A Design Approach for Asymmetric Coupled Line In-Phase Power Dividers with Arbitrary Terminal Real Impedances and Arbitrary Power Division Ratio

Abstract

In this paper, we first introduced asymmetric coupled lines (ACLs) into both the transmission path and isolation path in traditional in-phase Gysel power dividers and proposed the single-resistor asymmetric coupled line in-phase Gysel power dividers (ACPDs). Utilizing the decoupled branch-line model of ACLs, a generalized design approach for ACPDs with arbitrary terminal real impedances and arbitrary power division ratio was innovatively proposed. Design formulas relating terminal real impedances, power division ratio, and image impedances of ACLs, for simultaneously satisfying the perfect port isolation and match conditions, are presented. ACPDs achieved a large in-phase power division ratio of 100:1 (20 dB) and offered significant advantages, including impedance transformation, high design freedom, and miniaturization. To automatically determine accurate initial values of geometric parameters for ACLs, a solution software based on MATLAB-HFSS co-simulation and multi-layer perception neural networks was developed, significantly reducing subsequent optimization iterations. To verify the proposed analysis theory and design approach, three ACPDs with different power division ratios of 1:1 (3 dB), 10:1 (10 dB), and 100:1 (20 dB) were implemented. Comparisons of the measured and simulated results showed great accordance, and the three ACPDs achieved good frequency bandwidth, high isolation, excellent port match, and compact size.

1. Introduction

With the development of wireless technology, improving design freedom and reducing optimization iterations have become critical issues for engineers. In practical design, increasing the number of adjustable geometric parameters of components is an effective method to achieve higher design freedom. Consequently, compared to symmetric structures, the importance of asymmetric structures, such as asymmetric coupled lines, is gradually emerging.

As a fundamental unit in the field of RF and microwave engineering, coupled lines possess a compact structure and broad bandwidth, making them widely utilized in couplers [1], power dividers [2], filters [3], antennas [4], power amplifiers [5], etc. Compared to the extensively adopted symmetric coupled lines (SCLs), the asymmetric coupled line (ACL) structure [6] offers two significant advantages.

• Firstly, ACL provides an additional line width variable, allowing for more design freedom.
• Secondly, the terminal real impedances of ACL devices can simultaneously take on different values, endowing them with the capability of impedance transformation and expanding the application space of devices.

Although the analysis and design of ACLs are relatively complex due to the difficulty of decoupling and the lack of empirical formulas, we have successfully established an effective branch-line decoupled model of ACLs using the method of N-port network equivalence in [7].

As a classic type of power divider, Gysel power dividers boast higher power capability than Wilkinson power dividers by introducing two short-ended resistors to transfer heat to the ground plane. In modern RF circuits, key concerns for a Gysel power divider include achieving arbitrary and large power division ratios (PDRs) [8,9], impedance transformation, compact size, multi-band [10], and broad bandwidth. Various methods, such as phase inverters [11], electromagnetic bandgap (EBG) structures [12], coupled lines [10,13,14], and open or short stubs [15,16], have been employed to meet some of these requirements. In [13], the proposed in-phase Gysel power divider with one isolation resistor saves circuit space by eliminating the 180° electrical length transmission line and incorporating a quarter-wavelength SCL section. However, this design cannot accommodate impedance transformation, achieves a relatively small PDR, and still requires two isolation resistors in the equal power division case. Generally, achieving a large in-phase PDR is challenging, and an in-phase PDR of more than 10 dB is rarely reported in the literature.

In this paper, to explore the advantages of ACLs for impedance transformation, large PDR, and miniaturization of power dividers, we innovatively proposed the asymmetric coupled line in-phase Gysel power dividers (ACPDs). The ACPDs require only one isolation resistor for both equal and unequal power division cases. Through detailed analysis of the isolation path and power dividing path in an N-port network, combined with the decoupled branch-line model of ACLs at the central frequency, the relationships between terminal real impedances, image impedances (defined by the decoupled model of ACLs), and PDR to meet the conditions for perfect port match and isolation are established. To reduce the design difficulty, an automatic solution software developed using MATLAB (R2020b)-HFSS (2021R1) co-simulation and multi-layer perception neural networks to determine accurate values of geometric parameters for ACLs was developed. For verification, three single-resistor ACPDs with PDRs (dB) of 3 dB, 10 dB, and 20 dB were implemented. For the first time, our proposed ACPD achieved a large PDR of 20 dB. The remarkable consistency between simulated and measured data strongly supports the validity of the proposed design approach and highlights the flexibility and performance advantages of ACLs.

This paper is organized as follows: Section 2 introduces the theoretical analysis and design methodology for ACPDs. The principle and processing procedures of the automatic solution software are presented in Section 3. The performance of the three fabricated ACPDs is shown in Section 4. Section 5 discusses the application of devices with arbitrary terminal real impedances and arbitrary power division ratios. Section 6 concludes the whole paper.

