Model and formulation of the proposed MTSS

The proposed MTSS is presented in Fig. 1(b). This MTSS is equivalent to a quarter-wavelength transmission line with a characteristic impedance of ${Z_T}$ (Fig. 1a) at four operating bands. The proposed MTSS is developed by adding two series coupled lines to the conventional dual-band T-shaped structure [Reference Sahu, Shamaileh, Aaen and Abu-shamleh28]. The MTSS is composed of two transmission lines, two coupled lines, and a short stub. Two transmission lines are the same with characteristic impedance ${Z_1}$ and electrical length $\theta $. Two coupled lines have the same even and odd characteristic impedances ( ${Z_e}$ and ${Z_o}$) and the same electrical length $\theta $. The short stub has characteristic impedance ${Z_2}$ and electrical length $\theta $.

Figure 1.

(a) A quarter-wavelength transmission line. (b) A proposed modified T-shape structure.

To determine the parameters of the proposed MTSS, the methodology from [Reference Althuwayb24] was applied. The ABCD matrix of the proposed MTSS is the product of ABCD matrices of sections:

(1)\begin{equation}M = \left[ {\begin{array}{*{20}{c}} A&B \\ C&D \end{array}} \right] = {M_C}{M_1}{M_2}{M_1}{M_C}\end{equation}

In which, ${M_C}$, ${M_1}$, and ${M_2}$ are ABCD matrices of coupled line, transmission line ( ${Z_1}$, $\theta $), and short stub ( ${Z_2}$, $\theta $), respectively. These matrices are determined as follows:

(2)\begin{equation}{M_C} = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{\frac{{j({Z_e} + {Z_o})\sin \theta }}{2}} \\ {\frac{{j2\sin \theta }}{{({Z_e} + {Z_o})}}}&{\cos \theta } \end{array}} \right]\end{equation}

(3)\begin{equation}{M_1} = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{j{Z_1}\sin \theta } \\ {\frac{{j\sin \theta }}{{{Z_1}}}}&{\cos \theta } \end{array}} \right]\end{equation}

(4)\begin{equation}{M_2} = \left[ {\begin{array}{*{20}{c}} 1&0 \\ { - \frac{{j\cot \theta }}{{{Z_2}}}}&1 \end{array}} \right]\end{equation}

Besides, the ABCD matrix of the quarter-wavelength transmission line with characteristics impedance ${Z_T}$ has the following form:

(5)\begin{equation}{M_T} = \left[ {\begin{array}{*{20}{c}} 0&{j{Z_T}} \\ {\frac{j}{{{Z_T}}}}&0 \end{array}} \right]\end{equation}

To make the proposed MTSS equivalent to the quarter-wavelength transmission line, the following relation must be satisfied:

(6)\begin{equation}{M_C}{M_1}{M_2}{M_1}{M_C} = \pm {M_T}\end{equation}

Hence,

(7)\begin{equation}{M_1}{M_2}{M_1} = \pm M_C^{ - 1}{M_T}M_C^{ - 1}\end{equation}

By substituting Equations (2)–(4) and (5) into Equation (7), the following equation is obtained:

(8)\begin{align} &\left[ {\begin{array}{*{20}{c}} {\left( {1 + \frac{{{Z_1}}}{{{Z_2}}}} \right){{\cos }^2}\theta - {{\sin }^2}\theta }&{j\left( {{Z_1} + \frac{{{Z_1}^2}}{{2{Z_2}}}} \right)\sin 2\theta } \\ {j\left( {\frac{{\sin 2\theta }}{{{Z_1}}} - \frac{{{{\cos }^2}\theta \cot \theta }}{{{Z_2}}}} \right)}&{\left( {1 + \frac{{{Z_1}}}{{{Z_2}}}} \right){{\cos }^2}\theta - {{\sin }^2}\theta } \end{array}} \right] \nonumber \\ & \quad = \pm \left[ {\begin{array}{*{20}{c}} {\left( {\frac{{{Z_C}}}{{4{Z_T}}} + \frac{{{Z_T}}}{{{Z_C}}}} \right)\sin 2\theta }&{j\left( {{Z_T}{{\cos }^2}\theta - \frac{{{Z_C}^2}}{{4{Z_T}}}{{\sin }^2}\theta } \right)} \\ {j\left( {\frac{{{{\cos }^2}\theta }}{{{Z_T}}} - \frac{{4{Z_T}{{\sin }^2}\theta }}{{{Z_C}^2}}} \right)}&{\left( {\frac{{{Z_C}}}{{4{Z_T}}} + \frac{{{Z_T}}}{{{Z_C}}}} \right)\sin 2\theta } \end{array}} \right] \end{align}

