310 BELL SYSTEM TECHNICAL JOURNAL 



Applications to Electric Railroad Networks 

 A.-c. electric railroad networks in one-line diagram are predominantly 

 of the ladder type. The series elements of sides 1 and 2 represent, for 

 two-wire networks, impedances of sections of transmission lines and 

 traction circuits, respectively; the shunt elements represent transformer 

 impedances. For three-wire networks, the series elements may repre- 

 sent trolley-feeder (or feeder-rail) and trolley-rail impedance elements, 

 the shunt elements autotransformer impedances. 



The theorem may be used for representing portions of a network or a 

 whole network of ladder form,^ when the series impedances satisfy the 

 condition of the theorem. As the circuits are linearly extended this is 

 almost always the case except where the traction circuits change 

 character, from two to four tracks, for example. For approximate 

 purposes the H networks may be used even in these cases provided 

 that the parameter v is properly chosen. In many cases the transfer 

 impedance Zx2:u is negligible and a value of v may be associated with 

 each pair of terminals; the values for the sections immediately ad- 

 joining the terminal pairs 1-2 and 3-4 (sections 1 and n on Fig. 1^) 

 are of dominant importance and serve for rough purposes. If the 

 transfer impedance is not negligible a mean of these values may be 

 sufficiently accurate. 



In two-wire networks, generator circuits are connected directly to 

 the transmission line (side 1), and the short circuits of chief interest 

 (grounding points on the one-line diagram) are those on the traction 

 circuits (side 2). Thus, for a single generating point the network is 

 energized between points on sides 1 and 2, such as 1 and 4, for example; 

 if the impedance in the generator connection is Zg and the impedance of 

 the short circuit is zero, the short-circuit driving-point impedance and 

 the traction circuit currents are as follows: 



Z{) ^ Zg -\- Zu:i4 



= Zg-^Z -^ p-'Zn-.ii + (1 - J^yZu:u + 2K1 - ^)Zi2..u, (9) 



h= L^-i- VU:l2 + (1 - V)4:34]-^ , (10) 



where E is the generator voltage. The impedance may be obtained 

 immediately from either of the H networks — as the sum of the self- 



' For multiple transmission-line two-wire networks the ladder form is obtained 

 when aU transmission lines are bussed at all generating stations and substations or 

 when the generators, step-up transformers and substation transformers connected to 

 each line are of similar impedances and are similarly connected. When these condi- 

 tions are not met the network is of multiple-side ladder form, for the representation of 

 which an extension of the theorem would be required. Similar remarks apply to 

 three-wire networks. 



