IEEE MTT-V031-I05 (1983-05)

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IEEE TRANSACTIONS ON MICROWAVE THEORY AND

TECHNIQUES,

VOL.

M’IT-31, NO. 5, M4Y 1983

373

Folded l?abry-Perot Quasi-Optical Ring Resonator Diplexer: Theory and Experiment HERBERT

M. PICKETT

AND ARTHUR

Abstract —Performance of folded Fabry-Perot quasi-opticaf ring resonator diplexers with different geometries of reflecting surfaces is investigated both theoretically and experimentafly. Design of optimum surface geometsy for minimnm diffraction, together with the figure of merit indicating improvement in performance, are given.

E. T. CHIOU

PORT III

PORT IV (O”TFWTOM,<

,61.NA,INW

MESH —-

------

I.

A

FOLDED parallel

FABRY–PEROT reflectors,

which

ing the noise and diplexing energies

into

the

mixer,

resonator

serves the purpose

was

described

resonator

is tuned

by moving

nant

band

of the cavity

FIXED MIRROR

of filtertested

by

L--__ ---__ --_-’

in Fig. 1. The

the mirror

block

so that

from port I is at resonant

peak of

the cavity and gets transmitted into input at a slightly different frequency

[

plane

and signal

and

[1], [2]. The basic idea is illustrated input

with

the local oscillator

Gustincic

the local oscillator

— —--------

INTRODUCTION

MESH

port IV. The signal lies in the anti-reso-

and gets reflected

from

port

‘ORTI~ (L. O. lNP~)

PORT II (FREE) ‘“pv,Ew

\

III

into port IV. The transmission characteristic of the FabryPerot cavity is the well-known Airy Function [3]. The advantage of the Fabry-Perot ring resonator diplexer over the two-beam mission that

port

signal

The

advantage

Section

type

slab Fabry-Perot

by Nakajima II,

effect

and introduce Mathematical formance,

of this

loss is eliminated.

were described diffraction

the Finesse F >>1,

for the local

its

we give limits

a simple

for the

Other

types

and Watanabe

[7].

of the problem

underlying

assumptions,

V, we conclude

by summarizing

Schematic

diagram

of

Fabry-Perot

ring

resonator

a description on how they can be applied to estimate the figure of merit of the diplexer with a curved reflector as compared

to one with plane reflectors.

II.

EFFECT OF DIFFRACTION AND THE RELATED DESIGN PROBLEM

diffraction.

on diplexer detailed

a folded diplexer.

on how

per-

analysis,

and theoretical results are given in Section III. Some of the mathematical details are given in the Appendix. Experimental results at 100 GHz are presented in Section IV. In Section

1.

of the diplexer

for minimum

formulation

Fig.

analyzed

of diplexers

treatment

the performance

SIDE VIEW

of Fabry-Perot

[6] is that geometri-

qualitative

solution

oscillator

band

resonator

et al. [5] and by Goldsmith

cal walk-off In

with

factor

[4], whose trans-

nature, lies in the fact

and also a much Wider reflection

port.

by Arnaud

resonator,

noise rejection

1’ cavity over the infinite / ,/

diplexer

is of sinusoidal

the Fab~–Perot

has a better input

interferometer

characteristic

our results

with

Manuscript received June 7, 1982; revised January 11, 1983. This work was supported by NASA under a contract with Caltech Jet Propulsion Laboratory. H. M. Pickett is with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109. A. E. T. Chiou is with the IBM San Jose Research Laboratory, San Jose, CA.

0018-9480/83/0500-0373

For applications regions,

performance

in millimeter-

and submillimeter-wave

of the diplexer

is’ essentially

limited

by diffraction effects. Diffraction in the vertical direction results in energy loss of the system, since the top and bottom of the cavity are open,, while diffraction in the horizontal direction couples port III and port IV together so that a significant fraction of the local oscillator input energy is distributed into port III. This can be visualized by conceptually propagating the input beam through a lattice of vertical cavities, as is illustrated in Fig. 2. Diffraction effects can be controlled by replacing one of the plane reflectors signed

with

a curved

mirror

so as to phase match

$01.00 @1983 IEEE

with

surface properly

a Gaussian

beam

with

deap-

374

IEEE

TRANSACTIONS

principle

2.

Horizontal

@ PORTTI

reflecting

PORTU

a schematic

transmission

geometries.

However,

mirrors.

illustration.

-.-...

‘. ‘\

,’

difficult

“’;”” .+=7.5,.

imposed

case is formally

by Arnaud

and here by the

equiva-

if the single detector The curved mirror

because edge diffraction

in terms of its resonant

-T d2=5.2Scm ‘\

one

of the flat-wall

conditions

of the cavity becomes significant. . .>,-.

Rhotiz z 62 cm

improvement

the results of the walk-

The flat mirror

lent to the case treated more

2w=2.5 cm A:.

MTT-31, NO. 5, MAY 1983

in that paper are not applicable

replaced by an array of detectors.

2w=2. 5 cm < “~

VOL.

how much

because of the special boundary

Y\ PORT1

effect:

to predict

off analysis presented

v’ r-l

diffraction

TECHNIQUI?S,

used to predict

curved-wall PORTI!z

AND

should expect by introducing the phase-matching curved reflector. The formalism used by Amaud et al. [5] can be in

‘//

Fig.

THEORY

is far too simple

—. J41!fP

d

ON MICROWAVE

from

is

case is

the aperture

Description

of the cavity

modes is more convenient

in the

high finesse case where edge diffraction is important because the effects are included implicitly. An analysis based on a modal expansion will be described in the next section.

.-L

F

(a)

III.

MATHEMATICAL

FORMULATION AND ANALYSIS

Exact three-dimensional analysis of performance of the diplexer with a curved reflector is fairly complicated. For practical

. Fig.

3.

Reflector

surface

design for minimum (b) Side view.

diffraction.

(a) Top view.

propriate beam waist at the plane reflector, as shown in Fig. 3. A toroidal surface with horizontal radius of curvature twice that of the vertical

application

wave regions,

turns out to be a very good

approximation. Steps leading to a solution of the design problem follow. 1) Choose the free spectral range (FSR) of the cavity so that it is twice the intermediate

frequency

(i.e., FSR = 2~1~). The condition

~1~ of the system

above fixes the width

W’

in the millimeter-

where

sume that diffraction

the” curvatures

and submillimeterare mild,

effects in the vertical

we can as-

and horizontal

directions can be decoupled. The original problem is thus resolved into two simpler problems, namely, a one-dimensional

infinite

vertical

strip

diffraction

problem

resonator

problem

for

treatment

loss, and a two-dimensional

for treatment

of horizontal

diffraction

effect. The

two are decoupled in the sense that the solution from first part enters only as a parameter into the second. A. Infinite

Str@ Resonator

of

waveguide the

Model

and length L of the square cavity to W= L = c/4@1~ where c is the velocity of light in free space. 2) The appropriate beam waist radius WOand the Raleigh length (ZR ) are given by WO= W/3ti, Z~ = n w~/A, where

For the plane resonator diplexer, the infinite strip plane resonator model of Barone [7] is used to approximate the

A is the free-space wavelength.

