analysis of waveguide junction discontinuities … of waveguide junction discontinuities using...

NASA Contractor Report 201710

Analysis of Waveguide JunctionDiscontinuities Using Finite ElementMethod

Manohar D. Deshpande

ViGYAN, Inc., Hampton, Virginia

Contract NAS1-19341

july 1997

National Aeronautics and

Space AdministrationLangley Research Center

Hampton, Virginia 23681-0001

https://ntrs.nasa.gov/search.jsp?R=19970023023 2018-06-11T21:00:16+00:00Z

Contents

List of Figures

List of SymbolsAbstract

1. Introduction

2. Theory3. Numerical Results

4. Conclusion

References

2

3

5

5

9

13

20

20

Figure l(a)

Figure l(b)

Figure1(c)

Figure2

Figure3

Figure4

Figure5(a)

Figure5(b)

Figure6

Figure7

Figure8

Figure9(a)

Figure9(b)

List of Figures

Rectangular waveguide junction with misalignment in the x-direction

(waveguide flanges at the junction are not shown)

Rectangular waveguide junction with misalignment in the y-direction

Rectangular waveguide junction with an air gap

Geometry of rectangular waveguide junction discontinuity and an air gap

Top view of H-plane discontinuity in rectangular waveguide (Waveguide I,

Wxl = 2.286cm, Wy =1.02cm, Waveguide II Wx2 -- 1.5 cm, Wy = 1.02 cm)

Comparison of magnitude of reflection and transmission coefficients calculated

using Finite Element Method and Mode Matching Technique [19]

Geometry of E-plane ridge discontinuity in rectangular waveguide.

Cross sectional view of E-plane ridge discontinuity in a rectangular waveguide

( W -- 0.1016 cm, h = 0.7619 cm, L = 0.508 cm, a = 1.905 cm, b = 0.9524 cm)

Transmission coefficient of E-plane ridge discontinuity in a rectangular

waveguide shown in figure 5(a)

Concentric step in a rectangular waveguide; input waveguide dimensions

( a 1 - 1.58 cm, b 1 = 0.79 cm), output waveguide dimension ( a2 = 2.29 cm,

b 2 = 1.02 cm)

SI1 [ and $21 [ parameters for the concentric step discontinuity in a rectangular

waveguide shown in figure 7

Geometry of concentric rectangular waveguide with an inductive junction.

Input reflection coefficient of concentric inductive rectangular waveguide junc-

2

tionshownin figure9(a).

Figure 10(a)Geometryof concentricrectangularwaveguidewith acapacitivejunction.

Figure 10(b)Inputreflectioncoefficientof concentriccapacitiverectangularwaveguidejunc-

tionshownin figure10(a).

Figure 1l(a) Geometryof offsetrectangularwaveguideinductivejunction.

Figure 1l(b) Inputreflectioncoefficientof offsetinductiverectangularwaveguidejunction

shownin figure 11(a).

Figure 12(a)Geometryof offsetrectangularwaveguidecapacitivejunction.

Figure 12(b)Inputreflectioncoefficientof offsetcapacitiverectangularwaveguidejunction

shownin figure 12(a)

Figure 13(a)Geometryof offsetrectangularwaveguidejunctionwith x- andy-offset.

Figure 13(b)Inputreflectioncoeffcientof offsetrectangularwaveguidejunctionshownin fig-

ure 13(a).

Figure 14 Inputreflectioncoefficientof inductivejunction in aS-bandrectangular

waveguide.

Figure 15 Inputreflectioncoefficientof capacitivejunction in aS-bandrectangular

waveguide.

Figure 16 Inputreflectioncoefficientof x- andy-offsetjunction in S-bandrectangular

waveguide.

Figure 17 Inputreflectioncoefficientof S-bandrectangularwaveguidejunctionwith anair

gap.

a o

ap

bm

ro]

b o

bP

EloverS I

E over S 2

E (x,y,z)

-)11

E (x,y,z)

--)111

E (x,y,z)

elO

elp

e2o

e2p

Hx, y,z)

--)!1

H (x,y,z)

-) 111

H x,y,z )

_.....)

hlo

÷

hip

h2p

]

ko

[Sel

List of Symbolsdominant mode reflection coefficient

thcomplex waveguide modal amplitude of p mode in region I

complex amplitude associated with tetrahedral basis function

column matrix

dominant mode amplitude in region HI

thcomplex waveguide modal amplitude of p mode in region HI

tangential electric field vector over the plane PI

tangential electric field vector over the plane P2

transverse electric field vector in region I

electric field vector in region II

transverse electric field vector in region 11I

dominant vector modal function for region I

rectangular waveguide vector modal function for pth mode for region I

dominant vector modal function for region III

rectangular waveguide vector modal function for pth mode for region III

transverse magnetic field vector in region I

magnetic field vector in region II

transverse magnetic field vector in region UI

dominant vector modal function for region I

threctangular waveguide vector modal function for p mode for region I

threctangular waveguide vector modal function for p mode for region I

free-space wave number

global finite element matrix

element matrix for single tetrahedral element

S 1

$2

$11

$21

..)T(x,y,z)

