GROUND RETURN IMPEDANCE 95 



a' = a^^, 



h' = h-yja, 



a = 47rXaj where X is the conductivity of the ground in 

 elm. c.g.s. units, co is 27r times the frequency, 



i = ^l- 1. 

 KQ{a'i->il) is the Bessel Function of the second kind; it is equal to 

 y Ho''^\a'i-yJt) where Ho'^^'> is the Hankel function as defined by 



Jahnke u. Emde in their Funktionentafeln. Denoted by ker a' + 

 i kei a' the function KQia'i^yfi) has been computed and tabulated by 

 the British Association. The only restriction on formula (2) is that 

 the radius a is supposed small compared with the depth h. 



Now the ground conductivity X lies between 10"^'* and 10"^^^ while 

 the depth h will not in practice exceed a few meters {h < 10^). Under 

 such circumstances, at ordinary frequencies, h' will be exceedingly 

 small compared with unity, and a' still smaller. Consequently in 

 evaluating the infinite integral in (2), it is permissible to take g-s^i'V^^+i 

 as unity, since ior (x > 2, the rest of the integrand converges as il^ix^. 



Now we have 





and hence c of formula (2) becomes 



1 1 



c = - 



2 Ko(a'^^|i) 



Furthermore since a' by hypothesis is very small compared with 

 unity, we can replace Ko by its limiting form for vanishingly small 

 arguments which is approximately 



log d/a'). 



We thus get, finally, the approximate formula, valid for most practical 

 applications, 



The interesting and somewhat surprising feature of this formula is 

 that the value of the correction term 1/2 log (I /a') likely to occur in 