This chc)ice gives a Gaussian

each

has a waist parameter

Gaussian

amplitude

on the mesh which

equal

by

to 1/3 of the cavity width. 3) The horizontal toroidal

and vertical

radii

of curvature

for the

Following

input

beams

(plane

overlap

eigenfunction

integral

of

the

wave

and

sizes) is evaluated

input

function

of each mode. For the cylindrical

and

the

and toroidal

the steps listed above, we have constructed

an

of Boyd and Gordon

based

direction,

designed

on the following

a resonator

to control

loss has also been constructed tal models

various

resonator diplexers, diffraction loss in the vertical direction is estimated by the infinite strip cylindrical resonator model

model

the horizontal

the

by

beams with various beam-waist

RO

are determined

parameters:

~,, = 1 GHz, W= L = 5.30 cm, WO= 1.25 cm, A = 0.3 cm, R~ =62 cm, and RO=31 cm. In or(der to separate the effect of diffraction in the vertical direction from that in reflector

mode

by R~ = 2L + Z; /L,

surface

= R~/2. experimental

eigenfunction and the diffraction loss associated with each mode in the vertical direction. The excitation efficiency of

also include

only

with

the vertical with

of the actual cavity

a cylindrical B. Two-Dimensional

diffraction

and tested. Our experimen-

a diplexer

[8]. Dimensions

and the associated models are illustrated in Fig. 4, with Fresnel Numbers and round-trip amplitude diffraction losses of the fundamental modes given in the lower part of the figure.

plane reflectors

so

that the total effect of diffraction in both directions can also be observed. Preliminary experimental results were reported in [6]. Although the cavity model described above leads us to the optimum curvature with minimum diffraction effect, it

Mathematical guide

problem,

Waveguide

formulation together

with

Problem

,

of the two-dimensional the appropriate

wave-

coordinate

system, are illustrated in Fig. 5. Dependence on the y coordinate is ignored, since the effect of field variation in the y direction was separately taken into account in Section III-A, as explained above. The electric and magnetic fields at z = O and z = W can be expanded in terms of the

PICKETT

AND

CEEEOU: FOLDED

FABRY–PEROT

QUASI-OPTICAL

PLANEFitSfX4ATOR

RING

RESONATOR

CYLINDRICALFWONATOR

DIPLEXER

the

dl=5.2Ecm

375

eigenvalue

problem

R*31cm(FOR

2w=2.5cm)

the

waveguide

can

then

2a =4.11 cm L

where Ua and A.

Iml

are the eigenvectors

and the eigenvalues

of the waveguide. The procedure for determination matrix will be described in the Appendix. In general,

TOP VIEW ‘2”51mKxl

KIl /“’’.\4L?74cmcm

f

11111

I

[ !

J

I

I

I

Twill

be a nonsymmetric

eigenmodes

will

exp ( + i+),

representing

Fig.

4.

Estimation

and

and the given by

backward-going

Note that if we represent

2a =4.11 cm

the eigenvectors

in the basis set

as ( Ula \

Nf, ,.32, AL,!= ~

0.34

0. m2

of vertical diffraction finite strip resonator

eigenvalues

forward-

~;:3;yJ

AL’ =3.8

DIFFRACTIONLOSS (ROUNOTRIPI

real matrix

appear in pairs with

[:1 Uza

Ua = N’=a2/

FRESNfLNO.

of T

waves.

J_

1-0 MODfL ,,=,=374,”,

be (3)

Tua = Aaua = exp (i+a)ua

-tSIDEVIEW

of

expressed as

the associated

loss by one-dimensional model.

in-

eigenfunctions +.=

(4)

+. will then be given by

Xu..w

(5)

sin(nnx/LJ.

n

Consider a linearly polarized input beam from port I with the electric field vector given by E = Iio(x ) y at z = O. If we ignore mismatch 1

L ,0

Eo-

the magnetic

between

expand the incoming basis functions as

;:

part

and also the impedance

free space and the waveguide,

we can

field in terms of the complete

set of

EO(X) = ~S~~sin(nx/L,)

(6)

n

~—,,—+ Fig.

5.

Dimensions

where

and coordinates associated sional wavegnide analysis.

with

the

two-dimen-

S. =~~Lgsin(nx/L~) o

Eo(x)

(7)

dx.

.EO(x) cart also be expressed in terms of the eigenfunctions complete

basis set as Ey(z

as (8)

(la)

= O) = ~e~~sin(nrx/Lg)

a

n

where Hx(z=

(lb)

O) = ~h~~sin(nmx/LJ n

EY(z=

W) = ~e~~sin(nnx/Lg) n

HX(Z =W)

(lC)

= ~h~~sin(n~x/LJ

the prime

is used to symbolize

the fact

that

the

summation should run only over the “E-parts” of the eigenfunctions with eigenvalues representing the forwardgoing waves. Using (5), we can rewrite EO(X) = ~qa~a~

(id)

(8) as

sin(n~x/Lg)

(9)

an

n

where all the e:, h:, en, and h ~ are complex Symbolically,

we can express the expressions

above as

(:)==0= (;) beam linearly

polarized

V is the submatrix the forward-going

of u which

eigenmodes.

couples

the E field

By comparing

(6) and

(9), we have a.

in they

(if)

or ‘n. =

modes are

Presume we have a T matrix that transforms M fields at z = O to those at z = W, so that

the E and

Each eigenmode propagates independently through the Fab~–Perot, with amplitude transmittance and reflectance given by [3] t.=

(2)

(11)

E Va.’sn . n

direction,

we consider only the TE modes, since the TM not significantly coupled to our input beam.

(f)=T(;)

with

(le)

(:)Z=W=K) For an input

where

in general.

qat2AJ(l

ra= qa[r

- r2A~)

+ t2A~/(1–

(12a) r2A~)]

(12b)

376

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-31, NO. 5, MAY 1983

where r and t are the amplitude

reflection

coefficient

and

the amplitude transmission coefficient of the interfaces. From (9), (1 1), and (12), the total transmitted and reflected amplitudes E,=

~

can be expressed as

sin(rzm-X/~g)[ui.

Va~SmKa~

and k;= The eigenvectors

(i:n)and(-;k.)

/(l - “AI)]

mm

(13a) E,=

~

va;’sm~a~

satisfy the eigenvalue

~i; ()

sin (~~@g)

awn

t2A~/(1-

Alternatively,

tor (receiver)

ik

,2A;)].

(13b)

and

amplitude

taking

distribution

_ ~.k

overlap

treating

integral

of

n

()

by a detec-

at an appropriate

with the normalized amplitude plane of the field to be detected. has the same geometry have, from (6) and (7)

received

by conceptually the

exp ( – ik.W)

analyzed into far-field

the amplitude

can be determined

as a transmitter emitted

(’.landz(-;kn)

= exp (ik~W)

=

The field E, and E, can be Fourier patterns.

equations

n

. [r+

(24)

(2~/X)2–(n~/L~)2.

it

as can be easily verified. For input plane wave from port I with incident

its

Eo(x)

plane

=~exp[ikOsinO(x

– L,/2)]

angle=

= ET(x)

distribution at the same If the detector at port IV

as the transmitter

at port

(25) E;(x)

I, we

From E:=

EO = ~S.~sin(n~x/L~) n

S.=

~~LgEj

8

=~exp[-

(26)

ikosin8(x-L~/2)].

(17) and (18) we have

(14)

and sin (nnx/L~)

“JLgsin(n~x/Lg) o

(15)

dx.

exP[i~osinO(x

- %“2)]

dx

(27)

o Amplitude

received by the detector

at port IV is then given

by

(27a)

By straightforward t4 =

Using

Qn=S:.

‘SE:E, Jo

(16)

dx.

S.=

Similarly,

~ V.;’fi.SJ. anm

– il/2[exp

(n~i/2)

– exp ( – n~i/2)

(12a), (13b), and (14), we get t,=

integration,

[t2AJ(l

– r2A~)]

.

(17)

Field

amplitude

be determined The rapid

if

sinc(kOLgsin sinc(kOLgsin

as “seen”

sin(qrx/L,)

(18)

dx

then t3 =

r2At)].

Va;l~aS~Q.[t2AJ(l-

~

(19)

enm

decay of”

sine” function

~ v::~asmsn[r a nm

r2 = ~

+ t2A~/(1

Va;lV.#~Qm[r

+

t2Ai/(1

(

sin(k.W)

(21).

with increasing

argu-

sums, over index n and m in

sum, over index

to the diagonal),

a, also converges

so that

rapidly

the

within

a

– r2A%)].

(21)

is thus carried via the mode q = kO Lg sin @/n and the adjacent modes. It is the interference between the q mode and adjacent

(22) sin(k~W)/kH cos(k.W)

modes which leads to tion of the q mode magnitude of - 1/2. contribute 0.405. For modes add, while for subtract.

This picture,

of interference (23)

–k.

at each port can

few terms in the neighborhood

.TH X...

