Characteristics of the SIR and UIR

Based on the selected F4B substrate, this section begins by introducing the SIR and UIR. The structure of the SIR is depicted in Fig. 2, where θ_i and Y_i (i = 1, 2) represent the electrical length and characteristic admittance of the microstrip line, respectively. According to transmission line theory, the input admittance of SIR is

(1)\begin{equation}{Y_{{\text{in}}}} = \frac{{2{\text{j}}{Y_2}({R_{\text{z}}}\tan {\theta _1} + \tan {\theta _2})({R_{\text{z}}} - \tan {\theta _1}\tan {\theta _2})}}{{{R_{\text{z}}}(1 \!- {{\tan }^2}\,{\theta _1})(1 \!- {{\tan }^2}\,{\theta _2}) - 2(1 - R_{\text{z}}^2)\tan{\theta _1}\tan {\theta _2}}}\end{equation}

where R _z is the impedance ratio, R _z = Y ₁/Y ₂. When R _z = 1, it is UIR. When Y _in = 0, the odd- and even-mode resonance conditions can be derived as

(2)\begin{equation}\tan\left[\left(1-a\right)\left(\theta_1+\theta_2\right)\right]\tan\left[\alpha\left(\theta_1+\theta_2\right)\right]=R_2\;\;\left(\mathrm{Odd}-\mathrm{mode}\right)\end{equation}

(3)\begin{equation}R_ztan\left[\left(1-\alpha\right)\left(\theta_1+\theta_2\right)\right]+\tan\left[a\left(\theta_1+\theta_2\right)\right]=0\;\;\left(\mathrm{Evee}-\mathrm{mode}\right)\end{equation}

Figure 2.

The structure of the stepped impedance resonator.

where α is the electrical length ratio, α = θ ₂/(θ ₁ + θ ₂). From Equations (2) and (3), it can be analyzed that for the SIR, both the odd- and even-mode resonance conditions are related to θ_i, α, and R _z. As illustrated in Fig. 2, the solid and dashed lines indicate the voltage distribution of the SIR under odd- and even-mode excitation, respectively, and the symmetry plane MN behaves as a shorted circuit under odd-mode excitation and an opened circuit under even-mode excitation, which ensures that the dominant resonance is excited exclusively by the DM signal and effectively suppresses the CM noise.

The SIR has an infinite number of resonant modes. The fundamental resonance occurs in the odd-mode, the second resonance in the even-mode, and higher-order resonances in that order. Here we discuss the first five resonance modes in detail. Figure 3 plots the variation of the f _o1 (the first-order odd-mode resonant frequency of the SIR) with the physical dimension L _t (L _t is the total physical length corresponding to the electrical length 2θ ₁ + 2θ ₂) when R _z is different. The results indicate that f _o1 gradually decreases as L _t increases, and when f _o1 is identical, the overall physical dimension decreases as the R _z decreases. Accordingly, the stepped impedance structure facilitates a compact design [Reference Chen, Tseng, Su, Li, Wang and Li6, Reference Roshani and Roshani7]. Figure 4 depicts the variation of the resonant frequencies ratio with α when R _z is different; it suggests that when R _z < 1, with the increase of α, the f _e1/f _o1 increases gradually, the f _o2/f _o1 and f _e2/f _o1 initially increase and then decrease, and the f _o3/f _o1 first increases, then decreases, and then increases again. Here, f _e1 represents the first-order even-mode resonant frequency of the SIR, f _o2 represents the second-order odd-mode resonant frequency of the SIR, f _e2 represents the second-order even-mode resonant frequency of the SIR, and f _o3 represents the third-order odd-mode resonant frequency of the SIR. And these high-order resonant frequencies increase as R _z decreases, as shown in Fig. 4. That is to say, by adjusting the R _z and α, the higher-order harmonics of SIRs can be positioned at different locations, which indicates that SIRs with different R _z and α will contribute to achieving wide DM stopband and wideband CM suppression. Besides, this design does not require tuning the resonant frequency ratio to its maximum; it only needs to ensure that the spurious harmonics of the used resonators are as widely spaced as possible. Through experimental verification, from the point of view of suppressing spurious harmonics and achieving a compact layout, the R _z of the SIRs are ultimately selected as 0.69 and 0.32 for the lower channel, and the R _z of the SIRs are ultimately selected as 0.69 and 0.48 for the higher channel. To simplify the simulation, the α for SIR with R _z = 0.69 is set to 0.51, while the α for R _z = 0.32 and R _z = 0.48 are both set to 0.43.