TElo

x, y, z

YIo

Ylp

_0

_to

Er

YlO

'Ylp

'2ptO

surface area over plane P1

surface area over plane P2

return loss in dominant mode at plane P1

transmission coefficient in dB at plane P2

vector testing function

dominant mode

excitation column vector

x-dimensions of rectangular waveguide

y-dimension of rectangular waveguide

vector basis function associated with tetrahedron

Cartesian Coordinate system

dominant modal admittance for region I

thmodal admittance of p mode for region I

thmodal admittance of p mode for region III

permittivity of free-space

permeability of free-space

relative permittivity of medium in region II

relative permeability of medium in region II

dominant mode propagation constant for region I

thpropagation constant for p mode for region I

thpropagation constant for p mode for region 11I

angular frequency

Abstract

A Finite Element Method (FEM) is presented to determine reflection and

transmission coefficients of rectangular waveguide junction discontinuities. An H-plane

discontinuity, an E-plane ridge discontinuity and a step discontinuity in a concentric

rectangular waveguide junction are analyzed using the FEM procedure. Also, reflection and

transmission coefficients due to presence of a gap between two sections of a rectangular

waveguide are determined using the FEM. The numerical results obtained by the present

method are in excellent agreement with the earlier published results. The numerical results

obtained by the FEM are compared with the numerical results obtained using Mode Matching

Method (MMM) and also with the measured data.

1.0 Introduction

A slotted rectangular waveguide array antenna is being proposed to be used in a

microwave scattrometer for soil moisture measurements. During the launch phase of such a

scatterometer, the slotted waveguide should be folded. After full deployment of the

scatterometer, the waveguide must be unfolded to its full length. Due to the mechanical

imperfection of the joints and hinges, there will be misalignments and gaps. These gaps and

misalignments will affect the performance of the slot array. It is the purpose of this report to

analyze the effects of the gaps and misalignments on the transmission line properties of

rectangular waveguide sections. However, the effect of these gaps and misalignments on the

performance of the slot array will not be considered here.

6

The types of misalignments and gaps that may occur after unfolding the various sections

of rectangular waveguide are shown in figure 1

Output Wave

I

X I/

Input Wa_

Figure 1(a) Rectangular waveguide with misalignment in x-direction.

(Waveguide flanges are not shown)

X

Input Wave

Figurel(b) Rectangular waveguide junction with misalignment iny-dimension.

7

Input Wave Output Wave

Gap Between Two Sections

Figure l(c) Rectangular waveguide junction with a gap.

For assessment of the effects of such discontinuities on the transmission and reflection

properties of rectangular waveguide, many analytical techniques can be used. The modelling of

waveguide junction discontinuities has been a subject that has been studied considerably in the

past. An equivalent circuit approach based on an electrostatic approximation and variational

principle [1-4] has been used to analyze these discontinuities. However, these approximate

techniques may not be accurate for electrically large discontinuities. Furthermore, only single

mode interactions are accounted in these simple representations. Higher order mode interactions

are taken into account by using the mode matching technique[5,6]. In the MMM, the fields in

each region across the junction are expressed in terms of infinite number of waveguide modal

functions. Application of continuity of tangential components of electric and magnetic fields

across the junction in conjunction with the Method of Moments (MoM) yields a matrix equation

with tangential fields over the junction as an unknown variable. From the solution of the matrix

8

equation, the reflection and transmission properties of the waveguide junction are determined.

The resulting code, although accurate, can be computationally inefficient. There are different

versions of the MMM reported in the literature such as generalized scattering matrix (GSM) [7]

techniques, multimodal network representation methods using admittance, or impedance matrix

representation [8-10]. The MMM is mostly applied to analyze zero-thickness discontinuities at

the junctions. For non-zero thickness discontinuties in the direction of propagation, the mode

matching techniques become quite involved. Furthermore, it is cumbersome to apply the MMM

when the waveguide junctions are loaded with three-dimensional arbitrarily-shaped

discontinuities. In such cases, a numerical technique such as the FEM [11-13] is more versatile

and easily adaptable to changes in the structures of discontinuities. In this report, a numerical

technique using the FEM is developed to analyze step discontinuities as well as three-

dimensional arbitrarily-shaped discontinuities present at the rectangular waveguide junctions.

In [11], reflection and transmission characteristics of metal wedges in a rectangular

waveguide were studied using H-field FEM formulation. It has been shown in [11] that the

vector edge based formulation eliminates the spurious solutions. In this report, the FEM using

the E-field formulation is developed to analyze rectangular waveguide junction discontinuities.

Because of the metal boundaries of rectangular waveguide, the E-formulation results in fewer

unknowns compared with the H-field formulation of [11 ].

