Shaped-multibeam antennas for the reception of TV-programsof broadcasting satellitesCitation for published version (APA):Buijnsters, J. L. M. (1996). Shaped-multibeam antennas for the reception of TV-programs of broadcastingsatellites. Technische Universiteit Eindhoven.

Document status and date:Published: 01/01/1996

Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Please check the document version of this publication:

• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.

If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne

Take down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.

Download date: 22. Nov. 2020

EINDHOVEN UNIVERSITY OF TECHNOLOGY

DEPARTMENT OF ELECTRICAL ENGINEERING

PROFESSIONAL GROUP: Electromagnetism

Shaped-multibeam antennas for the reception of TV-programs of broadcasting satellites.

by

ir. J.L.M. Buijnsters

EM-2-96

Philips GmbH Tuner Werk Krefeld Division Development TV Tuners & RF Units

Supervisors: R. Schiltmans (PTWK) dr. M.E.I. Jeuken (EUT)

Eindhoven, June 13th 1996.

@ Philips Electronics N. V. 1996

tlB

All rights are reserved. Reproduction in whole or in part is prohibited without the written consent of the copyright owner.

Acknowledgments

This report could not have been completed without the physical and moral support of various people. The cooperation with Philips GmbH TImer Werk Krefeld has been pleasant, fruitful and enlightening. I am particularly indebted to P.Burgstaller from Philips Krefeld for taking the initiative for this project and to R.Schiltmans also from Philips Krefeld for the stimulating discussions during many of the meetings. The work in this report has been performed during my one year stay at the Elektromagnetics division of the Eindhoven University of Technology. I am very grateful to all members of this group, particular to Dr. M.Jeuken for the discussions about the work and for his constructive comments on the draft version of this report.

1

2

Summ~ry

At the mOlrtent, the number of satellites for TV-broadcasting is increasing very fast. Because of this, th~re is a great demand for single reflector antennas for receiving TV-signals from two or more satellites simultaneously. To satisfy this demand a new special shaped-reflector

I antenna has been developed. This antenna can receive TV-signals from two or more satellites simultaneously, which are positioned up to 60 degrees apart from each other. The new shaped-reflector antenna is described comprehensively in this report. First, the shaped-refl~ctor surface and the optimum feed locus are derived both for the symmetric and the offset-d:~flector antenna. Second, the radiation pattern of the shaped-reflector antenna is described. :Then the calculations of the cross polarization, the antenna gain and the G/T are described. Finally, three specific designs of shaped-reflector antennas for simultaneous reception of to or three satellites are presented. These designs were made with a computer

I

programme developed by the author of this report. The four designs are: _

1. Astra ~ (19.2° east) and Eutelsat (13° east) 1. Astra ~ (19.2° east) and Astra 2 (28.2° east) 3. Astra 1 (19.2° east), Eutelsat (13° east) and Astra 2 (28.2° east) 4. Astra 1 (19.2° east), Eutelsat (13° east) and Telenor (1° west)

The advant~ges for the the first design of the shaped-reflector antenna compared to the offset parabola are moderated. The main reason of this is that the two satellites are positioned too close tokether to get some profit of the shaped-reflector antenna. For the second, third and fourth design, in which the satellites are positioned further appart from each other, the shaped-reflector antenna has great advantages when compared to the offset parabola. One of these advantages is that the antenna gain for the shaped-reflector antenna remain across the

I total field of view pretty much the same, while the gain of the offset parabola has a strong decrease for ,greater angles of reception. Furthermore for greater scan angles of reception the offset parabola does not meets the requirements of the European Telecommunication Stan­dard (E.T.SJ). For that case the shaped reflector should be preffered. An other antenna with the possibility to receive satellites which are positioned further away from each other is the torus. The main disadvantage of the torus compared to the shaped-reflector antenna is the limited space for the positions of the feeds, since the "foci" of the torus are positioned very close to eaclI other. The main conclusion is that the shaped-reflector antenna is superior to any other conventional­reflector antenna for the reception of TV-satellites which are standing approximately more then 10 degr~es apart from each other.

I

3

4

Contents

1 Introduction

2 Design methode of the shaped. reflector 2.1 Initial antenna geometry . . . . . . . . 2.2 Two dimensional profile optimization . 2.3 Derivation of the optimum feed locus .

3 Three dimensional optimization. 3.1 Dimensions of the reflector surface. . . . . . . . . . . . . . . . . . . 3.2 Calculations of the reference planes and reference distances ..... 3.3 Calculations of the reflector surface with an uniform illumination .. 3.4 Ca:lculations of the reflector surface with a non uniform illumination.

7

9 9

14 16

19 19 20 22 24

4 Symmetric-shaped reflector 29 4.1 Determining of the coefficients 29 4.2 Phase errors. . . . . . . . . . . 32

5 Offset-shaped reflector 35 5.1 Determining of the coefficients. 35 5.2 Phase errors. . . . . . . . . . . 39

6 Radiation patterns 41 6.1 Calculation of the far-field radiation pattern for the shaped-reflector in the

scaimed situation. ................................. 41 6.2 The optimum radiation pattern of the feed ................... 50 6.3 Calculations of the far-field radiation pattern for the shaped-reflector in the

unsanned situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53

7 Basic antenna parameters 59 7.1 Calculation of the gain. . . . . . . . 59 7.2 Calculation of the cross polarization 61 7.3 Calculation of the G/T ..... 62

8 Accuracy of the antenna design. 63 8.1 Accuracy of the feed position. . . . . . . . . . . . . . . . . . . . . . . 63 8.2 Accuracy of the antenna design for different locations inside Europe. 67

9 Results. 73

10 Conclusions 131

Bibliography 133

5

6

1 Introduction

Reflector antennas can be thought of as mathematical surfaces that redirect microwave rays from a point source feed to an aperture and beyond, into the far-field radiation pattern. In order for the surface to be useful as an antenna, it must have a special shape that reflects all rays so that they leave in a collimated beam and traverse the same total distance from the feed to a perpendicular planar wavefront. Only the parabola accomplishes these two constraints simultaneously. However, the greatest disadvantage of the parabola is the limited scan angle. When the f~ed source of a parabola is moved away from the focus on the axis of symmetry, the rays reflecting from the surface point in the direction of the scan angle, measured from the axis or boresight direction, are no longer exactly parallel. This leads to phase errors across the wavefront. As the feed displacement and scan angle increase, the aberrations begin to lower the peak gain and raise the sidelobe level of the antenna. The factors that effect the radiation pattern of a parabolic reflector for a particular scan angle are the ratio of the focal length to the reflector diameter (F / D) and the electrical size of the aperture. As the F / D ratio of a reflector increases, the surface curvature decreases, so the individual ray variations are not as great as for a deep, small F / D reflector. Thus, large F / D reflectors have large maximum scan angles. Also, as the electrical diameter D /). of the antenna increases, the geometrical errors are electrically magnified, resulting in greater phase errors and a smaller field of view. The conclus~on of the parabola with a general F / D ratio and electrical diameter is that scan angles of only a few degrees can be reached. This makes the parabola unsuitable for our purpose. The parabola with the feed in and out focus and the corresponding perpendicUlar wavefront is shown in fig. 1.

I I I

equi-phase plane

I I

,'equi-phase ./ plane

Figure 1: The parabola with the feed in and out focus.

7

The torus reflector is an other type of reflector antenna, which generates much better scanned beams. The torus reflector is circular in the plane of scan or azimuth plane and parabolic in the orthogonal or elevation plane. Since the radius of curvature of a parabola at its vertex is double its focal length, the torus azimuth circle radius is chosen normally be twice the elevation parabola focal length. The important condition for this reflector with circular profile is that each feed illuminates only a small portion af the reflector for which the circle does not deviate from a parabola too much. As with paraboloids, the geometrical difference between the actual circle and the desired parabola becomes increasingly significant with increased frequency. The greatest problem with circular profile reflectors is that for large scan angles the illuminated aperture portion in only a small fraction of the entire reflector. In other words, the aperture efficiency of the torus is very bad. The aperture efficiency of the torus is also inversely related to its field of view. An other disadvantage of the torus is the positions of the "focal points". Since these "focal points" are closely together, there is limited place for the different feeds. The torus with the illuminated portions of the reflector surface is shown in fig. 2.

Figure 2: The torus

This report present a new scanning reflector shaping method based on an idea proposed by Rappaport [1] and [2]. This principle is to shape the reflector to optimize simultaneously the aperture phase laws for the two different beams, one in the boresight direction and the other at the maximum angle of scan. The proposed method can roughly be summarized into three steps. Setting up of the initial geometry and optimization of the reflector surface and derivation of the optimum feed locus take place. In all the above optimization processes Geometrical Optics is applied.

8

2 Design methode of the shaped reflector

The shaped-reflector surface has to be optimized to minimize the phase errors for two beams. The first beam propagates along the axis of symmetry of the shaped reflector, known as the unscanned direction. The second beam propagates along an axis which is rotated a degrees with respect to the symmetry axis, known as the scanned direction. Purpose of this section is to find a' describtion of the shaped-reflector surface in the plane y = 0 and the locations and orientations of the feeds such that the phase errors of the scanned and unscanned beams are minimum. For this reason, the starting point is a paraboloid with focal lenght It and tilded by a degrees. The mathematical description of this paraboloid is given in the first paragraph. Then, in the second paragraph a two dimensional optimization at the plane y = 0 is described. And in the last paragraph a derivation of the locations and orientations of the feeds is given.

2.1 Initial antenna geometry

The starting point is a description of the tilted paraboloid with focallenght It and a being the maximum angle of scan. The tilted paraboloid has'to meet the following two requirements:

1. The orgin of the coordinate system has to belong to the tilted paraboloid. 2. The tilted paraboloid has to be a zero partial derivative over x at the orgin of the

coordinate system; g; = o.

To find the expression of the tilted paraboloid we start with the expression of the paraboloid, which is not tilted or translated in the XII, yll, Zll coordinate system. The symmetric paraboloid with the vertex at the orgin of the coordinate system and the z"-axis as the axis of symmetry can be expressed as:

1 zl/ = _(x,,2 + y"2)

4ft (1)

Extension of equation (1) gives an expression of the offset-paraboloid antenna:

Y"2 1 Zll = -4ft + 4ft (x"2 + (yll - yn

2) (2)

The parameter yf' represents the height between the centre of the reflector and the scanned feed which is called the offset-height. So, y/' = 0 corresponds to a symmetric paraboloid and y/' > 0 will be used for an offset paraboloid. It can be shown in fig. 3 that the vertex of the paraboloid y~' is moved along the y"-axis. The expression of the offset paraboloid (2) meets also to the two requirements mentioned above. An intersection of this paraboloid with the yll zit-plane and the XII zit-plane is shown in fig. 3 and fig. 4 respectively. The focus Ft in the (XII, y", z") coordinate system is given by:

- 0 - yf - It l!E

4ft

(3)

9

Z"-as

Zi' -. - - - - - - ...

Figure 3: Paraboloid in the plane x" = o. Z" .. as

<>

Figure 4: Paraboloid in the plane yll = O.

Making use of the property of a paraboloid that transmitted rays starting at a feed located at the focus will reflect and be collimated to form a planar aperture field without phase errors:

p+ pcosO - 2ft

J X"2 + (y" - yn2 + (Zll - zi')2 + (z:' - Zll) = 2ft

x"2 + (y" - y~')2 + 4ft(z~' - Zll) 4ff = 0

The plane Zll zf is chosen to be the equi-phase plane.