Figure 3.

Curves of the variation of f _o1 with the physical dimension L _t (Y ₁ = 0.0114 S).

Figure 4.

Curves of the variation of the resonant frequencies ratio with the α (Y ₁ = 0.0114 S). (a) f _e1/f _o1, (b) f _o2/f _o1, (c) f _e2/f _o1, (d) f _o3/f _o1.

Coupling routing scheme

Figure 5 illustrates the coupling topology of the proposed balanced diplexer. In this topology, S and L represent the differential source and load, respectively; solid lines denote strong coupling, dashed lines denote weak coupling, R ₁, R ₃, R ₄, and R ₆ correspond to fundamental resonant modes of the SIRs, and R ₂ and R ₅ denote fundamental resonant modes of the UIRs. Their resonant frequencies are f _R1, f _R3, f _R4, f _R6, f _R2, and f _R5.

Figure 5.

The coupling topology of the proposed balanced diplexer.

The lower channel is formed by coupling the three resonators R ₁, R ₂, and R ₃, while the higher channel is established by coupling R ₄, R ₅, and R ₆. Both channels exhibit a third-order bandpass response, enhancing channel selectivity. DM isolation between channels is improved by introducing out-of-band transmission zeros through cross-coupling between SIRs. And by adjusting K ₁₃ and K ₄₆, the transmission zeros can be controlled. Here, the control of the transmission zero of the lower channel is described as an example. The control mechanism of the transmission zero of the higher channel is similar to it. Figure 6 illustrates the curve of the transmission coefficients with the different values of K ₁₃. It can be observed that as the coupling coefficient K ₁₃ gradually increases, the transmission zero gradually moves closer to the passband, but the stopband suppression level deteriorates. Therefore, when adjusting the transmission zero, it is necessary to weigh the suppression level of the stopband, ultimately selecting an appropriate coupling coefficient K ₁₃. It is also worth noting that the lower and higher channels each possess independent transmission paths, allowing for separate control over the center frequencies and bandwidths. This ensures that the design of the balanced diplexer has greater flexibility.

Figure 6.

Curve of the transmission coefficients with the different K ₁₃.

Design of the balanced-to-balanced diplexer

The evolution of the bandpass filtering structures is shown in Fig. 7. Both structures use cross-coupling to introduce additional transmission zeros. For structure 1 with a symmetric configuration of SIRs, although an upper stopband up to 3.5f ₀ and CM suppression up to 2.3f ₀ are achieved by adjusting the impedance ratio of the SIRs to keep their second-order odd-mode and first-order even-mode harmonics away from the fundamental resonance frequency f ₀, they still affect the performance of the diplexer through the weak coupling path. For structure 2 with an asymmetric configuration of SIRs, in order to further broaden the upper stopband and CM suppression bandwidths, employ SIRs with different impedance ratios, such that the harmonics of the two SIRs and UIR are different from each other. When the passband center frequency is 2.0 GHz, the resonant-frequency spectra of SIR1, SIR2, and UIR are shown in Fig. 8, which shows that, except for the fundamental frequencies (i.e., first-order odd-mode resonances), the three resonators have different first-order even-mode harmonics, second-order odd-mode harmonics, second-order even-mode harmonics, third-order odd-mode harmonics, etc.

Figure 7.

The evolution of the bandpass filtering structures.

Figure 8.

The resonant-frequency spectra of SIRs and UIR.

The balanced diplexer is designed with the layout shown in Fig. 9. Port A serves as the differential antenna port, while Ports B and C function as the differential transceiver port. The transmission coefficients between the differential ports can be defined by the mixed-mode S-parameters [Reference Eisenstant, Stengel and Thompson28]. For example, the DM and CM transmission coefficients between differential Port A and differential Port B can be defined as |S _ddBA| and |S _ccBA|, respectively, which represent the transmission responses between differential ports.

