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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007 2365

Rigorous Mode-Matching Method of Circular to

Off-Center Rectangular Side-Coupled Waveguide

Junctions for Filter Applications

Jingliang Zheng and Ming Yu, Senior Member, IEEE

Abstract—An accurate and efficient method for analyzing cir-

cular to multiple off-center rectangular side-coupled waveguide

T-junctions is developed based on a rigorous mode-matching tech-

nique. The method is very general and not limited by symmetry or

other dimensional constraints to the junction. A new way to match

fields on the curved interface between subregions is described in

detail. The computer code developed based on this theory is highly

efficient. The numerical results match with the results obtained by

other methods including experimental ones. The method is used to

design a side-coupled circular waveguide dual-mode filter.

Index Terms—Circular waveguide T-junction, mode-matching

method.

I. INTRODUCTION

T

satellite splitting and combing networks because of its compact

size and excellent performance. Traditionally, this type of filter

is coupled at the ends of the circular cavities. In some cases,

this may not be possible since the ends may be inaccessible,

or may contain an adjustment mechanism for temperature

compensation or frequency tuning. In the new generation of

dual-mode waveguide filters introduced by Yu et al. [1], the

input/output coupling and/or coupling between the circular

cavities are realized at the sides of the circular cavities. In these

filters, there are rectangular irises and/or rectangular waveguide

sections on the sidewalls of the circular cavities (Fig. 1). To

design such filters more efficiently, an accurate and efficient

method to simulate circular to rectangular side-coupled wave-

guide T-junctions, as shown in Figs. 2 and 3, is desired.

The mode-matching technique [2]–[7] is a powerful electro-

magnetic (EM) modal analyzing method and is widely used for

simulating different waveguide circuits and solving other EM

problems.Forproblemswherethediscontinuityinterfaceispar-

allel to the coordinate planes at both sides of the discontinuity,

mode-matching formulas are straightforward to derive and the

method is extremely efficient and convenient. However, if the

discontinuity interface is not parallelto one side’s or both sides’

coordinate plane, the matching between the fields of both sides

becomes much more complicated. In order to obtain an efficient

HE DUAL-MODE circular waveguide filter using the

TE11n mode is one of the most important filter types for

Manuscript received May 16,2007; revised August 28, 2007.

The authors are with Com Dev Ltd., Cambridge, ON, Canada N1R 7H6

(e-mail: jingliang.zheng@comdev.ca; ming.yu@ieee.org).

Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2007.908662

Fig. 1. Side-coupled dual-mode filter.

Fig. 2. Side-coupled circular to single rectangular waveguide T-junction.

solution, the authors developed different ways to choose a con-

venient surface or region for matching the fields.

Melloni et al. [8] analyzed a waveguide bandpass filter using

the mode-matching technique. The main building blocks of the

filter are the cylindrical resonators coupled by rectangular irises

from the sidewalls. When matching the fields in the junction,

the curvature effect of the cylinder is neglected. Therefore, the

accuracy of this method will be questionable if the size of the

rectangular waveguide is large.

0018-9480/$25.00 © 2007 IEEE

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2366IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007

Fig. 3. Subregions and the geometry of side-coupled circular to multiple rect-

angular waveguide T-junction.

Krauss and Arndt [9] divided the whole T-junction structure

into six subregions, as shown in Fig. 2(a): two circular wave-

guide regions at two circular ports (outside of surfaces

), one rectangular waveguide region at the rectangular port

(outside of surface

), one circular cavity region between sur-

faces

and , and two additional regions, which are the “in-

homogeneous region” between surfaces

finitely thin artificial intermediate region on surface

“inhomogeneous region” contains the half-moon-shaped area.

The field in this region is calculated by the boundary-contour

mode-matching method [10]. The existence of the “inhomoge-

neous region” requires a minimum length of rectangular wave-

guide such that the rectangular port is not too close to the junc-

tion.

GentiliandMelloni[11]dividedthejunctionpartofthestruc-

ture into subregions using a similar approach to Krauss and

Arndt [9]. The generalized admittance matrix (GAM) has been

used and the infinitely thin artificial intermediate region on sur-

face

is no longer needed. Although minimal detailed deriva-

tion was given, a different method from [9] for the analysis of

thetransitionregionhasbeenemployed.Whiletherearenolim-

itations on the physical dimensions, the transition subregion be-

tween surfaces

andis still needed.