By means of:

we finally find:

x"2 + y"2 - 2y" y~' - 4/tZIl = 0

10

(4)

Now the paraboloid will be rotated by an angle -a degrees. However, the same result is obtained by rotating the coordinate system by an angle +a degrees. This rotation is shown in fig. 5. The matrix of the coordinate transformation is given by:

Z"·as

Z'·as I y"=y'=o I

X"·as

X'·as

Figure 5: Coordinate tronsformation

(

X" ) y" = Zll . (

cosa 0 Sina) 010

-sina 0 cosa (5)

Substitution of (5) in (3) gives:

X~ cos a + z~ sin a - 0

yi - x~ sin a + z~ cos a

The final result of the focus Ft in the (x', y', z') coordinate system can be written as follows:

x~ - /tsina (Uif -1) y~ - y~ (6)

z~ /tcosa(l-(fttf)

The expression ofthe paraboloid in the (x',y', z') coordinate system can be found by substi­tution of (5) in (4):

x,2 cos2 tt + 2x' z' sin a cos a + z,2 sin2 a + y'2 - 2y' y~ + 4ftx' sin a - 4ftz' cos a = 0 (7)

To meet the two requirements mentioned above, a translation of the paraboloid has to be executed. The same results can be obtained by a translation of the coordinate system, which is shown in fig. 6.

11

I y'=y=O I z-as

~

x'-as

x-as <} ............ -='--,.-'-:-. O.

Figure 6: Coordinate transformation.

To find the transformation (XI,y',Z/) -? (x,y,z) we have to look for the point, at the plane y' = y = 0, where the partial derivative over x, equals zero, or g~: = O.The first step is to rewrite (7) as a second-order polynomial in x'. Then we have to look for the value of Zl

with the discriminant equals zero. With the discriminant equals zero, there is precisely one solution of x' for this particular value of Z'. SO, the value of x, and Zl correspond with the orgin of the (x, y, z) coordinate system. Rewriting (7) as a second-order polynomial in x, ,gives:

ax'2 + bx' + c 0

with:

a - cos2 a b - 2Z' sin acosa + 4ft sin a c - zl2 sin2 a + yl2 - 2y' y~ - 4ftZ' cos a

The discriminant of this equation is:

D2 _ b2 - 4ac

- 16z' ft cos a - 4yl2 cos2 a + 16fl sin2 a + 8Y'Yi cos2 a

With the constrain that the discriminant equals zero at the plane y' = y = 0, gives:

• 2 Zl = -It sm a

cos a I sin2 a

z=z +ft-­cos a

The solution of (7) with the discriminant equals zero can be written as:

x' xi = x~ = -2b

= -ft sin a (1 ++) a cos a

===> x = x' + ft sin a (1 + +) cos a

12

(8)

(9)

The transformation of the (x', y', Z/) coordinate system to the (x, y, z) coordinate system can be found with the help of y' = y and the expressions (8) and (9):

Substitution of (10) into (6) gives:

. sin a Xt - ftsma- ft-­

cos2 a

Yt - y~

. ( cosa+ c~o: ) _ ft s1na 0

cos a . sma

z, = /tcosa(I-(i;J')

(10)

Finally the coordinates of the focus Ft of the tilted and translated paraboloid can be written as:

Xt . - ft sin a ( co;2 0: + (fj; f) Yt - Yt (11)

Zt = ft cos a (CO;20: - (?]; f) To find the ;general solution of Zti(x, y) we have to rewrite expression (7) as a function of z. First we have to apply the coordinate transformation to the equation (7). After the coordi­nate transformation the expression (7) can be written as:

az2 +bz+c= 0

with:

a sin2 a b - 2xsinacosa - -.ilL coso: C x 2 cos2 a + y(y - 2Yt)

The two solutions of this equation are:

-b± Vb2 - 4ac Zl,2 =

2a·

- Si!a [Sin~~:"a XC08a± (Sin~:'J +y(2y,-y) :!~X] With the constrain a~~2 = 0 at the plane y = 0, there remains only one solution:

1 [ 2ft Zt" = -- - xcosa-~ sina sinacosa ( )

2 ] 2ft 4ft

+y 2Yt- - --x sin a cos a ( y) sina

13

(12)

The tilted paraboloid Zti(X, y) is described in (12) and the coordinates of the corresponding focus F t are given in (11). The equations (11) and (12) are similar to expressions to be found in [1] and [2].

2.2 Two dimensional profile optimization

The next step is to approximate the tilted paraboloid Zti(X) by an even polynomial in x called Zpol(X) at the plane y = O. Since the paraboloid is symmertic about the YOZ-plane, the polynomial has only terms of even order:

(13)

On account of the symmetry about the YOZ-plane and the constraint that the orgin of the coordinate system has to be a point on the paraboloid, Zo is equal to:

Zo 0

At the moment we assume that Zpol (x) is the expression of the unknown profile of the reflector surface for y=O, so:

(14)

The coefficients Tl and T2 are determined over a specified x-range interval. The range of this interval depends on the maximum acceptable phase error. The x-range interval is shown in fig. 7.

The coefficients Tl and T2 of the polynomial will be found by means of the least-squares

z-as

" .Zpol

. zti x-as <;.J--------===--I--=------

o Xt -interval

Figure 7: x-range interval

method. The procedure of the least-squares method can be described as follows: . First the difference between the two functions Zpol(Xi) and Zti(Xi,O) is formed at N regularly spaced points. Then the differences at these points are squared and summed. To find the minimum values at Tl and T2 we have to differentiate the remaining equations with respect tOT! and TZ. Finally we obtain two lineair equations for Tl and TZ.

14

First we start with the following N expressions:

rlXY + r2xt - Zti(XI,O) - reSI rlx~ + r2x~ - Zti(X2, 0) - reS2

rlx; + r2X{ - Zti(Xi, 0) = reSi

rlxj., + r2xt - Zti(XN, 0) - reSN

res!, reS2, ... I reSi"", reSN are called the residues

Next we minimize that:

This means: ° N

a<p 2: (a<p ares i ) arl = . areSi' arl

t=1

Substituted into (15) gives:

{ f. reSi . x; = 0 i=1 N

2: reSi . X{ = ° i=l

N 2: res; = <p(rll r2) i=l

and

{ .J!!L -with: {Jres,

{JreSi -{Jrl

{ .J!!L -with: {JreSi

{}res. -{}r2

{ ,~ hx~ + r,xl - z.(x,. OJ) xi - 0

2: (rlx; + r2X{ - Zti(Xi, 0») Xl = 0 i=l

Rewriting these equations:

2 resi x?

$

2 reSi x1

$

(15)

(16)

(17)

We obtain a system of linear equations with rl and r2 as the unknown. To determine the coefficients rl and r2 with great accuracy, the value of N is equated to 100.

15

2.3 Derivation of the optimum feed locus

At this stage, only the location of the scanned focus is known. To find the location of the unscanned focus we start with the. expression of the symmetric untilted paraboloid with focal lenght / u and the z-axis as the axis of symmetry.

1 2 2 Zut(X,y) = 4/

u (x + y ) (18)

Extension of (18) gives an expression of the offset paraboloid:

() y; 1 (2 ( )2) Zut X, Y = 4/u + 4fu x + y - Yu (19)

The parameter Yu represents the height between the centre of the reflector and the unscanned feed. Equation (19) has to meet the two requirements from paragraph 2.1 also. With the help of fig. 8 we can rewrite (19) as follows:

Z-as

Zu .•.••••••••• ~Fu(O,Yu'ZJ I x=O I I I

yl u

V-as

Figure 8: Unscanned offset paraboloid.

Zut(X, y) = Zu

For small values of x, Zpol and zsur(x,O) can be approached as:

(20)

Zsur(X, O)lxR:iO = zpol(x)lxR:iO ::::: rlx2 (21)

Equation (21) describes in the plane y 0 a parabola. Comparing the equations (21) with equation (18) gives an expression for rl:

1 rl = - (22)

4fu

16

It should be possible to determine the position of the focus Fu(xu, Yu, Zu) by equated (20) and (21) together with their partial derivatives at the point x = Y = O.

Zut(O, 0) y~ Zu - fu + 4fu

(hut, _ 0 ox :z:=o-

OZut 1 __ Yu oy y=o- 2fu

Substitution of (23) into (22) gives:

Zu

From (24) follows:

Xu = 0

=

=

Zsur(O, 0)1:z:~0 = 0

oZsur(x, 0)1:z:~0 1

ox :z:=o= 0

OZ8ur(;~ 0)1:z:~0 Iy=o=?

(23)

(24)

(25)

(26)

(27)

The remaining task is to determine Yu' This is not possible with (25), since the partial deriva­tive of Zsur over y is unknown. However, this will be solved by replacing Zpol (13) with Zti (12) in equation (21), so:

And:

Zsur(X, O)I:z:~o = zpol(x)I:z:~o becomes: zsur(x, Y)I:z:=Y~o = Zti(x, y)

OZsur(X, Y)lx=y~o I _ OZti(X, y) I __ Yt cos a oy :z:=y=O- oy :z:=y=o- 2ft

Equation (25) becomes:

OZut 1 _ Yu oy y=o- -2fu

OZsur(X'Y)I:z:=Y~o I _ _ Ytcos a oY x=y=O- 2ft

By the use of equation (22) we finally find the expression of Yu'

1 Yu = Yt cos a-

4j tTl

Summarized the unscanned focus Fu:

{

Xu = 0

Yu = Yt cos a 4h.rl Zu = 4;1 - TIY~

(28)

(29)

At this stage the expressions of the reflector surface Zsur for Y = 0 and the feed coordinates of the scanned and unscanned foci are known. However, the orientations of the feeds are

17

still unknown. In other words, it is still unknown which parts of the reflector surface are considerd as the centre of the illuminated circles. These places can also be indicated as the places where: Ou = 0 for the unscanned situation and Ot = 0 for the scanned situation. Each feed is assumed to be at the origin of the corresponding (r, 0, ¢) coordinate system, with 0 = 0 as the axis of symmetry for the feeds. The middle of the Xt-range interval, indicated with Xct and the orgin of the (x, y, z) coordinate system are taken as the centre of the illuminated circles for the scanned and unscanned feeds respectively. The reflector surface Zsur for y = 0 and the locations and orientations of the feeds are shown in fig. 9. It should be noticed that fig. 9 is valid for symmetric as well as offset reflectors. The pa-

z-as

x-as<J----------+-~=-~--~----------

Xct 0

Figure 9: Locations and orientations of the feeds.

rameters Ct, it and Yt determine the shape of the reflector and the locations of the feeds. The offset height is determined by Yt with: Yt = 0 corresponds to a symmetric reflector and Yt > 0 will used for offset reflectors. In the next paragraph an extension of the reflector surface to three dimensions will be given.

18

3 Three dimensional optimization.

The extension of the plane Y 0 of the previous section to the 3 dimensional reflector surface zsur(x, y) is presented in this section. The derivation who describes this extension is valid for both the symmetric and offset reflector. The derivation starts with the groundplane of the reflector surface which will be divided into a grid of points (Xi, Yj). This grid of points is defined in the first paragraph. To calculate the phase errors in the scanned and unscanned direction we have to introduce a reference plane and a reference distance for both directions. The reference planes and distances are described in the second paragraph. In the third paragraph we calculate for each grid point the most optimum z-coordinate when we use theoretical feeds with uniform radiation patterns. The most optimum z-coordinate means that rays from both feeds have minimum phase errors. Also in the fourth paragraph we calculate for each grid point the most optimum z-coordinate but now for practical feeds with non uniform radiation patterns. Since each grid point is illuminated by a different amount of radiation energy from both feeds, we have to introduce weighting functions. At the end of this section we have calculated the total reflector surface into a grid of points (Xi, Y j, Zij ).

