Transients in Grounded Wires Lying on the 

 Earth's Surface* 



By JOHN RIORDAN 



Voltages during transient conditions in a grounded wire lying on the 

 earth's surface due to current in a second grounded wire also on the earth's 

 surface are formulated for types of transient currents ordinarily obtained 

 in a.-c. and d.-c. circuits. The fundamental formula is for voltage due to 

 a unit step current, that is, a current zero for time less than zero, and unity 

 for time greater than zero; curves are given for the function determining 

 this voltage for a wide range of values of its two parameters. The for- 

 mulas for other types of currents are not well adapted for numerical com- 

 putation, which should be more conveniently carried out by numerical 

 integration using the above curves. 



I 



A FORMULA for the mutual impedance of grounded wires lying 

 on the earth's surface has recently been published by R. M. 

 Foster.' 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 X, 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 "Mutual Impedances of Ground Return Circuits — • 

 Some Experimental Studies," by A. E. Bowen and C. L. Gilkeson; A. I. E. E. Trans., 

 Oct. 1930. 



' R. M. Foster: "Mutual Impedances of Grounded Circuits" (Abstract), BiiUetiu 

 of the American Mathematical Society, May, 1930, pp. 367-36iS; "Mutual Impedance 

 of Grounded Wires Lying on the Surface of the Earth," Bell Svstem Technical Journal, 

 Julv, 1931. 



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