A Transmission Zero Position Control for 28 GHz Rectangular Waveguide Cavity Bandpass Filter

This paper presents a method to control on single real-frequency transmission zeros (TZs) position in the reject band for third-order waveguide band pass filter (BPF). The external couplings are achieved by using a coaxial probes while the internal couplings are achieved by using short-circuit metallic posts. The TZs appear are generated by forming a triplet, i.e., multipath cancellation between non-adjacent resonators. The dimensions of the metal post between nonadjacent resonators are adjusted to control the position of TZs in the rejection band. A third order waveguide generalized Chebyshev BPF is simulated with HFSS at a center frequency of 28 GHz to validate the design method. The simulated insertion loss is 0.05 dB and the bandwidth (BW) is 500MHz. The simulation demonstrates the compatibility of the presented filtering structure with 5G applications.

designed to produce transmission zeros in the rejection band using upper mode cavities. Singlet block consisted of a resonators working in the TE201 was coupled to the source and load to produce the bypass couplings, and thus, transmission zeros.. This method can realise filters having transmission zeros equal to the number of reflection zeros. An example of a 3rd order filter has been designed at 30.75 GHz center frequency with NTZs above the passband. In [8], an integrated ceramic waveguide bandpass filter with N+1 finite transmission zeros has been designed by using the triplet technique. The inductive direct coupling has been realized by introducing a post between cascade resonators in order to produce TZs above the passband. Examples of TZs below the passband has been shown by introducing a capacitive cross-coupling between the first and third resonators by using partial metallic posts. The filter has been designed to operate at 1.7 GHz center frequency.
This paper presents a method to control the position of TZs in the rejection band based on a triplet routing scheme. The filter has been implemented in a rectangular waveguide technology. The filter configuration consists of cavity resonators having a conductivity of 5×10 7 S⁄m. The external couplings are achieved by metallic probes positioned at the first and last resonators. The internal couplings are achieved by metal posts positioned half-wavelength apart. The BPF was designed at 28 GHz with bandwidth(BW) of 500 MHz. The proposed filter is suitable for 5G applications.

DESIGN PROPOSED CAVITY RESONATOR
The cavity resonator is a section of rectangular waveguide has been short-circuited from both sides, as shown in Fig. 1. The frequency of resonator depends on the dimensions (width, height, length) of the rectangular cavity. The following dimensions are chosen for optimum out-of-band performance: a=2b= 7.122mm. The resonator operates at the dominant mode (TE101) having a frequency of 28 GHz. The length of resonator (d) can be calculated by equation [9] (1).
Where ε is the permittivity and μ is the permeability. l, n and m represent the halfwavelength variation of the electric field lines along the resonant dimensions: width (a), height (b), and resonant length (d). The resonant frequency for the dominant mode according to the change of the resonant length has been simulated by HFSS software as shown in Fig. 2. The relationship between the resonant frequencies and the length of the cavity is inversely proportional. The dominant frequency TE101 mode is a 28GHz occurs at d=8.1299mm. (2) = 0 + The effective wavelength is calculated:

EXTERNAL COUPLING
In order to achieve the external coupling in the rectangular waveguide filter, a coaxial probe is used, which is placed in the region where the electric field is most intense in the first and last resonators of the filter. The inner conductor of the coaxial probe acts as an antenna to excite the energy inside the waveguide. There are four variables to determine the specifications of the probe, which are the diameter of the probe, its distance from the outer edge of the resonator, the height of the probe inside the resonator, and its distance from the side wall of the resonator as shown in Fig. 3. The length and diameter of the feed probe are changed in order to reach the value of external coupling (Qe) through the following equation [2] = 0.5 * 0 ( ) * ( ) Where td is the group delay. The coupling factor was calculated with the change of the length and diameter of the probe as in Fig. 3. It is noted that the coupling factor is inversely proportional to the increase in the length and diameter of the probe. At a diameter of (r=0.5mm), the highest value of the external coupling factor when (Lf=0.25mm) is Qe=584.7 until quickly reaching the lowest value of Qe=38.06 when (Lf=1mm).

INTER RESONATOR COUPLING
The coupling between the resonators is achieved either by inductive coupling or capacitive coupling, both couplings were done by using posts. In general Chebyshev waveguide filters, the internal coupling is done by direct inductive couplings and other cross couplings are determined by specifying whether the TZ are above or below the pass band. If the TZ is the above of the pass band, then the cross coupling is done by inductive coupling.