The remainder of this report is organized as follows. The FEM formulation of the

waveguide junction problem using the weak form of the Helmoltz wave equation is developed in

section 2. Also in section 2, MMM formulation is presented to determine reflection and

transmission coefficients of transverse discontinuities in a rectangular waveguide. Numerical

results on the transmission and reflection coefficients for E-plane and H-plane step

discontinuities are given in section 3 along with earlier published results for comparison. Also in

section 3, the experimental results on some of the waveguide junction discontinuities measured in

the Material Measurement Laboratory of the Electromagnetics Research Branch are compared

with the results obatined using the present approach. The report concludes in section 4 with

remarks on advantages and limitations of the present technique.

2.0 Theory2.1 Finite Element Formulation:

In this section, the FEM will be used to determine the reflection and transmission

coefficients of the rectangular waveguide junction discontinuity shown in Figure 2.

Radiating _ Circilar Flang

perture

S 3

I

TE1o Mode I

I I atched LoadI I

Region _ Region H I Region lII/\

Z PlaneP_

---- Z=Zl

Figure 2 Geometry of rectangular waveguide junction with step discontinuityand gap.

It is assumed that the waveguide is excited by a dominant TEl0 mode from the left.

To analyze the junction discontinuity, the junction is assumed to be enclosed by two planes: P1

and P2- The planes P1 and P2 divide the waveguide region into three regions as shown in Figure

2. The air gap at the junction between two waveguide sections causes leakage of electromag-

10

netic energy which is accounted in the present formulation. Using the waveguide vector modal

functions, the transverse electromagnetic field in region I is expressed as [16]

-d _ -J Tl0 zE (x,y,z) = el0(x,y) e

+ Z apelp (x,y) / T,p z (1)

p=O

_1 _ -J T10zH (x,y,z) = hlO(X,y ) Y10e

") / Tip zZ aphlp (x,y) Ylp

p=O

(2)

In deriving equations (1) and (2), it is assumed that only the dominant mode is incident on the

interface P1 and the ap is the amplitude of reflected modes at the z=O plane. Ylp and T1p

appearing in equations (1) and (2) are respectively the characteristic admittance and propagation

thconstant for p mode and are defined in [16]. The unknown complex modal amplitude a may

p

be obtained in terms of the transverse electric field over the plane P 1 as follows

l+ao = ff EoverPl._lodS (3)S1

f f -E over P_ap = • _elpdS (4)$1

where S 1 is the surface area over the plane P1 •

Likewise, the transverse components of electric and magnetic fields in the region III

can be written as [16]

11

oO

+,'11 , J V_pzE (x,y,z) = Z be2p (x.y)e (5)

p=O

H (x,y,z) = E b_2p (x,y) Y2p e (6)

p=0

where b is the amplitude of transmitted mode at the z-z 2 plane, Y2p and T2p appearing in

equations (5) and (6) are respectively the characteristic admittance and propagation constant for

thp mode for output waveguide and are defined in [16]. The unknown complex modal amplitude

bp may be obtained in terms of the transverse electric field over the plane P 2 as follows

_ _ -E over P2 "bp = • e2pds

$2

where s 2 is the surface area over the plane P2-

The electromagnetic field inside region II is obtained using the FEM formulation [17].

The vector wave equation for the _11 field is given by

->I1_ { 2 "_+11(7)

Using the weak form of the vector wave equation and some mathematical mani

equation (5) may be written as

_er( + ( 1 -_1I_ { 2 "_-_II///VxTo/--VxE I-lk.e |E ,T dv) = 2jtO_toYoff_'i_o(x,y)ds

St

_ulation [17], the

p=O

Yip T°elp(X'y)ds •elp(X'y)ds

/\ S 1

12

where S 2

(/ //ss,"oe, 1)-JC0l't0 £ Y2p .ff_'e_2p (x'y) ds "_2p(X'y)ds

p = 0 $2 )\ $2

...)

-jcO_toI I T × h . Hapds

$3

is the cross sectional area at plane P2, $3 is the surface area of radiating aperture,

(8)

-->

and Hap is the magnetic field in the radiating aperture S 3 . In order to solve the equation (9),

the volume enclosed by region I/is discretized by using first-order tetrahedral elements. The

electric field in a single tetrahedron is represented as

6+!I __

E = £ bm. Wm (9)

m=l

where bm are the six complex coefficients of electric field associated with the six edges of the tet-

--)

rahedron, and wm (x, y, z) is the vector basis function associated with the m th edge of the tetra-

-->

hedron. A detailed derivation for the expressions for w m (x, y, Z) is given in reference [17].

