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Effect of Ground Permeability on Ground Return Circuits 



By W. HOWARD WISE 



The formulas for the self and mutual impedances of ground return 

 circuits are derived without restricting the ground permeability. Curves are 

 given to show the effect of a ground permeability 1.7 on the mutual imped- 

 ance between two parallel ground return circuits with the wires lying on the 

 ground. 



X account of the irregular and heterogeneous character of the 

 major portion of the earth's surface and the consequent difficulty 

 in choosing a conductivity to be used in a computation of ground 

 return circuit impedance it has heretofore been considered useless to 

 take into consideration the possibility of an earth permeability greater 

 than unity. However, since the permeability may sometimes be 

 known to be appreciably different from unity and it is always desirable 

 to reduce the probable error in a computation and since the inclusion 

 of the permeability in the formulas may sometimes lead to a better 

 agreement between the theory and experiments it seems worth while 

 to provide formulas which include the permeability. 

 The self impedance of a ground return circuit is 



Z = r: + iico log p":a + MP + iQ), 



where z + /2co log p"!a is the self impedance with a perfectly con- 

 ducting ground and 4w(P -f iQ) contains the effect of the finite con- 

 ductivity and permeability of the ground. Carson * has derived an 

 infinite integral and series expansions for P + iQ on the basis of unit 

 permeability. The infinite integral derived here is arrived at merely 

 by going through Carson's paper and writing in the permeability 

 wherever Carson has replaced it by unity. The reader will be expected 

 to have a copy of Carson's paper at hand as not all of the steps in his 

 paper will be here reproduced. 



Equations (23) and (24) respectively are the new infinite integral 

 formulas for self and mutual impedance. Equations (A) and (C) 

 respectively are the new asymptotic and convergent series formulas 

 for P and Q. The functions m and / occurring in equations (C) are 

 functions of the permeability. Since some of them are defined by 

 series and their computation is consequently rather laborious, enough 



1 John R. Carson, Bell System Technical Journal, Oct., 1926. 



472 



