LAMINATED TRANSMISSION LINES. II 1143 



d{ — pH^), dp = —ioienpI'Jr, 



, (367) 



dE.'dp = —{Ko/io)top)( — plf,:,), 



wiicrc Ihc propajiiatioii factor f""''^ *" has Ixmmi suppressed. On the otiier 

 hand, e([iiati()iis (270) I'oi' the fields in a stack of iiiliiiitesiniall\' thin 

 laminae \'iel(l 



d( — pll^)''dp = —gpEz, 



, (368) 



dEjdp = -iT]/gp)(-pH,), 



where Yr is gi\-en by (323). If wc neglect the displacement current in 

 the main (Helectric compared with the conduction current in the stacks, 

 replace T( by ix, express ko in terms of x I'y (3t)l), and wiite miv M for 

 e eu , we obtain the following eciuations for the fields in the coaxial 

 C'logston line: 

 p'or a ^ p S pi , 



d{-pH^'dp = -p(gE^), 

 d(gE.)/dp = (x'/p){-pH,); 

 wliil(> for pi ^ p ^ p_' , 



d{-pH,)/dp = 0, 



d{gE,) /dp = (piOC^/pp) ( - p//^) ; 

 and for P2 ^ p ^ h, 



d(-pH,)/dp = -p(gE.), 

 d(gE.)/dp = (x'/p)(-pH,). 



(3691) 



(369ii) 



(369iii) 



The ([uantities —pH^ and gEz must be continuous at pi amd po ; and 



the two-point boundary condition at the infinite-impedance surfaces 

 p ^ a and p — b, namely 



-aH^ia) '-= -bH^ib) - 0, (370) 



determines a sequence of eigen^^alues xi , xi , x^ , " " • , of which the low- 

 est corresponds to the principal mode. It is a routine matter to integrate 

 eciuations (369) in terms of Bessel fmictions and logarithms, and to show 

 that the continuity and boundary conditions lead exactly to equation 

 (364). 



If equations (369) are set up on a differential analyzer with adjustable 

 values of x , cl/^, Pi/b, pz/b, and mo/m, it is a simple procedure to make a 

 few runs with different choices of x", and so to locate the appi'oximate 



