Analysis and design of the quadplexer

The structure of the presented miniaturized high-isolation quadplexer has been shown in Fig. 1. It consists of a common multimode resonator and four channels using stepped impedance resonators (SIRs). The multimode resonator with an open-circuit branch and a short-circuit branch is adopted as a common resonator to replace matching network, which is good for miniaturization. The four bandpass filters using the structures of SIRs are coupled with the common multimode resonator and the SIRs are curved in order to get compact circuit size. It can be seen from Fig. 1 that the two passbands (port 2 and port 5) at the bottom of the circuit are realized by a short-stub loaded quarter-wavelength SIR and an open-stub loaded half-wavelength split-ring SIR, respectively. The other two passbands (port 3 and port 4) are realized by open-loop SIRs, which are coupling fed at the input feeder. The use of common resonator and the curved SIRs can further reduce the size of the circuit, making the overall circuit very compact.

Fig. 1. Structure of the proposed quadplexer.

Figure 2 illustrates the circuit topology of the quadplexer. The circle represents the resonator, and the solid line represents the coupling path between the resonators. The multimode resonator is coupled with the resonators of the four channels, respectively. It is obvious that the use of multimode resonator as the common resonator not only eliminates redundant resonators, but also eliminates additional matching circuits, which is helpful to achieve the miniaturization of the circuit.

Fig. 2. The circuit topology of the quadplexer.

The design procedure of the presented quadplexer can be summarized as three steps. Firstly, the common multimode resonator with an open-circuit branch and a short-circuit branch for desired operating frequencies is obtained by adjusting the lengths of the branches. Secondly, the two channels (port 2 and port 5) coupled with the short-circuit and open-circuit branches are realized by SIRs, and the central frequencies and relative bandwidths are 1.8/3.5 GHz and 5.1/5.7%, respectively. Finally, the other two channels (port 3 and port 4) are added at the input feeder by the coupling feed technology, and the central frequencies and relative bandwidths are 2.4/2.8 GHz and 2.6/2.5%, respectively. Figure 3 shows the structure of the stub-loaded multimode resonator, which is composed of an open-circuit stub and a short-circuit stub. The characteristic impedances are Z ₁, Z ₂, Z ₃, and the electrical lengths are θ ₁,θ ₂,θ ₃, respectively, as shown in Fig. 3. The input impedance of the resonator can be achieved:

(1)$$Y_{in} = \displaystyle{{\tan \theta _1-T_1\cot \theta _2 + T_2\tan \theta _3} \over {Z_1\lpar T_1\cot \theta _2\tan \theta _1-T_2\cot \theta _3\tan \theta _1\rpar }}.$$

Fig. 3. The structure of the stub-loaded multimode resonator.

When Y_in = 0, the resonant condition can be achieved:

(2)$$\tan \lpar \theta _1 \times R_n\rpar -T_1\cot \lpar \theta _2 \times R_n\rpar + T_2\tan \lpar \theta _3 \times R_n\rpar = 0\comma \;$$

where T ₁ = Z ₁/Z ₂, T ₂ = Z ₁/Z ₃, and R_n is the ratio of nth harmonic frequency f_n to fundamental frequency f ₀. Then, the fundamental frequency f ₀ and the first harmonic frequency f ₁ of the dual-mode resonator can be adjusted by changing the lengths of the short-circuit stub l ₂ and open-circuit stub l ₃, respectively. In addition, the resonant frequencies of the short-circuit branch and open-circuit branch are 1.8 and 3.5 GHz, respectively.

The relationship between resonant frequencies and lengths of loaded stubs is given by Fig. 4 (where T ₁ = Z ₁/Z ₂ = 32/41 = 0.78, T ₂ = Z ₁/Z ₃ = 32/84 = 0.38, l ₁ = 12.5 mm). It is obvious that the frequency f ₀ can be adjusted by changing the length of shorted branch l ₂ and there is almost no effect on f ₁. The frequency f ₀ will be smaller with the increase of l ₂. In addition, the frequency f ₁ can be adjusted by changing the length of open stub l ₃ without any influence on f ₀, and f ₁ will be smaller with the increase of l ₃. Thus, it is flexible to get desired operating frequencies by adjusting the lengths of the short-circuit stub l ₂ and open-circuit stub l ₃, respectively. In the proposed circuit, the resonant frequencies of the short-circuit branch and open-circuit branch are 1.8 and 3.5 GHz, respectively. It is also easy to achieve the desired return losses at the different channels by changing the widths of the short-circuit stub and open-circuit stub separately, because the width of the microstrip is related to the characteristic impedance of the microstrip.