Inductive coupling
For a rectangular waveguide filter, inductive internal coupling between the resonators can be achieved by placing circular post perpendicular to dimension (a) and parallel to dimension (b) of the waveguide. They are placed at intervals of half a wavelength, as shown in Where B is susceptance, λ g is e Guide wavelength and λ g0 is Guide wavelength at resonance frequency.

Capacitive coupling
This is achieved by using a partial post of diameter (r) perpendicular to dimension (a) and parallel to dimension (b) with depth (h) placed between two resonators of length, each of which is half-wavelength as shown in From the basic theory of circuits, the value of the capacitive can be written as [11]: It is also possible to calculate the value of the capacitance between two metal plates that has an area A separated by a distance of dsepration The dimensions of the inductive or capacitive posts are adjusted to achieve the inter-resonator coupling coefficients Mij.
The required coupling bandwidth between the filter resonators is calculated.
Fig . 6 shows the relationship of the inductive coupling bandwidth with the diameter of the post. It observed that, the coupling bandwidth is inversely proportional to the diameter of post. The largest simulated bandwidth has been recorded at r=0.25mm, while the coupling vanishes close to 1.5mm.

Design of 3rd order filter (TZ) below the passband
A third-order filter with a transmission zero below the passband was designed using a generalized Chebyshev rectangular waveguide with cross-coupling from resonant 1 to resonant 3 as shown in Fig. 8. Direct coupling between resonators 1 and 2 and between resonators 2 and 3 are achieved by placing a post in the intersection of the three resonators, with diameter (r) and depth (b) to provide inductive coupling between those resonators, while crosscoupling is achieved between resonators 1 and 3 By placing a post between these two resonators, with a diameter (r1) and a depth (h) for h<b, in order to achieve a capacitive coupling between the two resonators that works to achieve the transmission zero below the passband. The position of the (TZ) can be controlled by controlling the post diameter and depth. The filter specifications shown in the table below.  Fig. 9 Coronal shape of the coupling coefficients of a third-order generalized Chebyshev filter with (TZ) below the passband.
From the equations (1)-(4) and from the coupling coefficient was shown in Fig.(9). It can be calculated the length of each resonators according to European Standard WR28 (a=2b= 7.122mm). Also, the dimensions of inductive and capacitive couplings depending on the values that shown in Fig. 6 and Fig. 7.
The filter was simulated with the dimensions shown in table (2). The frequency response is shown in Fig. 10. The result shows that the return loss is 20dB at the passband and the bandwidth is 500MHz. The roll off speed on the side below the passband is large due to the presence of the (TZ) at the frequency of 27.1GHz.
To control the TZ position, the strength the cross-coupling bandwidth should be changed through change the dimeter (r) for post with optimize the other dimensions as shown in Fig. 11.

Design of 3 rd order filter (TZ) above the passband
A third-order filter with a TZ above the passband is designed using a generalized Chebyshev function in rectangular waveguide with cross-coupling between resonant 1 to resonant 3 as shown in Fig. 12.  13 Coronal shape of the coupling coefficients of a 3 rd order generalized Chebyshev filter with TZ above the passband. By using the equations (1)-(4) and the values of coupling in Fig. 13, the dimensions of filter can be determined as shown in table (3).  Fig. 14 shows the simulated frequency response of 3 rd order generalize Chebyshev BPF with a TZ above the passband. The results show that the return loss is 20dB at the passband and the bandwidth is 500MHz. The roll off at the above of the passband was large compared to the normal case due to a TZ at the frequency 29GHz. The position of the a TZ can be controlled by controlling the post diameter as shown in Fig. 15. The waveguide BPF filter is compared with other tlitertures as summarised in Table 4. It is shows that the proposed filter is competitive in terms of insetion loss and suppression.

CONCLUSIONS
The external couplings are realized by coaxial probes where the length of the probe determine the amount of coupling. The intra couplings of the filter have been realized by using inductive posts which is shorted between the top and bottom. A third order filter generalized Chebyshev waveguide was simulated by HFSS software. The center frequency and the bandwidth were 28GHz, 500MHz respectively which are meet the specification of 5G applications. TZ) position control has been achieved the change in the dimeter and height of post between the resonators 1 & 3.