(v) = f(p)WDW' = WZ,(p)W' 



is the matrix of a 2/i-pole N which can be obtained from Nj liy the use 

 of ideal transformers. C'learlj' Ni , and therefore N, contains exactly the 

 elements claimed in (B) of the theorem. 

 The dual theorem (C) needs no comment. 



3.11 Corollary: The conclusion (A) of 3.1 holds if the hypotheses on 

 j"(p) are replaced by "/(p) is PR." The same method of proof applies 

 but one must use the Brune theory to realize the impedances diif(p), 

 1 <i < r. 



3.2 The case (ii) of 3.1 shows that any physical system of coupled coils 

 can be realized \\ath a set of isolated (i.e., not coupled) coils, with ideal 

 transformers to supply the coupling [Cf. (I, 19.12)]. With this fact in 

 mind, we see that the method of network synthesis used in (I, 19) can 

 ])e simplified to the following: one starts with a finite collection of two- 

 poles: each one is a resistor, capacitor, or coil (inductor). These are then 

 appropriately connected to suitable ideal transformers. Viewed from 

 certain selected terminals of these transformers, this network is a 2n-pole 

 equivalent to the desired one. 



The difference between this process and that of (I, 19) is the minor 

 one that coupled coUs have been eliminated. We may then, however, 

 regard any finite passive network as made up solely of simple two-poles 

 (resistors, capacitors, coils) and ideal transformers. 



It is readily verified from (I, 19.2) that open and short circuits are 

 special cases of ideal transformers. 



If a network made up in this way uses / coils and c capacitors, we shall 

 call ^ -f c the number of reactive elements in the network (or used by, 

 or used in, the net\vork).* 



3.21 Lemma: The network described in the proof of 3.1 uses 8(Z) reac- 

 tive elements. This is obvious from 2.12, 2.15, and 2.16. 



IV. THE BRUXE PROCESS FOR A POSITIVE REAL MATRIX 



4.0 Let Z{p) be an n X n PR matrix. We can regard it as the impedance 

 matrix of a general 2n-pole N. In this section we shall describe the 



* By this definition, a reactive element is an energy storage element. Ideal 

 transformers are not reactive, by the very fact of their ideality. 