3.1 Dimensions of the reflector surface.

A front view of the reflector surface is shown in fig. 10. We distinguish the different parts of the reflector surface which are mainly illuminated by the different feeds. The feed which

y-axis

height x-axis

width

Figure 10: Groundplane of the reflector surface.

is placed in the unscanned focal point will mainly illuminate the circle in the middle of the reflector surface and the feed which is placed in the scanned focal point will mainly illuminate the outer circle. Areas where these circles overlap each other, will therefore be illuminated by both feeds. These areas play a prominent part in the use of theoretical and practical feeds, which will be treated in paragraph 3.3 and 3.4 respectively. Notice that the positions of the circles depend of the choice of the height and the width of the reflector surface. The centre of the scanned circle, indicated by Xct, can be calculated as follows:

width d' f h . I Xct = -2- - ra lUS 0 t e CIrc es

19

(

The groundplane of the reflector surface is divided into a grid of points (Xi, Yj). The aim is to find for each grid point the most optimum z-coordinate. The most optimum z-coordinate means that reflected rays from a grid point for both feeds have minimum phase errors. Since both the symmetric and offset reflector are symmertical with respect to the y-axis, we only have to calculate the grid points of the plane X > O. Therefore, the x-axis for X > 0 and the y-axis are divided into N points. The x values of the grid are indicated by Xi, with i = 1,2, .... N and the y values of the grid are indicated by Yj! with j = 1,2, .... N if X :5 Xct.

For X > Xct we have to determine for each Xi value the first value Ml and the last value M2 of the index j. Since Ml and M2 are dependented on the value of i, we can write for i and j:

i - 1,2, ... ,N

. M (.) M (.){ Ml(i) ?: 1 J = 1 Z I"" 2 z M

2(i):5 N

(30)

3.2 Calculations of the reference planes and reference distances.

The reference plane and reference distance in the unscanned situation are shown in fig. 11. The reference plane has to be perpendicular to the· reflected rays from the feed which is

z-axis

reference plane

dref u

x-axis <J----------<D-------o

Figure 11: Reference plane and distance in the unscanned situation.

placed in the unscanned focal point. For the unscanned reference distance drefu is chosen the distance from the unscanned focal point Fu to the orgin of the coordinate system and from the orgin to the reference plane. The orgin of the coordinate system is chosen because this is the centre of the illuminated circle in the unscanned situation. We assume that each reflected ray from the feed which is placed in the unscanned focal point is perpendicular to the corresponding reference plane. The reference distance in the unscanned situation can be calculated as follows:

drefu = v'ya + za+zpwith zp the distance from the reference plane to the plane z = 0(31)

The reference plane and reference distance in the scanned situation are shown in fig. 12. The reference plane in the scanned situation is perpendicular to the reflected rays from the feed which is placed in the scanned focal point. The expression for the reference plane in the scanned situation is:

z =xtana+Zt

20

x-axis <:]------+-""""'=::::-rd":-------

Figure 12: Reference plane and distance in the scanned situation.

Notice that the reference plane crosses the z-axis through the point Zt. For the scanned refer­ence distance dreft is chosen the distance from the focal point Ft to the point (xct, 0, Zpol(Xct))

of the reflector surface and from the point (xct, 0, Zpol(Xct» to the reference plane. The point (xct, 0, Zpol(Xct» is chosen because this is the centre of the illuminated circle in the scanned situation. We also assume that in the scanned situation each reflected ray is perpendicular on the corresponding reference plane. For the first part of the reference distance between the focal point Ft and the point (xct, 0, Zpol(Xct») we can write:

d1 =' V(xct - Xt)2 + Yl + (Zpol(Xct) - Zt)2

The distance between the point (Xct, 0, Zpol(Xct») and the reference plane can be calculated with the help of an extra plane. This extra plane has to pass through the point (xct, 0, Zpol(Xct))

and has to be parallel to the reference plane. This extra plane crosses the z-axis through the point C, which is shown in fig. 12. Point C can be expressed as follows:

Z - xtana+C

Zpol(Xct) = Xct tan a + C

::::? C = Zpol(Xct) - Xct tan a

The shortest distance d2 between both planes can be expressed as follows:

d'}, (Zt - C) cosO'

d2 = Zt cos 0'- Zpol(Xct) cos 0'+ Xct sin a

Finally, the total reference distance dreft = d1 + d2 can be expressed as:

dreft = V(xct - Xt)2 + yl + (Zpol(Xct) - Zt)2 + (Zt Zpol(Xct» cos a + Xct sinO' (32)

21

3.3 Calculations of the reflector surface with an uniform illumination.

In this paragraph we use feeds with an uniform radiation pattern. This means that within the illuminated circles, shown in fig. 10, the radiation energy uniform is. We assume that the radiated energy outside the illuminated circles equals zero. Now, we determine for each grid point (Xi,Yj) the optimum z-coordinate Zij. Therefore we have to compare for each grid point (Xi, Yj) the distance du,ij(Z) with the reference distance drefu in the unscanned situation and dt,ij(Z) with dreft in the scanned situation. The un­scanned and scanned situation are shown in fig. 13 and fig. 14 respectively. It can be shown

z·axis

(Xi'YI'?) '. a·

x.axis<l-----o""· '---0+-------

Figure 13: Calculations of the grid points in the unscanned situation.

x·axis <I----------=.,o~+o,,-------

Figure 14: Calculations of the grid points in the scanned situation.

from fig. 13 that the unscanned distance du,ij can be written as:

(33)

When we replace the point (xct, 0, Zpol(Xct» from (32) by (Xi, Yj, z), the scanned distance dt,ij can be written as:

22

(34)

The difference in path length between du,ij(Z) and dt.ij(Z) with the corresponding reference distances drefu and dreft can be expressed in the following two error functions:

erru,ij(Z) = (drefu - du,ij(Z»2

errt,ij(Z) = (dreft - dt.ij(z»2

(35)

(36)

Notice, that these error functions are only a function of z. Equations (35) and (36) are only valid for points of the reflector surface which are illuminated by the feed in the unscanned and scanned focus respectively. For points of the reflector surface which are illuminated by both both feeds, we can write the following error function.

errT.ij(Z) = erru,ij(z) + errt,ij(Z) (37)

The most optimum value of Zij can be found by minimizing the error functions (35), (36) and (37). This can be done by equating the derivatives of each error function to zero. For points of the reflector surface which are only illuminated by the feed which is placed in the unscanned focal point, we can write:

d (erru.ij(z» = -2 (drefu _ du ij(Z» d (du,ij(Z» = 0 dz ' dz

d (d . '(z» liM can be expressed as' d z •

d (du,ij(Z» = (z - zu) -1 d Z J(Xi - xu? + (Yj - Yu)2 + (z - zu)2

When we use the following definition:

disu = V(Xi - xu)2 + (Yj - Yu)2 + (z - zu)2

,equation (38) can be expressed as:

The definitions of drefu and du,ij(Z) can be found in the equations (31) and (33).

(38)

(39)

For points of the reflector surface which are only illuminated by the feed which is placed in the scanned focal point, we can write:

d (errt,ij(z» = -2 (dreft _ dt

i .(z» d (dt,ij(Z)) = 0 dz .J dz (40)

d (~!i; (z» can be expressed as:

23

When we use the following definition:

dis t V(Xi - Xt)2 + (Yj - Yt)2 + (z - Zt)2

,equation (40) can be expressed as:

(drelt - dt,ij(Z» . (Z~:tZ) + cos a ) = ° (41)

The definitions of drelt en dt,ij(Z) can be found in the expressions (32) en (34). For points of the reflector surface which are illuminated by both feeds, we can write:

d (errT,ij) = d(erru,ij(z» +d(errt,ij(z» = ° dz dz dz

(42)

Now, we are enable to calculate the reflector surface as a grid of points for feeds with an uniform radiation pattern. However, feeds with an uniform radiation pattern do not exist in practice. Therefor we calculate the reflector surface with more practical feeds in the next paragraph.

3.4 Calculations of the reflector surface with a non uniform illumination.

Now we are calculating the reflector surface by using feeds with non uniform radiation pat­terns. Suppose that a feed with a non uniform radiation pattern is placed in the orgin of a spherical coordinate system (r, 0, ¢) with 0 = ° the main direction ofradiation. The strength of the electric field for the feed with a non uniform radiation pattern decreases as the angle 0 increases. The power-current density S(T) of the feed is proportional to the squared absolute value of the electric field. The power-current density S(T), according [4], can be expressed as:

with Zo the wave impedance for free space. Fig. 15 shows the power-current density as a function of the angle O. The normalized radiation pattern of the feed can be expressed as follows:

( ) E(O,¢) n( ) I O,¢ = E(O,O) = cos 0

We assume that 1(0, ¢) has rotation symmetry and is therefore independent from ¢. Hence 1(0, ¢) = 1(0). We have to choose the value of n in such a way that we get an optimum between the spillover efficiency and the. aperture efficiency. An edge illumination of about -12dB appears to be the optimum. If the angle 0 at the edge of the illuminated circles, fig. 10, is marked with 00, then the value of n can be calculated as follows:

10 log (f(OO»2 - -12 dB -1.2

2 log ( cos (0)

In contradistinction to feeds with an uniform radiation pattern, each grid point of the re­flector surface will be illuminated by both feeds with a non uniform radiation pattern. Since

24

S(94) S(O,O) [dB]

-20

o

Figure 15: Power-current density of the feed.

z-axis

ZpoJ(xct)

x-axis <J-----_--.()-------o o

Figure 16: Determining of the weighting functions.

each grid point will be illuminated by a different amount of radiation energy by both feeds, we have to introduce weighting functions [3] for the calculations of the error functions. We distinguish weighting functions for the unscanned Wu,ij and for the scanned situation Wt,ij'

The total error function can be written as:

(43)

The definitions of erru,ij(z) and errt,ij(Z) can be found in the equations (35) and (36) re­spectively. As shown in fig. 16, we have to calculate the angles Ou and Ot for each grid point (Xi, Yj, Zij). These angles determine the values of the weighting functions according:

Wu cosnu Ou

Wt - cosnt Ot

(44)

(45)

The angles Ou and Ot can be calculated with the help of the cosine-rule. For calculations of

25

z-axis z-axis

x-axis <J-----==--~:.>::::_--===----- y-axis<J----=-::::::.cJ..---=-----

width height

Figure 17: Determining of the angle Ou,

z-axis

x-as <J---~==--+~==------XcI 0

x-axis 4~----=';---------". width

z-axis

y-axis<l-----====+-O..::===-----­

height

Figure 18: Determining of the angle Ot.

the angle Ou we use fig. 17:

a2 + b2 _ c2

Ou = arccos 2ab

where:

a Vya +z; b = .;r-x-; -+-C-Y-j ---y-u-)2-+-CZ---z-u-)2

c = J x; + yJ + z2

For calculations of the angle Ot we use fig. 18:

