New Wilkinson Power Divider Based on Compact Stepped-Impedance Transmission Lines and Shunt Open Stubs

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1 1 New Wilkinson Power Divider Based on Compact Stepped-Impedance Transmission Lines and Shunt Open Stubs Rohith Soman Abstract- The report presents the simulation of Wilkinson Power divider based on stepped impedance transmission lines and open stubs. Stepped impedance transmission line is used to reduce the size of conventional Wilkinson power divider and open circuited stubs used to have desired band rejection. This report discusses both equal and unequal power split cases. That is a power split of 1:1, 3:2, 2:1 are discussing here. The circuits are analyzed using even and odd mode analysis. In equal split case 33.33% size reduction is achieved. For all simulation it is assumed that the substrate is 1.56mm thick with refractive index 4.4 (FR4 substrate). The power divider is designed for a center frequency. All the simulation is done with the tool ADS Keywords: Wilkinson Power divider, even and odd mode analysis, stepped impedance transmission line. P I. INTRODUCTION ower dividers are widely used in various microwave, communication, and high frequency applications. In the conventional circuit design, the circuits for an even number of two or more output signals have been proposed. In 1960, Ernest J. Wilkinson provided a topology of power dividers with the equivalent amplitude and in-phase outputs [1]. The scattering parameters for the common case of a 2-way equal-split Wilkinson power divider at the design frequency is given by [2] S = j Inspection of the S matrix reveals that the network is reciprocal (S ij = S ji ), that the terminals are matched (S ii = 0), that the output terminals are isolated (S 23, S 32 = 0), and that equal power division is achieved (S 12 = S 13 ). The non-unitary matrix results from the fact that the network is lossy. An ideal Wilkinson divider would yield S 12 = S 13 = - 3dB. Network theorem governs that a 3 port divider cannot satisfy all three conditions (being matched, reciprocal and loss-less) at the registered for M.E. and are affiliated to Electrical and Communication Engineering Department. same time. Wilkinson divider satisfies the first two (matched and reciprocal), and cannot satisfy the last one (being lossless). Hence, there is some loss occurring in the network. No loss occurs when the signals at ports 2 and 3 are in phase and have equal magnitude. However, there are some obstacles to conventional circuit design, such as large circuit occupation, achieving multi-band response, and poor stopband rejections. Because of the same reason so many researches are going on in this field. The main reason for the circuit size of Wilkinson Power divider is the λ/4 length transmission line in each transmission path. Replacing these λ/4 transmission lines with different architectures makes them possible to achieve small circuit sizes. This report study a design in which the λ/4 branches of the conventional Wilkinson power dividers replaces with compact stepped-impedance transmission line sections with shunt-toground capacitors [3]. The shunt-to-ground capacitors in the proposed dividers can be formed easily by planner open stubs. With the appropriate lengths, these stubs can create additional transmission zeros to improve the required band rejections. II. METHODOLOGY Fig 1. Conventional Wilkinson Power divider As shown in Fig 1. Conventional Wilkinson Power divider is having two λ/4 length transmission lines (Za and Zb) and an isolation resistance (R). These two λ/4 length transmission lines increases the total size of the divider. To avoid these λ/4 sections we are using stepped impedance transmission line sections as shown in fig 2. In this design Za1 and Za2 and their electrical length θa1 and θa2 are different; similarly Zb1 and Zb2 and their electrical length θb1 and θb2 are also

2 2 different. For required band rejection two capacitors (C1 and C2) are provided, this can be implemented using distributed components (stunt open stubs). Isolation resistance is also provided to get proper isolation between output ports. The impedances (ZieR and ZieL) looking into both sides must be equal at the interface A for matching requirement. The resulting equation can be written as 1 Z0 = 1 + jωc Zie + jza2 tan θa2 Za2 Za2 + jzie tan θa2 Where 2Z0 + jza1 tan θa1 Zie = Za1 Za1 + j2z0 tan θa1 Rearranging and separating the real and imaginary parts leads to two independent equations. Fig 2.Porpossed Wilkinson Power divider III. ANALYSIS A. Equal Split Wilkinson Power Divider Even and odd mode analysis can be done to obtain the circuit parameters of equal split power divider. For equal split the two arms must be identical that is Za1 = Zb1, Za2 = Zb2, θa1= θb1, θa2 = θb2 and C1 = C2 = C as shown in fig 3 Za1 Za2 + Za1 2 2Za2 2 tanθa1 tanθa2 = ωcza1 Za2 (Za1tanθa1Za2 tan θa2) Za1 2 Za2 tanθa1 + Za1Za2 2 tanθa2 = 2Za2 2 ωcza2 Za1 tanθa1tanθa2 + 2 Z0 2 (Za2tanθa1 + Za1tanθa2) Fig 5. Odd mode equivalent circuit diagram Fig 3.Equal Split Wilkinson Power divider In even mode analysis apply equal voltage to both port two and port three and terminate port one with matched impedance. Equivalent circuit is shown in the fig 4. In odd mode analysis apply equal and opposite voltage port one and port two and terminate port one with a matched load. Equivalent circuit is shown in fig 5. In both even and odd mode analysis both the arms will be same. Fig 4.Even mode equivalent circuit diagram Similar to even mode analysis for matching requirements the impedances (ZioR and ZioL) looking into both sides must be equal 1 Z0 = 1 Zio + 2 R + jωc Where Za1 tan θa1 + Za2 tan θa2 Zio = Za2 Za2 Za1 tan θa1 tan θa2 Rearranging and separating the real and imaginary parts leads to two independent equations. R = 2Z0 Za2 Za1 tan θa1 tan θa2 ωc = Za1Za2 tan θa1 Za2 2 tan θa2 The above four independent equation can be used to design equal split Wilkinson power divider based on compact stepped impedance transmission line. B. Unequal Split Wilkinson Power Divider For unequal split divider the upper and lower arms should be different. Here also we can use even and odd mode analysis to get design equations. Fig 6 shows the proposed unequal split power divider with characteristic impedance and electrical length of each stub. Fig 7 and Fig 8 show the even mode equivalent circuit diagram of upper and lower arm

3 3 respectively. And Fig 9 and Fig 10 show the odd mode equivalent circuit diagram of upper and lower arm respectively. Zb1 2 Zb2 tanθb1 + Zb1Zb2 2 tanθb2 = Z1yZ3ωC2Zb2 Zb1 Zb2tanθb1tanθb2 + Z1yZ3(Zb2tanθb1 + Zb1tanθb2) Fig 6.Unequal Split Wilkinson Power divider Fig 9.Odd mode equivalent circuit diagram of upper arm For matching requirement upper arm the impedances (ZieRx and ZieLx) looking into both sides must be equal at the interface A. That can be written as Z2 = Rx Za2 Za1 tan θa1 tan θa2 ωc1 = Za1Za2 tan θa1 Za2 2 tan θa2 Fig 7.Even mode equivalent circuit diagram of upper arm For matching requirement upper arm the impedances (ZieRx and ZieLx) looking into both sides must be equal at the interface A. That can be written as Z1x Z2 Za1Za2 + Z2Za1 2 Z1xZa2 2 tanθa1 tanθa2 = ωc1za1 Za2 Z2 Za1tanθa1 + Za2 tan θa2 Za1 2 Za2 tanθa1 + Za1Za2 2 tanθa2 = Z1xZ2ωC1Za2 Za1 Za2tanθa1tanθa2 + Z1xZ2(Za2tanθa1 + Za1tanθa2) Fig 8.Even mode equivalent circuit diagram of lower arm For matching requirement lower arm the impedances (ZieRy and ZieLy) looking into both sides must be equal at the interface B. That can be written as Z1y Z3 Zb1Zb2 + Z3Zb1 2 Z1yZb2 2 tanθb1 tanθb2 = ωc2zb1 Zb2 Z3 Zb1tanθb1 + Zb2 tan θb2 Fig 10.Odd mode equivalent circuit diagram of lower arm For matching requirement upper arm the impedances (ZieRx and ZieLx) looking into both sides must be equal at the interface A. That can be written as Z3 = Ry Zb2 Zb1 tan θb1 tan θb2 ωc2 = Zb1Zb2 tan θb1 Zb2 2 tan θa2 As in the conventional unequal split power divider the power ratio (Port two to port three) k 2 can be written as k 2 = Z3 Z2 = Z1y Z1x = Ry Rx From the even and odd mode equivalent circuit R = Rx + Ry Z1 = Z1x Z1y The above equations can be used to design an unequal split power divider based on compact stepped impedance transmission line with power ratio k 2.

4 4 IV. DESIGN AND RESULTS DISCUSSION A. Equal Split Wilkinson Power Divider All the design parameters can be calculated from the equations given in analysis section. Here, the center frequency selected is and one of the solution set to the equations is system impedance Z0 = 50Ω, isolation resistance R = 100Ω, shunt to ground capacitor C = 0.75pF, characteristic impedances Za1 = 43.03Ω and Za2 = 129.3Ω, and the electrical length θa1 = 32.7 and θa2 = From the electrical length values it is clear that total electrical length is only 60.1, instead of 90 in conventional Wilkinson power divider. That is a size reduction of 33.33% is achieved by using stepped impedance transmission line section instead of λ/4 section. The capacitor is realized by equivalent open stub to produce additional transmission zeros in each path to improve the required stopband level. Length of the open stub in each transmission path is approximately λ/4 at 4GHz. All the structures are implemented on a substrate of thickness 1.56mm with refractive index 4.4 (FR4 substrate) i. Schematic diagram and Layout ii. Simulation Results Fig. 11 Schematic of equal split power divider Fig. 12 Layout of equal split power divider Fig 11 shows the schematic of equal split power divider with the designed values. Fig 12 shows corresponding layout. Fig. 13: S- Parameters of equal split Power divider

5 5 Power division Matching Isolation Simulated Results S 12 = S 13 = S 12 = S 13 = 3.85GHz S 11 = S 22 = S 33 = S 23 = S 23 = 3.85GHz Table 1: [S] Parameter of equal split power divider From Table 1 it is clear that equal power division is achieved by the circuit and there is good isolation between the output ports. And good band rejection at 3.85GHz B. Unequal Split Wilkinson Power Divider (k 2 = 3/2) This design is for an Unequal split power divider with power ratio 3:2. All the design parameters can be calculated from the equations given in analysis section. Here, the center frequency selected is and one of the solution set to the equations is Z1x = 83.33Ω, Z1y = 125Ω, Z1 = 50Ω, R = 100Ω, Za1 = 36.01Ω, θa1 = 35.4, Za2 = 126.7Ω, θa2 = 21.8, C1 = 0.96pF, Z2 = 40Ω, Z3 = 60Ω, Zb1 = 46.46Ω, θb1 =25.3, Zb2 = 129.5Ω, θb2 = 37.2 and C2 = 0.58pF. To realize capacitors C1 and C2 open transmission lines are used to have transmission zeros at 3.5GHz and 4.5GHz. λ/4 transmission lines must be added to the two output ports for matching requirements. i. Schematic diagram and Layout ii. Simulation Results Fig. 14 Schematic of unequal split power divider (k 2 = 3/2) Fig 15 Layout of unequal split power divider (k 2 = 3/2)

6 6 i. Schematic diagram Fig. 17 Schematic of unequal split power divider (k 2 = 3/2) with two stubs i. Simulation Results Fig. 16: S- Parameters of unequal split Power divider (k 2 = 3/2) Power division Matching Isolation Simulated Results S 12 = S 13 = S 12 = 4GHz S 13 = 4GHz S 11 = S 22 = S 33 = S 23 = S 23 = 4GHz Table 2: [S] Parameter of unequal split power divider (k 2 = 3/2) The simulation results shows that S 12 = dB, S 13 = dB at. So the power division ratio obtained is And very high band rejection at 4GHz. Isolation between output port obtained as dB at 2GHaz C. Unequal Split Wilkinson Power Divider (k 2 = 3/2) with two open stubs in each transmission path Here all the design values remains same, one change is two open stubs are used in each transmission path to have two transmission zeros in each path. The capacitors are designed for 4.5GHz and 5.5GHz.