Fig. 4. The relationship between resonant frequencies and lengths of loaded stubs. (a) Frequency f ₀ and f ₁ with the length of l ₂. (b) Frequency f ₀ and f ₁ with the length of l ₃.

Figure 5 gives the structures of a stepped-impedance resonator and the coupling feed topology, and the SIR is curved in the proposed quadplexer to miniaturize the circuit. As illustrated in Fig. 5 (a), Z ₁ and Z ₂ are the characteristic impedances of the high-impedance and low-impedance line, respectively, and θ ₁ and θ ₂ are the electrical lengths of the high-impedance and low-impedance line, respectively. The total electrical length of the SIR is θ _t. The input impedance Z_in of the SIR is derived as

(3)$$Z_{in} = jZ_2\displaystyle{{2T-0.5\lpar \cot \theta _2-\tan \theta _2\rpar \lpar \cot \theta _1-\tan \theta _1\rpar } \over {T\lpar \cot \theta _2-\tan \theta _2\rpar + \lpar \cot \theta _1-\tan \theta _1\rpar }}\comma \;$$

where T is the impedance ratio, which is defined as T = Z ₂/Z ₁. The resonance occurs whenZ _in = ∞, the resonant condition can be achieved:

(4)$$T(\cot {\theta _2} - \tan {\theta _2}) + (\cot {\theta _1} - \tan {\theta _1}) = 0.$$

Fig. 5. The structures of stepped-impedance resonator and the coupling feed topology. (a) Stepped-impedance resonator. (b) Coupling feed topology.

The length ratio β of the SIR is defined as β = 2θ ₂/(θ ₁ + 2θ ₂) = 2θ ₂/θ _t. In the proposed circuit, the symmetrical SIRs are adopted, so the length ratio β is 0.5. The fundamental frequency f ₀ is determined by the electrical lengths of the high-impedance and low-impedance line. In the proposed circuit, the electrical lengths of the high-impedance and low-impedance line θ ₁ and θ ₂ are both quarter-wavelength, which satisfies the resonant equation (4). Thus, the length of the open-loop SIR is half-wavelength. Figure 5(b) shows the coupling feed topology of the circuit, and it is obvious that the external quality factor is related to the gaps S ₁ and S ₂, and the desired return loss can be acquired by selecting appropriate gaps S ₁, S ₂ and appropriate widths of the resonator.

The channels (port 2 and port 5) coupled with the short-circuit and open-circuit branches are realized by SIRs, and the central frequencies and relative bandwidths are 1.8/3.5 GHz and 5.1/5.7%, respectively. The shorted quarter-wavelength SIR and open-loop half-wavelength SIR are used for the first and the second passbands, respectively. Based on the circuit performance index above, the external quality factor Q_e and coupling coefficient K of each channel filter can be obtained by the synthesis method in the filter design software, and the external quality factor and coupling coefficient of the presented circuit are shown in Table 1. After that, the next step is to extract external quality factor and coupling coefficient of the simulated circuit model in HFSS, which can be used to determine the physical sizes of the filters.

Table 1. External quality factors and coupling coefficients

Figure 6 shows the relationship between Q_e, l ₀, and g ₀, and the external quality factor Q_e of the common resonator is mainly related to the length of coupling line l ₀ and the coupling slot g ₀. In addition, Q_{e 0} and Q_{e 1} are the first- and second-channel external quality factors, respectively. The coupling coefficients between shorted/open-loop SIRs and common resonator are shown in Fig. 7. It can be seen that the coupling coefficient is mainly related to the coupling gap (g_a and g_b). Through the extraction of the external quality factor and the coupling coefficient, the initial values of the corresponding parameters can be determined: l ₀ = 11 mm, g ₀ = 0.15 mm, g_a = 0.95 mm, g_{b 1} = 0.25 mm, g_{b 2} = 0.4 mm. Finally, by optimizing the sizes of the circuit based on the above initial values of the corresponding parameters, the transmission response of the channel 1 and channel 2 is given by Fig. 8, and it can be seen that the performance of the circuit meets the requirements, verifying the correctness of the design.

