A shaped cylindrical double-reflector system for a broadcast-satellite antenna

A shaped cylindrical double-reflector system for a

broadcast-satellite antenna

Citation for published version (APA):

Verstappen, H. L. (1979). A shaped cylindrical double-reflector system for a broadcast-satellite antenna. (EUT report. E, Fac. of Electrical Engineering; Vol. 79-E-101). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1979 Document Version:

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by

E I N D H 0 V E NUN I V E R SIT Y 0 F T E C H N 0 LOG Y Department of Electrical Engineering

Eindhoven The Netherlands

A SHAPED CYLINDRICAL DOUBLE-REFLECTOR SYSTEM FOR A BROADCAST-SATELLITE ANTENNA by H.L. Verstappen TH-Report 79-E-IOl ISBN 90-6144-101-3 Eindhoven August 1979

Abstract

1. Introduction

2. The Prolate Spheroidal Wave Function

3. The Prolate Spheroidal Wave Function as Aperture Distribution

4. Elliptical Beams

5. The Cylindrical Two-Reflector System

6. Design of a Cylindrical Two-Reflector System

7. Conclusions Acknowledgement

References

Present address of the author: Ir. H.L. Verstappen,

Philips' Telecommunicatie Industrie B.V., Afd. Voorontwikkeling Radio,

HUIZEN, The Netherlands 1 2 7 16 29 48 56 68 69 70

1

-Abstract

In this report a study has been made of the design of an antenna which can produce a prescribed radiation pattern. By prescribed radiation pattern is meant here that the radiation pattern should fulfil a number of requirements given in the report of the World Administrative Radio Conference (Geneva, 1977).

Starting point in the investigations is the maximizing of the radiated power of an aperture antenna, in a certain area. The aperture distribution which causes this maximum appears to be a prolate spheroidal wave function. When using this aperture distribution i t is possible to fulfil the requirements with

respect to the level of sidelobes and the slope of the main beam, completely.

This aperture distribution has been applied in a double-reflector cylindrical antenna system. It has been shown that in such a way

it is possible to design an antenna which can produce an elliptical beam. The elliptical beam is caused in one plane by the line

source, which is the feed of the antenna system, while in the other plane the radiation pattern will be influenced by means of shaping the curvature of the reflectors of the antenna system. The first step in the shaping is the derivation of differential equations, which are determining the antenna system. These equations have been solved by means of high speed computations, with the shape of the cylindrical reflectors as a result.

In this way it is possible to design an antenna system producing an elliptical beam which fulfils the requirements given in the report of the World Administrative Radio Conference (Geneva, 1977). The final chapter gives a design of an antenna system fulfilling the requirements mentioned above.

Verstappen, Ii. L.

A SHAPED CYLINDRICAL DOUBLE-REFLECTOR SYSTEM FOR A BROADCAST-SATELLITE ANTENNA.

Eindhoven University of Technology, Department of Electrical Engineering, Eindhoven, The Netherlands. August 1979.

1. Introduction

During the past decade considerable research has been done on satellite communications. With the increase in the number of satellites several problems have arisen in the field of

satellite antennas and ground-station antennas. In both cases we are concerned with an interference problem. When decreasing the diameter and, consequently, applying ground-station antennas with less directivity, the interference potential increases, as

adjacent satellites and terrestrial radio relays may be operating in the same frequency band. For this reason requirements must be stipulated with regard to the radiation pattern of the satellite antenna as well as the ground-station antenna. Interference by the ground-station antenna with other satellites should be avoided

(see Fig. 1.1).

SAT 1

SAT 2

- -

-

-

-unwanted radiatioil

EARTH

Fig.1.1. Ground stations and adjacent satelies.

This means that requirements as regards the first sidelobes in the radiation pattern must be laid down.

a) .

b) .

3

-II ~ toO A

C(O) 32 - 2S log 0

where G (8) = gain relative to isotropic.

e

= angle in degrees off the main beam axis.

This formula applies only to the region beyond the first sidelobe peak, that is, at and beyond 8 (degrees) = 100 AID. In addition, it is never assumed that the reference antenna gain falls below -10 dB relative to isotropic.

o

< 100 A

G(e) = 52 - 10 log (D/A) - 25 log e.

The same remarks as in case a) can be made here.

On the other hand the satellite antenna should produce a radiation pattern which is optimized so that some area on earth will be illuminated as uniformly as possible. Outside this area there should be a minimum of radiated power. This means a high decay in the main beam and low levels of the sidelobes.

Several requirements concerning the radiation pattern have been stated in the WARe '77 Report [2]. A short summary is given here.

iii'

:s

c -;

..

~ c c II c ~ u >

~

'"

o

--10

- - - -

---20

--30

l.--"

I--" -40 -50 n,1 0.2 0,3

t---

_r-..

"

--\A

\

,

B

\

._-

_1,\

,

0,5 2 3 - ---- - f--- --

---I---- - -_.-_{l -} f---- f---- - -

---~

~=t-C-,.

_-

_1-1-

_f--

_-

I-'

r-5 10 20 30 50 100 Relative angle

(l)

Fig.1.2 Reference patterns for co-polar and

cross- polar components for satell ite

transmitting antenna.

Curve A: Co-polar component

-12 (..!..)2 8₀ for

o :: e

~ 1.58 80, -30 for 1.58 6 0 < 6 ~ 3.16 -[17.5+25 6 for 3.16 6 0 < 6 log (e)

1

a

after intersection with curve C: as curve C.

90/2

6 0,

5

-Curve B: Cross-polar component

-(40+40 log

I

Jl...

-8 0 1

I )

for

o

~ 8 ~ 0.33 80 -33 for 0.33 8 0 < 8 ~ 1. 67 80 -(40+40 log

I

Jl...

-8₀ 1

I )

for 1.67 80 < 8

after intersection with curve C: as curve C.

Curve C: Minus the on-axis gain.

It is usually a difficult problem to develop an antenna with a prescribed radiation pattern in two orthogonal planes. The

radiation patterns must satisfy the requirements given above. If this can be done an antenna with an elliptical beam can then be realized. A satellite antenna with an elliptical beam can

illuminate an elliptical area on earth, while with a ground-station antenna the interference potential can be reduced with an elliptical beam. The use of a cylindrical two-reflector system

(see Fig. 1.3) could offer a solution to the problem.

The system is composed of two cylindrical reflectors and a line source feeding the subreflector.

The great advantage of such a system is the possibility of shaping in two orthogonal planes.

In the first plane we can get an aperture field in reflector C which provides the desired radiation pattern by shaping the

reflectors. In the other plane the radiation pattern will be given by the radiation pattern of the line source.

Investigations on this point have been carried out earlier [3]. Another advantage of this reflector system is the fact that the aperture distribution will be synthetical impressed on the aperture of the main reflector. That is why it is possible to create almost any desired radiation pattern.

We will use a so-called spheroidal prolate wave function as aperture distribution. Investigations on this have been carried out by Borgiotti [4] and Mayhan [5] inter alia.

L----~-'-B

A= line source

B= cylindriCal reflector

( subreflector>

C=cylindrical reflector

(main reflector)

Rg.1.3. Cylindrical two- reflector system.

It is possible in this way to design an antenna which gives a radiation pattern with a prescribed form and low sidelobes. The principle can be applied to a satellite antenna as well as a ground-station antenna.

