Mutual Impedance of Grounded Wires Lying On or Above the Surface of the Earth

HE previous paper for mutual Tearth formula ofestablished in atothin wires wires lyingtheinthehorizontal impedance any grounded lying on surface of the has now been extended include 1 planes above the surface of the earth and grounded by vertical wires at their four end-points. As before, we assume the earth to be flat, semi-infinite in extent, of negligible capacitivity, of uniform resistivity p, and of inductivity v equal to that of free space. The air is also assumed to be of negligible capacitivity and of inductivity v equal to that of free space. All displacement currents are thus neglected both in the earth and in the air; this is the assumption which is ordinarily made as a first approximation at power frequencies for the shorter transmission lines. By the same general method of derivation as before, the extended formula is found to be: Z l 2 = where f f [ ^dSds'^ ° 0 3 +C S6Mr n h) dd Ss (>> ] > * A brief report of the principal theoretical result obtained in this paper was recently published in the Physical Review (2), 41, 536-537 (August 15, 1932). An error in the formula for N (r), as printed there, should be corrected: in the denominator of the fraction, r- should be r . R. M. Foster, "Mutual Impedance of Grounded Wires Lying on the Surface of the Earth," Bell System Technical Journal, 10, 408-419 (July, 1931); see also Bulletin of the American Mathematical Society, 36, 367-368 (May, 1930). 264 1 P(r,H,h) = P (r) + P,{r,H + h) - P*(r,H - h), 0 MUTUAL IMPEDANCE OF GROUNDED 2 WIRES 265 M(r,H,h) = Mo(r) + M^rJI + h) - M (r,H -h|), p , , iuv r j S 1 - f ^ r f c i + r » ) ! / » -- M i l r/ W = 47 J ( ; -- L (M + W + M J 1 0 2 ( r 2 + P , M ) = £ [ d log M (r)