FORMAL REALIZABILITY THEORY — II 



545 



Given 2n.-poles Ni and No , with impedance matrices respectively 

 Zi(p) and Zo(p), we know then that there is a general 2/i-pole N whose 

 impedance matrix is 



Z(p) = Z,{p) + Z,(p). 



We call this N the series combination of Ni and No . 



2.31 Designate the terminal pairs of Ni by (Sr , Sr), those of No b}^ (Tr , 

 Tr), 1 < /■ < n- It is evident that if Ni and No appear in a diagram so 

 connected that 



(i) Sr is connected to Tr , 1 < r < n; 



(ii) No other connections are made to these nodes; 

 then the device with terminals Sr , Tr is 'N. This follows at once from 

 Kirchoff's laws applied to the ideal graph (I, 4.11). 



2.32 Duallj'', if Ni and N2 have admittance matrices Yi(p), Yiip), then 



Y(p) = Y,{p) + Y,{p) 



is the matrix of a 2n-pole N defined as the parallel connection of Ni 

 and N2 . N is the device whose terminal pairs are formed by joining 

 Sr , Tr and also S'r , Tr , 1 < r < n. 



2.33 Fig. 1 shows the conventions to be used in indicating 2n-poles 

 (n = 4 in the Figure) with, respectively, impedance matrices and ad- 

 mittance matrices. Fig. 2 then shows the series connection of two im- 

 pedance devices and the parallel connection of two admittance devices. 

 In each case the terminals on the left are those of the composite device. 



2.4 The series and parallel connections just described are special ways 

 of combining 2n-poles needed for the generalized Brune process for 

 matrices. They have been introduced here on their merits, as new op- 





^■-t; 



^--v 



-— t; 



IMPEDANCE MATRIX ADMITTANCE MATRIX 



DEVICES WITH FOUR TERMINAL PAIRS 



Fig. 1 — Conventions used in representing 2N poles. 



