Dual Band Branch Line and Rat Race

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 5, MAY 2010 1213

Compact Dual-Band Branch-Line and Rat-RaceCouplers With Stepped-Impedance-Stub Lines

Kuo-Sheng Chin, Member, IEEE, Ken-Min Lin, Yen-Hsiu Wei, Tzu-Hao Tseng, and Yu-Jie Yang

Abstract—This study constructs stepped-impedance-stub linesfor a dual-band branch-line coupler design with improved designflexibility. The proposed structure demonstrates dual-band perfor-mance and a compact size due to additional stepped-impedancestubs to branches. The developed synthesis method has two de-grees of freedom which can be exploited to miniaturize circuit sizeand/or replace impractical impedances with more realizable ones.Observations also show the advantage of a wide-range realizablefrequency ratio of dual bands. The current work fabricates threeexperimental dual-band branch-line couplers, including a two-sec-tion coupler, and achieves a size reduction up to 21.7%, comparedwith conventional structures. The measured results validate gooddual-band performance at 2.4/5.8 GHz with enhanced bandwidthsup to 21% and 12%, respectively. This research also successfullyapplies the proposed circuit to synthesize a dual-band rat-race cou-pler.

Index Terms—Dual band, branch-line coupler, rat-race coupler,stepped impedance, two-section coupler.

I. INTRODUCTION

B RANCH-LINE couplers offer a 90 phase difference andequal/unequal power splitting, which is useful in various

microwave circuits, such as balanced mixers, data modulators,phase shifters, and power combined amplifiers. The traditionaldesign of branch-line couplers operates only at single band.Such couplers suffer from disadvantages of narrow bandwidthand large size. In modern communication systems, the needfor dual-band operation, wide bandwidth, and compactness im-poses new requirements. To fulfill these requirements, the con-ventional branch line coupler must be redesigned.

Previously published reports on bandwidth enhancement ofbranch-line couplers, include using a multisection cascadedstructure [1]–[6], a parallel-strip branch line with a swap atits center [7], and the elliptic patch [8]. However, achievingmore than two sections in the microstrip is difficult becausethe outside branch lines typically require very high impedancesexceeding the upper limits of a practical realization. Thus,novel circuit structures which reduce impedance levels areimportant for enhancing bandwidth.

Manuscript received September 21, 2009; revised December 14, 2009. Firstpublished April 19, 2010; current version published May 12, 2010. This workwas supported in part by the National Science Council, Taiwan, under GrantNSC 97-2221-E-182-015 and in part by Chang Gung University, Taiwan, underContract UERPD280061 and the High Speed Intelligent Communication Re-search Center.

The authors are with the Department of Electronic Engineering, Chang GungUniversity, Kwei-Shan, Taoyuan 333, Taiwan (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2010.2046064

Fig. 1. (a) Configuration of a conventional T-shaped line. (b) Proposed dual-band stepped-impedance-stub branch line. (c) Equivalent dual-band 90 line.

Research has proposed various planar structures for adual-band branch-line coupler design. One study [11] addeda pair of cross coupling branches to introduce more designfreedom. Lin et al. replaced branches with the com-posite right-/left-handed metamaterial transmission line [12].However, the losses caused by lumped elements should beconsidered. The study achieved wider operating bandwidthby adding additional microstrip stubs to branches [14]. Usingthe three-branch-line stretched structure implements the dualband and enhances bandwidth [15], requiring a third branchto the conventional branch line in coupler synthesis. Chengand Wong described a dual-band rat-race coupler based onthe tri-section branch line [16]. The structure using two shuntsusceptances attached to the two ends of a stepped-impedanceline was presented and validated in [17]. Recently, T-shapedlines were introduced for size reduction [18], broad bandwidth[19], and dual-band operation [20]–[22]. The T-shaped line,shown in Fig. 1(a), consists of a signal path tapped with auniform stub at its center, leading to great improvement incoupler performance. Designing dual-band couplers with morerealizable impedances, arbitrary frequency ratio, and a compactsize is an ongoing challenge.