2. Theoretical Analysis and Design Methodology

2.1. Two-Resistor APCD Structure

The prototype of the proposed ACPD is shown in Figure 1. This design features two sections of quarter-wavelength ACLs, each with different geometric parameter values and, respectively, constituting the transmission path and part of the isolation path of ACPD. To enhance the universality of the following analysis, it is initially assumed that the ACPD includes two isolation resistors R₁ and R₂. The terminal real impedances of ACPD are denoted as Z_T1, Z_T2, and Z_T3. Utilizing the decoupled branch-line model of ACLs at the central frequency proposed in [7], the decoupled circuit of the ACPD is presented in Figure 2. Here, Z_I1 to Z_I4 represents the image impedances of the left ACL section, while Z_I1r to Z_I4r are the image impedances of the right one.

For the ACPD shown in Figure 2, there is one ideal isolation condition between ports T₂ and T₃. Consider the 2-port subnetwork constituted by ports T₂ and T₃ as shown in Figure 3, which consists of three subcircuits A, B, and C in parallel. Their corresponding admittance matrices, Y_A, Y_B, and Y_C, are as follows:

$${\mathit{Y}}_A=\left[\begin{array}{cc}{\displaystyle \frac{1}{G_1{Z}_{I1r}^2}}& {\displaystyle \frac{-1}{G_1{Z}_{I1r}{Z}_{I4r}}}\\ {\displaystyle \frac{-1}{G_1{Z}_{I1r}{Z}_{I4r}}}& {\displaystyle \frac{1}{G_1{Z}_{I4r}^2}}\end{array}\right],$$

(1)

$${\mathit{Y}}_B=\left[\begin{array}{cc}{\displaystyle \frac{1}{G_2{Z}_{I2r}^2}}& {\displaystyle \frac{-1}{G_2{Z}_{I2r}{Z}_{I3r}}}\\ {\displaystyle \frac{-1}{G_2{Z}_{I2r}{Z}_{I3r}}}& {\displaystyle \frac{1}{G_2{Z}_{I3r}^2}}\end{array}\right],$$

(2)

$${\mathit{Y}}_C=\left[\begin{array}{cc}{\displaystyle \frac{{(Y_{I4}-Y_{I1})}^2}{Y_{T1}}}& {\displaystyle \frac{(Y_{I1}-Y_{I4})(Y_{I3}-Y_{I2})}{Y_{T1}}}\\ {\displaystyle \frac{(Y_{I1}-Y_{I4})(Y_{I3}-Y_{I2})}{Y_{T1}}}& {\displaystyle \frac{{(Y_{I2}-Y_{I3})}^2}{Y_{T1}}}\end{array}\right],$$

(3)

where G₁ = R₁⁻¹, and G₂ = R₂⁻¹. Then, the overall admittance matrix Y_T2T3 is obtained as follows:

$${\mathit{Y}}_{T2T3}={\mathit{Y}}_A+{\mathit{Y}}_B+{\mathit{Y}}_C=\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right],$$

(4)

where

$$\left\{\begin{array}{l}{Y}_{11}={\displaystyle \frac{1}{G_1{Z}_{I1r}^2}}+{\displaystyle \frac{1}{G_2{Z}_{I2r}^2}}+{\displaystyle \frac{{(Y_{I4}-Y_{I1})}^2}{Y_{T1}}}\\ Y_{12}={\displaystyle \frac{-1}{G_1{Z}_{I1r}{Z}_{I4r}}}-{\displaystyle \frac{1}{G_2{Z}_{I2r}{Z}_{I3r}}}+{\displaystyle \frac{(Y_{I1}-Y_{I4})(Y_{I3}-Y_{I2})}{Y_{T1}}}=Y_{21}\\ Y_{22}={\displaystyle \frac{1}{G_1{Z}_{I4r}^2}}+{\displaystyle \frac{1}{G_2{Z}_{I3r}^2}}+{\displaystyle \frac{{(Y_{I2}-Y_{I3})}^2}{Y_{T1}}}\end{array}\right..$$

(5)

If the perfect isolation between ports T₂ and T₃ is realized, the matrix Y_T2T3 should satisfy that [7],

$${\mathit{Y}}_{T2T3}=\left[\begin{array}{cc}{Y}_{11}& Y_{12}\\ Y_{21}& Y_{22}\end{array}\right]=\left[\begin{array}{cc}{Y}_{T2}& 0\\ 0& Y_{T3}\end{array}\right].$$

(6)