Substituting equation (10) into equation (9), integration over the volume of one tetrahedron

results in the element matrix equation

(10)

where the entries in the element matrices are given by

:- (Iss( :-->->))Sel (m, n) 1 V× • VxW n - koe.rW n • W m dv

_tr ' V

O<D

((ss )(ss-+' ))+ (jt'0l't0) Z Ylp Wm • elp (x, y) ds , W n • elp (x, y) ds

p = 0 S l S 1

13

OO

(Ors+.e:,'){rS+ ))+ (./c°i'to) _ Y2p Wm° (x, y) ds . W_ ° _2p(x, y) dsp = 0 S2 $2

:sL(-"<"r-;"'/}(-jo)l.l.o)k0 (WmX_) " _',,xhe i>,._>,.,ia,,a,S3\

4-_J'_( -Jk°(l>r->"1),rr 1}+ jo_laO["f__ V'(WmXh )"-+ V, ( w,, x h)->e I__>, _' ds$3

(11)

v(m)= 2 (jtOlXo) YloSSWm,_lO(X,y)ds

SI

These element matrices can be assembled over all the tetrahedral elements in the region H to

obtain a global matrix equation

(12)

IS_ Ib] : [vl (13)

The solution vector [b] of the matrix equation (14) is then used in equation (3) to determine the

reflection coefficient at the reference plane P1 as

_ [. E over P,a o = - 1 + • _lods (14)

S,

The transmission coefficient at the plane P2 is obtain as [18]

bo = 20 • _2ods (15)

The return loss and power transmitted through the rectangular junction are then calculated using

Sll = 20.log (la0l) (16)

$2, = 20.log (Ibol) (17)

The power transmitted through the junction can also be calculated using

S21 = 20.1ogt, 1-- 1_'- ap (18)

where the summation should be done over the propagating modes only.

2.2 Mode Matching Method:

14

In this section the MMM for the rectangular waveguide junction discontinuities is

presented. Since the modelling of air gap between the two waveguide sections using the MMM

is quite involved, the junction discontinuities of E- and H-plane steps types (as shown in fig. 3)

are only considered.

TElo Mode

__ Region I Region II

TEt0_Mode

To_l _¢/latched Load

Figure 3 Geometry of waveguide junction discontinuity without an air gap.

The transverse components of electric and magnetic fields in the region I and II can be written as

+IE (x,y,z) = _10 (x,y) e -j

Tl0 Z

oo

+ Z apelp (x,y)J Tip z (19)

p=O

-d _ -J T10zH (x,y,z) = hlo(x,y) Y10 e

_a

Z ap]llP (x,y) Ylp J "YlPZ

p=O

(20)

In deriving equations (1) and (2) it is assumed that only the dominant mode is incident from the

left and the at, is the amplitude of reflected modes at the z=O plane. Ylp and Ylp appearing in

equations (1) and (2) are respectively the characteristic admittance and propagation constant for

15

thp mode and are defined in [16]. The unknown complex modal amplitude a may be obtained

p

in terms of the transverse electric field over the z = 0 plane as follows

( 0. /1 + a 0 = _lodS (21)S

ap = =0S

(22)

where s is the surface area over the z = 0 plane.

Likewise, the transverse components of electric and magnetic fields in the region II

can be written as [16]

Oo

.tt , j V2p zE (x,y,z)= E bpe2p(X'y)e (23)

p=O

j zH°ll (x,y,z) = _ b]_2p (x,y) YEp e (24)

p=0

where b is the amplitude of transmitted mode at the z--0 plane, Y2p and ]t2p appearing in

equations (24) and (25) are respectively the characteristic admittance and propagation constant

thfor p mode for output waveguide and are defined in [16]. The unknown complex modal ampli-

tude bp may be obtained in terms of the transverse electric field over the z = 0 plane as follows

b = o" e2pds (25)S

where s is the surface area over the z = 0 plane.

In order to determine the unknown coefficients ap and bq we assume the tangential electric field

16

over the plane z = 0 as

where cr

oo

E z = y___, ' (26)= 0 Cre3r

r=O

is unknown complex coefficient, and e_3r is the vector mode function for a rectangular

waveguide having a cross section same as the cross section of the aperture. Substituting (27) in

(22), (23), and (26), the unknown coefficients ap and bq are obtained as

oo

l+ao= Z Crf _ e_3r• _elodS (27)r = 0 Aperture

ap : Z Crf S +3re _elpdS (28)

r = 0 Aperture

bq = Z Crf _ _3r • _e2qds (29)

r = 0 Aperture

Substituting (28)-(30) into (20),(21) and (24),(25), the transverse fields in the regions I and II are

obtained as

-"# '__, {'___ J" J' t _>E (x,y,z) = -2JD10 (x, y) sin (_/10 z) + Cr e3r • Dlpds elp (x,y) d T'" z (30)

p = 0 r = 0 Aperture

-->I _' _lpd$ + d Tip zH (x,y,z) = 2h10 (x,y) YI0 COS('Y10 z) -pZ Cr e3r • hip (x,y) Ylp

= 0 = Aperture

(31)

for the region I and

17

-)11E (x,y,z) { _ Cr_ _ _3r°_e2qds}_2q (X,y) d_lqz

q = 0 r = 0 Aperture

(32)