This work proposes stepped-impedance-stub branches forboth dual-band operation and compactness. Section II derivesthe design equations of stepped-impedance-stub lines and findstwo degrees of freedom for determining circuit dimensions.The degrees of freedom can be used to reduce circuit sizeand/or replace high-impedance lines of couplers so that theymay be physically constructed by microstrips. Section IIIdevelops the synthesis methods for single-section/two-sectionbranch-line couplers and rat-race couplers. Section IV presentsthe experimental results to demonstrate the proposed circuitstructure. Finally, Section V draws conclusions.

II. ANALYSIS OF THE DUAL-BAND BRANCH LINE

Designers typically construct traditional 3-dB branch-linecouplers using branches with impedances of 50and 50 . These branches have an electrical lengthof 90 at a single frequency that must be replaced for

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1214 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 5, MAY 2010

a dual-band operation. Fig. 1(b) presents the proposedstepped-impedance-stub line comprising a signal pathtapped with a stepped-impedance stub of andat its center. The stub line performs an equivalent electricallength of 90 and desired branch impedances at two operatingfrequencies. If , it becomes a T-shaped line, asshown in Fig. 1(a). Three advantages of the proposed structureattribute to its increased nonuniform impedances, resulting ina compact size, wide range of realizable frequency ratio, andmore realizable impedances.

Fig. 1(b) shows as the input impedance looking into thesection and derived as

(1)

The composite matrix of the stepped-impedance-stubbranch line is obtained by multiplying the matrices ofeach cascade component in Fig. 1(b) to give

(2)

Since the proposed dual-band branch line acts as a 90 line atand shown in Fig. 1(c), where and are center fre-

quencies of the first and second bands, respectively, the corre-sponding matrix is given by

(3)

where is the characteristic admittance of the 90 line.Equating the matrices of (2) and (3) yields

(4a)

(4b)

and (4c) and (4d), shown at the bottom of this page, wheredenotes the frequency ratio of the second band to the first

band, and , and are all specified at . The value ofis decided according to the required dual band specification.

Moreover, (4a)–(4d) can be simplified to solve for and as

(5a)

Fig. 2. Solutions to (6) for � of stepped-impedance-stub lines (� �

�� , and � � ��).

(5b)

For a compact dual-band branch line, should be selectedin (5).

Two degrees of freedom exist due to six unknowns( , and ) but only four equations (4a)–(4d)are available. Thus, the impedance ratio and theelectrical length ratio are chosen as free variables tosolve (4a)–(4d) simultaneously. The solutions of transcendentalequations (4a)–(4d) are not unique even though the parameters

and are fixed, as explained in the following. Substituting(5a)–(5b), , and into (4a) and (4b)eliminates , leading to a single transcendental equation

(6)

Equation (6) only has a single variable since the constants, and can be arbitrarily chosen. A graphical method plots

the graph of each side of (6), using their intersecting points tofind solutions. With , and asexamples (used to synthesize the two-section branch-line cou-pler in Section III-B), the solutions to (6) are indicated by dotsin Fig. 2. has multiple solutions as shown in Fig. 2, and

, 44.74 , 58.21 , and 93.44 are the first four so-lutions which have respective , and . Obviously, theshortest is desired for size reduction if its correspondingand are realizable as well. However, the longer may bebetter for reducing the impedances and . This underdeter-mined feature is very helpful in reducing size and/or satisfyingthe high-branch-impedance requirement by enabling a suitable

(4c)

(4d)


CHIN et al.: COMPACT DUAL-BAND BRANCH-LINE AND RAT-RACE COUPLERS WITH STEPPED-IMPEDANCE-STUB LINES 1215

Fig. 3. Schematic of the dual-band single-section branch-line coupler withstepped-impedance-stub lines.

Fig. 4. � and � of the dual-band stepped-impedance-stub line versus �.(a) 50-� line. (b) 50�

��-� line.

combination of and to be chosen. The following sectionsexplain this advantage according to the specified and .

III. DUAL-BAND BRANCH-LINE AND RAT-RACE COUPLERS

A. Single-Section Dual-Band Branch-Line Couplers

Here, we develop the synthesis method of the dual-bandbranch-line coupler with equal power division. Fig. 3 plotsthe proposed coupler schematic with stepped-impedance-stub

Fig. 5. � of the dual-band stepped-impedance-stub branch line versus � .