In Equation (8), we set ${Z_C} = {Z_e} + {Z_o}$. By solving Equation (8), the following relations are deduced:

(9)\begin{equation}{\tan ^2}\theta \pm 2\tan \theta \left( {\frac{{{Z_C}}}{{4{Z_T}}} + \frac{{{Z_C}}}{{{Z_T}}}} \right) - \left( {1 + \frac{{{Z_1}}}{{{Z_2}}}} \right)\end{equation}

(10)\begin{equation}{\tan ^2}\theta \pm 2\tan \theta \left( {{Z_1} + \frac{{{Z^2}_1}}{{2{Z_2}}}} \right)\frac{{4{Z_T}}}{{Z_C^2}} - \frac{{4{Z^2}_T}}{{Z_C^2}}\end{equation}

(11)\begin{equation}{Z_1} = \frac{{Z_C^3}}{{8Z_T^2}}\end{equation}

(12)\begin{equation}{Z_2} = \frac{{Z_C^5}}{{8Z_T^2(4Z_T^2 - Z_C^2)}}\end{equation}

(13)\begin{equation}\tan \theta = \pm \sqrt {\frac{{4Z_T^2}}{{Z_C^2}} + \frac{{{{\left( {Z_C^2 + 4Z_T^2} \right)}^2}}}{{16Z_T^2Z_C^2}}} \pm \frac{{Z_C^2 + 4Z_T^2}}{{4{Z_T}{Z_C}}}\end{equation}

From Equation (13), we deduce:

(14)\begin{equation}{\theta _1} = {\text{Arctan}}\left[ {\sqrt {\frac{{4Z_T^2}}{{Z_C^2}} + \frac{{{{\left( {Z_C^2 + 4Z_T^2} \right)}^2}}}{{16Z_T^2Z_C^2}}} - \frac{{{Z_C}^2 + 4{Z_T}^2}}{{4{Z_T}{Z_C}}}} \right]\end{equation}

(15)\begin{equation}{\theta _2} = {\text{Arctan}}\left[ {\sqrt {\frac{{4Z_T^2}}{{Z_C^2}} + \frac{{{{\left( {Z_C^2 + 4Z_T^2} \right)}^2}}}{{16Z_T^2Z_C^2}}} + \frac{{{Z_C}^2 + 4{Z_T}^2}}{{4{Z_T}{Z_C}}}} \right]\end{equation}

(16)\begin{equation}{\theta _3} = \pi - {\theta _2}\end{equation}

(17)\begin{equation}{\theta _4} = \pi - {\theta _1}\end{equation}

From Equations (16) and (17), the following relations are obtained:

(18)\begin{equation}\frac{{{f_2}}}{{{f_1}}} = \frac{{{\theta _2}}}{{{\theta _1}}}\end{equation}

(19)\begin{equation}\frac{{{f_3}}}{{{f_2}}} = \frac{\pi }{{{\theta _2}}} - 1\end{equation}

(20)\begin{equation}\frac{{{f_4}}}{{{f_1}}} = \frac{\pi }{{{\theta _1}}} - 1\end{equation}

Thus, based on the derived equations, MTSS can be designed.

