# Highly Unequal Three-Port Power Divider: Theory and Implementation.

1. IntroductionTwo commonly used power dividers are the Wilkinson and Gysel power dividers, invented in 1960 and 1975, respectively [1, 2]. The Wilkinson and Gysel power dividers have the similar performance for their scattering matrices, but there is a difference in their structures and the later permits higher power loads than the former. The Wilkinson power divider with a power division ratio such as 9: 1 is difficult to be physically realized in microstrip form because the characteristic impedance of one branch line is 273.86 [OMEGA]. A symmetric coupler with 3 dB coupling coefficient is known as a quadrature hybrid or a branch line hybrid [3]. We are also facing the same difficulty as to the Wilkinson power divider that a symmetric coupler with the coupling coefficient for more than 8 dB is impractical to be physically constructed utilizing the microstrip technology.

The Wilkinson power divider and the symmetric coupler are mostly applied with equal or low power division ratios. Wilkinson power dividers with parallel RLC components [4], complex isolation component [5], and two-section two-resistor [6] were designed and miniaturized for high-frequency applications. A typical application for highly unequal power division is the Doppler radar [7], as illustrated in Figure 1. The RF input signal is divided into two-way outputs, RF output 2 and RF output 3. RF output 2 is further transmitted through a circulator to the antenna and radiated. Assuming a target moving with a relative velocity [+ or -] v and with a radar cross section a, the RF signal will be scattered by the target and then received by the antenna. After passing through the circulator, it will enter the radar receiver front end where it is firstly amplified by a low-noise amplifier (LNA) module. The two signals, RF output 3 and RF output 4, are then further fed to a mixer. RF output 3 is used as a reference signal for comparison and frequency downconversion.

One major challenge of accurately processing the information carried by the scattered signal at RF output 4 is the requirement of a good balance between the power levels of the reference signal and scattered signal [8]. In fact, it is found out with a computation that the signal level of RF output 3 might be 30 dB higher than that of RF output 4, even if the power division ratio of the power divider is [[absolute value of [b.sub.2]].sup.2]: [[absolute value of [b.sub.3]].sup.2] = 10 : 1 and the gain of the LNA module is 30 dB.

A three-port power divider with a power division ratio of 9: 1 was reported, including about nine LC elements in its fabricated sample [9]. A three-port power divider with a power division ratio of 10 : 1 has been presented with a Gysel power divider and two coupling structures of filtering responses [10]. A single-section coupled line coupler can be designed with the performance of 20 dB coupling factor and 50 dB isolation, but it is best suited for weak coupling and is a four-port network [3].

There have been some other works reported in the literature concerning highly unequal power dividers, which would not be cited here one by one. We may conclude that a three-port power divider with power division ratios for more than 10: 1 would be hard to realize in a physical structure, and their research work would be difficult to implement in engineering designs.

Since the RF amplitude balance of the signals to the mixer, that is, RF output 3 and RF output 4 in Figure 1, must be considered in a system design, a highly unequal three-port power divider is proposed in this paper along with its design methodology. Moreover, assuming that a good isolation can be achieved, the proposed power divider could be used without the circulator in Figure 1, which can be used in the microstrip form [11]. The paper focuses on theoretical analysis and design of a highly unequal three-port power divider. To verify the developed theory and design, the power divider was fabricated and evaluated through measurements finally. The resulting demonstrator is very compact when implemented in the microstrip planar technology and can be integrated into the entire RF front end including the antenna.

2. Theory

A model of three-port divider with a feedback mechanism is proposed, which consists of a directional coupler, a Wilkinson power divider, and two transmission lines. The scattering matrix and the maximum power division ratio for the three-port power divider are then formulated.

2.1. Feedback Model. A schematic of the three-port power divider with a feedback mechanism is shown in Figure 2. The power divider consists of four parts, that is, a directional coupler, a Wilkinson power divider, and two transmission lines of TL1 and TL2. The numbers 1, 2, 3', and 4' are the input port, the through port, the coupling port, and the isolated port of the directional coupler, respectively. The numbers 1 and both 2 and 3 are the input port and the output port of the Wilkinson power divider. The phase delays of TL1 and TL2 are [[theta].sub.1] = 2[pi][L.sub.1]/[[lambda].sub.g] and [[theta].sub.2] = 2[pi][L.sub.2]/[[lambda].sub.g], where [L.sub.1] is the length of TL1, [L.sub.2] is the length of TL2, [[lambda].sub.g] is the guide wavelength, and their characteristic impedances are 50 [OMEGA]. Power supplied to the port 1 is coupled to the port 3 and then transferred to the port 1 through TL1. The output power at the port 2 is fed to the port 4 through TL2. The functions of TL1 and TL2 are to provide a feedback path, and a power division ratio for the three-port power divider is reallocated. In the meanwhile, the output ports of 2 and 3 are the isolated ports.

To take account of reference phases at all ports of the directional coupler, the scattering matrix of the directional coupler in Figure 2 is a reciprocal four-port network and rewritten as [S.sub.DC] [3]

[mathematical expression not reproducible], (1)

where [alpha] and [beta] are the positive reals and satisfy the condition of [[alpha].sup.2] + [[beta].sup.2] = 1, [[psi].sub.2] + [[psi].sub.3] - 2[[psi].sub.1] = (2n + 1)[pi], (n [member of] N), and N is an integer. The quantities characterized the symmetric coupler are defined as follows: the insertion loss L = -20 log [absolute value of [S.sub.21]] dB, the coupling coefficient C = -20 log [absolute value of [S.sub.31]] dB, and the isolation I = -20 log [absolute value of [S.sub.41]] dB. If [[psi].sub.1] = -[pi]/2 and [[psi].sub.2] = [[psi].sub.3] = -[pi], it is a symmetric coupler. If [alpha] = [beta] = 1/[square root of 2], [[psi].sub.1] = -[pi]/2, and [[psi].sub.2] = [[psi].sub.3] = -[pi], it is a quadrature hybrid. If [[psi].sub.1] = [[psi].sub.2] = -[pi]/2 and [[psi].sub.3] = -3[pi]/2, it is an antisymmetric coupler.

The scattering matrix of the Wilkinson power divider in Figure 2 is rewritten as on the same grounds

[mathematical expression not reproducible], (2)

[mathematical expression not reproducible], (3)

[mathematical expression not reproducible], (4)

where [K.sup.2] = [[[absolute value of [b.sub.3]].sup.2]/[[[absolute value of [b.sup.2']].sup.2] is a power division ratio of the Wilkinson power divider, according to the traditional notation. The transpose of a matrix [S.sub.4] is denoted by [S.sup.T.sub.4]. The shunt resistor is determined by R = [Z.sub.0] (K + 1/K) [3], where [Z.sub.0] is the characteristic impedance of transmission lines connected to the three ports, generally taken as 50 Q. If K =1, the Wilkinson power divider will be the equal-split (3 dB) case.

2.2. Formulation. Vectors of the incident and reflected waves at the ports of the directional coupler are defined as [mathematical expression not reproducible]. Similarly, two wave vectors of the Wilkinson power divider are defined as [b.sub.1'2'] = [[b.sub.1'2'].sup.T] and [a.sub.1'2'] = [[a.sub.1'[a.sub.2']].sup.T]. The scattering matrix of the three-port power divider is derived

[mathematical expression not reproducible], (5)

[mathematical expression not reproducible], (6)

[mathematical expression not reproducible], (7)

[mathematical expression not reproducible]. (8)

Equations (5) and (7) can be solved simultaneously to give the vectors of [a.sub.1'2'] and [a.sub.3'4']

[mathematical expression not reproducible]. (9)

From (6) and (9), the vector of [b.sub.12] and the variable of [b.sub.3] are found

[mathematical expression not reproducible]. (10)

Equation (10) can be simplified as the following forms:

[mathematical expression not reproducible], (11)

[mathematical expression not reproducible], (12)

[mathematical expression not reproducible]. (13)

From (11), the scattering matrix of the three-port power divider is determined with [S.sub.21] and [S.sub.31]

[mathematical expression not reproducible]. (14)

It shows that the scattering matrix (14) of the three-port power divider is similar to (2) of the Wilkinson power divider, where the conditions of S = [S.sup.T], [S.sub.ii] = 0 (i = 1, 2, 3), [S.sub.23] = 0, and [[absolute value of [S.sub.21]].sup.2] + [[absolute value of [S.sub.31]].sup.2] = 1 are simultaneously satisfied. The power division ratio for the three-port power divider in Figure 2 is defined

[mathematical expression not reproducible], (15)

[mathematical expression not reproducible], (16)

where [[theta].sub.1] + [[theta].sub.2] - [[psi].sub.1] - [[phi].sub.1] = (2n + 1)[pi], (n [member of] N) in (16).