7 7 D. Unequal Split Wilkinson Power Divider (k 2 = 2/1) This design is for an Unequal split power divider with power ratio 2:1. All the design parameters can be calculated from the equations given in analysis section. Here, the center frequency selected is and one of the solution set to the equations is Z1x = 75Ω, Z1y = 150Ω, Z1 = 50Ω, R = 100Ω, Za1 = 32.00Ω, θa1 = 37, Za2 = 125.0Ω, θa2 = 20.0, C1 = 0.96pF, Z2 = 33Ω, Z3 = 67Ω, Zb1 = 50.5Ω, θb1 =23.3, Zb2 = Ω, and θb2 = To realize capacitor C1 open transmission lines are used to have transmission zero at 4.0GHz. λ/4 transmission lines must be added to the two output ports for matching requirements. i. Schematic diagram Fig. 19 Schematic of unequal split power divider (k 2 = 2/1) ii. Simulation Results Fig. 18: S- Parameters of unequal split Power divider (k 2 = 3/2) with two stubs Power division Matching Isolation Simulated Results S 12 = S 13 = S 12 = 4.35GHz S 13 = 4.35GHz S 12 = 5.5GHz S 13 = 5.5GHz S 11 = S 22 = S 33 = S 23 = S 23 = 4.35GHz S 23 = 5.5GHz Table 3: [S] Parameter of unequal split power divider (k 2 = 3/2) with two stubs

8 8 Power division Matching Isolation Simulated Results S 12 = S 13 = S 12 = 3.3GHz S 13 = 4GHz S 11 = S 22 = S 33 = S 23 = S 23 = 4GHz Table 4: [S] Parameter of unequal split power divider (k 2 = 2/1) The simulation results shows that S 12 = dB, S 13 = dB at. So the power division ratio obtained is But as the ratio increases the width of transmission line in one arm reduces which make it impossible to fabricate for higher division ration. For 2:1 ratio minimum width is in the order of 0.2mm. V. EXPERIMENT RESULTS Wilkinson power divider based on compact stepped impedance transmission line for unequal power division is fabricated on a FR4 substrate for a power division ratio of 3:2. Fig 19 shows the fabricated device. The fabricated device is tested using a network analyzer and the results are given in fig 20. Fig 19. Fabricated unequal power divider with k 2 =3/2 Fig. 18: S- Parameters of unequal split Power divider (k 2 = 2/1)

9 9 Power division Matching Isolation Fig. 20: Measured S parameters of the designed power divider. Simulated Results S 12 = S 13 = S 12 = 4GHz S 13 = 4GHz S 11 = S 22 = S 33 = S 23 = S 23 = 4GHz Experiment Results S 12 = S 13 = S 12 = 3.6GHz S 13 = 4.36GHz S 11 = S 22 = S 33 = S 23 = S 23 = 4GHz Table 5. Comparison between simulated and measured s parameters with k 2 =1.5 A. Result Discussion The area occupied by the fabricated device is 3.6cm x 3.5cm. For all the experiment results plots frequency vary from 1GHz to 5GHz. The designed power division ratio is 1.5, but measurement shows that it is 1.58 that is only 5.1% deviation from the designed value. In the upper transmission path we are getting a transmission zero 3.6GHz and for lower transmission path we are getting a transmission Zero at 4.36GHz. But both were designed for 4GHz. The results show that matching at all the ports are good. And as expected we are getting good isolation between output ports at designed frequencies. VI. CONCLUSION This report analyzed 3 ports Wilkinson power divider to reduce the size of it by replacing the λ/4 transmission line with a compact stepped impedance transmission line. This report also discussed one method of band rejection using shunt to ground capacitor. The capacitor is implemented using

10 10 distributed component. This report also presented equal split and unequal split power dividers. The designed values and experiment results are matching to a good extend. ACKNOWLEDGMENT The author would like to thank all the Microwave Lab(ECE) staff members and Packaging Lab (CEDT) junior scientific assistant Mr. Antonysamy for the support and also would like to express gratitude to Prof. K. J. Vinoy for continuous guidance. REFERENCES [1] E. J. Wilkinson, An N-way power divider, IEEE Trans. Microw. Theory Tech., vol. MTT-8, pp , Jan [2] Pozar, D. M., Microwave Engineering, 2nd edition, Chapter 7, Wiley, New York, [3] Deng, Guo, kuo, new wilkinson power dividers based on compact stepped-impedance transmission linesand shunt open stubs, Progress In Electromagnetics Research, Vol. 123, 407{426, 2012.