dual-band branch-line hybrid with distinct power division ratio over the two bands

Dual-Band Branch-Line Hybrid with Distinct PowerDivision Ratio Over the Two Bands

Karun Rawat, Meenakshi Rawat, Mohammad S. Hashmi, Fadhel M. Ghannouchi

Department of Electrical and Computer Engineering, University of Calgary,T2N1N4 Calgary, Alberta, Canada

Received 15 October 2011; accepted 14 March 2012

ABSTRACT: This article presents a novel methodology for the design of transmission line-

based dual-band branch-line hybrid with distinct power division over any two specified fre-

quencies. These distinct power divisions at specified frequencies are achieved while keeping

the quadrature relation intact at both the frequencies. To demonstrate the effectiveness of the

proposed technique, a prototype of dual-band uneven branch-line hybrid operating at 1960

and 3500 MHz has been designed for use in Wideband Code Division Multiple Access

(WCDMA) and Worldwide Interoperability for Microwave Access (WiMax) applications. The

designed hybrid possesses equal power division in the WCDMA band and 3-dB unequal

power division in the WiMax band. VC 2012 Wiley Periodicals, Inc. Int J RF and Microwave CAE

00:000–000, 2012.

Keywords: dual-band/dual-impedance quarter-wave transformers; uneven power division; stub-

loaded line; dual-band; branch-line hybrid; microstrip; couplers; WiMax; WCDMA

I. INTRODUCTION

Dual band transmission line circuits are very useful in

high power reconfigurable transmitters due to their high

power handling capability as well as cost effectiveness.

Among these, dual-band branch-line hybrids are often

used in the design of dual-band circuits such as Doherty

power amplifiers (DPAs) [1–4]. These applications have

led to proposal of various topologies of dual-band hybrids

using transmission line sections [5–13].

Among various methodologies, most common

approach used in the design of dual-band component is to

replace each branch in a conventional single-band design

with a 2-port dispersive structure. These dispersive struc-

tures possess specific image impedance and phase charac-

teristic which corresponds to the required values of the

characteristic impedance and electric lengths at the respec-

tive two frequencies [7, 8]. There are two common techni-

ques to achieve dual-band quarter-wave operation from a

transmission line. In the first method, known as T-type, a

transmission line segment is loaded with a shunt suscep-

tance at the center [7, 13]; whereas the second method,

known as Pi-type, involves a transmission line segment

loaded with shunt sucseptance at its ends [8–12].

In common practice, dual-band hybrids designed using

T-type and Pi-type structures provide the same power di-

vision ratio at both frequencies and therefore find limited

usefulness in applications such as dual-band DPA, that

may require different power division ratios at the two fre-

quencies of operation [2–4, 14–16]. To overcome these

limitations and to achieve different power division ratios

at the two specified band, there have been proposed modi-

fications in dual-band hybrid design techniques such as

stub-loaded stepped impedance transformers [9, 10].

This article presents a novel technique to design a

dual-band hybrid with uneven power division ratio at two

frequencies in which each branch of a conventional

branch-line hybrid is replaced with a transmission line

loaded with multisection stub. Such loaded structure emu-

lates a quarter-wave transformer that has two different

characteristic impedances but the same electric lengths of

690� at the two frequencies.

A conventional Pi-type and T-type stub-loaded struc-

ture can also be used to design dual-band/dual-impedance

transformers as described in Refs. [4, 17]. However, it has

also been mentioned in Refs. [4, 17] that in common prac-

tice, to ensure such dual-band operation, a simultaneous

solution must be sought for the design parameters of the

loaded transmission lines and the open or short circuit

transmission line stubs realizing the required shunt suscep-

tance value [4, 17]. As in dual-band/dual-impedance

Correspondence to: K. Rawat; e-mail: [email protected]

VC 2012 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20655Published online in Wiley Online Library

(wileyonlinelibrary.com).