26

where:

a = V(Xi - Xt)2 + (Yj - Yt)2 + (Z - Zt)2

b - JCXct - Xt)2 + Yt + (Zpol(Xct) - Zt)2

c J(Xi - Xct)2 + yJ + (Z - Zpol(Xct))2

We see that both angles are only dependent on z, so also the weighting functions Wu en Wt are only dependent on z. Therefore, the total error function is only a function of z. To find the most optimum z coordinate we have to minimize the error function. This can be done by taking the derivative of the totale error function to z and equating this derivative to zero:

d -----=~ = 0

where:

d errT,ij dz

«() ( » d erru,ij(z) d wu(Ou(z» ( )

= Wu u z· d Z + d z . err u,ij Z +

«() ( ») d errt,ij(Z) d Wt«(}t(z)) ( ) Wt t z· d Z + d Z • errt,ij Z (46)

In equation (46) are wu«(}u(z», Wt(f1t(z)), erru,ij(z), errt,ij(Z), d er~"~ij(Z) and d er;t~j(z) are given by the equations (44),(45),(35),(36),(38) and (40) respectively. The remaining two terms which have to be calculated are d W.,(O.,(z» and d Wt{Ot{z» Using the rules of differentiation d z d z . for these two unknown terms, we get:

d wu(Ou(z» dz

d Wt«(}t(z)) dz

d wu(Ou(z» . d (}u(z) d (}u d z

_ d Wt«(}t(z» . d (}t(z) d (}t d Z

Substitution of (44) and (45) gives us:

d wu«(}u(z» d Ou

d wtCOt(z» dOt

- -nu· cos(nu-I) Ou . sinOu

-nt . cos(nt-I) (}t • sin Ot

Finally we have to calculate d :,,;2:) and d !tiz). Both equations of Ou(z) and Ot{z) have the form:

y = arccos X with the derivative equal to:

After some calculations we get:

= Zu . q + (z - zu) . p

q. J(Y~ + z~)· q - p2

dy = -1 d x ,,11- x2

(Zt - Zpol(Xct» • S + (z - Zt) . r

27

where:

p = 2 2 Yu + Zu - YjYu - zZu

q - Xl + (Yj - Yu)2 + (Z - Zu)2

r - x; + Y; + Z; - XiXt - XctXt + XiXct - YjYt - ZtZpol(Xct) + Z(Zpol(Xct) - Zt)

s - (Xi Xt)2 + (Yj Yt)2 + (Z - Zt)2

Since all terms of equation (46) are known, we can calculate for each grid point (Xi, Yj) of the reflector surface the optimum z-coordinate. A computer program has been written for the calculations of all these z-coordinates. The input parameters of the computer program are: - the maximum scan angle ct, (deg.) - the wavelength A, (cm.) - the focal distance It of the scanned focal point, (A) - the offset distance Yt, (A) - the height of the reflector surface, (A) - the width of the reflector surface, (A) The numerial method used for the calculations of the values Zij is a combination of the Koorden-Newton method and the Bisection method. The Bisection method ensures conver­gence, while the Koorden-Newton is a much more faster convergent method, see also [5].

The output of the computer program is a matrix with the elements Zij which describes the total reflector surface in a grid of points. Since this description of the reflector surface is not very usefull for further calculations, we are looking for a continuous function which describes the surface of the reflector. In the next two sections we determine these functions of the symmetrical and offset-reflector.

28

4 Symmetric-shaped reflector

The analytic expression of the symmetric-shaped reflector will be determined in this section. The method of calculations for both the symmetric-shaped reflector and the offset-shaped reflector are the same. The symmetric-shaped reflector will be treated separately from the offset-shaped reflector because the number of equations for the symmetric-shaped reflector is much smaller. Therefore it becomes easier for the reader to understand the method of calculations. If the method of calculations for the symmetric-shaped reflector is known, it is also very easy to understand the calculations of the offset-shaped reflector. The analytic expression of the reflector surface can be found by an extent ion of the 2-dimensional expres­sion of zsur(x, 0) (14) to a 3-dimensional polynomial. Since the shaped-symmetric reflector is symmetrical with respect to the XZ-plane and the YZ-plane the polynomial consists only of even terms of x and y. We finally find for the polynomial:

(47)

The coefficients rl and r2 are determined by means of the least squared method. They are already calculated in paragraph 2.2. The remaining coefficients P, Q, Sand R are also determined by means of the least squared method. This will be treated in the first paragraph. Since the shaped-reflector surface is not a real parabola, there are always phase errors in the aperture. Calculations of these phase will be treated in the second paragraph.

4.1 Determining of the coefficients

As mentioned before, the coefficients P, Q, Sand R will be determined by means of the least squared method. Therefore we make use of the calculated matrix of the previous section which consists of the grid of points (Xi,Yj,Zij). Equation (47) will be sampled on the same positions (Xi, Yj) of the grid. The difference between the sampled points and the exact values Zij are called the residues reSij. These residues can be defined as follows:

Zsym-sur(XO, YO) - Zoo = resOO Zsym-sur(XO, yd - ZOI = resOl

Zsym-sur(X2, Y3) - Z23 = res23

Zsym-sur(Xi, Yj) - Zij = reSij

resoo, reSOI, ... , reSij, ... , res N M-J(N) are called the residues

N M2(i) We have to minimize: 2::: 2::: resij

i=l j=Ml(i) N M2(i)

Now we write: 2::: 2::: resij = cp(P, Q, S, R) i=l j=Ml(i)

29

The definitions of the boundaries N, Ml(i) and M 2(i) can be found in paragraph 3.1. To minimize I.{) means:

81.{) = 0 8P

and 81.{) = 0 8Q

and 81.{) = 0 8S

and 81.{) = 0 8R

(48)

8 N M,(.) ( 8 8~~ij ) { ...!l!L - 2 reSij 8; == f:t .lto 8r~ij' with: areSij

areSij VJ 8P -

l- j- 1 ~

8", N M,(.) ( 8", 8~~ij ) with: { ...!l!L - 2reSij

8Q = L L 8res··· 8reSij 8reSii = x? V?

i=l j=Ml(i) tj I:JQ t j

8 N M,(.) ( 8 8~;ij ) { ...!l!L - 2 reSij

~=L L -cp. with: 8reSij

8S i=l j=Ml(i) 8reSij areSij = x 1 V~ as t j

8 N M,(.) ( 8 8~~ij ) with: {

...!l!L 2 reSij ~=L L. _I.{)_. 8reSij

8R . . (") 8reSij 8reSij - yj $=1 j=Ml t 8R

Substituted in (48) gives:

N M2(i)

2: 2: reSij VJ - 0 i=l j=Ml(i) N M2{i) 2: 2: reSij x; V~ - 0 i=l j::.:::Ml(i) j (49) N M2(i) 2: 2: reSij xl VI = 0 i=l j=Ml(i)

N M2(i)

2: 2: reSij yi - 0 i=l j=Ml(i)

Substitution of:

reSij = Zsym..sur(Xi,Yj) - Zij

- rlx; + r2X[ + (P + Qx~ + SX[)Y; + Ryj- Zij

in (49) gives:

30

After reorganization of the different terms we get the following matrix:

( :f~:~l :I~:;l :I~:~l :g:!l) ( ~ ) _ ( ~f~l ) m[3, 1] m[3, 2] m[3, 3] m[3,4] S - r[3]

m[4, 1] m[4, 2] m[4, 3] m[4,4] R r[4]

with the following definitions of the matrix elements m[i,j] and column eh~ments r[i]:

N M2(i) m[l,~] L L yj

i=1 j=Ml(i)

m[1,2] = m[2, 1] N M2(i)

= L L xl yj i=1 j=Ml(i) N M2(i)

m[1,3] = m[2, 2] = m[3, 1] = L L xl yj i=1 j=Ml{i)

m[l,~] = m[4, 1] N M2(i)

L L Y1 i=1 j=Ml(i) N Mz(i)

m[2, 3] = m[3, 2] L L xf yj i=l j=Ml(i) N M2(i)

m[2,4] = m[4, 2] = L L xl Y1 i=1 j=Ml(i) N M2(i)

m[3, 3] - L L x~ yj i=1 j=Ml(i) N M2(i)

m[3, 4] = m[4, 3] - L L Xl yJ i=1 j=Ml(i) N M2(i)

m[4,4] L L y~ i=1 j=Ml(i) J

N M2(i)

r[l] =, L L (Zij - Xr(r1 + r2x;)) yJ i=1 j=Ml(i)

,N M2(i)

r[2] =! L L (Zij - x;(rl + r2x;)) x; YJ ; 1.=1 j=Ml(i)

N M2(i)

r[3] =, L L (Zij - x;(rl + r2x;)) Xl yJ i=1 j=Ml(i)

i N M2(i)

r[4] -, L L (Zij - Xr(rl + r2Xr)) Yl '1.=1 j=Ml(i)

(50)

We obtain a system of lineair equations with the unknown coefficients P, Q, Sand R. Table 1 gives, by the way of example, the calculated values of the coefficients for a scan angle of 30

31

degrees. In addition the dimensions of the shaped-reflector antenna are also given in table 1. The defintions of the used quantities can be found in paragraph 2.1 and 3.1.

Table 1: Quantities of the antenna and the calculated values of the different coefficients.

Quantities a= 30u Yt = 0 A=3cm width=28.8 A ft = 23.75 A height=16 A

Calculated coefficients rl = 0.2372 Q = 0.8157 r2= 0.1149 S = -3.8142 P = 0.2594 R = -0.0874

4.2 Phase errors

Since the reflector surface is not a real parabola, there are always phase errors in the aperture. The phase errors for the unscanned and scanned situation can be defined as follows:

fasefoutu = drefu - du

fasefoutt - dreft - dt

The terms drefu and dreft represents the reference distances for the unscanned (31) and scanned situation (32) respectively. The terms du and dt represents the distances from the feed to an arbitrary point on the reflector surface and from that point to the reference plane for the unscanned (33) and scanned (34) situation respectively. For the calculations of the phase errors we use the shaped reflector from the previous paragraph. The phase errors in the unscanned and scanned situation, expressed in wavelengths, are shown in fig. 19 and fig. 20 . respectively.

32

SCANNED PHASE ERRORS

2 4 6 8

'I (lambda)

., -13.132

-13.04

-0.06

-0.08

-0.1

-13.12

-0.14

-0.16

Figure 19: Unscanned phase errors.

SCANNED PHASE ERRORS

'I (lambda)

Phase error {lambdal

Phase error ( lambda)

Figure 20: Scanned phase errors.

33

34

5 Offset-shaped reflector

The analytic expression of the offset-shaped reflector will be determined in this section. The offset-shaped reflector is only symmetric with respect to the YZ-plane. In this case we also have to add odd terms of y to the polynomial. The expression of the offset-shaped reflector surface is defined as follows:

Calculations of the coefficients happens on the same way as the symmetric-shaped reflector with the only difference that the number of coefficients of the offset-shaped reflector and with that also the number of equations is increased. Calculations of these coefficients will be treated in the first paragraph, while the phase errors will be treated in the second paragraph.