Cos(knw)

(28)

(20)

nm c-i

Tn =

– rzT/2)].

- r2A~)]

As an illustrative example, let us apply the procedure described above to analyze the simplest special case where both side walls of the waveguide are flat. The transformation matrix T is given by

T= Z’,XT2X-.

+ nn/2)

the expression for t~, t~, r2, and r,, have only a few terms (modes) adjacent to q = koL~sin8/m that have significant contribution. Furthermore, the V matrix turns out to be “almost diagonal” (i.e., the elements that are significantly different from zero are those along the diagonal, and in infinite

,1=

(0/2)

by the detector

some cases, those adjacent

For port I and port II, we have

(0/2)

by using (17) and (19) through

ment ensures that the infinite

Q.= ~~L’E$

we get

1

diplexer

with

one curved

the directivity. In (17), the contributo the sum by product S~S~ has a The two adjacent modes together

t4,the phase is such that these three t~ the phase is such that these modes in which directivity

between a curved wall,

of a = q. Most of the energy

most

modes, persists mirror.

For

of the energy

is a consequence to the case of the

the waveguide comes in

two

with ad-

PICXETr

AND

CHIOU:

FOLDED

FABRY–PEROT

QUASI-OPTICAL

RING

RESONATOR

Fig.

11.4(km-!

Fig.

6.

Phase dispersion L=

jacent

modes with

of four of the eigenmodes W=5.28cm, R=60cm.

propagation

for waveguide

phase factors

7.

Normalized

phase difference

8 of the two dominant

eigenmodes

versus LO/R. m = E L$ / k is the resonant order. Straight lines represent the fit by the empmcaf relation 8= exp (– 0.0283 m2/LO/R)/m.

with

separated

377

DIPLEXER

by

almost n-. Using curved

parameters mirror,

of the eigenmodes Fig. 6. Note that such

that

introduces

for

our

theoretical

model

with

a

curves for four

have been computed and are plotted in the eigenmodes are labeled by index q

Iu~al > Iu~al for some ambiguity

are almost equally labels the particular

experimental

phase dispersion

all

n. Although

this

scheme

when two or more basis modes

dominant, it is convenient because it basis mode that has maximum contri-

bution. Results

of our numerical

for optimal

Gaussian

the eigenmodes difference

calculations

beam input

are significantly

8 turns

out

to

also indicate

excitation,

only

that, two of

coupled

and their

phase

be a crucial

parameter

that

‘:~

determines the transmission characteristics. We define d as the fractional part of the phase difference measured in

0.5

units of r. Equivalently, IS is the frequency separation of the modes divided by the free spectral range. In Fig. 7, values of 8 are plotted parameters.

against

by

to note

the empirical

(straight

with

A is the free-space wavelength

of LO, R, and L~ are defined interesting

LO/R

that relation

m - 2L~/A

as

and the meaning

into port

1.5

2.0

2.5

3.0

3.5

~o

Fig. 8. Through-put (port IV) versus waist size of input Gaussian beam for different wall curvatures. Waist sizes such that the beam phasematched to the curved surface are indicated by arrows.

in the inset for Fig. 7. It is

the results

(circles)

8 = exp ( – 0.0283

fit

very

well

m2L0/R)/m

lines).

Transmission

1.0

IV, Ild 12, is plotted

against input

Gaussian beam waist size with radius of curvature of the waveguide as parameter in Fig. 8. The arrows in the figure indicate the waist sizes for which the input Gaussian beam is phase-matched to the curved surface illustrated in Fig. 3. The agreement with our waveguide analysis is excellent.

requirement.

Since the Finesse is the free spectral

range in

units of transmission linewidthi and d is the phase difference of the two dominant eigenmodes in units of the free spectral range, the product Fi3 can be interpreted as the ratio of phase difference of the two dominant eigenmodes to that of the transmission linewidth. Transmission with flat and curved into port IV, Itd I2, for diplexers reflectors

are compared

So far we have completely ignored the existence of metal meshes at the input and output planes or our waveguide.

C. Solutions

The reflectivity of the metal meshes determine the Finesse F of the diplexer. In practice, values of F are set by system

two-dimensional

in Fig. 9 for various values of F13.

to the Three-Dimensional

To convert

the solutions problems

Problem

(17), (19), (20), and (21) of the into

those of the original

three-

378

‘

IEEE

01 0.1

,

ON MICROWAVE

THEORY

AND

TECHNIQUES,

VOL.

MTT-31, NO. 5, MAY 1983

—1

o

L

TRANSACTIONS

10

F8 Fig.

9.

Transmission (Td ) for diplexers with Ilat and curved versus FcS.(— curved mirror, ---- flat mirrcm.)

dimensional

diplexer

effect of finite direction. taken

problem,

Diffraction

into

rby

simply

by

to vertical

diffraction

the

size can easily be

replacing

r~and the amplitude

where 1 – f is the amplitude

to consider

of the walls in the vertical

loss due to finite

account

reflectance

we have

size and curvature

reflectors

the

amplitude

traumnittance

attenuation

tby tfi,

per round

trip due

loss. The values of 1 – ~ awe given in

Section III-A. Under the conditions that loss factor is relatively smzdl (i.e., 1 – ~ Yo)}2

{f2(LY)-f2(~o>

Yo)}21

SHIBATA

et a!.:

CALCULATION

co=

OF THE

CAPACITANCE

w2/4[{fl(xo>

FOR THE

Yo)}2+{f2(~03

COAXIAL

y)=

a2y2+/(w-x-~y)2

y) = W2

387

LINE

f,(x,

YO)}2]

(w+x+/3y)2+ g(x,

RECTANGULAR

‘+BY l+g(x,

f2(x,

y)

=

ay

y)

l–g(x,

y)

+a2y2

2

/’[’ 2

(x+py)2+

- W*+

a’y*

{w* +(x+

/?y)2+

a*y2}2–4w*(x

+py)2

+ a

~

I

Equation (6) gives the potential in the region exterior to the plate conductor. Of course, the potential on the conductor is +..

The

derivable

electric

field

at any point

as the negative

Equation

gradient

(6) is also applicable

in the Z-plane

of

the potential

to an isotropic

take Cl/cl, =1, i.e., a=l, ~ =0. In Fig. 3, we show the distributions

is (6).

region if we

of equipotentials

and lines of electric flux in the region of Fig. 2(a) by using (6). Fig. 3(a) and (b) shows the field distributions with the parameters of c1 /cll = 1/9, 6 = m/6 (anisotropic), and c1 / 6,, = 1 (isotropic), respectively. To compare, the line charge and the plate conductor for both

regions,

the properties

of electric

clearly presented correspond

are placed

respectively.

on the same positions

Considerable

fields

for

differences

the two

regions

of are

in these figures. The curves in the Z-plane

to the following

circle groups:

(a)

2 w*ro(c–l) r– CW2– R;

{

) 2

+

~_

W%o(cyl) Cw2 –R;

{ _ —

Wfi(R&w2)

477fo&m

?

1

– %) for equipotential

A

{

2

CW2– R;

{

c = exp

}

lines

} (7) (b)

,_

roc(R~+w2)-so(R~

-w2)

2cR~

{

+

s–

{

flux

(w=

1, .xO = O, y. =

2).

region flux.

2

2cR;

2cR0

and lines of electric

(a) Anisotropic region (c ~ /6,, = 1/9, O== 7r/6). (b) Isotropic (~1 /{,, = l). ---------- Equipotential fines. — Lines of electric

}

(R;-w2)4~

3. Equipotentiak

Fig,

~o@t+w2)+’o(@-w2)

( .

2

}

2 7

If

we need the potential

function

at any point

for

a

conducting plate of width 2W charged with charge density A per unit length, it can be readily obtained by using the function g(x, y) in (6) as

1

c = tan y for lines of electric flux

q)(x, y)=r)o+

A

lng(x,

y).

(8)

fields

by an in-

4Tco& in the T-plane. Plotted in Fig. 3(a) and (b) are the curves for the parameters with 4mo~(@ – @o)/A = 0.7m (m

Equation

=0,1,

finite

--- ,15) andy=+~/2n

(n=l,2,.