Then, one set of equations is obtained from (4) to (6) as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{R_1}{Z_{I1r}^2}}+{\displaystyle \frac{R_2}{Z_{I2r}^2}}+{\displaystyle \frac{{(Y_{I4}-Y_{I1})}^2}{Y_{T1}}}=Y_{T2}\\ {\displaystyle \frac{R_1}{Z_{I4r}^2}}+{\displaystyle \frac{R_2}{Z_{I3r}^2}}+{\displaystyle \frac{{(Y_{I2}-Y_{I3})}^2}{Y_{T1}}}=Y_{T3}\\ {\displaystyle \frac{R_1}{Z_{I1r}{Z}_{I4r}}}+{\displaystyle \frac{R_2}{Z_{I2r}{Z}_{I3r}}}={\displaystyle \frac{(Y_{I1}-Y_{I4})(Y_{I3}-Y_{I2})}{Y_{T1}}}\end{array}\right..$$

(7)

Now, consider the power transmission between ports T₁, T₂, and T₃. When the input port T₁ is excited, in an ideal case, the input power is transmitted entirely to output ports T₂ and T₃, with no current flowing into resistors R₁ and R₂. Figure 4 shows the simplified circuit of ACPD for the power analysis, with the definition of node voltages and currents. Based on voltage–current relation, there are

$$\left\{\begin{array}{l}{V}_1={\displaystyle \frac{j}{Y_{I1}-Y_{I4}}}{I}_2={\displaystyle \frac{j}{Y_{I1}-Y_{I4}}}{\displaystyle \frac{V_2}{Z_{T2}}}\\ I_a=j(Y_{I1}-Y_{I4})V_2\end{array}\right.,$$

(8)

$$\left\{\begin{array}{l}{V}_1={\displaystyle \frac{j}{Y_{I3}-Y_{I2}}}{I}_3={\displaystyle \frac{j}{Y_{I3}-Y_{I2}}}{\displaystyle \frac{V_3}{Z_{T3}}}\\ I_b=j(Y_{I3}-Y_{I2})V_3\end{array}\right..$$

(9)

By the definition of the power division ratio R_p = P_T2/P_T3, as well as the law of power conservation, the following set of equations is obtained:

$$\left\{\begin{array}{l}{Z}_{T2}(Y_{I1}-Y_{I4}{)}^2+Z_{T3}(Y_{I3}-Y_{I2}{)}^2={\displaystyle \frac{1}{Z_{T1}}}\\ {\displaystyle \frac{1}{Z_{T3}{Z}_{T1}{(Y_{I3}-Y_{I2})}^2}}-1=r_p\end{array}\right..$$

(10)

Set Z_T2 = xZ_T1, Z_T3 = yZ_T1, Z_I1 = K₁Z_T1, Z_I2 = Z_I4 = K₂Z_T1, Z_I3 = K₃Z_T1, Z_I1r = K₄Z_T1, Z_I2r = Z_I4r = K₅Z_T1, Z_I3r = K₆Z_T1, R₁ = t₁Z_T1, and R₂ = t₂Z_T1, where x, y, t₁, and t₂ are the arbitrary real coefficients can be realized, and substitute these equations into (7) and (10), there are,

$$\begin{array}{c}{\displaystyle \frac{1}{K_3}}-{\displaystyle \frac{1}{K_2}}={\displaystyle \frac{1}{\sqrt{y(R_p+1)}}},{\displaystyle \frac{1}{K_1}}-{\displaystyle \frac{1}{K_2}}=\sqrt{{\displaystyle \frac{R_p}{x(R_p+1)}}},{\displaystyle \frac{t_1}{K_4^2}}+{\displaystyle \frac{t_2}{K_5^2}}={\displaystyle \frac{1}{x(R_p+1)}}\\ {\displaystyle \frac{t_1}{K_5^2}}+{\displaystyle \frac{t_2}{K_6^2}}={\displaystyle \frac{R_p}{y(R_p+1)}},{\displaystyle \frac{t_1}{K_4{K}_5}}+{\displaystyle \frac{t_2}{K_2{K}_3}}={\displaystyle \frac{1}{R_p+1}}\sqrt{{\displaystyle \frac{R_p}{xy}}}.\end{array}$$

(11)

To solve the equations in Equation (11), and relations are obtained as follows:

$$\begin{array}{c}{K}_1={\displaystyle \frac{1}{\sqrt{\frac{R_p}{x(R_p+1)}}+\frac{1}{K_2}}},K_3={\displaystyle \frac{1}{\frac{1}{\sqrt{y(R_p+1)}}+\frac{1}{K_2}}},K_4=\sqrt{{\displaystyle \frac{t_1}{\frac{1}{x(R_p+1)}-\frac{t_2}{K_5^2}}}}\\ K_5=\sqrt{t_{2}x(R_p+1)+{\displaystyle \frac{t_{1}y(R_p+1)}{R_p}}},K_6=\sqrt{{\displaystyle \frac{t_2}{\frac{R_p}{y(R_p+1)}-\frac{t_1}{K_5^2}}}}.\end{array}$$

(12)

So far, the relationships between Z_Ii (i = 1, 2, 3, 4), Z_Ijr (j = 1, 2, 3, 4), Z_T1, Z_T2, Z_T3, R₁, R₂, and R_p, for achieving perfect isolation, perfect port match, and power division requirement of the two-resistor ACPDs have been completely deduced and established. It is important to note that, in addition to the values of Z_T1, x, y, t₁, t₂, and R_p, the value of K₂ also needs to be specified according to the application requirements, introducing an additional degree of design freedom.

