Some properties of the corrugated elliptical waveguide. Part I

Some properties of the corrugated elliptical waveguide. Part I

Citation for published version (APA):

Thurlings, L. F. G. (1975). Some properties of the corrugated elliptical waveguide. Part I. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1975

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·', t l , " "i' , • 'f,

EIN rnOV]~N' ONrvERS~TY OF TECHNOLCGY illPAR'I'l-JENTOF 'EIECTRICAL ENGJNEERING

,Some properties of the corrugated elliptical t,r~veguide by ,"'" Part I

ET-5-1975

August 1975 ,.' / " "

, Ii

•

.'

- , -2_ Contents , I Part I Chanter 1 Introduction

4

2 ,1'he behaviour of the phasefactor of the anisotropic

3

4

waveguide

2.1 Introduction

5

2 .. 2 Disturbance of the coefficients of the'matriX ' 6

2.3 Increasing the order of the matrix 8

2.4 Calculation of the 'eigenvalues with rET,: 0 ,10

2 ..

5

Phasefactor for ecc ': 0.25 10

The, corrugated elliptical "iaveguide

3.1

Introduction 3.2 De nth of the grooves '3 .. 3 Cut;"'off frequencies 3~4 ' The nhasefactor

3.5

Radiationoatterns Conclusions

14

18 19 29 48 References 63,

65

-

-3-Acknov.rledgement

The author appreciates the stimulating discussions ;.rith Dr. N"E.J~ Jeuken. Ir·" V.J. Vokurka and 1<1' •. N.H.M .. Knoben. The author wishes to ex~ress his apnreciation to I-Ir.

I.e.

Ongers for his valuable. assistance in constructing many ,..,rograms.

•

..

..

-4-Some nronerties of the corrugated elliptical waveguide

Chapter I

Introduction

This report concerns the ''lork done in the period (

15/3/ '75 - 1/9/ '75).

There has been done a fair amount of i'lork on the corrugated ellipti-cal waveguide to obtain an antenna which is able to radiate circular-ly polarised "laves in any direction, with an asynnnetrical radiation-DatteI'll.. Jansen and Jeuken [1] started the study, and derived the main properties for an ideal situation the so-called ltanisotropic

elliptical waveguidell

• Here it was assumed that corrugations could

provide a good boundary condition, but there was no experience of the actual behaviour of the corrugations. Later the study \'1as

exten-ded

[2]

,to the corrugated elliptical waveguide by setting up the

theory that takes into account the influence of the grooves, but no numerical or experimental results were obtained.. Comparing the

ex-periJuental results of [

1]

and the numerical results of [ 2], we· could

conclude that the anisotropic elliptical ,-;aveguide indeed could be

made ,·lith a corrugated elliptical lfaveguide

[6],

although it l'faS still

a little bit mysterious hOl'l to choose the exact depth of the grooves ..

In this report '<Te shall m8ke clear hO"T \:'0 compute beforehand this

depth. Also we shall give phasefactors and radiationpatterns for the corrugated elliptical waveguide ..

I t appeared that the llmode_mystery" as explained in

[2] ,

p42 for the anisotropic case, also exists for the corrugated elliptical waveguide •

That ' .... as for us fa reason to investi~ate this problem for the

".

..

. The behaviour of the 1""Ihasefactor of the anisotropic tfaveguide

2.1 Introduction

As mentioned in [ 2] phasefactor for the

,p42,

there is a problem concerning the anisotropic elliptical waveguide. As an

i

•

o

Fig. 2.1

example the phasefactor is given

in Fig. 2,,1, l-,here the modes

e

EH II

and eH E.31 obviously do not cross ,,' This is somel-That peculiar,

because in the case of the

circu-lar anisotropic waveguide these

curves are really crossing

[71 .

In the anisotrooic elliptical

,·raveguide the behaviour is different. There is aka-interval 6 ,.

,.,here no mode can 1""Iropagate (i"e .. the phasefactor may be complex) ...

" .

