Design and performance of an unequally-spaced slot-antenna array

Design and performance of an unequally-spaced slot-antenna

array

Citation for published version (APA):

Sastradipradja, S. (1966). Design and performance of an unequally-spaced slot-antenna array. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1966

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TECHNICAL UNIVERSITY EINDHOVEN

DEPARTMENT OF ELECTRICAL ENGINEERING

.",::~?-,,~ ~~~1.~,=·.;>"~=t:·~"·-·-'

[fl.·.· stud.iet-JibUoLheel\

~

j Elektr ot8chnJel: \1

L . '_",'':'__'',.,,,.-,,._---,_.~':: k.~_..

DESIGN AND PERFORMANCE OF AN UNEQUALLY-SPACED

SLOT-ANTE1mA ARRAY

by

Sastyatmodjo Sastradipradja, B.E.

1 • 2.

3.

4.

CONTENTS. Introduction

The General Array Theory

The Radiation Diagram of a Linear Array. 2.1 The Uniform Array

2.2 The Visible and Invisible Region

2.3 The effect of exciting the elements with progressive phase shift.

An Array of N isotropic sources with unequal spacings. 3.1 The Analysis of the Array problem using Poisson

summation formula.

3.2 Normalisation of variables.

3.3

The lIu -diagramll•

The sinusoidally spaced a~ray with equal excitation.

Page. 1

3

3

4

5

8

8

10 12

5.

The Design and Construction of an Unequally spaced Slot-Antenna Array with Uniform Excitation.

5.4

5.5

5.6

5.7

5.8

The mathematical length p.

The shortest distance between slots: d k• The determination of d

k and the choice of the waveguide.

The conductance distribution between the slots. The length of the slot: 1 .•

~

The width of the slot: w. Transforming pieces.

The construction of the antenna array.

13

14

14

15

18

20 20 20 21

6.

7.

8.

9.

The Antenna Performance tests. Discussions Conclusions Acknowledgments References Appendix. 23

24

27

28

INTRODUCTION.

Lately Unequally Spaced Arrays have received a considerable amount of interest because they offer:

I The possibility of controlling the sidelobe level by manipulating the positions of the source.

II The possibUity of achieving high resolution with a reduced number of elements.

III The Unequally spaced array has a broad bandwidth.

Various methods are available to solve the unequally spaced antenna array problem. A~ong these are: the matrix method, the perturbation method, the array analysis with the aid of computers, the approximate integral technique, the analytical approach using the Poisson summation theory and the probabilistic approach.

The purpose of this research is:

a. To find out the problems connected with the realization of an unequally-spaced slot-antenna array based on the analysis using the Poisson

summation formula as proposed by Ishimaru [2J •

b. To study the practical performance of the above mentioned slot-antenna array.

1 •

1. The General Array Theory.

The far field of a linear array of N identical radiation elements with arbitrary excitations and spacings can be expressed in spherical coordinates as: N = C. F 1 (e,~) 1:

i=1

jks. sin

e

1. e' ~ ~

where C is a constant determined by:

1. the medium in which the wave is radiated,

2.

the frequency, and

3.

the distance of the field being considered from the array.

F1(

e,~) and F2(e,~) are the corresponding space factors of the radiating elements.

1.

~ is the excitation coefficient of the i th

element. This represents the relative amplitude and phase of the current in the i th element. is the position of the i th element with respect to a reference

point

O.

e

is the angle made by the perpendicular axis of the array and the direction of observation.

~

is

2~

, the wave number of the medium.

Distance of the i th slot

to a far~int in the direction

(e,~).

,

\ \ \ \ \ \ \ \ \ I~ ...;I:L - ) +

-j;=:===~S2.=-====~»)_Si.

---_1

o Fig. 1

2.

The expressions:

N

jks

i

sin 6

E'(6,<1»= F

1

(6,<1»

_{. 1}L;

I.e

₁ 1=

(1.2a)

N

E ' ( 6, <1»

= F2 (

e,

<1»

L;

i=1

jks. sin 6

1

I.e

1 (1

.2b)

represent the corresponding space factors of the array.

The second factor of the above expressions depends on the excitation and

configuration of the array and is called the "field array factor",

denoted by: <l>f(6 )

jks

i

sin 6

I.e

1

<1>1(6) is the field radiation pattern of a similar array made up by

isotropic sources.

From equations (1.2a) and (1.2b) we get the power pattern:

~

] IN

jks i

sine~2

p(e,<I»

_{=lIF1 (e,<1»!2 +IF2 (e,<1»\2 • i:1 lie}

1

(1.4)

=

P

(e, <1»

e N L;

i=1

jks i sin e

f2 .

I.e 1

P

(e

,<1»

is the power pattern of the individual elements, and the "power

e

array factor" is

I

N

ejkSi sine \'2

<I> (e)

=

L; I.

p .

=1

1

The power array factor gives the power radiation pattern of a similar

array from isotrppic radiating elements.

