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BELL SYSTEM TECHNICAL JOURNAL 



The results of the numerical evaluation of equation (279), with these 

 values inserted, is shown in Fig. 30. The voltage is identically 

 zero until z'/ = 100; / = 100/y is the time of propagation of the line. 



At that instant it jumps to the value e 



-pi _ - lOOp/: 



'' and then begins 



to die away rapidly due to the draining action of the inductance. 



95 (00 105 no 115 120 



Values or vt 



Fig. 30 — -Voltage across terminal impedance on smooth line 



The effect of secondary reflection is insignificant and therefore not 

 shown. The current in the terminal resistance is V/Ro so that it is 

 given by the same curve. 



I have reserved until the last the exposition of the expansion theorem 

 solution as applied to transmission lines with terminal impedances, 

 for the reason that it is the least powerful and the most restricted, 

 although most closely resembling the classical form of solution. Fur- 

 thermore, it does not represent the sequence of physical phenomena, 

 in fact it is not a wave solution, but a solution in terms of normal 

 or characteristic vibration. In practical application its usefulness is 

 restricted to the non-inductive cable. 



It will be recalled that the expansion theorem solution is formu- 

 lated as follows : — 



" A=l/Z{p) 



is the operational equation of the problem, then the explicit solution is 



n 



where pi,p2 ... are the roots of the equation Z(p)=0. 