H (x,y,z) Cr _3r • _2q dS ->= h2q (x,y)

q 0 = Aperture

(33)

Equating the tangential magnetic fields across the aperture yields an integral equation with cr

unknowns as

+ =Z°2hlO(x,Y) YIO = c r _3rO_lpd$ hlp(X,y)

r p = 0 Aperture

Ylp

+ Z Cr e3r • (x,y)h2q Y2q

r= 0 _q = O Aperture

(34)

Taking cross product of (35) with _ and selecting _e3r, as a testing function, the integral equation

in (35) yields the following set of simultaneous equations

2_ _ _3r,• _elodSYlo : Z Cr Z _ _ _3r• _elpd$_ _ _e3F• _elpdS Ylp

Aperture r = 0 p = 0 Aperture Aperture

Z Cr Z f f _3r" _e2qds_ _ _e3r'' _e2q d• Y2q

r = 0 q = 0 Aperture Aperture

(35)

where r' = 1, 2, 3 .... By terminating the infinite summations with respect to p to Np, q to Nq,

and r to N r equation (36) can be solved for c r . The reflection and transmission coefficients are

then obtained as

Sll -1+ E Cr _ _ _ o_ --" 3r e lOaS

r = 0 Aperture

(36)

18

$21

N r

r = 0 Aperture

3.0 Numerical Results

(37)

To validate the present technique, we first present numerical results on the reflection

and transmission coefficients for H-plane discontinuity in a x-band rectangular waveguide as

shown in Figure 3. This geometry has been solved by earlier researchers using the MMM and

CAD-oriented equivalent circuit modelling [19].

Plane P1Plane P2

Waveguide I

v

TEl0 Mode Incident

X

II

l I

I

I

E II

Waveguide II

TE 10 Mode

Figure 3 Top view of an H-plane discontinuity in a rectangular waveguide.

Waveguide I ( Wxl = 2.286 cm, Wy = 1.02cm), Waveguide II ( Wx2 = 1.5 cm,Wy = 1.02 cm)

For the present analysis, the plane P1 was selected at z = 0 and the plane P2 was selected at z = 1

cm. The junction was at z = 0.5 cm. The reflection and transmission coefficients calculated using

the present approach are shown in Figure 4 along with the results obatined by the MMM [19].

There is an excellent agreement between the results of two methods. The transmission curve

shown as a dotted line is calculated by S12 = 20log 10_(1- Reflected Power) Seven

hundred twenty eight tetrahedral were used to discretize the FEM region.

19

For further validation of the code, an E-plane ridge waveguide discontinuity in a

rectangular waveguide, as shown in Figure 5, is considered. The transmission coefficient in the

presence of the metallic ridge is calculated using the present code as a function of frequency and

is presented in Figure 6 along with the earlier published data. There is good agreement between

the results obtained by the present method and earlier published data. For the numerical

calculations, the planes P1 and P2 were assumed to be 0.2 cm away from the rectangular ridge.

The number of tetrahedra used to discretize the FEM region was 2718. The number of higher

order modes considered in the input as well as the output waveguides were 40.

C'

-5

-10

--,-15r._

r._

-20

-25

-308

S12c,,'cul us,n,,!.,n. lT, .......... SI1 calculated using eqn. (18) _

Mode Matching [19]S12 calculated usin_ eqn. (19) W _ J

, , , i I i i i i I i Ii i i I i i i i I i i i i J i i i i I i i

9 10 11 12 13 14Frecluency in GHz

Figure 4 Comparison of magmtude of reflection and transmission coefficients calcula!Finite Element Method and Mode Matching Techniques.

20

Metallic Ridge

X //

I

I

/

i

/

/

Z

Plane

P2

Input PlaneP1

Figure 5(a) Geometry of E-plane ridge discontinuity in a rectangular waveguide.

a/2

v

! hW

a

Ir

b

Figure 5 (b) Cross sectional view of E-plane ridge discontinuity in a rectangular waveguide(W= 0.1016cm, h = 0.7619 cm, L = 0.508cm, a = 1.905 cm, b = 0.9524 cm)

21

O _

D •

.g,._

_-30{/3

i20

Present Method _ ', ,,0H-formulation [11] \ I '

• Measured Ref i20] \ "-", ,_1_

', //

"50 = , , , I = i , , I , , = i I = = = , T = = I J I

10 11 12 13 14 15Frequency in GHz

Figure 6 Transmission coefficient of E-plane ridge discontinuity in a rectangularwaveguide shown in figure 5.

The third example considered for the validation of the present code is shown in Figure 7.

The reference planes P1 and P2 were assumed to be 0.5 cm away from the junction. The FEM

region was discretized into 2700 tetrahedra and the number of higher order modes considered in

each waveguide was 40. The reflection and transmission coefficients calculated using the present

code are presented in Figure 8 along with the earlier published data. There is good agreement

between the earlier published data and the numerical results obtained using the present code.