The above equations demonstrate that the space factor of an array can be

considered as a multiplication of the element factor and the array

factor. The radiation pattern analysis of a general array can thus be

simplified by considering separately:

a.

The radiation pattern of a similar array of isotropic sources, and

b.

The radiation pattern of the individual elements.

3.

2. The Radiation Diagram of a Linear array.

Consider a linear array of N isotropic sources, the spacings and excitations are both uniform.

The radiation pattern of this array is represented by: N E'(8)

=

l: 1. i=1 jkdi sin 8 e = N l: i=1 I ,eji<p <p

=

k.d.sin 8. Now " 2 .(N1 1 ) E(8 )= (1+ e J<p + e J <p + ••• +e J - <p ) Equation (2.1) can be rewritten as:

" 2 '(N 1)

E(e) = (1+eJ~+ e J <p+ ••• + e J - 'P)

where

E~e)= E'~e)

is the normalised IeJ<p field.-(Iej<P

~

0) (2.2) sin

2

2 , N<p s~n -2 11-e jN<p

I

11-e

j_<P

_I

-I·"\, -

...

-/ / / / I f d - " .\ Q. 1., Configuration of a uniform array.

The maximum values of the above ex-pression = N and occur at

Fig. 2

<p = .:!:. n.2lt , thus when:

We see that nulls occur when

sin 8=

I

sin

~<p

I

0,

I

sin

~

I

n.2lt kd n= 0,1,

2,3, •••••

i.e. when

T :::

N(J) .! m.1t or ::: +

4.

m.2n ._{N ,m::: 1, 2,}

_{3, •••..}

(N_{- 1 .})

The pattern between the nulls are called lobes.

-;u

Nain lobe

o

Grating

side lobe

1\

E(e) - ~ diagram of a uniform array, N :::

4.

Fig.

3

Th ll~-diagra~ll shown in Fig.

3

represents the radiation pattern as a

function of q>::: k.d sin

e.

The true radiation pattern (in space) can be derived by using the relation be tween ~ and

e •

2.2. The Visible and Invisible Region.

---~

In prac tice _900 ~

e

~ + 900 or -kd::;;

e

~+ kd is the relevant region of the array and is called the "visible region".

e

.

q>A. J ]>2nd A.

::: arc. s~n 2ltd • If q> A. or

Ie.

2nd

I

>

1 an imaginary value is obtained for

e.

This is physically unrealizable.

5.

The radiation pattern belonging

tol~»2~d

is then said to lie in the "invisible region" of the array. It is obvious that the size of the visible region is determined by the value of d/A • (See Fig.

5).

Tr"'v'eLU,,'3 Wt>.ve,

~

o _{d. - _ , , - -c:I.} - - - - i

-

----•

Equally spaced array with progressive phase shift feeding.

Fig.

4

Sometimes the radiating elements are fed by a travelling wave which propagates in the direction of the longitudinal axis of the antenna array. A slot-antenna array is an example of this practice.

If ~ is the propagation constant of the travelling wave, we can express the radiation pattern as

N

E(e) = ~ I e-j~diejkdi sin e i=1

N J'i(kdsine-Ad) N , . ,

= ~ I e t-' = ~ I eJl.<p

i=1 i=1

An examination of the expressions (2.2) and (2.3) indicates that both have a similar radiation pattern.

Ascp' ;;: k.d sine - ~d,

the principal maximum occur when: k.d sin e

-

j3d ;;:

o.

0 Thus

1L

sin e ;;:

Ji

;;: 0 k _f.. g

6.

(2.4)

o We notice that the principle maximum of (2.3) is rotated by e

o

with respect to the principal maximum of the array pattern as stated by (2.1). e is the t i l t angle of the main-lobe.

o

Fig. 5 illustrates the graphical method of deriving the polar radiation pattern from the cp -diagram. Fig. 5a shows the case when the sources are excited in phase, whereas Fig. 5b is the case when the array sources are excited by ~ travelling wave. Fig. 5b also shows that the boundaries of the visible region is situated at :

cp 1 ;;: (kd - ~d)

As Icp

21>

1CP11, the boundary of the visible region is set by

· IE

(&) \ :<.ll: I~' I ~ 1 I I I I I I I I 1I

4~~---:---_"':::::::::~~:::::=---"""''--li':'''\Vi ~

ibIe

Lr-V15ib

~;"~:=====::::::::::

__

kcL__- -_-_-"::"-i"J-i\'-l-"'-oot '('

et'

01"\

---1'---

r c-C}ib't\

r<!.jiOl'\ .~iJ' I I t{::4 a b

Derivation of the true radiation pattern from the 11~ -diagram. Fig.5

0.1a)

8.

3. An array of

N

isotropic sources with unegual spacings.

3.1 The analysis of the array problem using the Poisson summation

---formula.

