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Progress In Electromagnetics Research, PIER 36, 319335, 2002

MODE-MATCHING ANALYSIS OF WAVEGUIDET-JUNCTION LOADED WITH AN H-PLANEDIELECTRIC SLAB

Z. Jiang, Z. Shen, and X. Shan

School of Electrical and Electronic EngineeringNanyang Technological University

Nanyang Avenue, Singapore 639798

Abstract This paper presents a full-wave mode-matching analysis of a rectangular waveguide T-junction partially loaded with an H-planedielectric slab. First, the longitudinal section electric (LSE) and lon-gitudinal section magnetic (LSM) modes in an H-plane dielectric-lledrectangular waveguide are obtained. Second, the dielectric-loaded T- junction is then divided into four regions and their electromagneticelds are expanded into the summation of their modal functions usingthe resonator method. Finally, a mode-matching process for the tan-gential eld components across regional interfaces is carried out toderive the generalized scattering matrix of the waveguide T-junction.Using the cascading connection technique of generalized scatteringmatrix, various waveguide couplers and power dividers/combinerswith H-plane dielectric slabs can be easily analyzed and designed.Numerical results are presented and compared with those obtainedby measurement or Ansofts HFSS. Good agreement is observed.

1 Introduction

2 Formulation2.1 Modes in a Dielectric-lled Waveguide2.2 Waveguide T-junction

3 Numerical Results

4 Conclusion

References


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320 Jian, Shen, and Shan

1. INTRODUCTION

Rectangular waveguide T-junction is a basic passive microwave com-ponent and it has found wide applications in waveguide directionalcouplers [14], power dividers/combiners [5,6], lters [7] and multi-plexers [8] for modern radar and satellite communication systems.

Compared to the conventional air-lled waveguide T-junction,using dielectric lling can obviously reduce the size of the T-junctionand in turn reduce the dimensions of various waveguide passive compo-nents, i.e., couplers, power dividers/combiners. Employing dielectricloading can also provide some other useful characteristics compared tothe standard air-lled waveguide T-junction. For example, the powercarried along a dielectric-lled waveguide is conned primarily to theinterior of the dielectric slab. This can reduce the radiation loss andfacilitate a tight coupling from the main waveguide to the side arm of a waveguide T-junction when a dielectric slab is inserted into the partclose to the side arm.

The standard air-lled rectangular waveguide T-junction has beenextensively investigated by a number of researchers [613]. Althoughthe structure is potentially very useful, it appears that no study of theproblem of a rectangular waveguide T-junction loaded with an H-planedielectric slab has been reported in the literature, to the authors bestknowledge.

This paper presents a rigorous analysis of a rectangular waveguideT-junction having an H-plane dielectric slab partially lled in the mainwaveguide using the mode-matching method. The electromagneticelds in the T-junction are expanded into the summation of a seriesof eigen modes and the derived generalized scattering matrix of thewaveguide T-junction can be directly used in the synthesis of variouswaveguide components. Compared to other numerical techniques, themode-matching method is formally exact and has been proved to bevery efficient. The cross-section of the side-arm waveguide in the T- junction can also be arbitrary and the two coupling waveguides can bedissimilar [14].

This paper is organized as follows. In Section 2, we rst describethe modes in a dielectric-lled waveguide. Then, the derivationof the generalized scattering matrix is presented for an H-planedielectric-lled rectangular waveguide T-junction with an arbitraryside arm. The scattering matrices of various single-aperture couplingstructures such as waveguide couplers or power dividers/combiners canbe obtained directly by cascading two three-port T-junctions together.Cascading a number of single-aperture coupling elements togethercan also provide the scattering matrix of a multi-aperture coupling


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Mode-matching analysis of waveguide T-junction 321

structure. In Section 3, we rst verify the scattering parameter resultsfor two rectangular waveguide T-junctions with and without dielectricslab insertion, respectively. Then, three examples of dielectric-lled

waveguide couplers and power divider/combiner designed at X-bandusing rectangular apertures are presented. The calculated scatteringparameters by our mode-matching method are compared with theavailable measured or simulated results by Ansofts HFSS. Excellentagreement has been observed for all the cases considered.

