finite-element formulation

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900

IEEE TRANSACTIO NS ON MICROWAV E THEORY AND TECHNIQUES, VOL.

MTI

-3 3, NO.

la,

OCT08ER

Finite-Element Formulation in Terms of the

Electric-Field Vector for Electromagnetic

Waveguide Problems

MASANORI KOSHIBA, SENIOR t-tEMBER , IEEE , KAZUYA HA YATA,

AND

MICHIO SUZUKI, SEN I

OR

MEMBER, IEEE

A

ftruct

- A veclo r fin ite-element method for the analysis of anisotropic

\\'1lvegui

des wilh off-diagonal elements in Ihe permeability tensor is for

mulated in terms of all th ree COmponents of Ihe electric field. In this

approach, spurious, nonphyslcal so lutions do

rIOl

appear anywhere above

the

a

ir-line. The appli cation of this finite-element method to waveguides

with an abrupt discontinuity In the permittivjty is discussed. In particular,

we discu

ss bow

to use the bou

nd

ary co nditions of the electric field al the

[nt

.,.

rface

e t w ~ n

two

medi

a

with

different pennlttivilies. To

show

the

validity and usefulness of this formulation, exa

mpl

es are

co

mputed for

dielectric- loaded waveguides

Ilnd

fe

rn i

e-l

oa

ded waveguides.

I. INTRODUCTION

T

HE VECTOR · finite-element method is widely used

either in an axial-component E, - Hz) formulation

[1)- (4) or in a three-component (either th e electric field E

or the magne tic field H ) fo rmu lation

(5

), [6), which enables

one to compute accurately the mode spectrum of an el

ec

tromagnetic waveg

ui

de with arbitrary cross section: The

most serious diffi

cu

hy in u

si ng

the vector

fi

nite-element

analys is is the appearance of spurious, nonphysi

ca

l

so lu

tions [1)-[6]. Hano

[7

has presented a three-component

finite-element formulation with rectangular elements. In

his formulation, spurious so

lu

tions, except for zero e

ig

en

values, do not appear, but a diagonal permittivi ty tensor

and a d iagonal permeability ten sor are assumed. Rece ntly,

an improved finite-element method with triangular

ments has been fo

rm

ulated for the anal

ysis

of anisotropic

di e

le

ctric waveguides in terms of all thr

ee

components of H

[8) - [l1J. In dielectr

ic wavegu

ides, the permeability is al

ways assumed to

be that of free space. Therefore, each

ever,

t

IS difficult to apply this H-field formulation

tQ:

waveguides containing anisotropic media such as

f e r r i

because the tenso r permeability may vary from material to :;

materia

l.

In such cases, it

is

advantageous

to

solve for

j

rather than for H . .

In this paper, an imp roved finite-element method with.

triangular elements is formulated for the analysis of n i s ~

tropic waveguides with a nondiagonal permeabili

ty

t e n s o r

using all three components of E. II) ferrite-loaded wave-

guides, the permittivity is assu med to be constant in each-

•

material,

but

may vary

fr

om material

to

material.

At

abrupt discontinuity in the permittivi ty,

th

ere is an abrupt

change in

E.

In

thi

s work, the application

of

the

E-field

fo rmulation to wavegu ides with abrupt discontinuities

in

the perrrtittivity is discussed in det

aiL

In particula

r,

we

discuss how to use the boundary conditions of

E

at

th

e

interface between two media wit h different permittivities.

In this improved E-rield

fo

rmulation, no spurious solu

ti

onf'

appear anywhe

re

above the air-line. The appearance

of

spurious solutions is

limi

ted to the region / ko< 1 and

these solutions are equivale

nt to t h ~

TM modes of

hollow

waveguides. To show the validity and usefuln

ess

of this

formulation, ex.amples are computed for dielectric-loaqed

waveguides and ferrite-loaded waveguides.