Let us consider a linear array of N isotropic radiators with unequal spacings. The radiation pattern is represented by

N

jk s. sine E(e) = L: I..e ~ i=1 ~ jks(i) sin e e or by using N E(e)= L: I(i). i=1

f(i) = I(i).ejk.s(i) sin e

N

E(e)= L: f(i) • i=1

0.1b)

The Poisson summation formula is

00 i=-oo • dv • _00 Let us define: f(v)

=

I(v).e jk s(v) sin e such that i t is continuous and f(v) = f(i)

at i = 1, 2, ••••• , N.

k

·

<1

For - I(i) =

O.

In order to preserve the continuity ofI(v) we

ir~Ner

define that

. 0 .. 3)

0.4)

I(v)

=

0 { V

<1 -

C for v

>

N + c

o

<

c

<1

Applying the Poisson's summation formula and the above defined f(v) to 0.1) we ge t : E(e) = 00L:

J+C

f(v) ejmnv• dv m=-oo 1-c or 00

1+

C I ( v ) • e j { ks ( v ), sine

+

m1tV} E(e) = L: m=-= 1 C

O. 6a)

9·

The"source position function": s = s(v)

is

defi~ed

to give the position of the i th source when v=i;

i t is single valued and reversible.

so that the numbering of the i th source follows the relation: Hence and s. = sCi) ~ v = v(s) i = v(s.) • ~

The function v = v(s) is called the "source number function".

0.10)

v

s

4

s

₌

s(v)

3

v(s)

or

v ₌ 2

1

0.11) I(s)e jks sin

e

+ mnv(s)

~:.

ds • 00 E( e) = Fig.

6

As a function of s eqn. (3.6b) can also be written as:

s(N+d

f

m=-oo s (1

-d

This can further be written as:

00 where E(e) = L: E (e) m m=-oo s(N+d E (e) =

1

m s(1-d I ( )s • ej k. s. sin

e

+ m1tV(s) d vd s . ds . Equation (3.13) is the radiation pattern of a continuous line-source with an excitation distribution: I(s).ejm1tv(s)C 2J • (3.14) Equation (3.11) is thus a transformation of the radiation pattern of the unequally spaced array into a series radiation patterns from con-tinuous line sources.

3.2 Normalisation of variables.

To simplify the computation of (3.11) the variables "s" and "v" are replaced by normalised variables "x" and "y" respectively. These variables can be chosen such that the boundaries of the integration assume the values -1 and +1 j Le. -1 < x <+1 j -1 < y< +1. [6J

In order to meet the above conditions, the normalised variables are defined as follows: v

₌

(N-1 + _{c ) Y} + (N +1 ) 0.15) 2 2 s(N+d - s( 1-d s(N+d + s(1-d _0.16) s

₌

_{2 X+} 2 As

0<

c

<

1 , we may choose c

=

i

and get

and s

=

s (N

+i )

2-s (~~).x 0.18a) or s=px+6 where _p

₌

0= 2 s(N+-t) 2

- s(i)

+ s(i)

In a symmetrical array i t is convenient to choose x ::;: 0 to coincide with the centre of the array configuration, so that 6::;:

o.

2p

=

s(N+i) -

s(i)

is the "mathematical length" of the array, and I I I I I I I I 1,1;-1 "><1 X ef\~th I I

I

_ _ _ ..1

,-I I I I I I ,.) -1' ",

L ::;: s(N) - s(l) is the "true length" of the array.

:(

_ _ _ _ _ _ _~ L _

Using the above defined normalised variables x and y, c

=

t

and adopting

the notat~on for u

=

kp sin e

equation (3.13) becomes:

m=-oo /

1 d j{ux + mN~ y(x)}

I(x)

d~e

dx.(3.20)

-1

The contribution from the mth line source to the far field pattern "' <3.21) jux .e • dx • is then: E (u)=..1-

fI(x)~.e(jmN~y(x)}

m 2__1J dx

It can be demonstrated that the main contribution to E(e) in the vicinity ofe

=

0, Le. u

=

0, comes from Eo(e), and E..

1(e) and E+1(e) add small corrections. The maximum value of E (e) occurs near

m u = -mN ~.

For the rigorous demonstration to the above points, the readers are referred to the papers of Ishimaru [2Jand of Meyers [6J •

3.3 The radiation pattern E(u) as a function of u = k psin

a,

the

"u-diagram", can be calculated. The true space ~adiation pattern can be constructed from the u-diagram by the same method shown in Fig.

5,

paragraph 2, noting that the circle diameter D

f

2k~

When the array sources are excited by a travell~ng wave with propagation constant

~

, the excitation can be expressed

by~

I'(x)

=

I(x).e-j~px

• <3.22)

The far field from the mth source is then:

Evidently

E ( ) - ..1- / 1 I(

).'!l

jmNrcy(x) j

(u-~p

)xd

u - 2 X d .e .e x

m x

now the main maximum occurs when (u - ~p)

=

0, which means that the maximum is translated by ~p. This translation is to be compared with the translation ~d of the case discussed in paragraph 2.3.

12.

4.