2.3. Analysis. In (16), the item of [[alpha].sup.2]/[[beta].sup.2] = [[absolute value of [b.sub.2]].sup.2]/[[absolute value of [b.sup.3]].sup.2] is the power division ratio of the directional coupler. When the power division ratio for the directional coupler is [[absolute value of [b.sub.2]].sup.2] : [[absolute value of [b.sub.3']].sup.2] = 1.34 : 1 and that for the Wilkinson power divider is [[absolute value of [b.sub.2']].sup.2] : [[absolute value of [b.sub.3]].sup.2] = 1 : 1, the power division ratio for the three-port power divider will be achieved by [[absolute value of [b.sub.2]].sup.2] : [[absolute value of [b.sub.3]].sup.2] = 10 : 1. It is concluded that a feedback mechanism for two transmission lines of TL1 and TL2 is very important to improve the power division ratio for the three-port power divider.

A special case is considered that the directional coupler in Figure 2 is a symmetric coupler. We give a computation on the power division ratios as the following: the symmetric coupler with C =7dB ([[absolute value of [b.sup.2]].sup.2]/[[absolute value of [b.sub.3']].sup.2] = 4), [[psi].sub.1] = -90[degrees], and [[psi].sub.2] = [[psi].sub.3] = -180[degrees], and the Wilkinson power divider with [[phi].sub.1] = [[phi].sub.2] = -115.52[degrees], which is from the simulation result with Advanced Design System (ADS).

The computation results with MATLAB are shown in Figure 3 with 1/[K.sup.2] = 1, 2, 3, 1/2, 1/3, respectively. Because of the presence of the minimum length in the physical structure, the value of [[theta].sub.1] + [[theta].sub.2] is taken from 180[degrees] to 540[degrees], that is, the length of [L.sub.1] + [L.sub.2] should be from 0.5 Xg to 1.5 Xg. The results reveal that a region of [[theta].sub.1] + [[theta].sub.2] exists to ensure that the power division ratio of the three-port power divider is more than 10 in the listed values of 1/[K.sup.2]. The maximum of the power division ratio occurs at the value of [[theta].sub.1] + [[theta].sub.2] = 450[degrees] + [[phi].sub.1] = 338.48[degrees], and the region of [[theta].sub.1] + [[theta].sub.2] for the power division ratio more than 10 is in a wide range to be used. If the Wilkinson power divider is K = 1, the maximum of the power division ratio for the three-port power divider is 25.65.

3. Implementation

Based on the above theory developed for the highly unequal three-port power divider, the schematic and layout of a 5.8 GHz power divider were captured using ADS 2015.01, from Keysight Technologies. A comparison between (7a) and the circuit simulation with ADS is first given. To verify the above design, a test sample is fabricated, measured, and compared with the electromagnetic simulation.

3.1. Schematic. Rogers RO4350B substrate has the dielectric constant of 3.66, the dissipation factor of 0.0034 at 5.8 GHz (0.0037 at 10 GHz), the thickness of 1.524 mm, and the copper cladding of 35 [micro]m.

Both the schematic of the test sample and the basic principle of the following layout implemented in ADS schematic are provided in Figure 4, where Term, TL, and Bend are the stardard components in ADS pallete. The transmission line of TL1 (see Figure 2) consists of TL16, Bend4, and TL17. The transmission line of TL2 (see Figure 2) is realized with TL11, Bend1, TL12, Bend2, TL13, Bend3, and TL14.

The symmetric coupler with C =7 dB and the Wilkinson power divider with K =1 were automatically generated and optimized by the tool of passive circuit of Design Guide in ADS schematic, respectively. The generated circuit components of the symmetric coupler and Wilkinson power divider were integrated into a three-port power divider in a new schematic by the model in Figure 2, a new layout was done by generate/update layout in the ADS schematic, and three test ports of RF input/output were added to it by three SMA connectors, Johnson part number 142-0701-801 of End Launch Jack Receptacle.