Fig. 6. � of the proposed 50-� stepped-impedance-stub branch line normal-ized with � of the conventional T-shaped line.

lines where their circuit dimensions can be determined bysolving (4a)–(4d). Size reduction is better using in (4c)and in (4d), which gives and phase differencebetween output ports for the first and second bands, respec-tively. Fig. 4(a) and (b) plots and of the dual-band 50-and 50 - stepped-impedance-stub branch lines versus

for various , respectively, when the dual 2.4-/5.8-GHzbands are required for a WLAN. Here, Fig. 4does not include since it can be easily calculated using (5).Fig. 4 shows that both and monotonically increase asincreases. The smaller has higher impedances of and .Typically, only an impedance of 20–120 can be realized usingthe microstrip fabrication process, so the range of is limitedto for the 50- branch, while the value ofdepends on the chosen . When , which is the case of theT-shaped line, the impedances are 252.75 thatare not realizable. A similar situation applies to the 50 -branch with a limited range of .

Fig. 1(b) presents that and notablycorrespond to the length and width of stepped-impedance-stublines. From (5a), depends only on and cannot be easilyreduced since does not vary with and . Thus, the pro-posed stepped-impedance-stub line has the same width as theT-shaped line, i.e., . However, can be reduced tominiaturize circuit size.



Fig. 7. Design curves of � �� , and � . (a) Dual-band 50-� branch line.(b) Dual-band 50�

��-� branch line.

Fig. 5 plots as a function of for various . The line(T-shaped line) correspond to a constant electrical length

of . As Fig. 5 shows, is always less than105.36 when , validating the length is reduced be-cause of the increased nonuniform impedances. The curves of

drop as and decline, reaching their lowest point around. Since a small may be associated with

unrealizable impedance values, the optimal values of andof stepped-impedance-stub branch lines must be considered to-gether to minimize and to ensure realizing the impedances

and . The curves in Figs. 4 and 5 reveal thatand (0.3, 0.2) are suitable for synthesizing dual-band

50- and 50 - branches, respectively. Fig. 6 plots ofthe 50- stepped-impedance-stub branch line, normalized to the

Fig. 8. Variation of bandwidth with � and � .

vertical length of the T-shaped line with respect to . Fig. 6presents that the normalized is always less than one, reducingto 0.79 when , revealing that the proposed structurehas reduced size. A large reduction ratio for couplers can beachieved if a small impedance ratio is chosen.

Fig. 7(a) and (b) indicates the design values of dual-band50- and 50 - branches, respectively. When a frequencyratio is specified, it is easy to find the related dimensions

, and from Fig. 7. Substituting andinto (5) also yields , and 38.12

and 26.96 for dual-band 50- and 50 - branches, re-spectively. The detail dimensions are listed in Section IV.

Fig. 7 and (5) are very useful for synthesizing dual-band cou-plers. If the desired is too high or low, and may becomeunrealizable. Therefore, Fig. 7 provides various combinations of

to support a wide achievable range of .Fig. 7 also plots the impedance of the T-shaped line for com-parison, where is not shown. The T-shaped line pro-vides a very limited realizable range of forthe 50 line and for the line, since ithas no degree of freedom to change and . Notably, the pro-posed structure has lower impedances than T-shaped lines for afixed . This situation becomes more obvious as the requiredbranch impedance becomes higher, helping to alleviate the dif-ficulty resulting from impractical high impedances.

Fig. 8 investigates the variation of bandwidth with and .Since compactness and realizable impedances are the most im-portant considerations, only the ranges and

are examined. The fractional bandwidths donot change significantly with and and are 11%–13% and4%–7% for the first and second bands, respectively.