Rong and Zaki [12] solved a more complicated, but symmet-

rical problem, where two circular cylinders with circular dielec-

tric bodies at their centers are connected by a rectangular wave-

guide. The cross-sectional geometry of the structure is symmet-

ricalinboththe -and -directions[seeFig.2(b)].Atthecurved

connecting interface between circular and rectangular parts, no

additional region was used and the fields were matched directly

ontheinterface.Usingthesymmetrycondition,theauthorssim-

plified the problem by putting a perfect electric/magnetic con-

ducting wall on both symmetry planes.

Wuetal.[13]solveda1-Dsymmetricalproblem,asshownin

Fig. 2(c). No additional region was used at the curved interface,

and the rectangular region was extended into the cavity region.

Instead of using the interfaces

artificial interface

. In order to avoid the numerical integra-

tions where the integrand includes Bessel functions, the finite

plane-waveseries-expansion technique [14] wasused toexpand

the modes in the cavity region. Since the boundary conditions

on the half-moon-shaped area between the circular waveguide

region and rectangular waveguide region were ignored, the ac-

curacy becomes poor if the rectangular waveguide size “b” is

large.

and

and , and an in-

. The

, the fields are matched on the

Zheng and Yu [16] uses the mode-matching method to ana-

lyze a side-coupled circular waveguide to multiple rectangular

waveguide T-junctions without any symmetry, as shown in

Fig. 3. The rectangular waveguides may have different dimen-

sions and not necessarily be centered on the circular cavity.

Since the longitudinal field components and the longitudinal

factor of the transversal field are included in the equations

obtained from the boundary conditions, choosing an appro-

priate weighting function is an essential part of the solution

process. Theweighting functions chosenin thisstudy ledto two

important features. First, the fields are matched directly on the

curved interface

with neither artificial intermediate region,

nor transition region being used, which reduces the complexity

of the problem. Second, there are no Bessel functions in the

integrand of the numerical integrations so that the numerical

computation is much faster. The approach of selecting the

weighting function in this study is applicable to either flat or

curved interfaces.

Thispaperdiscussesindetailthemethodforanalyzingaside-

coupled circular waveguide to multiple rectangular waveguide

T-junctionsdescribedbrieflyin[16].Theresultantformulations

are listed in the Appendix. The convergence problem is also

discussed. The method is used to design a side-coupled circular

waveguide dual-mode filter, and is validated by numerical and

experimentalresults.Itisdemonstratedthatthemethodishighly

efficient, and has minimal limitations on the geometry of the

waveguide sections.

II. FORMULATIONS

A. Sub-Regions

A few rectangular waveguides with different cross sections

connected to a circular waveguide from its side are shown in

Fig. 3. Angle

andcan be any reasonable value where the

subscript indicates the th rectangular waveguide. Fig. 2(a) is

again used to represent the 3-D view of Fig. 3, although only

one rectangular waveguide branch is shown. The whole T-junc-

tion is divided into three types of subregions, i.e.: 1) circular

waveguideregions

;2)rectangularwaveguideregions

3) cavity region

. There are two circular waveguide regions,

i.e.,

and, from interfaces

respectively. The number of rectangular waveguide regions

which span from interface

to one rectangular port, is equal

to the number of rectangular waveguides connected to the junc-

tion. The cavity region

is in the middle of the structure, con-

necting with the circular waveguide subregions at the interface

andand the rectangular waveguide subregions at the in-

terface. All circular and rectangular waveguide subregions

do not connect with each other directly. The EM fields in dif-

ferent subregions are expressed by their own modal function se-

ries. The fields in neighboring subregions are matched on their

common interface

,, or

;and

andto two circular ports,

,

.