In particular, consider cases of equal power division and equal terminal real impedances. Substitute R_p = 1 and x = y into (12), there are,

$$\begin{array}{c}{K}_1=K_3={\left({\displaystyle \frac{1}{\sqrt{2x}}}+{\displaystyle \frac{1}{K_2}}\right)}^{-1}\\ K_4=K_5=K_6=\sqrt{2x(t_1+t_2)},\end{array}$$

(13)

then we have Z_I1 = Z_I3 and Z_I1r = Z_I2r = Z_I3r. It is noted that, for a physically achievable SCL or ACL section, the value of Z_I2 (Z_I2r) is always greater than the values of Z_I1 (Z_I1r) and Z_I3 (Z_I3r) [7], meaning K₂ > K₁, K₂ > K₃, K₅ > K₄, and K₅ > K₆. The calculation formulas of Z_I1 (Z_I1r), Z_I2 (Z_I2r), and Z_I3 (Z_I3r) are listed in Section 3 as (20). Therefore, under conditions of perfect isolation and match, it is not possible to simultaneously achieve both equal power division and equal terminal real impedances of the two output ports. However, within a reasonable range, some impedance match and isolation performance can be sacrificed to achieve equal power division and equal terminal impedance at the same time.

2.2. Single-Resistor APCD Structure

Next, consider the structure of ACPD with only one isolation resistor R₁, as shown in Figure 5. Compared with the two-resistor ACPD structure in Figure 1, this ACPD structure reduces design complicity and saves circuit space. Substitute t₂ = 0 into (12), there is,

$$\begin{array}{c}{\displaystyle \frac{1}{K_3}}-{\displaystyle \frac{1}{K_2}}={\displaystyle \frac{1}{\sqrt{y(R_p+1)}}},{\displaystyle \frac{1}{K_1}}-{\displaystyle \frac{1}{K_2}}=\sqrt{{\displaystyle \frac{R_p}{x(R_p+1)}}}\\ {\displaystyle \frac{t_1}{K_4^2}}={\displaystyle \frac{1}{x(R_p+1)}},{\displaystyle \frac{t_1}{K_5^2}}={\displaystyle \frac{R_p}{y(R_p+1)}}.\end{array}$$

(14)

By solving Equation (14), relationships are obtained as follows:

$$\begin{array}{c}{K}_1={\displaystyle \frac{1}{\sqrt{\frac{R_p}{x(R_p+1)}}+\frac{1}{K_2}}},K_3={\displaystyle \frac{1}{\frac{1}{\sqrt{y(R_p+1)}}+\frac{1}{K_2}}}\\ K_4=\sqrt{t_{1}x(R_p+1)},K_5=\sqrt{{\displaystyle \frac{t_{1}y(R_p+1)}{R_p}}}.\end{array}$$

(15)

So far, the relationships between Z_Ii (i = 1, 2, 3, 4), Z_Ijr (j = 1, 2, 3, 4), Z_T1, Z_T2, Z_T3, R₁, and R_p, for achieving perfect isolation and port match, and the power division requirement of the single-resistor ACPD have been fully deduced and established. It is noted that, in addition to the values of Z_T1, x, y, t₁, and R_p, the values of K₂ and K₆ also need to be specified, providing two additional degrees of design freedom.

For the cases of equal power division and equal terminal real impedances, substituting R_p = 1 and x = y into (15), there are,

$$K_1=K_3={\left({\displaystyle \frac{1}{\sqrt{2x}}}+{\displaystyle \frac{1}{K_2}}\right)}^{-1}$$

(16)

$$K_4=K_5=\sqrt{2x{t}_1}$$

(17)

Clearly, it is physically impossible for both ACLs and SCLs to satisfy Equation (17). Therefore, similar to the two-resistor ACPD analyzed in Section 2.1, under conditions of perfect isolation and match, simultaneously achieving equal power division and equal terminal real impedances cannot be realized by the single-resistor ACPD.

In addition, for the cases of unequal power division, the values of R_p can either be greater than or less than 1. If R_p > 1, since the difference between the values of x and y is generally small, Equation (15) suggests that K₅ < K₄, which is also physically impossible. Conversely, if R_p < 1, it is straightforward to achieve K₅ > K₄. Therefore, practically speaking, for the single-resistor ACPD in Figure 4, under conditions of unequal power division, R_p must be less than 1, indicating that the power transmitted to port T₂ is smaller than that transmitted to port T₃.