7

-2. The Prolate Spheroidal Wave Function

As stated in the introduction, investigations on this area have been carried out by Borgiotti [4] and Mayhan [5].

In this chapter a prolate spheroidal wave function as an aperture distribution applied to the system, mentioned in the introduction, will be discussed.

This means a one-dimensional aperture with size t.

We are dealing with the following situation (fig. 2.1.):

j,

L ...

I I I

I

r

I~

z

•

Fi9.2.1. One - dimensional aperture.

All the phenomena will occur in the plane as sketched in fig. 2.1. We introduce (as usual):

with

t .

e

u ~ 1T

I

sw

A ~ wavelength.

(2-1)

The strategy we want to be followed is to develop an antenna with a radiation pattern which shows a peak in a certain direction and has low sidelobes in addition. In other words we want to maximize the energy in the main beam of the radiation pattern.

The radiated power in a certain direction can be noted as +1

P ~

f

du [

f

f(x)e jux dx

[2

(2-2)

Here the aperture of the antenna is limited to - 1 ~ x ~ + 1. The total radiated power is given by

+1

Pt =

J

1 f(x) 12dx -1

(2-3)

Now we require the main beam to contain the greatest ratio of the radiated power. This means that we are looking for that aperture distribution f(x) which maximizes

+1

J

du 1

J

f(x) e jux dx 12 C -1 +1

f

If(x)1 2dx -1

The solution of this problem is given in [6]. [7] and [8]. Here we give a short summary of this solution.

From u = kJI, sin 8 with k =

I

£ = aperture-size. we derive du

=

k£ cos 8 d8. We want to maximize This leads to +8₀ kJI,

J

cos 8 d8 1 -8₀

the radiated power

+1

J

f(x) e j H -1 +1

J

1 f (x) 12dx -1 sin in the 8.x dxl 2 region -8_o t>St> = (2-4) (2-5) (2-6) +6 0,

9

-+8

0 +1 +1

kJl,

J

_{cos 8 d8}

J

f(x) ej Hx sin 8 dx

J

f(y). e-j Hy

-8 ₀ -1 -1 = +1

J

If(x)1 2dx -1 +1 +1 +8₀

kJl,

I

dx.f(x)

J

fey) dy

f

_{cos 8.e j k£(x-y) sin 8d8}

-1 -1 -8 0 =---~---Now we put K(x-y) +8 _o = H

J

ej -8 _o H(x-y)

= f

eju(x-y)du C sin 8 cos

e de "

sin 8 dy = (2-7) (2-8)

This means that we have to look for the aperture-distribution f(x) which maximizes

+1 +1

J

dx

J

dy K(x-y) fex) fey) -1 -1

This is written as

dy.Ks(X-y) F(x) F (y) /

f

I FeY) 12dy

R

(2-9)

(2-10)

and we have to look for the maximum of this expression. It follows from [6J, [7] and [8] that the solution of this problem is the highest

eigenvalue of the integral equation

A

~(x)

=

f

Ks(x-y)

~(y)

dy

R

Now we have to deal with an eigenvalue problem in which

We have the three following theses:

(2-11)

1. If the kernel Ks(x-y) is symmetrical then the eigenvalues

A.

1 are real.

2. If the kernel K (x-y) is bounded then the eigenvalues A.

5 1

are bounded too.

3. If the kernel K (x-y) is positive definite then the eigen-5

values A. are positive: A. > O.

1 1

In that case each eigenvalue A. has a function '1'. with the

1 1

,~haracteristics :

a. The functions '1'. are complete. This means we can write 1

b. The functions 'l'i are orthogonal.

On the three theses the following queries arise

1. Is K (x-y) symmetrical? s

To answer this question we determine K (x-y). s +8₀ K(x-y)

=

J

eju(x-y)cos

e

d8 -8₀ with u = k~ sin 8. This leads to (2-12) (2-13)

11

-+e o

K(x-y) =

f

ej

k~

sin e.(x-y)cos e de = -eo

e=+e_o

=

t~

f

e j

k~

sin

e.(x-Y)d(k~

sin e) =

e=-e _o 1 = H 1 K (x-y) = k~ +k~ sin

f

eo ju(x-y)d e u -H

[

e~U(X-Y) ] u = + H J(x-y) u

= _

k~ = = ..,-,;--;;-;1;-.-___ {e j H(x-y)sin eo _ e- j j H(x-y) We find K(x-y) = 2 sin 6 o sin{k~(x-y)sin eo} k~(x-y)sin eo From this expression it is clear that

K(x-y) = K(y-x) because sin(x-y) = sin(y-x) x-y y-x H(x-y)sin eo} (2-14) (2-15) (2-16) (2-17) (2-18)

Thus K(s-y) is symmetrical, which means that the eigenvalues A. 1 are real.

2. Is K (x-y) bounded? s

In that case we have to deal with the question whether the expression

K(x-y) 2 sin 8 0 is bounded. Because sin 8₀ ~ 1 and sin x x .. 1 sin{kl(x-y)sin 8_o} k-Q,(x-y) sin 8₀

KCx-y) is indeed bounded. In that case the eigenvalues A. are 1 bounded too.

~. Is Ks(x-y) positive definite? +1 +1

J

dx

f

dy.Ks(x-y).fCx) fey) = -1 -1 +1 +1 =

f

dx

f

dy·f ejuCx-y)du.fCx) -1 -1 C +1 fCy) = +1 f du f dx f(x) ejux f dy.fCy) e-uy = C -1 -1 +1 =

f

du

₁

f

dx.e jux fCx)

12

> 0 C -1 +1 whenever

f

I

f(x)

12

dx > 0 . -1 = (2-16) (2-19)

13

-Now we will prove the highest eigenvalue to be the maximum. K (x-y) F(x) F(y) _s

IF(y)1 2dy

R R

= : : . . . - - . . : . . : . . . . - - - =

n

a. a_j

f

dx

f

dy K (x-y) '1'. (x) 'I' . (y)

i j 1 S 1 J R R =

I

I

L

a. 'I' . (x) 12dy j J J R

Because of the orthogonality

n:

a. a. A.

I

dx '1'. ex) 'I' . (x) i j 1 J 1 1 J R =

L

a.2 . J J LA; a.2

L

a. 2 j J. J j J = ~

_\

=

_\

I

a. 2

L

a.2 j J j J

This means that the required aperture distribution f(x) is the solution of the integral equation

A 'I'(x)

=

f

Ks(x-y) 'I'(y) dy .

R

(2-20)

(2-21)

Solutions of these equations form a class of "prolate spheroidal wave functions". These functions are denoted by S (c,x).

mn

IEteresting cases of the aperture distribution are Soo(c,x). In. this expression c is a parameter which gives the part of the radiation pattern with the highest ratio of energy.

We can write c = kR. sin 8

0 (2-23)

If we change the value of the parameter c, the level of the first sidelobe in the radiation pattern will also change.

The parameter x denotes the co-ordinate over the aperture and is limited by - 1 , x , + 1, as already stated.

In literature there are tables giving the value of Soo(c,x). We mention in this respect Flammer [9).

From [9) we get a table for Soo(5, cos $), which gives table 2.1. with x = cos $ and c

=

5.