II . FUNCTIONAL FORMULATION

We

co

nsider an anisotropic waveguide with a

tensa

permeability and a scalar pe·rmittivity. With a time depen


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el Ill. : ELECTROMAGNETIC WAVEGUIDE PROBLEMS

-

,

2( 0 11 0

(4)

functional

[12], (13)

for (3) is known to be

- k

J L ( , E . E d n (5)

where

n

represents the cross section of the waveguide and

asterisk denotes complex conjugation. In the finite-el

e

analysis using (5), spurious solutions appear scattered

the propagation diagram (51-112] , [141. 115]. These

SP

' ' '; '

). s solutions belong to

tw

o distinct cat

ego

ri

es [11)

,

The first one (SI) can be characterized as follows:

V

x E =O

'\7'(,£ + 0

o r k

~

6)

,· 1,, second group (S2) can be characterized as follows:

vxE*'O 'C'

rE-.l-O

for k5 > O.

(7)

In order to eliminate the spurious solutions SI and S2

we

propose the following functional according to the H-field

fo rmulation (8]- (11) :

(8)

For the functional (8), the first variation

aft

is given by

~ t - o E

. [ X

([ X

X E) - ( , , •,E) -

k <

,E] dO

-

I r ~ E [ n

x

r X

E)

- n(,,·.,E)]dr

(9)

where f represents the contour of the region 0 ,

n

is the

outward unit normal vector to

f ,

and the term

n

X

([

p,

,1-

1

v X E) correspond s to the tangent

ia

l components

of the magne tic

fi

eld H on

f .

The stationary requirement

901

Fi

g.

l . Interface with an abrupt discontinuity in the permittivity.

obey the following equations:

€,E - v1/;

( \7

2

+

k5)'

-= 0

in

region n

(12a)

(12b)

' _ 0 on perfect electric conductor (12c)

af an -= 0 on perfect magnetic conductor

(12d)

where , is the scalar field. The electric field E of (12)

satisfies the stationary requirement

8F

- 0, but the diver

gence of

€,E

is not ze ro. Therefore, in the finite-element

analys

is

using (8), spurious solut

io

ns

S

3'

which are not

included in (5), do appear. The solutions S3 are equivale

nt

to the TM modes of hollow waveguides (replace' in

(12b)- (12d) with E

z

) and the appearance is limited to the

region f3

/

ko < 1. They do not appear anywhere above the

ai r

-line.

III.

FINITE-ELEMENT DISCRET IZATION AN D

BOUNDARY CONDITIONS

Dividing the cross section n of the waveguide into a

number of second-order triangular elements

as

shown in

Fig. 1 and using the finite-elemenLmethod on (8), we can

write the functional fo r the whole reg ion 0 in the form

t -

Lt

.

(Il)

•

t. -

(E [AJ.(E

).

14

)

[AJ . -

[SJ. +[UJ k5[TJ . 15)

wh

ere

{ E } is

the electric-field vector

co

rr

es

ponding to the

nodal points within each element, thy matrices


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902

IEEE TRANSACTIONS ON MTCROW.WE THEORY AND TECHNIQUES, VOL, MIT-33, NO. 10, OCTOBER

I f the functional for € is used in its origirial fo rm (14),

we

should modify the functional

for

e

2

in order

to

satisfy

the boundary

conditions

of the electric field Eon f , For

e

2

•

the functional (14) can be rewritten as

A x x ]

lA,,)'

[AI,

[A.xll

[Ax'xli

IA ,L

[Az'xL

F , ~ E ) i l A I , E ) ,

(16a)

(EL

lA,,),

IA,,],

lA,,],

lA",],

lA",),

lA,,],

(E, ),

(E,) ,

(E,),

(E,,),

(E

),

(E,,),

[A.i:zL

lA,, ],

[A,,] ,

[A

",],

lA",],

[A

u'

],

IA,,),

[A u' )'

[A

x

'>;'