The sinusoidally spaced array with equal excitation.

Let the position of the sources be sinusoidally distributed according to: y = x + 2A sin 'ft x (4.1 ) +1 'mN ( 2 A . ) . =

-t

I

(1 + 2A cos 'ft x) e J ,'ftx-+--:jt s~n

we

e JUx • dx. -1 E (u) m

If the sources are excited equally, I(x) = 1.

The contribution of the mth line source to the radiation pattern becomes:

For this type of array having 21 elements, Meyers made several numerical evaluations of E (u) end E(u) for different values of

m

A.

[6J

An unequally spaced slot-antenna array with equal excitation and a 2 x 0.1

source distribution of: y

=

x + sin 'ftX, is constructed for 'ft

the following purposes:

1. To study the possibilities and problems of constructing this type of unequally spaced array, and

5.

The Design and Construction of an Unequally Spaced Slot-Antenna Array with uniform Excitation.

The designed and constructed slot-antenna array has the following specifications:

- The array should be constructed on a waveguide and thus is fed by a travelling wave.

The source distribution function is y

=

x + 2A sin Ttx

- The value of A is chosen to be 0.1, so that 0.2 .

y

=

x + - - Sln Tt x.

11

- The array is uniformly' excited, so lex)

=

1.

The radiation patterns E (u), E 1(u) and E 1(u) obtained from the

0 +

-computation of equation

(3.25)

with A= 0.1, are presented in Fig. 14. Based on these pattern our array was designed.

(A.5 I usa.

-

U

invisible re1lot)

I,

\r'\v',sible 1"'e~l,,1"\

I

I

I I I _I

I

I

I

I I I

I

I

I

_I

I I I I I I I 1 _I I

Relation between mathematical length pand the boundary of the visible region u •

14.

Let the boundaries of the visible region be u

s1 and us2(Fig.8). The mathematical length p is determined by the value of u_s1 "

Introduce the constants C

1 and C2 which fullfil the relations:

for u s1 = -c1N1t

e

= -

90

0 and for u s2 = c2N 1t

e

=

+

90

0 jc 1N1t

1

>

jc 2N1t

I·

As the array is fed with a travelling wave whose propagation constant = ~ , we have k P sin

90

0 + ~p := c 1Nit yielding c₁N1t p= k +~ (5.4)

With k := -,-;21t ₌ 21t sin e A ( _{e is the tilt angle of main} o=A.g

1\.

Ag

0

lobe) • (5.4) can be rewritten as:

c

1N sineo p =

2

"

1 +sin e A. g

0'

The shortest distance between slots: d "

--- k-From equation (5.6) (4.17) we have: N N + 1 v

=

2.

y + 2 Y = ~ (v _ N + 1) N 2 or 2(. N+1) Yi

=

N

~

-

2 •

Also from (5.1) and (5.6) we get so that

Xi + 2: sin 1t xi

=

~

(i _ N ; 1) •

As x := 0 is the centre of the array, the smallest value of x. is

~

x

_k

=

x

N+1

( 2

+ 1) Combine (5.7) and (5.8): for N = odd

~

[N+1 + 1 _ N +

1J

~ N 2 2 (5.8) 2 = N N

=

odd.

15.

We see that

The shortest distance

1 + sin e A.g • 0 in

(5.11)

gives sin e o Applying

(5.5)

and

(5.9)

d k := c,{

1

AN (7tx k ) } (

1

) A.

(5.12)

sin

1

7t

₁

g + sine~

If we want to eliminate the first grating-lobe, then c

<

1.

Also AN

1

-

sin(7tX k )

<

1 •

7t and

₁

0

1

<

~

1

+ sine 0 Hence

-

d k <~ A.g•

(5.13)

5.3

The determination of d and the choice of the waveguide.

---

k---Distances between the slots.

Fig.

9

The freedom of determining the shortest distance between the slots

is also restricted by the following physical and electrical requirements. 1. The physical length of each slot in resonance is about 0.49 A •

In order that the slots do not overlap one and another i t is necessary that

16.

2. To aV~id electrical interaction between the slots d

k has to provide adequate electrical separation. Let this requirement be formulated by

3. By considering the equations from (5.13) i t can be shown that for this type of array with N = 21 and 0

<.e

<

900, the minimum slot

o

This sets an upper limit to the choice d

k

values of 2(r-) with (qN) as parameter.

AA , we will 1ind the top limit g (Fig.16). distance d k

<

0.45 Ag (5.14) implie s tha t

..1..

<

2 ( dk) • of

.2L

Ag Ag

A •

g

Using

e

as variable plot the o

Superimpose on it the graph of

A

for

x=

_g for a particular

(sN).

undesirable as i t makes the array to approach

A

On the other hand

A--

should not be chosen too a fixed

A

means thagt a smaller waveguide should be A large value of

~

is . g an end-f~re character. A s~all.

