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An ultra wideband Wilkinson powerdividerAli Reza Hazeri aa Young Researcher Club, Kermanshah Branch, Islamic AzadUniversity, Kermanshah, IranPublished online: 15 Nov 2011.

To cite this article: Ali Reza Hazeri (2012) An ultra wideband Wilkinson power divider, InternationalJournal of Electronics, 99:4, 575-584, DOI: 10.1080/00207217.2011.629227

To link to this article: http://dx.doi.org/10.1080/00207217.2011.629227

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International Journal of ElectronicsVol. 99, No. 4, April 2012, 575–584

An ultra wideband Wilkinson power divider

Ali Reza Hazeri*

Young Researcher Club, Kermanshah Branch, Islamic Azad University,Kermanshah, Iran

(Received 14 January 2011; final version received 30 July 2011)

In this article, an ultra wideband power divider is proposed, analysed anddesigned. The design approach of the proposed power divider is derived fromeven/odd mode analysis. The design approach is validated, and the power divideris simulated by two full-wave electromagnetic simulators (ADS and Sonnet) andfabricated on a substrate with a thickness of 0.635mm and relative constant of10.2. Measured results of the proposed power divider show equal power split,excellent insertion loss and good return loss at all the three ports, and a goodisolation between the two output ports is achieved over the specified 3.1–10.6GHz ultra wideband range. In the input port return loss, there are twotransmission poles around 5.2 and 9.5GHz. The overall size of the proposedpower divider is just 5:8� 4:3mm2.

Keywords: microstrip transmission line; ultra wideband; Wilkinson power divider

1. Introduction

Power dividers/combiners are one of the important elements in communication circuits.The most practical power divider is the Wilkinson power divider that has the features ofequal amplitude and phase outputs as well as reciprocal operation. The Wilkinson powerdivider is utilised for antenna polarisation, balanced amplifier, high-power transmitters,impedance matching and phase control in phased array antenna. Figure 1(a) shows theconventional microstrip Wilkinson power divider. It is constructed using two quarter-wavetransmission lines and a resistor (which is located between the output ports). Ultrawideband (UWB) radio technology has attracted the attention of those working in the fieldof wireless communication and other applications, since the FCC announced its decisionto allow the unlicensed use of the bandwidth of range 3.1–10.6GHz (First Report andOrder 2002). The Wilkinson power divider operates only at one design frequency and at allof its odd harmonics. However, its narrow bandwidth is a problem for UWB applications.Recently, wideband band Wilkinson power dividers (Cohn 1968; Lee, Kim, Choi, Park,and Ahn 2001; Abbosh, Bialkowski, and Mazierska 2006; Chiu, Yum, Xue, and Chan2006; Oraizi and Sharifi 2006; Abbosh 2007, 2008; Abbosh and Bialkowski 2007;Bialkowski and Abbosh 2007; Dib and Khodier 2008; Ou and Chu 2008; Wong and Zhu2008; Song and Xue 2010) and multi-band Wilkinson power dividers (Avrillon, Pele,

*Email: alirezahazeri@gmail.com

ISSN 0020–7217 print/ISSN 1362–3060 online

� 2012 Taylor & Francis

http://dx.doi.org/10.1080/00207217.2011.629227

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Chousseaud, and Toutain 2003; Srisathit, Chongcheawchamnan, and Worapishet 2003;Wu, Yilmaz, Bitzer, Pascht, and Berroth 2005; Wu, Sun, Yilmaz, and Berroth 2006; Chengand Wong 2007; Cheng and Law 2008; Park and Lee 2008; Zhang and Xin 2008; Wu, Liu,and Li 2009; Faraji, Nosrati, and Hazeri 2011) have been proposed to improve itsbandwith. Multi-section transmission line (Cohn 1968; Lee et al. 2001; Chiu et al. 2006;Oraizi and Sharifi 2006; Dib and Khodier 2008) and multi-section transmission line withtwo open-ended stubs (Ou and Chu 2008) have been applied to build a wideband powerdivider. In the study of Dib and Khodier (2008), Lee et al. (2001), Oraizi and Sharifi(2006), Chiu et al. (2006), Cohn (1968) and Ou and Chu (2008), several resistors were usedin order to have good isolation and return loss at the output ports. However, lumpedelements are not easy to design and fabricate at high frequencies. In the study ofBialkowski and Abbosh (2007), Abbosh (2007, 2008), Abbosh and Bialkowski (2007),Abbosh et al. (2006) and Song and Xue (2010), multilayer substrates were introduced tomake a wideband power divider. However, increasing the number of substrate layers is notacceptable for some printed circuit boards and their scattering parameters are not suitableover UWB range. In the study of Wong and Zhu (2008), two stepped-impedance open-ended stubs and parallel coupled lines in two output ports were used to build a widebandpower divider.

