Circular SIW cavity resonator

The basic dual-mode resonator in a circular SIW cavity is based on the utilization of two degenerate TM₁₁₀ modes which are perpendicular to each other. The resonant frequencies of two degenerate TM₁₁₀ modes in a circular cavity can be calculated by [Reference Pozar15].

(1)$$f_{TM_{110}} = \displaystyle{{0.610c} \over {R\sqrt {\mu _r\varepsilon _r} }}$$

where c is the velocity of light in the vacuum, R is the equivalent radius of the SICC, μ_r and ε_r are relative permeability and permittivity of the substrate, respectively. Figure 1 shows the electric field distribution of the TM₁₁₀-similar modes in a SICC using the full wave simulator Ansoft HFSS.

Fig. 1.

Electric field distributions of TM₁₁₀ degenerate modes in a circular SIW cavity simulated by full wave simulation software Ansoft HFSS.

Directional filter design

Figure 2 shows the layout of the proposed DF based on the SICC resonators, where port 1, 2, 3, and 4 are the input, through, coupled, and isolated port, respectively. At each port, the microstrip is connected with the HMSIW via a taper-shape transition [Reference Deslandes and Wu6]. The coupling structure between the HMSIW section and the circular SIW cavity consists of one continuous aperture, and a waveguide step is used to improve matching at the operating frequency band.

Fig. 2.

Layout of the proposed two-circular cavity SIW DF.

The design parameters for DF are specified in terms of the bandpass frequency response at the coupled port. This bandpass response is related to the coupling coefficients k_{i ,i+1} between the resonators, external quality factor Q_e, and the response of a low-pass prototype filter. Hence, the element values of the low-pass prototype filter should be chosen first. Then, the design parameters are obtained by [Reference Matthaei, Young and Jones16]:

(2)$$\lpar Q_e\rpar _A = {{g_0g_1{{\omega }^{\prime}}_1} / W}$$

(3)$$K_{i\comma i + 1}\vert {_{i = 1\;to\;n} } = {W / {{{\omega }^{\prime}}_1\sqrt {g_ig_{i + 1}} }}$$

(4)$$\eqalign{& \lpar Q_e\rpar _B = {{{{\omega }^{\prime}}_1g_ng_{n + 1}} / W} \cr & W = \lpar \omega _2-{{\omega _1\rpar } / {\omega _0}} \cr & \omega _0 = {{\lpar \omega _1 + \omega _2\rpar } / 2}} $$

where w′ ₁ is the cut-off frequency of the low pass filter, (Q_e)_A and (Q_e)_B are the external quality factors of the resonators at the input and output, respectively. λ′_g is the wavelength of the guided wave and w ₀ is the center frequency.

(5)$$C_{01} = {1 / {\sqrt {\displaystyle{{{\lpar Q_e\rpar }_A} \over {2\pi n}}{\lpar \displaystyle{\lambda \over {{{\lambda }^{\prime}}_g}}\rpar }^2 + \displaystyle{1 \over 2}} }}\,$$

(6)$$ {C_{i\comma i + 1}} \vert _{i = 1\,to\,n-1} = n\pi K_{i\comma i + 1}\lpar \displaystyle{{{{\lambda }^{\prime}}_g} \over \lambda }\rpar ^2$$

(7)$$C_{n\comma n + 1} = {1 / {\sqrt {\displaystyle{{{\lpar Q_e\rpar }_B} \over {2\pi n}}{\lpar \displaystyle{\lambda \over {{{\lambda }^{\prime}}_g}}\rpar }^2 + \displaystyle{1 \over 2}} }}$$

where (C_{i ,i+1})² is the power coupling factor, n is the number of half wavelengths in the resonators, and ω ₁, ω ₂ are the low and high cut-off frequencies. The quality factor is inversely proportional to BW. Then we should optimize the physical circuit to match the required coefficients and the dimensions of the DF can be determined. The proposed DF has three couplings of which two are identical, the bandwidth of the bandpass can be adjusted slowly. The maximally flat low-pass prototype parameters are g ₀ = g ₃ = 1 and g ₁ = g ₂ = 1.4142. Having obtained the low-pass parameters, we can obtain the design parameters for the SICC DF with the center frequency at f = 12.3 GHz and the bandwidth of BW = 400 MHz as follows: C ₀₁ = C ₂₃ = 0.94, and C ₁₂ = 0.7934. As shown in Fig. 2, the proposed SICC DF is formed by two HMSIW sections and two SICC resonators. The coupling coefficient between two circular SIW cavities can be controlled by using two identical slotlines etched on the top and bottom planes. Figure 3 shows the simulated frequency response of the designed DF with different values of L_s. It can be seen that the bandwidth of the bandpass can be adjusted slowly by altering L_s. BW changed about 110 MHz for ΔL _s = 0.4 mm. Finally, a fine-tuning procedure is often used to optimize the entire structure with Ansoft HFSS software. Power incident at port 1 emerges from port 3 and it looks like a band-pass filter and remaining power emerges port 2, looks like a band stop filter.

Fig. 3.

Simulated S-parameters of the proposed DF for different values of L_s.

The E-field distributions of the optimized structure are shown in Fig. 4 at different frequencies. Figure 4(a) shows the distribution of E-field in the pass-band (at 12.3 GHz). According to Fig. 4(a), the input power at port 1 is transmitted to port 3. Figure 4(b) shows the distribution of the E-field out of band (at 13.3 GHz), that is given the input power at port 1 is not transmitted to port 3.

Fig. 4.

Electric field distribution for the (a) resonant frequency of SICC at 12.3 GHz and (b) far away from the resonant mode at 13.3 GHz.