544 BELL SYSTEM TECHNICAL JOURNAL 



unit length. (With small error this may usually be taken as the resist- 

 ance per unit length of the wire.) The axial electric intensity at the 

 surface of the wire is then zl. Equating this to the axial electric in- 

 tensity at the surface of the wire as given by (18) and replacing 9/93 

 by — r, we have 



zl= -4ccl r {.\/7+i-ij)e~^"'''diJi 



•^^ (19) 



-i2a;J.log(p'Va)+rF. 



Writing F=(2/Cand 



iu:Q = TI-GV=TI-^Q, 



where G is the leakage conductance to ground per unit length, we 

 have, solving for V, 



r^ = (G+?coC)[s+f2a;. log (p"/a)l+4co H (VTTi- fJ^)e-^"'>'dfi). (20) 



Writing this in the usual form 



r = (R+iX){G+io:C), (21) 



the characteristic impedance is given by 



^2^R±iX^ (22) 



G-\-io)C 



and the series impedance per unit length of the circuit is 



i?+,-X = Z = 2+i2co. log (p'Va)+4co r {\/7+i-t^)e-^'''>'dn. (23) 



It will be observed that the first two terms on the right hand side of 

 (23) represent the series impedance of the circuit if the ground is a per- 

 fect conductor; the infinite integral formulates the effect of the finite 

 conductivity of the ground. 



The mutual impedance''' Zx-i between two parallel ground return 

 circuits with wires at heights hi and //o above ground and a separation 

 X between their vertical planes is given by 



^ (Vm^+/-m)^~^'''+''^'' ^os -^'^ ^^' (24) 



'= It will be noted that the mutual impedance is equal to the axial electric intensity 

 at the axis of the second wire due to the varying magnetic field of unit current in 

 the first wire and its accompanying distribution of ground current. 