] 2

lA ,, ],

[A

,

),

' [A ,,

),

(16b)

lA,,),

IA

",

],

IA

",),

I

A

, ],

lA",,],

lA ,,),

Using (17), (16) can be transformed as follows:

[Ax:'L

lA ,],

[A ,

),

[A

,,

),

IA

, ] ,

[A ,,),

F , ~

(E)i XI,(E),

(E),

(E,\,

(E,

),

(E,

)2

(E,,),

(E

),

(E,,),

[AX •

J2

lA

,,],

[Au]2 [Ad],

I

A,,]

,

[A , ),

IA,,],

IA,,], IA,,], IAp),

IA ,],

I

A

,),

h ~

[A,J,

lA,,),

[A,,),

[Au'],

I

A",],

[A ,]'

[A

,,'],

lA,,),

[A ,],

[AN]'

IA,

,,,],

[AN]'

IA",),

IA ,I,

I

A

,I,

lA ,

, ,

IA""I,

IA

I,

[A

J,

lA ,),

[A ,],

[A

i',' ],

IA ,,),

[ A

where

the

components

of

the

{E

j

}

2

vector are the values

of

the eiectric field Ej l

=

x, y, z) at the nodal points within

,

the element e

z

except P, the components of the {Ei'h

vector are the values of

E;

t

the nOda l points

on

f

, ,

where

[Ax'x'b = q';x[A

X

X']2+ qxAxy([ A

x

,y

·12+

[A

y;x' ]2)

+q';y[Ay·y,12

[A X'Y'] 2

=

qxxqxy

(A x'x']2+

qxxqyy [

AX'y']2

19b)

16c)

19c)

(20a)


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KOSHIBA el al.: ELECTROMAGNETIC W

AV

EGUIDE PROBLEMS

•

•

-

b , . ; t"QI2

( 1.5

Fi

g.

2. Dispersion characteristics of a h

alf-fi 1ed

dielectric waveguide.

By using the original functional (14) fo r e

1

and 'the mod

ified functional (19) for e

2

• the boundary conditions of the

electric field E at the interface with an abrupt disco

nt

inu

ity in the permittivit

y

are sat

is

fied.

IV. NUMERICAL RE

SU LT S

A. Dielectric- Loaded Waveguides

Fi

rs

t, let us consider a rectangular waveguide half-

fil

led

wfth

a

dielectric

of

permittivity (\ (relati

ve

permittivity

f

,\

-

( 1 / (

0 ) ' .

We subdivide one half of the cross section into second

order triangular elements

as

shown in

th

e insert in Fig. 2,

where

(

1 - 1.5, the

pl

ane of

sy

mm etry

is

assumed

to

a

perfect magnetic conductor, 36 elements (N£) are used,

and the number of the nodal points N

p

is 91. Computed

res

ults (solid lines) for

th

e LSM

n

a

nd L S E m ~

modes agree

well with the exact results [16J. Spurious solutions S, and

S2

which are included in (5), do not appear. Spurious

solutions S3 (dashed lin es) corresponding to the sol utions

of (12) appear on

ly

in the region Il/ko <

1.

The solutions

903

•

,

,

•

,

•

.

•

,

,

,

,

,

,

100

_0

15

-0

F

ig. 3.

p-depcndence for the spurious

s o l u t i o n s ~ .

becomes F. For

(2

1),

(12)

is reduced to

( p 2 V 1 k ~ ) 1 f _ in

regi

on 0 (22a)

If - a on perfect electric conductor

(22b)

on per

fect

magnet

ic co

nductor.

(22c)

The appearance of the solutions of (22) is limited to the

region Pl ko i p and the cutoff frequencies of these

so lutions

va

ry in proportion to the value of

p.

Fi

g.

3 shows the p-dependen

ce for

the

so

lutions S3 in

the same waveg

uid

e

as

sh

ow

n in Fig.

2.

So

li

d and dashed

lines in Fig. 3 correspo

nd to

the TMIl and TM

I2

modes in

Fig.

