Figure 4. Measured (solid lines) and simulated (dashed lines) insertion loss characteristics of a tri-post section acting as a pseudoelliptic resonator (a) and a frequency response transformation at the centered post height hC varying (b).

Pseudoelliptic symmetric post-based resonator

It is observed that changing the height of the centered post affects a coupling of the third mode with other modes of the same symmetry that results in pseudoelliptic response as well. It should be noted one more that all known post-based designs were achieved with a mandatory broken symmetry.

There is a set of geometrical parameters providing a frequency response with two attenuation poles located symmetrically at both sides of the frequency of total matching. Here, it is implemented if the centered post is essentially higher than the pair of lateral symmetrical posts.

The simulated and measured frequency responses of the section with the posts of the heights of hL,R = 7.2 mm and hC = 9.4 mm are plotted in Figure 4(a) by the black curves labeled 1. The total matching frequency is 8.942 GHz with the insertion loss of 0.33 dB and two TZs are located at 7.999 and 9.756 GHz, respectively. The unloaded resonator quality factor Qun 850.

A possibility to generate a pseudoelliptic response in a different frequency range (up to 16.5 GHz) is illustrated in Figure 4(a) by the curves labeled 2 and 3. The measured data (labeled 2) are presented for the section with hL,R = 6.23 mm and hC = 8.46 mm. The response is intentionally asymmetric. Two TZs are detected at 9.09 and 10.97 GHz. The TP frequency is 10.3 GHz and the insertion loss is 0.2 dB. Besides, the measured data (labeled 3) are presented for the section with hL,R = 4.5 mm, hC = 5.9 mm. Here, two TZs positioned at 12.166 and 14.29 GHz are separated by the frequency interval where the insertion loss is less than 0.6 dB. Two TPs are detected at 12.9 and 13.5 GHz. The measured insertion loss is 0.2 and 0.05 dB, respectively.

Study of the transversal electric field distributions at the resonant frequencies was performed. The field distributions at the frequencies f (TZ2) and f (TP) are symmetric with respect to the vertical median line of the section. They have three variations along the section broad wall similar to the field distribution of the third mode. The field distribution at the frequency f (TZ1) is symmetric as well and it has one variation along the broad wall. Thus, as in the case of asymmetric resonator, symmetric resonator exploits the third mode as a resonant one in order to provide a filtering function as well.

Figure 4(b) shows how the centered post height affects the section performance. The section geometry has the dimensions a = 23 mm, b = 10 mm, tx = 0.5 mm, t = 0.5 mm, dx = 2.0 mm, hL,R = 7.2 mm. The parametric curves of f (TP), f (TZ1) and f (TZ2) are plotted in Figure 4(b) by solid lines. The cut-off frequencies of the first and the third modes are plotted by dashed lines. Frequency responses those are similar to ones of asymmetric resonator are obtained. The only difference is the height of the centered post grows.

It is seen from Figure 4(b) that as hC increases ( h grows):

•transmission zero TZ1 remains at the response and its frequency f (TZ1) shifts down monotonically;

•as the resonator symmetry remains undisturbed, only one singlet pair of TZ2 and TP appears in the lower frequencies;

•behavior dynamics of the resonant frequency f (TP) is similar to the one of the third mode cut-off frequency fcut(3). The resonant frequency f (TP) shifts down linearly and synchronously with the third mode cut-off frequency fcut(3) shifting down over whole range of the varying parameter;

•newly appeared transmission zero frequency f (TZ2) shifts down;

•a singlet-type response is achieved at a certain hC and a symmetric pseudoelliptic response is achieved at a different hC The latter is indicated by vertical dotted line in Figure 4(b). An appropriate response is shown in Figure 4(a) by the curves labeled (1).

Note, a singlet response is generated if the centered post height decreases as well but a symmetric pseudoelliptic response is not achieved. Appropriate data are omitted in Figure 4(b).

For given configuration of the resonator, the lower frequency f (TZ2) can be estimated by the frequency of well-known quarter-wavelength resonance of the centered post, whereas the higher frequency f (TZ1) can be estimated by the frequency of the quarter-wavelength resonance of the lateral posts.

Symmetric tri-post section can be used as a frequency-tunable resonator. Simultaneous changing of the heights of all the posts for an equal value results in a shifting of the response. However, the shifts of the pole TP and zeros are not equal. The frequency separation between the lower frequency zero TZ2 and the pole TP is changed whereas the frequency separation between the higher frequency zero TZ1 and the pole TP remains almost unchanged. In order to keep both the frequency separations equal, the heights hL,R of the lateral posts and the centered post height hC have to be changed for a different value in such a manner as it is shown in Figure 5. Frequencies of TZ1 and TZ2 and TP are presented in the figure. Case (1) from Figure 4(a) is marked by the vertical dashed line. The values of the centered post height and appropriate values of the height of the lateral posts are marked along the bottom axis and the upper axis, respectively. Other dimensions of the resonator are: a = 23 mm, b = 10 mm, tx = 0.5 mm, t = 0.5 mm, dx = 2.0 mm.

Higher order filters

Higher order filters can be obtained by cascading the resonators. A two-section bandpass filter is presented as the simplest example. The filter consists of two equal symmetric post-based resonant sections separated by a quarter-wavelength rectangular waveguide

Figure 5. Symmetric tri-post section as a frequency-tunable pseudoelliptic resonator. Shifting of the frequencies f (TZ1), f (TZ2), f (TP) at the heights hC and hL,R of the posts varying.

Figure 6. Insertion loss characteristics of multi-pole multi-section bandpass filters: (a) simulated and measured performances of a two-section filter; (b) simulated performances of a three-section filter and their sections (solid and dashed curves, respectively).

section. Taking into account phase incursions, the rectangular waveguide section length is l = 9.58 mm. Individual resonator response is presented in Figure 4(a) by the curve labeled 1. The resonators of the filter have the following geometry: a = 23 mm, b = 10 mm, tx = 0.5 mm, t = 0.5 mm, dx = 2.0 mm, hL,R = 7.2 mm and hC = 9.4 mm. The simulated and measured frequency responses of the filter are plotted in Figure 6(a). Measured insertion loss in the bandwidth of 150 MHz (with the center frequency of 9 GHz) does not exceed 0.5 dB. Here, two lower frequency TZs are detected at 7.75 and 8.04 GHz and one higher frequency TZ is detected at 9.79 GHz.

It is possible to change the frequency separation of TZs by varying the lateral posts locations. As the second example, a three-section filter is considered. The first and the third resonators of the filter have the same geometry as in the previous example: a = 23 mm, b = 10 mm, tx = 0.5 mm, t = 0.5 mm, dx = 2.0 mm, hL,R = 7.2 mm and hC = 9.4 mm. The second resonator with dx = 1.0 mm, hL,R = 7.65 mm and hC = 8.95 mm provides two TZs located at 8.672 and 9.449 GHz. The resonators are separated by the waveguide section of 10.75 mm long. Simulated frequency responses of individual resonators and three-section filter are plotted in Figure 6(b) by dashed and solid curves, respectively. As it is shown in the figure, the filter provides a narrowband pseudoelliptic filtering function with three pairs of TZs.