Then, consider the cases of a large power division ratio. Substitute R_p ≈ 0 into (15), there are,

$$\begin{array}{c}{K}_1\approx K_2,K_3={\left({\displaystyle \frac{1}{\sqrt{y}}}+{\displaystyle \frac{1}{K_2}}\right)}^{-1}\\ K_4=\sqrt{xt},K_5\approx +\infty .\end{array},$$

(18)

It is known that the larger the line spacing, the greater Z_I2 (Z_I2r) [7], and the larger K₂ (K₅). Thus, to satisfy conditions of K₁ = K₂ and K₅ ≈ +∞, the line spacing of the left ACL section is required to be as small as possible, and the line spacing of the right one is required to be as large as possible. The above analysis can be verified by the fabricated 20 dB ACPD in Section 4.

Next, the specific design procedures for the proposed ACPDs are summarized as follows:

• Determine the targeted power division ratio R_p, terminal real impedance values Z_T1, Z_T2, and Z_T3, self-specified values of K₂, K₆, R₁ (for the single-resistor structure), and self-specified values K₂, R₁, and R₂ (for the two-resistor structure).
• Calculate values of the eight image impedances Z_Ii (i = 1, 2, 3, 4) and Z_Ijr (j = 1, 2, 3, 4) by the formula (12) or (15).
• Obtain the initial geometric parameter values of the two ACL sections by the solving software introduced in Section 3, from Z_Ii (i = 1, 2, 3, 4) and Z_Ijr (j = 1, 2, 3, 4).
• Fine-turn the ACPD model to realize better performance.

3. Automatic Solution Software for ACLs

At present, the design of ACLs faces challenges due to the lack of robust commercial tools and software, which restricts their widespread application and development. To address these challenges and streamline the design process for coupled line devices, we developed an automatic solution software called CLS. This software tool leverages MATLAB (R2020b)-HFSS (2021R1) application programming interfaces (APIs) and multi-layer perception (MLP) neural networks to accurately determine the initial geometric parameters of coupled microstrip lines, thereby minimizing design complexities and reducing the number of optimization iterations.

The operational workflow of CLS is detailed in Figure 6 and can be summarized into three main components.

Firstly, it utilizes APIs between MATLAB and HFSS to automate tasks such as model creation, adjustment of geometric parameters, and export of S-parameters. Figure 7 illustrates the asymmetric coupled microstrip line model created in HFSS, highlighting parameters such as linewidths W₁, W₂, line spacing S, and line length L. The initial value of length L at the central frequency is estimated from the resonant peak of the simulated S-parameter curve of coupled lines.

Subsequently, CLS extracts distribute parameters R_ij, G_ij, L_ij, and C_ij (i, j = 1, 2) from S-parameters [17], and calculates four mode characteristic impedances of ACLs as follows [6]:

$$\begin{array}{c}{Z}_{c1}={\displaystyle \frac{(R_{11}+j\omega L_{11})(R_{22}+j\omega L_{22})-{(R_{12}+j\omega L_{12})}^2}{{\gamma }_c[(R_{22}+j\omega L_{22})-(R_{12}+j\omega L_{12})T_c]}}\\ Z_{c2}={\displaystyle \frac{T_c[(R_{11}+j\omega L_{11})(R_{22}+j\omega L_{22})-{(R_{12}+j\omega L_{12})}^2]}{{\gamma }_c[(R_{11}+j\omega L_{11})T_c-(R_{12}+j\omega L_{12})]}}\\ Z_{\pi 1}={\displaystyle \frac{(R_{11}+j\omega L_{11})(R_{22}+j\omega L_{22})-{(R_{12}+j\omega L_{12})}^2}{{\gamma }_{\pi }[(R_{22}+j\omega L_{22})-(R_{12}+j\omega L_{12})T_{\pi }]}}\\ Z_{\pi 2}={\displaystyle \frac{T_{\pi }[(R_{11}+j\omega L_{11})(R_{22}+j\omega L_{22})-{(R_{12}+j\omega L_{12})}^2]}{{\gamma }_{\pi }[(R_{11}+j\omega L_{11})T_{\pi }-(R_{12}+j\omega L_{12})]}}\end{array}.$$

(19)

where γ_c and γ_π are mode propagation constants, and T_c and T_π are mode voltage ratios. Then, the four image impedances Z_Ii (i = 1, 2, 3, 4) are calculated as follows:

$$\begin{array}{l}{Z}_{I1}={\displaystyle \frac{Z_{c2}{Z}_{\pi 2}(R_c-R_{\pi })}{R_c{Z}_{c2}-R_{\pi }{Z}_{\pi 2}}}\\ Z_{I2}=Z_{I4}={\displaystyle \frac{Z_{c1}{Z}_{\pi 1}(R_c-R_{\pi })}{Z_{c1}-Z_{\pi 1}}}\\ Z_{I3}={\displaystyle \frac{Z_{c1}{Z}_{\pi 1}(R_c-R_{\pi })}{R_c{Z}_{c1}-R_{\pi }{Z}_{\pi 1}}}\end{array}$$

(20)

Clearly, each set of W₁, W₂, and S corresponds to a set of Z_Ii (i = 1, 2, 3, 4). Thus, the sample data pairs between Z_I1, Z_I2, Z_I3, Z_I4 and W₁, W₂, S can be formed.

