. The Bell System technical journal . NA TKI) TRANSMISSION LINKS. I Si)3 TluMi from (33) and (84) the bouiulary coiiditioiis are AIo{koPi) — BKo{koPi) _ ry ^ s .4/i(koP2) + BKiiKopi) m\ Avhero KO — (ffo 7 /(tp Ko go + ?coe(i r7o(l — y/(Tu)\ (37. If e(iuatious (36) are to he satisfied by values of A and B which are notboth zero, it is easily shown tluit a necessary and sufficient condition is r]i)f,Ki,{Kupi) + Zi{y)Ki{Kopi) 77op/vo(^oP2) — ^2(7) Al(^■oP2) VOpIo(Kopi) — Zi(y)Ii(koPi) rjOp/o(KoP2J + Z2(y)Il{KQP2) (38) and (38) is a transcendental e(iuation for the determination of the propa-gatio
. The Bell System technical journal . NA TKI) TRANSMISSION LINKS. I Si)3 TluMi from (33) and (84) the bouiulary coiiditioiis are AIo{koPi) — BKo{koPi) _ ry ^ s .4/i(koP2) + BKiiKopi) m\ Avhero KO — (ffo 7 /(tp Ko go + ?coe(i r7o(l — y/(Tu)\ (37. If e(iuatious (36) are to he satisfied by values of A and B which are notboth zero, it is easily shown tluit a necessary and sufficient condition is r]i)f,Ki,{Kupi) + Zi{y)Ki{Kopi) 77op/vo(^oP2) — ^2(7) Al(^■oP2) VOpIo(Kopi) — Zi(y)Ii(koPi) rjOp/o(KoP2J + Z2(y)Il{KQP2) (38) and (38) is a transcendental e(iuation for the determination of the propa-gation constants of all the circular magnetic modes in the coaxial ill the discussion of the parallel-plane line, we shall confine ourattention to the principal mode and shall assume forthwith that the walllosses are small. Since for the principal mode we expect that 7 will benearly equal to ctq , we may write 70 for o-q and eyaluate Zi and Z2 at 70 ;and we may replace the modified Bessel functions in (38) by their ap-. Fig. 2—Transmission line bounded l)y coaxial inii)edance cylinders. ^ J. A. Stratton, Electrumniinctic Theori/, McGraw-Hill, New York, 1941, , gives a similar treatment of the jMiiicipal mode in an ordinary coaxialcable with solid metal walls. 894 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 proximate values for small argument. From the series given in , , , and , we have loix) ^ 1, h{x) ^ ir, Koix) ^ -( + log ir) = -log ;, (39) Ai(.r) ^^- -\- ^x log ,-, X for I X I <3C 1, where log represents the natural logarithm. If we putthese approximations into (38) and if we suppose that the wall im-pedances are so small that I cropiZi(7o)/ 2r?o | « 1, | (Top2Z2{Yo)/2rio | « 1, (40) we obtain, after a little algebra, 2 2 2 (To[Zl{yo) / pi + Z2(.yo)/p2] /.,x Ko = (To — y = — T J—^-^ . 141 j rjn log (po/pi) Now further assuming that 1 Zi(yo)/pi + Z2{yo)/p2 ^ ^ /^2) 8 (Tor/o log (p2/pi)
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