Transients in Grounded Wires on the Earth's Surface

I FORMULA for the mutual impedance of grounded wires lying 011 the earth's surface has recently been published by R. M. Foster. 1 The object of the present paper is to derive formulas for the voltages during transient conditions in one such grounded wire due to current in a second for types of transient currents ordinarily obtained in a.-c. and d.-c. circuits, and particularly for the voltage due to unit step current, zero for time less than zero, unity for time greater than zero. The voltage due to unit step current is expressed in closed form for straight parallel wires; closed form expressions have not been obtained for straight parallel wires for the exponential forms of current for a.-c. and d.-c. transients. While the integrals might be evaluated numerically, or transformed to asymptotic expressions, it appears more desirable in practical calculation to use the curves given for the unit step voltage directly; a single integration is necessary to find the voltage for current of arbitrary wave form, from the unit step result. The fundamental physical assumptions upon which the steady-state formula is based are as follows: The surface of the earth is assumed flat, the earth semi-infinite in extent, of uniform conductivity A, unit * A brief report of the results in this paper was given at the Summer Convention of the American Institute of Electrical Engineers, Toronto, Ontario, Canada, June 23-27, 1930, in Discussion of " M u t u a l Impedances of Ground Return Circuits-- Some Experimental Studies," by A.