**June, 1968**

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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.**Design procedure**

*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, λÊ¹_{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.

## Design formulas for end-coupled filters

_{1}, and two shunt capacitances, C

_{2}(Fig. 4). Following are approximate analytical relations, which relate the normalized susceptances, b

_{1}and b

_{2}, associated with C

_{1}and C

_{2}, to

*S*,

*D*and

and

Equations (1) and (2) are accurate for *W/D* > 1.2. For *S/D* < 0.2, | b_{1} | > 10 | b_{2} |, and for *S/D* < 0.1, | b_{1} | > 75 | b_{2} |. For the preliminary analysis of the end-coupled filter, *S/D* is assumed small enough that | b_{2} | may be neglected.

The normalized susceptance of the (*i* + 1)-th gap of an end-coupled filter with *n* stages, b* _{i,j+1}*, may be expressed as follows:

where

and

where

*f*_{1} and *f*_{2} are the lower and upper cut-off frequencies, respectively, and

where *f*_{0} is the center frequency of the filter and *g* is the normalized value of a low-pass prototype element.

Equation 1 can be rearranged into a form in which *S/D* is expressed as an explicit function of the series susceptance, b_{1}. From (1),

and utilizing the identity

substituting for b_{1} from Eq. 3, and omitting the *i* subscripts, Eq. 8 becomes

The value of *W/D* can be found from the equation given by Cohn for the characteristic impedance, *Z _{o}*, of a symmetrical strip line (Fig. 3):

from which it follows:

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

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 λÊ¹

_{0}/

*D*≥ 6.

_{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}|.

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 λÊ¹_{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 λÊ¹

_{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 â b = b ~ b_{1}. For b_{1} λÊ¹_{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 ≥ λÊ¹_{0}/*D* ≥ 2 and 1 > *X* > 0 (Figs. 7 and 8) and *W/D* as a function of Ïµ_{r} for Z_{o} = 50 Ω (Fig. 9).

_{0}/

*D*is selected; because, if λÊ¹

_{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.

## 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 Ïµ

_{r}have been given by Cohn (Ref. 4) and may be expressed as follows:

**Design formulas for side-coupled filters**

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 Ïµ_{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*, Ïµ_{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 Ïµ

_{r}and for

*Z*= 50Ω.

_{o}- 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).