Finally, CLS incorporates an MLP neural network model trained with prepared sample data pairs. During its operation, CLS takes the calculated targeted image impedances Z_I1t, Z_I2t, Z_I3t, and Z_I4t as input and automatically generates predicted accurate values for the geometric parameters W_1p, W_2p, and S_p.

The accuracy of CLS is ensured through two key aspects.

• Firstly, the theoretical foundation of CLS lies in the analytical solution of telegraph equations, which provides an accurate description of coupled line behavior without relying on approximate simplifications. This ensures that the physical characteristics of coupled lines are faithfully represented in CLS.
• Secondly, CLS employs a stable and accurate mapping relationship constructed by an MLP neural network. This network effectively correlates the desired characteristic impedances with the geometric parameters of coupled microstrip lines. By training on ample sample data, the MLP neural network learns to predict these parameters reliably, thereby enhancing the overall precision and reliability of CLS in determining the geometric parameters of coupled lines.

4. Experimental Implementation of ACPDs

4.1. Design Processes and Circuit Fabrication

To verify the validity and accuracy of the proposed analysis and design approach for single-resistor ACPDs in Section 2.2, we fabricated three single-resistor ACPDs with different terminal real impedances and different PDRs (dB) of 3 dB, 10 dB, and 20 dB.

The targeted design parameter values of the three ACPDs are calculated and listed in

Table 1. Targeted parameter values of the three ACPDs.

Parameters	x	y	Rp	t	K2	K6	K1	K3	K4	K5
3 dB	0.8	2.8	1	3	3	1	0.8898	1.3229	2.1909	4.0988
10 dB	0.6	1.7	0.1	0.72	3.6	1.5	1.4992	0.9910	0.6893	3.6693
20 dB	0.86	1.26	0.01	0.6	3	3.445	2.2695	0.8198	0.7219	8.7382
Parameters	ZT1 (Ω)	ZT2 (Ω)	ZT3 (Ω)	R (Ω)	ZI1 (Ω)	ZI2 (Ω)	ZI3 (Ω)	ZI1r (Ω)	ZI2r (Ω)	ZI3r (Ω)
3 dB	50	40	140	150	44.49	150	66.145	109.545	204.94	50
10 dB	50	30	85	36	74.96	180	49.55	34.465	183.465	75
20 dB	50	43	63	30	113.475	150	40.99	36.095	436.91	172.25

, all of which are finally well satisfied. In the following, we use t to represent the coefficient t₁. It should be noted that the values of x, y, t, K₂, and K₆ are arbitrarily selected for all ACPDs. Consequently, the value of the patch resistor is also specified artificially. The parameters of the substrate material are listed in

Table 2. Parameter values of the substrate material adopted.

Parameters	L (mm)	H (mm)	T (mm)	TanD	εr
Values	28	0.787	0.017	0.0009	2.2

. For the convenience of the test, two quarter-wavelength impedance transformers were introduced to the output ports T₂ and T₃ of ACPD to transform both terminal impedances Z_T2 and Z_T3 to 50 Ω. A 50 Ω transmission line with a fixed length of 5 mm was connected to the input port T₁ to facilitate test connector connections.

The initial geometric parameter values of the three ACPDs, obtained by CLS, are listed in

Table 3. Initial geometric parameter values of the three ACPDs determined by CLS.

Parameters (mm)	Wc1	Wc2	S1	Wc3	Wc4	S2
3 dB	2.14	0.89	0.15	0.16	0.87	0.15
10 dB	1.04	1.88	0.15	3.35	0.37	0.21
20 dB	0.16	2.41	0.14	3.79	0.24	1.98

(see Figure 8 for geometric parameter definitions). Subsequently, the parameters were fine-turned to further optimize the S-parameters. The fabricated geometric parameter values are listed in

Table 4. Geometric parameter values for fabrication of the three ACPDs.

Parameters (mm)	Wc1	Wc2	Wc3	Wc4	S1	S2	Lc1	Lc2	WT1	LT1
3 dB	2.1	0.93	0.15	0.94	0.15	0.15	29.49	28.51	2.4	5
10 dB	0.97	1.95	3.4	0.36	0.15	0.21	28.69	27.77	2.4	5
20 dB	0.15	2.38	3.8	0.24	0.15	2	29.62	25.59	2.4	5
Parameters (mm)	WT2	LT2	WT3	LT3	W	L	D1	D2	F	Rv
3 dB	2.83	25.1	2.4	27.07	55.35	70	0	0	1	0.15
10 dB	3.46	31.73	2.4	26.45	61.25	70	2.8	2.43	1	0.15
20 dB	2.52	26.75	2.4	25.28	54.71	70	4.72	3.65	1	0.15

. It is evident that the initial values are highly accurate, as the difference between the initial values and the fabricated values is minimal. Pictures of the three fabricated ACPDs are shown in Figure 9a–c. It is noted that in Figure 9c, due to the large difference in the value of line spacing between the two ACL sections, the bottom line in the left ACL section is chamfered to facilitate connection.

