628 BEI.L SYSTEM TECHNICAL JOURNAL 



practicable steady-state received current in bridge duplex operation 

 the bridge arms being of equal resistance.' 



The method of calculating curves ii, ii, and iz Fig. 2, will be discussed 

 presently along with certain other mathematical considerations. 



In setting up the equations for the received current and line current, 

 certain practical operating conditions of the circuit are assumed; first, 

 that the circuit as a whole be kept symmetrical by using the same 

 amount of regulating resistance at each station and second, that the 

 duplex sets be maintained in a state of balance for direct currents. 

 Line leakage will be neglected. 



To express the line and received currents as direct, or explicit 

 functions of the regulating resistances under the assumed condition 

 of the circuit requires the use of unusually cumbersome equations 

 which may to some extent be avoided in the early part of the solution 

 without sacrificing accuracy. The complicated nature of the equa- 

 tions is due largely to the intricate relation between the regulating 

 resistances and the overall network resistance of the duplex set from 

 the terminal of the line to ground and, in turn, the relation between 

 this network resistance and the two currents. With the exception 

 of one step in the present investigation the work has been shortened 

 by representing this network resistance by a parameter, or second 

 independent variable, r which is itself a quadratic function of the 

 regulating resistance, represented by R. The required values of r are 

 then computed from the equation connecting it to R. In the ex- 

 pressions for the ratios of received current to line current all de- 

 terminants which cannot be readily reduced to the second order 

 cancel out so that considerable work is avoided by using these ratios 

 rather than the explicit line current equations for calculating the 

 line currents. 



The equations expressing the relation between the received currents, 

 ii, «2 and iz and the regulating resistance in the three locations (1), 

 (2) and (3) respectively, are as follows : 



. ^ oE 



*2 ('7"J_/C_L4'-^ ('op^ J_^^_L^p _L E> 2 (2) 



'' {T+Rz + G-^rz){2a+b)+ab+a'^ ^^^ 



^ For the case of unequal bridge arms see Heaviside's "Electrical Papers," Vol. I, 

 p. 24, 