A

smaller ~ at g

used. This in turn will present the difficulty in bringing the slots accurately on the waveguide as small variations of the slot displacement

2 Ttr will greatly affect the conductance (see par. 5.4 Y=sin ( - )

r a

9250

MHz.,

the standard X-band waveguide

To obtain

~p.;

waveguide is

A

for ~

=

0.7 suppression only starts

g

at u

>

21Tt • This is objectionable as c ,

>

1.

Sl

If we select qN

=

20, i.e. suppression starts at u

=

20Tt , we will

A

~

be able to achieve it if

X-

0.59.

g

0.59 at f

=

0.9250

MHz.,

the most appropriate standard IEC: R-120.

For the operating frequency f =

(IEC : R-100) gives

~g

=

0.7. Examine Fig. 15. It shows that

The cross-section dimensions are: a

₌

19.050 mm b

₌

0.9525 mm A

₌

38.10 mm c f

₌

_7868.57

_MHz.

c

For the applied freq. f

=

9250 MHz.: A.

₌

32.410 mm A. 0.52568

T

=

g A.

₌

61.65348 mm g b Waveguide dimension. Fig.10 As sin

e.

=

0.52568 ~

e

=

320• o 0

Using (5.14) and (5.15) we get n

=

0.52568 ~ 0 26

'C(, 2 , . . . . , . .

Then d

k

=

0.26 A.g• This value is accepted as a compromise.

The value- of p can now be evaluated from (5.5): p = 223.05136 mm.

The positions of the slots: d. = x .• p (mm), where

~ ~

x. is the root of:

~ 2A sin It X. , y. =

x.

+ ~ ~ It ~ 2e- N + 1 ) y. =

_N

_~ 2 • ~

18.

For

A ; 0.1

and

N ; 21.

the smallest distance is determined by:

i.e.

; (0.0795

x

223.05136)

17.73258

mm

mm

The length of the slot ~ 0.484 A ; 15.7 mID. This means that the smallest distance between the slot edges ~ 2 mm. From the point of view of

machining the slots in the waveguide the value, of d

k found in (5.18) is acceptable.

The positions of the slots in the array is shown in table 1. of the appendix.

5.4 The conductance distribution between the slots.

---I"<,~--.e. ~,

4:-.

---4"*~-"-w

,.. ::::t:""l'

--"1'"

The slots are to be brought on the broadface of the waveguide. If r is the displacement of the slot from the waveguide broadface centre line, then the normalised conductance of each slot is according Stevenson's formula

[14J

a n.r, J. . 21 .sJ.n \ . -2 nr, ; G • sin (-..2). a a Fig.

11.

Slot dimension

The values of r, can be computed if

G.

is known.

J. th J.

Let Yi be the leakance of the i slot.

Y. can be computed by considering the array as a transmission line J.

with the slots act as shunt loads

[14J •

Assume that the attenuation in the waveguide is negligible. Let the array be closed by a matching load L.

- - . , . . - - - r - - -

-_ -_....l.--_--J.. _

y.

l.

L

The radiation efficiency ~ is defined by: p rad 11 - x 1000,,6 ,

P. -P

_L

~n where

p. is the input power;

~n

Transmission line equivalent of the array.

Fig. 12.

P d is the total radiated power in slots ra

P

L is the power dissipated in the load L.

Gi

Suppose p. is the radiated power from the i th slot. The required leakance of the i tE slot can be computed from:

p. Y1·(2:.) P1 P1 Y i = L: Pi

As each slot is supposed to radiate the same amount of power, (5.21) becomes:

Y1

Y~ ;:: ( ) (see ref.

[3J

)

... 1- Y1 i-1 The choice of P

L will influence the individual values of Yi. A small P

L and large Yi can raise objections because:

1. The position of the slot with respect to the waveguide centre-line will require a large transverse displacement, so that the slot will be too close to the waveguide wall.

2. Large y. will cause undesirable reflections in the waveguide, which

~

can cause a matching problem.

However if P

L is too large, the variation of the slot conductances, is too smallt hence the variation of the slot displacements is also

too small. This will present manufacturing difficulties. This last point is especially objectionable if the number of slots, N, is large.

20.

Thus the choice of P

L is clearly governed by the dimensions of the waveguide used and the number of slots: N.

For our array we decided to dissipate 1~~ of the totale power in the load; hence

P

L = 1O'~ P.~n

n

= 90% • (5.24)

Using (5.24) and (5.23) the values of r. are computed from (5.19).

~

The results are presented in table 2 of the appendix.

5.5. The length of the slot: 1 . .

---

~

we can find the appropriate slot length:l .•

~

table

3

of the appendix. The resonant length of each slot, 1., varies with.its displacement from

~

the waveguide centreline. Empirical curves showing the relations between

r '

1. and (~) are to be found in various Microwave Antenna handbooks.

~ a

Having known the values of r. ,

~

The slot resonant lengths are tabulated in

5.6.

The width of the slot: w.