Figure 8 shows geometric parameter definitions of the whole fabricated ACPD circuit. For the left ACL section, W_c1 and W_c2 are line widths, S₁ is the line spacing, and L_c1 is the line length. For the right ACL section, W_c3 and W_c4 are line widths, S₂ is the line spacing, and L_c2 is the line length. For the 50 Ω transmission line connected to T₁, in the 3 dB, 10 dB, and 20 dB ACPDs, the width W_T1 is equal to 2.4 mm, and the length L_T1 is equal to 5 mm. For the quarter-wavelength impedance transformer connected to port T₂, its width and length are defined as W_T2 and L_T2. For the one connected to port T₃, its width and length are defined as W_T3 and L_T3. The small blue square on the right of Figure 7 represents the place for the patch resistor, whose package is 0402. The side length F of the ground square pad is equal to 1 mm. Two metal vias, with the radius R_v equal to 0.15 mm, are adopted to realize grounding.

It is noted that for the 10 dB and 20 dB ACPDs, we chamfered the upper line of the right ACL section. The chamfered part has been marked with the dashed box and its amplifying circuit is shown in the upper right corner of Figure 8, with two cutting distances, respectively, defined as D₁ and D₂. To visualize the effect of chamfering on device performance, Figure 10 shows comparison curves of the simulated S-parameters before and after chamfering, of the 20 dB ACPD. It is observed that chamfering mainly affects the resonant frequency of S₂₂. Figure 11 gives the S₂₂ curves under different values of D₁ and D₂. Clearly, with the increase of D₁ and D₂, the resonant frequency moves to the higher frequency. It can be inferred that when the value of W_c3 is large, the impedance transformer at T₂ will produce a large coincidence area with the upper line of the ACL section on the right side, thereby increasing the electrical length of this ACL section, making the resonant frequency deviate from the central frequency to the lower frequency. Chamfering can reduce the electrical length to a certain extent, making the resonant frequency close to the central frequency.

4.2. S-Parameter Performance Analysis

The observation frequency band is from 1 GHz to 3 GHz, with a central frequency of 2 GHz. Figure 12a–c present magnitude curves of the simulated and measured S-parameters of the above three ACPDs. Great consistency between the simulated and measured results is achieved. The 3 dB ACPD has a fractional bandwidth of 13% (1.84 GHz to 2.10 GHz). The 10 dB ACPD has a fractional bandwidth of 25% (1.74 GHz to 2.24 GHz). The 20 dB ACPD has a fractional bandwidth of 23% (1.84 GHz to 2.3 GHz). The definitions of S-parameter values for the fractional bandwidth calculation of the three ACPDs are listed in

Table 5. Definitions of S-parameter values for the fractional bandwidth calculation of the three fabricated ACPDs.

S-Parameters	S11/S22/S23/S33 (dB)	S12 (dB)	S13 (dB)
3 dB	<−15	−3 ± 0.2	−3 ± 0.2
10 dB	<−15	−10 ± 0.5	−0.4 ± 0.2
20 dB	<−20	−20 ± 1.5	−0.03 ± 0.1

.

The frequency deviation of curves and differences between the simulated and measured results are related to objective errors such as fabrication inaccuracy, the effects of SMA connectors, substrate and conductor loss, layout discontinuity, board deformation, as well as the welding of patch resistors, to some extent. The line width and line spacing of PCB can be controlled in the order of 25 um. Figure 13 shows the simulated S-parameter curves of the 10 dB ACPD before and after changing the line spacing with the maximum fabrication error of 25 um. It is observed that the fabrication inaccuracy mainly influences S₂₂ and nearly has no impact on the other S-parameters. It could be inferred that the inconsistency of S₂₂ in Figure 12 is mainly caused by machining errors. In the operation frequency band, the insertion loss and reflection loss of SMA connectors can be controlled at a relatively low level (−0.1 dB, −20 dB). The values of the dielectric constant and dissipation factor of the material show little change.

Figure 12d shows the simulated and measured phase difference (PD) between the two output ports of three power dividers. The measured PDs at the central frequency (defined as PDC) are, respectively, equal to −3° (3 dB), 1.9° (10 dB), and 2.4° (20 dB), verifying the good in-phase characteristic. When the fractional bandwidth (FB) is defined as |PD − PDC| ≤ 10°, then the FBs of the three ACPDs are, respectively, equal to 17.5% (1.82 GHz to 2.17 GHz, 3 dB), 17.5% (1.84 GHz to 2.19 GHz, 10 dB), and 10.5% (1.9 GHz to 2.11 GHz, 20 dB). As the frequency deviates from the central frequency point, the PD increases gradually, which means the in-phase characteristics deteriorate. The reason for this phenomenon may be attributed to the increased difference between the c-mode and π-mode phase velocities, as the frequency deviates from the central frequency.

