Ground Return Impedance : Underground Wire 

 with Earth Return 



By JOHN R. CARSON 



Synopsis: In certain transmission problems principally those relating to 

 induction and interference phenomena, it is necessary to know the trans- 

 mission characteristics of a circuit composed of an underground wire with 

 earth return. These can be evaluated by well known engineering formulas 

 provided the ground return impedance is known. The present paper gives 

 the mathematical solution of this problem and shows that the ground return 

 impedance is substantially independent of the depth of the wire below the 

 surface. 



THE object of this note is to give the solution of a problem of 

 considerable interest and practical importance which does not 

 appear to have been solved heretofore; this is the "ground return" 

 impedance, per unit length, of a circuit composed of an underground 

 wire or cable with earth return. 



The physical system and the problem may be more explicitly 

 described and explained as follows: An underground wire or cable 

 parallel to and at depth h below the surface of the ground is surrounded 

 by a concentric dielectric cylinder of external radius a. The earth 

 then forms the return path for currents flowing in the wire. The 

 ground return impedance Zg is then defined as the ratio of the mean 

 axial electric intensity at the external surface of the dielectric sheath 

 to the current flowing in the wire. 



When the earth extends indefinitely in all directions about the 

 wire so that circular symmetry obtains, the problem is quite simple, 

 and the formula for the ground return impedance, denoted in this 

 case by ZJ^, is well known. In practice, however, we are interested 

 principally in the case where the wire is close to the surface of the 

 earth, so that the distribution of return current in the ground is 

 anything but symmetrical. For this case the formula for the ground 

 return impedance, which it is the object of this note to state and 



discuss, is 



Z, = (1 + c)Zg\ (1) 



Here the correction term c, which takes care of the departure from 

 circular symmetry, is given by 



c = ' ■ r ^gzj'-" . c:Srf^. (2) 



Koia'i^^i) Jo Vm^ + i -\- fi -^iy? + i 

 In formula (2), 



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