B. Two-Section Dual-Band Branch-Line Couplers

In advanced communication systems, branch-line couplersnot only require dual-band operation but also desire wide band-width. However, the single-section dual-band coupler only hasa very narrow bandwidth, especially 4%–7% at the high pass-band. Fortunately, using multiple cascading sections improvesthe coupler. Since the proposed dual-band lines can be designedseparately, the easy way for a two-section coupler design is tofind required branch impedances first from the single-band syn-thesis equations and then replace each with the correspondingdual-band stepped-impedance-stub line. Fig. 9(a) shows the



Fig. 9. Configurations of: (a) a conventional two-section branch-line coupler(single band) and (b) the proposed dual-band two-section branch-line couplerwith stepped-impedance stubs.

single-band two-section branch-line coupler with uniformimpedances of , and and electrical lengths of .

Using the even–odd-mode analysis technique, the currentwork determines characteristic impedances by

(7a)

(7b)

where is the coupling coefficient of two output ports. Thevalue of is chosen arbitrarily, and, generally,gives maximum bandwidth [1]. When equal power division isrequired , from (7), we obtain 120.7 and

50 . For such a high impedance of 120.7, the required of the T-shaped line is 610.14 (for

) which is not realizable in practice. Ifis still chosen for the proposed dual-band line, then will be290 , also exceeding the fabrication limit. Fortunately, the so-lutions of the transcendental equations (4a)–(4d) are not unique,as mentioned in Section II, allowing the choice of different so-lutions for more realizable impedances at the cost of longer .

In Section III-A, the goal is to find the shortest with thesmallest . However, we now change the priority to sacrificeslightly to find a more realizable and . Therefore, the valueof increases to find suitable solutions of (4a)–(4d). Fig. 10(a)plots and of the 120.7- stepped-impedance-stub branchline versus for . Fig. 10(a) shows20 when . These curves indicate different impedancevarying with , compared with Fig. 4. In our studies, when

, the tendency reverses and the bigger correspondsto higher impedance. Therefore, is chosen for con-sidering impedance realization. Fig. 10(b) plots as a func-tion of , in which the smaller and obtain a shorter .

Fig. 10. (a) � and � of the dual-band 120.7-� stepped-impedance-stub lineversus �. (b) � of the 120.7-� stepped-impedance-stub line versus � .

TABLE IDESIGN TABLE OF DUAL-BAND 120.7-� STEPPED-IMPEDANCE-STUB

LINES VERSUS �

TABLE IIDESIGN VALUES FOR BRANCHES WITH HIGH IMPEDANCES

120–155 ��

Fig. 10(a) and (b) concludes that have re-alizable impedances and a shorter . The calculated results



Fig. 11. Configuration of the proposed dual-band rat-race coupler.

Fig. 12. (a) Simulated and measured �� of dual-band branch-line coupler �with �� and (0.3, 0.2). (b) Photograph of coupler �.

are 21.2 106 87.32 , and58.21 . Fig. 2 notes that 58.21 is not the shortest solutionof (6) because we sacrifice to relax the high-impedance re-quirement. Fig. 9(b) plots the proposed two-section dual-bandbranch-line couplers with stepped-impedance stubs.

Fig. 13. Experimental results of coupler � with � � �� and � � ��.(a) Measured �� . (b) Simulated and measured � � � .

Table I lists the circuit dimension for the dual-band 120.7-lines for various . Different combinations of are ar-ranged to support a wide achievable range of from 1.7 to2.7. As mentioned above, the proposed circuit structure offersdesign flexibility in realizing impractical branch impedances.Table II lists the design values of stepped-impedance-stub linesfor branches with high impedances (120–155 ) which may berequired in three-section (143 in [2]) and more multisectionbranch-line coupler designs.

C. Dual-Band Rat-Race Couplers

The rat-race hybrid ring is another widely used coupler forproviding 0 and 180 phase shift outputs, consisting of several

sections between ports 1–4 around the top half of the ring.The bottom half of the ring is 3 in length between ports 2and 4. For equal power division, the impedance of the entire ringis fixed at or 70.7 for a 50- system. Because the pro-posed stepped-impedance-stub line is capable of providing 90electrical length and desired impedances at two center frequen-cies, we also can use it to design dual-band rat-race couplersfor compactness. The current work determines circuit dimen-sions of the dual-band 70.7- lines in the same way as discussedabove and does not repeat them again for purposes of concise-ness. Fig. 11 plots the configuration of the dual-band rat-racecoupler with stepped-impedance stubs.