1

transformer, the 90� electric length is desired along with

two different characteristic impedance at two frequencies,

sometimes, this methodology results in the physically

unrealizable solutions [17]. Commonly, these solutions are

in terms of the design parameters (characteristic imped-

ance and electric length) of the open and short circuit

transmission line stubs realizing the shunt susceptance, for

a given design parameters of a loaded line [17]. For

example, a very high value of characteristic impedance

results in narrow transmission lines, presenting difficulties

during fabrication [7, 8, 17]. The technique presented in

this article provides an analytical solution of the design of

dual-band/dual-characteristic impedance transformer utiliz-

ing the multisection stub loaded architecture instead of

open and short circuit transmission line stubs. In principle,

the presented technique can realize any values of suscep-

tance at the two specified frequencies provided these

selected frequencies are uncorrelated [3, 4, 17].

For the proof of concept an uneven branch-line hybrid

has been designed to operate at 1960 and 3500 MHz, for

WCDMA and WiMAx applications. The hybrid provides

equal-power division in the first frequency, whereas a 3-dB

unequal power division is targeted in the second frequency.

The design methodology is described in Section II, the experi-

ment and design considerations are presented in Section III.

Section IV describes the (electromagnetic) Electromagnetic

(EM) simulated and measured performance of the designed

branch-line hybrid. Section V presents the conclusion.

II. DESIGN METHODOLOGY

Figure 1 shows schematic of the proposed hybrid which

utilizes the transmission line loaded with dual-band multi-

section stubs at two edges.

For any arbitrary power division a, the required value

of characteristic impedances of a branch-line coupler is

given as [9, 10, 18]:

ZT;Aðf Þ ¼Z0

a11þa1

� �0:5

@f1

Z0a2

1þa2

� �0:5

@f2

8><>: (1a)

ZT;Bðf Þ ¼ Z0 a1ð Þ0:5 @f1Z0 a2ð Þ0:5 @f2

�(1b)

where ZT,A and ZT,B are the characteristic impedances of the

branches and a1 and a2 are the respective power division ratio

(PA/PB) between port 2 and port 3 in Figure 1 at f1 and f2.Z0 is the reference impedances of 50 X terminating the

ports. In this article, the design example has an equal power

division at f1 and a 3-dB unequal power division at f2 hencethe values of a1 and a2 are chosen as 1 and 0.5, respec-

tively. The corresponding required values of ZT,A(f1) and

ZT,A(f2) are calculated as 35.35 and 28.87 X, respectivelyusing Eq. (1a). Similarly, the values of ZT,B(f1) and ZT,B(f2)are obtained as 50 and 35.35 X using Eq. (1b).

The design methodology is divided into two sections.

The first section is related to the design of stub loaded 2-

port structure. In this section, the design parameters of the

stub loaded 2-port structure of Figure 1b are retrieved by

solving its corresponding ABCD matrix. The second sec-

tion deals with the realization of the multisection stubs

which load the transmission line section in Figure 1b.

A. Stub Loaded 2-Port StructureIn general, for a stub-loaded structure as shown in Figure

1b, if we consider BS(f) as the shunt susceptance loading

Figure 1 Dual-band hybrid architecture (a) overall schematic, (b) schematic of single branch, and (c) equivalent of stub-loaded structure.

[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

2 Rawat et al.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 000, No. 000, Month 2012

the transmission line, the ABCD parameter of the stub

loaded structure is given by [8]:

A B

C D

� �

¼ coshS � BSZS sinhS jZS sinhSj sinhSZS

1 � Z2SB

2S þ 2BSZS coths

� �coshS � BSZS sinhS

" #

(2)

The Pi-type structure of Figure 1b is equivalent to a

transmission line of the characteristic impedance ZT and

electric length hT as shown in Figure 1c. Thus, the expres-

sion for the ABCD matrix of the Pi-type structure of Fig-

ure 1b can be used to obtain respective overall electric

length and the characteristic impedance as:

cosðhTÞ ¼ Aþ D

2¼ cos hs � BSZS sin hs (3)

ZT ¼ffiffiffiffiB

C

r¼ ZS

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1� Z2SB

2S þ 2ZSBS cot hs

s(4)

where ZS is the characteristic impedance of a series line

of electric length, hS, loaded by a susceptance of BS, as

shown in Figure 1b. Referring to the condition of a 90�

transformer in Eq. (3), we have:

BS ¼ 1

ZS tan hs(5)

Inserting Eq. (5) in Eq. (4) gives another condition:

ZT ¼ Zs sin hs (6)

For the structure in Figure 1b, Eqs. (5) and (6) together

guarantee the overall 90� electric length.

