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J Electromagn Eng Sci > Volume 21(1); 2021 > Article
Yoon and Kim: Unequal Power Dividers Using Uniform Impedance Transmission Lines with Stubs

Abstract

This study proposes an impedance control method in transmission lines using open- or short-circuit stubs for unequal power dividers. The proposed method is based on the conversion of a two-port to a three-port transmission line, which is equivalent to multiplying the impedance at the internal transmission line by a scaling factor and then connecting open- or short-circuit stubs in parallel to each port on the three-port transmission line. To verify the effectiveness of the proposed method, Wilkinson and Gysel power dividers with splitting ratios of 9:1 and 5:1, respectively, using uniform impedance transmission lines with open- or short-circuit stubs at an operating frequency of 2 GHz were designed. The experimental measurements of the two power dividers agree well with those of the simulation.

I. Introduction

Power dividers are passive components essential for configuring wireless communication and equipment. Recently, many studies have been conducted regarding the application of asymmetric splitters to circuits with various uses, such as Doherty amplifiers [1] and antenna feeders [2]. Conventional asymmetric splitters have employed quarter-wavelength transmission lines of low and high impedances according to splitting ratios. However, the transmission line width of asymmetric splitters is either extremely large or extremely small to be implemented using microstrip technology.
To overcome implementation problems such as that of the planar structure microstrip, a thick material with a low dielectric constant [35], a suspended strip structure [6], or a defected ground structure (DGS) [7] can be used to realize a high-impedance transmission line. In addition, because low-impedance transmission lines have very large line widths, a structure with open-circuit stubs connected in parallel to the transmission line can be used to implement a suitable low-impedance transmission line [8]. However, the methods that use thick material or mechanical structures for low- or high-impedance transmission lines are applicable under limited conditions only.
In this study, to implement high- or low-impedance transmission lines, we propose an impedance control method for converting a two-port transmission line into a three-port transmission line. This method uses a three-port transmission line with an adjusted scaling factor that increases or decreases the internal impedance of the converted transmission line. In addition, it operates the equivalent circuit as one that connects open- or short-circuit stubs in parallel to each port on the three-port impedance line.
Using this method, an asymmetric power divider was designed by converting its low- and high-impedance transmission lines into suitable uniform impedance transmission lines that were connected in parallel to open- or short-circuit stubs. As an example, a 9:1 asymmetric Wilkinson power divider and a 5:1 Gysel power divider [911] were designed and measured at an operating frequency of 2 GHz.

II. Theory of Impedance Control Method

Fig. 1(a) depicts a transmission line with a characteristic admittance Yo1 and electrical length θo. This transmission line can be modified with port 2 inserted in the middle, as shown in Fig. 1(b) [12, 13]. The electrical length of the modified transmission line is
(1)
θo=θo1+θo2
In this case, the admittance parameters of the transmission line to which port 2 is added are
(2)
(Y)=jYo1(-cot θ01csc θ010csc θ01-cot θ01-cot θ02csc θ020csc θ02-cot θ02).
Without affecting the termination impedance of port 1 and 3, the internal impedance of port 2 can be adjusted by multiplying row 2 and column 2 of (2) by a scaling factor β that describes the ratio of internal impedance to terminal impedance. The result obtained after multiplying the scaling factor is given by (3).
(3)
(Y)=jYo1(-cot θ01β·csc θ010β·csc θ01-β2·(cot θ01+cot θ02)β·csc θ020β·csc θ02-cot θ02).
The internal impedance converted circuit can be implemented as a transmission line circuit in which stubs are connected in parallel to each port. Its equivalent circuit is illustrated in Fig. 1(c). The admittance parameters of this equivalent circuit can be expressed as
(4)
(Y)EQ=j(-Yo1Tcotθ01TYo1Tcscθ01T0Yo1Tcscθ01T-Yo1Tcotθ01T-Yo2Tcotθ02TYo2Tcscθ02T0Yo2Tcscθ02T-Yo2Tcotθ02T)+j(B1000B2000B3).
Because (3) and (4) should be the same as those for equivalent circuits, the following expression can be obtained.
(5a)
B1-Yo1Tcotθ01T=-Yo1cotθ01
(5b)
Yo1Tcsc θ01T=β·Yo1csc θ01
(5c)
B2-(Yo1Tcot θ01T+Yo2Tcot θ02T)=-β2·Yo1(cot θ01+cot θ2)
(5d)
Yo2Tcsc θ02T=β·Yo1csc θ02
(5e)
B3-Yo2Tcot θ02T=-Yo1cot θ02
The susceptance Bi (i = 1, 2, 3) can be implemented in open-or short-circuit stubs with admittance values and electrical length indicated in (6).
(6a)
Bi=Yopen-stubtanθopen-stub
(6b)
Bi=-Yshort-stubcotθshort-stub
In (5b) and (5d), if the conditions satisfy the relation,
(7)
β>sin θ01sin θ01T         and         β>sin θ02sin θ02T,
the admittances Yo1, Yo1T, and Yo2T will have the following relationship:
(8)
Yo1T>Yo1,Yo2T>Yo1.
This means that if condition (7) is satisfied, the transmission line of the equivalent circuit in Fig. 1(c) can be converted to obtain an impedance value lower than the original one.
In addition, if the conditions satisfy the relation,
(9)
β<sin θ01sin θ01T         and         β<sin θ02sin θ02T,
the transmission line of the equivalent circuit can be converted into a line with an impedance value higher than the original one.
To obtain B1, B2, and B3, the conditions of (5b) and (5d) are substituted into (5c), and the result is as follows:
(10)
B1=Yo1Tcot θ01T-Yo1cot θ01
(11)
B2=β·Yo1[cos θo1T·csc θo1(1-β·cos θo1·sec θo1T)+cos θ02T·csc θo2(1-β·cos θo2·sec θ02T)]
(12)
B3=Yo2Tcot θ02T-Yo1cot θ02
Fig. 2(a) displays the variation of B1, B2, and B3 when θo1T and θo2T change from 0°–45° under characteristic admittances Yo1 = 0.003663 S and Yo1T = 0.01254 S, with electrical lengths θo1 = θo2 = 45°. Furthermore, Fig. 2(b) shows the variation of B1, B2, and B3 when θo1T and θo2T change from 0°–45° under characteristic admittances Yo1 = 0.03289 S and Yo1T = 0.01254 S, with electrical lengths θo1 = θo2 = 45°.

