264 BELL SYSTEM TECHNICAL JOURNAL 



1.5 Direct Strokes — Lightning Voltages Along Cable 



It was shown above that with only a mmor error the impedance z may be 

 taken equal to the direct-current resistance of the sheath and that the 

 propagation constants may be taken as 



r = {iu:a)\ To = {io:^y 



With this modification expression (10) becomes: 



^W = T ^ ioc .e - 13 e \{io)) (21) 



2 a — f3 



The corresponding function S' is 



The voltage due to a surge current J{t), as obtained from (1), may be 

 expressed as: 



^(•^^' ^) = 0/ ^ ^, [«-g(«--v, - ^gi^x, /)] (23) 



2{a - 13) 



where, with a = a'.v or j8\v, 



gia,t) = f J{t-r)(^\ e'-'^'d. 



Jo XTTT"/ 



(24) 



For a current as given by (19), the latter integral may be expressed in 

 terms of error functions of complex arguments, for which, however, no tables 

 are available at present. Curves for the function g, as obtained by numer- 

 ical integration are shown in Fig. 4. 



When the core conductors are connected to the sheath at .v = 0, the con- 

 stants .4 and B of (6) and (7) are obtained from the following boundary 

 conditions: At :v = 0, 1^0) = so that .4 = -B. As before, B = -()( '^ ). 

 The voltage between core and sheath is then given by: 



— IP 



' 2 r^ - r! 



(25) 



JRa' r -(ioict)ix _ -(lO}l3)ix-l 



o \& e- 



r - r 



In this case the derivative of the voltage due to unit step current is: 



g-'^^''^'] (26) 