So' the auestion arises' 'tfhether the calculation of the ohasefactor _{... .} ~

is correct or not. It seems to be reasonable that there are no severe errors in the calculations because of tl1e good agreement.·

J bet'feen the numerical and experimental results[2J , .

althO~gh

this

is not

a

proof.

In this chapter we have done some numerical experiments, to

inves-tigate the numerical stability of the calculations:r

2.2 disturbance of the coefficients of the matrix

2 ..

3

increasing the order of the matrix

2.4

calculation of the eigenvalues ·with IET=O

2.5

nhas.e'f~lCtor _{for ecc=0.25}

The investigations ,.,ere done for the anisotronic case f')r those

.

•

..

1-,re are dealing I'vith:

(2.1)

(2 .. 2)

ko..

::

2-

{g'

e

_A

_{- 1}

,

?rj'k

,r

~

~

::. J, 0 (~,,4)

Thus \Ve compute the eigenvalues of eq. (2.1) as a function of qc " and transform this (A ,qc) solution to the ..

(r',

ka) solution with the formulas (2.3Land (2.4) •

?2 Disturbance of the coefficients of the matriX

vfnen computing the coefficients of the matriX AB. there may arise errors, i.e. rounding~off errors and truncation errors. The eigen-de~

1

ep&

values are the zeros of the determinant of

(A

B - )..

2 )

,so by the errors involve? the 'distance between t1rro eigen-values may become smaller ,(see Fig.

2.2).

'Therefore 1-,re disturbed the coefficients

;"1'ith an arbitrary value eps. and deter-mined the relative error 4) >./). for the

e EUu and e~EJI mode (Fig. 2.3). 'tIe con-clude that the relative error /J>).A is nroportional to the numerical error _{( , - -}ens~ and that the critical-level _,

..

..

..

-If 10 -Fig .. 2 ..

3

-7-','d:> .•. • .l \0 _2-10 ~ EPS

..

•

A

T

..

(i.e. where the actual eigenvalues of the two modes coincide) arpear

for large values of ~ ( ep~

>

1;1 ).

~cause of t!Je order of Tuagnitude of eus 1.;e may conclude that the

roundin?,-off and truncation errors do not cause a severe derivation of the exact eigenvalue. The following remark sUDPorts this.conclusion.

The matrix AB has been comouted and disturbed for a certain value

o~---~---_~\

q,= q, (see Fig. 2.4). But the distance between two values of higher modes is much smaller than the distance for the

modes

ee.~11

and

e

HE..l1 • So, i f

~

l'lould have a remarkable influence then this should appear for the smallest

eigenvalues and not for the highest ones.

S.S'

5.1 .5':0

Fig. 2.4

2.3 Increasinr the order of the matrix

The eigenvalue nroblem - eq. (2.1) - has a matrix of infinitely order, so

we wonder if this matrix may be auuroximated by a matrix of finite order.

Thus ,~ have to investigate the magnitude of the eigenvalues as a function

of the order of the matrix. ~e have done this for the dominant modes for

eccentricity e:O.916;0 at q=qc=-4.20 and q =q<: =l~.14 ( Figs. 2.5, ;2 ..

6

and,

2.7 • See also Figs .. .3.11 and 3.19) .. Vie observe that stability occurs for

N= 4. Usually all the computations have been carried out ,dth N= 10, so

for that order the eigenvalues are stable, even the Complex eigenvalues.

/

'I 7. 3 'of S

ecc

=

0.9180

qc:: 4.2

10

.,

...

'" . 'f~'O

Rt{A\

,

I

'Uo

-9-e=O.9l00 q =1~.14 c ~'~0r---

__

--~--

__

----~--

__

---~---~---

________ _

I.Y! 1· 3 10

Fig. 2.6 Real part of eigenvalue versus order of the matrix

1 J 10

e= 0 .. 9180

~::. 4,.14

•

•

-10-2.4 ,Calculation of the eigenvalues with rET= 0 \

,Next we vrant to know if 'the method used for the computation of the eigenvalues of an asymmetrical matrix ( procedure NONSYMEI(}ENV AHJES,

see

[2]),

calculates the eigenvalues ,<lith a sufficient accuracy.

