274 BELL SYSTEM TECHNICAL JOURNAL 



distances from the cable, although the resistivity near the surface up to 

 ciepths of say 50 ft. is only moderately high. 



It is seen, however, that when the earth resistivity of the lower layer is 

 low, failures due to strokes to ground not arcing to the cable are rather 

 unlikely, even when the earth resistivity near the surface is rather high. 

 On account of the higher surface resistivity, however, a greater number of 

 strokes would be expected to arc to the cable for a given equivalent re- 

 sistivity, than when the conductivity is uniformly distributed. On the 

 other hand, many strokes which would arc to the cable if the earth were 

 uniformly conducting may channel through the surface layer to the good 

 conducting lower layer, so that the incidence of direct strokes is reduced 

 on this account. Experience indicates that the latter factor tends to 

 predominate, so that lightning damage is not ordinarily severe when the 

 resistivity is low at depths beyond 20 ft. or so. 



In the case of discharges between clouds the coupling between the light- 

 ning channel and the cable depends, in the frequency range of importance, 

 to a great extent on the resistance of the lower layer. Thus, when the 

 resistivity of the lower layer is very high the voltages may possibly give 

 rise to insulation failures in the case of small cables, while this is not likely 

 to occur when the resistivity of the lower layer is small or when the earth 

 structure is uniform and of moderately high resistivity. 



1.10 Cables with Insulated Sheaths 



Assume that a short length A.v of insulated sheath is placed on the ground 

 and that a voltage is applied between the sheath and a remote ground. 

 When the applied voltage is greater than the breakdown voltage of the 

 insulation, arcing to ground will take place at numerous equidistant points, 

 provided the insulation and the earth are assumed to be uniform. The 

 voltage between the sheath and adjacent ground increases from zero at a 

 point where arcing takes place to a maximum value midway between two 

 points at which arcing occurs, the maximum value being equal to the 

 breakdown voltage of the insulation. Midway between two arcing points 

 the potential in the earth (referred to infinity) may with negligible error 

 be calculated as though the leakage current through the numerous arcs 

 were uniformly distributed along the sheath. This potential in the ground 

 would then be AI/GAx, where A/ is the total leakage current and 1/GA.Y 

 the resistance to ground of the sheath without insulation, G being the unit 

 length leakage conductance. Midway between the arcing points the 

 potential of the sheath to a remote ground is then: 



V = Vo + M/AxG (54) 



