400 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



The total impedance at a (inward + outward) for which we require an 

 equivalent network is 



Z = Z,ia) + Za 



where Za is the impedance of the central plug to axial current, viz., 



„ r]h /o(o-a) . . 



27ra /i(o-a) 



To evaluate the constants A and J5, the following boundary conditions 

 are imposed at radii a and h: 



at a: F = F(a), a given voltage 



at h: V = I{b)Z, 



where Zb is the impedance of the other "short circuit," comprising the 

 outer cylindrical wall. It is given by 



„ 7]h Koiab) , . 



ZtO K\{ab) 



Except for ignored small deviations of the field around the corners 

 of the cavity, the above expressions are exact. The process of finding 

 the singularities of Z by successive approximations results in expressions 

 that are too long to wTite down here. To obtain results sufficiently com- 

 pact for engineering use, we resort to the following asymptotic approxi- 

 mations for the Bessel functions: 



Jo{z) ~ { — I cos (2 — 7r/4) 



J^{z) '-'(—) cos {z - 37r/4) 



N^{z) ^ (^— j sin (2 - 7r/4) (3-11) 



Nr{z) ^ f- j sin (2 - 37r/4) 



Uz) ^ Koiz) ^ 



/i(2) ' K,{z) 



Also, with an error on the order of 10"*, 



Zo{r) - ^ = Ko(r) = l///o(r) 

 2irr 



