. The Bell System technical journal . simple example of a uniform type which can physically be made tohave properties equivalent to those of the ladder type. Its formulaefor propagation constant and characteristic impedance in terms ofthe series and lattice impedances, \z\ and 222, are known to be 2s icosh r = 1 -f- -—. 7, 4z2 — z\ (23)and K^VziZ,. A comparison of these formulae with those of the ladder type in (1)shows that when r = r, and K = KU Zi=Zi, (24)and 22 = 421 + 22; 20 BELL SYSTEM TECHNICAL JOURNAL and that when r = I\ and K = K2, 1 and (25) Zi = Z<i. In both cases it is apparent
. The Bell System technical journal . simple example of a uniform type which can physically be made tohave properties equivalent to those of the ladder type. Its formulaefor propagation constant and characteristic impedance in terms ofthe series and lattice impedances, \z\ and 222, are known to be 2s icosh r = 1 -f- -—. 7, 4z2 — z\ (23)and K^VziZ,. A comparison of these formulae with those of the ladder type in (1)shows that when r = r, and K = KU Zi=Zi, (24)and 22 = 421 + 22; 20 BELL SYSTEM TECHNICAL JOURNAL and that when r = I\ and K = K2, 1 and (25) Zi = Z<i. In both cases it is apparent that for equivalent results the latticetype requires more elements than the ladder type and is, therefore,not as economical. Part II. Design of Low-and-Band Pass Wave-Filters andReduction to Wave-Filters of Lower Class The foregoing theory of design can be applied separately to thedesign of wave-filters of each class in general use, which classes arethe low pass, high pass, low-and-high pass, and band pass. However, Lik o OXftO^. L2k=Uc 2k Fig. 7—Constant k Low-and-Band Pass Wave-Filter instead of such individual treatment designs will first be derived forlow-and-band pass wave-filters which are wave-filters of higher classthan these four classes and include the latter as particular cases. Thesimplifications in structure and formulae which result upon theirreduction to the lower classes will be considered later. Low-and-Band Pass Wave-Filters The structure of the constant k low-and-band pass wave-filteras derived from the attenuation requirements has the form of Fig. this form may be obtained from that given in Fig. 2 by assumingthe critical frequency, /3, in the latter to be infinite, we may underthis assumption refer to Fig. 2 for the impedance and attenuationcharacteristics corresponding to Fig. 7. The series impedance zVz expressed as a function of frequency is hk = i2irfLlk(l + l-4r2PrLlkClk ), (26) THEORY AND DESIGN OF WAVE-FILTERS 21 where r is the ratio b
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