MUTUAL IMPEDANCE OF GROUNDED WIRES 



169 



pedance measurements.^ For a given frequency the complete group 

 of curves of which Figs. 2 and 4 are examples may be put in more con- 

 venient form by plotting the equivalent conductivity as dependent on 

 the other parameters of the problem. 



The d.-c. mutual inductance of wires 6" and s, as given at the head 

 of the corresponding column in Table I, is: 



L^ss = N°s, + N°{Aa) - N°(Ab) - N°iBa) + N°(Bb), 



where i\''°ss is the mutual Neumann integral of the wires and is the 

 main term in the formula. ■* The small contribution arising from the 

 remaining terms is as shown on Fig. 5 always less than A = — Aa 



0.8 



0.6 



0.4. 



r k 0.2 



z| 







-0.2 



-0.4 



-0.6 



0.01 0.05 0.1 



''- r 

 Fig. 5 — D.-C. mutual inductance, exclusive of mutual Neumann integral of wire 

 paths, of wires on surface of two-layer earth; 



N\A-Bna-b) = N°{Aa) - N°{Ab) + N°{Bb) - N°{Ba). 



-{• Ah -\- Ba — Bh. Figure 5 shows values of N°{r)lr for values 

 of ^ = Ihjr from .01 to 10 and for a range of values of /3 = (Xi — 

 X2)/(Xi -f X2) from - 1 to -f 1. 



The d.-c. mutual resistance, of course, also varies between the 



' Extensive results of such measurements are published in the following papers: 



A. E. Bowen and C. L. Gilkeson: "Mutual Impedances of Ground Return 

 Circuits," Trans. A.I.E.E., 49, 1370-1383 (Oct. 1930), and Bell System Technical 

 Journal, 9, 628-651, Oct. 1930. 



G. Swedenborg: "Investigations Regarding Mutual Induction in Parallel Con- 

 ductors Earthed at the Ends." The L. M. Ericsson Review, English Ed. No. 7-9, 

 1931, pages 189-204. 



H. Klewe: "Gegeninductivitats Messungen an Leitungen mit Erdruckleitung," 

 Elektrische Nachrichten Technik, 1929, page 467, and 1931, page 533. 



J. CoUard: " Measurement of Mutual Impedance of Circuits with Earth Return," 

 The Journal of The Institution of Electrical Engineers, Vol. 71, No. 430 (Oct. 1932), 

 pages 674-682. 



^ The only formal result in the evaluation of the Neumann integral known to us 

 is that for arbitrary straight paths published by G. A. Campbell, "Mutual Induc- 

 tances of Circuits Composed of Straight Wires," Phys. Rev., 5, pp. 452-458 (June 

 1915); see also his "Mutual Impedance of Grounded Circuits," Bell System Technical 

 Journal, 2, pp. 1-30 (Oct. 1923). 



