44 BELL SYSTEM TECHNICAL JOURNAL 



E 



$ {D) in = -K COS CO / 



i $ (D) i,, = [-^ + 2^n + (I + 2L„) D~^ I cos CO / 



where 



'J>(Z))=LC"o(| + 2L„)L>3+[c:'o(rL + 2L„r + 2r„L)+g'oL(^+2L„)Jz)2 



+ [g'o (^L + 2L„r + 2rnL) + Co (^ + 2rr„) + L + 2L„ 1z) 



+ g'o (^ + 2rr„) +r+2r„, 

 Co = C + C, g'o = g + g'. 



To solve the equations, the cubic equation <!> {D) = must be solved. 

 An algebraic solution would be so cumbersome as to be impracticable. 

 The following numerical values of the constants have therefore been 

 inserted, as what is desired is a numerical solution representing the 

 effect in a practical case: 



g = 0.37 X 10-« mho g' = 0.18 X 10-« mho 



C = 0.55 X 10-« farad C = 0.30 X 10-« farad 



Ln= 6.4 henries r„ = 200 ohms 



L = 0.022 henry r = 2.0 ohms 



E = V2 X 26,400 = 37,350 volts co = 377 

 With these assumptions 



$ (D) = 2.4 X 10-7^)3 + 29.4 X IQ-^D^ + 12.8 D + 402, 



of which the roots are 



- 31.4, - 45.5 +i7,300, - 45.5 - j7,300 



which may be denoted by —a', —a -\- jb, — a — jb respectively. 

 The resulting equations for iig and i„ are 



iig = Pe-"'^ + Qe-"^ sin {ht + 9) + gE cos co t, 



in = P'e-"'^ + Q'e-'^^ sin {bt + 6') + ^ sin (cot + 4°.7). 



The relations between the two sets of arbitrary constants may be 

 obtained by inserting these solutions in the following differential 

 equation connecting iig and i„: 



iijg -[r/2 + 2r„ + (L/2 + 2L„) D]i„ = 0, 



and the three independent arbitrary constants so found are determined 

 by the following conditions when ^ = (it is assumed that breakdown 



