SOME RESULTS ON CYLINDRICAL CAVITY RESONATORS 435 



An investigation needs to be made of the behavior of the formulas as 



77 — > before any conclusion may be drawn regarding their blending 



into those for the cylinder. For TE modes with ^ = 0, the term involving 



jj 



— disappears, hence no question arises. Consider then / > 0, and let 



X = Tjr for the discussion following. From expansions given in McLachlan, 

 it is easy to show that, for small x 



J({x) = 



2i{( - 1) ! 



y'lir) y'ti-nr) Y({x) 

 it is found, upon substitution of the approximations given above: 



That is, Zt{x) '~ x^ and hence — > as x ^ 0. Furthermore Zt{r) remains 

 finite as t? -^ 0. Hence H -^ 0^^ and — '^ x^~^. Therefore, for / > 0, 



n 



— — > as 77 — > 0. 



Hence, the expression for Q - for the coaxial structure reduces to that for 



the cylinder, for any value of (, in the TE modes. 



For the TM modes, and for ^ > 0, an entirely similar argument shows 



that H' remains hnite as 7? — > 0. Hence, the expression for Q - for these 



A 



modes also reduces to that for the cylinder. 



For the TM modes, and with / = 0, we have 



Zo(x) = -7i(.r) + 7o(-t) 



F,(x-) 



For X — )■ 0, /i(.v) — > and Jq{x) -^ 1, hence for small x, 



yo{x) 



