634 BELL SYSTEM TECHNICAL JOURNAL 



A more complete analysis of the results of both groups of tests is 

 given in connection with the discussion of theory which follows: 



Equivalent Ground-Plane Theory. The equivalent ground-plane 

 method of computing the mutual impedances of ground-return circuits 

 utilizes a very simple formula and has been in common use for a number 

 of years. A derivation and discussion of the formula together with 

 some experimental results are given in the report published by the 

 California Railroad Commission in 1919.^ 



This method assumes that the returning earth current may be con- 

 sidered as flowing in a hypothetical plane surface of perfect conduc- 

 tivity located some distance below the actual surface of the earth. 

 This surface is usually termed the "equivalent ground-plane." The 

 depth of the equivalent ground-plane below the actual surface of the 

 earth varies in different locations from about 50 ft. to 5000 ft. or more, 

 depending upon the character and resistivity of the earth and the fre- 

 quency. 



This method is subject to the objection that it fails to represent com- 

 pletely the observed phenomena. For instance, the method represents 

 the mutual impedance only with a reactive term, while the experimental 

 results indicated a substantial resistance component, particularly at the 

 wider separations and higher frequencies. Furthermore, no attempt 

 is made to explain the phenomena observed in the neighborhood of the 

 ground electrodes. However, in one respect the theory leads to results 

 comparable to those observed; the magnitudes of the mutual imped- 

 ances as observed under conditions in which end effects are negligible 

 can be checked reasonably well with a suitable choice of ground-plane. 

 Comparisons demonstrating this point are made on Fig. 7, where it 

 will be seen that the curve of experimental mutual impedance for a 

 frequency of 60 cycles can be fitted very well by a calculated curve 

 with a ground-plane depth of 8vS5 ft. That the depth of the equivalent 

 ground-plane depends on the frequency is seen from the fact that to 

 fit the experimental curve at 500 cycles requires the use of a ground- 

 plane depth of 385 ft. 



Method Assuming D.-C. Distribution in the Earth. For an earth of 

 uniform conductivity, the distribution of the current in the earth for a 

 ground-return circuit energized from a d.-c. source has been employed 

 by G. A. Campbell ^ to derive formulas for the mutual resistance and 

 inductance of ground-return circuits. The mutual resistance is ex- 

 pressed by a very simple formula which involves only the earth resis- 

 tivity and the distances between the points of ground connection on the 



^ See Bibliography. 

 2 See bibliography. 