Fig. 6. Relationship between Q _e, l ₀, and g ₀.

Fig. 7. Coupling coefficient K. (a) Coupling coefficient between shorted SIR and common resonator. (b) Coupling coefficient between open-loop SIR and common resonator.

Fig. 8. Transmission response of channel 1 and channel 2.

Based on the above circuit, the other two channels (port 3 and port 4) are added at the input feeder by the coupling feed technology, and the central frequencies and relative bandwidths are 2.4/2.8 GHz and 2.6/2.5%, respectively. The two open-loop half-wavelength SIRs are used for the third and the fourth passbands, respectively. The design procedure of the added channels is similar to that of the above circuit. Firstly, the external quality factor and coupling coefficient of each channel can be obtained by synthesis method, and the external quality factor and coupling coefficient of the presented circuit are shown in Table 1. Secondly, the initial values of the physical sizes can be obtained by extracting the external quality factor and coupling coefficient. Finally, the final physical sizes of the quadplexer can be determined by optimization.

The extracted external quality factors of the expanded channels are shown in Fig. 9, and Q_{e 3} and Q_{e 4} are the third- and fourth-channel external quality factors, respectively. Similarly, the coupling coefficients of the expanded channels are shown in Fig. 10, where K ₃ is the coupling coefficient between channel 3 and the common resonator, and K ₄ is the coupling coefficient between channel 4 and the common resonator. By extracting the external quality factor and the coupling coefficient, the initial values of the circuit can be determined: gc ₁ = gc ₃ = 0.15 mm, gc ₂ = 0.5 mm, gd ₁ = gd ₃ = 0.1 mm, gd ₂ = 0.5 mm.

Fig. 9. External quality factors of the expanded channels.

Fig. 10. Coupling coefficients of the expanded channels.

Finally, the transmission response of the channel 3 and channel 4 is given by Fig. 11, and it can be seen that the two extra channels added at the input feeder have little effect on the existing channels. In other words, the implementation of the common input will not change the characteristics of the channel filters. In addition, the performance of the circuit meets the requirements, verifying the correctness of the design. Therefore, the quadplexer can be realized by adopting this structure.

Fig. 11. Transmission response. (a) Triplexer. (b) Quadplexer.

Implementation and measurements

Through the above analyses, the compact high-isolation quadplexer is designed and fabricated with the substrate Taconic RF-35. The related parameters of this substrate are as follows: dielectric constant ε_r of 3.5, a thickness of 0.508 mm, and a loss tangent of 0.0018. The structure is simulated and optimized in Ansys-HFSS. Table 2 shows the final physical sizes of the presented quadplexer circuit. Figure 12 shows the fabricated miniaturized high-isolation quadplexer. The total size of the fabricated quadplexer is 0.36 λg × 0.42 λg. What's more, all the ports are connected by the type-SMA connectors.

Fig. 12. Photograph of the fabricated quadplexer.

Table 2. The final physical sizes of the presented quadplexer circuit (unit: mm)

The simulated and measured results of the proposed quadplexer are shown in Fig. 13. The measured central frequencies and 3 dB relative bandwidths of the quadplexer are 1.8/2.4/2.8/3.5 GHz and 5.1/2.6/2.55/5.74%, respectively. The measured return losses of four passbands are all greater than 12 dB, as shown in Fig. 13(a). Moreover, Fig. 13(b) shows that the measured insertion losses of the four passbands are 2.4/2.6/2.8/2.5 dB, respectively, including the losses of connectors and the error of the fabrication. The measured isolation of the entire frequency band is all greater than 40 dB, as shown in Fig. 13(c). Figure 14 shows the measured wideband response of the proposed quadplexer. It can be seen that the 20 dB upper stopband of the presented quadplexer is up to about 12 GHz. In addition, the measured results show a reasonable agreement with the simulated ones over the operating frequency range. Table 3 gives a comparison with some prior works. It can be seen that the proposed quadplexer has the advantages of compact size, wide out-of-band rejection, high isolation, low insertion loss, and good impedance matching.

Fig. 13. Simulated and measured results. (a) Return loss. (b) Insertion loss. (c) Isolation.

Fig. 14. Wideband response of the proposed quadplexer.

Table 3. The comparison with some prior works