KENNELLY. — BUILDING UP VOLTAGE AND CURRENT. 711 



phase. The wave after reaching A and going to ground returns back 

 on the circuit towards B, but reversed in sign. The voltage at A is 

 thus not affected by the return of the wave, because although it tends 

 to be increased by A'c-^ia volts on the first return it is immediately 

 increased by —Ef.-"^^"- volts on the recoil from A, these two impulses 

 being e(|ual and opposite. 



The wave reflected with inversion at A now runs back towards B for 

 the second time. When it reaches B it will have run a total distance of 

 3 L kilometers from the start, and its magnitude will be —Ei~^^"^ volts. 

 At the open end B, it is reflected without reversal, thus adding a sud- 

 den jump of —2 Efr'^^'^ to the voltage previously existing at B. Back 

 it comes to A in the condition — 7i'e~*-^*. It is again reflected from 

 A and ground with reversal, making no change in the voltage im- 

 pressed on A, but starting out for B with the value E<r-^^'^. It 

 reaches B in the condition E^~^^°-, adding a sudden jump of 2 Ef.—^^"^ 

 to the voltage existing at that moment, and again sets out for A. 



The time of one passage along the line is — seconds, where L is the 



length of the line and v is approximately -—:= kilometers per second. 



vie 



In overhead aerial wires v is nearly 3()(),0()0 kilometers per second or the 



light speed in air; so that over a circuit 300 kilometers long, the time 



of flight of a wave would be about toVo second, and the time of one 



return trip jAo second. In one second of time after applying the 



alternator to A there would have been completed nearly 500 return 



trips over the circuit. 



From time ^ = to ^ = 0.001 second, the voltage at B = 



From time t = 0.001 to 0.003 second, the voltage at B = 2 E e-^^'^ volts 



From time t = 0.003 to 0.005 second, the voltage at B = 



From time t = 0.005 to 0.007 second, the voltage at B = 



2 Ee-L'^ — 2 E^-^^o- + 2 ii'e-5-^^« 



and so on indefinitely, until the succeeding jumps of voltage at B 

 become infinitely small, or the wave has been attenuated away to 

 nothing. The total final voltage at B after the steady state is finally 

 reached is : 



^^ = 2 E{c-i^^ — e-3^« -f e-si-^ — €-'^i« -t- volts (38) 



= 2 E^^''{\ — €-'-^^" -f- c--*^''^ — €-G^" -f " (39) 



-La 



2 E E 



~ ^ ^ 1 -f €-2^- '~ £^« + £-^- ~ cosh La " *^^^^ 