5.1 Determining of the coefficients.

The coefficients N, T, U, P, Q, S, V, Wand R from equation (51) will be determined by means of the least squares method. Since this method has been already explained for the symmetric-shaped reflector, we will only present some interim and final results. Equation 51 will be sampled on the points (Xi, Yj). For these points we can express the residues reSij as follows:

N M2(i)

We have to minimize: L: L: restj i=l j=Ml(i) N M2(i)

Now we write: L: L: res;j = <p(N, T, U, P, Q, S, V, W, R) i=l j=Ml(i)

To minimize ;p means:

8cp = 0 8N

and 8cp = 0 aT

and 8cp = 0 8U

and 8cp = 0 8P

and

and 8cp = 0 8S

and 8cp = 0 8V

and 8cp = 0 8W

and

35

8cp = 0 8Q

8cp = 0 8R

(52)

Equation (52) can be written as:

N M2(i)

-GN 2L; L; reSij Yj = 0 i=1 j=Ml(i) N M2(i)

~ = 2L; L; reSij x; Yj = 0 i=1 j=Ml(i) N M2(i)

~ = 2L; L; reSij xt Yj = 0 i=1 j=Ml(i) N M2(i)

U = 2 L; L; reSij yJ = 0 i=l j=Ml(i) N M2(i)

~ 2L; L; reSij xl yJ = 0 i=l j=Ml(i)

N M2(i)

~ = 2 L; L; reSij xt yj = 0 i=l j=Ml(i)

N M2(i)

Gv = 2 L; L; reSij yj = 0 i=l j=Ml(i)

N M2(i)

iW = 2 L; L; reSij xl yj = 0 i=l j=Ml(i) N M2(i)

~ = 2 L; L; reSij yj = 0 i=l j=Ml(i)

Now we have nine equations with nine unknown coefficients. After substitution of reSij and some reorganization we remain the following matrix:

N T U P

M[ij] Q = R[i] (53) S V W R

36

with the following definitions of the matrix elements m[i,j]:

m[!,I]

m[l~ 2] = m[2, 1]

m[I,3] = m[2,2] = m[3, 1]

m[I,4] = m[4,l]

m[l,S] = m[2,4J = m[4, 2J = m[5, IJ

m[l,6] = m[2, 5] = m[3,4] = m[4,3] = m[5, 2] m[6, 1]

m[l,7] = m[7, 1] = m[4,4]

m[l, SJ = m[2, 7J = m[4, 5] = m[5, 4] = m[7, 2] = m[S, IJ

m[I,9] = m[4, 7] = m[7, 4] = m[9, 1.]

m[2,3] = m[3, 2]

m[2,6] = m[3, 5] = m[5, 3] = m[6, 2]

m[2, S] = m[3, 7] = m[4, 6] = m[5, 5] m[6,4] = m[7, 3] =

m[S,2]

m[2,9] = m[4, S] = m[5, 7] = m[7, 5] m[S,4] = m[9, 2]

m[3,3]

m[3, 6] = m[6, 3]

m[3, SJ = m[5, 6] = m[6, 5] = mrS, 3]

m[3, 9] = m[5, 8] = m[6, 7] m[7,61 = mrs, 5] = m[9, 3]

m[4, 9] = m[7, 7] = m[9, 4]

37

N M2(i) mr5,9] = m[7, S] = mr8, 7] = m[9, 5] = I: I: x; yj

i=l j=Ml(i) N M2(i)

m[6, 6] I: I: x~ yj i=l j=Ml(i) N M2(i)

m[6, 8] = mrS, 6] = I: I: x~ YJ i=l j=Ml(i) N M2(i)

m[6,9] = mr8, s] = m[9, 6] - I: I: xi y*'! i=l j=Ml(i) J

N M2(i)

m[7,9] = m[9, 7] = I: I: Y] i=l j=Ml(i) N M2(i)

m[S, 9] m[9, 8] = I: I: x; y] i=l j=Ml(i) N M2(i)

m[9, 9] I: I: yJ i=l j=Ml(i)

The definitions of the column elements r[i] are:

N M2(i)

r[l] = L L (Zij - x;(rl + r2x;)) Yj i=l j=Ml(i)

N M2(i)

r[2} = L L (Zij - x;(rl + r2x;)) xt Yj

i=l j=Ml(i)

N M2(i)

r[3] - L L (Zij - xt(rl + r2x;)) xi Yj

i=l j=Ml(i)

N M2(i)

r[4} - L L (Zij x;(rl + r2x;)) Y;

i=l j=Ml(i)

N M2(i)

r[5] - L L (Zij - xt(rl + r2xt)) xt Y;

i=l j=Ml(i)

N M2(i)

r[6] - L L (Zij - XtCrl + r2x;)) xi Y;

i=l j=Ml(i)

N M2(i)

r[7] - L L (Zij - xtCrl + r2xt)) Y; i=l j=Ml(i)

N M2(i)

r[8] - L L (Zij - xtCrl + r2xt)) x; yJ i=l j=Ml(i)

N M2(i)

r[9] - L L (Zij - x;(TI + r2xt)) yj i=l j=Ml(i)

38

Again we have a system of lineair equations, but now with the coefficients N, T, U, P, Q, S, V, Wand R. Table 2 contains the calculated values of the coefficients for a scan angle of 30 degrees. The quantities of the antenna are also given in table 2.

Table 2: Quantities of the antenna and calculated values of the coefficients.

Quantities 0; = 30u Yt = 9.33..\ ..\=3 em width=28.8 ..\ It = 23.75 ..\ height=16..\ Calculated values of the coefficients

T! 0.2372 Q = 0.8394 r2 = 0.1149 S = -3.9110 N = 0.1706 V = 0.0094 T = 0.1545 W = -0.5229 U = -0.2542 R = -0.0742 P = 0.2604

5.2 Phase errors

Also for the offset situation the phase errors are calculated and shown in figures (21) and (22). The phase errors for the unscanned and scanned situation can be defined as follows:

fasefoutu - drefu du fasefoutt - dreft dt

The terms drefu and drelt represents the reference distances for the unscanned (31) and scanned situation (32) respectively. The terms du and dt represents the distances from the feed to an arbitrary point on the reflector surface and from that point to the reference plane for the unscanned (33) and scanned situation (34) respectively. For calculations of the phase errors we use the reflector from the previous paragraph. The phase errors in the unscanned and scanned situation, expressed in wavelengths, are shown in fig. 21 and fig. 22 respectively.

39

UNSCANNED fASE fOUTEN

0.12

1'1.1

1'1.1218

fasefoui {lambda)

1'1.1'14

1'1.132

1'1 2 4 6 9

'( !lambda)

Figure 21: Unscanned phase errors.

SCANNED rASE ,OUTEN

o 2 " 6 B

'( (lambda)

1'1

-0.02

-121.04

-121.06

-0.08

-0.1

-0.12

-0.14

-0.16

-121.18

Figure 22: Scanned phase errors.

40

fasefout (lambda)

6 Radiation patterns

In the previous sections we have investigated the shape of the reHector. The next task is to calculate the radiation pattern of the shaped reHector if a feedsystem is known. To that end we have to find the induced currents on the reHector. These currents act then as sources for the radiation field. The purpose of the first pargraph is to find a general expression of the radiation pattern which is valid for an arbitrary shaped-reHector surface. A proper describ­tion of the feedsystem is given in the second paragraph. Since the calculation of the radiation pattern of the reflector antenna in the unscanned situation differ some how from the scanned situation, this will be treated in the last paragraph of this section.

6.1 Calculation of the far-field radiation pattern for the shaped-reflector in the scanned situation.

The most general calculation of the far-field radiation pattern for the shaped-reHector occurs in the scanned-situation. The angle a between the reHected rays and the z'-axis is called the scan-angle. The shaped-reHector with the cartesian coordinate systems is shown in fig. 23.

Figure 23: The shaped-reflector with the cartesian coordinate systems

41

In fig. 23 we distinguish the following cartesian coordinate systems:

(x',y',z')

(XI,yl,Z")

(x,y,z)

The coordinate system of the feed. The origin of the coordinate system is in the focal point with the zs-axis pointed in the direction of the feed main beam.

The primary coordinate system of the shaped-reflector.

A new translated and rotated coordinate system of the shaped-reflector with the z" -axis pointed in the scan-direction.

The coordinate system of the far-field radiation pattern for the shaped­reflector.

To calculate the far-field pattern of the scanned beam we have to transform the far-field pat­tern of the feed, which is expressed in the cartesian coordinate system (xs, Ys, zs), into the cartesian coordinate system (x", y", Zll). Hereby we make use of a translation along s (fig. 23) and three rotations along the Eulerian angles Ca, /3, ,). The cartesian coordinate systems (xs,ys,zs) and (x",y",Z'/) with the Eulerian angles (a,/3,,) are shown in fig. 24

Figure 24: Eulerian angles

42

Definition of the Eulerian angles:

x* The x*-axis is generated by the intersection of the xsYs-plane and the x"y"-plane.

a Agle a describes a couterclockwise rotation about the z"-axis which brings the x"­axis to the x* -axis.

(3 Angle (3 defines a rotation about the x* -axis in a counterclockwise sense which brings the z"-axis to the zs-axis.

'Y Agle'Y describes a couterclockwise rotation about the zs-axis which brings the x*-axis to the xs-axis.

We can derive the transformation matrix (CB AC") which transforms the cartesian coordinate

{Cll} into {cs } .The transformation matrix is defined as:

And:

(AU AI' A13) A21 A22 A 23

A31 A32 A33

C~' sin 'Y DO - s~n'Y cOS'Y

0

An cos 'Y cos a - sin'Y cos (3 sin a

A12 cos'Y sin a + sin 'Y cos (3 cos a

An - sin 'Y sin (3

A21 -

A22; =

A23

- sin 'Y cos a - cos 'Y cos (3 sin a

- sin 'Y sin a + cos 'Y cos (3 cos a

cos 'Y sin (3

A31 . - sin (3 sin a A32 . - - sin (3 cos a

A33 - cos(3

0 o )( cosa sin a D (54) cos (3 sin (3 - sin a cos a sin (3 cos (3 0 0

(55)

The far-field pattern of the feed expressed in the cartesian coordinate system (XSl YSl zs) which is transformed to the cartesian coordinate system (x" l y" l Zll) can be expressed as follows:

(

sin Os cos 4>s . sin Os sin 4>s

cos Os

cos Os cos 4>8 cos Os sin 4>8

-sinOs

43

(56)

.. .. .. .. .. ..

x'

z'

...

zit

Figure 25: The reflector surface E with the projected circle (J"

A matrix who describes one or more rotations of a cartesian coordinate system is always orthogonal. This means that:

(57)

To express the spherical coordinate system of the feed (rs,()s, ¢s) into the spherical coordinate system of the shaped-reflector (r", ¢", Oil) we use the following transformation:

{

rssinosCOS¢s} (An A12 A13) {r"SinO"cos¢" 81 } r s sin Os sin ¢s = A21 A22 A 23 rtf si: 0" si~ ¢" - 82

rs cos Os A3! A32 A33 r cos () - 83

For further calculations of the far-field radiation pattern we make use of fig. 25. The reflector surface E can be written as follows:

z" = fex",y") = j(p",¢") with

j means an other function as the function f. (J" the surface of the projected circle.