III. Simulation and Experimental Results

Fig. 3 presents the schematics of a Wilkinson power divider and a Gysel power divider with splitting ratio of 9:1 and 5:1, respectively. The impedance values of each quarter-wavelength transmission line are provided in Tables 1 and 2.
To design high- and low-impedance transmission lines in Tables 1 and 2 using the impedance control method, a Rogers printed circuit board with a dielectric constant of 3.48, a dielectric thickness of 0.762 mm, and a copper thickness of 0.035 mm was used.
The simulation was performed using Microwave Office software version 13 developed by Cadence Design Systems Inc.
In the Wilkinson power divider, first, for the transmission line of 273.8 Ω with θo1 = θo2 = 45°, a microstrip line width capable of implementing this line was selected as 0.7 mm (characteristic impedance of 79.7 Ω), and the electrical lengths were determined to be θo1T = θo2T = 36°. The scaling factor of this design condition was β = 4.13348, and its susceptance values were calculated as B2 = −0.09024 and B1 = B3 = 0.01362.
Because B2 is negative, the short-circuit stub was implemented at a line width of 0.4 mm (characteristic impedance of 99.8 Ω) and an electrical length of 6.3° using 0.4 mm via holes. Further, because the B1 and B3 values are positive, the line width of the open-circuit stub was implemented at 3.5 mm (characteristic impedance of 30.5 Ω) and an electrical length of 22.5°.
Second, for the transmission line of 30.4 Ω with θo1 = θo2 = 45°, a microstrip line width capable of implementing this line was selected as 0.7 mm (characteristic impedance of 79.7 Ω), while the electrical lengths were determined to be θo1T = θo2T = 31.5°. The scaling factor of this design condition was β = 0.51666, and its susceptance values were calculated as B2 = 0.0234 and B1 = B3 = −0.01239.
Using the B2 value, the line width of the implemented open-circuit stub was implemented at 3.5 mm (characteristic impedance of 30.5 Ω) and an electrical length of 35.5°.
In addition, using the B1 and B3 values, the line width of the short-circuit stub was implemented at 0.4 mm (characteristic impedance of 99.8 Ω) and an electrical length of 38.9° with 0.4 mm via holes.
At the intersection point of Zo1 and Zo2, the open-circuit stub susceptance with B1 = 0.01362 at Zo1 and the short-circuit stub susceptance with B1 = −0.01239 at Zo2 meet; therefore, we can implement the difference between the two values, i.e., susceptance Bx = 0.00123. For this susceptance, the open-circuit stub has a line width of 1.2 mm (characteristic impedance of 60.9 Ω) and an electrical length of 4.3°.
In the Gysel power divider, first, for the transmission line of 183.1 Ω with θo1 = θo2 = 45°, a microstrip line width of 0.7 mm was selected, which was capable of implementing this line (characteristic impedance of 79.7 Ω), and the electrical lengths were determined to be θo1T = θo2T = 31.5°. The scaling factor of this design condition was β = 3.10962, and its susceptance values were calculated as B2 = −0.06465 and B1 = B3 = 0.01501.
Because B2 is negative, the short-circuit stub was implemented at a line width of 0.4 mm (characteristic impedance of 99.8 Ω) and an electrical length of 2.4° using 0.4 mm via holes. Further, because the B1 and B3 values are positive, the line width of the open-circuit stub was implemented at 3.5 mm (characteristic impedance of 30.5 Ω) and an electrical length of 39°.
Second, for the transmission line of 111.8 Ω with θo1 = θo2 = 45°, a microstrip line width capable of implementing this line was selected as 0.7 mm (characteristic impedance of 79.7 Ω), and the electrical lengths were determined to be θo1T = θo2T = 31.5°. The scaling factor of this design condition was β = 1.89833, and its susceptance values were calculated as B2 = −0.02352 and B1 = B3 = 0.01153.
Because B2 is negative, the short-circuit stub was implemented at a line width of 0.4 mm (characteristic impedance of 99.8 Ω) and an electrical length of 23.1° using 0.4 mm via holes. Further, because the B1 and B3 values are positive, the line width of the open-circuit stub was implemented at 3.5 mm (characteristic impedance of 30.5 Ω) and an electrical length of 35.1°.