This means that Soo(5, x) is only known in 37 non-equidistant points on. the interval - 1 , x , + 1.

However, we will use this function Soo(5,x) below as an aperture-distribution to compute the radiation pattern.

16

-3. The Prolate Spheroidal Wave Function as Aperture Distribution

As is well known, the far-field pattern of an antenna can be computed from

+1

F(u) :

~

I

f(x) ejux dx -1

with

~

the aperture size, u : IT

i

sin 6 and f(x) the aperture-distribution.

(3-1)

As aperture-distribution we want to take the prolate spheroidal wave function Soo(c,x).

The level of the first sidelobe in the radiation pattern should be -40 dB approximately. That means in this instance taking the case c : 5.

In the expression for c C : k~ sin 6

0 (3-2)

however, there are still two parameters which can be varied, namely

~ and 6

0• We want to regard k and hence the wavelength A as constants.

When fixing 6₀ we have to adjust the value of ~ , as the relation c : 5 must apply.

Concerning the aperture distribution we note that the function to be used is only known in 37 non-equidistant points at the interval - 1 , x , + 1.

Interpolation, however, is a method which can give many more points of Soo(5,x). This gives tabel 3.1. and fig. 3.1.

<p [0

1

x Soo(5,x) 0 1.0000 0.05024 5 0.9962 0.05224 10 0.9848 0.05845 15 0.9659 0.06942 20 0.9397 0.08609 25 0.9063 0.1079 30 0.8660 0.1419 35 0.8192 0.1893 40 0.7660 0.2380 45 0.7071 0.3046 50 0.6428 0.3839 55 0.5736 0.4748 60 0.5000 0.5742 65 0.4226 0.6773 70 0.3420 0.7776 75 0.2588 0.8673 80 0.1736 0.9383 85 0.0872 0.9842 90 0.0000 1.0000 Table 2.1.

17 -x Soo(5,x) x S 00 0.00 1.000 0.26 0.863 0.01 1.000 0.27 0.854 0.02 0.999 0.28 0.846 0.03 0.998 0.29 0.835 0.04 0.996 0.30 0.824 0.05 0.995 0.31 0.814 0.06 0.993 0.32 0.804 0.07 0.991 0.33 0.792 0.08 0.98B 0.34 0.777 0.09 0.983 0.35 0.769 0.10 0.979 0.36 0.756 0.11 0.974 0.37 0.744 0.12 0.968 0.38 0.731 0.13 0.963 0.39 0.718 0.14 0.958 0.40 0.706 0.15 0.951 0.41 0.692 0.16 0.945 0.42 0.678 0.17 0.938 0.43 0.665 0.18 0.932 0.44 0.650 0.19 0.925 0.45 0.636 0.20 0.917 0.46 0.624 0.21 0.909 0.47 0.611 0.22 0.900 0.48 0.599 0.23 0.890 0.49 0.585 0.24 0.881 0.50 0.574 0.25 0.871 Table 3.1.

x Sooe5,x) x Sooe5,x) 0.51 0.560 0.76 0.246 0.52 0.546 0.77 0.235 0.53 0.532 0.78 0.224 0.54 0.519 0.79 0.214 0.55 0.505 0.80 0.206 0.56 0.492 0.81 0.196 0.57 0.477 0.82 0.187 0.58 0.465 0.83 0.176 0.59 0.452 0.84 0.166 0.60 0.438 0.85 0.156 0.61 0.425 0.86 0.146 0.62 0.411 0.87 0.138 0.63 0.398 0.88 0.130 0.64 0.385 0.89 0.122 0.65 0.375 0.90 0.114

I

0.66 0.361 0.91 0.106 I , 0.67 0.350 0.92 0.099 , , 0.68 0.337 0.93 0.092 0.69 0.325 0.94 0.086 0.70 0.313 0.95 0.079 0.71 0.302 0.96 0.072 0.72 0.290 0.97 0.065 0.73 0.280 0.98 0.058 0.74 0.269 0.99 0.053 0.75 0.257 1. 00 0.050 Table 3.1.

Fig.3.1.

Aperture -distribution.

I

SoJc,x) C:5 Q3

cw

Q.5 Q.6 Q7 Q.8 Q.9 1.0

•

x

As already said, we want to use this aperture-distribution for computing the radiation pattern in the far field. This will be done by means of Fourier integration.

For the parameter c we take c = 5, as already mentioned.

The aperture-dimension is R, = 201. with A as the wavelength. This will give fig. 3.2. as the radiation pattern.

From this radiation pattern it is immediately clear that the level of the first sidelobe is about - 36 dB, and not the - 40 dB which we tried to realize. The difference may be explained from

inaccuracies in computing the radiation pattern.

WE' will compare the radiation pattern from fig. 3.2. with the requirements given in the WARe '77 Report [2].

From fig. 3.2. we get the following as 3 dB-beam width (see fig. 3.3.). 8₀

/2

= 1. Sa

or 8

0 = 3.6 0

The same figure shows as first zero in the radiation pattern 8, = 5.20

We get the ratio

8,

e

_a = 1.44 In [2] the requirement 8

,

1.58

e=

a

Thus the slope of the is

main beam is better then required in [2]. (3-3)

(3-4)

21 -o

Flg.3.2. Radiation pat tern.

-10

C=5

....

- 20

I:

-30 - 4 0 -50 o 2 3 4 5 7 8

_"

10 11 12 13 14

---0 ()

(degrees)

Fig. 3.3.

-3dB-point.

Concerning the sidelobe level we note that this, too, is much better than required.

A level has been computed from about - 36 dB.

The requirement in [2] is - 30 dB. This means that on this point also the WARC '77 Rep?rt [2] requirement has been fulfilled.

23

-The radiation pattern caused by the prolate spheroidal wave function has been computed by means of approximations as the function 5₀₀(5, x) is given only at a number of nonequidistant points. Therefore the computed radiation pattern will also be an approximation of the real radiation pattern. Since we want to know how good or bad this approximation is, we computed the radiation pattern caused by the function cos (~x) as aperture distribution. This has the advantage that besides computing the radiation pattern in the same way as the prolate spheroidal wave function, we can now compute the radiation pattern more exactly.

We now write for the aperture distribution

with -l~x~+l

The radiation pattern can be expressed as

with +1 F(u)

= f

f(x) ejuxdx -1 U = 11" £ sin

e

A and we get +1

F(u) =

f

cos (IT;) ejuxdx -1

After solving this integral we have cos u F (u) = - IT

--='::":::'-=---:,....

_{IT IT} (u - 2) (u + 2) (3-6) (3-1) (3-7) (3-8)

The radiation pattern caused by cosCIT 2

x

) as aperture distribution can now be computed in two different ways. The first is the same as with the prolate spheroidal wave function as aperture distribu-tion. This means that the radiation pattern from expression (3-1) is computed by means of interpolation of the aperture distribution

I

U> Q9 QB Q,7 Q.6

as

CIA Q.3 IU

'...;.0 ~ Il9 -qs -03 -0.6 -0.5 -CIA -0.3 -IU -Q.1 0.0 OJ Q2 0.3 1M

as

Q.6 Q7 lUI Q9 1.0

•

o -10

co

u w Q) - 20

I!

....

