A Dual-Band Quarter-Wave Transform and Its Applications to Directional Coupler Design ^†

Abstract

The quarter-wave transformer is a useful circuit for impedance matching. In this paper, we use three equal-length transmission lines to design dual-band quarter-wave transformers. Closed-form design equations are derived. The proposed structure is found to be suitable for dual-band operation with a frequency ratio greater than 5. Numerous microwave passive components are composed of quarter-wave transformers. For these components consisting of quarter-wave transformers, the use of dual-band quarter-wave transformers can inherently result in dual-band operation. The proposed structure is, therefore, a simple and effective element for designing dual-band microwave passive components with a frequency ratio greater than 5. Because the existing techniques for designing dual-band circuits are mostly suitable for frequency ratios lower than 5, the proposed structure, therefore, complements the existing techniques. To demonstrate the applicability of the structure, two directional couplers, namely, a dual-band branch-line hybrid and a dual-band rat-race hybrid, are designed and fabricated on a RO4003C substrate. Measurement results validate the applicability of the proposed structure.

1. Introduction

Impedance transformers or impedance matching networks are essential for RF and microwave circuit design, and quarter-wave transformers (QWT) are fundamental elements for numerous microwave passive components such as impedance matching network, resonators, power dividers, and directional couplers, etc. [1]. Note that a QWT is an impedance transformer, but not vice versa. A QWT must have a phase delay of 90° to be an impedance inverter [1].

For impedance matching using transmission lines, it is well known that a quarter-wavelength line can match a real load impedance at a single frequency [1], and two quarter-wavelength lines in cascade, known as the Monzon transformer [2], can match a real load impedance at the two frequencies f₁ and kf₁, where k is the frequency ratio. Note that both lines of the Monzon transformer are a quarter-wavelength long at the center frequency, i.e., f_c = (1 + k) f₁/2.

The Monzon transformer finds many practical applications [3,4,5,6,7,8]. However, the phase delays of the Monzon transformer are not exactly 90 degrees at the two midbands. In other words, the Monzon transformer is not a dual-band QWT. A dual-band QWT must have not only the right magnitude responses, but also the right phase responses. For example, in [7,8], the dual-band Wilkinson power dividers based on the Monzon transformer, therefore, reactive elements, i.e., a capacitor and an inductor, need to be used in the isolation networks to compensate for the phase responses of the Monzon transformers.

In contrast to the two-section Monzon transformer that has only a unique solution to a given dual-band matching problem, a three-section transformer brings in extra freedom, and can have infinitely many solutions [9]. Thus, we could choose an appropriate one from the solutions to achieve the 90° phase delays for realizing a dual-band QWT.

Microwave passive components are essential parts for the realization of wireless systems [1,10], and dual-band design is a current trend [10,11,12,13,14,15]. In this paper, inspired by the Monzon transformer, we study using three equal-length transmission lines to design dual-band QWTs [16]. Numerous microwave passive components are composed of QWTs [1]. For these components consisting of QWTs, the use of dual-band QWTs can inherently lead to dual-band operation.

In the literature, various methods have been developed to implement dual-band QWTs. These methods are, respectively, based on the composite right-/left-handed transmission lines [17], the π-type structures [18,19,20,21], the T-type structures [22,23,24], the bridged-T coils [25], and coupled microstrip lines [26].

Compared to those in [17,18,19,20,21,22,23,24,25,26] that use dual-band QWTs as the constituent elements to implement dual-band circuits, there are still many techniques for dual-band designs such as three-branch-line design for branch-line couplers [27], the port-extension technique for branch-line couplers [28], and the interesting topology for power dividers in [29]. Dual-band circuits like those in [27,28,29] rely on the specific topology as a whole to realize the dual-band operation.

As a notable point, the existing techniques for designing dual-band circuits are mostly suitable for frequency ratios lower than 5 [10], except for [26]. In [26], a dual-band QWT with wide frequency ratio is designed based on the coupled microstrip lines and open stubs.

Because the proposed dual-band QWT in this paper is simply a transmission-line step impedance structure, it is a simple and effective element for designing dual-band circuits with frequency ratios greater than 5. The proposed dual-band QWT, therefore, can complement the existing techniques [17,18,19,20,21,22,23,24,25,26,27,28,29].