The slot displacement from the centre line is dependent on the dimension of the broad face dimension of the waveguide. The width of the slots has to be chosen such that the slot is entirely situated on one side of the waveguide centreline. For this particular antenna we choose the slot width: w

=

1.00 rom •

Note: The slot width normally used with a standard X-band waveguide is about 1.59 mm. The use of a smaller slot width will only have a very negligible effect on the resonant length of the slots. The effect will appear in a smaller bandwidth of the slots.

(

A

'[

(l,'4 _Q).1

- -

-

-

_{- -}

-

_-

f -

-\

The transforming piece •

21.

The antenna is employed in a system using standard X-band waveguide. For this purpose two transformers of waveguide pieces are required to connect the antenna to the load and feeding waveguide respectively. These transforming pieces should have the right length to prevent the introduction of any mismatch in the system. The length of the trans-forming pieces can be calculated by making use of the following:

[1 2]

[13J

[1

8 J •

D. n.(a 2 - a1) ;;: 2a 2 2a1

arC.COS{4

2 2a 1 2a2 ( - - - ) ( 1 + - - • A - ) Ag2 Ag1 a1a2 Ag1 g2 a

1 is broad dimension of antenna wavegu;i.de

A is A

_"

_"

_"

gl g

a

2 is broad dimension of X-band waveguide.

A is A g

"

"

"

g2 n is an integer n ;;: 1 , 2

,

3

...

With n ;;:

4

we get D.;;: _{9.52 cm.}

5.8 The construction of the antenna array.

---The antenna waveguide is built from brass plates of 1.25 mm thickness. The slots are cut by means of a fraising machine. The ends of the antenna are provided with

RG

52/U waveguide connecting flanges.

Foto nr. 2

Mounting the antenna on the turntable.

22.

Foto nr. 1

The constructed unequally spaced slot-antenna array assembled with the mixer (left) and the matching load (right).

Foto nr

3

23.

6.

The Antenna Performance tests.

The radiation pattern measurement is done by operating the antenna under test as a receiving antenna. A transmitter placed at a suitable distance from the receiving antenna acts as a source. The antenna is rotated and the relative amount of signal received by the antenna at the various positions are recorded by a pattern recorder.

To reduce any stray pick-up the antenna is mounted on the turn-table through a wooden frame.

The azimuthal radiation pattern measured is shown in Fig. 18 and Fig.19 shows the antenna pattern for Variable elevation angles and constant azimu th (

e

= 320) .

