#### 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 side-coupled filters

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**Design formulas for side-coupled filters**

The equations which related the even and odd mode impedance, *Z _{oe}* and

*Z*, respectively, to

_{oo}*W/D*,

*S/D*and ϵ

_{r}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 ϵ_{r}, *Z _{oo}* and

*Z*:

_{oe}A further set of equations relating *Z _{oe}* and

*Z*to the bandwidth parameter,

_{oo}*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 *Z _{oe}* and

*Z*from Eqs. 24 and 25 into Eq. 23, and eliminating the

_{oo}*i*subscripts and simplifying,

As the maximum permissible value of *X* is 0.5 (Ref. 3), the term *X ^{6}* can usually be neglected.

Substituting for *S/D* from Eq. 28 into Eq. 20, and for *Z _{oe}* 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 *X ^{2}* and

*X*terms can be neglected in Eq. 28 for narrow bandwidths.

^{6}Hence, ratios *S/D* and *W/D* can be found for side-coupled filters as a function of the parameters *X*, ϵ_{r}, and *Z _{o}*. Additional graphs have been prepared (Figs. 10-12) in which

*S/D*and

*W/D*are plotted as functions of

*X*for values of ϵ

_{r}and for

*Z*= 50Ω.

_{o}**References**

- E.G. Bradley, “Design and Development of Strip Line Filters,”
*IRE Trans. MTT,*(April, 1956). - S.B. Cohn, “Parallel Coupled Strip Line Resonator Filters,”
*L’Onde*[France], (October, 1957). - J.K. Richardson, “Gap Spacing for End Coupled and Side Coupled Band Bass Filters,”
*IEEE Trans. MTT.*(June, 1967). - S.B. Cohn, “Shielded Coupled Strip Transmission Line,”
*IRE Trans. MTT*, (October, 1957). - E.G. Bradley,
*op.cit.* - S.B. Cohn, “Direct Coupled Resonator Filters,”
*Proc. IRE*, (February, 1957). - H.M. Altschuler and A.A. Oliner, “Discontinuities in the Centre Conductor of Symmetric Strip Transmission Line,”
*IRE Trans. MTT,*(May, 1960). - G.L. Matthew, L. Young and E.M.I. Jones,
*Microwave Filters, Impedance-Matching Networks, and Coupling Structures*, (McGraw-Hill Book Co., 1964). - E. Rubins, W. Fromm and H. Keen, “New Techniques for High Q Microwave Components,”
*IRE Convention Record, Part 8*, (1954).

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