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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 , the wavelength in the dielectric medium:


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, bi,j+1, may be expressed as follows:




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, Zo, of a symmetrical strip line (Fig. 3):

from which it follows: