Abstract—In this paper, the design of miniaturized unequal-split Wilkinson power divider (WPD), with 1:2 split ratio, using non-uniform transmission lines (NTLs) is presented. To achieve compactness, the uniform transmission lines of the conventional WPD are substituted by their equivalent NTLs. Two extra compact NTLs transformers are incorporated in each arm of the divider for output ports matching purposes. To prove the validity of the design procedure, the proposed divider is simulated using full-wave simulators, fabricated and tested. Both simulation and measurement results are in good agreement. Using NTLs, a size reduction of 22.2% is achieved, besides odd harmonics suppression.
Keywords—divider; Wilkinson power divider; non-uniform transmission line
I. INTRODUCTION Microwave power dividers are essential components in modern microwave applications, such as antenna feed networks, phase shifters, and frequency mixers. Since its invention back in 1960 [1], the Wilkinson power divider (WPD) has been considered as one of the most important dividers in microwave circuits. Recently, WPDs have been notably addressed by researchers in many different aspects, such as reducing the size of their overall circuit area. The use of non-uniform transmission lines (NTLs) as one of the miniaturization techniques was presented in many papers [2-6]. In [2], an equal-split WPD was miniaturized using NTLs, and a size reduction of 52% was achieved. In [3], and as an extension to what was done in [2], a dual band WPD was proposed with 26% reduction in size (compared to the conventional dual-band WPD). In [4] and [5], NTL-based Bagley power divider and branch line coupler were presented, respectively. A general design procedure for NTLs-based compact multi-band equal-split WPD was proposed in [6]. Moreover, many miniaturization techniques were introduced in the literature to accomplish compactness, such as the use of stubs. In [7, 8], dual band compact WPDs were proposed in which stubs were incorporated to gain a significant size reduction of the circuit area. In [9], WPD has been miniaturized using stubs too, where artificial TLs have been used to accomplish the design. In [10], a stepped impedance interdigital coupling element has been used to achieve the compactness for a single band WPD and to suppress the odd
harmonics. In this paper, based on NTLs theory, compact unequal-split WPD, with 1:2 split ratio, is presented. This is in contrast to [2, 6], where an equal-split WPD was considered. To achieve compactness, the conventional uniform arms of the divider are replaced by their equivalent NTLs at a specific design frequency. The proposed divider is then, simulated using two full-wave simulators to prove the validity of the design procedure. Moreover, the designed unequal-split WPD is fabricated and measured, and both simulation and measurement results are in good agreement.
II. DESIGN OF CMPACT NTLS The key step in designing compact NTLs is to find an equivalent NTL for a certain uniform transmission line (UTL) at a specific design frequency, keeping in mind that the length of the equivalent NTL (d) should be less than the UTL length (d0), as illustrated in Figure 1.
As shown in Figure 1, the equivalent NTL has a varying characteristic impedance Z(z), and propagation constant β(z), compared to the conventional uniform TL, that has a constant characteristic impedance Z0, and propagation constant β0. The ABCD matrix of the UTL is given as follows [11]:
Figure 1. Uniform TL and its equivalent NTL
2011 IEEE Jordan Conference on Applied Electrical Engineering and Computing Technologies (AEECT)
where θ0 is the electrical length of the UTL at the design frequency. In order to characterize the NTL section, it is firstly subdivided into K uniform electrically short sections. The overall ABCD matrix of the whole NTL can be obtained by multiplying the ABCD matrices of these uniform sections as follows [11]: … … … … 2
where the ABCD parameters of the ith section can be expressed as follows [12]:
cos 3. 12 ∆ sin ∆ 3. b
∆ sin ∆ 3.
where Δ 2 ∆ 2 ∆
Then, the normalized characteristic impedance is expanded as follows [10]: ln cos 2 4
So, an optimum designed compact length NTL has to have its ABCD parameters as close as possible to the ABCD parameters of the UTL at a specific frequency. Therefore, the optimum values of the Fourier coefficients Fn’s can be obtained through minimizing the following error function [12]: 14 | | | | | | | | 5
This error function should be restricted by some constraints, such as reasonable fabrication and physical matching, as follows:
( ) maxminZ Z z Z≤ ≤ (6.a)
( ) ( )0 1Z Z d= = (6.b)
So, the goal is to find the Fourier coefficients values (Fn’s) that give an NTL that has its ABCD parameters approximately equal to those of the UTL by minimizing the above error function at a specific design frequency. To solve the above constrained minimization problem, the MATLAB function “fmincon.m” is used.
III. 1:2 NTL-BASED WPD The conventional unequal-split WPD parameters can be calclulated using the following equations [11]: 1 7. a
1 7. b
7.
where is the power ratio between ports 3 and 2, i. e., . Since the designed divider is of unequal-split type,
output port 2 has an impedance of , while output port 3 has an impedance of [11].