Therefore we compute the eigenvalues with an indeoendent method. namely the determination of the zeros of the determinant:

, As an exam~le we give here the results for a waveguide with eccen-tricity e=0.919O for several values of qc; (Fig. 2.8). Itapryeared

(2.5)

,that the rela.tive error in the largest eigenvalue is smaller than 10'8 ,

so this vlay of finding the eigenvalues is also Norkable. although it

is some.<lhat slO\<T6r. than the above mentioned. procedure .. l'iealso found' that· th'e determinant has no zeros in the II

3.11

and 3.19), so we can concluqe that the

~interval (see Figs.

2.1,

proca<iu.re NONS'YllElGENVALUES find all the eigenvalues. with ~naccuracy of at least ₁₀ -8

2.5 Phase factor for eccentricity e= 0.25

In the reports [1] and [2]it was assumed ( 1.<lithotit proof) that the curves of the phase factors were crossif1.g. Now' we begin to believe that this is not 50. In

[2]

i'ie had no proof· ,,[hether . the curves were crOSSing or not for

small ecc,entricities .. He have now reryeated the calculations for e

=

0.25,' i.e. major axiS/minor axis

=

1.033 (Figs. 2.9 and 2 .. 10) .. vie may conclude from't..l"ese results that the feature IInon-crossing!! is a function of the eccentricity,' even for small values ..

'.

cl'~'l""'; __ l

r

.'

,

10 of 10

..

10 ""00 '100

"0

0 1 -1',.

_'''0

_,o)u,) ~ -\\) f -to 6

-,0

'.

BETR

1

+ +

*

+

..

.. .9

.8

.7 I

.6

~ I r

r

~

.

.

... ' .4 I l > - '

-2

1

2

3

4

c::

w '6

7

8

Q . ..j 10 1 1

1

2 1 31 .~ 1 5 1 6 . 1 71 8 1 9 2

0

BETR

.765

"64

,. ;' I,

.763

"'62

" /"" ,

.761

.76

3,G1

'.

/

3·62

3.63

KR

[XC =0.2500

RNISOTROPIC

..

-14-Chapter

3

The corrugated elliptical waveguide

3.1

Introduction

As we have seen in [2 ] , the solution of

t~e

Naxwell equations in

the corrugated elliptical waveguide can be descriood by t\'10

in-denendent classes of modes, namely the~modes and the odd

modes. Moreover, each of these two classes can be split into

modes of even order and odd order. For reasons mentioned in [1 ] ,

p 46 tM even orders of the even and odd modes have not been

in-vestigated.

-In

general it holds that the electric field lines of the dominant

even mode is parallel with the minor axiS, which is depicted

b.Y

and for the dominant odd modeparallei to the major axiS, thus

-,

-15-( see [2 ] , p146 ):

[cBA - CD) -

X

(! -

<;,.Q)]

b:

0

(3.1) -1

A -

[~(~I'~C)J

VT

Se(~1.1,)

(3.2) -1

B -

6

S

e'

(~I,~J]V

G.

(~,,~c)

(3.3) , -1

C -

[Se'

(~.,~J] ~T

N)

(~h~~.~,J

(3.4) ... 1

D-"=

~(~1>1'~ yiN(~"q,,~O)

(3.5)

N '. N-'.

~Se (~,~). ~'( ~",.)-

S.'

(~,,~.). ~ 4~ '4-~

lI!

[~.(t.'l.)· ~(q.)- 5~(~

..

~.). ~

(,.,<J.)r

(306) , j , , -1

Q. -

t

[C&'(~

..

~.~

yT

C;e(~,,'l')

.Q

(3.7)

\lfith the a;erturefields ( see

[1 ] ,

p28):

Er. _

,

-

-16-%.'

E~

=

-j

ko

s,J

-

ys" .

(3.9) • %.

- jko

..

₂

0

,\-1\

=

-r

53

S"

(3.10) . 1:.

Zo

~~

=

-I'

5,

...

j

leo

51.

(3.11) , eO

Si

=

L

a. ...

