A LADDER NETWORK THEOREM 



313 



Trolley-Feeder, Trolley-Rail Base 



A . A dual Network Connections 

 \ 



3T-F 



<)4 T-R 



B. Equivalent Network for Trolley- Rail Short Circuits 

 1 



^Z|2;l2 + 0-V)Z,2:34 



Zl4:24 



Cl-v)(Z|2:i2-Z|2:34) 

 Z|4:24 =Z-V(l-V)(Z|2;|2-Z.i2:34)-*-(^'"'^^ (234:34-^12:34) 



C. Equivalent Network for Trolley- Feeder Short Circuits 



^(Z|2.I2-Z|2:34) 



Z|3:23 



1.^ 



(l-V)Z|2:12 + ^Z|2;34 



Z|3:23 = Z-v(l-V)(Z|2:|2-Zi2:34) + ^^ (234:34-2,2:34) 



T 



Fig. 4 — Equivalent networks for electrified railways; three-wire system, single source, 

 trolley-feeder, trolley-rail and feeder-rail, trolley-rail bases. 



Thus the complete set of short-circuit currents (trolley-rail, trolley- 

 feeder and feeder-rail short circuits) may be made from a single 

 determination of the transducer impedances and current distributions 

 on either of the two bases, whenever the theorem is applicable. 



For multiple generator three-wire systems, and for three-wire 

 systems with auxiliary transmission lines, the theorem may be used to 

 represent portions of the network, possibly broken as in the two-wire 

 cases illustrated above, four-terminal representation being necessary in 

 general. The application follows the lines indicated above. 



Proof of Theorem 



For energization between terminals 1 and 2, the sum of the currents 

 in sides 1 and 2 at any point on the ladder is zero; the current ik-.u may 

 be taken as flowing in a mesh made up of the ^-section sides and its 

 terminating shunt impedances. For unit current supplied, the 



