Mutual Impedances of Grounded Circuits

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 frequency and extent of the circuits are not larger than occur in many practical applications. 1. INTRODUCTION T HE i m p o r t a n t discovery of the possibility of using the earth as the return conductor for electric telegraphic c o m m u n i c a t i o n was a n n o u n c e d by Steinheil in the Comptes Rendus of September 10, 1838, a n d t h r o u g h o u t the entire d e v e l o p m e n t of telegraphy grounded circuits have been extensively e m p l o y e d . Considering the extensive application of such a capital discovery extending over a period of 85 years, it is surprising t h a t so little is k n o w n q u a n t i t a t i v e l y a b o u t grounded circuits. W e have, however, long k n o w n t h a t conditions are n o t of the extreme simplicity pictured under the early view t h a t the earth acts as a reservoir presenting no resistance to the return current and i n t r o d u c i n g no interference between parallel returns.