NETWORKS FOR IMPEDANCE FUNCTIONS 399 



Tf T^ is the voltage between the flat faces of the cavity at a radius r and 

 1 the total current in the lower face at this radius, we have 



'^ = -IZ(r) 

 ar 



^= -VYir) 

 ar 



(3-3) 



By differentiating, 



d'V _ jdZ rydl _ (i dZ\dV 

 But 



dr- dr dr \Z dr / dr 



ZY = (2, + i.,h) '^ = 7^ 

 h 



which is a s(iuared propagation constant, independenl of r, and 



1 rfZ ^ _1 

 Z dr r 



Therefore, 



i^+-f- y-v = (3-5) 



dr^ r dr 



is the differential ecjuation for the voltage. The usual solution of this 

 equation is a linear combination of Io(yr) and Ko{yr) but since, in this 

 case, the arguments will be almost purely imaginary, it is more con- 

 venient to employ the pair of functions, Jo( — iyr) and Noi — iyr). 



The solution for the voltage between the upper and lower surfaces at 

 radius r is 



V(r) = AJ,{-iyr) + BN,{-iyr) (3-6) 



and, from this, the total radial current in the lower surface, at that 

 radius, is 



/(/•) = -i '^ = -iY,{r)[AJ,{-iyr) + BN,{-iyr)] (3-7) 

 Z dr 



where 



}'„(/•) = l,'Zo(/-) = [Y{r)/Z{r)r 



The impedance at the inner radius a, looking outward, is then 



Ziia) = J-—- = tZoia) -r^j- — -. — . , p„ , — -. — -^ (3-8) 



I {a) AJx{ — iya) -f BNi{ — iya) 