G~ (~,~)

ce;;,GH)

(3 .. 12 ) 00

S .. ::

L

b",

Se ..

(~,~)Se~(~'1)

, (3.13)

-5

_{3 ;"-}

L

Q. ...

G

...

(~,~)_cl(l~"J

(3 .. 14) 00

S, ::

~

b

...

Se~ (~'l).

se ...

(1.'l)

Yh;','

and the relations:

~~

:

'~O(

1

-f>

(3.16)

~

(3 .. 17)

. '"

-17-The index 1 stands for the values of . the inner ellips, the index 2 for the bottom of the groove ( vrhich is also an eLlips).

The odd modes

are

des cribed by ( [2] , p156):

[(8B-

Er)->-(

~

-

EF)]

a

=

0

(3 .. 19)

-1

E

=

~'(~

..

'JJ].

z.T

[j'

(~

..

~,,~o)

(3.20)

.. · - 1

{1",

[G

(Ii ..

~~

Z

T

M

(~, .~

..

~.J

(3.

21)

M'.

c{

= [

G(~s.). &:"(t.~.) -c:(~

..

'j..).

&::'(~

..

1o)] ..

[C.(~

..

1.1.

!;(~,.'l.)-G(~,>~.}.

f ..

~(~

..

~.j

(3,22)

(3 .. 23)

The aperture fields are no'" defined by:

\'lith these formulas He are able to. compute all the desired- properties of the corrugated elliptical \"aveguide.

3.2 Depth

of

the

grooves

The first problem we meet is to determine the depth of the grooves in such a l-raythat for a frequency ka the waveguide becomes an

ani-sotropic wavelmide for both even and odd modes. Then .. re have to

re-quire:

H

,.

=0 .

E

: : 0 ~

_~

':

~1

For 'the even modes this leads to ([2],

(3 .. 29)

p

150):

(3.30)

~,

..

...

..

"

-19-The relationS (3.30) and (3.31) are unmanageable because of the

infinite sununations. Even a truncation to a certain length N i'lould

not help us for .. ra 'do not know the coefficients

d ...

and. Ct\o\...

Therefore we use - for the dominant mode,(see

[2])_

th~

relations,'

based on the first term:

'0.32)

(3.33)

Thus for a given e~centricjty e1, of the inner-ellips.

,-te

,can now

determine the eccentricity e₁ of the outer-elJips as a function of

the frequency ka~ , for the even and for the odd modes.

In

Fig. (3.1)

,

,ore have done this for e,,::: 0.m6S' and in Fig. (3.2) for e :0.9100.

.... , 1

\

- Also the values of the corresponding minor to major axis ratio (~/bl.)

have been given. We .observethat the ,·;aveguide never becomes

aniso-" I:' ' ~,

tropic. HOliever, the larger the irequen'cy ka the more the waveguide behaves like an anisotropic pipe.

3.3 Cut-off freguencies

In

the following we shall consider the three antennas:

(a) e_{t :} 0"O,76A / e~: 0.80

(b) e,,:: 0.i576.~ / e₁ ..: 0.Q4

'.

e2,

_e

₁

=

0.8

_1'8

1

M ;:. ₁ \

e'l..

0.8so _-~-

_...

_.°2./1>1

•

/-A/ \

. / .

. v

'

// 1 ; ) '

-(I

0)

(j/ EVG. ... 10

\or.

e.,:: o.

&:,.68

IS' 2.0 t.97

..

,

It') I. B

1.6

20

-~)k0:1

..-

I.

7

Q.1/b

'&.

r

.

-

..

t.20

N ': 1

•... , 2.10

0.850

. 0.800

\ , - " - ...;.&;. ___ """--Jlr-_______

\~I

--,.---.F""""---r--,---S \ .,. 10 ' . IS

___

·_F_\_.a~ ... "

_3_,

2. ______

{)e;...;..fl ...

,h ...

o...:f ___

~

... h ...