2. FORMULATION

This section presents the theoretical formulation for the generalized

scattering matrix of an H-plane dielectric-slab-loaded waveguide T- junction. First, we give the LSE and LSM modes in an H-planedielectric-lled rectangular waveguide. Then. The mode matchingmethod is used to derive the generalized scattering matrix of thedielectric-lled waveguide T-junction.

2.1. Modes in a Dielectric-lled Waveguide

Figure 1 illustrates the cross-section of an H-plane dielectric-lled

rectangular waveguide.

Figure 1. Cross-section of an H-plane dielectric-lled waveguide.

It is known that the normal modes of propagation for a waveguideloaded with dielectric slabs, as shown in Fig. 1, are not, in general,either E or H modes, but combinations of an E and an H mode. Asthe boundary conditions at the interface ( y = t) can be satised byE or H modes alone at xz -plane, we know that the basic modes of propagation for slab-loaded rectangular guides may be derived frommagnetic and electric types of Hertzian potential functions, havingsingle components directed normal to the interface of two dielectriclayers [15, 16]. The derived modes are called longitudinal section


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electric (LSE) and longitudinal section magnetic (LSM) modes, whichcan be written in the following forms.

LSE modes: h = Ah cosnx

a sin(ky1y) 0 y tB h cos

nxa

sin[ky2(b y)] t y b(1)

LSM modes: e =Ae sin

nxa

cos(ky1y) 0 y t

B e sinnx

acos[ky2(b y)] t y b

(2)

where (h,e ) are the potential functions for LSE and LSM modes,respectively. A(e,h ) and B (e,h ) are the modal amplitude coefficients,which can be related to each other during the mode-matching process.ky1 and ky2 can be derived from

k2y1 = k2r 1 + 2

na

2(3a)

k2y2 = k2r 2 + 2

na

2(3b)

where is the propagation constant of the LSE or LSM mode.r 1 and r 2 are the relative permittivities for Region I and RegionII, respectively. k is the wavenumber in the free space. Theelectromagnetic elds in Fig. 1 can be obtained from

LSE modes: E = j0 h H = h

(4)

LSM modes: E = e H = j (1 ,2) e(5)

where the magnetic-type Hertzian potential for LSE modes is h =ay h (x, y )e

z and the electric-type Hertzian potential for LSM modesis e = ay e (x, y )e

z with the assumption that z is the propagationdirection.

2.2. Waveguide T-junctionFigure 2 illustrates the waveguide T-junction partially loaded withan H-plane dielectric slab. The cross-section of the side arm can bearbitrary.


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Figure 2. Waveguide T-junction partially loaded with an H-planedielectric slab.

The whole waveguide T-junction is divided into four regions: 1, 2,3, and 4. Regions 1 and 2 are dielectric-lled semi-innite rectangularwaveguides, where the expansion expressions of the electromagneticelds can be easily obtained from subsection 2.1. Region 3 is ahomogeneous semi-innite waveguide of arbitrary cross-section, whosemodal functions and cutoff wavenumbers are available. Interface S

1is

at z = 0 , S 2 at z = c, and S 3 is at y = b (0 < x < a, 0 < z < c ).The electromagnetic elds in the above three semi-innite wave-

guides can be expressed as [6]:

E (v) =1

j A(v)e + A

(v)h (6a)

H (v) =1

j A(v)h + A

(v)e (6b)

where v = 1 , 2, 3, Ah and Ae are the vector potentials for LSE, LSMmodes (v = 1 , 2) or TE, TM modes ( v = 3), respectively. For Regions1 and 2, we can have the following forms for A(1 ,2)h and A

(1 ,2)e :

A(1 ,2)h = azN h

i=1N (1 ,2)hi

(1 ,2)hi [A

+1,2hi e

1 , 2 hi z + A1,2hi e+ 1 , 2hi z ]

A(1 ,2)e = azN e

i=1N (1 ,2)

ei(1 ,2)

ei[A+

1,2eie 1 , 2ei z A

1,2eie+ 1 , 2ei z ]

(7)

where A+1i and A1i are the modal amplitude coefficients of the incident

(+) and reected ( ) waves in waveguide Region 1; A2i and A+2i are


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the modal amplitude coefficients of the incident ( ) and reected (+)modes in waveguide Region 2.

