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it is worthy to note in the derived Green’s function, the term expð cjz z0jÞ leads to considerably fast convergence, so that with increasing the distance between source and field elements, only with a few modes, the double series converges.

3 Examples and results

3.1 First example

In this example, a third-order waveguide band-pass filter (N = 3) with the center frequency of 8.7 GHz, fractional bandwidth (FBW) of 6 %, using WR90 (a = 22.86, and b = 10.16 mm) is designed. For the given design parameters, circuit element values are obtained, and based on the explained procedure; one can design the resonators in the form of longitudinal patterned planes. Figure 4 shows the proposed designed patterns, and corresponding scattering parameters versus frequency, obtained by method of moment, which is compared to the results of HFSS simulator.

Fig. 4 The proposed designed pattern and corresponding scattering parameters versus frequency a for resonators 1, and 3 b for resonator 2

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Impedance inverters, also, are replaced with quarter-wavelength hollow waveguide, whose length is computed in center frequency. So we have a structure as it is shown in Fig. 1a and b that each of patterned planes has the length and width of L = W=10 mm. Patterned planes are inserted in WR-90 waveguide, and the gaps between them are d = 8 mm, Also S is equal to 0.08 mm. Finally whole filter length is 46 mm.

Figure 5 shows the frequency response of proposed designed band-pass waveguide filter. The performance of the proposed filter is compared with that of conventional one (Konishi and Uenakada 1974), and periodically loaded one (Goussetis and Budimir 2003). It can be deduced the proposed band-pass filter is more selective, than others. Since, the coupling effect between subsequent resonators, varies the transmission and reflection coefficient, as well as desired bandwidth in the filter design, so it would not exactly lead to the target filter. Therefore, the coupling level should be strictly controlled. Since here are quarter wavelength sections, resonating at the higher frequencies, the proposed filter, as well as two other filters, has spurious resonance, which is a deficiency of the designed filter by this manner.

3.2 Second example

In the first example, a third-order waveguide band-pass filter for given specifications, has been designed, so that, the weak coupling effect between the subsequent resonators, has been partially controlled by changing the distances between them. In this example, a filter with the specification the same as periodically loaded one (Goussetis and Budimir 2003), has been re-designed, using an integrated (only one patterned plane) structure, in order to strictly take into account the coupling effect of the whole structure. For this purpose, the optimization has been done for the entire structure, meaning one patterned plane. Figure 6

Fig. 5 Frequency response of the proposed band-pass waveguide filter compared with conventional one, and periodically loaded one

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shows the proposed designed pattern, and corresponding scattering parameters versus frequency, obtained by method of moment, which is compared to the results of HFSS simulator. Furthermore, new design method improves the out of band performance of the filter, since there aren’t quarter wavelength sections, resonating at the higher frequencies. The patterned plane has the length and width of L = 30 mm, W = 10 mm, S = 0.08 mm, so the length of proposed filter in this example is shorter than the one in first example. For computation time and resources, the patterned plane has been assumed to have symmetry in x, and z directions. Figure 7 shows the frequency response of the designed filter, including only one patterned plane, and the performance of the proposed filter is compared with the periodically loaded filter, proposed in (Goussetis and Budimir 2003). It can be deduced the proposed band-pass filter has better performance in the stop band.

Fig. 6 a The proposed designed pattern and b corresponding scattering parameters versus frequency, obtained by the proposed approach and HFSS

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Fig. 7 Frequency response of the proposed band-pass waveguide filter compared with periodically loaded filter

3.3 Third example

In this example, a third-order elliptic band-stop filter is designed. The center frequency of 9.5 GHz, pass bandwidth of 1000 MHz, stop bandwidth of 333.3 MHz, and equal ripples of 0.1 dB in the pass band are supposed. First of all, as well as first example, we find the equivalent circuit of the filter, then its resonators were designed in the form of longitudinal patterned planes.

Impedance inverters, also, are replaced with 3kg=4 hollow waveguide, whose length is computed in center frequency, measured from center to center of the resonators. So we have a structure as it has been shown in Fig. 8, in which each of patterned planes has the length L = 8 mm and width of w = 10 mm, that is inserted in WR-90 waveguide, and the gaps between them are d = 25 mm. Furthermore, S is equal to 0.08 mm. Finally whole filter length is 107 mm. The frequency response of the designed waveguide band-stop filter, obtained by HFSS is illustrated in Fig. 9. As, there is a little difference between ideal desired and obtained filter, caused by coupling effects, in this case it could be performed another whole structure optimization, to obtain a filter with more desired specification that is neglected in this example.

Fig. 8 Side view of designed band-stop filter

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