22

OutputReferencePlaneP2

TE 10Mode Out

b2

TE 10ModeIncident

Input Reference PlaneP1

Figure 7 Concentric step discontinuity in a rectangular waveguide; input waveguidedimension (a 1 = 1.58 cm, b 1 = 0.79 cm), output waveguide dimensions

( a2 = 2.29 cm, b 2 = 1.02 cm)

23

1.0

0.8

0.6

0.0

O_- t- Q-O t-- tP

• ISll H-Formulation [11]

.... A- .... ISlll Ref. [ 21]

--O-- IS211 H-Formulation [11]

A,/i Is,,I PresentMethod

Figure 8 $11 and

i , I i J i i I i , , i i , , , i 1 , i , i I

0.6 0.8 al 1.0 1.2 1.4

IS2]1 parameters for the concentric step discontinuity in

rectangular waveguide shown in figure 7

The discrepency in the results at higher frequencies may be due to the same discretization used for

lower and higher frequency range. This causes the results to be less accurate at higher

frequencies.

Other discontinuities considered in this report are shown in Figures 9 - 13. The input

reflection coefficients calculated using the FEM and MMM as described in the previous section

24

are presented in Figures 9 -13. Number of higher order modes considered for MMM were 100

modes in input and output waveguides. The number of modes considered (in all of the problems

shown here) on the aperture were 20. For FEM the number of elements used were approximately

1200. The numerical results obtained using the FEM and MMM agree well with each other.

b = 1.0 cm

a' = 1.0 cm

I b' = 1.0 cm

Ia=2.286

Figure 9(a) Geometry of concentric rectangular waveguide with an inductive junction.

25

1.0

0.50

=1,-4

8

_i 0.0O

_ -0.5

-1.0

Mode Matching Method

..... FEM

i _ i i I I i I I I i i = J [ i i h * I

e 9 Frequenc_O(GHz) 11 12Figure 9(b) Input reflection coefficient of concentric inductive rectangular waveguide

junction shown in figure 9(a).

b =l.0c_

a=2.286

a' = 2.286 cm

' = 0.6 cm

Figure 10(a) Geometry of concentric rectangular waveguide with a capacitive junction.

26

cD°,.._

c.)

8L)

O

cD

o_

0.5

O.O

-0.5


FEM

8 9 10 11 12

Frequency (GHz)

Figure 10(b) Input reflection coefficient of concentric capacitive rectangular waveguidejunction shown in figure 10(a).

b = 1.0c_

a=2.286

a' = 2.286 cm

I b' = 1.0 cm

Figure 11 (a) Geometry of offset rectangular waveguide inductive junction.

27

0.5

Oo 0.0

8

0

¢:: -0.5


FEM

Imaginary Part

Real Part

, , i , I , i i , I i , I i l , i i , I

8 9 10 11 12

Frequency (GHz)

Figure 11(b) Input reflection coefficient of offset inductive rectangular waveguide

junction shown in figure ll(a).

a' = 2.286 cm

b' = 1.0 cm

b =1.0 cm I

0.4 cm

a=2.286

Figure 12(a) Geometry of offset rectangular waveguide capacitive junction.

28

1.0

0.5

._,._

0

0 0.0

-0.5

-- Mode Matching Method

FEM (No. of Elements 1206)


Real Part

...... -_-.- - _ -_ _ _ ___ __________

-1.0 _ _ _ _ I _ _ L _ I L _ _ _ L , , _ L I

8 9 10 11 12

_.. Frequency (GHz)..Figure 12(b) Input reflection coemclent or orIset capacmve rectangular waveguide

junction shown in figure 12(a).

0.5 cm

a = 2.286 cm

b = 1.0 cm I

Ib = 1.0 cm

a =2.286 cJ

cm

Figure 13(a) Geometry of offset rectangular waveguide junction with x- and y-offset.

29

1.0

0.5

• i,,,_

8

O

t=

0_

0.0

-0.5


FEM ( No. of Elements 1258)


_Real pi =-_

-1.0 , _ L L l , _ J _ I , , , _ I _ , _ , I8 9 10 11 12

Frequency (GHz)

Figure 13(b) Input reflection coefficient of off-set rectangular waveguidejunction shown in figure 13(a).

The numerical results shown so far were related to the misalignments in the X-band

waveguide. For a S-band waveguide, some typical misalignments analyzed using the FEM and

MMM and compared with the measured data taken in the measurement laboratory of ERB are

shown in Figures 14-16. The measurements were done using the HP 8510 Network Analyzer.

For the FEM analysis the reference planes P1 and P2 were selected at 0.5 cm away from the junc-

tion. The number of elements used in all cases were around 1258. For the MMM the number of

modes consided in both waveguides were 100, and the number of modes used to represent aper-

ture field were 20.

3O

-5

-10

0


FEM

Measured

-15

_-20 , ....0

e oOr..)

