#### 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

**June, 1968**

Strip-line band-pass filters can be constructed either of half-wavelength strips capacitively coupled end-to-end as shown in Fig. 1, or using parallel coupling of the half-wavelength strips as shown in Fig. 2. The advantage of parallel or side coupling over end coupling is that the filter length is reduced by approximately half, and a symmetrical frequency-response curve is obtained. The advantage of end coupling over side coupling is that the width of the filter is much less and the widths of all resonator strips are the same. The gaps between adjacent strips may be greater for side coupling but not necessarily so. If the gaps are greater, the gap tolerance for a given bandwidth is less; also, a broader bandwidth for a given tolerance can be achieved. Cohn has derived formulas which permit side-coupled filters to be accurately realized for bandwidths up to about 20% for a maximum flat response, and 30% for an equal ripple response. Other formulas are available to design end-coupled filters up to approximately the same bandwidths. The equations by Bradley and Cohn in the reference cited, are used here to construct graphs for determining band-pass filter dimensions as a function of normalized bandwidth. These graphs are for symmetrical strip line in the form shown in Fig. 3.

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**Design procedure**

In designing these filters, first decide upon the required frequency response and the rate of attenuation beyond cut-off; then calculate the number of resonators required and the values of the equivalent low-pass prototype elements.

*For the end-coupled filter*, Fig. 1, it is necessary to determine the susceptance of the capacitative gaps between resonant elements; 1. as a function of strip-line geometry and permittivity of the dielectric between ground planes, and 2. as a function of the required frequency response and normalized bandwidth. Next, by eliminating the susceptance values from the two sets of equations, it is possible to obtain expressions which explicitly relate the rations *S/D* and *W/D* (See Figs. 1-3) in terms of bandwidth and frequency response. The spacings between strips will differ from one resonator to another, being least for the first and last sections.

*For the side-coupled filter*, Fig. 2, a similar procedure is adopted, except that instead of susceptances it is necessary to evaluate even- and odd-mode characteristic impedance of the coupled resonator strips. By eliminating the impedance values from two further sets of equations, the ratios *S/D* and *W/D* are obtained as a function of bandwidth and frequency response. As with end-coupled filters, the spacings between resonator strips will be smallest for the end sections, but the strip widths differ from one section to another. However, for bandwidths less than 1%, the value of *W/D* does not significantly differ from that obtained for the terminal strips.

A difference exists between the electrical length of the resonator strips, λʹ_{0}/2, and the physical length for both filter types. Due to fringe fields, the electrical length is greater than the physical one, and a reduction in the latter is essential if the filter is to have an accurately positioned center frequency. Unfortunately, the formulas available to determine the necessary reduction in physical length are only approximate and have not been given in this article.