MUTUAL IMPEDANCES OP GROUNDED CIRCUITS 



15 



of parallel filaments which do not start at a common perpendicular, 

 which may thus be derived from Fig. 3. 



The mutual Neumann integral between any two parallel filaments 

 may also be obtained by means of the formula 



Nj?r = i[ - N (A a) + N (Ab) + N iB a) ~ N (B b)], 



(7) 



where N (Aa) now stands for the Neumann integral between the pro- 

 jections of Aa on -ft and r, extended if necessary, the projections 



HjO -&9 -OB -0.7 -0.6 -0.5 -0.4 -02 -02 -0.1 



0.1 02 03 0.4 OS 6 07 0.8 09 



Fig. 6 — Contour lines of the mutual Neumann integral between a counter-clockwise 

 small loop on the surface of the earth, per unit area, and a straight grounded filament 



AB of unit length 



having the same or opposite positive directions in agreement with 

 .?? and r. This formula for the mutual Neumann integral presents the 

 advantage of requiring only a single entry diagram, which is supplied 

 by Fig. 4 and on a logarithmic scale by Fig. 5. 



The mutual inductance may be required between a small, closed 

 loop lying upon, but insulated from, the surface of the earth and a 

 straight grounded conductor. The value depends upon the location, 

 area and assumed positive direction around the loop, but is inde- 

 pendent of the shape of the loop. Contour curves for the mutual 

 inductance per unit area of the loop are given by Figs. 6 and 7; the 



