Design for Strip-Line Band-Pass Filters

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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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, λʹ₀/2, 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

The equivalent circuit of a series gap in an end-coupled filter (center line representation) comprises a series capacitance, C₁, and two shunt capacitances, C₂ (Fig. 4). Following are approximate analytical relations, which relate the normalized susceptances, b₁ and b₂, associated with C₁ and C₂, to S, D and

and

Equations (1) and (2) are accurate for W/D > 1.2. For S/D < 0.2, | b₁ | > 10 | b₂ |, and for S/D < 0.1, | b₁ | > 75 | b₂ |. For the preliminary analysis of the end-coupled filter, S/D is assumed small enough that | b₂ | 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

f₁ and f₂ are the lower and upper cut-off frequencies, respectively, and

where f₀ 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, b₁. From (1),

and utilizing the identity

substituting for b₁ 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)

Referring to the equivalent circuit of the series gap (Fig. 4), parameter X may be expressed as a function of both b₁ and b₂:

where

Therefore,

where

From Eq. 3, b = X/(1-X²).

Substituting from Eq. 15 and providing

Substituting for b₁ and b₂ 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 λʹ₀/D and compared with b₁ obtained from Eq. 1 as a function of the same parameters. Graphs have been prepared with the ratios b/b₁ and b₁/b plotted in Figs. 5 and 6 as functions of λʹ₀/D for 1.0 ≥ S/D ≥ 0.1. Note that for S/D < 0.5, b > b₁ and for S/D > 0.6, b₁ < b.

The effect of the difference between b and b₁ on the value of strip spacing can be determined for relatively small differences by differentiating S with respect to b₁ in Eq. 10. Upon simplification, the following result is obtained:

where ∆ b = b ~ b₁. For b₁ λʹ₀/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, b₂) may be determined.

Unfortunately, it is not possible to obtain an explicit expression for S/D in terms of X, accounting for b₁ and b₂, 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 ≥ λʹ₀/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 λʹ₀/D is selected; because, if λʹ₀/2 is less than D, higher modes will be generated, and loss by lateral radiation takes place. And from Eq. 1, b₁ 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_oo, respectively, to W/D, S/D and ϵ_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 ϵ_r, Z_oo and Z_oe:

A further set of equations relating Z_oe and Z_oo 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 Z_oe and Z_oo 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 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 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, 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_o = 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).
5. E.G. Bradley, op.cit.
6. S.B. Cohn, “Direct Coupled Resonator Filters,” Proc. IRE, (February, 1957).
7. H.M. Altschuler and A.A. Oliner, “Discontinuities in the Centre Conductor of Symmetric Strip Transmission Line,” IRE Trans. MTT, (May, 1960).
8. G.L. Matthew, L. Young and E.M.I. Jones, Microwave Filters, Impedance-Matching Networks, and Coupling Structures, (McGraw-Hill Book Co., 1964).
9. E. Rubins, W. Fromm and H. Keen, “New Techniques for High Q Microwave Components,” IRE Convention Record, Part 8, (1954).

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