C1l Q) L. - 30 - 40 -50 o 1 _{2 3} 5 II 25 -Fig.3.5. Radiation Computed pattern.

by

the approximation method.

1=20.1

7 8 10 11 12 13 - - -... () (degrees) 14

Computed from formula (3-8).

1,

=

20.1

-10 ~ CD "0

...

(J) - 20

I;

..

_CO (J)

...

- 30 -40 -110 o l' 2 ·3 4 5 7 8 10 11 12 13 14

e

(degrees)

27

-over a large number of points in Fig. 3.4. Fig. 3.4. shows the aperture distribution f(x) = COS(~x). Calculation of the radiation pattern gives Fig. 3.5.

On the other hand the radiation pattern can be computed from formula (3-8). This is a more exact calculation because now a computation can be made at each point, whereas with the method described above, it can only be done at a number of points whose aperture distribution is known. The radiation pattern calculated from expression (3-8) can be seen in Fig. 3.6.

It is immediately clear that there are differences between the two radiation patterns. However, these differences are not large enough to constitute a reason for not using the approximation method. For calculation of the radiation pattern, especially main beam and first sidelobe, the approximation method is certainly suitable.

We can also compare the results obtained by the approximation method with those of Silver [11

J .

From [I1J we see that the half-power beam width should be

On the other hand, for the angular position 8_j of the first zero we can write

e

= 1.5 ),JR-.

I

As we are working here with an aperture size R-

=

20 A we obtain

and

and

It is thus permissible to use this approximation for computing the radiation pattern.

Considering the radiation pattern from the approximation method viewpoint,the following observations can be made.

The first zero is at 8₁

=

4.25°. For the 3dB beam width we get

8 /2 = 1.62°.

°

This leads to 8 1 4.25°

~

= 2*1.62° or 1.31

The WARC '77 Report [2] requirement was 1.58. In other words, this requirement is fulfilled. However, the requirement

concerning the level of the first sidelobe is -30dB and in our case we get a level of about -23dB. Thus, no fulfilment of the requirement and much worse than with the prolate spheroidal wave function as aperture distribution.

29

-4. Elliptical Beams

In the WARe '77 Report [2] several beam widths are given in two planes. Most of these beam widths refer to an elliptical beam of the antenna. The advantage of an elliptical beam has been

discussed in the introduction. Here we mention again the fact that with a satellite antenna producing an elliptical beam it will, under certain circumstances, be possible to illuminate a certain area on earth very sharply, while outside this area the radiated power would be minimized. Therefore some radiation pattern requirements have been given in the WARe '77 Report [2].

In chapter 3 we saw that it is possible to fulfil these requirements by using a prolate spheroidal wave function as aperture distribution of a reflector antenna. Here we shall give some examples of elliptical beams as have been mentioned in the WARe '77 Report [2].

In those cases we make use of a cylindrical two-reflector system, as mentioned in the introduction, and which will be discussed more extensively in the following chapters.

Because we use a cylindrical reflector system and a line source as feed for this antenna system, the radiation pattern of the antenna in one plane will be completely prescribed by the radiation pattern of the line source. This means that in that plane the beam width will be equal to that of the line source. Therefore the radiation pattern of this line source has been given in Fig. 4.1. From Fig. 4.1 it is clear that the half-power beam width of the line source is 5°. It should be stated here that some investigations into the line source have already been carried out

(see [10]).

When the beam width in the plane of the line source is known it will be possible to create an elliptical beam by shaping the radiation pattern, and hence the beam width in the other plane, orthogonal to the first one. This can be done by using the prolate spheroidal wave function for the case c = 5 as aperture

I I I I I. I ~ 20)'

it

0)

a:

- 2 - 1 0 1 2 3 4 5 () (degrees) Fig.4:1. 8eamwidth of the line source ( H-plane).

31

-In that case the beam width will be varied, too.

It is possible to deduce experimentally a relation between the aperture size and the beam width produced by that aperture size. In fact this can be done with each antenna and aperture distri-bution. It will be done here for a reflector antenna with the prolate spheroidal wave function in the case c

=

5 as aperture distribution.

Denoting the half-power beam width by 6

3dB we assume the relation

where A

=

wavelength ~

=

aperture size

and p is a constant to be determined.

We take an aperture size ~

=

48 cm which is the same as ~ = 14.8002 A because A = 3.2432 cm. Computing the radiation pattern we find Fig. 4.2 which shows a beam width of

(4-1)

Substituting this, together with the aperture size, into (4-1) gives

4.9

=

p.1/14.8002 (4-2)

From this it follows that

p = 72.52 (4-3)

This leads to the relation 8

3dB = 72.52 A/~ (4-4)

Now we will compute several elliptical beams which can be produced by a prolate spheroidal wave function as aperture distribution. The starting point is the beam width obtained from the Taylor distribution which is the distribution of the line source in the plane of this line source. The beam width in that plane is

-10 - 20 - iO - 40 -50 o

-

CO "0

....

....

I:

_-

tti 0;'

_....

1

=48cm.14.8002

C

-5

2 3 II 7

•

10 11 12 13 14

- - - " (J

(degrees)

33

-(4-5)

It is assumed that we want to realize an elliptical beam with a ratio 1.25 : 1. This means that we require a beam width 8

3dB

=

6.25° in the other plane. Substituting this in relation (4-4) gives

6.25 = 72.52 ),/2,

which leads to the aperture size

2 = 11. 6032 ), or

2 = 37.6315 cm

This method will be applied in the subsequent cases.

We shall compute the radiation patterns for elliptical beams with a ratio of the axes

a. b. c. d. e. 1.72 2.5 1.15 1.23 2.15 1 1 1 1 1 (4-6) (4-7) (4-8)

Note: a ratio of the axes has been taken here and not the angle size, since the beam width in one plane is 5° constant.

Now it is possible to compute the aperture size of the reflector for the various elliptical beams.

a· 1.72 1 8 3dB

=

1.72 x 50

...

83dB

=

8.6 ° 8 3dB 72 .52 A/2 2/>-72.52

=

...

=

_8.6 2/),

₌

8.43

...

2

₌

27.35 cm

We also can calculate ~ as follows: 8 3dB

=

5°/1.72

...

83dB

=

2.91° 8 3dB

=

72.52 A/~

...

~/A 72.52

=

'2":91

~/A

₌

24.92

...

~

=

80.82 cm.

In the same way we find: b. 2.5 : 1 8 3dB

=

12.5°

...

~/A

=

5.80 ~

=

18.82 cm 8 3dB

=

2°

...

~/A

=

36.26 ~

=

117.60 cm c. 1.15 1 8 3dB

=

5.75°

...

~/A

=

12.61 ~

=

40.90 cm 8 3dB

=

4.35°

...

~/A

=

16.67

...

~

=

54.07 cm d. 1. 23 1 8 3dB

=

6.15°

...

~/A

=

11. 79 ~

=

38.24 cm 8 3dB

=

4.07° .;- ~/A

=

17.82 ~

=

57.79 cm e. 2.15 1 8 3dB

=

10.75° .;- ~/A

=

6.75 ~

=

21.88 cm 8 3dB

=

2.33°

...

~/A

=

31.12 ~

=

100.94 cm

Computing the radiation patterns for aperture sizes 19, 22, 27, 38, 41, 54, 58, 81, 100, 101 and 118 cm we find respectively the

-. -.