..,5).

line

(8) corresponds charge

A on

to the electric the origin

in

the

T-plane.

The

388

IEEE TRANSACTIONS

ON MICROWAVE

THEORY

AND

TECHNIQUES,

M’IT-31,

NO.

Pc ~

Pc 1

1983

x ~k+f

k

PC

*C i

Pcn

Pck+l x x

n~o %x

5, MAY

;Jxxxx+k

~lxxxx

Ann

VOL.

~

x

x

F(x,y) Pcnl

x km

‘L–, ,! 4.

Fig.

Field

distribution

by a conducting electric flux.

in anisotropic

plate.

----------

region

(c ~ /6,, = 1/9,

Equipotentiaf

lines.

—

Xq

8 = m/6) Lines

of

based on (8) are shown

in

Xxxx

o

distributions

of electric

Fig.

parameters

4 &ith

formula

fields

c1 /cll = 1/9

and

O = r/6.

XXXXM

xxx

CONTOUR

‘-{i-

xLINE

POINTS

CHARGES

The

for the case of a = 1, /3 = O is given in [21]. III.

APPLICATION TO CHARGE SIMULATION METHOD

In this section, can be applied the method, conductor infinity.

we show that the potential

to a charge simulation the distributed

are replaced

surface

(6)

[17], [18]. In

line charges at

course the difference between those potentials must lessened. The capacitance Ca per unit length is obtained

of the conduc-

follows:

charges on the outer

by discrete fictitious

Those charges are arranged

function

method

outside

Fig. 5. Arrangement of line charges and contour points for the calculation of the field potentiaf between outer and inner conductors by charge simulation method.

be as

tor, as shown in Fig. 5. Now we assume that the number and plate conductors these charges,

of pairs of line charges

is n. To determine

n contour

points

the magnitude

are chosen

on the outer

conductor. By applying the superposition of the potential function (6) at every contour point, the system of n linear as equations for n line charges is obt;

4?1

Am,

&

[lnF(xi,

Yi;

xj,

Yj)]

The accuracy

i.i

error and the location

points.

coordinates on

magnitudes potential conductors That is

The

are the potential

pairs

of contour

By solving tion

. . . n)

+c~ (i=l,.

(xi, y;)

and

outer

(xj, yj)

and of charge points,

the system of (9) under

the

values

conductor, (x, y)

the boundary

the inner

analytically

1

@(x, y)=n@o+

4~~0~

condi-

@ci = *CO ( = const),

between

can be calculated

are the

respectively.

of those charges [ Aj] are determined. at a point

on the

~

the

So, the

IV.

are correct

or not. The judgment at a number

the outer conductor

with

the calculated

is performed

the given boumdary

error (11) will

NUMERICAL

the contour the locations

for

of the structure be discussed

a

No

rectangular

shown in Fig. in

1

the following

EVALUATION

OF CAPACITANCE

points

are shown in Fig. 5. In Fig. 5, we select

for the line charges and the contour

points

as

follows. a) b) c)

y; Xj, ~).

of check points

using

however,

11.6, c1 ==9.4) [24] by using the method presented in the previous section. The arrangement of the line charges and

charges

potential.

points

are equally

placed

on each side

of rectangular. The line charges are equally arranged on the straight lines which parallel the sides of the rectangular. The lines equal

to those of the

The h4X and h~Y, the distances

e)

lines and the sides; are chosen as shown in Fig. 5, respectively. Moreover, the charges are horizontally slid for each

on Of

of line charges n is chosen to be 36.

d)

by comparlocated

The number The contour

lengths of the straight respective sides.

(lo)

ing the potential

as well as shape of the boundary.

presented,

We calculate in this section the capacitances of the structure shown in Fig. 1 with a sapphire dielectric (q, =

j.1

By using (10), we can check whether

on the placement

and outer

by superposition.

AjhlF(x,

not only

section.

(9)

contour

was

calctdatecl

in where

of (11) depends

and the number of charges but also on the method of the system of equations in (9). For these problems, Murashima et al. [22], [23] discussed the properties of the potential

boundary. The capacitance

o

(11)

J

discussion

\: +n[@O]=

~oo : ~co ,:, Xj .

Ca =

of

between

the straight

SHIBATA

et a[.: CALCULATION

OF THE

CAPACITANCE

TABLE

FOR THE

COAXIAL

OF THS CAPACITANCES

Ca/co~

LENGTH

FOR THE ZERO-OFFSET

STRUCTURES

389

LINE

method

I

COMPARISON

SAPPHIRE

RECTANGULAR

(method

B) as

PER UNIT WITH

K’(k) — K(k)

Ca = 4co~

DIELECTRIC

I

(12)

I hx=O .0 b T 0.5

1“0 2.0

w T

hy=O. Ca/c

~ Method

0.1

‘“2 0.6

where

O 0=

B

Method

2.37875 0 x/2

2.37875

2.56979

2.56979

0

2.59301

2.59301

n/2

2.70460

2.70460

0

4.63194

4.63198

Tr/2

4.65335

4.65329

TABLE

k =sn{~K(k,),

A

K’(~g)

—

PER UNIT

pared

LENGTH FOR THE OFFSET STRUCTURES WITH SAPPmR8 DIELECTRIC hx=O .0

iy

b

b

z’

w

05 0.25

C

Chang

I Method

0.2

3.30091

3.31655

6.66683

6.66888

1.0

:“:

;“:;:; .

;“:;:;:

2.0

:::

: “::;::

:“::::;

0.2

3.71425

3.78599

7.87796

7.88577

2.86054

2.86346

5.51966

5.52019

[6]

of method

~=

0.6

.

those calculated

and

capacitances

Ca/E Method

a=

A

0.50

1.0 Z.

;“; 0.2

2.53589

2.53593

0.6

4.75462

4.75539

Al=

outer

conductor.

(for f3=7r/2)

by the improved

(method

C).

and by substituting

(for O = O)

~

(for 6 = ~/2)

(13)

for (24) in [6].

The distances

For the offset structure,

width

errors

on

each side are averaged. Both x~ and y~ are zero when 8 is either O or 7r/2. of this method

based on

shown in Fig. 1, the

for the arbi-

of the strip w/a

for b/a

= 1

in Fig. 6 are design curves for of 0,5. Both curves are

with

tively.

In Fig. 6, the values of capacitances by method

A C. of

per unit length

shown obtained

the parameters

accuracy

Ca/cO~

zero offset and an offset ratio hy /b

the potential

the results of method

yield a small difference when compared to method Therefore, the capacitances calculated by the method

are shown in Fig. 6. Plotted

For the structure

as

–1

~~”

the coefficient

versus the normalized

We have to check the accuracy

of the

Comparing methods A and B, the capacitances of method A for the symmetrical structure obviously have a good

and

the above locations.

formula

calculated

the coefficient

ab~ Sinh — a

this paper seem to have satisfactory

By the slide,

We

C by improving

trary angle 8. The capacitances

respectively.

“

{

accuracy. side of rectangular x~ and y~ are

(~

(xhyv ab~ cosh — – cosh — a a

a= 0.5

(for O= O)

(

with

Tippet

0=0.0

z

a;

(~”

K and K’ are complete elliptic integrals of the first kind. The formula for a = 1 is given in [2]. Also in Table II, Ca/cO~ of the offset structure by method A is com-

II

COMPARISON OF THE CAPACITANCES Ca/eO~

b

.