4.3. Performance Comparison

Table 6. Performance comparison of the fabricated three ACPDs with other power dividers.

Refs.	TRI	MPDR	RN	Phase Relationship	FBW (%)	Circuit Element Type	Occupied Area (λg2)
[11]	Fixed	2:1	2	In-phase	137.5	Branch lines with one slotline phase inverter as the isolation path	0.072
[13]	Fixed	4:1	2/1 *	In-phase	57.69	Branch lines with one SCL section as the isolation path	0.114 1
[18]	Fixed	8:1	2	In-phase	66.67	Branch lines with one coupled line section as the isolation path	0.045
[19]	Arbitrary	2:1	2	In-phase	20.96	Five sections of branch lines	-
This work	Arbitrary	100:1	1	In-phase	34/42/60 2	Two sections of ACLs	0.033
[20]	Arbitrary	1000:1	1	Out-of-phase	100	Branch lines with one SCL section as the transition path	0.046
[21]	Arbitrary	12:1	2	Out-of-phase	120	Parallel-strip ring with one swap as the isolation path	-
Contribution		Impedance transformation, miniaturization, high in-phase power division ratio

TRI, terminal real impedance; MPDR, the maximum achievable power division ratio; RN, number of the required isolation resistors; FBW, fractional bandwidth when isolation performance is better than −20 dB; *, only for unequal power dividing cases, one resistor is needed. For the equal power dividing case, two resistors are still required: ¹, not directly provided in the paper. -, not presented in the paper; ², 34, 42, and 60, respectively, correspond to the FBW of fabricated 3 dB, 10 dB, and 20 dB ACPDs proposed in this paper.

presents the comparison results of the measured three ACPDs with other in-phase and out-of-phase power dividers. The comparison criteria include terminal real impedances, maximum achievable power division ratio, number of required isolation resistors, phase relationship of output ports, fractional bandwidth when isolation performance is better than −20 dB, type of circuit elements, and occupied area of main circuits.

Clearly, compared to other in-phase power dividers, our proposed ACPD features a simpler structure and smaller size. It also offers impedance transformation capability and achieves an exceptionally large power division ratio of 100:1, surpassing other power dividers in this aspect. Moreover, utilizing ACLs for the transmission path allows flexible placement of the two output ports, accommodating various application scenarios. It is worth noting that out-of-phase power dividers, commonly used in scenarios requiring out-of-phase outputs, typically achieve a large PDR easily. Therefore, their performance cannot be directly compared with that of in-phase power dividers.

5. Application of Devices with Arbitrary Terminal Real Impedances and Arbitrary Power Division Ratio

The design approach for arbitrary terminal real impedances and arbitrary power division ratio has wide application scenarios and significance importance. Although 50 Ω is a standard value for the characteristic impedance of coaxial transmission lines, the optimal input impedance for packaged antennas, amplifiers, and filters in RF integrated circuits is not necessarily 50 Ω. For conventional power dividers, all three terminals are typically matched to 50 Ω, requiring additional impedance transformers to achieve port match when connecting non-50 Ω devices. Additionally, different circuit systems have diverse input power requirements for terminal components connected to power dividers. The ACPDs we propose can simultaneously achieve impedance transformation, port match, and power dividing functions. This is beneficial to reduce circuit size and enhance the overall performance of the system.

6. Conclusions

For the first time, ACLs are introduced into the design of in-phase Gysel power dividers. By the simplified analysis of N-port networks and the decoupled branch-line model of ACLs, the design formulas are derived and provided in detail, for satisfying the perfect match and isolation simultaneously. From theoretical analysis, the limitations of the proposed ACPDs are pointed out. Moreover, to reduce the subsequent optimization iterations of devices, an automatic solution software is developed to obtain accurate initial values of ACL geometric parameters. For verification, three ACPDs with different PDRs of 1:1 (3 dB), 10:1 (10 dB), and 100:1 (20 dB) were implemented.

The proposed ACPD has several key advantages:

• ACPD possesses the capability of flexible impedance transformation.
• ACPD requires only one isolation resistor for both equal and unequal power division cases, and resistor values can be flexibly determined.
• ACPD can easily achieve a large in-phase PDR. ACPD realized the large in-phase PDR of 100:1 (20 dB) and offers a broad in-phase PDR range from 1:1 (3 dB) to 100:1 (10 dB).
• ACPD is compact and the positions of the two output ports can be arranged as needed.

The proposed approach in this paper for designing ACPDs offers enhanced design flexibility to RF engineers. It allows for specifying terminal real impedances, achieving any desired power division ratio, and placing the positions of the two output ports. These advancements significantly contribute to automating device design processes, thereby advancing the level of design intelligence in RF engineering.