#### What is in this article?:

- Design for Strip-Line Band-Pass Filters
- Design formulas for end-coupled filters
- Design formulas for end-coupled filters (continued)
- Design formulas for side-coupled filters

## Design formulas for end-coupled filters (continued)

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However, it has been assumed that b_{2} is negligibly small in comparison with b_{1}; yet for *S/D* ≈ 0.46, | b_{1} | = | b_{2} |, and for larger values of *S/D*, | b_{2} | becomes increasingly greater than | b_{1} |.

The following analysis, which accounts for b_{1} and b_{2}, shows that Eq. 10 may be used for *S/D* as large as 0.6 accuracy of better than 1% providing λʹ_{0}/*D* ≥ 6.

Referring to the equivalent circuit of the series gap (Fig. 4), parameter *X* may be expressed as a function of both b_{1} and b_{2}:

where

Therefore,

where

From Eq. 3, b = X/(1-X^{2}).

Substituting from Eq. 15 and providing

Substituting for b_{1} and b_{2} 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 λʹ_{0}/*D* and compared with b_{1} obtained from Eq. 1 as a function of the same parameters. Graphs have been prepared with the ratios *b/b _{1}* and

*b*plotted in Figs. 5 and 6 as functions of λʹ

_{1}/b_{0}/

*D*for 1.0 ≥

*S/D*≥ 0.1. Note that for

*S/D*< 0.5,

*b*>

*b*and for

_{1}*S/D*> 0.6,

*b*.

_{1 }< bThe effect of the difference between b and b_{1} on the value of strip spacing can be determined for relatively small differences by differentiating *S* with respect to *b _{1}* in Eq. 10. Upon simplification, the following result is obtained:

where ∆ b = b ~ b_{1}. For b_{1} λʹ_{0}/*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, b_{2}) may be determined.

Unfortunately, it is not possible to obtain an explicit expression for *S/D* in terms of *X*, accounting for b_{1} and b_{2}, 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 ≥ λʹ_{0}/*D* ≥ 2 and 1 > *X* > 0 (Figs. 7 and 8) and *W/D* as a function of ϵ_{r} for Z_{o} = 50 Ω (Fig. 9).

A lower limit of two for λʹ_{0}/*D* is selected; because, if λʹ_{0}/2 is less than *D*, higher modes will be generated, and loss by lateral radiation takes place. And from Eq. 1, b_{1} 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.