LAMINATED TRANSMISSION LINES. II 1159 



wliore 



' g K,(Vta)h(V,p,) - /vi(r,p,)/i(r,a) ' ^ ^^ 



7 = r, /vo(r,p2)/i(r,6) + /vi(r,6)/o(r,p2) , . 



' g /Ci(r,p,)/:(r,6) - iCi(r,6)/i(r,p,) * ^ ^ ^ 



Physically it is clear that the modes of the coaxial cable must be of tlie 

 same general types as the modes of the parallel-plane line, and so in 

 seeking the roots of equation (417) we shall be guided by the results 

 which we have already found for the plane structure. 



The dielectric modes in the cable may be located, to a first approxi- 

 mation, by setting Zi and Zo equal to zero, whence (417) becomes 



Ko{koPi) Ko(koP2) 



(420) 



The substitution 



/o(koPi) Io{koP2) 



kI = -Ii\ Ko = ill, (421) 



transforms (420) into 



Jo{hp,)N-o(hp-^ - Jo(hp2)No(hp,) = 0. (422) 



Equation (422) has an infinite number of real roots hi , hi , Jh , • • • , 

 hm , • ■ ■ , of which the inih one may be "UTitten in the form" 



h^ = "^^^-^P^/P^^ , (423) 



P2 — Pi 



where jPm(pi/ P2) is a function which increases from slightly less than 

 unity at pi/p2 = to unity at pi/p2 = 1. From equations (330) and (423) 

 we have, approximately, 



r, = /. V^ = '^ +|7-^-<''-(^'> ^i , (424) 



y we V2 (,P2 — pi) y (^e 



imirF m{pl/ pi) (,^rs 



Ko = ihm = -— , (.425) 



Pi — Pi 



and the fields of the mtli mode are given by substituting these expressions 

 into equations (340) to (344) of the preceding section. 



From equation (388) the propagation constant of the mth dielectric 



" Reference 18, pp. 204-206. What we call mwFm{pi/p2) is tabulated bv Jahnke 

 and Emde, pp. 205-206, as ik - l)x["'\ where k = ps/pi . 