··~__=9_r'_oo_v_e~...:. ~fo_\-

e l :

0.9

180

-22-Hodel (a) and (b) are improvements of the so-called antenna TV [1]. Antenna (b) should have a good nerformance at 11.5 GHz ( ka

1:1S.S).

fintenna (a) should be anisotropic for kai~lO,thus the frequency is

f

=

6.11

GHz. This model has not been constructed yet,but now 'We can,

,determine the influence of the grooves by comparing the numerical results 01 model (a) with model (b).

f:~odel

(c) is an improvement of the antenna III [1 ] . With a :: 29 :mm

and f ..:.11.5 GHz we now have ka

=6.5,

and '\tIe observe from Fig. 3.2 that

1 ' ,

the depth for the odd modes is very different from the one required

for the even modes. lie chose e~=

0.006.

In Fig.

3 .. 3

and

3.4

l'fe have

given the geometries for the models (b) and (c), which are the

im-nroved models of antenna TV and III resp.

Ive

now comntit'e the cu.t-off frequencies of'the lowest' modes for the

models a, band c and determine the mode-classification, (' [2], pl4.7

and p157 ). When we compare the mode-classification used here -'lith

the one used in

[4],

p91~'we

obtain:

Thurlings AI-Harriri

'C'

(~.>

KE'

Eit

e,

_~

"Se.

(~t)

e

Ell

'

0

KE

Set

(~,)

0

HE

0

EH

','Ct

(~~)

0

EH'

e

tts.

The difference is caused by the fact that our definitions are based

on the symmetry

of

the fieldlines while' those of AI-Harriri are

based ~n the symmetries of the Mathieufunctions involved ..

'- ... ""-,

'.-The cut-off;frequencies of the eME and oHE modes are related with;

So 10 10 10

-23-so e

=

0.fr168 e

=

0.84

e 0.9100 e

=

0.006

.

-24-..

ell!:

0.80

-EWt'l~

_{.. '}

_G~7.)

_::..

_Cz.

_(f).go)

0 q ka:. ka.l oENu

_3.35

_'4.58

E~ll 0

14 .. 63

9.56

&"i,sl 0

8.66

7.36

EH.ll 0

15.54

9.86

ERS! . o ,

16.78

10 .. 24

~

EHn~

_{S~ (4,.)}

':::

5e

(o.8o)

e

e

E _1-1 II

_{5 .. 43,}

_5.83 . EI-I,I.' "";'::-~--".~- ,.~-~-.... , ..

-.

e

19.48, .

1l.03

e

EM-,.

_10.80

_8.22 EJ.tsi e

18.64

. 10 .. 79 .

Table 301

..

-25-,~ HE,,~

G'~,

)

=

Cc.' (

o.8~&O

)

q ka -ka - 1

e

~E"

_0.675

_1.87

e

HEft

12.66

8.12

e

HE.lI

_4.69

_4.94

e HE.12

21.22

10 .. 51

WEsl

_12.05

_7.92

~ e eHEn

32.93

13.09

0

H

EhW\.

~'~I)

=

£e' ( ()

'&1 6

&)

. 'HE II 0

2.58·

.

3~67 HEI1. o .

_20.37

_{10 .. 29}

, oHE~i

_6.83

_5.96

'0 HColt

30.10

12 .. 51

oHEs'

13.90

8.50

*E52..

31 .. 36

12.77

0 /

Table 3.2

EH",

_(A

(;,.1

-

C.

Co.

_{8ft )}

o

-

-q ka:: ka~ o eUIl

4.07

4.00

<:>

eU'

1

_18.93

_10.36

OEHJI

10803

7.54

oEHJt

_{15 .. 54}

9.39

o1H1S' I

19.14

10.42

•

_E"'11

_Sf!

(~'")

_S(

L

_0.8'1)

e

~

--• ~

e

E." II

7.15

6.37

e

£"31

13840

8.72

Table 3.3

-27-EU",

Ct(~~)

-

~(o.8o'J

o

WI.

-q _{ka = ka~} 0 E.~II

3.45

4.606

oE"',t

_15.17

_9.66

0 e~l'

_{8 .. 85}

_7.38

e".Jt.

0

_15.53

_9.78

E".ll 0

23.68

12 .. 08

EHS\ 0

17.11

10 .. 26

• _EHs2.