Fig. 14. (a) Simulated and measured � parameters of two-section coupler � .(b) �� . (c) � � � . (d) Photograph of fabricated coupler � .

IV. EXPERIMENTAL RESULTS

A. Single-Section Dual-Band (2.4/5.8 GHz) Branch-LineCouplers

To demonstrate the design concept, this study fabricatestwo dual-band signal-section branch-line couplers and

TABLE IIIBANDWIDTHS OF THE THREE EXPERIMENTAL DUAL-BAND

BRANCH-LINE COUPLERS

on an Arlon 25N substrate with a dielectric constant of 3.38and a thickness of 0.762 mm. Both couplers are designedto operate at 2.4/5.8 GHz, except they use different pairs of

for comparison. The circuit dimensions of the 50-branch of coupler are 23.8 14119 69.5 38.12 , and 105.36 ,with , and, for the 50 - branch,

32 14.5 106 74 26.96 ,and 105.36 , with . In practice, thediscontinuity effects at branch corners and stepped-impedancestubs may require optimizing electrical lengths of branches.

Fig. 12(a) plots simulated and measured results of coupler .The performance clearly shows the existence of two passbandscentered at 2.36 and 5.7 GHz, respectively. The simulated andmeasured results agree closely with each other. Detailed datareveal that the measured insertion losses of the through and cou-pled ports are 3.87 and 3.41dB with a phase difference of90 at 2.36 GHz and 4.374 and 4.074 dB with a phase dif-ference of 89 at 5.7 GHz, respectively. Both andare less than 38 dB at the first center frequency and 29 dB atthe second center frequency. If a 0.5-dB magnitude imbalanceis used to define the usable frequency range, then the measuredbandwidths of the first and second bands are 11% and 5%, re-spectively. Fig. 12(b) displays a photograph of coupler . Theheight of the 50- stepped-impedance-stub line is 82.5 . Ifusing the T-shaped line in design, then 105.36 . Fig. 6exhibits that the proposed coupler achieves a size reduction of21.7%.

The other coupler is designed with and .Fig. 13(a) shows experimental results of coupler , in which thecircuit dimensions of 50 - branch are 22.8525.83 76.17 64.58 26.96 , and

105.36 . The dimensions of 50- branch are32.31 25.83 107.7 64.58 38 ,and 105.36 . The insertion losses are 3.57 dBand 3.62 dB at 2.4 GHz and 4.06 dB and

4.23 dB at 5.74 GHz, respectively. The measuredbandwidths of dual bands are 13% and 7%, which are slightlywider than coupler . Coupler also performs better than theother couplers in [12], [14], and [21] in terms of bandwidth.Fig. 13(b) plots the phase difference between the coupled andthrough outputs of the two passbands. A phase difference of

between these two ports is maintained over a widebandwidth of the dual bands.



Fig. 15. (a) Simulated and measured � parameters of the dual-band rat-racecoupler. (b) � � � and � � � . (c) Photograph of the rat-racecoupler.

B. Two-Section Dual-Band (2.4/5.8 GHz) Branch-LineCouplers

To increase bandwidth, the current work designs couplerwith a two-section cascaded structure at 2.4/5.8 GHz. The

circuit parameters of the 50 - branch are the same ascoupler , but 21.2 87.32 10658.21 92 , and 105.36 are used for the 120.7-

branch. Fig. 14(a) shows the results of coupler with en-hanced dual bandwidths of 21% and 12%. The measuredis shifted to 2.23 GHz, and is shifted to 5.58 GHz. Themeasurement shows that, at 2.23 GHz, 3.73 dB,

4.11 dB, and 91.3 , and, at5.58 GHz, 4.54 dB, 4.97 dB, and

90.9 . Fig. 14(b) and (c) plots the magni-tude imbalance and phase difference, respectively. Fig. 14(d)shows the photograph of coupler . Table III summarizes thebandwidths of couplers , and .

