312 BELL SYSTEM TECHNICAL JOURNAL 



difference; the parts of the original network thus formed are represented 

 by F-connected impedances which may be derived from the first H 

 network. On Fig. d)B the network is broken at the intermediate 

 generator point and similarly represented. A similar process may be 

 followed for any number of generator points but in some cases it may 

 be expedient to superpose additional generators; the network imped- 

 ances required for superposition may be formulated in the manner 

 followed in the proof of the theorem. 



The solution of these reduced networks supplies the currents /i, I^, 

 Ii\ //, Iz , li , etc., from which the current in branch k of side 2 of any 

 of the ladder sections may be found from equation (2). Thus, for 

 example the current Ik in the ^th section of the ladder network with 

 terminals 1, 2, 3 and 4 on Figs. 2) A and d>B is formulated as follows: 



Ik = [l' + 1^4:12 — vik:u']l2, + \iv -\- viic-n + (1 — I')4:34]/4, (H) 



which follows from equation (2) with I2 — 0; — Ii — I3 -{- li- Similar 

 formulas apply to the other two ladder networks on Fig. 3. 



In three-wire networks, generators are usually connected to the 

 traction network by three-winding transformers which may be repre- 

 sented on the network diagram by three impedances connected in star. 

 The traction network may be represented on a trolley-feeder, trolley- 

 rail or a feeder-rail, trolley-rail base and it is well known that the three- 

 winding transformer equivalent impedances for the two bases are 

 related. Using the notation shown on Figs. AA and 4^', with primes 

 distinguishing the feeder-rail, trolley-rail base, the relations are as 

 follows: 



VtfZa = VtfZj + VtrZt', 



VtfZ, = VfrZb', (12) 



Vtf'Z. = Vsr^Z: - V,rVs,Z^, 



where Vtr, Vt/ and V/r are the trolley-rail, trolley-feeder and feeder-rail 

 circuit voltages, respectively. 



From Figs. 4B and AB' showing the reduced networks for trolley-rail 

 short-circuits on the two bases for a single source feed, it is apparent 

 that the impedances involved in the equivalent networks must be 

 similarly related. The relations are found to be as follows: 



(13) 