To obtain 1:2 split ratio (k2 = 0.5), the unequal-split WPD parameters are found to be: Z02=51.5 Ω, Z03=103 Ω, R=106.06 Ω, R2=35.36 Ω and R3=70.71 Ω (considering a reference impedance Z0= 50 Ω). Finally, to match the output ports to 50 Ω, quarter-wavelegnth matching transfromers are needed. The characteristic impedances of these matching transofrmers are calculated as follows: for port 2: √35.355 50 42.045 Ω; and for port 3: √70.71 50 59.46 Ω. Considering an FR-4 susbtrate (with a thickness of 1.6 mm and dielectric constant of 4.6), and a design frequency of 1 GHz, a schematic diagram for the conventional 1:2 WPD (using Ansoft Designer [13]) including the output ports matching transformers is shown in Figure 2.
Figure 2. Schematic diagram of the 1:2 WPD including the output ports matching transformers. The rectangles represent uniform microstrip lines with W as the microstrip line width
and P its physical length.
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Now, each uniform microstrip line section in Figure 2 is replaced by its equivalent compact NTL. Table I shows the parameters used in the optimization process. It should be pointed out here that the optimized NTL WPD arms lengths were originally chosen as 26.87 mm and 30 mm, respectively. A frequency shift in the design frequency appeared due to the discontinuities effects, and to overcome this frequency shift, the arms’ lengths were shortened such that each NTL arm becomes equivalent to a uniform arm at the design frequency. Figure 3 shows both sections before and after shortening the arms’ lengths. Two output ports NTL matching transformers have been also designed. The length (d) of these transformers is 24.3 mm and 25.7 mm. Figure 4 shows the the layout of the compact NTL-based 1:2 WPD.
TABLE I. PARAMETERS OF THE UTL AND NTL USED IN THE OPTIMIZATION
The optimized Fourier coefficients of the NTL-based WPD main arms and output ports matching transformers are shown in Tables II-V.
TABLE II: OPTIMIZED FOURIER COEFFICIENTS FOR ARM 1
-0.1837 -0.9998 0.3825 0.5502 -0.0350 -0.0310
0.1897 0.1413 -0.1153 0.0471 0.0541
TABLE III. OPTIMIZED FOURIER COEFFICIENTS FOR ARM 2
-0.0974 -0.7283 -0.4308 -0.0796 0.2060 0.3467
0.3431 0.2484 0.1337 0.0496 0.0086
TABLE IV. 42.045 Ω TRANSFORMER FOURIER COEFFICIENTS
-0.2123 -0.9423 0.6728 0.1704 -0.0912 0.2111
0.1328 -0.0429 0.0783 0.0432 -0.0199
TABLE V. 59.46 Ω TRANSFORMER FOURIER COEFFICIENTS
-0.1816 -0.9822 0.2789 0.3590 -0.1426 -0.1584
0.1206 0.1952 0.1683 0.2078 0.1350
The proposed WPD is simulated using IE3D [14] and HFSS [15] full-wave simulators. Simulation results are shown in Figure 5 which validates the design. Specifically, at the design frequency (1 GHz), S11= -27.88 dB using IE3D, while it equals -25.55 dB using HFSS. S21 equals -2.52 dB and -2.4 dB using IE3D and HFSS, respectively, which are close to the theoretical value of -1.76 dB. S31 equals to -4.76 dB in IE3D and -4.77 dB using HFSS, which are almost equal the theoretical value of -4.77 dB. S22 and S33 are -27.94 dB and -21.54 dB, respectively, using IE3D; and around -25.1 dB and -24.94 dB, respectively, using HFSS. Finally, S23 equals -33.49 dB using IE3D, and -23 dB using HFSS. The differences between the theoretical and simulation results are due to dielectric losses, coupling effects, and discontinuities effects.
UTL NTL Constraints WPD section1
51.49 Ω 40.31 3.4376
50 26.87
0.3 128.53 2.496 15 15.2880.297
WPD section2
103Ω 42.28 3.1234
50 30
0.3 128.53 1.248 15 15.2880.1484
Figure 3. The two NTL arms of the WPD before and after length reduction (a) upper arm (b) lower arm.
Figure 4. The proposed NTL-based WPD layout. (Dimensions are in mm)
2011 IEEE Jordan Conference on Applied Electrical Engineering and Computing Technologies (AEECT)
For verification purposes, the NTL-based WPD is fabricated and measured using an Agilent Spectrum Analyzer (with a built in tracking generator extending
from 0-1.5 GHz). Figure 6 shows the measured results of the 1:2 WPD, while Figure 7 shows a picture of the fabricated WPD. Experimental results show an acceptable agreement between both simulated and measured results. The small discrepancies in the measured results could be due to conductor and dielectric losses, the use of the connectors and the errors in the measurements, keeping in mind that a spectrum analyzer (not a network analyzer) was used.
IV. COMPARISON BETWEEN CONVENTIONAL AND COMPACT WPD
Figure 8 shows the layout of the conventional 1:2 WPD structure and Figure 9 shows the simulated S-parameters using IE3D. Using NTLs instead of UTLs, two main advantages are obtained: (1) the size reduction and (2) the odd harmonics suppression. A total size
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Figure 5. S-parameters of the proposed NTL-based WPD using IE3D and HFSS
Figure 6. Measured S-parameters of the fabricated NTL-based WPD
Figure 7. Fabricated NTL-based 1:2 WPD
(c)
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reduction of almost 22.2% is achieved as shown in Figure 10.
