circular cylindrical for the posts.. Also, the equations for K inverters presented in the preceding text have limitations; those do not take into consideration the frequency dependence of the waveguide discontinuities.

III. THE FULL-WAVE THREE-DIMENSIONAL MODELING BASED APPROACHED WITH DISTRIBUTED ELEMENT K-INVERTERS OF LEVY AND RHODES

A viable option is the full electromagnetic modeling of an off-centered round post discontinuity in a rectangular waveguide. A number of papers have appeared in the literature that deal with such analysis w12x. At the same time, today, there exist a number of very accurate full electromagnetic analysis based commercially available software which can be used to generate the design model for the K inverters for the filter configuration shown in Figure 1. The transmission line matrix analysis based software MICROSTRIPE w4x was used to develop the model for the K inverters for the round rod discontinuities shown in Figure 4. We will also calculate the required K-inverter values for our filters using the distributed element prototype network by Levy w13x

and Rhodes w14x. According to w13x and w14x the required K-inverter values for a filter are obtained as described in the following text:

1. The order of the filter, N, will give the number of resonators and N q 1 will be the number of rod couplers needed to meet the design specifications. N is determined from the ripple bandwidth D f, the isolation bandwidth D fi , the passband return loss Lr , and the stop band isolation Ls of the filter using the following equation for a Chebyshev type response. The filter order is given by w14x. Thus,

where g is the ratio D firD f.

2. The midband guide wavelength lgo is determined by solving w12x,

lgL sin pllgLgo / q lgH sin pllgHgo / s 0, 8b.

where lgL and lgH are the guide wavelengths in the resonator section at the lower and upper cutoff frequencies, respectively,

of the filter. For a narrow-band case,

A suitable numerical method is applied for solving eq. 8b..

3. A scaling parameter is given by

4. The impedance Zn of the distributed elements and the impedance inverter values kXn, nq1 are given by

and « is the factor that determines the passband ripple level of the filter.

5. The normalized K-inverter values which are to be realized by the waveguide rod discontinuities for the filter are defined by

Once the required K-inverter values have been calculated from the preceding equations, they can be physically realized in terms of discontinuities in a rectangular waveguide. However, as mentioned, although the approach is quite general for any type of discontinuity in a waveguide, we have chosen single and double round rod discontinuities of all types in the present work. Figure 4 shows four round rod discontinuity types, generally used in waveguide post filters and Figure 3 shows their equivalent circuits. The equivalent circuits are used to determine the K-inverter values and the values of the associated phase angle w, using the following equations,

Using the previous equations, uj in Figure 3 can be calculated as

The physical distance between two adjacent K inverters is given by

Si j are the computed scattering parameters of the K-inverter forming discontinuity shown in Figure 4c.

Figure 5 shows the computed K inverter and phase w. values, as functions of the normalized distance from one of the side walls of the waveguide with normalized frequency as a parameter,

Figure 5. Design curves for constant diameter and single rod filters.