44

(58)

(59)

We assume that the reflector surface:E is a perfect conductor. Then we can write the induced current J on the· reflector surface as follows:

J = 2ft x Hs(f ")

The unit normal of the reflector surface is given by:

A IV . h N- [ 81 A 81 A A] n = N WIt = - a;;uax ll - fiijiayll + azll

(81)2 (81)2 and N = 8x" + 8y" + 1

The general equation for the scattered H-field may be expressed as:

H(T) = V x A

with the vectorpotential A expressed as:

- J -e-jkjr-r "I II

A = J 47r1f _ fllids r:

Introducing the usual far-field approximation:

the phase term : If f "I ~ r - f 1/ • f

the distance : If - f 1/1 ~ r

We get for the magnetic far-field:

e-jkr ... H(T) V x -4-T(O, t/J) with

7rr T(O,</» J J(f")eJkrll.fdsll

r:

The rotation of the H-field in spherical coordinates is defined as:

H(f') = ~ [8(AI/>SinO) _ 8Ao] + 0,0 [_1_ 8Ar _ 8(rAI/»] r sin 0 80 8</> r sin 0 8</> 8r

+ 0,1/> [8(rAo) _ 8Ar ]

r 8r 80

In the far field we get:

H(f') = _ 0,0 8(rAI/» + &1/> 8(rAo) + 0 (~) r 8r r 8r r2

with

and

45

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

Finally we arrive at the expression for the magnetic far-field: -jkr

H(f} = jk~4 (Tq/Lo - Toa</» 7fr

The electric far-field can be expressed as:

, f!0 with Zo = -EO

B(T) - -Zoar x H(T) -jkr

- -jkZo~4 ar x (T</>ao - Tea</» 7fr

-jkr

- -jkZo~4 (T</>a</> + Teae) 7fT

(69)

(70)

(71)

The integration on the reflector surface ~ can be transformed into an integration over the projected circular region u. By this transformation a small surface element of the reflector ds" can be expressed as:

ds" = 1~p'1 x ~lIldp" d¢" (72)

For the cartesian coordinates of the reflector surface expressed in spherical coordinates we get:

x" - Til sin e" cos ¢" = p" cos ¢" y" Til sin e" sin ¢" = p" sin ¢" Zll - j(p",¢") (73)

The cross-product is given by:

( 8XP) (8XII) 8pll {J</>II £JL:.. £JL:.. ~pll X ~/I - {Jpll X {J</>II {JZII {JZII {Jpll {J</>II

(

COS ¢" ) (-p" sin ¢" ) (Sin 4>" (If,) - p" cos ¢" (if, ) ) - sin 1" x p" co~ ¢" = _p" sin -h" (2L) _ cos -hll (2.L) .P..L 2.L 'I' tJpll 'I' {JcPJl

tJp'1 {J</>II pll

_ sin2 -h" ( 8j) 2 _ 2 sin -h" ( 8j ) p" cos -hIt ( 8j ) 'I' 8¢" 'I' 8¢" 'I' 8 p"

+P''' cos'~" ( :t.) , + p'" sin' ~" ( :t.) , + 2 sin -hit ( 8 j ) p" cos -h" ( 8 j) + cos2 -hit ( 8 j ) 2 + p,,2

'I' 8¢" 'I' 8 p" 'I' 8¢"

112 8j 8j 112

( _)2 ( _)2

P 8 pli + 8¢" + P

46

",,2 ( OJ) 2 ( oj ) 2 "'/2 I~pli X ~tli I-' a;;;; + 8¢i' + I-'

= p" ( OJ)· 2 /I -2 ( oj) 2 o p" + (p ) O¢" + 1 (74)

Finally we arrive at the expression for the small surface element ds":

ds" (75)

The transformation described here is called the surface Jacobian transformation with the Jacobian expressed as:

Of "-2 OJ ( _)2 ( _)2

o p" + (p ) O¢II + 1

In the following expressions we use JE as a notation for the Jacobian. We can rewrite (65) as follows:

T(O, ¢)

a 27r J J J(f")ejkrtlof JEP"dp"d¢"

o 0

(76)

(77)

The magnitude of the normal vector to the reflector surface in spherical coordinates can be expressed as:

N = I~ptl X ~tli = Of /I -2 of (

_)2 ( _)2 Op" + (p) . O¢" + 1

It can readily be shown that (61) and (76) are identical, so that:

N

Defining an equivalent current as: ... j(P",¢") - J(r"). JE = 2n x H(r"). N

- 2N X H(f"). N = 2N x H(r/l) N

,we can write(77) as:

a 27r

T(O, ¢) = J J J(p", ¢/I)dkr " 0

1' p"dp"d¢" o 0

with J(P'I,<jJII) = 2N x H(r")

47

(7S)

(79)

(SO)

(S1)

Now we are going to rewrite the exponential term of (81)

(

pit cos <//' ) r It = p" sin <p"

z" and

(

sin e cos <p ) f = sin hin <p

cose

r If • f - pit cos <pit sin () cos <p + pit sin <pit sin e sin <p + zit cos e p" sin e( cos <p cos <p" + sin <p sin <p") + z" cos 0

- pit sin e cos (<p" - <p) + z" cos e We can finally write (81) as:

a 211"

T( 0, <p) = J J ](P'I, <p")ejkzll cosO. ejkpll sin ()cos(if/, -4» . p" dp" d<p"

o 0

with .... i(P", <p") = 2N X it(r")

With the following transformation we find the spherical far-field components of T:

{ ~: } ~ (CO~:i:~¢ CO:~¢ -,:8) { ~ }

(82)

(83)

(84)

Using (71) we finally obtain the expression of the electrical far-field in spherical components.

-jkr

E(T) = -jkZo~4 (T4>a4> + T()a,) 7fT

(85)

The co-polar radiation pattern with the peak gain normalized at OdB is defined as:

F(e <p) = 10 log pee, <p) , P(O, 0) (86)

,with P(O,O) the maximum radiated power by unit of solid angle in the direction of 0 = ° and <p = 0. For the maximum electrical field Emax we find:

-jkr

Emaa: = E(e = O,<p = 0) = -jkZo~4 (T4>(O,O)a4> +T,(O,O)a() 7fT

(87)

The maximum power density Smax is expressed as:

Smax = S(O,O) = ~Zol [IE,(O, 0)12 + IE4>(0,0)12]

.... e-jkr with E()(O,O) = -jkZo-4-T,(0, 0)

7fT

.... e-jkr

and E4>(O,O) = -jkZo-4-T4>(O, 0) 7fT

(88)

48

So:

.... 2 2211 2 IEo(O, 0)1 - k Zo (47rr)2 To(O, 0)1

.... 2 2 2 1 12 IE.p(O,O)1 = k Zo (47rr )2IT.p(O, 0) (89)

For the maximum radiated power by unit of solid angle we find:

P(O,O) - r 28(0, 0)

= r2~ZOl [1.80(0,0)12 + IE.p(0,0)12] (90)

With (90) we finally obtain an expression for the normalized radiation pattern:

(91)

The radiation pattern (91) is normalized at OdB in the direction of 8 0 and 4> = O. With this are we enable to calculate the 2-dimensional radiation pattern in an arbitrary tj>-plane and across a certain 8-range. For the non-normalized radiation pattern we first have to calculate the antenna gain and then add this gain to the normalized-radiation pattern. The antenna gain will be calculated in section 7. In the next paragraph we derive an idealized feed pattern which gives a good approach of the Philips corrugated horn.

49

6.2 The optimum radiation pattern of the feed

As mentioned before the far-field pattern of the reflector cannot be determined without a proper description of its feed pattern. An idealized feed pattern which gives a good approach of the Philips corrugated horn (4 grooves) may be described as:

for O<li<~ - - 2 and zero otherwise

G.y polarized

(92)

The exponent L will be replaced by "nu" (44) or "nt" (45) in the unscanned and scanned situation respectively. By choosing L=O we get an uniform illuminated reflector surface .. However an uniform illuminated reflector surface will result in a great spillover-loss, because lots of energy will radiate along the reflector surface. To reduce the spillover-loss we have to taper the edge illumination of the reflector (L > 0). However reducing the edge illumination will also result in a reduction of the aperture efficiency. Therefore we have to look for a particular edge illumination where the multiplication of the aperture- and spillover-efficiency is maximum. This multiplication is called the efficiency factor 170 and is for the parabola defined as:

110 = cot' ~'" [i [G f("')l~ tan ~,pd'" r (93)

with Gf('I/;) : antenna gain function and W: antenna aperture angle

For a first estimation we can use (93) also for the shaped reflector. Finally for an accurate result we have to vary the value of L a little bit. We can write the radiation pattern of the feed as follows:

0< .1. < ~ -'f'-2

1f '1/;>-

2

(94)

(95)

Note that the radiation pattern of the feed is limited to t/> < 7r /2, and the relation between L (92) and M (94) is defined as:

M=2L

The gain Go of the antenna can be obtained by using the following expression:

21r 1r J J G /('1/;) sin 'l/;d'l/;de = 41f

o 0

50

(96)

2?f ?f J J Go COSM 'IjJ sin 'ljJd'IjJde = 47r

o 0

2?f

J (Go (M ~ 1») de = 47r o

2 ( Go (M ~ 1») 7r 47r

1 GO(M + 1) = 2

Go = 2(M + 1) (97)

We have to choose the value of M so that the antenna aperture angle W correspond with the maximum of the efficiency factor 110 (93).

Example of calculation: Calculate the value of L and the corresponding edge taper for the antenna given in fig. 26. First we have to calculate the antenna aperture angle W with the help of fig. 26. The antenna aperture angle W is equal to:

0.25 0 ( ) W = arctan 1.033 = 13.6 98

The efficiency factor 110 as a function of the antenna aperture angle W is for M=90 plotted in fig. 27 We can see in fig. 27 that for M=90 the antenna aperture W correspond with the maximum of the efficiency factor 110' With the help of (96) the value of L is equal to:

L=M/2=45

The edge-taper of the reflector for L=45 is equal to:

-20 log (COS45 (13.6°») = 11.1dB (99)

The edge-taper of the reflector is not only caused by the taper of the feed but also by the path loss. The path loss is caused by the path lenght difference between the feed to the center and the upper tip of the reflector. In our case the path loss is equal to:

20 log v'(0.25~~0:i1.033)2 = O.122dB (100)

This path loss is negligible with respect to the egde-taper of 11.1 dB caused by the feed taper.

51

effi- O.S ciency factor

0.6

0.4

0.2

FEED ~~--------~r---~ \

'I \ I I \

I I \ I I \

I I .. t i-"" \

" J ... T, \ I I r \

t I \ I I \

t I \ I \

I \ I \

I \ I \

I \ I \

I \ I \

I \

1.033 1.048

Figure 26: The antenna aperture angle 'Ill

......................... . . .

I I

... ~ •...• 1 .•. :- .• - ••...• :.

I

.~ .......... . ' . ......... . . .

. . . . . . ~ .. - .. - ... ~ - .. - - i' . -;- ... - .... -:- ... - . - - - -:- . - ...... -;- - . - - ... -

0~~---4------+---~-r-----4------r-----~----~

10 13.6 15 20 25 30 35 o 5

antenna aperture angle (degrees) •

Figure 27: The efficiency factor 'TJo

52

6.3 Calculations of the far-field radiation pattern for the shaped-reflector in the unsanned situation

Calculations of the radiation pattern in the unscanned situation are less complex than in the scanned situation. The reason for this is that for the derivation of the unscanned-radiation pattern less coordinate systems and thus less rotations will be used. Because of this differences we will treated the derivation of the unscanned-radiation pattern separately. Transmitted waves starting at the feed located at the unscanned focus reflect and propagate along the z-axis, known as the boresight, or unscanned direction. This makes the rotated coordinate system x"y" Zll from fig. 23 superfluous. The unscanned situation of the shaped reflector with the cartesian coordinate systems is shown in fig. 28. The equation for the symmetrical-reflector surface E is given by:

x(x') - - .. - - - . - ...... - .. ~ .. ~ . ~ ..... - . ~ .. - .. ---... -~ -

y(y')

I I

/

................ I··

, /

/

Figure 28: The shaped-reflector in the unscanned situation

far­field point

Ys z(z')

(101)

The coefficients b, all a2, P, Q, Rand S can be calculated with the techniques describes in the previous sections. To calculate the far-field pattern of the shaped reflector in the unscanned situation we have to transform the expression of the feed into the expression of the far-field pattern of the reflector. This transformation is obtained by three rotations about the Eulerian angles (a, j3, 'Y). These Eulerian angles are shown in fig. 29. Since the xsYs-plane coincide with the x'y'-plane, the line of intersection x* is chosen equal

to the x'-axis and xs-axis.