2, re

spectively. When

p

= 2, the

so

lutions

S3

appear in

th

e

re

gion /3 l k(j<

0.5

and the cutoff

fre

quencies of

th

e

solutions corresponding to the TMA) and TM Il

mod

es in

Fig. 2 become

koQ

- 2fi IT and 2v 5

IT,

respective l

y.

Wh en

p

=

0.5, the so lutions S3 appear in the region Plk

o

< 2 and

the cuto

ff

freq

uen

cies of the

sol ut io

ns corresponding to the

TM

ll and TM

12

modes in

Fi

g.

2 become

k o

Q

0.5fi IT

and

0.51S IT,

respectively. The p-dependence is very sm

all

for the physical solutions. For larger valu es of

p,

how ever,

the degree of accura

cy

for

th

e physical

so

lutions becomes

poorer. For smaller val ues of p. on the other ha

nd

, more


6/7

IEEE TRANSACTIONS ON

MI

CIlOWAVE THEOIlY AN D TECHNIQUES,

VOL

. MIT-B , NO.

10

,


7/7

[6]

[ ]

[8]

[9]

[IO[

[ll]

[ 2J

[13J

[14]

[15]

[16J

[17]

[18J

t'r al,:

ELECTROMAGNETIC WAVEGUIDE PROIJLEMS

M. Ikeuch

i, H.

Sawami, and H.

Nih

Analysis of open-type

dielectric waveguides by the finite·element iterative method, IEEE

Trans, Microwave Til/wry Tech" vol.

MTI-29, pp. 234 - 239 , Ma

r.

I n l

N. Mabaya, P. E. Lagasse, and P. Vandenbukke, Finite element

analysis

of

optical waveguides,

IEEE Trans. Microwave Theory

Tech., vol.

MIT-29, pp. 600- 605, June 1981.

K. Oyamada and T. Okoshi, Two-dimensional finite-element

calculation

of

propagation characteristics of axially nonsymmetrical

op

tical fibers,

Radio Sci., vol.

17, pp. 109- 116,

Jan.-Feb

. 1982.

A. Konrad, High-order triangular finite elements for electromag

netic waves in anisotropic media,

IEEE Trans. Miuowaue Theory

Tech.,

vol. MIT-25, pp.

353

- 360. May 1977.

B. M. A Rahman and J. B. Davies, Finite-element analysis of

optical and microwave waveguide problems,

IEEE Tram. Micro

wave Theory Tech., vol. MIT-n

pp.

20

-

28,

Jan. 1984.

M. Hano, Finite-element analysis of dielectric-loaded waveguides,

I

Trans.

Microwave

Theory

Tech ..

vol.

MIT-32, pp. 1275-1279,

.Oct. 1984.

M. Koshiba, K. Hayata, and M. Suzuki, Vectorial finite-element

formulation without spurious modes for dieleclric waveguides,

Trans. lnsl. Eleclron. Commull .

Eng.

Japall, vol.

E67,

pp.I91

- 196,

Apr. 1984.

M.

Koshiba,

K.

Hayata, and

M.

Suwki, Vectorial finite-element

method without spurious solutions for dielectric waveguide prob

lems, Elec/ron. Lell.,

vol. 20,

pp. 409 - 410, May 1984.

B.

M. A Rahman and

J.

B. Davies, Penalty function improvement

of waveguide solution by finite elements,

IEEE Truns. Mi{'rQwuve

TheaI) Te,·h.,

voL

MIT

-

n

pp. 922-

928,

J\ug. 1984,

M.

Koshiba, K. Hayata, and

M.

Suzuki, Improved finite-clement

formulation in terms of the magnetic-field vector for dielectric

waveguides, IEEE Tram. Microwave Theory

Tech.

, vol. MTT-33,

pp. 227- 233, Mar. 1985.