3.2. Layout. The microstrip layout of the test sample was designed in ADS layout at the center frequency of 5.8 GHz, see Figure 5. The dot-dash lines indicate the microstrip paths of TL1 and TL2. Reference planes of the symmetric coupler are [A.sub.1] and [A.sub.2] and that of the Wilkinson power divider are [B.sub.1] and [B.sub.2]. The full size is 33 mm x 46 mm fed by 50 [OMEGA] microstrip, and the parameters of the three-port power divider are listed in Table 1.

3.3. Circuit Simulation. Simulation results of the symmetric coupler and Wilkinson power divider using ADS in the schematics are shown from Figures 6-8.

The parameters of the symmetric coupler are [absolute value of [S.sub.21]] = -0.93 dB, [absolute value of [S.sub.3'1]] = -7.01 dB, [[psi].sub.1] = -89.81[degrees], and [[psi].sub.1] = [[psi].sub.1] = -182.34[degrees]. We have noticed that there was a phase shift of 5.06[degrees], because of [[psi].sub.2] + [[psi].sub.3] - 2[[psi].sub.1] = -185.06[degrees]. The reason is that all simulations in both Figures 6 and 8 were conducted with the conductor (Cu) copper conductivity of 5.8 x [10.sup.7] S/m. The parameters of the Wilkinson power divider are [absolute value of [S.sub.2'1']] = [absolute value of [S.sub.31']] = 3.07 dB and [[phi].sub.1] = [[phi].sub.2] = -111.52[degrees]. The identical phase shift of TL1 and TL2 is [[theta].sub.1] + [[theta].sub.2] = 403.66[degrees]. The scattering matrix of the three-port power divider is [absolute value of [S.sub.21]] = -0.35 dB and [absolute value of [S.sub.31]] = -12.89 dB. It is found that the power division ratio for the three-port power divider is 1/[K.sup.2.sub.TWD] = 17.92 at 5.8 GHz in the ADS simulation. The computation results with MATLAB in Figure 3 are 1/[K.sup.2.sub.TWD] = 18.50 with [[theta].sub.1] + [[theta].sub.2] = 403.66[degrees]. Compared with the ADS simulation, the relative error of 1/[K.sup.2.sub.TWD] is equal to (18.50 - 17.92)/17.92 = 3.2%, although the above theory is in the ideal conditions without any conductor loss.

In fact, the lengths of two transmission lines of TL1 and TL2 in the test sample are arbitrarily chosen in the range of power division ratios being more than 10, only for connection with both the symmetric coupler and Wilkinson power divider and can be adjustable to specific applications.

3.4. Experiment. The test sample of the three-port power divider with C = 7 dB and K =1 was fabricated as shown in Figure 9. All scattering parameters of the above test sample of the three-port power divider are provided in Figures 10 and 11, where the prefix of "Sim" represents the electromagnetic simulation with ADS Momentum Microwave and that of "Mea" means the measurement by a Rohde & Schwarz vector network analyzer. Power division ratios for the above test sample are also obtained for both the electromagnetic simulation with ADS and measurement results in Figure 12. A fairly good agreement between the simulation and measurement is seen. It is also shown in Figure 12 that the -10 dB bandwidth of the three-port divider is more than 1 GHz.

4. Conclusions

Theory of the highly unequal three-port power divider with a feedback mechanism is presented. The three-port power divider can provide with a power division ratio of more than 10. A high isolation and impedance match are achieved at all ports. Moreover, it is compact in the planar form. The presented theory agrees well with the simulated results using ADS and the measurement results. The three-port power divider can be realized in the microstrip form without difficulty and easily integrated into radio frequency front ends. This work has completely solved the problem of limited power division ratios associated with three-port power dividers.

Data Availability

All the data used to support the findings of this study are included within the article.

https://doi.org/10.1155/2018/9141964

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors would like to thank Gustav Knutsson, in the research group of Communication Electronics, Linkoping University, for fabrication of test samples. This work was supported in part by the Natural Science Foundation of Guangdong Province under Grant no. 2017A030310063.

References

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[6] X. Wang, Z. Ma, and M. Ohira, "Theory and experiment of two-section two-resistor Wilkinson power divider with two arbitrary frequency bands," IEEE Transactions on Microwave Theory and Techniques, vol. 66, no. 3, pp. 1291-1300, 2018.

[7] V. C. Chen, F. Li, S.-S. Ho, and H. Wechsler, "Micro-Doppler effect in radar: phenomenon, model, and simulation study," IEEE Transactions on Aerospace and Electronic Systems, vol. 42, no. 1, pp. 2-21, 2006.