20.98 ,

11.36

o . ~

. E"

S~

(;J:

Sc

(0.806)

e

\"10k

e

E"'"

5.64

5.89

~e~l~

_20.28

_11.1S

&HJJ ~ .

11.14

8,,28

.

E"j~ Q ..

29.38

13.45

~ EUs1

19 .. 13

10.85

Ta,ble 3.4

e,

:0.9180

ez:

0 .. 806

","Eft.,...

e

.

G.

f

(q, )::

G,.'

(0.9"0)

q ka ... ka ... 1· e"'Eu

_0.743

_1.f378

e

~E.,1..

18.93

9.48

HE:iot

e

.

_5.23

_{4 .. 98}

e

~E.J1

15 .. 54

8.59

e.

HE.u

29.29

11.79

~

I+

E

5I

_13.58

_8.0.3

• _{Q. HEst}

20.98

9 .. 98

".

"

t+e..,lM.,

_Se'(~,

_}:

Sel

_{(o.Cjc80 )}

0 0

}4E

II

4.05

4.38

0 I+E.,JI

9 • .30

6.64

t+E:! I 0:., "

17.66

9.16

Table

3 .. 5

-29-Remember that the cut-off,frequencies kac:obtained from· the zeros of

the }I!athieufunctions

Se(~,J

and

C't

(~'1)

are equal to

(kGl),:,(k4

'l.)'f

sothat

The cut-off frequencies of model (a) and (b) have been given in

tables

3.1

to

3.3.

of model (c) in tables

3.4

and

3.5.

3.4

The phase factor (\1:

~/i("

As .may be well-kno .... 'n \'fe find the phasefactor

~f:

(\!ko

by solving

an eigenvalueoroblem, for instance for the, even modes

~~-<J2)

-

AO. -

<;.P)

J

b

=

0 (3.1)

as a . function of the frequency ka [2] • First of· all i'Ie have to

remark thHt the matrixAB is a function of the :factor q,,:::;~o('-f)',

and the matr;ix CD of

(k

and~o. Comparing this "Jiththe aniso-'

tronic case. we see that we do not have a plire eigenvalue~roblem

]::)eca.use weh~v.e> to ,knQl'lf the eigenvalue " t o comnute

<t,

for a

given ~o ( or qofor 8 given ~( ). So we cannot make use of the

numerical nrocedure NaJSYMEIGlliNVALUES as has been done. in the

anisotropic case (

[2],

p99), but only by computing the zeros of

the determinant.

\'lhen lre have found these eigenvalues we :1ave t9 transform them to

'. more understandable figures .vith the relations (3.16), (3 .. 17) and

0.18):

~

-30- .

(3.39)

In l1"ig.

3.5

we have shol'm the transforrnation to the, ( ~f , ka)-plane

of some straight lines in the (

A

,q)-plane.

, c: 1 ~ ~~---=~---o+-~----__ ~ ______________ _ ,0 CIO s· o 10

_"

. --t

kh.

"100

-

---

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A

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t

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-

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----+

- - --

).:. 100 1 ₀ 0 1 10 S 10 IS" 0

---"'i.

~kh. Fig.

3 .. 5

-31-,

-~'dth these illustrations we liould demonstrate the complexity of the

trans~onnation. Curves bending t,o

p':

1 in the (

p',

ka) plane are

des cended from comnlete different tyoes of curves in the, ( ).. ,q) ulime. For- the antennas (a). (b) and (c) we have comnuted the eif!8nvalues and their transformations to the nhasefactors for the lo\,rest modes • Vie have also given the solutions for the anisotropic case (Figs. 3.6 to 3.20 ).

Antenna (a) Figs. 3.15 and 3.16

-'This antenna has been designed to be anisotropic for ka!Y10 ( Fig. ,3 .. 1). 1r,nen l-re compare'the phasefactors, for the even and odd' ciominant modes

( Figs. 3.15 and 3.16 ) idtheachother and ldth ,the phasefactor for the anisotropic case ( Fig. 3.14 ), "fe observe ,that anisotropy is reasonably good aChieved. The curves of the even and odd mode cut each other at ka~ 10, but they also cut the anisotropic curve at that frequency. So we 'may expect good results for the p:r:'onerties of the radiationoatterns.

