Figure 1. A photograph (a) and a schematic view (b) of proposed three-post resonator.

well. Two-post-based waveguide sections having the capability to provide both a transmission pole (TP) and a transmission zero (TZ) and thus acting as singlets were reported in [11,12]. Later on, the filters based on L-shaped inserts in a rectangular waveguide and in a substrate integrated waveguide were reported in [13,14]. According to the nonresonating node (NRN) approach proposed by S. Amari and U. Rosenberg [15,16], known post-based resonators exploit the fundamental nonresonating mode to provide a bypass coupling and to generate a TZ. The second mode is exploited as the resonant one in order to provide a filtering function. Breaking the section symmetry is the mandatory condition as the second mode is not excited in a symmetric section.

A resonator of a different geometry is presented in the paper. It is a tri-post-based waveguide section. Its photograph is shown in Figure 1(a). The basic concept of such a tri-post resonator was proposed in [17] with the aim to implement a waveguide singlet. The resonator exploits another higher mode as a resonating one. It is the next higher (the third in the order) mode having a symmetric electric field distribution. It is shown in this paper that a tri-post resonator enables both a bandstop response and a pseudoelliptic response with two attenuation poles located symmetrically at both sides of the frequency of total matching. It is demonstrated that changing the height of only one post provides a transformation of the resonator response from one-pole bandstop characteristic into a pseudoelliptic one. This transformation is realized both with and without breaking the resonator symmetry. Besides, it is shown that the asymmetric resonator can act as a switching waveguide filter providing the ON and the OFF states by changing the height of one lateral post to allow/block the signal propagation in a narrow vicinity of a given frequency. A parametric numerical study is presented in order to demonstrate a transformation of the resonator response. The height of a post is the varying parameter in order to model a tuning unit. Finally, a possibility to design multi-pole multi-section filters is discussed.

The resonator design and characteristics

The section proposed in the paper is constructed by three E-plane rectangular partialheight posts inserted along a rectangular waveguide cross-section. One centered post is mounted on a broad wall of the waveguide whereas a pair of posts is mounted oppositely and symmetrically with respect to the vertical median line of the section. The distance between the posts must exceed a quarter of the waveguide width. Schematic view of the section is shown in Figure 1(b).

Figure 2. Insertion loss characteristic of an equal-post resonator acting as a single TZ generating section: heights of the posts hL,C,R = 7.2 mm (1), 8.0 mm (2), 6.2 mm (3) (a) and a comparison of a single-post, a dual-post and a tri-post sections (b).

Scattering parameters of a tri-post resonator are examined using the software based on the scattering matrix technique and the mode-matching technique for the waveguides with the piece-wise boundaries. A lossless model was used in the calculations and all the numerical data were obtained for a perfect conductor, whereas the measurements were performed with the brass samples. All the data presented below are obtained for rectangular foot-print posts of equal width tx = 0.5 mm and the thickness t = 0.5 mm inserted into the rectangular waveguide of the width a = 23 mm and the height b = 10 mm.

A bandstop equal-height post-based section

As the first example, an equal-height post-based section is examined. The simulated and measured transmission characteristics of the section with the posts of the heights hL = hC = hR = 7.2 mm and dx = 2 mm are plotted in Figure 2(a) by the curves labeled 1. As seen in the figure, the section provides a bandstop frequency response. A single TZ is detected at the frequency f (TZ1) = 10.906 GHz.

Numerical investigations show that a simultaneous variation of the heights of the posts is effective to control a TZ location. In the same manner, as for a section with two identical posts [18], the shorter are the posts the higher is the frequency of TZ. As an illustration, simulated data for the sections with the posts of the heights of hL,C,R = 8.0 mm and hL,C,R = 6.2 mm are plotted in Figure 2(a) by dashed curves labeled 2 and 3, respectively.

Besides, a comparison of calculated reflection frequencies and quality factors for a tripost section, a double-post section and a section with one off-centered post is presented in Figure 2(b). The height of the post(s) h(= hL,C,R, hL,R, hL, respectively) is the varying parameter. a = 23 mm, b = 10 mm, tx = 0.5 mm, t = 0.5 mm, dx = 2 mm for all the sections. As seen in the figure, a lower quality-factor reflection is achieved by using a three-post section.

A study of the transversal electric field distribution at the TZ frequency was performed in order to identify the mode dominating in a TZ generation. The field distribution at the frequency f (TZ1) is symmetric with one variation along the broad wall. It is similar to the lowest mode field distribution. Thus, the section exploits the lowest mode as a nonresonating mode to generate a transmission zero. A filtering function is not provided. The second mode has antisymmetric electric field distribution with two variations along

Figure 3. Measured (solid lines) and simulated (dashed lines) insertion loss characteristics of a tri-post resonator acting as a switching filter (a); and a frequency response transformation at the lateral post height hR varying (b).

the section broad wall. It is not excited in a symmetric structure. The third mode is symmetric. Nevertheless, a resonant coupling through the resonator via this mode fails to occur.