Figures 4.3 up to and including 4.13.

We note here that a size of 1, = 100 em has also been included in thE' calculations.

35

-From the radiation patterns we can again deduce the beam width and afterwards recalculate the ratio of the beam widths. With the results we can construct table 4.1.

In this table we have: 8

3dB

=

72.52

A/!!-with !!-/ A from the same table Ratio)

₌

6 3dB /5 i f 63dB ₎ > 5° Ratio)

=

s/6_3dB i f 8 3dB < 5° ) ) 8

3dB comes from Figures 4.3 up to and including 4.13 Ratio 2

=

83dB _2-/5 i f 83dB > 5° 2-Ratio2

=

5/6 3dB ₂ i f 63dB < 5°. 2

-3dB -10

.-

_co

u

.-I

....

Q) ~ 0

c.

~

-20

~

Q)

....

-30

1=19cm

C=5

-50+-__ . ______

~----~~~j-7-·~---~---~r_--~----T_--o 4 8 12 111 20

•

() (degrees)

37 -o

Fig.4:4.Radiation pattern.

-10 ~

i=22cm

co

'C

1

~

...

C=5

Q) ~ 0

a.

.~

...

t -20 (1l Q)

_...

I -30 I 15.33°

-~+---~---L---r---~---~~~---'-~

_{o 4 8 12 111 20}

•

(J

(degrees)

-3dB ,-10

1=27cm

C=5

--

Ql

"'CJ

.-CD

r

C) (:l. -20 CD

-

.

I . ::- ...-(0

,

CD I

"--10 -40 I I I -110 I 4.33° o 1 2 3

..

II II 7 II D 10 11 12 13 14

- - - . . . ()

(degrees)

o

...,...----3dB -10 ~

en

"0

I

-

~ Q) ~ 0

a

.~

--20

co

Qi ~ :-30 -40 1_{3 .09'} 1 39

-Fig.4.6. Radiation pattern.

i.38cm

C .. 5

-~+---r---~--~---r---~--LL----r--o 2 4 e 8 10

•

(J

(degrees)

-3d8

-10 ~ III '0 ~

1

...

Q) ~ 0 a

.~

-20

10

Ql

...

I I 2.82° I

i=41cm

C=5

-504-__ . ________

,_--~---,_---,_---~~---,_--o 2 4 8 10

•

fJ

(decrees)

0+-_ _

-3dB

-10

-

!Xl 'U

-I

...

Q) ~ 0

a

~

.--20 ctI Q)

...

-30 t

I

12.20·

41

-Fig.4B. Radiation pattern.

1=54cm

C ... 5

-~+---r---T~----~----~~---r---~L----T---~

_{5 7 8} o 1 2 3 4 8

•

(J

(degrees)

-3dB

-10 ~

i"s8cm

CD "0 ~

I

...

Q)

e·o

~ 0 a Q) .~

...

-20 ttl Ql

_...

-30 -40 o -~+-

____

- r ____ ~~2=._O_2

__

~

____

- r ____ ~ ____ ~r-____ ' -__ ~~ o 7

•

() (degrees)

43

-0;-_ _

_

Fig.4.10.Radiation pattern.

-3dS

r -10 ~

1 ...

81 em III '0 ~

1

...

_{C .. 5}

Q) ~ 0 a ~

...

-20

'"

Ql

_...

-30 -40 • 0

.1.43

-50+-____ ~ ____ - r __ --~--__ ~ __ - -__ ~----r_----~----~LL--_r----_r----~L-Qp 1.11

•

() (degrees)

-3dB

~ "£=100cm

co

"0

-I

....

C .. 5

Q) == 0 0.

.~

-20

1il

Qj

....

-30 -~~---,~----~-i----r---T---,---~--~-r---' 0.0 1.0 2.0

() (degrees)

45

-0-t-_ _

_

Fig.4.:12.Radiation pattern.

-3dB

-

!Xl

₁₌₁₀₁

_em

'0 ~

I

....

Q)

C=5

~ 0 a ~

--20

co

a;

_....

I -~+---~r---r~----~----~~----~---'---~~~---r OJJ lJJ 1,5 21J

•

(J

(degrees)

-3 dB

-10 ~

i,"118Cm

III 1J ~

I

...

Q)

C=5

~ 0 c. Q) .~

...

-20 ell Q)

...

-30 -40 I I

: 1.00·

-50 0.0 0.5 1.0 \,5 3.0

•

o

(degrees)

47 -R,[cm] R,/ ).. 6 3dB ₎ Ratio) 63dB Ratio 2 2 19 5.86 12.38· 2.48:1 12.34· 2.47:1 22 6.78 10.69" 2.14:1 10.66' 2.13:1 27 8.33 8.71' 1. 74: 1 8.66' 1. 73: 1 38 11. 72 6.19· 1. 24: 1 6.18' 1. 24: 1 41 12.64 5.74' 1.15:1 5.64· 1.13:1 54 16.65 4.36· 1.15: 1 4.40' 1.14:1 58 17.88 4.06· 1. 23: 1 4.04' 1. 24: 1 81 24.98 2.90' 1. 72: 1 2.86' 1.75:1 100 30.83 2.35' 2.13:1 2.36' 2.12:1 101 31.14 2.33' 2.15:1 2.28' 2.19:1 118 36.38 1.99' 2.51:1 2.00' 2.50:1 Table 4.1

There is no need to verify whether the requirements of the WARe '77 Report [2] are fulfilled, as this has already been done in chapter 3. The only difference from the results in chapter 3 is the fact that now the half-power beam width differs from that in chapter 3. The form of the radiation pattern, however, is still the same.

5. The Cylindrical Two-Reflector System

In this section the purpose is to derive the equations of the cylindrical two-reflector system. These equations make i t possible to compute the shape of the two cylindrical reflectors. As already stated in the introduction, investigations have already been

carried out in this field [3]. The further considerations here will exist partly of an expansion and partly of a specific case taken from the discussion in [3].

The specific case is the fact that we are now concerned with a un.iform phase distribution in the aperture of reflector C, while the case dealt with in [3] was the more general one of an

arbitrary phase distribution.

The expansion consists of a decrease in the horizontal direction of the reflector system (see fig. 5.1). The assumptions made in

[~;] are applied here again. This means that both reflectors are

assumed to be large and have a radius of curvature which is large compared with the wavelength.

The following optical principles are used in developing the system equations:

1. Snell's law,

2. Conservation of energy flow along the ray trajectories, that is the energy flow must be conserved in any solid angle bounded by ray trajectories,

3. Surfaces of constant phase are normal to the ray trajectories and this normal congruence is maintained after any number of reflections (theorem of Malus).

For this case we use the configuration of the system given in fig. 5.1. Snell's law at reflector B gives

with

e - e

tan ( 2 2

j)

x. + x

e

_{= arctan}

(

j 2 ) 2 -o.+S-Yj+Y2 (5-1) (5-2)

49

-Reflector C

x

'r1

~

8

₂ 0

Y2

~ma. Y2

Reflector B

I I I I

I

---

X

,

_...

x

1ma•

X

₁

I •

• I

I I I

••

_~

.. I

I I _I

We assume Cl and S to be constants. Sn'ell's law at reflector C gives

dY2 62

- = - tan (-)

dX₂ 2

Application of the conservation of energy principle leads to

where I] (6])

=

power density of the primary illumination.