53

Y'

Figure 29: Eulerian angles in the unscanned sitation.

By this choice the Eulerian angles are equal to:

a _ 00

fJ - 1800

'Y - 00

For the elements of the transformation matrix (co ACIl

) we find:

All - cos 'Ycos a - sin'Y cosfJsin a = 1

A12 - cos 'Y sin a + sin 'Y cos fJ cos a = 0

Al3 = sin 'Y sin fJ = 0

A21 - sin'Ycosa - cos'Y cos fJsin a = 0

A22 = - sin'Y sin a + cos 'Y cos f3 cos a = -1

A 23 = cos 'Y sin fJ = 0

A31 - sin fJsin a = 0

A32 - sin fJ cos a = 0

A33 - cosfJ =-1

And:

(102)

(103)

It was also possible to derive the transformation matrix immediately from fig. 29, since the xl-axis and y'-axis coincide with the xs-axis and ys-axis respectively and the z'-axis is rotated about 1800 from the zs-axis. The magnetic field of the feed expressed in the cartesian components of the reflector is given

54

by:

{ H~(r')} C 0 ~ ). ( sinO.cos</>, cos Os cos ¢s -sin</>. ) Hs (f ') = Hyl (fl) = 0 -1 sin Os sin¢s cos Os sin¢s cos¢s

Hzt(fl) 0 0 -1 cos Os sin Os 0

( 0 ) 1 -j~. . -cos¢s . -F(Os)' _e __ with F (Os) = cosnu Os . ¢ Zo 47rrs sm s

0 0

~J' ( -cosO,cos'</>. - sin'</>. ) - -1 cos Os sin ¢s cos ¢s + sin ¢s cos ¢s

0 sin Os cos ¢s

.{ 0 }. ~, e-jkr• -cos¢s (Os), -4-sin¢s 7rrs

(

- cos Os cos2 ¢s - sin2 ¢s ) 1 cos Os sin ¢s cos ¢s - sin ¢s cos ¢s . z

- sin Os cos ¢s 0

-< (-<') Hs r = (104)

The next step is to express the spherical coordinates Os, ¢sand r s into 0', ¢', r' with the help of (58).

o o ) { r' sin 0' cos ¢' - 81 } o r' sin 0' sin ¢' 82

-1 r' cos 0' - 83

(105) o

The components of the § vector of the shaped-symmetric reflector in the unscanned situation are given by:

81 = 0 82 = 0 and 83 = z~nsc =/: 0

with: z~nsc the z'-coordinate of the unscanned focal point

This results in the following three expressions:

rs sin Os cos ¢s = r' sin 0' cos ¢I

rs sin Os sin¢s = -r'sinO'sin¢'

O I 0' I rs cos s = -r cos + z1:msc

Formula (107) divided by formule (106) gives an expression for ¢s:

r s sin Os sin ¢s -r' sin Of sin -'-~-:---...:.,.;- = r s sin Os cos ¢s r' sin 0' cos ¢'

-+ tan¢s = -tan¢'

Formula (107) divided by formule (108) gives an expression for Os:

rs sin Os sin¢s rs cos Os

= -r' sin 0' sin ¢'

-r' cos 0' + z~nsc

55

(106)

(107)

(108)

¢s = -¢' (109)

tan Os --r' sin 0' sin <//

sine -4/)( -r' cos 0' + z~nsc)

Os = arctan -.---:---:-:-----'------:-( -r' sin 01 sin <// )

sm( -4>')( -r' cos ()I + z~nsc) (110)

Substition of formula (110) into formula (108) gives an expression for rs:

, (jl + 1 -r COS!7 zunsc rs =

cos Os (111)

With the aid of fig. 28 we can find an expression for r':

cos( 1T - 0') = l:1 r'

IZII rl COS(1T - 01)

(112)

The different cartesian components of the magnetic field are given by:

1 e-jkr• HXI(f I) = --Z (cos Os cos2 4>8 + sin2 4>s)F«()s) . -4--

o 1frs (113)

(114)

(115)

,for ¢s, Os and rs we substitute (109), (110) and (Ill) respectively. The cartesian components of the unit normal are given by:

_ aj = -(2alX' + 4a2x,3 + 2Qx1y,2 + 4Sx,3y,2) ax'

The equivalent current on the reflector surface can write as:

In cartesian components:

fx, 2(Ny'Hzl - Nz,Hy')

iy, - 2(Nz,Hx' - Nx,Hzl)

.lx, 2(Nx1Hyl - Ny,Hx')

56

(116)

(117)

(118)

(119)

(120)

Integration over the projected circular region sigma with a substitution of the equivalent current can be expressed as:

a 211"

T( (}, </» = f f] (p', </>')e?kz ' cos (J , ejkp' sin (J cas( <I/-q,) , p' dp' d</>' o 0

with ... j(p', </>') 2N X Her ') (121)

To find the spherical far-field components of T, one then simply uses the following transfor­mation:

{ T(J } = ( cos (} ,cos </> cos () sin </> - sin () ) { ~x }

Tq, - sm </> cos </> ° i The spherical far-field components of E may be expressed as:

-jkr

E(T) = -jkZo~4 (Tq,a,p + T(Ja(J) 7rT

We finally obtain an expression for the normalized radiation pattern:

P(lJ, </» F(B,</» = 10log P(O,O)

[IE(J(B, </»12 + IEq,«(), </>w]

10 log 2::----------=-[IEeeo,o)12 + IEq,(O, 0)12 ]

57

(122)

(123)

(124)

58

7 Basic antenna parameters

Antenna parameters are very important to design reflector antennas. Therefore, a few basic . antenna parameters will be treated in this section. A comprehensive describtion of the an­

tenna gai~ is given in the first paragraph. Then a short describtion of the cross polarization is given and finally the G IT-ratio will be treated on the basis of an example in the last paragraph.

7.1 Calculation of the gain

An useful measure. describing the performance of an antenna is the gain. The gain of an antenna ,in given direction is defined as "4 7r times the ratio of the radiation intensity in that direction . P( 0, cp) to the net power accepted by the antenna from a connected transmitter Pt". Wh~n the direction is not stated, the gain is usually taken in the direction of maximum radiation. Thus, in general:

(125)

For the calculation of the net power accepted by the antenna from a connected transmitter we take the radiated power of the horn feed. Furthermore, we assume that the horn feed only radiates in the forward direction. The radiation pattern of the feed, described in section (??), is defined as:

The squared absolute value of (126) can be written as:

... 2 !Es(r's)!

!Es(f's)12

The power density § can be expreesed as:

S.. 1Z -liE" ( .. )12 A - 2" 0 s rs • rs

1 -1 1 ( )2L A

2"Zo (47rrs)2 cosOs . rs

Using (130) we finally obtain the expression of the radiated power of the feed:

59

(126)

(127)

(128)

(129)

(130)

1 1 l1f /2121f - 2Z0-1(411")2 _ _ (cosOs)2LsinOsdOsd¢s

08-0 4>8-0

1 -1 211" r/2 2L . = 2Zo (411")2 }08=0 (cos Os) smOsdOs

1 -1 211" {1f/2 2L - -2Zo (411")2 }o.=o (cos Os) d(cosOs)

1 -1 211" [ -1 2L+1] 1f/2 - 2Z0 (411")2 2L + 1 (cos Os) 0

= !Z -1~ 1 __ 1_ . -1. 1 2 0 (411")2 2L+ 1 - 1611" Zo 2L+ 1

The radiated intensity in a given direction can be expressed as:

with IEo(O, ¢)1 2

IE4>(O, ¢)12

2 2 1 2 = k Zo (411"r)2ITe(0, ¢)I

2 2 1 2 = k Zo (411"r)2IT4>(0,¢)1

So, for the power density § we find:

.... 1 1 2 2 1 [I 2 2] S(O, ¢) = 2Zo - k Zo (411"r)2 Te(O, ¢H + IT.p(O, ¢)I

Now, the radiation intensity in the direction (0, ¢) can be expressed as:

The radiation intensity in the main direction (0 = 0, ¢ = 0) can be written as:

with Te(O,O) - Tx(O, O)

T.p(O, 0) - Ty(O, O)

The physical optics radiation integral in the x~direction can be expressed as:

60

(131)

(132)

(133)

(136)

The physical optics radiation integral in the x-direction can be expressed as:

with Jyu(pll,l/l') == 2 (NZIIHzlI - N;r;IIHzlI)

The antenna gain in the main direction (0 = 0, ¢ = 0) can be written as:

G(O, 0) = 4 P(O,O)

7r Pt

47rk Zo -1 k2 Z02 (4;r)2 [lTo(O, 0)12 + IT4>(O, 0)12] - 1 Z-l 1

1611" 0 2L+l

So:

GeO,O) = 2Z02 k2(2L + 1) [ITe(O, 0) 12 + IT</> (0, 0)12] G(O,O) - 10l0g {2Z02k2(2L + 1) [lTe(O, 0)12 + IT4>(0,0)12]}

7.2 Calculation of the cross polarization

The rectangular and spherical components are related by:

az = Ar sin 0 cos ¢ + Ae cos 0 cos ¢ - A4> sin ¢

ay . = Ar sin 0 sin ¢ + Ae cos 0 sin ¢ + A</> cos ¢

(137)

(138)

(139)

(140)

(141)

Since the cross-polar radiation pattern is maximum for small angles of 0, we can reduce (140) and (141) to:

az = cos ¢ie - sin ¢i4>

ay = sin ¢ie - cos ¢i.p

For a polarization in the y-direction, we can express the co-polar R(O, ¢) and the cross-polar C(O, ¢) as:

R( 0, ¢) - E· ay = sin ¢ie + cos ¢i4>

C(O, ¢) - E· ax = cos¢ie - sin¢i4>

For a polarization in the x-direction, we can express the co-polar R(O, ¢) and the cross-polar C(O, ¢) as:

R( 0, ¢) - E· az = cos ¢ie - sin ¢i</>

C(O,¢) - E.liy = sin¢ie + cos¢i4>

. 61

7.3 Calculation of the G/T

The derivation of the G/T-ratio will be given on the basis of an example. The requirement of the G/T-ratio according the contract is:

G T > 13.0 dB/K @ 11 Ghz @ 30° elevation angle

For the calculation of the G /T we assume that the antenna is mounted on a wall of infinite height. In this worst case situation we assume that the temperature of the wall and the earth are both 300 K. For the edge illumination of -12 dB we have a spillover efficiency of about 93%. This means that 93% of the radiated energy will fall on the reflector surface and 7% will fall on the "hot" wall. The system-noise temperature (Ta) exists of the noise temperature of the antenna (Ta) and the noise temperature of the LNB (Tc). Furthermore we assume that the noise figure of the LNB equals 1.1 dB (=1.29*). Now the noise temperature of the LNB will be:

Tc = (F 1) . To = (1.29 -1) . 300 = 86.5K

with To : the temperature of the surroundings

The noise temperature of the antenna exists for a part of the noise temperature of the sky and for the other part of the noise temperature of the earth/wall. According measurements of the noise temperature of the sky at a frequency of 11 GHz and at an elevation angle of 30 degrees in Eindhoven it appears that the noise temperature of the sky can be taken at 50 K [?J. The noise temperature of the wall/earth will be:

Tal = 300 . 0.07 = 21 K

The total noise temperature of the antenna will be:

Ta Tal + Ta2 50 + 21 = 71 K

The system-noise temperature will be:

Ts = Ta + Tc = 71 + 86.5 = 157.5 K

Ifwe assume that the antenna gain is equal to 39 dB (=7943*), then the G/T-ratio will be:

G G 7943 157.5 = 50.43 = 17.03 dB/K

It appears that even in the worst case situation that the G/T-ratio meets the requirements.