Figure 9.

Layout of the proposed balanced diplexer.

Compared to the parallel-coupled single-band and dual-band bandpass filters using the SIRs with different dimensions [Reference Kuo and Shih29, Reference Kuo and Lin30], the coupling topology and the input/output feed method of the filtering structures in this paper and the principle of harmonic suppression are different. The principle of harmonic suppression in Kuo and Lin [Reference Kuo and Lin30] is to collocate the transmission zeros generated by the tap feed method [Reference Kuo and Shih29] and inherent transmission zeros of open-circuited parallel-coupled lines [Reference Kuo and Lin30], but the principle of harmonic suppression in this paper is based on the excitation principle. Specifically, based on the layout and coupling topology of the proposed balanced diplexer shown in Fig. 9 and Fig. 5, due to the symmetrical feed ports exciting the SIRs, the even- and odd-mode harmonics of the SIRs only can be excited by common- and differential-modes, respectively; thus, the DM signal can pass only when the three resonators (SIR1, SIR2, and UIR) have the same odd-mode resonant frequency, and when the odd-mode resonant frequencies of the three resonators are different, especially when the odd-mode resonant frequencies of resonators R ₁ and R ₃ are different, the DM signal will be blocked to achieve a wide DM stopband. Similarly, when the even-mode resonant frequencies of resonators R ₁ and R ₃ are different, the CM noise will also be blocked to achieve wideband CM suppression. In this design, the three resonators have the same resonant frequency only at the fundamental frequency, while the even- and odd-mode spurious harmonics of the three resonators cannot resonate at the same frequency in the high-frequency band. Therefore, the proposed diplexer can result in wideband harmonic suppression. For example, between differential Port A and differential Port B, the comparison of the DM upper stopband (which can be expressed by |S _ddBA|) and CM suppression (which can be expressed by |S _ccBA|) simulation results for two different filtering structures is shown in Fig. 10.

Figure 10.

The comparison of the stopband and CM suppression simulation results. (a) |S _ddBA|, (b) |S _ccBA|.

It can be seen from Fig. 10(a) that for the structure 1 (resonators R ₁ and R ₃ use the same SIR1, the second-order and third-order odd-mode harmonics of which resonate at about 7.1 GHz and 11 GHz, respectively, as shown in Fig. 8), the DM out-of-band suppression near 7.1 GHz and 11 GHz is only about 12 dB (i.e., |S _ddBA| is about − 12 dB), but when resonators R ₁ and R ₃ use the different SIRs (i.e., the structure 2 with asymmetric configuration of SIRs), resonators R ₁ and R ₃ (i.e., SIR1 and SIR2) have different resonant frequencies of second-order and third-order odd-mode harmonics, the DM out-of-band suppression near 7.1 GHz and 11 GHz has been significantly enhanced, achieving a wide 20-dB DM stopband. It can be seen from Fig. 8 that the first-order and second-order even-mode harmonics of the SIR1 resonate at about 4.7 GHz and 9 GHz, respectively; thus, the CM suppression for the structure 1 near 4.7 GHz and 9 GHz is less than 20 dB (i.e., |S _ccBA| > − 20 dB), as shown in Fig. 10(b). Similarly, by using an asymmetric configuration of SIRs, different even-mode resonant frequencies of SIR1 and SIR2 result in 20-dB CM suppression up to 13.4 GHz, achieving wideband CM suppression.

The design specifications for the lower channel of the balanced diplexer are as follows: The center frequency f _L is 2.0 GHz with a 3-dB fractional bandwidth (FBW) of 7.5% and a return loss (RL) of 15 dB. The coupling matrix can be synthesized as

(4)\begin{equation}{{\mathbf{M}}_{\mathbf{L}}} = \left[ {\begin{array}{*{20}{c}} 0&{0.794}&0&0&0 \\ {0.794}&0&{0.647}&{0.227}&0 \\ 0&{0.647}&{ - 0.285}&{0.720}&0 \\ 0&{0.227}&{0.720}&0&{0.959} \\ 0&0&0&{0.959}&0 \end{array}} \right]\end{equation}

