A Ladder Network Theorem 



By JOHN RIORDAN 



The theorem of this paper gives four-terminal representation of 

 ladder networks satisfying a prescribed condition on the side 

 impedances, in terms of the three parameters specifying the net- 

 work connected as a transducer, the driving-point impedance 

 between short-circuited transducer terminal pairs, and an im- 

 pedance ratio involving the side impedances only. This mode of 

 representation has a special advantage in applications to electric 

 railway networks in that the transducer parameters which alone 

 involve the ladder shunt impedances (under the stated conditions) 

 may be calculated in a relatively simple fashion, and extensive 

 networks reduced to manageable form. The theorem is stated 

 and proved, and its applications are sketched in some detail. 



'T^HE theorem of this paper gives a four-terminal representation of 

 -*• ladder networks satisfying a certain condition with respect to the 

 side impedances. Ladder networks appearing in transmission and 

 filter theory generally are connected as transducers, that is, such that 

 the entry and exit terminals on the ladder sides are associated in pairs; 

 the networks are two-terminal pairs. As is well known, passive 

 transducers may be completely specified by three parameters (as is the 

 case for three-terminal networks, with which transducers are similar in 

 some, though not all, respects), the choice of which has been the 

 occasion for much study and ingenuity.^ The present theorem does 

 not assume transducer connection and is thus quite distinct from 

 earlier work; indeed it arose outside the communication field in the 

 problem of the calculation of short-circuit currents and network 

 current distribution of electric railway networks, where at present it 

 seems to have chief application. 



This paper gives a statement of the theorem, an indication of its 

 applications, and finally its proof. 



The Theorem 



A ladder network, composed of any number of arbitrary shunt imped- 

 ances forming sections whose side impedances Zi^^^ Z2^*^ and Z^^''^, k = \, 



1 Five types of equivalent networks by which a transducer may be replaced, 

 including T, it, transformer and artificial line networks, and their interrelations are 

 given on Table I of "Cisoidal Oscillations" by G. A. Campbell, Trans. A. I.E. E., 

 30, pp. 873-909 (1911). The most significant addition to the table would appear to be 

 the image impedance representation due to O. J. Zobel. 



303 



