MUTUAL IMPEDANCE OF GROUNDED WIRES 267 



P{r,IIJi) = -^ + ^^ - // log ! [r^ + (// - hYJ'''- + 11 -h 

 Zivf 47r 



- // log j [/-2 + (// - /O^]''- - 11 + h\ 

 + [r^ + (// - /j)2]i/2 - r 

 + F{H,h,T) +0(rMogr)}, 



(2) 



The function F{H,h,T) is of no consequence, since it does not 

 involve r; it contributes nothing to the value of the impedance. The 

 remaining terms are infinitesimals of order (T^ log F) for infinitesimal 

 values of T; they are thus of higher order than T itself. 



By means of equation (2) we can now show that the first three terms 

 in the expansion of Zn for low frequencies and for any heights are 

 given by 



P / 1 1 1 , 1 \ , iojv 



^'' = 2^[Aa~Ab~B^^Bb^4^'''^''-''''-'' 



+ -7 — i-^r-) ABabcosd+ •'-, (3) 

 Ox \ Ip J 



where N(s-E)(s-e) is the mutual Neumann integral between the two 

 circuits formed by the wires 5 and s, lying in planes at heights H and h 

 above the earth, grounded by vertical wires at their four end-points, 

 and with earth returns, — the four grounding points on the surface of 

 the earth being A,B and a,h, respectively. The angle between the 

 straight lines AB and ah is designated by Q. N(s-E){s-e) is equal to 

 N ss, the mutual Neumann integral between the two wires 5 and s, 

 augmented by terms which depend only on the arithmetical distances 

 between eight points, — the four end-points and the four grounding 

 points. 



The first two terms in the expansion (3) are precisely the direct- 

 current mutual impedance as given ten years ago by G. A. Campbell.^ 

 The third term is independent of the heights of the wires; it is thus 

 identically the same as the third term previously found for wires on 

 the surface. 



The leading term in the expansion of Zn for a long straight wire 5 

 and any wire 5 located near the midpoint of S, for any heights, is 



/( 



-;^ log 



27r ^ [x2 + {H- h)^y^ 



H V o I T^2M/-> I COS Xfx dfi cos e ds, (4) 



■ G. A. Campbell, "Mutual Impedances of Grounded Circuits," Bell System Tech- 

 nical Journal, 2, (no. 4), 1-30 (October, 1923). 