A

Konrad, Vector variational formulation of electromagnetic

fields in anisotropic media, IEEE Tral/-$. Microwave Theory Tech.,

vol. MIT;24,

pp. 553-559, Sept. 1976.

A.

D. Berk, Variational principle for

el

ectromagnetic resonators

and waveguides, IR Tran$. Amellnas Propaga/.,

vol.

AP-4, pp.

104-111, Apr. 1956.

J.

B.

Davi

es

,

F. A.

Fernandez, and G. Y. Philippou, Finile element

analysis of aU modes in cavities with circular symmetry, I

Trall$. Microwave Theory Tech

..

voL

MTT-30, pp. 1975- 1980, Nov.

1982.

M.

Hara

. T. Wada, T. Fukasawa, and

F.

Kikuchi,

A

three dimen

sional analysis of

RF

electromagnetic fields by the finite element

method,

IEEE Trall . MagI ., vol.

MAG·19, pp. 2417-2420, Nov.

1983.

N. Marcuvitz,

Wuveguide Halldbook.

New York: McGraw-Hill,

1951.

Z.

J. Cscndcs and P. Silvester, Numerical solution of dielectric

loaded waveguides:

II

- Modal approximation technique

,

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Trans. Micro

wave

Theory Tech., vol.

MTT-19, pp. 504- 509, June

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new ferrite modes in rectangu

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, pp. 1184- 1185, Sept. 1955.

905

Masanori Koshiba

(S

M'84)

was

born in Sapporo,

Japan, on November 23,. 1948. He received the

B.S.,

M.S.

, and Ph.D. degrees in electronic en

gineering from Hokkaido University, Sapporo,

Japan, in 1971, 1973, and 1976, respectively.

In 1976 , he joined the Department

of El

ec

tronic Engineering, Kitami Institute

of

Technol

ogy, Kitami. Japan. Since 1979,

he

has been an

Assistant Professor

of

Electronic Engineering at

Hokkaido University. He has been engaged in

research on surface acoustic waves, dielectric

optical waveguides, and applications

of

finite-element and boundary-ele

ment methods to field problem's.

Dr

. Koshiba is a member of the Institute of Electronics and Communi

cation Engineers of Japan, the Institute

of

Television Engineers of Japan,

the Institute

of

Electrical Engin

ee rs of

Japan, the Japan Society for

Simulation Technology, and Japan Society for Computational Methods

in

Engineering

KazllJ a

Hayata was born in Kushiro, Japan,

on

December 1,1959. He rece

ived

the

B.S.

and M

,S.

degrees in electronic engineering from Hokkaido

University, Sapporo, Japan, in 1982 and 1984,

respectively.

Since 1984, he has been a Research Assistant

of Electronic Engineering at Hokkaido Univer

sit

y. He

has

been

engaged in research on dielec

lric optical waveguides and surface acoustic

waves.

Mr . Hayata

is

a member of the Institute

of

Electronics and Communication

n g i n ~ r s

of Japan.

Miehio Suzuki (SM'57) was born in Sapporo,

Japan, on November 14,

1923.

He received the

B.S.

and Ph.D. degrees in electrical e n g i n ~ r i n g

from Hokkaido University, Sapporo, Japan, in

1946 and 1960, respectively.

From 1948 to 1962, he was an Assistant Pro

. f

es

sor of Electrical Engineering at Hokkaido

University. Sin

ce

1962,

he

has ~ n a Professor

of Electronic Engineering at Hokkaido Univer

sity. From 1956 to 1957, he

was

a Research

Associate at the Microwave Research Institute

of

the Polytechnic Institute of Brooklyn, Brooklyn, NY.

Dr. Suzuki

is

a

m m ~ r

of the Institute

of El

ectronics and Communi

cation Engineers of Japan, the Institute of Electrical Engin

ee

rs of Japan,

the Inst itute 'of Television Engineers of Japan, the Japan Society of

Information and Communication R

ese

arch, and the Japan Society for

Simulation Technology.

finite-element formulation

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