TRANSIENTS IN GROUNDED WIRES 421 



permeability and negligible dielectric constant. The air above the 

 earth is of zero conductivity, unit permeability, and negligible dielectric 

 constant. Because of the assumption of negligible dielectric constant, 

 the formulas for voltages during transient conditions do not hold 

 strictly for small values of the time, that is, during the initial stages of 

 the transient. The wires are of negligible diameter, lying on the 

 surface of the earth, and insulated from it except at the ends, where 

 there is point contact. 



In using the steady-state solution as the basis of transient solutions, 

 the Heaviside operational calculus is employe d af ter replacing 7co, 

 where co = lirf is the radian frequency and i = V— 1, by p = d/dt, the 

 time differentiator, since (rf"/W/") (exp icot) = (/w)" exp ioot, where n is 

 integral. 



II 



The mutual impedance of grounded wires lying on the surface of 

 the earth and insulated from it except at the ends is given by the 

 following formula: ^ 



The integration is extended over the two wires S and s, having 

 arbitrary paths, r and e are the distance and angle, respectively, be- 

 tween differential elements dS and ds, and 7 = {'iTrXiooY'^; X is the 

 ground conductivity and w = 27r/ is the radian frequency. 



Replacing ico by p = d/dt in 7, the resulting forms to b e evaluated 

 are exp ( — a>[p) and ^ip exp ( — aVp) where a = rV-ivrX. The first 

 of these is known and, following Heaviside,^ may be developed as 

 follows. 



Expressing the exponential in series form: 



exp(- ay^p) = 1 - ^^^+_i^_-^^+ .... 



Integral powers of p are neglected, since (omitting the discontinuity 

 at / = 0) the operand is unity and the derivative of a constant is 

 zero. Then: 



exp (— a4p) = 1 — a\^ 



<iP a-^ 



^ i\ ^ 51 ^ 



The bracketed terms may now be assumed to operate on y[p = (x/) "^ 



2 Foster, loc. cit. 



3 Heaviside: "Electromagnetic Theory," Vol. II, pp. 49-51, equations (4) and (12). 



