Characteristics of QMSIR

Figure 2 shows the reported QMSIR with transmission line model (TLM). Its corresponding electrical lengths θ ₁, θ ₂, θ ₃, and θ ₄. The design conditions are assumed to be (θ ₁ + θ ₂) > (θ ₃ + θ ₄). The reported resonator is symmetrical, analysis based on even-/odd-mode method is employed. Its corresponding even-/odd-mode equivalent circuits are given in Fig. 3, respectively. At the resonance frequencies, the input impedance (Z _in) of each mode will be equal to zero. When plane A-A′ is applied the resonant frequencies can be derived as

(1)\begin{equation} \tan\theta_{2}\tan\theta_{1} - R_{\text{Z1}} =0\to f_{1}\end{equation}

(2)\begin{equation} R_{\text{Z1}}(\tan \theta_{4}+R_{\text{Z3}}\tan{\theta}_2)-\tan \theta_{1}(\tan \theta_{2}\tan\theta_{4}-R_{\text{Z3}})=0\to f_{2}\end{equation}

Figure 3. Even-/odd-mode equivalent circuits. (a) and (b) Odd-/even-mode equivalent circuit, respectively, when plane A-A′ is applied. (c) and (d) Odd-/even-mode equivalent circuit, respectively, when plane B-B′ is applied.

Similarly, when plane B-B′ is applied, the resonant frequencies are derived as

(3)\begin{equation}\tan\theta_{3}\tan\theta_{4} - R_{\text{Z1}} =0\to f_{3}\end{equation}

(4)\begin{equation}R_{\text{Z2}}(\tan \theta_{1}+R_{\text{Z4}}\tan{\theta}_3)-\tan \theta_{4}(\tan \theta_{1}\tan\theta_{3}-R_{\text{Z4}})=0\to f_{4}\end{equation}

where R _Z1 = Z ₁/Z ₂, R _Z2 = Z ₄/Z ₃, R _Z3 = Z ₄/Z ₂, and R _Z4 = Z ₁/Z ₃ are the impedance ratios. By defining δ ₁ = |f ₂ − f ₁| and δ ₂ = |f ₄ − f ₃|, the mode splitting is studied as shown in Fig. 4. As it can be seen in Fig. 4, δ ₁ decreases and δ ₂ increases as θ ₁/θ ₄ is increased. At θ ₁/θ ₄ equals zero, δ ₁ tends to large number and δ ₂ becomes zero. It suggests that the two resonant frequencies of a f ₃ and f ₄ will merge. At θ ₁/θ ₄ tends to large number, δ ₁ becomes zero and δ ₂ tends to large number. It suggests that two resonant frequencies of a f ₁ and f ₂ will merge. At point S, δ ₁ and δ ₂ are equal. By adjusting θ ₁/θ ₄, the quad-mode SIR can provide more freedom to exhibit different characteristics for quad-mode designs. The solutions for θ ₁, θ ₂, θ ₃, and θ ₄ are dependent on the choice of impedance ratios R _Z1 and R _Z2, and electrical ratios β ₁ = θ ₂/θ ₁ + θ ₂ and β ₂ = θ ₃/θ₃ + θ ₄. By properly choosing β ₁ and β ₂, θ ₂ and θ ₃ become a function of θ ₁ as illustrated in following expressions

(5)\begin{equation}\theta_{2}=\frac{\beta_{1}}{1-\beta_{1}}\theta_{1}\end{equation}

(6)\begin{equation}\theta_{3}=\frac{\beta_{2}}{1-\beta_{2}}\theta_{1}\end{equation}

Figure 4. Variation δ ₁ and δ ₂ with different electrical length ratio θ ₁/θ ₄ under the condition of θ ₂ = 11.6°, θ ₃ = 27.2°, R _Z1 = 5.95, and R _Z2 = 5.25.