B. Field Modal Functions in Subregions

The modalfunctions for EMfields incircular and rectangular

waveguide can be found in standard textbooks, which are re-

peated here for completeness. In order to stay concise, only the

longitudinal magnetic field component

for TE modes and

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ZHENG AND YU: RIGOROUS MODE-MATCHING METHOD OF CIRCULAR TO OFF-CENTER RECTANGULAR SIDE-COUPLED WAVEGUIDE JUNCTIONS2367

the longitudinal electric field component

expressed in this paper. All transverse field components can be

derived from longitudinal field components by the formula

for TM modes are

(1)

where

junction,

mode in the waveguide, and

ator

isthewavenumberofthemediumfillingthewaveguide

is the propagation constant of the

is the transverse gradient oper-

(Cartesian coordinate)

(cylindrical coordinate).

(2)

Theupperandlowersigninsymbols

formulasinthispaperalwaysindicatethewavepropagatingfor-

wards and backwards, respectively.

Incircularwaveguideregion

and in(1)andother

,thefieldmodalfunctionsare

(3)

where

are the th zero of the

tive, respectively,

and

indicate the TE and TM mode, and

- and-

coordinate factor.

Incircularwaveguideregion

almost the same as (3). The only difference, which is needed for

added convenience during field matching, is that the coordinate

in (3) has to be replaced with

between interfaces

and.

In rectangular waveguide regions

is the Bessel function of order

th-order Bessel function and its deriva-

is the radius of the circular waveguide, and

are the normalization factors. Subscripts

, and

and

and represent the

mode. Subscript indicates the transverse

,thefieldmodalfunctionsare

, whereis the distance

,

(4)

where

are local coordinates in the th rectangular waveguide with

as the longitudinal coordinate and

circular cylinder in the structure.

andare the normalization factors., , and

parallel to the axis of the

andare the dimensions

of the th rectangular waveguide in the

spectively.

There are three different sets of modal functions in circular

cylindricalcavityregion

.Thefirstmodalfunctionsetcontains

the modes for the circular cylinder shorted at interface

and

- and-direction, re-

(5)

where the transverse coordinate factors

and

, their normalization factors and, and

the propagation constants

as they were in (3) for the circular waveguide region since the

radius is the same for both regions.

The second modal function set in the cavity region includes

themodesinthecircularcylindershortedatinterface

and. Its expressions can be obtained from (5) directly by

substituting

with .

The third modal function set in the cavity region is the mode

set for the space between two parallel plates at the interface

and, respectively,

andare exactly the same

(6)

Each mode normalization factor

determined by forcing the vector product between

andof that mode over the entire cavity side surface

to be 1, while all other normalization factors in (3)–(5)

are determined by the vector product over the cross-sectional

surface of the corresponding waveguide.

The propagation constants in (3)–(6) are

or in (6) will be

(7)

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2368IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007

C. Field Matching Procedure

Arranging the mode sequence in each modal function set ac-

cording to their cutoff frequencies, the double mode indices

and can be replaced by the mode sequence’s number in the

formulas. In this paper, , , , ,

and TM-mode indices for different mode sets instead of

.

The fields in each subregion are the superposition of the

modal functions in that subregion. Taking only the first finite

terms, the superposition can be expressed in the following

matrix expressions. Since the electrical and magnetic fields

have a similar form, only the

-field expressions will be shown

here. In subregion

and, the total field is

and will be used as TE-

and

(8)

where superscript

can beand

...

...

...

...

(9)

and aresimilarto

, and the coefficient matrix

and, hasasimilar

form to

(10)

and are similar to and, has a similar

form to

In subregion

.

, the fields have similar expressions

(11)

where

...

...

...

...

(12)

(13)

and coefficient matrix

(12) and (13).

In cavity region

have expressions similar to

, the fields contain three groups of modes

(14)

where

pressions, as shown in (9) and (10).

The matrices

cient matrices, which will be determined by the boundary con-

ditions on all ports and all interfaces between subregions. There

are

unknown coefficient matrices altogether,

where

indicates the number of rectangular waveguides in

the structure and

is the total port number.

matrix equations are needed to obtain the relationship between

the incoming and outgoing wave modes on all ports, which may

bepresentedintheformofageneralscatteringmatrix.Fromthe

electrical boundary conditions on the interface between subre-

gions, three matrix equations can be obtained. Another

equations will be derived from the magnetic boundary condi-

tions.