The required coupling coefficients K ₁₂, K ₁₃, and K ₂₃ and the lower channel loaded quality factors (Loaded-Q) Q _1S and Q _3L are calculated as [Reference Hong and Lancaster31]:

(5)\begin{equation}K_{12}=\text{FBW}\cdot m_{12},\,K_{13}=\text{FBW}\cdot m_{13},\;K_{23}=\text{FBW}\cdot m_{23}\end{equation}

(6)\begin{equation}Q_{1\text{S}}=\frac1{\text{FBW}\cdot m_\text{1S}^2},\;\,Q_{3\text{L}}=\frac1{\text{FBW}\cdot m_\text{3L}^2}\end{equation}

The required fundamental resonant frequency of the UIR is determined by the following equation [Reference Hong and Lancaster31]:

(7)\begin{equation}{f_{{\text{R}}2}} = {f_{\text{L}}} \cdot \left( {1 - \frac{{{m_{{\text{22}}}} \cdot {\text{FBW}}}}{2}} \right)\end{equation}

(8)\begin{equation}{{\mathbf{M}}_{\mathbf{H}}} = \left[ {\begin{array}{*{20}{c}} 0&{0.887}&0&0&0 \\ {0.887}&0&{0.650}&{0.350}&0 \\ 0&{0.650}&{ - 0.445}&{0.625}&0 \\ 0&{0.350}&{0.625}&0&{0.829} \\ 0&0&0&{0.829}&0 \end{array}} \right]\end{equation}

Based on Equations (5)–(7), the calculated parameters for the lower channel are K ₁₂ = 0.0485, K ₁₃ = 0.0170, K ₂₃ = 0.0540, Q _1S = 21.15, Q _3L = 14.50, f _R1 = f _R3 = 2000.0 MHz, and f _R2 = 2021.4 MHz. Figure 11(a) shows the curves of coupling coefficients K for the lower channel as the gap s changes. Figure 11(b) illustrates the curves of loaded-Q as the gap s changes. To meet the design specifications, the coupling coefficients K and the loaded-Q must match the calculated values. This can be achieved by carefully selecting the relevant physical dimensions, as indicated in Fig. 11, to satisfy the design requirements. And the resonant frequencies of the resonators are adjusted by tuning length.

Figure 11.

The curves of coupling coefficients K and loaded-Q for the lower channel as the gap s changes. (a) Coupling coefficients K. (b) Loaded-Q.

The center frequency f _H, 3-dB FBW, and RL of the higher channel of the balanced diplexer are 3.5 GHz, 8.0%, and 15 dB, respectively, with the coupling matrix given by Equation (8). Similarly, the required coupling coefficients K, loaded-Q, and fundamental resonant frequencies are calculated as follows: K ₄₅ = 0.0520, K ₄₆ = 0.0280, K ₅₆ = 0.0500, Q _4S = 15.89, Q _6L = 18.19, f _R4 = f _R6 = 3500.0 MHz, and f _R5 = 3562.3 MHz. Figure 12(a) illustrates the curves of the higher channel coupling coefficients K as the gap s changes. Figure 12(b) shows the relationship between the higher channel loaded-Q and the gap s. By carefully selecting the relevant physical dimensions based on Fig. 12, the desired coupling coefficients K and loaded-Q can be achieved to meet the design specifications. And the self-resonant frequencies are adjusted by tuning length.

Figure 12.

The curves of coupling coefficients K and loaded-Q for the higher channel as the gap s changes. (a) Coupling coefficients K. (b) Loaded-Q.

Design procedure

Based on the above analysis, the design procedure for the proposed novel planar balanced-to-balanced diplexer with high isolation and harmonic suppression can be summarized as follows:

Step 1: Define diplexer center frequencies, along with the 3-dB FBW and RL, and synthesize the coupling matrix.

Step 2: Using the synthesized coupling matrix, determine the center frequencies of the SIRs and the UIR, and employ eigenmode analysis in HFSS to select the appropriate impedance ratio for the SIRs and sequentially simulate the different gaps according to coupling coefficients K and the loaded-Q. Ensure that both channels exhibit great third-order filtering responses.

Step 3: Combine the two channel structures and perform simulations. Optimize the physical dimensions of the overall structure to address potential issues arising from the discontinuity and the interaction.