Mutual Impedance of Grounded Wires Lying On or Above 

 the Surface of the Earth * 



By RONALD M. FOSTER 



This paper presents a formula for the mutual impedance of any insulated 

 wires of negligible diameter lying in horizontal planes above the surface of 

 the earth and grounded by vertical wires at their four end-points. The 

 formula holds for frequencies which are not too high to allow all displace- 

 ment currents to be neglected. Tables and curves are given to facilitate 

 numerical computation by means of the formula. 



In the expansion of this formula for low frequencies and for any heights 

 the first two terms give the direct-current mutual impedance; the third 

 term is independent of the heights, thus being identically the same as that 

 previously found for wires on the surface. The mutual impedance for 

 wires at any heights H and h, with separations large in comparison with 

 these heights, is found to be approximately equal to the mutual impedance 

 for wires on the surface multiplied by the complex factor [1 -\- T{H -\- h)~\, 

 where V is the propagation constant in the earth. 



THE formula established in a previous paper ^ for the mutual 

 impedance of any grounded thin wires lying on the surface of 

 the earth has now been extended to include wires lying in horizontal 

 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 



'-// 



j-T-j + COS € MirJIJi) 



dSds 



dSds, (A) 



where 



P(r,H,h) = Po{r) + Pi{r,H + h) - P2{r,\II - h\), 



* 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 A^o(''), as printed there, should be corrected: in the denomi- 

 nator of the fraction, r- should be H. 



' 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 

 oj the American Mathematical Society, 36, 367-368 (May, 1930). 



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