The vector product of functions

are defined by the following integration:

,, and,,have similar ex-

in (8), (11), and (14) are unknown coeffi-

andon the surface

(15)

The electrical boundary condition on the interface

is

on(16)

where

,

functions, and taking the vector product on both sides of (16)

on

, it is derived that

is the unit vector in the -direction. Choosing

,, andas the weighting

(17)

where

larly,from theelectricalboundaryconditionontheinterface

the following equation is obtained:

is the diagonal matrix shown in the Appendix. Simi-

,

(18)

On the interface

, the electrical boundary condition is

on(19)

where

cavity. Using the weighting functions

, and

both sides of (19) on

containsthecompletesidesurfaceofthecylindrical

,,

, and taking the vector product on

, (20) is obtained as follows:

(20)

where matrices

Appendix. Most integrations included in the elements of ma-

trices

, except for a few special cases, cannot be solved an-

alytically. However, since the integrands do not contain Bessel

functions, numerical integration can be done quickly. That is

one of the advantages of choosing a cylindrical surface as the

and diagonal matrixare shown in the

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ZHENG AND YU: RIGOROUS MODE-MATCHING METHOD OF CIRCULAR TO OFF-CENTER RECTANGULAR SIDE-COUPLED WAVEGUIDE JUNCTIONS 2369

interface between the cavity and rectangular waveguide subre-

gions.

The magnetic boundary condition on the interface

shows

on(21)

Using weighting functions

and

(21) on the interface

,,,

, and taking the vector product on both sides of

, it is derived

(22)

Similarly, from the magnetic boundary condition on the in-

terface

, using similar weighting functions, it is obtained

(23)

The magnetic boundary condition on the interface

shows

on (24)

where

rectangular waveguide region.

The choice of weighting functions for performing the vector

productson bothsides of (24)is important.The weightingfunc-

tions should have the characteristics of both the forward and

backward propagating waves. If the interface is a flat plane such

as

or , and is parallel to the tangential field components,

field expression factors having the longitudinal coordinate are

constant on the interface, and thus the forward and backward

propagating waves differ by a constant factor. In the case of a

curvednonparallelinterface, fieldexpressionfactorshavingthe

longitudinal coordinate are not constant over the interface so

the difference between the forward and backward propagating

waves is no longer described by a constant factor. A possible

way to achieve the goal is to use the sum of the forward and

backward propagating waves as the weighting functions

is the interface between the cavity region and the th

(25)

Performing thevectorproduct toboth sidesof(24) withthese

weighting functions on

yields

(26)

where superscript

tion(17),(18),(20),(22),(23),and(26)aretogether

matrix equations. Combining these equations, unknown coef-

ficient matrices

, , and

desired matrix equations are obtained. These con-

tain

unknown coefficient matrices

representing the magnitude and phase of the forward and

backward propagating wave modes at each port. The general

scattering matrix for the structure can be obtained from these

equations.

Selecting the weighting function and taking the vector

product in order to get equations from the boundary conditions

indicatesthetransposeof thematrix.Equa-

can be eliminated and

, , and

is one of the essential steps of the mode-matching technique.

Actually, (25) is a general way to select the weighting func-

tions. It is valid for cases where the interface is parallel or is not

parallel to the tangential field components in the corresponding

subregions. Following this approach, the weighting function

for boundary condition (21) should be

(27)

Since the interface

cular waveguide,

is parallel to the cross section of the cir-

on

on (28)

By combining (27) and (28), the weighting function used to de-

rive (22) can be obtained.

III. CONVERGENCE PROBLEM

For practical purposes, all the field modal functions have to

be truncated in numerical calculation. It is well known in mode-

matching techniques that the choice of the number of the modes

and the ratio of these numbers between different subregions can

strongly influence the accuracy of the simulation. In this study,

a critical frequency is used to determine the mode numbers. All

modes that have cutoff frequencies lower than the critical fre-

quency must be taken into account.