Thus, for different characteristic impedance at two dif-

ferent frequencies, Eqs. (5) and (6) can be written as:

ZTðf Þ ¼ ZS sin hSð Þ @f1ZS sin nhSð Þ @f2

�(7)

BSðf Þ ¼ ZS tan hSð Þð Þ�1 @f1ZS tan nhSð Þð Þ�1 @f2

�(8)

where, n is the frequency ratio f2/f1.If ZT(f1) and ZT(f2) are the two required characteristic

impedances at two frequencies f1 and f2, respectively, andtheir ratio is denoted by j, one can deduce following

expression from Eq. (7):

j ¼ ZT f2ð ÞZT f1ð Þ ¼

sin nhSð Þsin hSð Þ

(9)

The ratio of two sine functions can be solved graphi-

cally or analytically as given in appendix to obtain hS for

a required value of impedance ratio j.Figure 2 give range of the impedance ratios that can

be achieved by choosing different values of hS for a given

frequency ratio n for this topology.

Using Figure 2, the value of hS is obtained for a given

value of n and j. This value hS along with the known values

of ZT(f) and n are then used in Eq. (7) to obtain the corre-

sponding value of ZS. These values of ZS and hS once final-

ized according to Eq. (7), the corresponding required values

of BS(f) at the two frequencies can be obtained from Eq. (8),

hence guaranteeing the simultaneous existence of Eqs. (7)

and (8).

Figure 2 Range of the impedance ratio ZT2/ ZT1 obtained from proposed methodology for various frequency ratios n. [Color figure can

be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Dual-Band/Dual-Impedance 90� Transformer in Hybrid Coupler Design 3

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

It is worth mentioning that the shunt susceptance and

series transmission line parameters are obtained by solving

Eqs. (7) and (8) simultaneously, but the susceptance val-

ues are realized using multisection stubs independent of

the design of loaded transmission line.

B. Multisection Stub DesignFigure 3 shows multisection stub that provides different

susceptance values at its input for two distinct frequen-

cies. The design of multisection stub can be divided in

two sections, as shown in Figure 3a, where the first step

is the realization of the required susceptance, BS(f2), at theinput, which is accomplished as in section 1 of Figure 3a.

Here, f1 is the lower frequency between f1 and f2. InFigure 3a, section 1 is depicted as a transmission line

short circuit at one end at node A. This short circuit is

realized by a 90� open circuit transmission line designed

at f2, and hence, section 1 can be realized as shown in

Figure 3b. Once the short circuit at node A is realized by

open circuit transmission line of quarter-wave length

at frequency f2, adding any further section beyond this

point will not affect the input admittance at frequency f2.Thus, section 2 is designed as an admittance that, when

terminating section 1 at node A, emulates susceptance

BS(f1) at the input of section 1. This terminating admit-

tance is denoted as YB(f1) in Figure 3a that can be further

realized by either an open or short circuit transmission

line.

For designing section 1 in Figure 3a, if the designer

chooses a certain realizable value of characteristic imped-

ance ZC1, to achieve the desired input admittance values

of jBS(f2), the electric length hC1 can be calculated as:

hC1ðf2Þ ¼ tan�1 �1

ZC1BSðf2Þ �

(10)

where BS(f2) can be positive or negative depending on the

required susceptance value. The value obtained for hC1 is

considered at f2, to calculate the physical length. Referring

to Figure 3b, once the short circuit at node A is realized

by open circuit transmission line of quarter-wave length at

frequency f2, the value of susceptance YB can be synthe-

sized for frequency f1. This requires synthesis of admittan-

ces YA(f1) and YC(f1), as shown in Figure 3a such that

YBðf1Þ ¼ YAðf1Þ � YCðf1Þ (11)

where, YC(f1) is the admittance at the input of the 90�

transformer in Figure 3b at f1 and is given as:

YCðf1Þ ¼ j1

ZC;OCtan

p2

f1f2

�(12)

However, YA(f1) can be synthesized by de-embedding

section 1, with a known value of BS(f1). This can be

obtained using a standard transmission line impedance

equation as follows:

YAðf1Þ ¼ j

ZC1

ZC1BSðf1Þ � tan hC1 f2ð Þ f1f2� �

1þ ZC1BSðf1Þ tan hC1 f2ð Þ f1f2� �

24

35 (13)

where BS(f1) can be positive or negative depending on the

imaginary value of the required susceptance to be seen at

the input of the multisection stub at frequency f1.

