This discontinuity of the waveguide bifurcation will be treated as a typical multiport network if the GSM method is applied. To enhance the computational efficiency, here we adopt a different method that is a combination of the modematching and the Ritz variational methods. First, the standard mode expansion is used to describe the fields in the two regions. When the symmetrical plane T0 is assumed to be open or short, the incident and reflected wave fields can be added to form the total fields in each region, respectively. By matching the tangential electric and magnetic fields at the reference plane T1, and then by expressing the fields of the dominant mode by higher order modes, a variational expression for the input admittance can be obtained. In order to find the minimum of the expression, the Ritz variational method is used to obtain a group of linear equations. By solving the equations group, the input admittance for the open and the short can be obtained.

From the viewpoint of circuit equivalence, the discontinuity can be regarded as a symmetrical T-circuit shown in Figure 1Žb.. From the above two input admittances, the equivalent-circuit parameters can be obtained directly by using the following equation Ž1.:

where yŽ o. and yŽ s. represent the input admittance when assuming the magnetic wall Žopen circuit. and the electric wall Žshort circuit. for the symmetrical plane, respectively. Both xs and xp are normalized to the characteristic impedance of the standard waveguide.

The circuit works as an impedance inverter of ratio K with a certain associated electric length . The relationship is as follows:

For filter design, the standard waveguides between the adjacent metal inserts work as half-wavelength resonators. Their length lk should satisfy the following equation:

lk g 0 2 Ž k 1, k 2 k , k 1 2. g 0 2 ,

k 1, 2, . . . , n Ž3.

where g 0 is the waveguide wavelength of the center frequency, and n is the filter order.

3. COMPARISONS

In this section, we will give some numerical examples to show why the new structure can be used for wideband filters. Generally, the first and last inserts are of the smallest width to reach the highest required ratio for wideband filters. So we will confine the width to be a reasonably small value, say 0.1 mm, for our comparison. When the filling is air, or r is equal to 1, the result is the same as that calculated from 13 . This proves that the method is valid.

The calculated results of the inverter’s ratio with different dielectric permittivity values are listed in Table 1. It can be concluded from Table 1 that the new structure can offer large ratio impedance inverters, thus enabling the wideband

TABLE 1 Equivalent Parameters with Increase of Dielectricr , Ka-Band, a = 7.112 mm, b = 3.556 mm, t = 0.1 mm,

W = 0.1 mm, F = 29.0 GHz, 15 Modes

filters. The larger the dielectric constant, the larger the ratio is.

An interesting example would be the combination of offsetting the metal insert and partially filling the region. The calculated results of the inverter’s ratio with different offsetting values are listed in Table 2. It can be seen that the ratio can be effectively increased by the use of an offset metal insert.

From both tables, it can be seen that the equivalent serial impedance is almost unchanged with the changes of the permittivity values or the offset values, while the parallel impedance can be increased significantly. This is the reason why the impedance ratio can be increased.

4. FILTER PERFORMANCE

In this section, a wideband filter at the lower end of the Ka waveguide band is designed to cover the frequency range of 26 30 GHz. A numerical simulation is done which confirms the theory and practical availability of the new structure.

The equal ripple ŽChebyshev. model is applied to the filter synthesis. To satisfy the specification requirements, the filter order n and the equivalent admittance gk for the low-pass filter model can be easily calculated. And then the ratio K of the impedance inverter can be calculated. Because the relationship between the width of the metal insert and the ratio K is monotonous, it is easy to find the proper width to realize the expected ratio of the impedance inverter.

After obtaining the actual dimensions of the bandpass filter from the model and Eqs. Ž2. and Ž3., a computer simulation can be done by cascading all of the discontinuities and the resonant waveguide to determine the actual performance. Shown in Figure 2 are the results for the reflection

TABLE 2 Equivalent Parameters Offset from the Center, Ka-Band, a = 7.112 mm, b = 3.556 mm, t = 0.1 mm,

W = 0.1 mm, F = 29.0 GHz, r = 2.5, 15 Modes