Third, for the transmission line of 36.6 Ω with θo1 = θo2 = 45°, a microstrip line width capable of implementing this line was selected as 0.7 mm (characteristic impedance of 79.7 Ω), and the electrical lengths were determined to be θo1T = θo2T = 27°. The scaling factor of this design condition was β = 0.71577, and its susceptance values were calculated as B2 = 0.02127 and B1 = B3 = −0.00268.
Because B1, and B3 are negative, the short-circuit stub was implemented at a line width of 0.4 mm (characteristic impedance of 99.8 Ω) and an electrical length of 23.7° using 0.4 mm via holes. Further, because B2 is positive, the line width of the open-circuit stub was implemented at 3.5 mm (characteristic impedance of 30.5 Ω) and an electrical length of 33°.
Fourth, for the 22.4 Ω transmission line with θo1 = θo2 = 45°, a microstrip line width capable of implementing this line was selected as 0.7 mm (characteristic impedance of 79.7 Ω), and the electrical lengths were determined to be θo1T = θo2T = 27°. The scaling factor of this design condition was β = 0.43697, and its susceptance values were calculated as B2 = 0.03217 and B1 = B3 = −0.02010.
Because B1, and B3 are negative, the short-circuit stub was implemented at a line width of 0.4 mm (characteristic impedance of 99.8 Ω) and an electrical length of 23.7° using 0.4 mm via holes. Further, because B2 is positive, the line width of 3.5 mm of the open-circuit stub was implemented (characteristic impedance of 30.5 Ω) and an electrical length of 44.5°.
Fig. 4 depicts a photograph of the devised unequal Wilkinson power divider after the optimization process. Because there is no standard resistance value, the isolation resistance is implemented by connecting 150 Ω and 15 Ω in series.
Fig. 5 shows the S-parameters of the simulated and measured unequal Wilkinson power dividers. The insertion losses of |S21| and |S31| were −0.96 dB and −10.01 dB, respectively. Furthermore, the isolation of |S32| was greater than 35 dB. The input return loss of |S11| was less than −25 dB and the output return losses of |S22| and |S33| were less than −20 dB at a center frequency of 2 GHz. The measured −15dB bandwidth of |S11| was in the range of 1.88 to 2.08 GHz, featuring a fractional bandwidth of 10%. In addition, the phase difference between the output ports was measured within ±15°.
In addition, Fig. 6 depicts a photograph of the fabricated unequal Gysel power divider after the optimization process. Because there is no standard resistance value, the isolation resistance is implemented by connecting 100 Ω and 47 Ω, in series, and 22 Ω.
Fig. 7 displays the S-parameters of the simulated and measured unequal Gysel power dividers. The insertion losses of |S21| and |S31| were −1.92 dB and −7.74 dB, respectively. Furthermore, the isolation of |S32| was greater than 15 dB. The input return loss of |S11| was less than −25 dB, and the output return losses of |S22| and |S33| were less than −10 dB at a center frequency of 2 GHz. The measured −15dB bandwidth of |S11| was in the range of 1.91 to 2.03 GHz, featuring a fractional bandwidth of 6%. In addition, the phase difference between the output ports was measured within ±4°.
It can be observed that these measurement results are almost identical to the simulation results in Figs. 5 and 7. Table 3 shows a comparison of the proposed divider to the conventional unequal divider.

IV. Conclusion

This study applied an impedance control method to the design of a Wilkinson and a Gysel power divider with splitting ratios of 9:1 and 5:1, respectively. In this method, the high impedance line was separated into two lines, and open- and short-circuit stubs were connected to three ports in parallel. This method is considerably more convenient than the conventional one using thick materials to implement high-impedance lines.
The method presented here, which can easily implement high-impedance lines, can be used for the design of various parts.