Figure 3 Schematic of Multisection stubs used to realize susceptance values at two frequencies. [Color figure can be viewed in the

online issue, which is available at wileyonlinelibrary.com.]

4 Rawat et al.


The synthesized value of YB(f1) can be obtained using

Eqs. (12) and (13) in Eq. (11) and can be realized by an

open or short stub of characteristic impedance ZC2 and

electrical length hC2, which is represented as section 2 in

Figure 3a. If the designer chooses a certain realizable

value for the characteristic impedance of ZC2 for this stub,its electric length can be given by:

hC2ðf1Þ ¼tan�1 ZC2imag YBðf1Þð Þð Þ for open stub

tan�1 �1ZC2 imag YBðf1Þð Þ

� �for short stub

((14)

The choice of using an open or short stub and ZC2 in

Eq. (14) depends on the realizability of YB(f1) with a min-

imum stub length. The imaginary value of YB(f1) in Eq.

(14) can be positive or negative, depending on the calcu-

lated results in Eq. (11).

III. EXPERIMENT AND DESIGN CONSIDERATIONS

Photograph of a dual-band hybrid designed and fabricated

on RT5870 is given in Figure 4. This prototype is

designed for the equal-power division (a1 ¼ 1) at 1960

MHz and half power division (a2 ¼ 0.5) at 3500 MHz.

The overall dimensions are 2.8 � 3 inch2. The substrate

has dielectric constant of 2.33, height of 20 mil, and loss

tangent of 0.0012.

For a given frequency ratio of n ¼ 1.786, the value of

series transmission line of Figure 1b has been obtained

using design curve given in Figure 2. It is evident from

Figure 2 and sine series expansion given in appendix, that

the maximum value of the impedance ratio j given in Eq.

(9), which can be achieved with the proposed methodol-

ogy, is equal to n. Thus, for a particular frequency ratio, a

certain level of uneven power division (corresponding to

the value of a) can only be achieved with the proposed

methodology. Thus, it is an important design consideration

while deciding the feasibility of the given 2-port structure

for achieving a given value of power division a for a

given frequency ratio n. It is also worth mentioning that

while designing multisection stub of Figure 3a, there is no

such limitation that section 1 should always be designed

for f2. As the two frequencies are assumed to be

Figure 4 Photograph of fabricated circuit. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

TABLE I Design Parameters for the Dual-Band/Dual Impedance Transformers

Electrical Design Parameters

Value Physical

Design

Parameters

Value Calculated

(mm)

Value Optimized

(mm)

ZT,A ZT,B ZT,A ZT,B ZT,A ZT,B

ZS (X) 37.13 51.42 wS 2.25 1.4 2.22 1.46

hS @f1 (�) 72.21 76.51 lS 21.63 23.29 19.18 19.48

ZC1 (X) 50 50 wC1 1.47 1.47 1.47 1.47

ZC,OC (X) 50 50 wC,OC 1.47 1.47 1.47 1.47

ZC2 (X) 50 50 wC2 1.47 1.47 1.47 1.47

hC1@f2 (�) 42.56 44.15 lC1 7.24 7.51 6.9 8.33

hOC @f2 (�) 90 90 lOC 15.32 15.32 15.2 15.2

hC2 @f1 (�) 39.41 35.27 lC2 11.98 10.72 11.81 10.08



uncorrelated, section 1 can also be designed for f1 rather

than f2, depending on the value of susceptance BS(f) to be

realized at a particular frequency. For such a case, f1 is

replaced by f2 and vice versa in the design equations

(10)–(14). The choice of designing section 1 at f1 or f2depends on the configuration resulting in the overall

smaller size of the dual-band stub. In the present design

case, where the value of BS(f2) is much closer to the short

circuit than that of BS(f1); therefore, section 1 is designed

to realize BS(f2) and section 2 is designed to realize

BS(f1), to achieve an overall smaller size of the multisec-

tion stub.