° 25• i,_ o

-30

-35

7.21 cm

-" 7.21 cm "-

3.4 cm

et 0

3.0 3.5Frequency (GHz)

Figure 14 Input reflection coefficient of inductive junction in a S-band rectangularwaveguide.

I

4.0

31

B

-5

-10

-15

-20

r_

. .25

-30

-35

-402.5

0


FEM

Measured

t

0 0 0.635 cm

_ iii!iii!!iiil¸ !i̧ !!i!̧!_!_ !il i il ?i ! _

3.4 cm

7.21 cm

J i I I i I i I I I i I i I

3.0 3.5 4.0

32

_

-10

Oc.)_-30O

°l,._

-40

-50


FEM 7.21 cmram..._

3.4c 5 34cm

. .

7.21 cm

\\\\0_0 /

\\ _ I/

XX I I

I J

I I

I I

I I

w I i t I i I I_/I f I i I I

3.0 Frequency (GHz) 3.5 4.0

Figure 16 Input reflection coefficient of x- and y-offset junction in a S-band rectangular

waveguide.

33

For application of the FEM to analyze gap between two rectangular waveguide sec-

tions we consider S-band rectangular waveguide junction as shown in Figure 17.. The reflection

coefficient obtained using the present FEM procedure is presented in figure 17 along with the

measured results.

0

-5

_-10

,1.,_

8r.)= -15O

¢dlo

_.-2o

-25

FEM ( No Radiation)

.......... FEM ( With 2 M) ," ",,/ X

FEM ( With M) / _ Curve A

• Measured Data _ \_(

/;. , \ ,y or,,o -"77/-". "\f"

/ /,' ",, k ',

,/,, \,ii \"

/// L'. \ \

/ N \

// ' // Curve-_B _'\'\.\. \' \\

/*/ /,/{_/'/" / / " \'x*'N'N. N \_N'xNN N

I I _ S-Band RectangularWaveguide (7.21 cm x 3.4 cm)

-30 , , i , I = , , , I , i , , I , , i , I , , i , I , , , i I , i

2.6 2.8 3 0 3 2 3 4 3.6 3.8• _ Frequeiicy (GHz) "

Figure 17 Input reflection coetticient of S-band rectangular waveguide junctionwith an air gap.

34

The curve A in Figure 17 shows the input reflection coefficient obtained assuming there

is no radiation through the gap aperture. In this simulation the gap opening was not terminated

into metal boundary. The curve B in Figure 17 was obtained by assuming the radiating gap aper-

ture is backed by an infinite ground plane. The presence of infinite ground plane is taken into

__>account by considering magnetic current source M of amplitude twice that of magnetic current

present in the aperture. The curve C in figure 17 shows the input reflection coefficient obtained

without an assumption of infinite ground plane. In that case there was no factor two involved in

the amplitude of magnetic current source. From the comparison of calculated and measured

results in figure 17 it may be concluded that not assuming the presence of infinite ground plane is

more appropriate.

4.0 Conclusion

A FEM procedure has been presented to determine complex reflection and

transmission coefficients of rectangular waveguide junction discontinuities. The discontinuities

that can be analyzed using the present procedure can be E-plane, H-plane, or both. The present

procedure can also handle the air gap that may be present between the junctions of two

rectangular waveguides. The numerical results obtained from the present method are compared

with earlier published results. An excellent agreement between the numerical results obtained by

the present code and the earlier published data validates the present method and the code

developed.

References

[1] L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters Impedance Matching Net-

work, and Coupling Structures, New York, McGraw-Hill, 1964.

35

[10]

[2] N. Marcuvitz, Waveguide Handbook, Radiation Laboratory Series, Vol. 10, New York,

McGraw-Hill, 1951.

[3] H.A. Bethe, "Theory of diffraction by small holes," Phys. Rev., Vol. 66. pp. 163-182,

1944.

[4] S.B. Cohn, "Electric polarizability of apertures of arbitrary shape, "Proc. IRE, Vol. 40, pp.

1069-1071, 1952

[5] R. Mittra and W. W. Lee, Analytical Techniques in the Theory of Guided Waves, New York,

Macmillan, 1971.

[6] T. Itoh, Ed., "Numerical Techniques for microwave and millimeter wave passive structures, "' New

York, Wiley, 1989

[7] R.R. Mansour and J. Dude, "Analysis of microstrip T-junction and its applications to the

design of transfer switches," 1992 IEEE-MTT-S Dig., pp. 889-892.

[8] E Alessandri, et al, "Admittance matrix formulation of waveguide discontinuity problems:

computer-aided design of branch guide directional couplers, " IEEE Trans. Microwave

Theory and Techniques, Vol. 36, No. 2, pp 394-403, Feb. 1988.

[9] M. Guglielmi and C. Newport, "Rigorous, multimode equivalent network represntation of

inductive discontinuities," IEEE Trans. Microwave Theory and Techniques, Vol. 38, No.

11, pp 1651-1659, Nov. 1990.