For Region 3, we can have the following forms for A(3)h and A(3)e :

A(3)h = ayM h

j =1N (3)hj

(3)hj [A

+3hj e

3 hj y + A3hj e+ 3hj y ]

A(3)e = ayM e

j =1N (3)ej

(3)ej [A

+3ej e

3 ej y A3ej e+ 3ej y ]

(8)

where A+3 j , A3 j are the modal amplitude coefficients of the reected (+)

and incident ( ) waves in the side arm; h and e are the propagation

constants of the TE and TM modes, respectively; i and j are the modeindexes; N (v)(he ) are the normalized factors such that the power carried

by each mode is 1 W. (v)(h,e ) is the modal function for different regions.

For Regions 1 and 2, the modal functions (1 ,2)(h,e ) are given in subsection

2.1. For Region 3, the modal functions (3)h and (3)e will depend on

the waveguide cross section of the side arm considered.For Region 4, the resonator method [17, 18] is employed to obtain

its electromagnetic elds by superimposing three suitably chosenstanding-wave solutions:

A(4)h,e = A(4)(1)h,e + A

(4)(2)h,e + A

(4)(3)h,e (9)

where solution A(4)(1)h,e is obtained when the boundary planes S 2 and S 3in Fig. 2 are short-circuited and S 1 is left open; solutions A

(4)(2)h,e and

A(4)(3)h,e are obtained in a similar way.Once all the eld expressions for all the four regions are available,

we then match the tangential electric and magnetic elds along thethree regional interfaces S 1, S 2, and S 3. The relationship betweenthe modal amplitude coefficients ( A+(v) and A

(v) , v = 1 , 2, 3) of the

three semi-innite waveguides can be derived to result in the desiredgeneralized scattering matrix of the waveguide T-junction. For therectangular waveguide T-junction having a side arm of arbitrary shape,as shown in Fig. 2, the nal scattering matrix can be written as:

[S T junction ] =S 11 S 12 S 13S 21 S 22 S 23S 31 S 32 S 33

=A B CB A DE F G

1

I (10)


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where

C =12

H 1 M A (11)

D = 12

H 2 M A (12)

G =12

(I Y 13 MTA Y 4 ,II G P M A ) (13)

E = 12

(W A W B L 21 )( I L21 )

1 (14)

F =12

(W A W B )L 1 (I L 21 ) 1 (15)

withAi,j =

11 e 2 1 i C

i,j (16)

B i,j =e 1 i C

e 2 1 i C 1i,j (17)

L1i,j = e 1 i C i,j (18)

H 1,ik =

S 1 ,I

e1it,I 2e 4k,II (t b)

P 1GS {[ cosh( 4k,I y)] hx 4k,I x

+ [sinh( 4k,I y)] hy4k,I y} ds + S 1 ,II e1it,II e 4k,II (y b)

Q1e 2 4k,II (t b)

P 1e 4 k,II (y b) hx4k,II x

+ e 4k,II (y b) +Q1e

2 4k,II (t b)

P 1e 4 k,II (y b) hy4k,II y

dsGS

(19)M A,kj = S 3 ( e4kt e3 jt )ds = S 3 (y e4kt ) (y e3 jt )ds (20)H 2,ik = ( 1)n

(4) +1 H 1,ik (21)

W A,ji =1

Y 3 j S 3 y e3 jt e 1 i z [hz 1i,II z + hx1i,II x] y= b2 ds (22)W B,ji =

1

Y 3 j S 3

y e3 jt e 1 i z [hz1i,II z hx 1i,II x]y= b2

ds (23)

andP 1,k=sinh( 4k,I t) cosh( 4k,I t)

