Hemispherical shunt resonators

As shown in Table 1, the HCR has the highest Q/volume, and the first spurious resonance is farther than other resonators. In addition, the rational configuration of HCRs may lead to a more compact filter structure. In this paper, the in-band zero method is proposed to design a dual-band filter [Reference Gao, Zhou and Huang18, Reference Vaitukaitis, Rao, Nai and Hong19]. It can be realized through a compound resonator topology (CRT). In Fig. 1(a) and (b) [Reference Macchiarella and Tamiazzo20], two HCRs are face-to-face parallel connected through a coupling iris, which presents two parallel and one series resonances. It can be used to realize a dual-frequency filter. In this compound resonator coupling topology, it is possible to realize N transmission zeros in each guard-band, where N is the order of each passband [Reference Vaitukaitis, Nai and Hong21].

Table 1.

Comparison of four types of cavity resonator

Figure 1.

The compound resonator. (a) Compound resonator implemented by HCRs. (b) Prototype circuit of compound resonator [Reference Macchiarella and Tamiazzo20].

Dual-band filter design

Based on the above-mentioned analysis, a compact dual-band hemispherical cavity BPF is proposed by using three CRTs. The BPF operates at 9.1 and 9.77 GHz, with respective bandwidths of 210 and 200 MHz, and a passband return loss (RL) of less than 20 dB. The 3D geometric configuration of the proposed dual-band filter is shown in Fig. 2, and the filter’s physical dimensions are summarized in Table 2. It mainly consists of three sets of CRTs, four rectangular and three pairs of semi-cylindrical coupling irises. The coupling of three CRTs is achieved through rectangular iris structures, and the coupling strength is dependent on the iris’s length and width. In the CRTs, the coupling is realized by a pair of semicircular iris components; thus, the coupling value is decided by the irises’ radius and the spacing between them.

Figure 2.

Geometric configuration of the dual-band BPF.

Table 2.

Main parameters of 3 order dual-band filter

The coupling topology is shown in Fig. 2. The calculated N + 2 coupling matrix [Reference Brand, Meyer and Geschke22]:

(1)\begin{equation}\left[ {\begin{array}{*{20}{c}} 0&{0.7412}&0&0&0&0&0&0 \\ {0.7412}&{ - 0.0306}&{0.4830}&0&{0.7276}&0&0&0 \\ 0&{0.4830}&{ - 0.0306}&0&0&{0.7276}&0&0 \\ 0&0&{0.4830}&{ - 0.0306}&0&0&{0.7276}&{0.7412} \\ 0&{0.7276}&0&0&{0.0576}&0&0&0 \\ 0&0&{0.7276}&0&0&{0.0576}&0&0 \\ 0&0&0&{0.7276}&0&0&{0.0576}&0 \\ 0&0&0&{0.7412}&0&0&0&0 \end{array}} \right]\end{equation}

In the Electromagnetic (EM) design, the external quality factor (Qe) is controlled by the area of the coupling apertures (ts $ \times $hs), while the coupling coefficient (M) between cavities is tuned by varying the size of the coupling iris (t $ \times $h). The Qe and M values for the filter can be extracted using the following equations [Reference Hong and Lancaster23]:

(2)\begin{equation}M = \frac{{f_2^2 - f_1^2}}{{f_2^2 + f_1^2}}\quad \end{equation}

(3)\begin{equation}{Q_e} = \frac{{\pi {f_{{\tau _{max}}}} \cdot {\tau _{max}}}}{2}\end{equation}

where ${f_1},{f_2},{\tau _{max}}$, and ${f_{{\tau _{max}}}}\,$are the resonant frequencies, the maximum group delay, and the frequency at ${\tau _{max}}$, respectively. The targeted Electromagnetic simulations were conducted to determine the required Q_e and coupling coefficients M. Figure 3 displays the extracted Q_e and M under different design parameters. Figure 3(a) exhibits M₁₂ and M₂₃ decline with increasing of h while increase with increasing of t. Similarly, as shown in Fig. 3(b), M₁₄, M_25, and M₃₆ increase with increasing radius of semi-cylindrical coupling iris (vr) and length of two semi-cylindrical (vd). Figure 3(c) and (d) demonstrate an inverse correlation between Q_e and rectangular coupling window dimensions, with Q_e decreasing as length (ts) and width (hs) increase.

Figure 3.

Coupling coefficients and external quality factor under different parameters. (a) Lengths t (h). (b) Lengths vr (vd). (c) Lengths ts (hs). (d) Lengths hs (ts).

As shown in Fig. 4, the matrix and EM simulations are in good agreement. There are two passbands centered at 9.1 and 9.77 GHz, with in-band return losses below −20 dB and −30 dB, respectively. Due to the CRT, three TZs are introduced between the two passbands.

Figure 4.

S-parameters of extracted coupling matrix and simulation.

Enhanced out-of-band rejection

The initial design demonstrates an ideal in-band performance, but its out-of-band rejection still needs improvement. Introducing TZs is an effective strategy to improve out-of-band rejection. Unlike [Reference Vaitukaitis, Nai and Hong13, Reference Vaitukaitis, Rao, Nai and Hong19], as shown in Fig. 5(a) and (b), a novel resonant discontinuity in hemispherical resonators is proposed, it consists of a metal pillar, which not only can place the transmission zero very close to the passband but also much smaller.

Figure 5.

HCR with Metal Pillar Incorporation. (a) Main view. (b) Side view.

The electric field and magnetic field energy density distribution of this resonant discontinuity structure at the resonant frequency is shown in Fig. 6(a) and (b). The structure’s electric field energy is concentrated at the front end of the metal column, which can be equivalent to a capacitor; the magnetic field energy is concentrated at the tail end of the metal column, which can be equivalent to an inductor. From a circuit perspective, therefore, the resonant discontinuity can be represented by Fig. 7. The resonant frequency f_z in the figure can be obtained using the following formula:

(4)\begin{equation}{X_s}\left( f \right) = - Im\left\{ {\frac{1}{2}\frac{{{S_{12}}}}{{{S_{11}}}}} \right\},{X_s}\left( {{f_z}} \right) = 0\end{equation}

Figure 6.

Energy density of the resonant discontinuity structure. (a) The electric field. (b) The magnetic field.

Figure 7.

The circuit of the resonant discontinuity structure.

where S₁₁ and S₂₁, respectively, represent the return loss and insertion loss in the filter’s transmission response. When the length d of the metal pillar in Fig. 5 is changed, the equivalent capacitance value and equivalent inductance value in Fig. 7 will also change. Therefore, the resonant frequency of this structure can be adjusted by modifying the length d of the metal pillar, and the position of the transmission zero introduced by this structure will shift accordingly. This change is clearly shown in Fig. 8(a) and (b).

Figure 8.

Effect of the metal pillar length d on TZ. (a) The left side of the passband. (b) The right side of the passband.

Based on the above, it optimizes the existing dual-band filter design and proposes the HCR dual-band filter with enhanced selectivity shown in Fig. 9. The original filter is modified by adding two pillars in cavity 1 and one in cavity 3. By optimizing the position and length of the three metal pillars, two TZs appear in the upper stopband and one in the lower. As shown in Fig. 10, the improved design demonstrates better S11 in-band, and its out-of-band rejection is also significantly improved.

Figure 9.

Geometric configuration of the optimized dual-band filter. (a) Main view. (b) Section view.

Figure 10.

S-parameters of improved filter and unimproved.