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.