Y 4k,I Y 4k,II

(24)


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Q1,k = sinh( 4k,I t) + cosh( 4k,I t)Y 4k,I Y 4k,II

(25)

GS,k = 1 +e 2 4 k,II (t b) Q1,k

P 1,k (26)

G p,k =1

e 2 4k,II (t b)Q1,kP 1,k

1 +e 2 4k,II (t b)Q1,k

P 1,k

(27)

hx 4k(I,II ) =

l (4) a 2 + n (4) c 2 k2r 1,2 j

l(4) a

N (4)ln sinl(4) x

acos

n (4) zc

TE-modes

j r 1,2

l (4) a 2 + n (4) c 2 k2r 1,2

n (4) a

N (4)ln sinl(4) x

acos

n (4) zc

TM-modes

(28)

hy4k(I,II ) =

l (4) a

2+ n

(4) c

2

j N (4)ln

cosl(4) x

acos

n (4) zc

TE-modes

0 TM-modes

(29)

hz1i,II =

( 1i ky2) sin(ky1t)sin(ky2(b t))

cos(ky2(b y)) cos n(1) x

a N LSE LSE-modes

( j r 1)n (1)

acos(ky1t)

cos(ky2(b t))

cos(ky2(b y)) cos n(1) x

a N LSM LSM-modes

(30)


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hx1i,II =

n (1)

a ky2

sin(ky1t)sin(ky2(b t))

cos(ky2(b y)) sinn (1) x

a N LSE LSE-modes( j r 1) 1i

cos(ky1t)cos(ky2(b t))

cos(ky2(b y)) sinn (1) x

a N LSM LSM-modes

(31)

In (13), Y 3 is the diagonal admittance matrix for the side arm

waveguide, which is dened as Y 3i = 3i /j forTE

modes andY 3i = j/ 3i for TM modes. Y 4( I,II ) are the admittance matrices forRegion 4(I) and Region 4(II), respectively. i,j is the Kronecker delta,which is dened as i,j = 0 for i = j , and i,j = 1 for i = j . Thematrix M A the E-eld mode-matching matrix at interface S 3 and thesuperscript T denotes the transpose operation. e3 jt is the normalizedtransverse component of the electric eld for the j th mode in the sidearm, which has the form of:

e3 jt = y t N h3 j h3 j TE-modes

t N e3 j e3 j TM-modes(32)

e4kt is the normalized transverse component of the electric eld for thekth mode in Region 4 and can be expressed using the same way as (32)in terms of its potential functions. e1it ( I,II ) is the normalized transversecomponent of the electric eld for the i th mode in Region 1(I,II) andRegion 2(I,II), which can be derived directly from the modal functionsgiven in subsection 2.1.

It is noticed that equations (10)(32) give a complete solutionfor the scattering parameters of an H-plane dielectric-slab-loadedrectangular waveguide T-junction having a side arm of arbitrary shape.For waveguide side arms of different shapes, the only difference will bein equations (20), (22), (23) due to the different e3 jt for different crosssections of the side arm.

The scattering parameters for a single-aperture coupler can bederived from the two scattering matrices of the basic T-junctions bycascading two three-port networks together. The diagonal transmissionmatrix of the side arm waveguide has its diagonal element of:

L i = e 3 i t = e

t k2c,i

k 2 (33)


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with k2 = 2 and t being the thickness of the aperture. Theeffects from higher-order mode interaction are automatically includedin the calculation when full sets of TE and TM modes in the side-arm

waveguide are employed in the analysis.

3. NUMERICAL RESULTS

In order to verify the validity of our mode-matching formulation,ve examples are considered: an air-lled rectangular waveguide T- junction; an H-plane dielectric-lled waveguide T-junction; two cou-plers employing transverse and longitudinal rectangular apertures,respectively, between two parallel dielectric-lled rectangular waveg-

uides; and a 3 dB power divider designed for X-band slot antenna ar-rays. The results obtained by our method are then compared with theavailable measured data or those obtained by Ansofts HFSS. Excellentagreement is achieved for all the cases considered.