Next, we present an illustrative example of designing an impedance-matching circuit between 75 $\Omega $ and 100 $\Omega $ at four frequency bands, utilizing the proposed MTSS. It assumed that the first and final frequencies are ${f_1} = 1{\text{ GHz}}$ and ${f_4} = 4{\text{ }}$GHz, respectively. The process of design includes the following steps. First of all, we calculate the characteristic impedance ${Z_T}$ of the quarter-wavelength transmission line for matching between $75{ \Omega }$ and $100{ \Omega }$, ${Z_T} = \sqrt {75.100} = 86.6{ \Omega }$. Next, the quarter-wavelength transmission line ${Z_T}$ is replaced by the proposed MTSS. Parameters of the proposed MTSS are calculated by applying the closed-form equations (11)–(20). ${\theta _1}$ is determined by using Equation (20), ${\theta _1} = {36^o}$. By substituting ${\theta _1}$ and ${Z_T}$ to Equation (14), ${Z_C}$ is given as $120{ \Omega }$. By using Equations (15) and (18), we obtain ${f_2}$ = 1.965 GHz. From Equation (19), ${f_3}$ is calculated as 3.035 GHz. Impedances ${Z_1}$ and ${Z_2}$ are determined from Equation (11) and (12), ${Z_1} = 28.8{ \Omega }$, ${Z_2} = 26.6{ \Omega }$.

Finally, the impedance matching circuit is simulated to verify the calculation method. Figures 2 and 3 illustrate the ideal simulated parameters |S ₁₁|, |S ₂₂|, and |S ₂₁| of the circuit.

Figure 2.

The ideal simulated scattering parameters |S ₁₁| and |S ₂₂| of the impedance matching circuit between 75 $\Omega $ and 100 $\Omega $.

Figure 3.

The ideal simulated scattering parameter |S ₂₁| of the impedance matching circuit between 75 $\Omega $ and 100 $\Omega $.

In Fig. 2, it can be observed that the reflection coefficients ∣S ₁₁∣ and ∣S ₂₂∣ exhibit identical responses and reach their minimum points at the four designed frequencies 1, 1.965, 3.035, and 4, with values lower than −53 dB. Additionally, Fig. 3 shows that the transmission coefficient ∣S ₂₁∣ approaches its maximum value of 0 dB at these four design frequencies. These results indicate that the impedance-matching circuit performs effectively, confirming the validity of the design method. The following section will present the design approach for a quad-band RRC based on the proposed MTSS.

Design method of quad-band RRC

The conventional single-band RRC is presented in Fig. 4 [Reference Pozar29].

Figure 4.

The conventional single-band rat-race coupler [Reference Pozar29].

Figure 4 illustrates a conventional single-band RRC, in which transmission lines are connected to form a circular or square ring. The RRC consists of four ports, including input, output, coupled, and isolated ports. In the RRC, three transmission lines with characteristic impedances ${Z_{T1}},{\text{ }}{Z_{T2}}$, and ${Z_{T3}}$ are quarter-wavelength (λ/4) lines, and one transmission line with characteristics impedance ${Z_{T4}}$ is a three-quarter-wavelength (3λ/4) line. All of the transmission lines have the same characteristics as follows:

(21)\begin{equation}{Z_{T1}} = {Z_{T2}} = {Z_{T3}} = {Z_{T4}} = {Z_T} = {Z_0}\sqrt 2 \Omega\end{equation}

where ${Z_0}$= 50 $\Omega $.

A proposed quad-band RRC is depicted in Fig. 5. To design the quad-band RRC, the quarter-wavelength transmission lines ${Z_{T1}},{\text{ }}{Z_{T2}}$, and ${Z_{T3}}$ in the conventional RRC are replaced by the proposed MTSS blocks, which are, respectively, symbolized as E₁, E₂, and E₃. Since the transmission line ${Z_{T4}}$ has an electrical length of 3λ/4, it is divided into three quarter-wavelength transmission lines with the characteristic impedance Z_{T 4}. Then, these three quarter-wavelength transmission lines are replaced by MTSS blocks E₄₁, E₄₂, and E₄₃.

Figure 5.

The proposed quad-band RRC.