[8] L. Chioukh, H. Boutayeb, D. Deslandes, and K. Wu, "Noise and sensitivity of harmonic radar architecture for remote sensing and detection of vital signs," IEEE Transactions on Microwave Theory and Techniques, vol. 62, no. 9, pp. 1847-1855, 2014.

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[10] K. X. Wang, X. Y. Zhang, and B. J. Hu, "Gysel power divider with arbitrary power ratios and filtering responses using coupling structure," IEEE Transactions on Microwave Theory and Techniques, vol. 62, no. 3, pp. 431-440, 2014.

[11] R. S. Adams, B. O'Neil, and J. L. Young, "Integration of a microstrip circulator with planar Yagi antennas of several directors," IEEE Transactions on Antennas and Propagation, vol. 56, no. 11, pp. 3426-3432, 2008.

Duolong Wu, (1) Adriana Serban, (2) Magnus Karlsson, (2) and Shaofang Gong (2)

(1) School of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou 510006, China

(2) Department of Science and Technology, Linkoping University, Campus Norrkoping, 601 74 Norrkoping, Sweden

Correspondence should be addressed to Duolong Wu; edlwu@gdut.edu.cn

Received 21 April 2018; Accepted 26 June 2018; Published 19 July 2018

Academic Editor: Flaminio Ferrara

Caption: Figure 1: Simplified block diagram of the front end of a monostatic Doppler radar.

Caption: Figure 2: Schematic of the three-port divider with a feedback mechanism.

Caption: Figure 3: Computation on the power division ratio with C = 7 dB, [[psi].sub.1] = -90[degrees], [[psi].sub.2] = [[psi].sub.3] = -180[degrees], and [[phi].sub.1] = [[phi].sub.2] = 115.52[degrees]. The power division ratio is 1/[K.sup.2.sub.TWD] and the TL1 and TL2 phase delay is [[theta].sub.1] + [[theta].sub.2].

Caption: Figure 4: Schematic of the three-way divider with C = 7 dB and K = 1 in ADS.

Caption: Figure 5: Layout of the three-way divider with C = 7 dB and K = 1. The ADS layout exported a DWG format file, which was inserted and processed in Microsoft Visio.

Caption: Figure 6: Amplitudes of the scattering parameters of the symmetric coupler simulated in ADS schematic with C = 7 dB.

Caption: Figure 7: Amplitudes of the scattering parameters of the Wilkinson power divider simulated in ADS schematic with K = 1 and R = 100 [OMEGA].

Caption: Figure 8: Phases of the scattering parameters of the symmetric coupler (SC) and Wilkinson power divider (WPD) simulated in ADS schematic with C = 7 dB, K =1, and R =100 [OMEGA].

Caption: Figure 9: Fabricated sample of the three-way divider with C = 7 dB, K = 1, and R = 100 [OMEGA].

Caption: Figure 10: Comparisons of [absolute value of [S.sub.11]], [absolute value of [S.sub.22]], [absolute value of [S.sub.33]], and [absolute value of [S.sub.32]] between the electromagnetic simulation in ADS and measurement.

Caption: Figure 11: Comparisons of [absolute value of [S.sub.21]] and [absolute value of [S.sub.31]] between the electromagnetic simulation in ADS and measurement.

Caption: Figure 12: Comparison of power division ratio (PDR) between the electromagnetic simulation in ADS and measurement. The measured PDR is 13.59 at 5.8 GHz.

Table 1: Parameters of the three-port power divider. Objects Parameters Dimensions (mm) TL1 [L.sub.1] 5.90 TL2 [L.sub.2] 27.90 RF input [L.sub.3] 12.85 [W.sub.1] 3.40 Symmetric coupler [L.sub.4] 9.40 [W.sub.2] 13.06 [L.sub.8] 4.97 [W.sub.4] 7.70 Wilkinson power divider [L.sub.7] 13.17 [W.sub.3] 5.12 [W.sub.5] 1.84 RF output 2 [L.sub.5] 10.75 RF output 3 [L.sub.6] 12.81

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Title Annotation: | Research Article |
---|---|

Author: | Wu, Duolong; Serban, Adriana; Karlsson, Magnus; Gong, Shaofang |

Publication: | International Journal of Antennas and Propagation |

Date: | Jan 1, 2018 |

Words: | 3901 |

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