The input Voltage Standing Wave Ratio

of

the antenna is measured at

---different frequencies. The result is presented in Fig. 20. The measurement is done with conventional technique using a standing wave detector.

~~~_~~~~ of the antenna is obtained by comparing the performance of

the antenna under test with the performance of a standard horn which has a gain of 22 dB at a frequency of 9250 MHz.

~.

7.

Discussions.

The measured radiation pattern shows that the maine lobe occurs at ~

60

=

32°. This t i l t angle agrees very well with the predictions.

The first side-lobe level being -12 dB is higher than the anticipated side-lobe level of -16.5 dB.

The elevation pattern variation appears to be of a few dB only (Fig.19). The actual radiation pattern deviates form the expected one.

The apparent discrepancies can be attributed to various factors. One of the factors arises from the manufacturing limitations.

Deviations in the slot conductances exist as a result of the departure in the slot dimensions inherent to the inaccuracies of the manufacturing process. Consequently a variation of the slot illumination exists and this kind of variation is found to have a marked effect on the resulting radiation pattern [7J •

We may note that the designed slot-lengths and slot-displacements for our antenna array differ very little one from the other. This is especially so for the first ten slots.

Another factor causing the apparent discrepancies is the fact that the slots do not lie in a line parallel to the longitudinal centreline of

the waveguide. The peak of the radiation pattern of a slot is shifted further from the normal as the slot is further displaced from the longitudinal centreline of the waveguide. As a result the contribution given by each slot is not entirely equal to the theoretically desired value [17J •

The assymmetry occuring on both sides of the main-lobe is partly caused by the slot element factcr. The radiation pattern of a typical slot has been measured. Using this result as the element factor we can derive the expected antenna from the theoretically calculated array pattern.

Further factors contributing to the deviation of the antenna radiation pattern is the presence of mutual coupling between the slots. The antenna array is designed on the assumption that no mutual coupling exist. It has to be admitted that the value of d

k

=

0.26

A

g was accepted rather optimistically.

25·

The e ffec t of mutual coupling tends to heigh ten the sidelobes [

8 ] •

The measured radiation pattern shows that this phenomena occurs in the

constructed antenna.

The smallest distance between the slots, d

k, appears to have a deciding influence in the design of the unequally spaced antenna array. A larger value of d

k is desirable from the point of view of manufacturing convenience. A proposal to meet this aim is the employement of slots filled or covered with a dielectric material.

Experiments on dielectric covered slots have been done by author and an indication of the shortening of the resonant lengths is observed. As many more experiments should be done in order to collect more datas no definite conclusion on the characteristics of the dielectric covered slots Can be presented now.

It is necessary to investigate the effects on the mutual coupling if dielectric filled or covered slots are used for the array.

Another proposal to improve the antenna design is to employ "complex" slots. It is known that the phase and the amplitude of the radiated electromagnetic wave from a rotated slot is related to the angle of rotation of the slot [10].

This fact can be used as a means to control the phase of the slot illumination. It is then possible to get an unequally spaced array fed by a travelling wave and yet having a uniform phased excitation. For a normally travelling wave fed array the boundary of the visible

region occurs whenkP+ ~p

=

u

s1 • Since for a uniform phased array the term ~pdisappears, i t is possible to obtain a larger value for p , than those of an array with non-rotated slots and having tte same value for u

s1• Hence, a larger dk can be obtained.

Some possibilities offerred by the use of "complex" slots are:

1. The smallest distance between the slots can be made larger and hence reduce the effect of mutual coupling and at the same time provide more convenience to the manufacturing process.

26 •

3.

The boundary of the visible region can be chosen at a smaller value of u.

On the other hand the field radiated by a "complex" slot will have a polarisation which depends on the amount of slot rotation. This may present an additional problem to the "complex" slot array design. This problem can be solved by attaching a pair of parallel plates on

the broadface of the waveguide [11] • (Fig.14).

The result of the input V.S.W.R. measurement proves that the unequally spaced antenna array has a reasonable bandwidth.

Fig. 14

8.

Conclusions.

The possibility to conGtruct an unequally spaced slot-antenna array based on the Poisson summation analysis and using longitudinal shunt-slot is verified. Taking the influencing factors, such as manufacturing limitations and the presence of mutual coupling, into consideration the result of the antenna test agrees qUite well with the theoretically result.

The first sidelobe of this antenna appears rather high. The restricted manufacturing ability and the limitations in finding a suitable value

for d

k are major problems met when designing an antenna by this method. These problems are a consequence of the fact that longitudinal shunt-slots have been used in this array. The use of "complex" shunt-slots seems to offer a better alternative. Further investigations are necessary to obtain a solution to these problems.

