Potential Coefficients for Ground Return Circuits 



By W. HOWARD WISE 



This paper is concerned with the effect of the finite conductivity and dielectric 

 constant of the earth on the potential coefficient for a 1-wire ground return 

 circuit. It has been customary to say that the potential coefficient V IQ is 



pvi = c^2 log p^/p', elm units per cm. 



It is generally realized of course that this is just a good approximation to the 

 true pu . To see that it is just an approximation one has only to imagine the 

 earth turning into air, in which case the distance p'' will eventually cease to 

 have significance. The object of this paper is to derive the complete expression 

 for pi2 . 

 It turns out that 



Pi2 = cV log pVp' + 4(M + iN)] (7) 



where 



r 



e-»+2)fV5< cos 



a = 4irao}, as in Carson's work on Zu and in mine on Zn at high frequencies^-^ 

 fe^" = Vl + i{€ - l)/2cX<r = s 



e = dielectric constant in electrostatic units 

 or = conductivity in electromagnetic units 



= 10~^3 to 10~" in ordinary soil 

 X = wavelength in centimeters 

 c = velocity of light, in cm per sec. 



M + iN vanishes as/— > 0,/-^ oo^ e -^ oo ora — > <». 



Ordinarily 4(if + iN) will not be e n important correction to 2 log p''/ p'\ but if the 

 frequency is high and h or z is small it can be a worthwhile correction. For 

 example, if a ,02535 inch wire be thrown out on the ground to be a 2 mc antenna 

 and we assume that<r = 10~", e = 15 and A = 3 cm. then, with a for wire radius, 



pn = c2[2 log Ihia + 4{M + iN)\ 

 = c2[10.455 + .152 + i .319]. 



1/^11 is the capacity to ground. If there were two parallel wires the scalar 

 potential at the first wire would be Vi = pnQi + PviQz - 



Derivation of the Formula 



WE BEGIN with the wave-function for an exponentially propagated 

 current in a straight wire parallel to a flat earth. The wave-function 

 for a horizontal current-element dipole has been formulated as an infinite 



ijohn R. Carson: "Wave Propagation in Overhead Wires with Ground Return" 

 Bell Sys. Tech. Jour. 5, pp. 539-554, 1926. 



2W. Howard Wise: "Propagation of High-Frequency Currents in Ground Return 

 Circuits," Proc. I. R. E. 22, pp. 522-527, 1934. 



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