Next, the design constraints of the proposed quad-band RRC are analyzed. When the characteristic impedance of a transmission line is too high or too low, the line width becomes excessively narrow or wide. A narrow line is challenging to fabricate, whereas a wide line significantly increases transmission loss. Therefore, an important design constraint is related to the standard PCB technology, which requires all characteristic impedances ${Z_1}$ and ${Z_2}$ of the circuit to be in the range from 15 Ω to 120 Ω. From Equations (11) and (12), the following conditions are deduced:

(22)\begin{equation}{15_{}}\Omega \leq \frac{{Z_C^3}}{{8Z_T^2}} \leq {120_{}}\Omega \end{equation}

(23)\begin{equation}{15_{}}\Omega \leq \frac{{Z_C^5}}{{8Z_T^2(4Z_T^2 - Z_C^2)}} \leq {120_{}}\Omega \end{equation}

Solving the system of inequalities (22) and (23) with the condition Z _T = 70.71, we obtain

(24)\begin{equation}92.7 \Omega \leq Z_C \leq 120.9 \Omega \end{equation}

By substituting condition (24) into Equations (14) and (15), the following conditions are derived:

(25)\begin{equation}{28.13^o} \leq {\theta _1} \leq {38.12^o}\end{equation}

(26)\begin{equation}{68.66^o} \leq {\theta _2} \leq {71.37^o}\end{equation}

By substituting Equation (25) into Equation (20), and Equation (26) into Equation (19), we have

(27)\begin{equation}3.72 \leq \frac{{{f_4}}}{{{f_1}}} \leq 5.39\end{equation}

(28)\begin{equation}1.52 \leq \frac{{{f_3}}}{{{f_2}}} \leq 1.62\end{equation}

Thus, from the above analysis, it can be concluded that the proposed quad-band RRC can operate at four operating bands, provided that the condition f ₄/f ₁ ≤ 5.39 is satisfied. This constraint ensures the feasibility of fabricating the quad-band RRC using conventional PCB technology.

From the design equations presented earlier, it also can be seen that when f ₁ and f ₄ are selected, the frequencies f ₂ and f ₃ are also determined based on the values of f ₁ and f ₄. Next, the dependence of the four fractional bandwidths (FBWs) of the quad-band RRC, namely ${\Delta }$f ₁, ${\Delta }$f ₂, ${ \Delta }$f ₃, and ${\Delta }$f ₄ on the frequency ratio ${\text{ }}{f_4}/{f_1}$, is investigated. The bandwidth is defined as the range where |S ₁₁| is less than −15 dB. Figure 6 illustrates the simulated dependence of these four FBWs on the frequency ratio f ₄/f ₁ .

From Fig. 6, it can be observed that as the frequency ratio f ₄/f ₁ increases, the FBW of the first band (Δf ₁) exhibits a significant upward trend, indicating a notable expansion in bandwidth. The second band (Δf ₂) also shows a moderate increase, though with a much slower rate compared to Δf ₁. Meanwhile, the third (Δf ₃) and fourth (Δf ₄) bands remain nearly constant with minor variations, implying minimal dependence on the frequency ratio f ₄/f ₁ . These trends highlight the impact of the frequency ratio f ₄/f ₁ on the four FBWs of the proposed quad-band RRC.

Figure 6.

The dependence of four bandwidths of the quad-band RRC on the frequency ratio ${f_4}/{f_1}$.

Finally, the process of designing a quad-band RRC includes the following steps:

1. 1. Separate the conventional RRC into $\lambda /4$ transmission lines with the same characteristic impedance, ${Z_T} = 70.71{ \Omega }$. The $3\lambda /4$ transmission line is separated into three $\lambda /4$ transmission lines.

2. 2. Replace the quarter-wavelength transmission lines with equivalent MTSS blocks. Parameters of the MTSS are calculated as below.

   1. 2.1. Select the minimum and maximum operating frequencies, ${f_1}$ and ${f_4}$, these frequencies must satisfy the condition (27).

   2. 2.2. From Equation (20), determine the electrical length ${\theta _1}$.