K(%)

k,}

6 of O, T/4,

and 7r/2, respecfor w/a>

A have an error. The arrangement

0.8 of

charges which was used to obtain the results of Fig. 6 is shown in Fig. 5. Also, the capacitance error based on the arrangement

of charges shown in Fig. 5 gradually

as the hy increases from O to b. This fact indicates

increases that the

analysis for the cases of arbitrary angle O has not been presented. However, the capacitances for the cases of par-

arrangement of charges shown in Fig. 5 is not all-powerful. Therefore, for w/a >0.8 and hy + b, the arrangement of

ticular angles, i.e., 0 = O and n/2, can be obtained by bestowing the slight improvements in the conventional methods [2], [6]. Those methods will be shown presently in

charges must be changed.

this section. Therefore, we first compare the values of capacitances using the method based on (11) with the analytical values for O = O and m/2. The comparison of the capacitances of the rectangular coaxial lines is shown in Tables I and II. In Table I, the numerical results of Ca/~O~ of symmetrical lines by

of this method s~fficiently compensates for this weakness. The usefulness wdl be especially shown for the structures with arbitrary angles O where the analytical solutions for the capacitance cannot be obtained. The practice is presented on the curves of both hy /b = O and 0.5 with d = r/4 in Fig. 6 and the curves in Fig. 7. Fig. 7 shows Ca /t ~fi

the method

versus /3 for the structures

of this paper

those calculated

by authors

(method

A) are compared

using the conformal

with

mapping

Moreover,

we have to check over

again whether the new arrangement of charges is correct or not. It is a disadvantage of this method, but the usefulness

of the capacitances

with zero-offset

with O increase as b/a

strip. The slopes decreases.

390

IEEE

hx=O.O bla-1.O

TRANSACTIONS

ON M2CROWAVS

THEORY

(2%)

,,

TECHNIQUES, VOL. M3T-31, NO. 5, MAY 1983

AND

m

x

I

I

x

20

x

x

x 8

Ca &OJE-G

[

x x

7

lo-

-x

x

6 ~.—-T-

:76-

x

x

OFFSET ( by/b=

x

0.5)

,x

54-

ZERO-OFFSET ( by/b =0.0)

-

3-

~—

2d

oCONTOUR

POINTS

hq=

f

—

A

Method

1+ 0.00.1

0.2

0.3 0.40S

0.60.7

0.00.9

—----_’ X LINE

CHARGES

0.5d

Fig. 8. Cross section of a coaxial line with circulm outer conductor having an anisotropic medium and the arrangement of line charges and contour points (n = 36).

4 tO

w/a Fig.

6.

Capacitances rectangular

Ca/cofi

per

unit

length

coaxiaf lines with sapphire

versus

w/a

for

the

Ca

dielectric.

Eo~l 4.5

hy/b=O.O

,

W/a

=

W{d = 0.5

0.2

4.0

w

Ca comL 3.5 “

b/a =0.5 = 0.2

w/d

2.5

I

i

3.0-

2.0L_---_l

b/a =1.0

o

2.5

11/2

—--e

b/a=2.O Fig.

9.

Capacitances structure

Ca/c ~~

per

unit

length

shown in Fig. 8 with sapphire

versus

O for

the

dielectric.

2.0 ~

o Fig.

7.

Capacitances rectangular

Ca/co~ coaxial

—e per

Tr/2

unit

length

lines with sapphire

versus

O for

the

dielectric.

TABLE III COMPARISON OF mm CAPACITANCES Co/c ~ PER UNIT LENGTH FOR mm STRUCTURES WITH A CIRCULAR OUTSR CONDUCTOR SHOWN IN FIG. 8

Another considerable method to improve the accuracy of the capacitance is to augment the number of charges by

t

comparison with one used in this paper. By the augmentation, the capacitance error is generally decreased, because the capacitance value theoretically approaches the true value if we use the arrangement of the discrete charges which has much the same effect as the true distribution of charge density

on the outer conductor.

have to use great numbers

However,

we present the application

2.72899

0.5

4.55873

I

I

2.72899 4.55873

we shall

of charges for the purpose.

A merit of the method of this paper is that it can be applied to the structures with arbitrary outer conductors. As an example,

0.2

1

1.0

EL=

al=

by a conformal co=

to the structure

with a circular outer conductor as shown in Fig. 8. Table III shows the comparison of the capacitance CO/cO for the structure with zero-offset strip having the medium of c1 /cll =1. The capacitances of method D in Table III are given

mapping

Table Ca/co~ strip.

III

technique

K’(k) 460— K(k)

shows versus

very

k=

[2] as l–(w/d)2 l+(w/d)2”

good

O for

agreement.

the structures

Fig. with

9 shows zero-offset

SHIBATA

et al.: CALCULATION

OF THS

V. We function

have

FOR THS

RECTANGULAR

analytically

presented region.

to the charge simulation

electric

potential

The potential

the

function

method.

The potential

function

(6) is applicable

[19]

The numeri-

cal results of capacitances for two structures with sapphire dielectrics are also presented. This method is useful for the numerical analysis of the coaxial lines where the analytical formulas for the capacitance cannot be obtained. source method

COAXIAL

[18]

CONCLUSION

in the anisotropic

was applied

CAPACITANCE

[20]

[21]

to the equivalent

[25], but it is not described

in this paper.

[22]

ACKNOWLEDG~NT [23] The Technical

authors

would

College

for

like

to thank

helpful

K.

Karnata

at Ibaraki

discussions.

REFERENCES

[24]

[1] M. L. Crawford, “Generation [2]

[3]

[4] [5]

[6]

[7]

[8]

[9]

[10] [11]

[12]

[13]

[14] [15]

[16]

[17]

of standard EM fields using TEM transmission cells~ IEEE Trans. Electronragn. Compat., vol. EMC16, pp. 189-195, NOV. 1974. W. Maguns and F. Oberhettinger, “Die berechnung des wellenwiderstandes~ Arch. Elektrotech,, 37 Bund, Heft ‘8, pp. 380-390, 1943. G. M. Anderson, “The calculation of the capacitance of coaxird cylinders of rectangular cross-section,” AZEE Trans., vol. 69, pp. 728-731, Feb. 1950. S. B. Cohn, “Shielded coupled-strip transmission liner IRE Trans. Microwaoe Theay Tech., vol. MTT-3, pp. 29-38, Oct. 1955. T. Chen, “Determination of the capacitance, inductance, and characteristic impedance of rectangular lines,” IRE Trans. Microwaue Theo~ Tech., vol. MTT-8, pp. 510-519, Sept. 1960. J. C. Tippet and D. C. Chrmg, “Characteristic impedance of a rectanguhu coaxial line with offset inner conductor,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. 876-883, Nov. 1978. H. J. Riblet, “The characteristic impedance of a family of rectangular coaxiaf structures with off-centered strip inner conductors;’ IEEE Trans. Microwave Theory Tech., vol. MTT-27, pp. . . 294-298, Apr. 1979. B. T. Szentkuti, “Simple anrdysis of rmisotropic rnicrostrip lines by a transform method,” Electron. Left., vol. 12, pp. 672–673, Dec. 1976. E. Yamashita, K. Atsuki, and T. Mori, “Application of MIC formulas to a class of integrated-optics modulator analyses: A simple transformation: IEEE Trans. Microwave Theory Tech., vol. MTT-25, pp. 146-150, Feb. 1977. S. Kusase and R. Terakado, “Mapping theory of two-dimensionaf anisotropic regionsfl, Proc. IEEE (1.ett.), vol. 67, pp. 171-172, JarL 1979. M. Kobayashi and R. Terakado, “New view on an rmisotropic medium and its application to transformation form anisotropic to isotropic problems,” IEEE Trans. Microwave Theory Tech., vol. MTT-27, pp. 769-775, Sept. 1979. H. Shibata, S. Minakawa, and R. Terakado, “A method for equalizing the even- and odd-mode phase velocities of shielded coupled-strip line with an smisotropic medium; Trans. Inst. Elect. Eng. Japan, vol. 102-A, pp. 149-154, Mar. 1982 (in Japanese). H. Shibata, S. Minakawa, and R. Terakado, “Analysis “of the shielded strip transmission line with an anisotropic mediumfl IEEE Tram. Microwave Theory Tech., vol. MTT-30, pp. 1264-1267, Aug. 1982. T. KoImo and T. Takuma, Numerical Calculation Method of Electric Field. Tokyo: Corona-sha, 1980 (in Japanese). H. Shibata, S. Minakawa, M. Kobayashi, and R. Temkado, “Upper and lower bounds of the resistance value of two-dimensionaf anisotropic compound regions by supposing the potentird distnbution~ Trans. Inst. Elect. Eng. Japan, vol. 98-A, pp. 494, Sept. 1978 (in Japanese). M. Panizo, A. Costellanos, and J. Rivas, “Finite-difference operators in inhomogeneous anisotropic media; J. Appl. Phys., vol. 48, no. 3, pp. 1054–1057, Mar. 1977. H. Steinbigler, “ Digitale berechnug elektrischer feldeq” Elektotechnische Zeitschrift, ETZ-A, Bund 90, pp. 663–666, 1969.