Fig. 1
(a) Arbitrary transmission line, (b) transmission line with port added in the middle, (c) equivalent circuit to (b).
jees-2021-21-1-44f1.jpg
Fig. 2
(a) Variation of B1, B2, and B3 according to θo1T, and θo2T at admittances Yo1 = 0.003663 S and Yo1T = 0.01254 S, and electrical lengths θo1 = θo2 = 45°. (b) Variation of B1, B2, and B3 according to θo1T, and θo2T at admittances Yo1 = 0.03289 S and Yo1T = 0.01254 S, and electrical lengths θo1 = θo2 = 45°.
jees-2021-21-1-44f2.jpg
Fig. 3
Schematic of (a) unequal Wilkinson power divider and (b) unequal Gysel power divider.
jees-2021-21-1-44f3.jpg
Fig. 4
Photograph of the proposed Wilkinson power divider.
jees-2021-21-1-44f4.jpg
Fig. 5
Simulated and measured S-parameters of the proposed Wilkinson power divider: (a) |S11|, |S21|, |S31|, and phase difference (∠S21–∠S31) and (b) |S32|, |S22|, and |S33|.
jees-2021-21-1-44f5.jpg
Fig. 6
Photograph of the proposed Gysel power divider.
jees-2021-21-1-44f6.jpg
Fig. 7
Simulated and measured S-parameters of the proposed Gysel power divider: (a) |S11|, |S21|, |S31|, and phase difference (∠S21–∠S31) and (b) |S32|, |S22|, and |S33|.
jees-2021-21-1-44f7.jpg
Table 1
Impedance values of unequal Wilkinson power dividers with a 9:1 splitting ratio
Parameter Value
Zo1 (Ω) 273.8
Zo2 (Ω) 30.4
Zo3 (Ω) 86.6
Zo4 (Ω) 28.8
R (Ω) 167
Table 2
Impedance values of unequal Gysel power dividers with a 5:1 splitting ratio
Parameter Value Parameter Value
Z1 (Ω) 183.1 Z6 (Ω) 22.4
Z2 (Ω) 36.6 Z7 (Ω) 74.8
Z3 (Ω) 111.8 Z8 (Ω) 33.5
Z4 (Ω) 22.4 R1 (Ω) 111
Z5 (Ω) 111.8 R2 (Ω) 22
Table 3
Comparison of the proposed divider to the conventional unequal divider
Ref. Center freq. (GHz) Dividing ratio Implement method Insertion loss (dB)
[7] 1.5 1:4 DGS −1.0 / −7.0
[9] 0.8–2.27 1:2 Microstrip/slotline −2.09 / −5.41
[10] 2 1:10 Coupled lines −0.75 / −10.3
[11] 1 1:10 Capacitive loaded TL −0.77 / −9.7
This work 2 1:9 Open/short-stub −0.96 / −10.1
1:5 −1.9 / −7.7

References

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9. F. Lin, QX. Chu, Z. Gong, and Z. Lin, "Compact broadband Gysel power divider with arbitrary power-dividing ratio using microstrip/slotline phase inverter," IEEE Transactions on Microwave Theory and Techniques, vol. 60, no. 5, pp. 1226–1234, 2012.

10. B. Li, X. Wu, and W. Wu, "A 10:1 unequal Wilkinson power divider using coupled lines with two shorts," IEEE Microwave and Wireless Components Letters, vol. 19, no. 12, pp. 789–791, 2009.
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12. Q. Wu, Y. Yang, Y. Wang, X. Shi, and M. Yu, "General model for loaded stub branch-line coupler," In: Proceedings of 2016 IEEE MTT-S International Microwave Symposium (IMS); San Francisco, CA. 2016; pp 1–4.
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Biography

jees-2021-21-1-44i1.jpg
Young-Chul Yoon received his B.S., M.S., and Ph.D. in electronics engineering from Sogang University, Seoul, South Korea in 1978, 1982, and 1989, respectively. In 1987, he joined the Department of Electronics Engineering, Catholic Kwandong University, Gangneung, South Korea, where he is currently a professor. His areas of interest are the design of high-power amplifiers for the ISM band, and RF and microwave circuit analysis and design.

Biography

jees-2021-21-1-44i2.jpg
Young Kim received his B.S., M.S., and Ph.D. in electronics engineering from Sogang University, Seoul, South Korea in 1986, 1988, and 2002, respectively. He developed cellular and PCS linear power amplifiers at Samsung Electronics Co. Ltd. In 2003, he joined the School of Electronics Engineering, Kumoh National Institute of Technology, Gumi, South Korea, where he is currently a professor. His areas of interest are the design of high-power amplifiers and linearization techniques, and RF and microwave circuit analysis and design.
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