I 2(x_{2 )}

₌

power density flow normal to the aperture of main reflector.

(5-3)

(5-4)

the

It is chosen arbitrarily except for the required normalization

From 6] max x2 min

J

I] (6]) d6] =

J

_{I 2}(x_{l )} dX₂ 6] . _m1n (5-4) we d6 _ _ I ~ -dX 2 x 2 max derive 12 (x₂) = f I] (6]) ] The path length is given by

wHh

C(6)) is defined by the primary illumination and can be regarded as a phase-correction term.

(5-5)

(5-6)

(5-8)

(5-9)

51

-From fig. 5.1 it is evident that x _{1 max} -x ₁ S - Yl This leads to We get - (S - y ) tan e + x 1 1 1 max This gives Using - (S - y ) I dX 1 aXI dX

z

= aYI dy I _{aX 1} de I dX

z

+

ae

l • dX

z

together with (5-14) and (5-15) leads to

tan

e

I

We have already found

Substitution yields dx _ 1 = dX

_z

de 1 dx

_z

(5-11) (5-12) (5-13) (5-14) (5-15) (5-16) (5-17) (5-6) (5-18)

We know that dY l dY l dX l dX 2

=

dXl dX2 From (5-1) we get dy 1 8 - 8 dX l = tan ( 2 2

1)

with 8₂

=

arctan Xl + x₂ C,,+S-Y l +Y} This leads to d

1

arctan( x! + X2 ) Yl -X+~-Yl+Y2

d

= tan Xl . 2

Substitution of (5-20) into (5-19) gives

Now substituting (5-17) into (5-21) we find dY_l

- = tan 8

dX

2 1

From this expression we derive dY l -_dX (1 - tan 8₁ " _{f 2)} = - (S - Y_l) sec 2_{8 1 "} flf2 2 which leads to dY l -(S - yl)sec 281

"f/

z dX 2 = 1 - tan 81 "f 1 f 2 We have already found

(5-19) (5-1) (5-2) (5-20) (5-21) (5-22) (5-23) (5-24)

53 -This gives p _Z

=

R - r ₁ - P - C ₃ or p = B - P 2 3 with B = R - C - r ₁

Raising expression (5-26) to a square leads to

p~

=

B2 - 2BP3 + P~ We already knew and P3

=

Y2 This gives P 2 = (x + x ) 2 = (- a + ~ - YI + Y,2 ) 2 2 I 2 or

P~ = (xI +x2)2 + (-a+B -y₁)2+ 2Y2(-a+~-Yl) + Y~ Comparing with expression (5-28) leads to

(xI +x2)2 + (-a+B _y l)2 + 2y

z

(-a+B-yl) +Y~

=

= 82 - 2By 2 + y~ (5-25) (5-26) (5-27) (5-28) (5-9) (5-10) (5-29) (5-30) (5-31 )

From this equation it is possible to derive an expression for Y2:

All the equations for computing the shaping of the reflector system have now been derived.

(5-32)

The main lines of the solution of the problem will be as follows. From

we get

(5-34) This means that,by varying 81 it is possible to compute x2 as a funct ion of 8₁ 0

(5-35)

Then we substitute into

dYl -(S-Yl)sec281oflf2

dX₂

=

I-tan 8₁of₂ (5-24)

The value of the constant

a

as well as the expressions

(5-36) (5-37) (5-13) (x + x ) 2 + (_ " + S - YI) 2 _ S2 1 2 (5-32) (5-38) which gives (5-39)

Now, with (5-35) we can compute

55

-This can be shown schematically

order of coming 6 j Xj Yj x₂ Y2 solution from formula x _{1 (5-34)} x ₂ (5-39) x ₃ (5-13) x 4 (5-32)

6. Design of a Cylindrical Two-Reflector System

In this chapter we shall discuss the design of an antenna system which can produce an elliptical beam in the radiation pattern. We use the two-reflector system that has been discussed in chapter 5. As has already been said in chapter 5 we use a line source as a feed for our antenna system. This line source has been described in more detail in [10].

For a cross-section of this line source see Fig. 6.1.

g~

~tes

57

-Some characteristics of this line source are flare angle

_!

ex

=

15° frequency f

₌

9.25 GHz depth of groove

₌

9mm width of groove

₌

2.5 mm width of dam

₌

2.5 mm width of plates a

=

15.5 cm length of system

₌

60 cm

For the most part this line source is the same as the one described in [3]. The great difference, however, is the flare angle which is 30° in [3].

The smaller flare angle in our case gives the advantage of a smaller aperture of the first reflector, too. In Fig. 6.2, for the aperture sizes in the two cases we find

x

=

60° x

=

30° t B60 = r,l3 t B30 = r/,I3

Since we assume r to be constant, it is obvious that the case x

=

30° is preferable.

In [3], however, we made use of the 60° case. This was done because of the relatively high edge-illumination. With the new equations (see chapter 5), however, the feed with x

=

30° can be used.

I I I I I I

R

I 8₆₀ I

,

..

,

"

j,

8₃₀

--

-

-

-r r

59

-This is possible because we introduced in chapter 5 a phase correction term. In other words, we no longer consider the phase distribution over the subreflector as constant, but take in account the deviations from a linear phase distribution. This means that it will now be possible to get a low edge-illumination at the first reflector, because the subreflector can be extended over the edge of the feed and hence a lower edge-illumination can 'be obtained than at the edge of the line source (see Fig. 6.3).

On the other hand, for the power distribution of the feed, we now take the actually measured distribution for 11 (9 1),

Line_source

,

I -...-..

J,C."'---,

I I I I

"

I I I I I

"

"

"

I

"

"

I

Fig._6.3. Line source combined with subreflector.

Therefore a cylindrical two-reflector system can now be designed with a low edge-illumination and consequently a low spillover and a low sidelobe level in the radiation pattern.

As already said, use has been made of the measured values of the power distribution as well as the phase distribution of the feed.

8 [0] I II (81)[dB] lIH8 I )[0] 8 I [0] II (8 1) [dB] LlH8 1)

n

0 0.0 + 4 0 0.0 + 4 1 - 0.1 + 3

-

1 - 0.1 + 5 2 - 0.1 + 2

-

2 - 0.2 + 6 3 - 0.2 + 4 - 3 - 0.3 + S 4 - 0.4 + 5 - 4 - 0.5 +10 5 - 0.7 + 6 - 5 - O.S +13 6 - 1. 0 + 8 - 6 - 1.2 +16 7 - 1.4 +11

-

7

-

1.6 +lS 8 - 1.9 +13

-

S - 2.0 +22 9 - 2.4 +16 - 9 - 2.5 +24 10 - 2.9 +18 -10 - 3.0 +27 11 - 3.4 +22 -11 - 3.5 +29 12 - 3.9 +23 -12 - 4.1 +32 13 - 4.4 +25 -13 - 4.7 +34 14 - 5.1 +26 -14 - 5.4 +35 15 - 5.7 +26 -15 - 6.2 +36 16 - 6.4 +27 -16 - 7.0 +37 17 - 7.2 +26 -17 - 7.S +37 18 - 8.0 +25 -lS - S.7 +37 19 - 8.9 +24 -19 - 9.6 +37 20 - 9.9 +22 -20 -10.6 +36 21 -10.9 +19 -21 -11.6 +35 22 -11. 9 +16 -22 -12.6 +34 23 -12.9 +12 -23 -13.7 +32 24 -13.9 + 7 -24 -14.9 +30 25 -14.9 +11 -25 -16.0 +27 26 -15.8 - 5 -26 -17.1 +23 27 -16.6 -11 -27 -lS.l +19 28 -17.4 -19 -2S -19.1 +13 29 -18.2 -26 -29 -19.9 + 7 30 -18.S -23 -30 -20.5 0 Table 6.1

61

-By indicating the power distribution by I) (e)) and the phase correction by d~(e)) we can construct Table 6.1.