A. A. Melcon, et al, "New simple procedure for the computation of the multimode

admittance or impedance matrix of planar waveguide junctions," IEEE Trans. Microwave

Theory and Techniques, Vol. 44, No. 3, pp 413-416, March 1996

36

[11] K. Ise, et. al, "Three-dimensional finite element method with edge elements for electro-

magnetic waveguide discontinuities, "IEEE Trans. Microwave Theory and Techniques,

Vol. 39, No. 8, pp. 1289-1296, August 1991.

[12] J.F. Lee, et. al, "Full wave analysis of dielectric waveguides using tangential vector finite

elements, "IEEE Trans. Microwave Theory and Techniques, Vol. 39, No. 8, pp. 1262-1271,

August 1991.

[13] L. Zhou and L. E. Davis, "Finite element method with edge elements for waveguides

loaded with ferrite magnetized in arbitrary direction, "IEEE Trans. Microwave Theory and

Techniques, Vol. 44, No. 6, pp. 809-815, June 1996.

[14] L.T. Tang, M. S. Nakhla and R. Griffith, "Analysis of lossy multiconductor transmission

lines using the asymptotic waveform evaluation technique, "IEEE Trans. Microwave

Theory and Techniques, Vol. 39, pp. 2107-2116, December 1991.

[15] L.T. Pillage and R. A. Rohrer, "Asymptotic waveform evaluation for timing analysis,"

IEEE Computer Aided Design, pp. 352-366, 1990.

[16] R. E Harrington, Time-harmonic electromaneticfields, McGraw-Hill Book Company,

New York, 1961.

[17] J. Jin, The finite element method in electromagnetics, John Wiley & Sons, Inc., New

York, 1993.

[18] R.E. Collins, Foundation for microwave engineering, Chapter 4, McGraw-Hill Book

Company, New York, 1966.

[19] A. Weisshaar, M. Mongiardo, and V. K. Tripathi, "CAD-oriented equivalent circuit

modelling of step discontinities in rectangular waveguides, "IEEE Microwave and Guided

Wave Letters, Vol. 6, No. 4, pp. 171-173, April 1996.

37

[20]

[21]

R. Mansour, et al., "Simplified description of the field distribution in finlines and ridge

waveguides and its application to the analysis of E-plane discontinuities," IEEE Trans.

Microwave Theory and Techniques, Vol. MTT-36, pp. 1825-1832, Dec. 1988.

H. Patzelt and E Arndt, "Double-plane steps in rectangular waveguides and their

application for transformers, irises and filters, "IEEE Trans. Microwave Theory and

Techniques, Vol. MTr-30, pp. 771-776, May 1982.

38

REPORT DOCUMENTATION PAGE Form ApprovedOMBNo. 0704-0188

P_i¢ reposing burdln for thi= cobctlon of INommlkm Ii ulirmaKI to mintage 1 hour per ms_oonse, In_ the tJrm fm rev_ Imtruczlonk uarcNng eKillng dala =oumm,gallw_g ancl rnm/ntaJning the daU needed, amd oomoTmlng =ml revlm_ng the ooMctbn of intormmtlon. Stud ¢omrnem regaling thi= burden mVrr,_e or any o_'w roped of thbcoil=orion CdI_ k'=duding=ugge=_m for mduc_ rid=buRkm, to Wihk_ I._uaJle_ ,.Rw_=o Dk,eom_te f_ _ _ _ _ 1215 _ _Highway. _ 1204, Atl_gton. VA 222(:Q-4302. _ to the Olfice of Manai_ and Budget. Papefwoek Reduction Project (0704-0188), Wlmhlngton. DC 20603.

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Analysisof Waveguide JunctionDiscontinuitiesUsingFinite Element C NAS1-19341Method

6. AUTHOR(S)

Manohar D. Deshpande

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ViGYAN, Inc.

Hampton, VA 23666-0001

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National Aeronautics and Space Administration

Langley Research CenterHampton, VA 23681-0001

522-33-11-02

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!13. ABSTRACT (Maximum 200 words)

12b. DISTRIBUTION CODE

A Finite Element Method (FEM) is presentedto determine reflection and transmissioncoefficientsof rectangularwaveguide junctiondiscontinuities. An H-plane discontinuity,an E-plane ridgediscontinuity,and a stepdiscontinuityin a concentric rectangularwaveguide junctionare analyzed usingthe FEM procedure. Also,reflectionand transmissioncoefficients due to presence of a gap between two sections of a rectangularwaveguide are determined using the FEM. The numerical results obtained by the present method are in

excellent agreement with the earlier published results. The numerical resultsobtained by the FEM arecomparedwith the numerical results obtained usingthe Mode Matching Method (MMM) and also withthemeasured data.

14. SUBJECT TERMS

FiniteElement Method; Mode Matching Method; Waveguide discontinuity

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analysis of waveguide junction discontinuities … of waveguide junction discontinuities using...

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