Mutual Impedance of Grounded Wires Lying on the 

 Surface of the Earth * 



By RONALD M. FOSTER 



This paper presents a formula for the mutual impedance between two 

 insulated wires of negligible diameter lying on the surface of the earth and 

 grounded at their end-points. The formula holds for frequencies which are 

 not too high to allow all displacement currents to be neglected. For any 

 two elements dS, ds of the two wires the mutual impedance is obtained 

 from their direct-current mutual impedance by introducing the complex 

 factor 2(7?-) -2 [1 - (1 + 70e-'>"'] in the reactance term, 7 being the propaga- 

 tion constant in the earth, and r the distance between the elements dS 

 and ds. 



THE mutual impedance of grounded circuits may be derived from 

 certain results obtained by A. Sommerfeld/ who has developed 

 formulae for the electric and magnetic fields in the earth and in the air 

 due to horizontal and vertical electric and magnetic antenucC situated 

 at the surface of the earth. For our present problem we use his formu- 

 Iffi for the electric field in the earth due to a horizontal electric doublet, 

 since this doublet may be regarded as a short element dS of a wire of 

 negligible diameter carrying a finite current. At the end of this 

 present paper we shall show how the same formula for the mutual 

 impedance may be obtained directly from first principles. 



Sommerfeld uses rectangular coordinates {x, y, z) and the corre- 

 sponding cylindrical coordinates {r, (j), z), the surface of the earth, 

 assumed flat, being the xy plane, and the s axis extending upward into 

 the air. The doublet is at the origin, and its axis along the x axis. 

 Then the components of the Hertzian vector - in the earth (s < 0) from 

 which the electric field is determined are ^ 



(1) n.= C*4£:^.w.-rv„ 



(2) II, = 0, 



* Presented by title at the Eugene, Oregon meeting of the American IMathemat- 

 ical Society, June 20, 1930, as "Mutual Impedances of Grounded Circuits." 



lA Sommerfeld, " Uber die Ausbreitung der Wellen in der drahtlosen Tele- 

 graphic," Aunalen der Physik, (4), 81, 1135-1153 (December 1926). This paper is a 

 summary and an extension of earlier work by Sommerfeld and von Hoerschelmann, 

 references to which will be found in the paper. _ 



2H. Abraham and A. Foppl, "Theorie der Elektrizitiit," 5th ed., Leipzig and 

 Berlin, 1918; Vol. I, § 79, page 331. . . , 



a A. Sommerfeld, loc. cil., pages 1145 and 1146, introducing the constant factor 

 defined on page 1152. 



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