Pseudoelliptic asymmetric post-based resonator

A tri-post section provides a more complicated response if its cross-section symmetry is broken. Here, a symmetry breaking is achieved by changing the height of one lateral post. As an example, the simulated and measured frequency responses of the section with the lateral post height of hR = 5.9 mm (hR = hC − h, h = 1.3 mm) and all other parameter unchanged are plotted by dashed and solid curves marked 1 in Figure 3(a).

The section exhibits pseudoelliptic frequency response with two attenuation poles TZ1 and TZ2 located at both sides of the frequency of total matching f (TP) = 10.966 GHz with the insertion loss of 0.62 dB. The poles are detected at the frequencies f (TZ1) = 11.669 GHz and f (TZ2) = 10.522 GHz, respectively. Unloaded resonator quality factor Qun can be calculated from the measured loaded quality factor Q and insertion loss S21 at the total matching frequency. Here, Qun 700.

A detailed numerical study was performed to understand such a transformation of the response. Figure 3(b) shows how the height of one lateral post affects the filtering function. The parametric curves of f (TP), f (TZ1) and f (TZ2) are plotted in Figure 3(b) by solid black lines. The cut-off frequencies of the first three modes are plotted by dashed lines. As it can be seen, transmission zero TZ1 still can be observed at the response if the height hR decreases. Its frequency f (TZ1) shifts up monotonically. As soon as the resonator becomes asymmetric ( h =0), two singlet pairs of a TZ and a TP appear at the lower frequencies. An extremely high-quality singlet pair TZL/TPL is detected in a close vicinity of 9.15 GHz. The field distributions at the frequencies of f (TPL) and f (TZL) are asymmetric and have two variations like the second mode distribution. Besides, the effect of the height hR on the behavior dynamics of the pair TZL/TPL is similar to the one of the second mode cut-off frequency fcut(2). The location of the singlet pair TZL/TPL and a frequency separation between the pole and zero is weakly affected that is why it is excluded from further investigations. Another singlet pair of a zero and a pole is in the focus of our investigations. It appears in a close vicinity of 9.5 GHz.

For example, if hR = hC − h = 6.95 mm ( h = 0.25 mm), then f (TZ2) = 9.5575 GHz and f (TP) = 9.5675 GHz, respectively.

It is seen from Figure 3(b) that as hR decreases ( h increases):

•the resonant frequency f (TP) of disturbed resonator increases linearly and synchronously with the third mode cut-off frequency fcut(3) growing over all the range of the varying parameter;

•newly appeared transmission zero frequency f (TZ2) grows having the frequency f (TZ1) for the equal-post resonator as a limit. The latter is marked in Figure 3(b) by the horizontal dotted line;

•the higher frequency TZ1 is slightly affected at first and then shifts up.

At a different value of h, a pseudoelliptic response is generated. Its pole frequency is equal to the rejection frequency of the equal-post rejection section. An appropriate h is marked in Figure 3(b) by the vertical dotted line. Thus, a switching waveguide filter providing the ON and the OFF states by changing the height of the lateral post to allow/block the signal propagation is realized. Here, if the switch is initially in the OFF state it means that heights of all three posts are equal and the input signal is reflected at a given frequency. Switching in the ON state is performed by changing the height of the lateral post and it results in passing of the input signal at the same given frequency. It is illustrated by Figure 3(a), where the response of the asymmetric resonator is plotted by the curve labeled 1 and the response of equal-post symmetric resonator is plotted by the curve labeled 2. In known designs (see for ex. [19]), a quarter-wavelength post or an insert of the waveguide height b are used to block the signal propagation. The post or the insert should be removed out of the waveguide in order to switch in the ON regime. Here, a switching between the states OFF and ON is realized by changing the lateral post height for 1.35 mm ( h/6 andb/8) only.

It is clear that the posts of circular foot-prints are preferable to use in the tuning designs. If the areas of rectangular and circular foot-prints are equal, the waveguide switch performance remains unchanged.

A study of the transversal electric field distributions at the frequencies of TZs and TP was performed. They are shown schematically in the inset in Figure 3(a).

The field distribution at the frequency f (TZ1) of the higher frequency zero remains symmetric and has one variation similar to one of the first mode.

The field distribution at the pole frequency f (TP) is symmetric and has three variations like the third mode distribution. Thus, the resonator exploits the third mode as a resonant one in order to provide a filtering function.

Transversal electric field distribution at the transmission zero frequency f (TZ2) is antisymmetric with respect to the vertical median line of the section. Is has two field variations along the section broad wall similar to the second mode distribution.

It should note that the field distribution at f (TZ2) has three variations for all other similar designs. It is discussed further. Here, the resonance TPL seems to be affecting the field distribution at f (TZ2) badly.

Note, a singlet response is generated if the lateral post height increases as well but a symmetric pseudoelliptic response is not achieved. That is why appropriate data are omitted in Figure 3(b).