In this article, an UWB power divider is proposed, simulated and fabricated on asubstrate of thickness 0.635mm and relative dielectric constant 10.2.

2. Design of the proposed UWB power divider

The circuit topology of the proposed UWB power divider is depicted in Figure 1(b) for adivider with equal power division. The proposed UWB power divider consists of twomicrostrip transmission lines with characteristic impedance Z1 and electrical lengths � and�, a parallel coupled microstrip transmission line with characteristic impedance Z2 and

Figure 1. Circuit topologies of (a) conventional microstrip Wilkinson power divider, (b) proposedUWB power divider, (c) equivalent even-mode circuit of proposed power divider and (d) equivalentodd-mode circuit of proposed power divider.

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electrical length �, and a resistor, which shunts the output ports. Since the proposed UWBpower divider is symmetrical, the even- and odd-mode analyses can be utilised todetermine the proposed UWB power divider parameters. Circuit symmetry allowsdecomposition into the even- and odd-mode circuits by applying proper voltageexcitations at the output ports (Pozar 2005).

2.1. Even-mode excitation analysis

For the even-mode excitation analysis, two signals of the same magnitude and phase areapplied to ports 2 and 3 of the circuit shown in Figure 1(b). No current flows through theplane of symmetry. The circuit is bisected at the mid-plane, as shown in Figure 1(c). Whileno current flows through the plane of symmetry, the resistor (R) can be eliminated. Theimpedance at port 1 is doubled on the half-circuit. Mathematically, the ABCD matrix ofthe even-mode circuit can be written as follows (Pozar 2005):

cos� jZ1 sin�

ð j sin �Þ=Z1 cos�

� �1 1

ð j tanð�=2ÞÞ=Z1 0

� �

�cos� jZ2,e sin �

ð j sin�Þ=Z2,e cos�

� �¼

A B

C D

� �ð1Þ

where subscript ‘e’ denotes the even-mode excitation. The input impedance of the half-circuit may thus be derived as follows (Pozar 2005):

Zin,e ¼ 2Z0 ¼AZ0 þ B

CZ0 þDð2Þ

If the network is assumed to be reciprocal and lossless (A and D are real numbers; Band C are imaginary), from Equation (2), the input impedance matching condition can besatisfied by imposing the following condition to the proposed UWB power divider:

A ¼ 2D ð3Þ

B ¼ 2C ð4Þ

2.2. Odd-mode excitation analysis

For odd-mode excitation analysis, the two signals applied to ports 2 and 3 have the samemagnitude but are of opposite phases; there is a voltage null along the middle of the circuitshown in Figure 1(b). Therefore, this circuit is bisected by grounding the mid-plane, asshown in Figure 1(d). According to the transmission line theory, the output impedance ofthis half-circuit is obtained as follows (Pozar 2005):

Z0

odd ¼1

1jZ1 tan�

þ 1jZ1 tanð�=2Þ

ð5Þ

Z00

odd ¼ Z2Z0

odd þ jZ2,o tanð�Þ

Z2,o þ jZ0

odd tanð�Þð6Þ

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Zout ¼ Z0 ¼1

1Z00

odd

þ 2R

ð7Þ

where subscript ‘o’ denotes the odd-mode excitation. By combining Equations (5)–(7) and

separating the real and imaginary parts, the following equations can be expressed:

R ¼ 2Z0 ð8Þ

1

Z00

odd

¼ 0 ð9Þ

Finally, at the centre frequency ( f0), after some algebraic manipulation, the values of

circuit parameters are approximately calculated: Z1 ¼ 1:66, Z2,e ¼ 1:24, Z2,o ¼ 0:92,� ¼ 98�, � ¼ 70� and � ¼ 20�, where all characteristic impedances are normalised to the

port impedance (Z0). Theoretically, there are numerous values of Z1, Z2,e, Z2,o, �, � and

� satisfying (1)–(9). However, due to the limitation in the bandwidth, circuit size and

photolithography, such values are impractical.

3. Discussion and results

In this section, in order to verify the design approach, the proposed UWB power divider is

simulated and fabricated at f0 ¼ 7GHz. Two commercial full-wave simulators are

employed for simulation. Full-wave simulations are carried out by Agilent’s ADS/

Momentum and Sonnet simulators. Agilent’s ADS/Momentum uses frequency-domainmethod of moments technology to simulate complex electromagnetic (EM) effects

accurately including coupling and parasitic (Help Advanced Design System 2008). The

Sonnet simulator employs a rigorous method of moments EM analysis based on Maxwell’s

equations (Help Sonnet). The proposed UWB power divider is measured by HP8722ES

network analyser. Choosing Z0 ¼ 50�, the values of the characteristic impedances are

calculated as Z1 ¼ 1:66� 50 ¼ 83�, Z2,e ¼ 1:24� 50 ¼ 62� and Z2,o ¼ 0:92�50 ¼ 46�. The simulated results of the scattering parameters of the proposed UWB

power divider by ADS/Momentum and Sonnet simulators and measured results are

exhibited in Figures 2–4. According to Figure 4, the simulated insertion losses of the

proposed UWB power divider (jS12j and jS13j) are better than 4 dB, and the return loss atport 1 (jS11j) is greater than 15 dB over the whole UWB band. At port 1, 20-dB return is

from 3.7 to 7.48GHz and from 8.9 to 10GHz. The isolation between output ports is better

than 13 dB. The output return losses (jS22j and jS33j) are larger than 12 dB. The simulated

phase difference between ports 2 and 3 is 0� � 5� across the UWB.There are two resonance frequencies around f1¼ 5.2 and f2¼ 9.5GHz ( f0¼ ( f1þ f2)/2).

The characteristic impedances can be rewritten as Z2,e ¼ Z1ðffiffiffiffiffiffiffiffiffiffif1=f2

pÞ and Z2,o ¼ Z1ð f1=f2Þ.

Here, � is 90� at the second resonance frequency.In order to investigate the effect of resistor tolerances on the isolation between the

output ports, and output ports impedance matching, the simulated results using different

values of Rð100� 10Þ by ADS simulator are plotted in Figure 5. Figure 5(a) shows that theoutput return loss of the proposed UWB power divider becomes poor as R is larger.

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Figure 5(b) shows that when R is larger, the isolation of the proposed UWB power dividerbecomes better.

Figure 6 displays the layout and photography of the proposed UWB power divider.The optimised parameters of the proposed UWB power divider are L1¼ 0.84mm,L2¼ 3.2mm, R¼ 1.1mm, S¼ 0.64mm, W1¼ 0.6mm, W2¼ 0.19mm and W3¼ 0.52mm.The overall size of the proposed UWB power divider is 5:8� 4:3mm2.

A parametric analysis is performed to study the effect of geometrical dimensions on theproposed UWB power divider performance. The effect of the L2 is depicted in Figure 7. Itis observed that the optimum value of L2 is 3.2mm. As L2 is larger, jS22j and jS33j aremoved to lower frequency sides.

A comparison of the power divider performances operating over UWB band issummarised in Table 1.

4. Conclusions

In this article, a novel UWB microstrip power divider with just one resistor and withouttransmission line stubs and multilayer substrate has been proposed and designed.

112

–30

–20

–10

–40

0

Frequency (GHz)

Mag

nitu

de (

dB)

dB(S(1, 1))

dB(S(1, 2))

dB(S(1, 3))

dB(S(2, 2))

dB(S(2, 3))

(a)

3 4 5 6 7 8 9 10

63 4 5 7 8 9 10 112

–100

0

100

–200

200

Frequency (GHz)

Phas

e

phase(S(1, 2))

phase(S(1, 3))

(b)

Figure 2. S-parameter results of the proposed UWB power divider obtained by ADS/Momentumsimulator: (a) magnitude and (b) phase.

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112

–30

–20

–10

–40

0

Frequency (GHz)

Mag

nitu

de (

dB)

dB(S(1, 1))

dB(S(1, 2))

dB(S(1, 3))

dB(S(2, 2))

dB(S(2, 3))

(a)

3 4 5 6 7 8 9 10

53 4 6 7 8 9 10 112

–100

0

100

–200

200

Frequency (GHz)

Phas

e

phase(S(1, 2))phase(S(1, 3))

(b)

Figure 3. S-parameter results of the proposed UWB power divider obtained by Sonnet simulator:(a) magnitude and (b) phase.

4 5 6 7 8 9 10−5

0

5

10

15

20

Frequency (GHz)

Phas

e D

iffe

renc

e (°

)

(b)

4 5 6 7 8 9 10−50

−40

−30

−20

−10

0

Frequency (GHz)

Mag

nitu

de (

dB)

S11 S12 S13 S22

(a)

Figure 4. Measured results of the fabricated UWB power divider: (a) magnitude and (b) phasedifference.

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The proposed UWB power divider has excellent insertion loss and equal power dividing,

good return loss and isolation through the UWB. There are two transmission poles in the

input port return loss. Actually, this UWB microstrip power divider can be extremely

useful for wideband applications.

R=90

R=110

R=100

3 4 5 6 7 8 9 10 112

–30

–20

–10

–40

0

Frequency (GHz)

Out

put r

etur

n lo

ss (d

B)

3 4 5 6 7 8 9 10 112

–30

–20

–10

–40

0

Frequency (GHz)

Isol

atio

n (d

B)

R=90

R=110

R=100

(a)

(b)

Figure 5. Results of the proposed UWB power divider as a function of R (90, 100 and 110�):(a) output return loss and (b) isolation.

Figure 6. (a) Layout and (b) photography of the proposed UWB power divider.

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Table 1. Performance summary of the published power dividers and this study.

References

S-parameters (3.1–10.6GHz)Type of sub-strate layerjS11j420 dB jS11j415 dB jS22j415 dB jS23j410 dB

This work 3.7–7.48 and1.9–10GHz

2–10.6GHz 6–10.6GHz 3.7–10.1GHz Single layer

Ou and Chu(2008)

– 3–4.2, 5.5–6.4and 9.2–10.5GHz

3–7.5GHz 3–9.8GHz Single layer

Bialkowski andAbbosh(2007)

– – – 8.5–10.2GHz Multilayer

Abbosh (2008) 4.4–8.8GHz 3.6–9.5GHz – 3–3.5 and 10–10.5GHz

Multilayer

Abbosh andBialkowski(2007)

8.8–10.5GHz 3.9–10.8GHz – 4–8.8GHz Multilayer

Abbosh et al.(2006)

5.9–6.4, 7.2–7.6, and 9–10.2GHz

3.8–11GHz – 3–11GHz Multilayer

Song and Xue(2010)

– 4–5GHz 2.8–3.2, 5.8–7,and 9.6–10GHz

0–15GHz Multilayer

Wong and Zhu(2008)

3.9–4.1, 6.1–6.3, and 8.9–

9.1GHz

3.85–4.2, and5.8–9.8GHz

7–8GHz 0–14GHz Single layer

L2=2.7L2=3.2L2=3.7

|S12|

122

–40

–30

–20

–10

–50

0

–3.6

–3.4

–3.2

–3.8

–3.0

Frequency (GHz)

dB(S

11) dB

(S12)

(a)

122

–40

–30

–20

–10

–50

0

Frequency (GHz)

dB

(S2

3)

L2=2.7L2=3.2L2=3.7

4 6 8 10

4 6 8 10 4 6 8 10 122

–30

–20

–10

–40

0

Frequency (GHz)

dB(S

22)

L2=2.7L2=3.2L2=3.7

(b) (c)

Figure 7. Effect of L2 on the proposed UWB power divider performance: (a) jS11j and jS12j,(b) jS23j and (c) jS22j.

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Acknowledgements

This study was supported by Young Researcher Club, Kermanshah Branch, Islamic AzadUniversity. The author thank Engineers T. Faraji, S. Mohammadpour and M. Akhlagh-pasandi fortheir assistances.

References

Abbosh, M. (2007), ‘A Compact UWB Three-way Power Divider’, IEEE Microwave and Wireless

Components Letters, 17, 598–600.Abbosh, M. (2008), ‘Multilayer Inphase Power Divider for UWB Applications’, Microwave and

Optical Technology Letters, 50, 1402–1405.Abbosh, M., and Bialkowski, M.E. (2007), ‘An UWB Planar Out-of-phase Power Divider

Employing Parallel Stripline-Microstrip Transitions’, Microwave and Optical Technology

Letters, 49, 912–914.Abbosh, M., Bialkowski, M.E., and Mazierska, J. (2006), ‘An UWB Planar Out-of-phase Power

Divider Employing Microstrip-Slot and Parallel Stripline-Microstrip Transitions’, in IEEE,

Proceedings of Asia-Pacific Microwave Conference, pp. 905–908.Advanced Design System (2008), Help of Advanced Design System 2008. Available at http://www.

home.agilent.com/agilent/product.jspx?cc=IR&lc=eng&ckey=1887116&nid=-33748.0.00&id=1887116&cmpid=zzfindeesof-momentum.

Avrillon, S., Pele, I., Chousseaud, A., and Toutain, S. (2003), ‘Dual-band Power Divider Based onSemiloop Stepped-impedance Resonators’, IEEE Transactions on Microwave Theory and

Techniques, 51, 1269–1273.Bialkowski, M.E., and Abbosh, A.M. (2007), ‘Design of a Compact UWB Out-of-phase Power

Divider’, IEEE Microwave and Wireless Components Letters, 17, 289–291.Cheng, K.-K.M., and Law, C. (2008), ‘A Novel Approach to the Design and Implementation of

Dual-band Power Divider’, IEEE Transactions on Microwave Theory and Techniques, 54,

487–492.Cheng, K.-K.M., and Wong, F.-L. (2007), ‘A New Wilkinson Power Divider Design for Dual Band

Applications’, IEEE Microwave and Wireless Components Letters, 17, 664–666.Chiu, L., Yum, T.Y., Xue, Q., and Chan, C.H. (2006), ‘A Wideband Compact Parallel-strip 180

Wilkinson Power Divider for Push-Pull Circuitries’, IEEE Microwave and Wireless Components

Letters, 16, 49–51.Cohn, S.B. (1968), ‘A Class of Broadband Three-port TEM Mode Hybrids’, IEEE Transactions on

Microwave Theory and Techniques, MTT-16, 110–116.Dib, N., and Khodier, M. (2008), ‘Design and Optimization of Multi-band Wilkinson Power

Divider’, International Journal of RF and Microwave Computer-Aided Engineering, 18, 14–20.Faraji, T., Nosrati, M., and Hazeri, A.R. (2011), ‘A Novel Miniaturized Dual-band Power Divider

Using Microstrip Coupled Transmission Lines’, Microwave and Optical Technology Letters, 53,

676–679.Federal Communications Commission (FCC), ‘Revision of Part 15 of the Commission’s Rules

Regarding Ultra-Wideband Transmission Systems’, First Report and Order, ET Docket 98-153,FCC 02-48; Adopted: February 2002; Released: April 2002.

Lee, S.W., Kim, C.S., Choi, K.S., Park, J.S., and Ahn, D. (2001), ‘A General Design Formula ofMulti-section Power Divider Based on Singly Terminated Filter Design Theory’, Microwave

Symposium Digest, 2001 IEEE MTT-S International, 2, 1297–1300.Oraizi, H., and Sharifi, A.-R. (2006), ‘Design and Optimization of Broadband Asymmetrical

Multisection Wilkinson’, IEEE Transactions on Microwave Theory and Techniques, 54,

2220–2231.Ou, X.P., and Chu, Q.X. (2008), ‘A Modified Two-section UWB Wilkinson Power Divider’’, in

Proceedings of IEEE, ICMMT, pp. 1258–1260.

International Journal of Electronics 583

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ded

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nive

rsity

Lib

rary

] at

18:

30 0

6 O

ctob

er 2

013

Park, M.-J., and Lee, B. (2008), ‘Wilkinson Power Divider With Extended Ports for Dual-bandOperation’, Electronics Letters, 44, 916–917.

Pozar, D. (2005), Microwave Engineering (3rd ed.), New York: Wiley.Song, K., and Xue, Q. (2010), ‘Novel Ultra-Wideband (UWB) Multilayer Slotline Power Divider

With Bandpass Response’, IEEE Transactions on Microwave Theory and Techniques, 20, 13–15.Sonnet pro 11.54 (2007), Help of Sonnet pro 11.54. Available at http://www.sonnetsoftware.com/

products/sonnet-suites/how-EM-works.html.

Srisathit, S., Chongcheawchamnan, M., and Worapishet, A. (2003), ‘Design and Realization ofDual-band 3 dB Power Divider Based on Two-section Transmission-line Topology’, IEEElectronics Letters, 39, 723–724.

Wong, S.W., and Zhu, L. (2008), ‘Ultra-wideband Power Divider With Good In-band Splitting andIsolation Performances’, IEEE Microwave and Wireless Components Letters, 18, 518–520.

Wu, Y., Liu, Y., and Li, S. (2009), ‘Dual-band Modified Wilkinson Power Divider Without

Transmission Line Stubs and Reactive Components’, Progress In Electromagnetics Research,PIER 96, 9–20.

Wu, L., Sun, S., Yilmaz, H., and Berroth, M. (2006), ‘A Dual-frequency Wilkinson Power Divider’,IEEE Transactions on Microwave Theory and Techniques, 54, 278–284.

Wu, L., Yilmaz, H., Bitzer, T., Pascht, A., and Berroth, M. (2005), ‘A Dual-frequency WilkinsonPower Divider: For a Frequency and its First Harmonic’, IEEE Microwave and WirelessComponents Letters, 15, 107–109.

Zhang, H., and Xin, H. (2008), ‘Designs of Dual-band Wilkinson Power Dividers With FlexibleFrequency Ratios’, in IEEEMTT-S International Microwave Symposium Digest, pp. 1223–1226.

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