Directional couplers are important devices. Their ability to sample either the forward or reverse direction of signal propagation allows for a wide range of applications in measurement, power and signal monitoring, and feedback and control. To demonstrate the applicability of the proposed structure, a branch-line hybrid operating at 0.4 GHz and 2.4 GHz and a rat-race hybrid operating at 0.4 GHz and 2.3 GHz are designed and fabricated on a 20-mil RO4003C.

This paper includes the theoretical and experimental treatment of the dual-band QWTs, and is organized as follows. Section 2 analyzes the three-section transformer and presents the design equations. A design example based on the ideal lossless transmission line is discussed in Section 3. Section 4 presents the experiment results of two directional couplers. The concluding remarks are given in Section 5.

2. Theoretical Analysis

Figure 1 illustrates the three-section transformer, which is a cascade of three equal-length transmission lines. The two midbands are, respectively, at f₁ and kf₁, where k is the frequency ratio. Note that z₁, z₂, and z₃ are the normalized characteristic impedances. Like the Monzon transformer [2], each section is a quarter-wavelength long at the center frequency f_c = 0.5 (1 + k) f₁. As an example, for f₁ = 0.4 GHz and k = 6, each section is 90° at 1.4 GHz and is 25.7143° at 0.4 GHz. The design equations are derived as follows.

2.1. Conditions for Impedance Matching

In Figure 1, the input impedance can be derived by an iterative use of (2.44) in [1] as

$$z_{in}={\displaystyle \frac{\alpha +j\beta }{\kappa +j\rho }}$$

(1)

$$\alpha =r_L{z}_1\left[\left(z_2^2+z_1{z}_2+z_1{z}_3\right){\tan }^2\theta -z_2{z}_3\right]$$

(2)

$$\beta =z_1{z}_3\tan \theta \left(z_1{z}_3{\tan }^2\theta -z_1{z}_2-z_2{z}_3-z_2^2\right)$$

(3)

$$\kappa =z_3\left[\left(z_2^2+z_2{z}_3+z_1{z}_3\right){\tan }^2\theta -z_1{z}_2\right]$$

(4)

$$\rho =r_L\tan \theta \left(z_2^2{\tan }^2\theta -z_1{z}_2-z_1{z}_3-z_2{z}_3\right)$$

(5)

where $\theta =0.5\pi f/f_c$ and f is the operating frequency. For the purpose of impedance matching, we should design the matching sections to meet the conditions of $\alpha =\kappa$ and $\beta =\rho$ to make $z_{in}=1$. The two conditions can be expressed in two equations as

$$F_{11}=\alpha -\kappa$$

(6)

$$F_{21}=\left(\beta -\rho \right)/\tan \theta$$

(7)

Note that $\tan \theta \ne 0$ is a common factor of $\beta$ and $\rho$, and therefore can be factored out in (7). It can be seen that (6) and (7) are functions of z₁, z₂, z₃, r_L, and ${\tan }^2\theta$. For a dual-frequency matching problem, F₁₁ = 0 and F₂₁ = 0 need to be satisfied at f₁ and kf₁. The electrical length of each section is θ₁ = π/(1 + k) at f₁ and θ₂ = kπ/(1 + k) at kf₁.

Because ${\tan }^2\theta ={\tan }^2\left(\pi -\theta \right)$ and θ₂ = π-θ₁, we can see that if F₁₁ = 0 and F₂₁ = 0 are satisfied at f₁, they will also be satisfied at kf₁. In other words, we only need to consider (6) and (7) at f₁ where θ₁ = π/(1 + k). As a result, for a given set of r_L and k, we now have three unknowns z₁, z₂, and z₃ in two Equations (6) and (7). We may, therefore, find infinitely many solutions [9]. Due to the extra freedom, we can impose one more condition during designing impedance transformers.

2.2. Condition for a Phase Delay of 90°

A QWT is an impedance inverter. It inverts the load impedances [1]. For a dual-band QWT, z_in in (1) should approach infinity at f₁ and kf₁ if r_L = 0. It means that κ in (4) must equal to zero at f₁ and kf₁ if r_L = 0, because inherently ρ = 0 when r_L = 0, as indicated (5). The third equation can then be written as

$$F_{31}=\kappa$$

(8)

Because (8) enforces κ = 0, (6) should be rewritten as

$$F_{11}^{\prime }=\alpha$$

(9)

Again, due to tan² θ = tan² (π − θ), we only need to deal with (8) and (9) at f₁.