H. Singer, H. Steinbigler, and P. Weiss, “A charge simulation method for the calculation of high voltage fields,” IEEE Trans. Power App. Syst., vol. PAS-93, pp. 1660-1668, 1974. S. Murashima, “Application of conformaf mapping to charge simulation method; Inst. Elect. Eng. Japan, ED(PE)-79-3, pp. 21-30, 1979 (in Japanese). K. J. Biros and P. J. Lawrenson, Analysis and Computation of Electric and Magnetic Field Problems, New York: Pergamon, 1973, pp. 184-186. K. Tsuruta and R. Terakado, ‘ O), which leads to both the resonant

~. and the intrinsic

Q value Q. due to radiation

are given explicitly

by

fo = lW277 Q.=

frequency loss. These

(6)

lQ1/20i.

the boundary III.

surface, that is; n X (E, – E z) = O

and n X (O-M~ – HI z ) = O (n is the unit vector normal surface). However, as mentioned before, the infinite in (2) should be truncated to a finite number of Such approximated n = N for practical calculations. never satisfy the above type of boundary condition. therefore

coefficients

by means

n + 1/2, respectively. The characteristic

contour

we obtain

We minimize

(!2, >0,0,

Here, J.+ ,,2 and H(2J n+ 1/2 are ‘he ‘irst tion and the second kind of Hankel

(l)–(3)

numerically,

i. P~m(cos O)

proach.

J .+,,2(k1r),

substituting

integration

phase

is given by

to the

at an arbitrary q coordinate, as shown in Fig. 2, and E ~1,O-O ~1(i= 1, 2) denote the field components tangential to 17. After

In

can be reduced

line integral:

this

to the series terms fields We

boundary

condition in the sense of least-squares [18]. For this purpose, we introduce the mean-square error E in the boundary

NUMERICAL

RESULTS

Since the pillbox resonator considered here has a plane of symmetry with respect to the r – q plane at O = r/2, symmetric and antisymrnetric modes to this plane can exist independently. Then, the r – rp plane at O = 7/2 can be replaced with a magnetic wall without affecting the field distribution (or

about

Hw-component

antisymmetric

which

the Eq-component

is antisymmetric).

(or HW is symmetric),

is symmetric

Similarly, an electric

if Eq is wall

re-

394

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-31, NO. 5, MAY 1983

T EOISmode

1.0-

—

Present Method

———

Van B[adel’s Method

—-—

Magnellc Wall Model

x

0.8 -

Komshi’s

--

----

—-—

--—-—

-

Er=35 -_ -—

----—-—

.—-—

I 0.8

0.6

---—-

------

______

-----

-----

---------

I

1.0

‘0

__ &r=35

20 4

0.4

-----

Er=88

--Er=88

0.2 -

0

--------

---

40 -

.—-—

Van BladeCs Method

G

0.4 -~ -—-—

Present Method —

200 -

Melhed

1000 x

—

TEOIS mode

I

I

0.4

0.6

1

1

0.8

1.0

bla

blcr

(b)

(a) Fig. 3. Comparison of the numerical results for the TE016 mode between the present method and different methods. (a) Normalized resonant frequency. (b) Intrinsic Q value.

places that plane. boundary contour kind

Then,

contour

it is enough

in the first

of (5) if we utilize

associated

Legendre

to consider

quadrant

the following function

for relation

P#(cos

only

the

the integral

TABLE I NORMALIZED RESONANTFREQUENCIESAND INTRINSIC Q VALUES

CALCULATEDFORTHE DIFFERENTNUMRER N OF THE ExPANs1014TmMs

of the first

0) at tJ = 7r/2: b/a=l. N

(

P:+q(())=

(– 0q’2”(q+2m o,

-l)!!,

q=o,2,4,

...

q=l,3,5,

.-.

where 1, s!!=

S(S–2). ( s(s–2).

Now,

for

S=–l,

o

. . 43.1,

S = 1,3,5,...

. . .4.2,

s = 2,4,6,...

the case of m = O, the field

(8)

components

pressed by (1) and (2) split into two independent

ex-

groups of

Er= 80

k. a

(7)

O

Er.35

1

0474

Q.

Q.

k. a

o. f)43xlo2

0.304

0 1 56x103

2

0.473

0.433X102

0,303

0 1 52x103

3

0,470

0.401

0.300

o.lf$o

4

0.469

O.4OOX1O*

0.299

0.1 39X103

X102

Mo3

5

0467

0.395X102

O 299

0.1 38x103

6

0,667

0.395X102

0.298

0.1 38x103

7

0,467

0.393X102

0.298

0.1 37X103

8

0.467

O.393X1O2

0.298

0.1 37X103

of the correction term to the dominant one is proportional to l/&. Q. is compared with those by Van Bladel’s

(~,, He, Eq) and (-% EO, HY). The former group expressed by ~,, only become TE modes in the sense of E,= O, while the latter expressed by +~i only become TM

method in Fig. 3(b), and his results are found to be slightly lower than our results. This discrepancy may be caused by

modes in the sense of Hz = O. Such q-independent

becomes more noticeable

modes

are extensively used in most practical applications, and we hereafter investigate such modes only. First, we compute the complex resonant frequency of the

the same reason

as mentioned

above.

In fact,

this effect

in Fig. 4(a) and (b) which

shows

koa and Q. as a function of the dielectric constant c,. The solid lines indicate the present results and the dashed lines indicate figures

Van

Bladel’s

results.

that his results approach

It

is obvious

from

these

our results with increasing

q-independent TE Olo model (m = O). Table I shows the calculated results for different numbers N of expansion terms in (2) in order to investigate the convergence of both

e,, but the accuracy is poorer when c, is lower.

~. and Qo. These calculations are performed for the structure with c, = 35, 88, and b/a = 1. It is clear from this table that both the normalized resonant frequency /coa and the intrinsic Q value Q. due to radiation loss almost converge

Cp-independent mode, i.e., the TMola mode. Fig. 5(a) and (b) show k. a and Q. calculated for N = 7, where the solid lines indicate the results by the present method and the cross marks indicate

for N >5.

Van Bladel’s results. Unlike the TE 018 mode case, agreement between both methods is very good, even for lower c,. This feature can be understood from [15], since, when Van BladeI’s method is followed for calculations of the TM018

Also, in Fig. 3(a) and (b), koa and Q. calculated

for N = 7 are compared seen in

with Fig.

those obtained 3(a),

the

resonant

by different

methods.

As

calculated those by

by the present method agrees very well with Konishi’s method [11], which gives agreement

frequency

with experimental results to within 1 percent. Van BladeI’s method, indicated by the dashed line, gives satisfactory results for the resonant frequency in case of c, = 88, but its accuracy becomes worse for t, = 35 because the magnitude

We have discussed ‘TEoIa

mode.

we

so far the numerical

next

mode, the first correction is of order 1/c,, instead

investkate

results

for the

another

term in the asymptotic expansion of order l/~ in the case of the

TEold mode. Hence, his method gives more accurate results, especially for Qo, for the TM016 mode than it does for the TE018 mode.

TSUJI et al.: COMPLEX RESONANT FREQUENCY OF OPEN DIELECTRIC RESONATORS

395

~

TEoIt mode bla=l.O

100 -

Present Method

—

—

Present

———

Van Bkxtel’s hlelhcd

&o-

MettKKI

o

4

;

20 -

10-,

1.0-

40.5 2-

/

1,

‘241O

20

40

100

&r (a)

(b)

Fig. 4. Resonant characteristics of the TEOla mode as a function of the dielectric constant c,. (a) Normalized resonant frequency. (b) Intrinsic Q value.

