MUTUAL IMPEDANCE OF GROUNDED WIRES 413 



direct-current mutual impedance as given by G. A. Campbell. i° 

 Introducing this factor, which is a function of yr only, into the re- 

 actance term for the direct-current mutual impedance between two 

 elements dS and ds gives the general expression for their mutual im- 

 pedance corresponding to the propagation constant 7. It is interesting 

 also to determine, for any given value of 7, the variation of the factor 



(22) for increasing values of r. This is shown very clearly in Fig. 1, 

 where the real and imaginary parts of (22) are plotted for increasing 

 values of / = [7^! = (47rXa))^/V. The real part, we note, decreases 

 rapidly from the initial value unity as r' increases, while the imaginary 

 part is always negative, decreasing from zero to a minimum value 

 (approximately — 0.3 for r' = 1.5) and then increasing towards zero, 

 although it does not approach zero so rapidly as the real part does. 



The first three terms in the expansion of Z12 for low frequencies 

 are given by 



(23) Z^.. = ^l^-^--l+^)+io.Ns. 



lirX \Aa Ab Ba Bb 



-r (1 - i)^(S ttXoo'Y'- A B ab cos d -\- •••. 



where Nss is the mutual Neumann integral between the two wires S 

 and 5 of arbitrary form but with end-points A, B and a, b respectively; 

 d is the angle between the straight lines AB and ab. The first two 

 terms in this expansion are precisely the direct-current mutual im- 

 pedance as given by G. A. Campbell. 



The first term in the expansion of Zn for a long straight wire S and 

 any wire s located near the midpoint of 5 is 



(24) 



/ 



-T— , — Ai(7.v, 



TTAA'- TTAA 



COS e ds, 



X being the positive distance from ds to 5, and e the angle between ds 

 and S. Kilz) = - ^tHi^^^iz) is the Bessel function of the second 

 kind for imaginary argument as defined by G. N. Watson." In ob- 

 taining (24) from (21) we use the derivative with respect to x of the 

 integral 





-yr 



dz ^ Ko{yx), 



which is a special case of the integral used above in evaluating Q, with 

 X assumed positive. 



^°G. A. Campbell, "Mutual Impedances of Grounded Circuits," Bell System 

 Technical Journal, 2, 1-30 (October 1923). 



" G. N. Watson, op. cit., page 78. 