The convergence of the example shown in Fig. 4 is studied. It

is worth pointing out that the aperture at the junction is large in

the circumferential direction of the cylinder. The

of the junction are simulated at frequency

different critical frequencies. Table I shows the mode numbers

and the simulated data versus the critical frequency. When the

critical frequency is selected as six times the frequency of

the simulated

-parameters converged. Although S11sin (the

return loss of sin mode at the circular port) varies rapidly in the

frequency band near

(see Fig. 4), its simulated results at

showed little change, while the critical frequency changed from

to.

-parameters

GHz using

,

IV. NUMERICAL RESULTS

A computer program based on the presented theory is de-

veloped to simulate side-coupled circular waveguide to mul-

tiple rectangular waveguide T-junctions. A series of examples

of such structures with different dimensions and different sym-

metries is simulated by the program and compared with the

measurements and other simulation tools. Three examples are

shown in Figs. 4–6.

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2370 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007

Fig. 4. Magnitude of ?-parameters of a circular to single rectangular wave-

guide T-junction.

Fig. 4 shows an example of a circular to single rectangular

waveguideT-junction, where

in.Thetwocircularportsaresettobeports1and2,whilethe

rectangular port is port 3. This figure shows the

of the first two modes at the circular ports, the TE11-cos and

TE11-sin mode, and the first mode atthe rectangular port TE10.

The inflection point at approximately 12.1 GHz is caused by

the TM01 mode in the circular waveguide. Its cutoff frequency

is 12.116 GHz. On a 3.6-GHz P4 computer with 3.5-GB RAM,

thecomputationoftheproposedmethodtakesapproximately1s

for one frequency point and approximately 2 min for the entire

,inand

-parameters

TABLE I

CONVERGENCE OF THE SIMULATION DATA

Fig. 5. Magnitude of ?-parameter of a circular cavity coupled to a rectan-

gular waveguide with a longitudinal slot at its side and connected to rectangular

waveguides at both its ends (plot from [16]).

Fig. 6. Magnitude of ?-parameters of a circular to two off-center rectangular

waveguide T-junction with shorted ends at two circular ports (plot from [16]).

frequency band, while HFFS uses over 40 min for the whole

frequency band.

Fig.5showsanexampleofacircularwaveguidecavitywitha

longitudinal slot iris coupled to a rectangular waveguide. Both

cavity ends are connected to rectangular waveguide junctions.

,,

is 2.2 in, iris thickness is 0.014 in, and the WR75 waveguide

in,, cavity length

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ZHENG AND YU: RIGOROUS MODE-MATCHING METHOD OF CIRCULAR TO OFF-CENTER RECTANGULAR SIDE-COUPLED WAVEGUIDE JUNCTIONS2371

TABLE II

? ? ? COUPLING MATRIX OF THE FOUR-POLE FILTER

is used at all three ports. The simulation results match with the

measurement.

Fig. 6 shows an example of a circular to two off-center rect-

angular waveguide T-junction, where

, in,

. The two circular ports are shorted at the

ends. The length of each circular waveguide section on both

sides of the T-junction is 1.5 in. This figure shows the -param-

eters of the first mode at the rectangular ports TE10 mode. Both

themode-matchingresultsandAgilentTechnologies’HighFre-

quency Structure Simulator (HFSS) results show glitches at ap-

proximately 10.4 and 12.2 GHz, which correspond to the reso-

nant frequency of the TE113 and TE115 modes in the circular

cylindricalcavityrespectively.Computationusingtheproposed

method takes approximately 1 s for one frequency point and ap-

proximately 3 min for the entire frequency band, while HFFS

uses over 80 min for the entire frequency band.

,

, , and

V. SIDE-COUPLED DUAL-MODE FILTERS

Aside-coupledcircularwaveguidedual-modefour-polefilter

with a trifurcated iris, as shown in Fig. 1, is designed. The ge-

ometry of the filter is described in [1]. The design is performed

by using the space-mapping method [15]. The fine model com-

putation is performed using computer code (engine) based on

the presented theory. In addition to the iris models given earlier,

tuning screws are modeled using the 2-D finite-element method

and 3-D mode-matching method. Unlike [1], which used HFSS

as a fine model engine, the mode-matching-based engine runs

30 times faster. Since the EM code is called many times in the

space-mapping process, the presented technique improved total

design cycle time by a factor of ten. The coupling matrix used

to design the filter is shown in Table II. Fig. 7 shows the per-

formance of the filter where the simulation results match with

the measurement results quite well. Typical computation time

for one frequency point is approximately 4 s. Only a few min-

utesarerequiredwhenafull bandresponseis desiredfortypical

filter design problems.

