Design formulas for end-coupled filters (continued)
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However, it has been assumed that b2 is negligibly small in comparison with b1; yet for S/D ≈ 0.46, | b1 | = | b2 |, and for larger values of S/D, | b2 | becomes increasingly greater than | b1 |.
The following analysis, which accounts for b1 and b2, shows that Eq. 10 may be used for S/D as large as 0.6 accuracy of better than 1% providing λʹ0/D ≥ 6.
Referring to the equivalent circuit of the series gap (Fig. 4), parameter X may be expressed as a function of both b1 and b2:
From Eq. 3, b = X/(1-X2).
Substituting from Eq. 15 and providing
Substituting for b1 and b2 from Eqs. 1 and 2 into Eq. 16, and simplifying, ψ may be expressed directly in terms of S/D:
Utilizing equations 17 and 18, b can be obtained as a function of S/D and λʹ0/D and compared with b1 obtained from Eq. 1 as a function of the same parameters. Graphs have been prepared with the ratios b/b1 and b1/b plotted in Figs. 5 and 6 as functions of λʹ0/D for 1.0 ≥ S/D ≥ 0.1. Note that for S/D < 0.5, b > b1 and for S/D > 0.6, b1 < b.
The effect of the difference between b and b1 on the value of strip spacing can be determined for relatively small differences by differentiating S with respect to b1 in Eq. 10. Upon simplification, the following result is obtained:
where ∆ b = b ~ b1. For b1 λʹ0/D ≤ 0.5, Eq. 19 has a maximum error of approximately 4%. By considering Figs. 5 and 6, in conjunction with Eq. 19, the error in gap spacing obtained by using Eq. 10 (which does not account for the shunt susceptances, b2) may be determined.
Unfortunately, it is not possible to obtain an explicit expression for S/D in terms of X, accounting for b1 and b2, because of the transcendental nature of Eq. 18. Nevertheless, utilizing Eqs. 12, 15 and 18, a set of graphs has been prepared giving X as a function of S/D for 100 ≥ λʹ0/D ≥ 2 and 1 > X > 0 (Figs. 7 and 8) and W/D as a function of ϵr for Zo = 50 Ω (Fig. 9).
A lower limit of two for λʹ0/D is selected; because, if λʹ0/2 is less than D, higher modes will be generated, and loss by lateral radiation takes place. And from Eq. 1, b1 is always positive if S/D > 0; while from Eq. 3, b is positive only if X < 1. Consequently, for Eqs. 1 and 3 to be consistent, 1 > X > 0.