An air-lled rectangular waveguide T-junction with the sidearm also being a rectangular waveguide has been analyzed by otherauthors [6, 9]. The example of a rectangular waveguide (WR-137:a2 = 34 .85mm, b2 = 15 .799mm) T-junction with a side rectangularwaveguide of reduced size (WR-90: a1 = 22 .86mm, b1 = 10 .16mm)

located at the center of the broad wall, as shown in Fig. 3, is consideredhere rst for comparison. By setting the relative permittivities r 1 andr 2 (in Fig. 2) to 1 for both Region I and Region II, the dielectric-lled T-junction shown in Fig. 2 will reduce to the conventional air-lled waveguide T-junction. The scattering parameters obtained byour mode-matching method are compared to the measured data givenin [9]. Good agreement between them is observed. Only |S 11 | and|S 21 | are given in Fig. 3 because |S 31 | and |S 33 | can be readilycalculated from |S 11 | and |S 21 | based on the unitary property of a

lossless network. In our computation, 50 modes are considered in themain waveguide, 20 modes are used in the side arm and 35 modes areretained in Region 4.

Another waveguide T-junction of the same dimensions partiallylled with an H-plane dielectric slab in the main waveguide isillustrated in Fig. 4. It is noticed that the behavior around the cutoff frequency of the rst mode in the side arm (about 6.56GHz in Fig. 3) isgreatly altered after inserting the dielectric slab. The loading dielectricslab reduces the resonant frequency from 6.56 GHz to 5.55 GHz, as we

expected. The relative permittivity r 2 for the dielectric slab used inFig. 4 is 2.5 and the thickness of the dielectric slab is ( b t) = 8 .299 mm.Our mode-matching results are compared to those obtained by AnsoftsHFSS. It is seen that the agreement between them is very good.


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Figure 3. Scattering parameters of an air-lled rectangular waveguideT-junction with a side arm of reduced size waveguide ( a

2= 34 .85mm,

b2 = 15 .799 mm, a1 = 22 .86mm, b1 = 10 .16 mm).

By cascading two dielectric-lled waveguide T-junctions shown inFig. 2, we can easily obtain a single-aperture waveguide coupler loadedwith dielectric slabs, as shown in Fig. 5. It is clearly seen from Fig. 5that the dimension of a waveguide coupler can be greatly reduced byinserting dielectric slabs. The relative permittivity r 2 for the twoidentical dielectric slabs used in Fig. 5 is 3.5 and the thickness of thedielectric slabs is ( b t) = 5 .0 mm. The thickness of the apertureis th = 1 .0 mm. The aperture lies at the center of the broad walls of two identical dielectric-lled waveguides. 30 modes are employed in theanalysis for the rectangular aperture. Comparing the calculated resultsusing different r 2, it is easy to see that a tighter coupling through theaperture can be readily realized by employing a slab of larger r 2. Thescattering parameters obtained by the mode-matching method are ingood agreement with those calculated by Ansofts HFSS.

A similar dielectric-lled rectangular waveguide coupler usinglongitudinal rectangular aperture is presented in Fig. 6. The off-center aperture is employed to facilitate a tight coupling. It isnoticed that the same coupling can be realized with an aperture of reduced length by inserting the dielectric slabs in the main waveguides,


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Figure 4. Scattering parameters of an H-plane dielectric-slab-loadedrectangular waveguide T-junction ( a = 34 .85mm, b = 15 .799 mm,a1 = 22 .86mm, b1 = 10 .16mm, t = 7 .5mm, r 1 = 1, r 2 = 2 .5).

compared to the conventional air-lled waveguides. The dimensionsfor the aperture used in Fig. 6 are: a1 = 5 .0mm, b1 = 9 .4mm andthickness th = 1 .0 mm. The center-to-center offset for the aperture isX C = 4 .5 mm. The relative permittivity r 2 for two identical dielectricslabs used in Fig. 6 is 3.5 and the thickness of the dielectric slabsare (b t) = 1 .5 mm. 80 modes are employed in the analysis of the

dielectric-lled rectangular waveguides [19]. The computed scatteringparameters by the mode-matching method are in good agreement withthose calculated by Ansofts HFSS and it is seen from Fig. 6 that thenarrow-banded nature of the longitudinal slot coupling between twoparallel rectangular waveguides can be alleviated by inserting dielectricmaterials.