By normalizing the f ₂ to the f ₁ and f ₄ to the f ₃, two normalized resonant frequencies of the QMSIR are obtained as

(7)\begin{equation}\frac{f_2}{f_1}=\frac{\theta_{1}+\theta_{2}+\theta_{4}}{\theta_{1}+\theta_{2}}\end{equation}

(8)\begin{equation}\frac{f_4}{f_3}=\frac{\theta_{1}+\theta_{3}+\theta_{4}}{\theta_{3}+\theta_{4}}\end{equation}

By introducing the electrical ratios β ₁, β ₂ and α = θ ₁/θ ₄ in equations (7) and (8), the normalized resonant frequencies f ₂/f ₁ and f ₄/f ₃ become the function of β ₁, β ₂, α, R _Z1, and R _Z2 as shown in Fig. 5. As it can be seen in Fig. 5, when the electrical lengths (θ ₁ and θ ₄) are fixed, varying the impedance ratios (R _Z1 and R _Z2) and electrical ratios (β ₁ and β ₂), the resonant frequency ratios (f ₂/f ₁ and f ₄/f ₃) decrease as the electrical length ratios (β ₁ and β ₂) and impedance ratios (R _Z1 and R _Z2) increase. Based on these characteristics of quad-mode SIR, the design procedures to obtain the quadruple-mode of quad-mode SIR are summarized as follows: (1) Fix the electrical ratio α. (2) Choose the electrical length ratios β ₁ and β ₂ according to the resonant frequency ratios (f ₂/f ₁ and f ₄/f ₃) and determine the impedance ratios (R _Z1 and R _Z2). (3) Fix the value of θ ₁ to compute θ ₂, θ ₃, and θ ₄ using equations (5), (6) and electrical length ratio α = θ ₁/θ ₄. Then the electrical length parameters are found.

Figure 5. Normalized resonant frequencies of QMSIR against impedance ratios: R _Z1 with varied β ₁ and R _Z2 with varied β ₂. α is fixed to 1.2.

Design of quad-band BPFs

Based on the above analysis, two second-order quad-band BPFs are designed according to the configuration and coupling routing topology shown in Fig. 6. Multiple coupling technique is employed to implement the four passbands. The couplings ${{ \theta }}_2^1$, ${\text{ Z}}_1^1$, and ${\text{ Z}}_1^1$ are used to generate the passbands f ₁ and f ₂ while the couplings ${\text{Z}}_2^1$, ${{\theta }}_4^1$, and ${{\theta }}_3^1$ are used to generate the passbands f ₃ and f ₄. By considering f ₁₁ = 2.5 GHz, f ₂₁ = 3.3 GHz, α = 1.2, and β ₁ = 0.1, with f ₂₁/f ₁₁ = 1.32 and R _Z11 = 3.3, and f ₃₁ = 5.2 GHz, f ₄₁ = 6.57 GHz, α = 1.2, and β ₂ = 0.2 with f ₄₁/f ₃₁ = 1.26 and R _Z21 = 5.1 for the first quad-band BPF, and f ₁₂ = 3.85 GHz, f ₂₂ = 4.65 GHz, α = 1.2, and β ₁ = 0.1 with f ₂₂/f ₁₂ = 1.21 and R _Z12 = 7.9, and f ₃₂ = 5.6 GHz, f ₄₂ = 7.2, α = 1.2, and β ₂ = 0.2 with f ₄₂/f ₃₂ = 1.28 and R _Z22 = 5.5 for the second quad-band BPF, the initial electrical length parameters of the two quad-band BPFs are determined as: (1) for the first quad-band BPF, by taking $\theta _1^1 = 90^\circ $ with $R_{{\text{Z1}}}^1 = 3.3\Omega $, the initial values are $\theta _2^1 = 10^\circ $, $Z_1^1 = 87.5\Omega $, $Z_2^1 = 26.5\Omega $, $\theta _4^1 = 75^\circ $ and $\theta _3^1 = 18.75^\circ $, ${\text{ Z}}_1^1 = 105.8\Omega $ and ${\text{ Z}}_2^1 = 20.75\Omega $. (2) For the second quad-band BPF, by taking $\theta _1^2 = 87.3^\circ $, $R_{Z1}^2 = 7.9\Omega $, the initial values are ${{ \theta }}_2^2 = 9.7$, $Z_1^2 = 82.99\Omega $, $Z_2^2 = 10.5\Omega $, ${{ \theta }}_4^2 = 72.75$, ${{ \theta }}_3^2 = 18.18$, ${\text{ Z}}_1^2 = 114.61\Omega $, and $Z_2^2 = 20.83\Omega $. The required coupling coefficients and the external quality factors for BPF I and II are listed in Table 1. Figures 7 and 8 show the coupling coefficient and external quality factor for four passbands of the BPFI and BPFII, respectively. The coupling coefficients and external quality factors are determined by the parameters of the resonators and feeding lines. As it can be seen in Fig. 7, the coupling coefficient decreases as the spacing S ₁ and S ₂ increase. At a fixed coupled length L _C3, the stronger the coupling leads to mode splitting when the spacing S ₁ is smaller as shown in Fig. 7(a) and (c). The larger spacing S ₁ makes proper coupling for the passbands corresponding to ${\text{ M}}_{12}^{1,{\text{I}}}$, ${\text{ M}}_{12}^{2,{\text{I}}}$, ${\text{ M}}_{12}^{1,{\text{II}}}$, and $M_{12}^{2,{\text{II}}}$. At a fixed length L _C4, the weaker the coupling leads to mode splitting when the spacing S ₂ is larger, as shown in Fig. 7(b) and (d). The small spacing S ₂ makes proper coupling for the passbands corresponding to ${\text{ M}}_{12}^{3,{\text{I}}}$, $M_{12}^{4,{\text{I}}}$, ${\text{ M}}_{12}^{3,{\text{II}}}$, and${\text{ M}}_{12}^{4,{\text{II}}}$. As it can be observed in Fig. 8, by setting the widths of feeding lines for BPFI and BPFII to 0.3 mm and 0.2 mm, respectively, Q _e for four bands will be increased as the coupled lengths L _C1 and L _C2a increase. For the coupled length L _C2b, Q _e for passband 3 will be increased as the L _C2b increases while Q _e for passband 4 will be decreased as L _C2b increases. As shown in Fig. 9, simulation results of the two quad-band BPFs are obtained. The simulation of the two quad-band BPFs were carried out using Taconic RF-35(tm) tangent of 0.0018. The simulation results for the two substrate with a thickness of 0.508 mm, relative dielectric constant of 3.5, and loss tangent of 0.0018. The simulation results for the two quad-band BPFs show that each quad-band BPF presents four passbands and eight transmission zeros. According to the standard design procedure given in paper [Reference Denis, Song and Zhang16], the four transmission zeros (TZ2, TZ3, TZ4, and TZ5) are created due to the coupling coefficients MSL1 and MSL2 while the four transmission zeros (TZ6, TZ7, TZ8, and TZ9) are created due to the coupling coefficients MSL3 and MSL4. The extra transmission zero TZ1 in lower stopband is created due to S ₄ and L ₅. It is easy to see that around 7.5 GHz, BPF1 has an excess parasitic passband, since the proposed four-bandpass filter BPF1 is based on a cascade of two QMSIRs, it may generate a parasitic passband due to inappropriate size design. However, the filter size is optimized based on the frequency of the four passbands, so the parasitic passband of BPF1 is difficult to eliminate. However, the parasitic passband is eliminated after the cascade of two four-BPFs, mainly because the passband is slightly offset after the cascade, and the overall size is optimized again, thus eliminating the excess parasitic passband.