These results have been measured at a distance of 20 cm from the phase centre.

The d~(e)) in this table corresponds to the factor C(e)) in

chapter 5.

Some parameters are required for computing the shape of the reflector system. Before giving the values of these parameters some further discussion is felt to be desirable.

The values apply to the distance between the line source and the first reflector. This distance for a value of 6) from 0° can be computed as follows (see Fig. 6.4).

I I I I I _I I

...t

I I /

_I

I

'"

I

'"

I

'"

d l I / I

'"

I

_r _

-1

£jg.6.4.

Calculation of rl.

From this Fig. 6.4 we can derive tan Ct

=

dlr'

or

d = r'tan Ct'

On the other hand cos S

=

r'lr or

r'

=

r cos S We have to deal with

S

=

a'-a ₂

In our case, this gives S

=

60· - 30· ₂

or

The values given in Table 6.1 have been taken at a distance

r = 20 cm from the phase centre. Hence here too we take r = 20 cm. This leads to

r' = 20 cos IS· or

r' = 19.32 cm.

For the aperture size of the subreflector we thus get d

=

19.32 tan 60·

or

d = 33.46 cm.

When considering a system with a subreflector and a main reflector we have to take into account that there must be a reasonable ratio between the aperture size of main reflector and subreflector.

63

-Because we found a dimension of d = 33.46 cm for the subreflector we have to think about a dimension of the main reflector of

£

=

100 cm. This means a ratio of about 1 : 3, which sounds reasonable.

As is well known, the beam widthinthe radiation pattern of an antenna is prescribed by the aperture size of the antenna. As can be seen from chapter 4, abeam width of 8

3dB = 2.36° will be realized by an antenna of this dimension.

In the other plane the beam width will be prescribed by the radiation pattern of the line source. Measuring the radiation pattern shows a beam width of 8

3dB = S°.

This means that with this antenna we could realize an elliptical beam with a ratio of the axis of 2.12 : 1 when taking an aperture size of £

=

100 cm (see chapter 4).

Knowing the various dimensions of the antenna system, we are able to compute the shape of the reflectors of the cylindrical two-reflector system. This computing was done with the help of a high-speed machine. The computer program was written, taking into account the equations derived in chapter S.

This means that we still have to introduce a number of parameters, which we are now able to give. We get the distance ~

=

19.32 cm, x

1max = 33.46 cm. For the aperture size of the main reflector we

already found £ = 100 cm. Again we used a frequency of f = 9.25

GHz. This means a wavelength A = 3.24 cm. For the distance S we take S

=

2a. We also not here x

2max

=

£

=

100 cm. When normalizing these parameters with respect to x

2 max we get a = 0.19319 S

=

0.38637 x_1max

=

0.33461 x_2max

=

1 Sf

₌

_S

- a = 0.19319 Af _{= A/£ = 0.03243.}

Furthermore, as functions of 6

1 we need the power intensity of the feed II _{(6 1) as well as the phase distribution lI<j>(6 1) of that} sa.me feed. These values are given in Table 6.1. The aperture distribution of the main reflector has been used too. This distribution was given in chapter 3 Table 3.1. With all these pa.rameters it is possible to compute the shape of the cylindrical two-reflector system. When this is done we get Fig. 6.5 a.s a result. In Figures 6.6 and 6.7 the sub- and main reflectors are shown, respectively.

Fig

6.5.

Shape

of the

reflec-tor

system.

•

_.y

2

Q3 65 -0.4 0.1 lID

Q;lD

Q;l8

67 -OJ .1 ~~ ____ ~ ______ ~ ______ ~ ____ ~~ ____ - T ______ ~I~O OJ OD

..

7. Conclusions

A study has been made about the search for an aperture distri-bution, with which it is possible to maximize the radiated power within a certain area. The aperture distribution which can do this appears to be a prolate spheroidal wave function. After this the radiation pattern caused by this function as an aperture distribution has been studied theoretically. This has been done.

with regards to the requirements as has been given in the WARe '77 Report [2]. The conclusion is that it is possible to fulfil these requirements with respect to the sidelobe level and the slope of the main beam in the radiation pattern. The results are even better than these requirements.

Furthermore a design has been given to develop an antenna which possesses such an aperture distribution and therefore can produce

a radiation pattern fulfilling the requirements as given in [2]. This has been done by means of shaping of a two-reflector system fed by a line source. The shape of the reflector system has to be calculated by means of high-speed computations. The conclusion of this investigation is that it is indeed feasible to design such an antenna. Then it is possible to shape the radiation pattern in one plane. In the other plane the radiation pattern will be defined by the line source which gives a Taylor distribution. Accommodation of the line source however, will give another shape of the

69

-Acknowledgement

The author wishes to thank Dr. M.E.J. Jeuken for giving him the opportunity to carry out the investigations described in this report.

References

[1] CCIR Study Groups,

Special Preparatory Meeting (WARC-79) Chapter 5; § 5.4.1.3.

Geneva: International Telecommunication Union, 1978.

[2] Final Acts of the World Administrative Radio Conference for the Planning of the Broadcasting - Satell i te Service in Frequency Bands 11. 7 - 12.2 GHz (in regions 2 and 3) and 11. 7 - 12.5 GHz (in region I), pp. 102 -103.

Geneva: International Telecommunication Union, 1977. [3] H.L. Verstappen,

Shaping of a Cylindrical Double-Reflector System; Part 1. Ir. Thesis; Report ET-16-78. Dept. of Electrical Engineering, Eindhoven University of Technology, 1978.

[4] G. Borgiotti,

Design of Circular Apertures for High Beam Efficiency and Low Sidelobes.

Alta Frequenza; Vol. XL - N8 - Agosto 1971; pp. 652 - 657. [5] J.T. Mayhan,

A Linear Aperture Radiating Maximum Power Within Prescibed Angular Constraints.

IEEE Transactions on Antennas and Propagation. pp. 490 - 493. May 1974.

[6] D. Slepian and H.O. Pollak,

Prolate Spheroidal Wave Function, Fourier Analysis and Uncertainty - I.

Bell System Technical Journal; January 1961; pp. 43 - 64.

[7] H.J. Landau and H.D. Pollak,

Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - II.

Bell System Technical Journal; January 1961; pp. 65 - 84. [8] D. Slepian,

Prolate Spherical Wave Functions, Fourier Analysis and Uncertainty - IV: Extensions to Many Generalized Prolate Spheroidal Wave Functions.

Bell System Technical Journal; November 1964; Vol. XLIII; pp. 3009 - 3057.