Table I gives the electrical and physical design para-

meters of the design parameters of dual-band/dual-imped-

ance transformers used in Figure 1a calculated using the

proposed methodology of Section II.

IV. RESULTS AND DISCUSSION

The fabricated prototype has been measured using cali-

brated Vector Network analyzer. Figure 5 shows EM

simulated and measured results of return loss and isolation

of the fabricated branch-line hybrid. The measured return

losses are better than 10 dB over a band of 160 MHz

symmetrical around 1960 MHz frequency. The corre-

sponding isolation is around 15 dB over a band of 100

MHz symmetrical around 1960 MHz. Similarly the meas-

ured return loss is better than 10 dB over a band 120

MHz symmetrical around 3500 MHz. The corresponding

isolation is around 15 dB over a band of 100 MHz sym-

metrical around 3500 MHz. Because of some fabrication

error there is a slight frequency shift at both center fre-

quencies resulting into slightly lower bandwidth symmet-

ric around the two center frequencies of 1960 and 3500

MHz as shown in zoomed versions of graph in Figure 5.

Hence, if we ignore this symmetry about these center fre-

quencies, the first and second band has bandwidth of 200

MHz, where the return loss is around 10 dB.

Figure 5 Measured and EM simulated isolation and return loss of the fabricated branch-line hybrid. [Color figure can be viewed in the


Figure 6 Measured and EM simulated insertion loss of the fab-

ricated branch-line hybrid. [Color figure can be viewed in the


6 Rawat et al.


Similarly, the isolation is around 15 dB over a fre-

quency range of 120 and 180 MHz at the first and second

band of operation, respectively.

Figure 6 shows EM simulated and measured results of

insertion loss of the fabricated branch-line hybrid. The

equal power division is maintained with an error of 1 dB

over a frequency range of 120 MHz in the first band and

unequal 3 dB power division is maintained with 1 dB

error over a frequency range of 200 MHz in the second

band.

Figure 7 shows measured results of quadrature phase

relationship between split ports of the fabricated branch-

line hybrid.

Again, phase error of around 10� from the quadrature

relation has been achieved over the frequency range of

150 and 200 MHz in the first and second band, respec-

tively, as shown in Figure 7.

V. CONCLUSIONS

A dual-band/dual-impedance transformer has been real-

ized to design dual-band branch line hybrid with two

different power divisions at two frequencies of opera-

tion. The multisection stubs guarantees that any

required value of susceptance can be realized independ-

ent of series lines parameters and hence resulting into

robust solution. The electric design parameters calcu-

lated using the given formulae validate the proposed

methodology.

ACKNOWLEDGMENTS

The authors thank T. Bata for his technical assistance in

manufacturing the prototype. They also appreciate the valua-

ble support of the team of the iRadio Laboratory at the Uni-

versity of Calgary. This work was supported by the Alberta

Innovates Technology Future (AITF), The Natural Sciences

and Engineering Council of Canada, the Canada Research

Chair (CRC) Program.

APPENDIX

The ratio of Sine functions given in Eq. (9) can be

expresses as:

sin nhSð Þsin hSð Þ

� A0 þ A1h

2S þ A2h

4S þ A3h

6S þ A4h

8S

þ……Amh2mS (A1)

where; 0 < hS <2pn

Expanding sine series in each numerator and denomina-

tor of Eq. (A1) and rearranging

nhS � nhSð Þ33!

þ nhSð Þ55!

� nhSð Þ77!

þ :::

¼ A0 þ A1h2S þ A2h

4S þ ::::

� �hS � h3S

3!þ h5S

5!� h7S

7!:::

�(A2)

Figure 7 Measured quadrature phase performance of the fabricated hybrid. [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]



Comparing each term of the same order of hS in Eq. (A2),

values of coefficients can be obtained as

A0 ¼ n; A1 ¼ n� n3

3!; A2 ¼ n5 � n

5!þ A1

3!; A3

¼ n� n7

7!� A1

5!þ A2

3!; A4 ¼ n9 � n

9!þ A1

7!� A2

5!þ A3

3!(A3)

As the coefficients corresponding to higher values of �Sin Eq. (A2) tend to vanish in a fast manner; the approxi-

mation up to eighth order in right hand side of Eq. (A1)

(four coefficient terms) can give sufficient accuracy in

calculating �S.

