The Bell System Technical Journal 



October, 1923 

 Mutual Impedances of Grounded Circuits 



By GEORGE A. CAMPBELL 



Synopsis: Formulas are derived for the direct-current mutual resistance 

 and inductance between circuits grounded at the surface of the earth. For 

 circuits composed of straight filaments, the mutual inductance is reduced 

 to known Neumann integrals which involve only comparatively simple 

 expressions for the case of horizontal, coplanar conductors above, below 

 or on the surface of the earth. Numerical values for these integrals may 

 be readily obtained from new and accurate graphs for straight filaments 

 which meet at a point or start from a common perpendicular. It is shown 

 that these new results supply a useful first approximation to the actual 

 alternating-current mutual impedance of grounded circuits, when the fre- 

 quency and extent of the circuits are not larger than occur in many prac- 

 tical applications. 



1. Introduction 



THE important discovery of the possibility of using the earth 

 as the return conductor for electric telegraphic communication 

 was announced by Steinheil in the Comptes Rendus of September 10, 

 1838, and throughout the entire development of telegraphy grounded 

 circuits have been extensively employed. Considering the extensive 

 application of such a capital discovery extending over a period of 

 85 years, it is surprising that so little is known quantitatively about 

 grounded circuits. We have, however, long known that conditions 

 are not of the extreme simplicity pictured under the early view that 

 the earth acts as a reservoir presenting no resistance to the return 

 current and introducing no interference between parallel returns. 

 This view was expressed in 1857 by Bakewell as follows: "There is 

 no mingling of currents, the electric current of each battery being 

 kept as distinct as if separate wires were used both for the trans- 

 mitted and the return currents. It would indeed be as impossible 

 for the separate currents transmitted from the two batteries to be 

 mingled together as it would be for the written contents of two letters 

 enclosed in the same mail bag to intermix." 



Measurements made a few years ago of the mutual impedances 

 between grounded circuits which are restricted to a territory six miles 

 square, at frequencies of 25 to 60 cycles per second, showed that within 

 10 per cent, the mutual reactance increased in the same ratio as the 

 frequency. It was inferred that the effective inductance under the 

 conditions of these tests was approximately the same as for direct 

 current, or in other words, the incomplete penetration of the alter- 

 nating currents into the earth was not of controlling importance in 

 tests upon this scale. 



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