274 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



Let 



[it] = [ui , ■■- , u,] 



be the s-tiiple of mesh voltages. We suppose that j'l , • • • ,jn,Ui, ■ • • ,Un 

 refer respectively to the n external meshes. (Cf. 19.03.) 



19.06 The results of Synge^" can now be stated as follows: 



There exists a real constant matrix [Ci] of s columns and h -\- n rows 

 (having, in fact, elements which are +1, —1, or 0) such that for any [j] 



[C] = [Ci][j] (1) 



is a set of branch currents satisfying Kirchoff's node law, and for anj^ [iv] 



M = [C.]'{w] (2) 



is a set of mesh voltages satisfying Kirchoff's mesh law. Furthermore, 

 given any [I] which satisfies the node law, there is a [j\ such that (1) 

 holds. * 



19.07 If we interpret the [t], [j], etc., as representations in real bases 

 then [Ci] is real and [d]' = [Ci]*. 



19.08 The matrix [Ci] has the form 



[Ci] = 



w^here [C2] is an 71 X n diagonal matrix (having diagonal elements ±1, 

 in fact). 



Proof: By construction, ji , • • • , j„ are mesh currents in the external 

 meshes. These are then equal, save for sign, to the currents ^1 , • • • , /„ 

 in the generator branches. 



19.09 By 19.08, (1), and the definitions in 19.04, 



[k] = [C][j], [v] = [C]'[v], 

 and by 19.07, [C]' = [C\*. 



19.1 Let us suppose that we have enumerated the branches |S„+i , 

 • • • , /Sft in 19.02 in such a way that iS„+i , • • • , ^c are all the two poles in the 

 graph, iSc+i , • • • , /3d are all the branches containing coils Avhich are 

 magnetically coupled, and ^d+i , ■ ■ ■ , (3b the remaining ideal branches of 

 ideal transformers. 



Let [Zd{p)] be the (d — n) X (d — n) impedance matrix relating the 

 voltages across the branches /3„+i , • • • , /3d to the currents in them when 