   3. 2.3. Substitute ${Z_T}$ and ${\theta _1}$ into Equation (14), then calculate the value of ${Z_C}$.

   4. 2.4. From Equation (15), determine the value of ${\theta _2}$.

   5. 2.5. Use Equations (18) and (19) to calculate operation frequencies ${f_2}$ and ${f_3}$.

   6. 2.6. Determine the characteristic impedances ${Z_1}$ and ${Z_2}$ using Equations (11) and (12).

Experimental validation

For further demonstration, a quad-band RRC was designed, fabricated, and tested. The circuit was fabricated using a Rogers RO4003C substrate, which has a relative dielectric constant of 3.55, a thickness of 0.813 mm, and a loss tangent of 0.0027.

Applying the design procedure presented in the previous section, the parameters of the scheme are calculated. Firstly, we choose frequencies ${f_4}$ = 2.45 GHz and ${f_1}$ = 0.6 GHz. Then, these frequencies are substituted into Equation (20), and the electrical length ${\theta _1}$ is defined as ${\theta _1}$ = 35.4°. From Equations (14) and (15), the values of ${Z_C}$ and ${\theta _2}$ are computed as ${Z_C}$= 99.6 Ω and ${\theta _2}$= 70.6°. From there, Z _e = 51 Ω and Z _o = 48.6 Ω are chosen. From Equations (18) and (19), the operating frequencies ${f_2}$ and ${f_3}$ are defined as ${f_2}$ = 1.196 GHz, ${f_3}$ = 1.853 GHz. Substituting the values of ${Z_C}$ and ${Z_T}$ into Equations (11) and (12), we have ${Z_1}$ = 24.7 Ω and ${Z_2}$ = 24.3 Ω. The selected frequency bands 0.6, 1.196, 1.853, and 2.45 GHz correspond to various practical applications: 0.6 GHz for low-frequency wireless communication systems such as LTE Band 71 (600 MHz), which is currently used to provide extended coverage in 4G and 5G networks; 1.196 GHz for global navigation satellite systems (e.g., Galileo E5b, GPS L5); 1.853 GHz for mobile communication; and 2.45 GHz (ISM band) for WLAN and wireless sensor networks. These bands highlight the versatility of the proposed design for modern communication systems.

Figure 7 shows the layout and fabricated prototype of the proposed quad-band RRC. A Vector Network Analyzer is employed to measure the scattering parameters of the fabricated prototype.

Figure 7.

(a) Layout and (b) photograph of the fabricated quad-band RRC.

Simulated and measured performance of the designed quad-band RRC are presented in Figs. 8–11. The performance results demonstrate the presence of four passbands centered at 0.6, 1.195, 1.825, and 2.42 GHz. There is a slight shift in the center frequencies of each band, which becomes more pronounced at higher frequencies. This shift is primarily attributed to transmission line discontinuities, including T-junction and bend discontinuities. Additionally, at higher frequencies, the shorter wavelength increases the model’s sensitivity to etching inaccuracies, which can affect overall performance. Nevertheless, within the considered frequency range, the proposed model exhibits reliable performance, as validated by both simulation and experimental results. The results suggest that impedance mismatches caused by additional junctions in the proposed circuit have a limited impact within the studied range. However, this behavior may vary if the operating frequency range is extended, and further investigation is required to fully assess the impact under such conditions.

Figure 8.

Measured and simulated parameters S ₂₁ and S ₁₁ of the proposed quad-band RRC.

Figure 8 shows that the measured insertion losses |S ₂₁| at the four operating frequencies 0.6, 1.195, 1.825, and 2.42 GHz are 3.18, 3.03, 3.14, and 3.53 dB, respectively. Thus, compared to the ideal 3 dB level, the additional insertion loss is less than 0.53 dB. Meanwhile, the measured input return loss |S ₁₁| is better than 21.75 dB at all four central operating frequencies.