200 -

T MOICmode

3.0 -

100 2.5 “

TMo16mode

—

O

Present Methnd km BhrlelS Mettwl

x —

bla=l.

Presenl Methml 40 -

Van Blmiek Method

2.0 -

6

20 -

a ~ 1.5. 10 1.04 0.5 -

2 -

o~

10

4

m

40

I

,\iil

la)

4

10

20

40

100

&r (a)

: Fig. 5. Resonant characteristics of the TM016 mode as a function of the dielectric constant c,. (a) Normalized resonant frequency. -.. (b) Intrinsic Q value.

Finally, hybrid

in case of m *O,

ones. For

taken into account,

both

so that the number

cients to be determined in calculations

modes

become

+,, and ~,i must of unknown

be

coeffi-

becomes twice as much as that in

case of m = O. This point, ficulty

the resonant

such modes,

however,

does not cause any dif-

in the present method.

paper will present numerical discussions hybrid modes, as well as the experimental IV.

A succeeding

about important investigations.

CONCLUSION

A new analytical method has been presented for calculating accurately the complex resonant frequency of an open dielectric is presented accuracy

pillbox

resonator.

The numerical

for the TEO1a and the TM0,8

of the present

method

discussion

modes,

is confirmed

with

and the respect

to both good convergence of calculations with previously published approximate method which

is based on the Rayleigh the approximate

the Hehrtholtz satisfy

fields,

equation

the boundary

expansion

expanded

in the spherical

conditions

and comparison methods. This theorem

in

in the solution

of

coordinate

in the least-squares

system, sense.

Hence, the uniform convergence in the sequence of the truncated modal expansions such as in (2) can be assured mathematically [20]. In actual numerical calculations, however, one cannot always obtain precise solutions, in particular, for the problem with edge-shaped boundaries, though the method is complete in theory. This difficulty is due mainly to the slow convergence. Paying attention will be indeed necessary on this point

even in the present problem,

but special care is not taken into

account;

nevertheless

a

395

reasonable

IEEE TRANsACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-31, NO. 5, MAY 1983

convergence

is obtained

in calculations

in Table I. Readers will find the detailed the convergence expansions

‘

and the analytical

property

documents

as seen

[21]

about

of the Rayleigh

in [21] and [22].

[11 D. J. Masse and R. A. Pucel, “A temperature stable baudpass filter using dielectric resonators; Proc. IEEE, vol. 60, pp. 730-731, June 1972. [21 E. Fox, “Temperature stable low-loss microwave filters using dielectric resonators,” Electron. Lett., vol. 8, pp. 582–583, Nov. 1972. [3] W. Harrison, “A miniature high Q bandpass filter employing dielectric resonators: IEEE Trans. Microwaoe Theory Tech., vol. MTT- 16, pp. 218-227, Apr. 1968. [41 T. D. Iveland, “Dielectric resonators filters for applications in microwave integrated circuits: IEEE Trans. Microwave Theory Tech., vol. MTT- 19, pp. 643-652, July 1971. [51 M. A. Gerdine, “A frequency stabilized microwave band-rejection filter using high dielectric constant resonators: IEEE Trans. Microwaue Theory Tech., vol. MTT- 17, pp. 354-359, July 1969. [61 A. Karp, H. J. Shaw, and D. K. Winslow, “Circuits properties of microwave dielectric resonators,” IEEE Trans. Microwaue Theory Tech., vol. M’IT-16, pp. 818-828, Oct. 1968. [71 S. B. Cohn, “Microwave bandpass filters containing high Q dielectric resonators; IEEE Tram. Microwave Theory Tech., vol. MTT- 16, pp. 210-217, Apr. 1968. [31 H. M. Schlicke, “Quasi-degenerated modes in high c dielectric cavities,” J. Appl. Phys., vol. 24, pp. 187-191, Feb. 1953. [91 H. Y. Yee, “ Naturaf resonant frequencies of microwave dielectric resonators,” IEEE Trans. Microwaoe Theory Tech., vol. MTT- 13, p. 256, Mar. 1965. [101 A. Okaya and L. F. Barash, “The dielectric microwave resonatorsfl Proc. IRE, vol. 50, pp. 2081-2092, Oct. 1962. [11:1 Y. Konishi, N. Hoshino; and Y. Utsumi, “Resonant frequency of a TE018 dielectric resonator: IEEE Trans. Microwave i%eo~ Tech., vol. MTT-24, pp. 112–1 14, Feb. 1976. [121 T. Itoh and R. Rudokas, “New method for computing the resonant frequencies of dielectric resonators: IEEE Trans. Microwave Theory Tech., vol. MTT-25, pp. 52-54, JarL 1977. [13:1 M. W. Pospieszakki, “ Cylindncaf dielectric resonators and their applications in the TEM line microwave circuits,” IEEE Trans. Microwaoe Theoty Tech., vol. MTT-27, pp. 233-238, Mar. 1979. [141 Y. Garault and P. Guillon, “Higher accuracy for the resonance frequencies of dielectric resonators: Electron. Lett., vol. 12, pp. 475-476, Sept. 1976. [15:1 J. Van Bladel, “On the resonances of a dielectric resonator of very high permittivity~ IEEE Trans. Microwave Theory Tech., vol. MTT-23, pp. 199-208, Feb. 1975. [16:1 M. Verplanken and J. Van Bladel, “The electric-dipole resonances of ring resonators of very high perrnittivity,” IEEE Trans. Microwave Theory Tech., vol. MTT-24, pp. 108-112, Feb. 1976. [17:1 M. Verplanfren and J. Van Bladel, ” The magnetic-dipole resonances of ring resonators of very high perrnittivity,” IEEE Trans. Microwave Theory Tech., vol. MTT-27, pp. 328–333, Apr. 1979. M. Tsuji, S. Suhara, H. Shigesawa, and K. Takiyanm, “ Submilfime[18 ter guided-wave experiments with dielectric rib waveguides~ IEEE Trans. Microwave Theoiy Tech., vol. MTT-29, pp. 547–552, June 1981. Time Harmonic Electromagnetic Fields. New [191 R. F. Barrington, York: McGraw-Hill, 1961, ch. 6. K. Yasuura, “A view of numencaf methods in diffraction problems; [W in Progress in Radio Science 1966 – 1969. Brussels: URSI, 1971, pp. 257-270.

[22]

H. Ikuno and K. Yasuura, “Numerical calculation of the scattered field from a periodic deformed cylinder using the smoothing process on the mode-matching method:’ Radio Sci., vol. 6, pp. 937–946, June 1978. Y. Okuno and K. Yasuura, “ Nuruericaf algorithm based on the mode-matching method with singular-smoothing procedure for analyzing edge-type scattering problems,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 580-587, July 1982.

+ Mikio Tsuji (S’77-M82) was born in Kvoto. Japan, on September 10:1953. He reeeive~ the B.S. and M.S. degrees in electrical engineering from Doshisha University, Kyoto, Japan, in 1976 and 1978, respectively. Since 1981, he has been a Research Assistant of the Faculty of Engineering at Doshisha University. His research activities have been concerned with submillimeter-wave and microwave transmission lines and devices of open structures. Mr. Tsuji is a member of the Institute of Electronics and Communication Er@eers (IECE) of Japan.