2.3. Design Equations

The design of quarter-wave transformers is based on the equations $F={\left[\begin{array}{ccc}{F}_{11}^{\prime }& F_{21}& F_{31}\end{array}\right]}^T={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T$. We now have three unknowns, i.e., z₁, z₂, and z₃ in three equations, while the electrical length is already determined, i.e., θ₁ = π/(1 + k). Fortunately, the solutions in this case can be expressed explicitly in closed form.

Consider a dual-band QWT with equivalent characteristic impedance z_q. It can transform r_L = (z_q)² to 1 at f₁ and kf₁. We then replace r_L in the set of simultaneous equations with (z_q)², and solve the equations. The solutions are given by [16]

$$z_1=z_q\sqrt{{\displaystyle \frac{{\tan }^2{\theta }_1-2N-N^2}{N^2{\tan }^2{\theta }_1-2N-1}}}$$

(10)

$$z_2=z_1/N$$

(11)

$$N={\displaystyle \frac{\left(1-{\tan }^2{\theta }_1\right)\pm \sec {\theta }_1\sqrt{1-3{\tan }^2{\theta }_1}}{2{\tan }^2{\theta }_1}}$$

(12)

$$z_3=z_1$$

(13)

where θ₁ = π/(1 + k). Note that, in (12), the impedance ratio N has two solutions. By denoting the solutions as N₁ and N₂, we can figure out that N₁ = 1/N₂. Moreover, the dual-band QWT is a symmetrical structure, because z₃ = z₁, as indicated in (13). Note that we do not impose z₃ = z₁ in advance; it is the solution by solving the problem of three unknowns in three equations.

In summary, the dual-band QWT is a symmetrical structure, involving high- and low-impedance lines. There are two solutions for the line impedances z₁, z₂, and z₃. They can, respectively, be z_H, z_L, and z_H, or z_L, z_H, and z_L, where z_H and z_L stand for high and low impedances. Note that, though two solutions differ in physical structure, they behave identically in electrical properties.

Consider the achievable frequency ratios. It should be noted that, for the solutions to be valid, the square root in (12) must be a real number, i.e., 1 − 3tan² θ₁ ≥ 0. This condition can be written as

$${\theta }_1\le {\tan }^{-1}\left({\displaystyle \frac{1}{\sqrt{3}}}\right)$$

(14)

We can therefore deduce that the frequency ratio k must be greater or equal to 5, due to θ₁ = π/(1 + k) and ${\tan }^{-1}\left(1/\sqrt{3}\right)=\pi /6$.

Actually, when the frequency ratio is k = 5, we can find that the electrical length is θ₁ = 30° and the impedance ratio is N = 1. Accordingly, when k = 5, the proposed structure degenerates to a uniform 90° line, which has spurious bands at 3f₁ and 5f₁, etc. In other words, the uniform 90° line is a special case of the proposed structure. Figure 2 depicts the solutions for k from 5 to 8.

3. Design Example

In this section, we use the ideal lossless transmission lines to design a branch-line hybrid with k = 6, and compare its performance to the conventional design using uniform 90° lines. A branch-line hybrid is composed of two QWTs with z_q1 = 0.707 and two QWTs with z_q2 = 1 [1]. The solutions, for the 50 Ω characteristic impedance and k = 6, found by (10)–(13), are listed in

Table 1. The impedance values of the three-section transformers for branch-line hybrid with k = 6, i.e., θ₁ = 25.7143°.

Zq [Ω]	ZH [Ω]	ZL [Ω]
35.35	61	20.5
50	86.25	28.98

.

Figure 3a,b shows the comparison of transmission, coupling, reflection, and isolation. We can further scrutinize the performance via the amplitude balance and phase difference, as shown in Figure 3c,d. The amplitude balance and phase difference are defined as the ratio of S₂₁ to S₃₁.

As observed in Figure 3, the performance of the dual-band design is similar to that of the conventional one in the vicinity of f₁, and the performance reappears at 6f₁ for the dual-band design.

4. Microstrip Realization and Measurement

In this section, we present two experimental results, which are a branch-line hybrid operating at 0.4 GHz and 2.4 GHz, and a rat-race hybrid operating at 0.4 GHz and 2.3 GHz.