71

-[9] C. Flammer,

Spheroidal Wave Functions.

Stanford University Press; Stanford, California 1957. p. 149.

[10] F.M.J. Willems and W.J.W. Kitzen,

Circulair Gepolariseerde Straling m.b.v. een Axiaal Gegroefde Tweevlaksstraler.

Report ET-11-76. Dept. of Electrical Engineering, Eindhoven l!D,iversity of Technology.

[11] S. Silver(ed.),

Microwave Antenna Theory and Design; McGraw - Hill Book Company, Inc.

New York, 1949. MIT Radiation Laboratory Series, Vol. 12.

Reports:

I) Dijk, J., M. Jeuken anti E.J. I\laantlers

AN ANTENNA FOR A SATELLITE COMMUNICATION GROUND STATION (PROVISIONAL ELECTRICAL DESf(;N).

TH-Rcport61l-E-UI. 1961l. ISBN 90-6144-001-7 2) Veefkind, A., J.H. Blom and L.H.Th. Rieljens

THEORETICAL AND EXPERIMENTAL INVESTI(;ATlON OF A NON-EQUILIBRIUM PLASMA IN A MHD CHANNEL. Submitted to the Symposium on Magnetohydrodynamic Electrical ,power Generation, Warsaw, Poland, 24-30 July, 1968.

TH-Repon 68-E-02. 1968. ISBN 90-6144-002-5 3) Boom, A.J.W. van den and J.H.A.M. Melis

A COMPARISON OF SOME PROCESS PARAMETER ESTIMATlN(; SCHEMES. TH-Report 68-E-03. 1968. ISBN 90-6144-003-3

4) Eykhoff, F' .• P.J.M. Ophey, J. Severs and J.O.M. Oome

AN ELECTROLYTIC TANK FOR INSTRUCTIONAL PURPOSES REPRESENTING THE COMPLEX-FREQUENCY PLANE.

TH-Report 68-E-02. 1968. ISBN 90-6144-004-1 5) Vermij, L. and J.E. Daalder

ENERGY BALANCE OF FUSING SILVER WIRES SURROUNDED BY AIR. TH-Report 68-E-05. 1968. ISBN 90-6 I 44-005-X

6) Honllen, J.W.M.A. and P. Mnssee

MHD POWER CONVERSION EMPLOYING LIQUID METALS. TH-Report 69-E-06. 1969. ISBN 90-6144-006-8

7) Heuvel, W.M.C. van den and W.F.J. Kersten

VOLTAGE MEASUREMENT IN CURRENT ZERO INVESTIGATIONS. TH-Report 69-E-07. 1969. ISBN 90-6144-007-6

8) Vermij, L.

SELECTED BIBLlO(;RAPHY OF FUSES. TH-Report 69-E-08. 1969. ISBN 90-6144-008-4 9) Westeni>erg. J.Z.

SOME IDENTIFICATION SCHEMES FOR NON-LINEAR NOISY PROCESSES. TH-Report 69-E-09. 1969. ISBN 90-6144-009-2

10) Koop, H.E.M., J. Dijk and E.J. Maanders ON CONICAL HORN ANTENNAS.

TH-Report 70-E-10. 1970. ISBN 90-6144-010-6 I I) Veefkind, A.

NON-EQUILIBRIUM PHENOMENA IN A DISC-SHAPED MAGNETOHYDRODYNAMIC GENERATOR.

TH- Report 70-E-11. 1970. ISBN 90-6144-0 I 1-4 12) Jansen, J.K.M., M.E.J. Jeul.en and C.W. L::m:.rechtse

THE SCALAR FEED.

TH-Report 70-E-12. 1969. ISBN 90-6144-012-2 I 3) Tenling, D.LA.

ELECTRONIC IMAGE MOTION COMPENSATION IN A PORTABLE TELEVISION CAMERA. TH-Report 70-E-13. 1970. ISBN 90-6144-013-0

ElNLHIOVEN UNIVERSITY OF TECHNOLOGY THE NETHERLANDS

DEPARTMENT OF ELECTIHCAL ENGINEERING

Reports;

14) Lorencin, M.

AUTOMATIC METEOR REFLECTIONS RECORDING EQUIPMENT. TH-Report 70-E-14. 1970. ISBN 90-6144-014-9

15) Smets, A.S.

THE INSTRUMENTAL VARIABLE METHOD AND RELATED IDENTIFICATION SCHEMES. TH-Report 70-E-15. 1970. ISBN 90-6144-015-7

16) White, Jr., R.C.

A SURVEY OF RANDOM METHODS FOR PARAMETER OPTIMIZATION. TH-Report 70-E-16. 1971. ISBN 90-6144-016-5

17) Talmon, J.L.

APPROXIMATED GAUSS-MARKOV ESTIMATORS AND RELATED SCHEMES. TH-Report 71-E-17.1971./SBN90-6144-017-3

18) Kalalek, V.

MEASUREMENT OF TIME CONST ANTS ON CASCADE D.C. ARC IN NITROGEN. TH-Report 71-E-18. 1971. ISBN 90-6144-018-1

19) Hosselet, L.M.L.F.

OZONBILDUNG MITTELS ELEKTRISCHER ENTLADUNGEN. TH-Report 71-E-19. 1971. ISBN 90-61 44-019-X

20) Arts, M.G.J.

ON THE INSTANTANEOUS MEASUREMENT OF BLOODFLOW BY ULTRASONIC MEANS. TH-Report 71-E-20. 1971. ISBN 90-6144-020-3

21) Roer, Th.G. van de

NON-ISO THERMAL ANALYSIS OF CARRIER WAVES IN A SEMICONDUCTOR. TH-Report 71-E-21. 197 \. ISBN 90-6144-021-1

22) Jeul.en, P.I., C. Huber and C.E.Mulders

SENSING INERTIAL ROTATION WITH TUNING FORKS. TH-Report 71-E-22. 1971. ISBN 90-6144-022-X

23) Dijk, J., J.M. Berends and E.l. Maanders

APERTURE BLOCKAGE IN DUAL REFLECTOR ANTENNA SYSTEMS - A REVIEW. TH-Report 71-E-23. 1971. ISBN 90-6144-023-8

24) Kregting. J. and R.C. White, Jr. ADAPTIVE RANDOM SEARCH.

TH-Report 71-E-24. 1971. ISBN 90-6144-024-6 25) Damen, A.A.H. and H.A.L. Piceni

THE MULTIPLE DIPOLE MODEL OF THE VENTRICULAR DEPOLARISA TION. TH-Report 71-E-25. 1971. ISBN 90-6144-025-4

26) Bremmer, H.

A MATHEMATICAL THEORY CONNECTING SCATTERING AND DIFFRACTION PHENOMENA,INCLUDING BRAGG-TYPE INTERFERENCES.

TH-Report 71-E-26. 1971. ISBN 90-6144-026-2 27) Bokhoven, W.M.G. van

METHODS AND ASPECTS OF ACTIVE RC-FILTERS SYNTHESIS. TH-Report 71-E-27. 1970. ISBN 90-6144-027-0

28) Boeschoten, F.

TWO FLUWS MODEL REEXAMINED FOR ACOLLISIONLESS PLASMA IN THE STATIONARY STATE.

TH-Report 72-E-28. 1972. ISBN 90-6144-028-9