* Shigesawa (S’62-M63) was born in Hyogo, Japan, on January 5, 1939. He received the B.S., M. S., and Ph.D. degrees in electrical engineering from Doshisha University, Kyoto, Japan, in 1961, 1963, and 1969, respectively. Since 1963, he has been with Doshisha University. From 1979 to 1980, he was a Visiting Scholar at Microwave Research Institute, Polytechnic Institute of New York, Brooklyn, NY. Currently, he is a Professor at the Facufty of Engineering, Doshisha University. His present research activities involve microwave and submillimeter-wave transmission lines and devices of open structure, fiber optics, and scattering problems of electromagnetic waves. Dr. Shigesawa is a member of the Institute of Electronics and Communication Engineers (IECE) of Japan, the Japan Society of Appfied Physics, and the Opticaf Society of America (OSA). Hirosfsf

* Kei Takiyama (M’58) was born in Osaka, Japan, on October 20. 1920. He received the B.S. and Ph.D. degrees” in electrical engineering from Kyoto University, Kyoto, Japan, in 1942 and 1955, respectively. Since 1954, he has been a Professor of Electronic Engineering at Doshisha University, Kyoto, Japan, where he carried out research in the fields of microwave transmission lines and optical engineering. From 1957 to 1958, he was a Fulbright Scholar and a Research Associate at the Microwave Research Instit~te, Polytechnic Institute of Brooklyn, New York. Dr. Takiyama is a member of the Institute of Electronics and Communication Engineers (IECE) of Japan, the Institute of Electrical Engineers of Japan, and the Optical Society of America (OSA).

IEEE TRANSACTIONS ON MICROWAVE THSORY AND TECHNIQUES, VOL. MTT-31, NO. 5, MAY 1983

397

A Generalized Chebyshev Suspended Substrate Stripline Bandpass Filter CHRISTOPHER

Chebyshev

low-pass

realization

in suspended

A prototype

prototype

device,

such a filter

replace

more

TEM-mode

is used, resnfting

substrate designed

as an example.

reafizing

Resrdts

AND JOHN

in a convenient

conventional

form

with from

the aid of a computer this device

show that

types

of

program,

is of

and maybe

microwave

filter

suitable

reafized

structed

is suitable

but problems bandwidths

using

of less than degree

needed

selectivity,

and

problems

Fig. 1.

a true bandpass performance. will

allow

and high-pass

attempting about for

10 percent. the filters

structure

when

is large

Generalized Chebyshev low-pass prototype.

good

roll-off

at the

with

substrate

prototypes

performance

+----/$-

The impedance

hence

I I 1

0 0

generalized zero

L.1O

Fig. 2.

1

COS2 ( (n-l)

10&@+c2

Insertion

_, .0S

loss characteristic

J-2

(*) +‘Os-lO)) of generalized

Fig. 2 and has n – 1 transmission

proto-

as good as the same degree elliptic

II.

zeros at a frequency

@o

where n is the degree of the low-pass

LUMPED

CIRCUIT

DEVELOPMENT

produced,

a prototype

Chebyshev

low-pass

tech-

close to bandedge, prototype.

at infinity

Chebyshev

prototype.

Chebyshev

problems

u

“o

to be used

variation

to realize presents

---’

I

to designs derived

flat or conventional

one transmission

L

fractional

to achieve good narrow-

selective

attempting and

circuit realization. The odd degree nearly

1

I

This is due to

to achieve

The use of suspended highly

degree possible.

however,

with

I

wide bandwidths,

types. The exact elliptic function prototype will realize a given selectivity and passband ripple level with the minim-

kind,

L(1)

sections. This

to realize

associated

which can achieve superior the maximally

Loss,

and low be con-

band edges of the two discrete sections, causing large insertion losses and poor amplitude flatness. Considering these factors, it would seem necessary to

um

L(n-4)

I

substrate stripline discrete low-pass

for devices having

occur when

the high

from

L(.-2)

1

to

Imxrtmn

by cascading low-pass

technique

niques

IEEE

INTRODUCTION

and high-pass devices which have high selectivity insertion loss. Bandpass filters may therefore

band

FELLOW,

resonators.

XISTING DESIGNS of suspended filters are available [1] to produce

realize

RHODES,

m---n

of

the method

DAVID

L(n)

stripline.

is viable for many applications

I.

E

MOBBS

—A design metftod for narrow-band suspended substrate striphaving a true bandpass structure is presented. A generalized

Abstract fine filters

given

IAN

for

To produce

of this printed

prototype

formation prototype

the required

of (1) is applied of Fig. 1

structure, to

all

the bandpass the

elements

the

[2] (1)

has a selectivity function

transof

proto-

type and has a much smaller impedance change through the network. This generalized Chebyshev prototype (Fig. 1) was therefore used in the design. The frequency response of this prototype is shown in Manuscript received August 17, 1982; revised December 29, 1982. C. I. Mobbs was with tie Department of Electrical and Electronic Engineering at the University of Leeds and is now with Filtronic Components Ltd., 4/5 Acorn Park, Charlestown, Shipley, BD 17 7SW, England. J. D. Rhodes is with the Department of Electrical and Electronic Engineering at the University of Leeds and Filtronic Components Ltd.

where B=

Uz — al,

-G, uC—

and p=

ju

for sinusoidal

input. ti, and U2 are the bandedge frequencies of the bandpass network. This results in the network of Fig. 3, which has the frequency response indicated in Fig. 4. Examining this network, we see there are n complex shunt networks of the form of Fig, 5(a), Since these are very difficult to realize directly in suspended substrate stripline, a more convenient equivalent

circuit

must be found.

ture has been derived

001 8-9480/83/0500-0397$01

.0001983

IEEE

Such an equivalent

struc-

[3] and is shown in Fig. 5(b). We may

398

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MT’F31, NO. 5, MAY 1983 C(r)

L(r)

~(n)

r4fn’Y”; w t----------I

----

(a)

.-

Cz

.2

Fig. 3. Ins . Loss

Bandpass network.

1

I ;L t--

L~ --

---

I I

(b)

I I I I I I L 8

6.

Fig.

Equating

Introduction

of redundant series elements.

Y= and Y~ gives the identities c+=

/3+

C.=p.

. Ulul

Fig.

‘o

and

‘+

‘,

Response of generalized Chebyshev bandpass filter.

4.

L_=~

L+=%

(5)

P.

n-

$

L, c,

side of the passband. not physically

c

+

c,

L,

L

+

B

We now have pairs of series resonant shunt circuits which resonate at frequencies a+ and a_ located at either

form very difficult.

iE_-

(a) Fig. 5.

elements

(b)

Alternative

configuration

for shunt elements,

aration.

between

the values of L+,

section as follows.

L_,

C+, C_ by synthesizing

the

For Fig. 5(a), we have

c1p[L2c2p2+1]

y = a

(2)

What is required

is an extra element

resonant

will

allow

or

sep-

in the center of the

in a redundant

manner

networks of Fig. 6. The two-port impedance matrices Fig. 6(a) and (b), respectively, may be written as

into the for

1 + ap2 + bp4 Z(a+l)

az

aZ

az

[ a = LIC1 + LZC2 + L2C1

Z(b+d)

and

[Z21=

~bz

[ b = LlL2C1C2. (2) into partial

y=

fractions P+P

I+

+

fx+pz

Equating

gives the result B-P

p+=C1L2C2 –

+ c)

the terms of these two matrices

Applying nant circuit

1

(6)

-

gives the identi-

Clcq

LC circuit

c.

of Fig. 5(b) we

p

(4)

+

L_ C_p2+l

“

canceling

to give the network

result is that of introducing between

values are derived

of the network

c+ p

a similar procedure to the adjacent series resoand cascading the two networks results in the

1: m transformers The overall

C1LZC2 – C1a _ ~+_a —

L+ C+p2+l

m2Z(b

(3)

a_—a~

Considering the admittance can immediately write

mbZ

l+a_p2

a+,_ =; J@jG

P-=

1

ties

where

Yb =

are

in printed

the network to provide the separation as follows. This extra section may be derived by considering

[z,]=

a

realization

these two sections which

may be introduced

where

Splitting

sections, however,

thus making

A series LC circuit,

passband, obtain

These resonant

separated,

the shunt resonators. from the identities

of Fig. 7.

the required

extra

The new element

of (7). For a low-pass

prototype of degree n the above process is repeated (n – 1)/2 times for each of the pairs of shunt resonant sections in the circuit. The overall result is a network of the form of Fig. 8. To produce realizable element values,, the transformer ratio m must be positive and 1 —— -z ,,,

~>o, m ---

hence O