The polynomial expansion in Eq. (A1) with coefficient

given by Eq. (A3) can be used for solving �S for a

required value �. It has been used for plotting Figure 2.

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BIOGRAPHIES

Karun Rawat (IEEE M’08, S’09)

received his B.E degree in electron-

ics and communication engineering

from Meerut University, UP, India,

in 2002. He is currently pursuing his

Ph.D. in the Department of Electrical

and Computer Engineering, Schulich

School of Engineering, University of

Calgary, Calgary, AB, Canada. He worked as scientist in

the Indian Space Research Organization (ISRO) from

2003 to 2007. After that, he joined the iRadio Laboratory

of the Schulich School of Engineering, University of Cal-

gary, where he has been working as a student research as-

sistant. He is the reviewer of several well-known journals,

and his current research interests are in the areas of

microwave active and passive circuit design and advanced

transmitter and receiver architecture for software defined

radio applications. His research involvement has resulted

in more than 20 publications in journals and conferences.

Due to his active research activities and publications, he

has been recipient of research production award for the

three consecutive years from 2009 to 2012 by Department

of Electrical and Computers Engineering at University of

Calgary.

8 Rawat et al.


Meenakshi Rawat (IEEE S’09)

received her B. Tech. degree in electri-

cal engineering from Govind Ballabh

Pant University of Agriculture and

Technology, Pantnagar, Uttaranchal,

India, in 2006. She is currently pursu-

ing her Ph.D. in the Department of

Electrical and Computer Engineering,

Schulich School of Engineering, University of Calgary, Cal-

gary, AB, Canada. She was associated with Telco Construc-

tion Equipment Co. Ltd., India, from 2006 to 2007 and Hin-

dustan Petroleum Corporation Limited (HPCL), India,

during 2007–2008. She is now working with the iRadio Lab

of the Schulich School of Engineering, University of Cal-

gary, as a student research assistant. She is also reviewer of

several well-known journals and her current research interest

is in the area of digital signal processing, neural networks,

and microwave active and passive circuit modeling.

Mohammad S. Hashmi (IEEE S’04–

M’09) received his B. Tech. degree

from Aligarh Muslim University,

India, in 2001, his M.S. degree from

the Darmstadt University of Technol-

ogy, Germany, in 2004, and his Ph.D.

degree from Cardiff University, UK,

in 2009. He subsequently worked as a

postdoctoral fellow at Cardiff University. Since September

2009, he has been a postdoctoral fellow with iRadio Lab,

University of Calgary, Canada. He also worked at PhilipsSemiconductors and Thales Electronics in Germany, duringwhich time he was involved in the field of RF circuits andsystems. His current research interests are the characteriza-tion and linearization of power amplifiers for mobile and sat-ellite applications, microwave active and passive circuits andnonlinear microwave instrumentation. Dr. Hashmi was the re-cipient of the 2008 Automatic Radio Frequency TechniquesGroup (ARFTG) Microwave Measurement Fellowship.

Fadhel M. Ghannouchi (IEEE S’84,

M’88, SM’93, FIEEE’ 07) is currently

a professor and iCORE/CRC Chair in

the Department of Electrical and

Computer Engineering of the Schulich

School of Engineering at the Univer-

sity of Calgary and Director of the

Intelligent RF Radio Laboratory. He

has held numerous invited positions at several academic

and research institutions in Europe, North America, and Ja-

pan. He has provided consulting services to a number of

microwave and wireless communications companies. His

research interests are in the areas of microwave instrumen-

tation and measurements, nonlinear modeling of microwave

devices and communications systems, design of power and

spectrum efficient microwave amplification systems and

design of intelligent RF transceivers and SDR radio sys-

tems for wireless and satellite communications. His

research activities have led to more than 450 publications

and 10 US patents (three pending).



dual-band branch-line hybrid with distinct power division ratio over the two bands

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