Antenna (b) Figs. 3.17 and 3.18

,Antenna (b) has been deSigned for a nruch higher frequency. namely

ka~19.'I'hisiS

rather good demonstrated' by Figs. 3 .. 17 and.3.l8. Here,' the modes also coincide. but the anisotropic behaviour has not been pointed out convincingly due to a deficit of information in Fig. 3.14. Antenna (c) Figs" 3.:20 and 3.21

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:F'ig. 3.21 is n~t com"lete, we I"'..d.ss here the dominant oEHu mode. According

to table 3.4 the cut-off frequency of this mode is kat:' 4.04 (i.e. ~ =3.45),

but even very extensive computations around this frequency gave no result. Due to a lack of time-- the Burroughs computer was out of order in the

period 28/7/1975-18/8/1975-- it ,~s not possible to investigate this

pro-blem more profoundly.

For later research it is aqvisable to use the technique of Gershgorin's

diSks~]for finding the eigenvalues, which not only reduces the

proces-sing time but also gives intervals where the eigenvalues must be found.

Finally. observe that in Ileneral the _{• ..> e} EN .. and the ₀ EMil modes are the dominant

modes, and not the olH'1 _{mode as might be exnected from the anisotro...,ic}

study

[lJ ..

3.5 Radiation...,atterns

The preceeding chaoters shO'l<red that l'Ie obtain an almost anisotropic lfave-guide by designing a correct depth of the grooves. So 'fe may expect rather good results for the radiationpatterns.

Antenna (a)

In Figs. 3.22 and 3.23 we have given the radiationpatterns of the even and

,odd dominant mode, "lith kat'::: 10.64. We observe that the radiationpatterns

are not equal, although the deviations are small. This may be caused by the

following reasons:

(1) For this de1"lth of the grooves, the ,,,,ave guide becomes anisotropic at kat:: 9 .. 5 for the even modes and for the odd l"1odes at ka,:: 10.3. There is no frequency .... 'here for both the even and odd modes the wavejruide is

anisotronic ( see Fig. 3.1). vie have chosen ka,:::10.64. tihich is a little bit high, but a lower frequency causes difficulties because then 'fa

,

.

-49-nass a II crossingH with a hi5!.her mode.

(2) As we have seen in Figs. 3.1 and 3. 2, andv.rhat l'lill be confirmed

by follo'-!ing radiationl')atterns, the frequency for "rhich the even mode is anisotronic a"'1")roaehes the frequency for ",hieh the odd mode is anisotropic when ka is increased. Thus the larger the

,

aperture ( in wavelenghts) the better the anisotropic waveguide

is approximated. So we may cqnclude that ka

=

10 is still too small ..

In Fig. 3.24 ,.re have given the -10 dB angle of the E-plane and H-plane

for both modes as a function of the frequency. Of course, both inter-. sections should annear for the same frequency, which happens here at

Antenna· (b)

This antenna has the e,ccentricities

e

I '= O.~76g and e %." P,,$h. sothat

. . - .

it becomes anisotropic at kat "= 1~.9 for the even modes and atka, ': 19.8

for the odd modes. Thus for these frequencies the antenna is large (in wavelenghts).

In

Figs. 3.17 and 3.18 it appeared that the phasefactors for both,modes

are (allnost) equal for a rath~r. large frequency domain.

Figs. 3.25 and 3.26 show the radiationpatterns for the even and odd

modes respectively, at a ~requency ka,~18.8. Both diagrams are equal,

with a sidelobe level lower than ~25 dB.

Compared ,'lith the diagram for the anisotropic case ( Fig .. 3 .. 27) we observe that there is a good agreement between the anisotropic and the corrugated waveguide radiationpatterns, although there is a slight deviatton for the major-axis nlane.

In Figs .. 3.2$ a1).d 3 .. 2Q we have nlotted the radiationnatterns for ka,~ 17.A.