4.1. Branch-Line Hybrid with k = 6

Figure 4 shows the photograph of the fabricated dual-band branch-line hybrid, which is designed to have midbands at 0.4 GHz and 2.4 GHz, i.e., k = 6. Because high-impedance lines are thinner and easier to route and connect, we adopt the z_H-z_L-z_H topology, to design the constituent QWTs, and the solutions are listed in Table 1.

Because the circuit is symmetric and the PCB process variation is negligible, the scattering parameters S_i1 for i = 1~4 can fully characterize the circuit. We therefore show only the results of port 1 excitation. Figure 5a–f shows the comparison of simulation and measured results. The simulation results are obtained from full-wave simulations conducted using Keysight’s Momentum. The dual-band operation can be evidently seen from Figure 5a–f. Note that the amplitude balance and phase difference is defined as the ratio of S₂₁ to S₃₁.

4.2. Rat-Race Hybrid with k = 5.75

A rat-race hybrid is composed of six QWTs with z_q = 1.414 [1]. While branch-line couplers have 90° phase shift between the two output ports, rat-race couplers can have 0° and 180° phase shifts, and can be useful for certain applications, such as mixers and phase shifters [30,31].

The solutions, for the 50 Ω characteristic impedance, found by (10)–(13), are listed in

Table 2. The impedance values of the three-section transformers for rat-race hybrid with k = 5.75, i.e., θ₁ = 26.667°.

Zq [Ω]	ZH [Ω]	ZL [Ω]
70.71	113.5	44.05

. The electrical length of each section is 26.667° at f₁.

Figure 6 shows the photograph of the fabricated dual-band rat-race hybrid, which is designed to have midbands at 0.4 GHz and 2.3 GHz. The ports are numerated according to Figure 7.42(a) in [1].

Because the circuit is symmetric and the PCB process variation is negligible, the scattering parameters S_i1 and S_i2 for i = 1~4 can fully characterize the circuit. We therefore show only the results of port 1 and 2 excitations. When port 1 is excited, outputs at ports 2 and 3 are in phase. By contrast, when port 2 is excited, outputs at ports 1 and 4 are 180° out of phase.

Figure 7a–d shows the measured results. The dual-band operation can be evidently seen from the measured results. Note that there are two sets of the amplitude balances and phase differences. They are defined by the ratios of S₂₁ to S₃₁, and S₁₂ to S₄₂, corresponding, respectively, to port 1 and 2 excitations.

5. Conclusions

This paper proposed the design of dual-band QWTs using three equal-length transmission lines. Closed-form design equations were derived and presented in (10)–(13). The dual-band QWT is simply a transmission-line step impedance structure. For a given equivalent characteristic impedance Z_q, there are three design parameters to be determined. While the electrical length at the first midband is predetermined [2] as θ₁ = π/(1 + k), the impedance values of the high- and low-impedance lines, i.e., Z_H and Z_L, can be calculated using (10)–(13).

The dual-band QWT behaves like a 90° transmission line in the two midbands, as discussed in Section 3. Compared to the existing dual-band design techniques [10,17,18,19,20,21,22,23,24,25,26,27,28,29], the proposed dual-band QWT is a simple transmission-line structure suitable for dual-band operation with frequency ratio greater than 5. In principle, the dual-band QWT can be realized with all types of transmission lines [1]. The achievable operating frequencies, e.g., millimeter-wave applications, are therefore determined by the quality of the transmission lines. Meanwhile, the maximum achievable frequency ratio is primarily limited by the high-impedance lines. Depending on the PCB manufacturing process capabilities, high-impedance lines could be too thin to be realized.

The applicability of the proposed structure was verified by a dual-band Wilkinson power divider in [16], and was demonstrated in this paper by two directional couplers, namely, a dual-band branch-line hybrid and a dual-band rat-race hybrid. Design parameters of the directional couplers were, respectively, listed in Table 1 and Table 2.

As a dual-band QWT, the three-section transformer is a simple and effect element for the design of dual-band circuits with frequency ratio greater than 5. Because the existing techniques for designing dual-band circuits are mostly suitable for frequency ratios lower than 5, the proposed structure therefore complements the existing techniques [10,17,18,19,20,21,22,23,24,25,26,27,28,29]. If necessary, the high-impedance lines can also be meandered for circuit miniaturization [32].