STEADILY ALTERNATING CURRENT ON A LONG LINE. 357 



Consequently, if instead of taking the final voltage at P in the steady 

 state, we take the stage of voltage growth at P found after k reflec- 

 tions from B have passed P, the corresponding vector stage of Ep, 

 which may be denoted by Epi., is 



Epk = Ep^ (i - e--''-^^) = Ep, (26-''^ sinh ^5^) volts Z . (24) 



The vector coefficient within the brackets may be described as the 

 vector coefficient of growth, k being an integer, increasing by unit 

 steps. At ^ = 00 , this coefficient reaches unity. It is a real coeffi- 

 cient, or has no imaginary component, when the imaginary part of 

 8 ^ is a quadrant or any integral number of quadrants. Thus, if the 

 line with its load at B develops a quarter- wave length at A; so 

 that 8 J contains one imaginary quadrant, then e'-^'^"^ is a real num- 

 ber, and so is the coefficient of growth at all stages. The phase of 

 the final voltage at P will then be the same as that at the arrival 

 of the first wave. 



Current Waves or Magnetic Flux Waves. — If the e.m.f. im- 

 pressed on ^ is i.o Z 0° max. cy. volts, without splash, the initial 

 outgoing current at A is 1 Z. o°/sQ^E^yQ = lQ amperes Z, where 

 3'o is the surge admittance i/^o- The first current wave arrival at 

 P finds this attenuated to IqC'^' and the first arrival at B to I^e' 

 vector amperes. In each successive arrival, the value /q appears, 

 and in tabulating the wave progress, /q may be omitted as a common 

 multiplier throughout, until the summation is effected. We may 

 therefore prepare a schedule of reflections similar to that given for 

 the waves of e.m.f. The coefficients of transmission and reflection 

 of the current waves arriving at B are however different from 

 those for the voltage waves. The coefficient of current-wave trans- 



mission « IS 



n = -p j r- = — ,- numeric Z (25; 



To + 0- 



C-^0 



and the reflection coefficient is 



numeric Z , (26) 