C. Dual-Band (2.4/5.2 GHz) Rat-Race Couplers

This study designs the experimental dual-band rat-race cou-pler to operate at 2.4 and 5.2 GHz . Fig. 15(a)plots the simulated and measured responses of . The mea-sured results are dB, dB,

dB, and dB at 2.34 GHz, anddB, dB, dB, anddB at 5 GHz. Fig. 15(b) shows the phase differ-

ences and . The detailed data showthat the phase deviations are 1.12 and and

are 181.48 and at 2.34 GHz and 5 GHz,respectively. Fig. 15(c) shows the photograph of the fabricateddual-band rat-race coupler.

Although the experimental branch-line and rat-race couplersare all designed with equal power division, designers can alsoapply the proposed circuit structure and synthesis method to ar-bitrary power divisions, according to the required characteristicimpedances of branches.

V. CONCLUSION

This study presents a novel circuit structure of dual-bandcouplers. The proposed stepped-impedance-stub branch linegives engineers more design flexibility in coupler synthesisbecause its synthesis equations have two degrees of freedom.A detailed comparison of the proposed circuit with conven-tional structures is carried out. The current work achievesthree advantages of compact size, wide range of realizablefrequency ratio, and more realizable characteristic impedancesby choosing a suitable pair in the synthesis formulas.For multisection couplers, the high branch impedance levelsare always required. The tradeoff between size and realizableimpedances is discussed in detail, and the associated designvalues are provided. Four experimental dual-band circuits, in-cluding two single-section branch-line couplers, a two-sectionbranch-line coupler, and a rat-race coupler are demonstratedwith good performance.

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Kuo-Sheng Chin (S’05–M’06) received the B.S.degree in electrical engineering from Chung ChengInstitute of Technology, Taoyuan, Taiwan, in 1986,M.S.E.E. degree from Syracuse University, Syracuse,NY, in 1993, and the Ph.D. degree in communicationengineering from National Chiao Tung University,Hsinchu, Taiwan, in 2005.

From 1986 to 2005, he was with the ChungShan Institute of Science and Technology, Taoyuan,Taiwan, as a Research Assistant, becoming anAssistant Scientist and then an Associate Scientist.

In 2006, he joined the faculty of Chang Gung University, Taoyuan, Taiwan,where he is currently an Assistant Professor with the Department of Elec-tronic Engineering. His main research interests are microwave measurements,microwave circuits, radome and antenna designs, and electromagnetic pulse(EMP) research.

Dr. Chin was the recipient of the Medal of Excellent Efficiency, the Orderof Loyalty and Diligence, the Medal of Outstanding Staff, A Class, and theMedal of Army Achievement, A Class, all from the Ministry of National De-fense, Taiwan. He was one of the recipients of the Best Paper Award of the 2009International Conference on Electromagnetic Near Field Characterization andImaging, Taiwan. He supervised a student team to win first place at the 2009National Electromagnetism Application Innovation Competition, Taiwan.

Ken-Min Lin was born in Taipei, Taiwan. Hereceived the M.S. degree in electronic engineeringfrom Chang Gung University, Taoyuan, Taiwan, in2009.

His research interest is the design of dual-bandbranch-line couplers and microwave filters.

Yen-Hsiu Wei was born in Chiayi, Taiwan. He re-ceived the B.S. degree in electronic engineering fromChang Gung University, Taoyuan, Taiwan, and is cur-rently working toward the M.S. degree in electronicengineering at Chang Gung University.

His research interest is the design of compact mi-crowave couplers.

Tzu-Hao Tseng was born in Taipei, Taiwan. He iscurrently a Senior Student with the Department ofElectronic Engineering, Chang Gung University,Taoyuan, Taiwan.

His research interests include microwave circuitand filter design.

Mr. Tseng was one of the members of the studentteam that won first place at the 2009 National Elec-tromagnetism Application Innovation Competition,Taiwan.

Yu-Chien Yang was born in Taichung, Taiwan. Sheis currently a Senior Student with the Departmentof Electronic Engineering, Chang Gung University,Taoyuan, Taiwan.

Her research interest is the design of microwavedevices.

Ms. Yang was one of the members of the studentteam that won first place at the 2009 National Elec-tromagnetism Application Innovation Competition,Taiwan.


Documents

Dual Band Branch Line and Rat Race