A LADDER NETWORK THEOREM 317 



ik-.li = 4:13 + 4:34 = I' + 1^4:12 + (1 " 1^)4:34, 



4:23 = 4:13 — 4:12 = J^ — (1 — 1^)4:12 — I'4:34, (25) 



4:24 = 4:14 — 4:12 = I' — (1 — I')4:12 + (1 — J')4;34. 



It will be observed that only three of the six currents 4.i2, 4:i3, 4:i4, 

 4:23, 4:24 and 4:34 are independent; one independent set is 4i2, 4:34 and 

 4:13. Hence any arbitrary set of currents /i, I2, I3 and li flowing out 

 of the network at the terminals may be resolved into three flows, such 

 as those illustrated in the independent set above, which leads to 

 equation (2). 



The first H network may be obtained in the following manner. The 

 value of Zi2:i3, namely, pZi2:i2 — vZh-.za, in conjunction with the 

 condition Zi + Z2 = Zi2:i2, Zi and Z2 being the impedances of 

 branches to terminals 1 and 2, respectively, suggests the following 

 values of branch self and mutual impedances: 



Zx = I'Zi2:12, 

 Z2 = (1 — I')Zi2-i2, 

 Zi3 = vZi^.zi. 



The value of Zu is verified by inspection of Zu-.zi if 



Z3 = vZ 34:34. 



Similarly, by inspection of Zi2:i4 and Z34:32, the values of Z24 and Z^ 

 may be tentatively set at 



Z24 = (1 ~ v)Z\i:Zi, 

 Z4 = (1 — l')Z34:34. 



The impedance of the crossbar, say Zo, may be found from any of the 

 impedances Zi3:i3, Zu.u, Zi3:iz, Zzi-.n; e.g., 



Zo = Zi3:i3 — (Zi + Z3 — 2Z13) 



= Z — v{l — v)(Zi2:i2 + Z34:34 — 2Zi2:34). 



But the presence of seven elements, as already mentioned, suggests an 

 arbitrariness which may be put into evidence by adding aZi2:34 to Zi, 

 which entails a similar addition to Z3 and Z13, and a similar subtraction 

 from Z2, Z4 and Z24- 



Similar considerations apply to the derivation of the second H 

 network. 



The direct impedances may be found, in a well-known manner, by 

 energizing the network between one terminal and the other three 