#

-50-In

Fig. 3.30 we have given the -10 dB angle as a function of the

frequency ka:t ., The intersection of :the curves arynears at ka,~ 17., 9., ,

Antenna (c)

For reasons mentioned in section 3.4 the investigations of this antenna are not cornrylete. We can only cornnare the results of ' the even modes with

those of' the anisotropic case II

[1]

1

2].

In

Fig. 3.31 we have deoicted theradiationpattern for the even mode

e

E'-" _{. .} of the corrugated waveguide at ka _.

.

=7 and the safJJ.e has been done

for the anisotropic waveguide in Fig. 3.32 ( copied from[2]) .. The

pat-, " , - , . ..

terns equal, which could be expected according to sect~on 3.4.

For later studies we have given in Fig. 3.33 the -10 dB angle as a function. of the frequency ka1..

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-63-Conclusions

The study on the ideal - a.nisotropic - ellintical \'laveguide has been started. In [2

J

and

[6]we

have shown that the experimental results of [ 1 ] do agree with those of the anisotrO"'lic study. of [2

J ,

sothat a corrugated ellintical waveguide could become an anisotropic one when chosen the narameters correctly. Also. in [2] .. fe have extended the theory to the corrugated waveguide. but no numerical results were available. In this report we have worked, out the numerical part of the study and set up all the computer programs to compute the desired features. Also Vfe have studied the problem of "mode-crossing!! again. To the latter we may conclude that there is no reason to as-sume that the curves of the phase factor for the anisotropic and cor-rugated "laveguide are not correct. This means that the curves do not intersect, as vIe are common for the corrugated circular waveguide. To our oninion there is only one l'lay left to justify this statement, and that is to confirm the numerical results \dth eXneri!:1ental ones. Thus \~ should measure the nhasefactor for the corrugated elliptical Have ?'Uide ( see also

[8]

t p

113).

The main conclusions of the rem:dning nart of this re'ortare: (1) the corru,!!ated ep iY)tical ,·!aveguide should have the geometry

as sho't:rJ. in

[2].

thus the depth of the grooves is, not constant when measured nernendicular to the waveguide wall.

(2) the dominant modes in the corrugated elliptical waveguide are the e EH" and _o EHu mode _•

(3) both dominant modes become anisotropic with a depth of the grooves ,ofhich is aODroximately the same. This denth may be determined

previously from the first-term approximation.

(4)

the larger kat the better the anisotropic nature is established. For later research we recommend to extend this study to the

exper~men-i

• J

.\0

t

··-64';"

tal part by constructing model.s (a), (b) and (c) as ~ood as '!)ossible

\ .

and verify the nUlTI.erically obtained radiationnatterns. It is also imnortant to ascertain the differences for the even and odd modes in nt'actice •

i '

,

'!

-65-[lJ

JoK.H. Jansen and H.E.J. Jeuken

Prona.p.;ation and radiation nronerties of e111pticH1 Haveguide with anisotronic boundary

ESTEC contract No 1657/72HP [2] L.F .. G. Thurlings

E1lintical waveguide as a feed for circularly polarised ... laves Afstudeerverslag ET-2-1975

[3

J

Dlrroughs information BJ3/1430/1431

Library. ~pt. of Hathernatics, group Numerical Analysis,

University of Technology, Eindhoven

[4]

Al-Harriri

Low attenuation micrO'I'Jave waveguides

~.M

..

Co London 1974

,

J.H. Hilkinson

The a1p.abraic ei~enva1ue nrob1em

Clarendon Press. OXford 196$

[6] ivr.E.J. Jeuken and L.F.G. Thurlings The- corru,~ated elliptical horn antenna IEBE Symposium 1975, Illinois

[7] H.E. J. Jeuken

Frequency-independence and syrmnetry properties of corrugated conical horn antennas "'lith small flare angles

Thesis 1970.. Eindhoven University of Technology

[8],

r:".J. Al-Hakkak and Y.T. Lo

Circular .<laveguides and horns "'ith anisotropic and corrugated ' boundaries