A LADDER NETWORK THEOREM 305 



terminal pairs, and the constant 



The current in any branch of the network for any condition of energiza- 

 tion of the network terminals is a linear function of the currents in the 

 same branch for energization at sending and receiving transducer terminals. 



It should be observed that the result stated is independent of the 

 number or values of the shunt impedances (except as they are included 

 in the transducer parameters); hence in the diagram on Fig. \A 

 illustrating the ladder network in question, any of the shunt imped- 

 ances may be allowed to vanish or become infinite, and their number 

 w + 1 may be increased or decreased at pleasure provided that one 

 shunt remains (this excludes the trivial case in which, the sides being 

 completely insulated from each other, the network degenerates to a 

 pair of single impedances). 



When the impedances of the sides are linearly extended impedances, 

 as is the case in electric railway applications, the section impedances 

 may be written : 



Zo^^) = 5,22, 

 Zi2^^^ = SkZi2, 



where Zi, z^ and 212 are self and mutual impedances of the sides per unit 

 length. The condition, [Zi^ - Zi2(^)][Z2(^-) - Zn^*)]-^ = const., is 

 replaced by the condition that the shunt impedances connect corre- 

 sponding points on the sides. 



Since a four-terminal network requires six independent quantities for 

 its specification, the conditions (a) that the network be of ladder type 

 and (b) that the given section impedance ratio be constant may be 

 regarded, at least intuitively, as replacing two (or more) of the measur- 

 able impedances at the terminals.^ 



With the sending and receiving transducer terminal pairs short- 

 circuited, and with the side impedances satisfying the given condition, 

 no current flows in any of the shunt impedances and the driving-point 

 impedance required for the theorem is 



^ ^ Zi(^>Z2<^> - Zi2^^^^- _ EZiW^Zz^^) - (i:zi2^^0^ ... 

 C;Zi^*^ + Z2(*> - 2Z12W EZi« + EZ2(« - 2i:zi2(^)' ^ ' 



3 It is interesting to observe that, if short circuits between terminals are permitted, 

 there are 64 measurable impedances for a four-terminal network. The network may 

 be specified by any six of these which are independent; hence the number of ways of 

 specifying the network is something less than the number of combinations of 64 things 

 taken 6 at a time, which equals 74,974,368. The number of non-independent sets 

 which make up this large total appears at the moment to be the smaller part, and 

 possibly a very small part indeed. These remarks are inspired by Mr. R. M. Foster. 



