KBNNELLY. — BUILDING UP VOLTAGE AND CURRENT. 713 



After an indefinitely great lapse of time, the current at A is 



I^ = I„-2lo e---^" + 2/0 c-*^-^ — 2/0 e-*^^'' + amperes (41) 



= /o{l — 2 e--^'' + 2 6-4^« — 2 e-*^^-^ + " (42) 



= /o{l — 2 €-2/><^ (1 - e--'^-"' + ^-^^'"- — e-^^" + " (43) 



= ^^ = ^ 



COth Za Co COth Xa ^ V \ "^J 



The final result (45) thus agrees with the result of (19). 



If now the line be grounded directly at B, the final voltage at B is 

 nil, because each voltage wave is now reflected from that end with 

 inversion. The voltage at A also remains steady at E because the 

 voltage waves are reflected from the ground at A with inversion. 



The current arriving at B the first time is Iq e-^'^. It is reflected 

 directly from the ground there, thus producing at B a total increment 

 of 2 /o (.~^'^ amperes. On arriving back at A its value is I^ e— 2i". 

 Being reflected from the ground at A without inversion, it adds a total 

 increment of 2 Iq c— 2^'^ to I^ the vector current at A. When it reaches 

 B the second time, its value is /q €~^-^^ and it adds there an incre- 

 ment of 2 /(, e— 3i/a_ Reversal does not recur at either end of the line. 

 The final current strength at B after a sufficient lapse of time is 



/b =: 2 /o €-^« + 2 /o €-3^-^ + 2 /t-5^^« + amperes (40) 



= 2 /o e--^" (1 -f £-2^- + e-4^» + e-<^^" + " (47) 



_ I 1(^ € _ Iq I _ J() _ J^ ,, . . 



1 — e-^La. ^Lo. _ g-ia gi^Jj 2/a Co sinh La. ^ ' 



The result (48) thus agrees with Is in (21) or the final receiving-end- 

 impedance of the grounded circuit is c^ sinh La. ohms. 



Similarly, the final current strength at A will be the vector sum 

 of the initial outgoing current plus all the subsequently added reflected 

 currents, or 



/^ = /o -1- 2 /o £-2/- + 2 /o ^^"^ -f 2 /o e-6^<^ + . . . . amperes (49) 

 = /o{l -f 2 €-2^-(l -h €-2^« + e-^^« -F . . . . " (50) 



( 2 £— 2^°^ ) /I -f- £— 2i<i\ 



= ^0 1 1 + -{z:^^:. \ = ^"(rfTi^r.) = ^o coth La « (51) 

 = ,\ T " (52) 



Co tanh La 



The result in (52) thus agrees with the value for /^ in (21). 



If now the line be grounded through an impedance c^ ohms at B, 