Figure 9 describes the measured and simulated |S ₃₁| and |S ₄₁| parameters of the proposed quad-band RRC. The simulated and measured results show a close agreement. From the graph of Fig. 9, the measured insertion loss |S ₃₁| at four operating bands is lower than 4.4 dB. Meanwhile, the measured isolation level (|S ₄₁|) is higher than 21.6 dB.

Figure 9.

Measured and simulated parameters S ₃₁ and S ₄₁ of the proposed quad-band RRC.

In-phase response of the proposed quad-band RRC is depicted in Fig. 10. The detailed data indicate that at the four operating frequencies 0.6, 1.195, 1.825, and 2.42 GHz, the phase differences $\angle {S_{21}} - \angle {S_{31}}$ are 3.45^o, 4.56^o, 2.58^o, and 1.3^o, respectively. Figure 11 illustrates the out-phase responses $\angle {S_{24}} - \angle {S_{34}}$ of the RRC. The results show that at four operating frequencies, the phase differences are within 180^o ± 6.7^o.

Figure 10.

Simulated and measured phase difference between S ₂₁ and S ₃₁ of the proposed quad-band RRC.

Figure 11.

Simulated and measured phase difference between S ₂₄ and S ₃₄ of the proposed quad-band RRC.

Finally, to evaluate the effectiveness of the design approach, the measured performance of the proposed quad-band RRC is compared with other reported works. Comparison results are listed in Table 1.

Table 1.

Comparison performance of the proposed quad-band RRC and other reported works

In Table 1, NI indicates no information; PD is the phase difference; IRL, IS, and IL denote input return loss, isolation, and insertion loss at the four operating frequencies, respectively; BLC is the branch line coupler; λ_g is the guided wavelength at frequency f ₁; and FBW is the fractional bandwidth. The FBW is defined as the frequency range where the amplitude imbalance between the two output ports remains within 1 dB for both the proposed quad-band RRC and [Reference Piazzon, Saad, Colantonio, Giannini, Andersson and Fager30], whereas for [Reference Guoan, Shihao, Wenguang, Baoquan and Lamin31], it is within 1 dB for the first two bands and 1.5 dB for the last two bands.

Compared to the previously published works [Reference Papanastasiou25, Reference Piazzon, Saad, Colantonio, Giannini, Andersson and Fager30–Reference Liang, Zhang and Liao32], the proposed quad-band RRC exhibits a significant improvement in isolation, return loss, and insertion loss performance. Specifically, it achieves an isolation level exceeding 21.6 dB, return losses greater than 21.75 dB, and an insertion loss of less than 4.4 dB at four operating frequencies. Although the circuit in [Reference Velan, Kingsly, Kanagasabai, Alsath, Panneer and Subbaraj26] achieves better isolation, it exhibits a high insertion loss of up to 6.3 dB and significant phase deviation. Moreover, the proposed RRC design is simpler than those presented in [Reference Papanastasiou25, Reference Velan, Kingsly, Kanagasabai, Alsath, Panneer and Subbaraj26], as it features a planar structure, does not use lumped elements, and relies solely on coupled lines, transmission lines, and short stubs. Meanwhile, the quad-band RRC in [Reference Papanastasiou25] utilizes negative refractive index transmission lines with MIM capacitors and inductors, while the circuit in [Reference Velan, Kingsly, Kanagasabai, Alsath, Panneer and Subbaraj26] employs a laterally offset dual-ring (LODR) resonator backed by concentric dual-split-ring (CDSR) slots and is implemented in a two-layer structure. Thus, these comparisons validate the advantages of the proposed design method. However, despite these benefits, the proposed circuit also presents some drawbacks. The main drawback is its relatively large size compared to the circuits listed in Table 1. This size increase represents a trade-off for achieving a simpler structure and improved performance. In addition, the use of multiple cascaded λ/4 transmission line sections introduces another potential limitation. At higher frequencies, the multiple junctions between these sections can cause impedance mismatches, leading to undesired reflections, signal loss, or distortion. To mitigate this issue, further optimization through the use of a low-loss substrate or advanced fabrication techniques could improve performance at higher frequencies.