Waveguide power divider/combiner has wide applications in radarfeed systems for its merit of high-power tolerance [2022]. It is clearlyknown that a compact-structure can be realized by using dielectric

lling. This size reduction may be particularly important for somespecial applications such as the airborne radar system using slotantenna arrays. Fig. 7 gives an example of 3-port dielectric-lledwaveguide power divider. Compared to the conventional waveguide


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Figure 5. Scattering parameters of a dielectric-lled rectangular-aperture broad-wall coupler ( a = 16 .0mm, b = 7 .5mm, a1 = 9 .4mm,b1 = 5 .0mm, th = 1 .0mm, t = 2 .5mm, r 1 = 1, r 2 = 3 .5).

power dividers/combiners using the standard Magic-T, the dimensionsof the device have been greatly reduced and no tuning screws ordiaphragms are needed. Dissimilar waveguides are used here due tothe structural consideration and an almost even power distribution toport 1 and port 2 is achieved when the input power is from port 3. Asatisfactory return loss over the operating frequency band is obtainedat the input port ( |S 33 | in Fig. 7). The relative permittivities fortwo dielectric slabs used in Fig. 7 are both 3.5 and the thicknessesof two dielectric slabs used in Fig. 7 are ( b2 t1) = 1 .5mm and(b3 t2) = 5 .0 mm, respectively. The rectangular aperture used forcoupling lies at the center of the broad wall in the top T-junction andthe center-to-center offset for the aperture in the bottom T-junctionis X C = 4 .5 mm. The bottom T-junction is shortened at a distanceL = 10 .0 mm away from the center of the aperture. In the calculation,50 modes are considered in the two main waveguides, 20 modes areemployed for the rectangular aperture, and 180 modes are used forRegion 4 to obtain convergent results. The computed scatteringparameters by the mode-matching method are in good agreement withthose obtained by Ansofts HFSS.


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Figure 6. Scattering parameters of a dielectric-lled rectangular-aperture broad-wall coupler ( a = 16 .0mm, b = 5 .0mm, a1 = 5 .0mm,b1 = 9 .4mm, th = 1 .0mm, t = 3 .5mm, X C = 4 .5mm, r 1 = 1,r 2 = 3 .5).

Figure 7. Scattering parameters of a dielectric-lled waveguide powerdivider ( a2 = 16 .0mm, b2 = 5 .0mm, b3 = 7 .5mm, a1 = 5 .0mm,b1 = 9 .4mm, th = 1 .0mm, t1 = 3 .5mm, t2 = 2 .5mm, X C = 4 .5mm,L = 10 .0mm, r 1 = 3 .5, r 2 = 3 .5).


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By narrowing down the aperture, the power enters into the leftarm (port 2) and the right arm (port 1) will become closer.

4. CONCLUSION

This paper has described a rigorous full-wave mode-matching analysisof a rectangular waveguide T-junction partially loaded with an H-planedielectric slab. The longitudinal section electric (LSE) and longitudinalsection magnetic (LSM) modes in a dielectric-lled rectangular wave-guide are rstly derived and then used in the mode-matching analysis.Numerical results for the scattering parameters of the T-junction arepresented and veried by comparing with the available measured data

or those obtained by Ansofts HFSS. Examples for dielectric-lledwaveguide couplers and power dividers used in waveguide slot antennaarrays are presented. Excellent agreement has been observed for allthe cases considered.

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