Design formulas for side-coupled filters

The equations which related the even and odd mode impedance, Zoe and Zoo, respectively, to W/D, S/D and ϵr have been given by Cohn (Ref. 4) and may be expressed as follows:

and

Equations 20 and 21 are accurate to approximately 1% for W/D = 0.35, and become increasingly accurate for W/D > 0.35, and become increasingly accurate for W/D > 0.35.

Rearranging (20) and (21), and subtracting from the other, the ratio W/D can be eliminated and the following expression obtained:

Solving for S/D using a similar procedure to that used to obtain Eq. 10 from Eq. 1, S/D can be expressed explicitly as a function of ϵr, Zoo and Zoe:

A further set of equations relating Zoe and Zoo to the bandwidth parameter, X, have also been given by Cohn (Ref. 2), and may be expressed as follows:

and

where n > i > 0 X is defined in Eqs. 4 and 5 with

Substituting for Zoe and Zoo from Eqs. 24 and 25 into Eq. 23, and eliminating the i subscripts and simplifying,

As the maximum permissible value of X is 0.5 (Ref. 3), the term X6 can usually be neglected.

Substituting for S/D from Eq. 28 into Eq. 20, and for Zoe from Eq. 24, an explicit expression for W/D is obtained:

where

For very narrow bandwidths, corresponding to X < 0.01, W/D can be taken as equal to that of the terminating strips, as given by Eq. 15 or graphically in Fig. 9. Moreover, both the X2 and X6 terms can be neglected in Eq. 28 for narrow bandwidths.

Hence, ratios S/D and W/D can be found for side-coupled filters as a function of the parameters X, ϵr, and Zo. Additional graphs have been prepared (Figs. 10-12) in which S/D and W/D are plotted as functions of X for values of ϵr and for Zo = 50Ω.

References

1. E.G. Bradley, “Design and Development of Strip Line Filters,” IRE Trans. MTT, (April, 1956).
2. S.B. Cohn, “Parallel Coupled Strip Line Resonator Filters,” L’Onde [France], (October, 1957).
3. J.K. Richardson, “Gap Spacing for End Coupled and Side Coupled Band Bass Filters,” IEEE Trans. MTT. (June, 1967).
4. S.B. Cohn, “Shielded Coupled Strip Transmission Line,” IRE Trans. MTT, (October, 1957).