This experiment was carried out under the directorship of

Prof.dr.ir.A.A.Th.M. van Trier in the Mircrowave Antenna Laboratory of the Technical University of Eindhoven, the Netherlands.

I would like to express my thanks to Prof.dr.ir.A.A.Th.M. van Trier and the other academic and technical staff members of the Electrical Engineering Department who have given me their full cooperation

to make this experiment possible.

I am particularly grateful to drs. M.E.J. Jeuken for his constant interest in this experiment and for making himself available for valuable discussions and instructive consultations.

For her assistance in typing this report I am very indebted to Miss W.M. Kuylaars.

REFERENCES 29. 1 • 2.

3.

4.

5.

6.

8.

9.

Dion, A Ishimaru, A. Ramsay, J.F.

&

Popovich, B.V. Allen, J.L. Ishimaru, A.

&

Chen, Y.S.

Meyer, A. Bailin, L.L.

&

Ehrlich, M.J. Allen, J.L.

&

Delaney, W.P. Oliner, A.A.

"Non-resonant slotted arrays".

IRE Trans. on AntennaS and Propagation, vol. AP-6, Oct. 1958, pp 360-365.

"Theory of Unequally Spaced Arraysll ,

IRE Trans. on Antennas and Propagation, vol.AP-10, Nov. 1962, pp. 691-703.

"Series Slotted Waveguide Array Antennas ll •

IEEE Conv. Record on Antennas and Propagation, 1963, pp. 30-55.

I~rray antennas a new application for an old

technique ll •

IEEE Spectrum, vol. 1, Nov. 1964, pp. 115-130. "Thinning and broadbanding of antenna arrays by unequal spacings".

IEEE Trans. on Antennas and Propagation, vol.AP-13, January 1965, pp. 34-42.

"Lineaire configuraties met plaatsafhankelijke spatiiringen van de stralingsbronne~I,

Afstudeerverslag Technische Hogeschool Eindhoven, groep ETA,> April 1965.

"Factors affecting the performance of linear arrays"~

Proc. I.R.E., Febr.1953, pp. 235-241.

"On the :E:ffect of Mutual Coupling on Unequally Spaced Dipole Arrays",

IRE Trans. on Antennas and Propagation, vol. AP-10, Nov. 1962, pp. 784-785.

"The impedance Properties of Narrow Radiating Slots in the broad Face of Rectangular Waveguide",

IRE Trans. on Antennas and Propagation, vol. AP-5, Jan. 1957, pp. 4-20.

1 0. Maxim, B • J • 11. de Brouwer,J.~.M. 12. Migliaro, N.J. .13. Johnson, R.C. 14. Silver, S. 15. Jordan, E.C. 16. Krauss, J.D. 30.

"Resonant Slots with Independent Control of Amplitude and Phase",

IRE Trans. on Antennas and Propagation, vol. AP-8, July 1960, pp. 384-389.

"De resonant gleuf als koppel element tussen een rechthoekige- en een tweeplaats golfgeleiderll

,

Afstudeerverslag Technische Hogeschool Eindhoven, ETA-5, Jan. 1966.

"Designing Tapered Waveguide Transitionsll ,

Electronics, vol. 30, Nov. 1957, p. 183.

"Design of Linear Double Tapers in Rectangular Waveguidesll

, IRE Trans, vol. MTT-7, p. 374,

vol. MTT - 8, p. 458.

II Microwave Antenna Theory and designll ,

McGraw-Hill Book Co., Inc., 1949.

IlElectromagnetic Waves and Radiating Systems", Prentice Hall Co., New York, 1950.

IlAntennasll ,

McGraw-Hill Book Co., Inc. 1950. 17.

18.

Jasik, H.

Harvey, A.F.

IlAntenna Engineering Handbook ", McGraw-Hill Book Co., Inc. 1961. "Microwave Engineeringll

,

APPENDIX.

Positions of the slots.

2A . y. = :x:. + - s~n 11 :x: • ~ ~ it ~ A 1 •

N

=

21

A =

0.1

p

=

223. 05136 rom

Table 1

i

x.

d.

i

x.

d.

~ ~ ~ ~

1

-0.9406

-209.802

mm

11

0.0000

0.0000

mm

2

-0.8236

-182.882

12

0.0795

17.733

3

-0.7118

-158.768

13

0 .. 1598

35.644

4

-0.6065

-135.281

14

0.2419

53.956

5

-0.5078

-113.265

15

0.3265

72.826

6

-0.4148

- 92.522

16

0.4148

92.522

7

-0.3265

- 72.826

17

0.5078

113.265

8

-0.2419

- 53.956

18

0.6065

135.281

9

-0.1598

- 35.644

19

0.7

1

₁₈

_158.768

10

-0.0795

- 17.733

20

0.8236

182.882

21

0.9406

209.802

A 2.

APPENDIX

The normalised conductance: G. and the dis~lacement r. of the slots.

~ ~ P load :: 10% Pin P_rad = 90% P. ~n T) :: _0.9 T)P1 _0.9 0.0428 Y1

=

_L:

= 21

:: P 1

Power radiated by ~.th slot

Y i

=

.th

Power incident on ~ slot

Y 1 (P i /P1 ) 1

=

~-i =

1-Y1 L: (P

m/P1) 1 -Y1(i-1)

ill::1

The normalised conductance:

G

i

=

{Z.09

%

~

G . 2( n:r . )

= .s~n

-~-o a

The antenna waveguide is IEC: R 120:

and a :: 2 b f = 9.250

GHz.

Thus G = 3.655226 • o ==;==::=.=====;::;:::= 2 cos (~x 0.52568 ) 2

APPENDIX

A

3.

Table 2.

. 2 (

Ttr . r. i

G.

s~n ---2:) .2:) _r. ~

a

a

~

1

0.0429

0.01172

0.03453

0.658

mm

2

0.0448

0.01225

0.03534

0.673

3

0.0469

0.01282

0.03618

0.689

4

0.0492

0.01345

0.03700

0.705

5

0.0517

0.01415

0.03795

0.723

6

0.0545

0.01492

0.03897

0.742

7

0.0577

0.01578

0.04010

0.764

8

0.0612

0.1675

0.04130

0.787

9

0.652

0.01784

0.04265

0.812

10

0.0698

0.01909

0.04412

0.841

11

0.0750

0.02052

0.04575

0.872

12

0.081

0.02218

0.04757

0.906

13

0.0882

0.02414

0.04965

0.946

14

0.0968

0.02648

0.05201

0.991

15

0.1071

0.02931

0.05476

1.043

16

0.1200

0.03283

0.05799

1.105

17

0.1364

0.03731

0.06187

1.179

18

0.1579

0.04320

0.0665

1.270

19

0.1875

0.05130

0.07271

1.385

20

0.2308

0.06313

0.08085

1.540

21

0.3000

0.08207

0.09250

1.762

A

4.

APPENDIX

The resonant lengtr~of the slots.

The lengths of the slots are obtained with the aid of the empirical curve shown in Fig. 16.

Table 3. i l. ~ 1 15.663 mm 2 15.664 3 15.665 4 15.666 5 15.667 6 15.669 7 15.671 8 15.673 9 15.674 10 15.677 11 15.678 12 15.682 13 15.686 14 15.691 15 15.697 16 15.703 17 15.712 18 15.723 19 15.741 20 15.766 .:?1 15,806

· j 0"

0.0

o

-·~T~·

:i

1· i' _{i ' '~}

; •• A ~-~'i

__

._-l~

_

.0-f I -J.

I

I

₁

T 1 :--~.. -~ --~ I CHART

I

A.

I

....

I

,-f- • -

---Unequally Spaced Slot-Antenna Array

,~-

.

---~ l--·--·-~·'·'" [ ~ .1 j • j