Figure 6. (a) Configuration quad-band BPF using TLM and (b) coupling schematic topology of quad-band BPF.

Figure 7. Coupling coefficient BPF I and II. (a) and (c) are coupling coefficients for passband 1 and 2. (b) and (d) are coupling coefficients for passband 3and 4.

Figure 8. External quality factors of BPF I and BPF II (a) for passband 1 and 2, (b) for passband 3 and 4 with L _c2a, and (c) for passband 3 and 4 with L _c2b.

Figure 9. Simulated quad-band bandpass filters (a) for BPF I and (b) for BPF II.

Table 1. Design coupling coefficients and external quality factors for BPF I and BPF II

Design of eight-channel diplexer

Based on the results from the simulation of the two quad-band BPFs, an eight-channel diplexer is realized as shown in Fig. 1(a). The proposed diplexer consists of a distributed coupling feeding line (port 1), two pairs of resonators (R ₁ to R ₄), and output feeding lines (port 2 and port 3). Basically, there are two unit cells in the proposed diplexer, and each unit cell is a quad-band BPF. The center frequencies of the signals filtered out by each output port are 2.5/3.3/5.2/6.57 GHz and 3.9/4.65/5.6/7.2 GHz for port 2 and port 3, respectively. Each four channels are controlled by a quad-band BPF. The parameters of the proposed diplexer are shown in Table 2. More precisely, design procedures for the proposed diplexer are given as: (1) two quad-band BPFs are designed according to the given passband’s specifications. (2) By properly locating the resonators with respect to the distributed coupling feeding line and output feeding lines, the harmonic passband of each quad-band BPF is effectively suppressed to achieve good isolation. (3) Design is finished with some final adjustment in a full-wave simulation. The electrical lengths and impedances of the four parts of the microstrip resonators become asymmetrical after final adjustment to get a good match for the diplexer.

Table 2. Circuit dimensions for diplexer