A LADDER NETWORK THEOREM 315 



The transfer impedances with respect to the side terminals 1, 3 and 2, 4 

 are formulated as: 



(16) 



Zi2:24 = L (^2^ - Zi2(^>)4:12. 



From the condition [Zi^^^ - Zi2^'=>][Z2('=) - Zi2^*>]-i = const., a con- 

 stant J' may be defined such that: 



p= p,= [Z/« - Zi2W][Z:(*) + Z2(^-) - 2Zx2(«]-\ 



1 - , = [22^ - Zi2(«][ZiW + Z2(^> - 2Zi2<'=>]-\ 



and equations (16) may be combined with (15) to give: 



Zi2:13 = J'Zi2:12 ~ »'Zi2:34, 



Zi2:24 = — (1 — I')Zi2:l2 + (1 " I')Zi2:34. 



The remaining transfer impedances follow by superposition ; thus 



Zi2:l4 = Zi2:l3 + Zi2:34 = l'Zi2:i2 + (1 ~ v)Zi2:Zi, 

 Zi2:23 = Zi2:13 — Zi2:12 = — (1 — V)Zii:\2 — vZi^.zi- 



It may be observed that 



Zi2:24 = Zi2:13 4" Zi2:34 Zi2:\2- (1") 



Similarly 



(17) 



(18) 



Zzi:\Z ~ — vZzi-.Zi 4" vZ\2:Zi, 



Zzi-.U = (1 ~ v)Zzi:Zi + vZ\2:Zi, 



Zzi:23 = ~ vZzi:Zi "" (1 ~ v)Z\2:Zi, 



Zzi:2i = (1 — J')Z34:34 — (1 — J')Zi2:34- 



(20) 



These impedances, together with Z34:34 and Z34:i2, form a set of 12 

 impedances of which only five are independent. There are three 

 independent impedances determined by energization at each pair of 

 terminals, including Zi2:34 and Z34:i2, which are equal by the reciprocity 

 theorem; one independent set, for example, is Zi2:i2, Zi2;i3, Znm, 

 Zzi:zi, Zzi-.xz- Consequently the network may be completely specified 

 by the addition of a single impedance; for the set illustrated, the 

 impedance required is Zi3:i3. 



This impedance may be formulated as: 



