LIGHTNING PROTECTION OF BURIED TOLL CABLE 275 



Let dh/dx be the leakage current through the arcs and dli/dx the leakage 

 current due to capacity C between the sheath and the adjacent ground. 

 For sinusoidal currents the following equations then hold, when Z is the 

 unit length impedance and G the unit length leakance for a sheath in direct 

 contact with the earth: 



'^ + '^)^^Vo=V (55) 



dx dx / G 



- (7o + h)Z = ^ (56) 



-^^^ = Fo (57) 



tooC dx 



In the last equation it is assumed that the voltage between sheath and 

 ground is equal to the breakdown voltage of the insulation, although this 

 is not true in the immediate vicinity of the arcs. 



Eliminating I' the following equation is obtained: 



where : 



(t^^»^) + (tp-.^)- 



11,1 ,, io^CG 



= — + ^— , or: Y = 



(58) 



Y G ixoC ' G + ^wC 



Equation (58) is satisfied when: 



/o = Aoe~^'' + Boe 



(59) 



where V and Fi are the propagation constants for a sheath in direct contact 

 with the ground and for an insulated sheath without breakdown, respec- 

 tively. 



F = {GZy, Fi = {YZf 



For a sheath of infinite length the Bq and Bi terms vanish, so that: 



/(.v) = /„ + /i = A,e-^' -f .4ie-'^^ (60) 



The constants .lo and .li are obtained from the following boundary 

 conditions: 