VI. CONCLUSION

A rigorous and efficient mode-matching technique has been

presented for analyzing circular to multiple off-center rectan-

gular side-coupled waveguide T-junctions. The method is gen-

eral and does not have any symmetrical or dimensional limita-

tions at the junction. The computer code developed based on

Fig.7. Performanceofaside-coupledcircularwaveguidedual-modefilterwith

trifurcated iris.

this theory is accurate and uses less computer time than ex-

isting solvers, which makes it possible to analyze and optimize

a side-coupled circular waveguide dual-mode filter more effi-

ciently.

APPENDIX

In all the following,

trices:

,, , andare diagonal ma-

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2372 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 11, NOVEMBER 2007

whereand are mode indicesACKNOWLEDGMENT

The authors wish to thank Dr. X. Tian and B. Yassini, both

with Com Dev Ltd., Cambridge, ON, Canada, for their help on

checking this paper’s long and tedious equations.

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Page 9

ZHENG AND YU: RIGOROUS MODE-MATCHING METHOD OF CIRCULAR TO OFF-CENTER RECTANGULAR SIDE-COUPLED WAVEGUIDE JUNCTIONS2373

Jingliang Zheng received the B.S. degree from Bei-

jing University of Posts and Telecommunications,

Beijing, China, in 1982, and the Masters and Ph.D.

degrees in electrical engineering from Tsinghua

University, Beijing, China, in 1984 and 1988.

For one year, he was with the Beijing Design In-

stituteofTelecommunications,Beijing,China,where

he was involved with wireless communication. From

1989 to 1993, he was involved with EM field simula-

tion with the Swiss Federal Institute of Technology,

Zurich, Switzerland. From 1994 to 1998, he was in-

volved with antenna and antenna array design with DSO National Laboratories,

Singapore. He was then an Engineer with GHz Technologies Inc., Montreal,

QC, Canada, for two years. In 2000, he joined the Research and Development

Department, Com Dev Ltd., Cambridge, ON, Canada, where he is currently

a Senior Member of Technical Staff, involved with the development of com-

puter-aided design (CAD) software for design, simulation, and optimization of

microwave circuits for space applications.

Ming Yu (S’90–M’93–SM’01) received the Ph.D.

degree in electrical engineering from the University

of Victoria, Victoria, BC, Canada, in 1995.

In1993,whileworkingonhisdoctoraldissertation

part time, he joined Com Dev Ltd., Cambridge, ON,

Canada, as a Member of Technical Staff, where he

was involved in the design of passive microwave/RF

hardware from 300 MHz to 60 GHz. He was also a

principal developer of a variety of Com Dev Ltd.’s

design and tuning software for microwave filters and

multiplexers. His varied experience with Com Dev

Ltd. also includes being the Manager of filter/multiplexer technology (Space

Group) and Staff Scientist of corporate research and development (R&D). He

is currently the Director of R&D. He is responsible for overseeing the devel-

opment of RF microelectromechanical system (MEMS) technology, computer-

aided tuning and EM modeling, and optimization of microwave filters/multi-

plexers for wireless applications. He is also an Adjunct Professor with the Uni-

versityof Waterloo,Waterloo, ON, Canada.Hehas authored orcoauthored over

70 publications and numerous proprietary reports. He is frequently a reviewer

for IEE publications. He holds eight patents with four pending.

Dr. Yu is vice chair of the IEEE Technical Coordinating Committee 8 (TCC,

MTT-8) and is a frequent reviewer of numerous IEEE publications. He is the

chair of the IEEE Technical Program Committee 11 (TPC-11) for 2006 and

2007. He was the recipient of the 1995 and 2006 Com Dev Ltd. Achievement

Award for the development of computer-aided tuning algorithms and systems

for microwave filters and multiplexers.

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