Since both structures have the same ABCD paramters at the design frequency only, the NTLs WPD behaviour is completely different from the conventional one at other frequencies. As an illustration, Figure 11 shows the conventional WPD input port matching parameter (S11) in an extended frequency range. As expected, the conventional WPD operates at the design frequency (1 GHz) and its odd harmonics. The first odd harmonic is slightly above 3 GHz, and the second odd harmonic is slightly below 5 GHz. This slight frequency shift could be due to the T-junction, the right-angle bends, the step discontinuities, and the dependence of the effective permittivity on frequency. Figure 12 shows S11 for the NTL-based WPD in an extended frequency range. It is clear that the first odd harmonic has been suppresed completely while the second odd harmonic is partially suppressed. As mentioned before, both of the NTL-based WPD and the conventional WPD are equivalent at the design frequency only, which justifies the suppression or partial suppression of the odd harmonics. Furthermore a performance improvement is noticeable in the NTL-based WPD, since S11 is close to 0 dB at frequencies other than the fundamental frequency and the second odd harmonic, while in the conventional WPD, it is about -10 dB.
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Figure 8. Conventional 1:2 WPD layout
Figure 9. S-parameters of the conventional 1:2 WPD
Figure 10. Conventional WPD vs. NTL-based WPD (dimensions are in mm)
Figure 11. S11 of the conventional WPD in an extended frequency range.
2011 IEEE Jordan Conference on Applied Electrical Engineering and Computing Technologies (AEECT)
V. CONCLUSION In this paper, the design and analysis of a compact 1:2
unequal-split NTL-based WPD was presented. In order to achieve compactness, each uniform transformer was replaced by its equivalent NTL at the design frequency. Besides suppressing some of the odd harmonics of the design frequency, a size reduction of 22.2% was achieved compared to the conventional WPD. This work will be extended to design multi-band unequal-split WPD as was done in [6] for the equal-split WPD.
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Microwave theory and techniques, Vol. MTT-8, No. 1, pp. 116-118, 1960.
[2] F. Hosseini, M. Khalaj, A. Hosseini and M. Yazdani, “A Miniaturized Wilkinson Power Divider Using Non-uniform Transmission Line”, Journal of Electromagnetic Waves and Applications, Vol. 23, pp. 917-924, 2009.
[3] K. Shamaileh and N. Dib, “Design Of Compact Dual-Frequency Wilkinson Power Divider Using Non-uniform Transmission Line”, Progress In Electromagnetics Research C, Vol. 19, pp. 37-46, 2011.
[4] K. Shamaileh, A. Qaroot, and N. Dib, “Non-uniform Transmission Line Transformers And Their Applications In The Design Of Compact Multi-Band Bagley Power Dividers With Harmonics Suppression”, Progress In Electromagnetics Research, Vol. 113, pp. 269-284, 2011.
[5] F. Hosseini, M. Khalaj and M. Yazdany, “To Compact Ring Branch-Line Coupler using Nonuniform Transmission line”, Microwave And Optical Technology Letters, Vol. 51, No. 11, pp. 2679-2682, Nov. 2009.
[6] K. Shamaileh, A. Qaroot, N. Dib, and A. Sheta, “Design And Analysis Of Multi-Frequency Wilkinson Power Dividers Using Non-uniform Transmission Lines”, International Journal of RF and Microwave Computer-Aided Engineering, Vol. 21, No. 5, pp. 526-533, September 2011.
[7] L. Shao, H. Guo, X. Liu, W. Cai and L. Mao “A Compact Dual-Frequency Wilkinson Power Divider with open-ended stubs”, Signals Systems and Electronics (ISSSE), International Symposium,Vol. 1, pp. 1-4, 2010.
[8] Z. Wang, J. Jang and C. Park, “Compact dual-band Wilkinson power divider using lumped component resonators and open-circuited stubs”, Wireless and Microwave Technology Conference (WAMICON), pp. 1-4, June 2011.
[9] C.-H. Tseng and C.-H. Wu, “Compact planar Wilkinson power divider using pi-equivalent shunt-stub-based artificial transmission lines”, Electronics Letters, Vol. 46, pp. 1327-1328, 2010.
[10] P. Cheong, K. Lai, and K. Tam, “Compact Wilkinson power divider with simultaneous bandpass response and harmonic suppression”, IEEE MTT-S International Microwave Symposium Digest, pp. 1588-1591, 2010.
[11] D. Pozar, Microwave Engineering, New York: John Wiley, 3rd edition, 2005.
[12] M. Khalaj, “Nonuniform Transmission Lines As Compact Uniform Transmission Lines”, Progress In Electromagnetics Research C, Vol. 4, pp. 205-211, 2008.
[13] Ansoft Corporation, www.ansoft.com. [14] www.zeland.com , 2006. [15] HFSS: High Frequency Structure Simulation based on Finite Element
Method, V. 10, Ansoft Corporation, www.ansoft.com.
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Figure 12. S11 of the NTL-based WPD in an extended frequency range
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