62

8 Accuracy of the antenna design.

8.1 Accuracy of the feed position.

For the investigation of the feed-position accuracy, three different positions of the feed are chosen in relation to the original position. These three different positions are given in fig. 30. The radiation patterns which correspond to the different positions of the feeds are given in fig. 31 up to fig. 33. The peak gain, main-direction level and side-lob level for the shaped reflector are given in table 3.

x-axis

pos.3 t-----I I

::::: FiI~A original position ~ t -----of the feed :

z-axis

Figure 30: Accuracy of the feed position.

position of the feed peak gain (dB) main direction level (dB) side-lobe level (dB) position 1 38.93 (-0.5 deg) 38.21 -22.97 (2.8 deg) position 2 38.85 (-1.2 deg) 34.97 -22.57 (2.1 deg) position 3 38.76 (-1.8 deg) 28.10 -22.33 (1,4 deg)

Table 3: Antenna parameters as a function of the feed position.

63

RADIATION PATTERN 3.385 DEG. SCANNING (SHAPED REFLECTOR)

40~---------------------------------------.

313

20

-113

-2e+-,-.-.-.--r-,-.-.-.-+-.-.-.-.-+-.-.-,-.~ -113

a: = 3.385° >. 2.654cm It = 25.0>' width = 35.78>' height = 32.0>'

-5

of f.height = 16.0>'

a

THETA (DEG. I

5

Figure 31: Feed in position 1.

64

10

CO-POL --------- CROSS-POL

RADIATION PATTERN 3.385 DEG. SCANNING (SHAPED REFLECTOR)

40.-------------------~------------------_.

30

20

~ 10

-10

-2e+-.-.-.-.-~_._r_r_._+_._._._.~_._._._,~

-10 -5

a = 3.385° >. = 2.654cm It = 25.0>. width = 35.78>. height = 32.0>' oll.height = 16.0>'

o THETA mEG.)

5

Figure 32: Feed in position 2.

65

10

CO-POL -.-.----- CROSS-POL

RADIATION PATTERN 3.385 DEG. SCANNING (SHAPED REFLECTOR)

40~---------------------------------------,

30

20

-10

-20+-~~~-.-r-r-,-,-,-+-,-,-,-,-1-'-'-'r-~ -10 -5

a 3.3850

A 2.654cm It = 25.0A width = 35.78A height = 32.0A 011. height = 16.0'\

o

THETA (DEG.)

5

Figure 33: Feed in position 3.

66

10

CO-POL CROSS-POL

8.2 Accuracy of the antenna design for different locations inside Europe.

For calculation of the difference between two satellite positions, seen from an arbitrary position on earth, expressed in angle <P, we use fig. 34.

z-axis

e.g. EINDHOVEN Ai = 5.47° E.L --......... ~ = 51.45° N.W.

sat. 1 x-axis

Figure 34: The earth with two geostationary satellites.

Explanation of the used symbols:

r : radius of the earth (r = 6378.16km). h, : hight of the geostationary orbit with respect to the equator. S1, S2 : satellites I and 2. dl, d2 : distance of the earth position with respect to satellite I and satellite 2 respectively. da : shortest distance between satellite I and satellite 2. AI, A2 : postion of satellite 1 and satellite 2 respectively with respect to the meridain.

67

z-as

y-as

Figure 35: A segment of the earth.

Ai) tpi : arbitary position on earth.

In the triangle with the sides d1 ) d2 and da we can calculate angle cP with the help of the cosine rule. To find expressions for dt, d2 and da we use fig. 35. For calcuating d1 we first have to find an expression for angle tpl.

With the help of fig. 35, it can be shown that:

b = r COStpi

COS(AI - Ai) (142)

• r cos tpi a = sm(Al - Ai) . (A A ) = r cos tp • tan(Al - Ai) (143)

cos 1- i

c - y'(r sin tpi)2 + (rcostpi' tan(Al - Ai))2

r· y'(sintpi)2 + (COStpi' tan(Al - Ai»2

With the help of the cosine rule we find for angle tpl:

r2 + b2 _ c2

COStpl = 2rb

68

(144)

(145)

Substitution of (142), (143) and (144) in (145) gives us:

cos !PI r2 + ( r cos <P. )2 _ r2 (sinlll.)2 + (cosu') .. tan(Al Ai»2) C08(>'I->"} T~ Ti

2r. TCOS<Pj C08(>'I->")

1 + (C08(~:.'>'i}r - (sin!Pi)2 + (cos !Pi • tan(Al - Ai»2) 2 . cos <P.

ooS(>'1->'.}

( COSte. )2 2 (COStl?. sin >'1->'. )2

1 ooS(>'1->'.) (sin !Pi) Tl cos >'1->'. - 2. COS¥'i + 2.. COS(fi 2.. C05<Pi 2.. COSY'i

ooS{>'I->") COS(>'1->'.) <:OS(>'I->'.) COS(>'I->'.)

COS(AI - Ai) cos !Pi (sin !Pi)2 . COS(AI - Ai) cos cpi(sin(Al - Ai))2 -~'----"- + ---;-:-'--,---::--:-

2 cos CPi 2COS(Al - Ai) 2 cos !Pi 2(COS(AI - Ai»2

_ 1 COS(AI - Ai) (1 _ (sin !pi)2) + ! cos !Pi (1 sin(Al Ai)2) 2 cos !Pi 2 COS(AI - Ai) 1 1

- 2" cos !Pi COS(AI - Ai) + 2" cos !Pi COS(AI - Ai)

cos !PI - COS !Pi COS(AI - Ai) (146)

We can find an expression for d1 by drawing a perpendicular from the position on earth to the line between the centre of the earth and the position of satellite 81.

d12 - (rsin!pl)2+(r+h-rcos!pt}2

- r 2(sin!Pl)2 + (r + h)2 - 2r(r + h) COS !PI + r2(cOS!pl)2

- (r + h)2 + r2 - 2r(r + h) cos CPl

- (r + h)2 + r2 - 2r(r + h) cos !Pi COS(AI - Ai)

The distance d2 can be find on the anology of dl.

d2 = (r + h)2 + r2 - 2r(r + h) cos CPi COS(A2 - Ai)

For deriving the the distance d3 we use fig. 36.

d3/2 d3 -r+h 2(r+h)

2(r + h) sin (A2 ; AI)

With the help of the cosine rule we finally find the expression of the angle .p.

d1 2 + d22 - d32

cosq, = 2dld2

69

(147)

(148)

(149)

(150)

d/2

Figure 36: derivation of the distance d3

Example of calculation

Given

Asked

Solution a.

Solution h.

: Position of the satellites 81 and 82.

: Difference between the two satellites 81 and 82 seen from two different places on earth, expressed in angle el>. Places: a. Eindhoven h. Dublin

: Coordinates of Eindhoven: Ai = 5.47°E.L., 'Pi = 51.45°N.W. d1

2 = 1.486 . 109 ~ d1 = 38551.1km d2 2 = 1.493 . 109 ~ d2 = 38637.9km d3 = 4560.4km cos el> = 0.993 ~ el> = 6.77°

: Coordinates of Dublin: Ai = 6.29°E.L., 'Pi = 53.34°N.W. d1 2 1.5155 . 109 ~ dl = 38929.5km d22 = 1.5286 .109 ~ d2 = 39097.6km d3 = 4560.4km eosel> = 0.993 ~ el> = 6.70°

70

The angle between two different satellite positions, seen from different places in Europe, is shown in table 4.

Eutelsat/ Astral Astra1/ Astra2 Eutelsat/ Astra2 Amsterdam 6.76 10.09 16.85 Dublin 6.70 9.98 16.67 Helsinki 6.63 9.95 16.58 Madrid 6.90 10.26 17.16 Oslo 6.64 9.94 16.58 Paris 6.80 10.15 16.96 Warsaw 6.77 10.16 16.93

Table 4: The difference in angle between two different satellites.

71

72

9 Results

This section contains the results of calculations carried out for seven shaped-reflector an­tennas. Each shaped-reflector antenna is compared with a corresponding offset-parabola. Corresponding means that both the shaped reflector and the offset parabola have the same size. However t the contour of both reflector surfaces are not the same, the front view of the shaped reflector is about elliptical and the front view of the parabola is circular. The seven designs presented in this report are:

1 Reception of the satellites Eutelsat (13° E) and Astral (19.2° E). offset-shaped reflector (width * height = 95 * 85cm) offset parabola (diameter = 91.1cm)

2a Reception of the satellites Astral (19.2° E) and Astra2 (28.2° E). offset-shaped reflector (width * height = 75 * 57cm) offset parabola (diameter = 60cm)

2b Reception of the satellites Astral (19.2° E) and Astra2 (28.2° E). offset-shaped reflector (width * height = 80 * 75.6cm) offset parabola (diameter = 78.4cm)

3a Reception of the satellites Eutelsat (13° E) Astral (19.2° E) and Astra2 (28.2° E). offset-shaped reflector (width * height = 75 * 57cm) offset parabola (diameter = 60cm)

3b Reception of the satellites Eutelsat (13° E) Astral (19.2° E) and Astra2 (28.2° E). offset-shaped reflector (width * height = 80 * 75.6cm) offset parabola (diameter = 78.4cm)

3c Reception of the satellites Eutelsat (13° E) Astral (19.2° E) and Astra2 (28.2° E). offset-shaped reflector (width * height = 110 * 85cm) offset parabola (diameter = 99.7cm)

4 Reception of the satellites Eutelsat (13° E) Astral (19.2° E) and Telenor (1° W). offset-shaped reflector (width * height = 110 * 85cm) offset parabola (diameter = 99.7cm)

73

Design 1: Reception Eutelsat (130 E) and Astral (19.20 E).

dimensions shaped reflector:

dimensions parabola:

wavelength = 2.65cm

FjD = 0.74 (approx.)

scan-angles 2 * 3.3850

offset-angle = 37°

height of the aperture = 85cm width = 95cm offset height of the aperture = 42.5cm

diameter of the aperture = 91.1cm offset height of the aperture = 45.6cm

(frequency = 11.3 GHz)

(with respect to Amsterdam)

elevation angle astral = 29.71° (with respect to Amsterdam)

elevation angle eutelsat = 30.68° (with respect to Amsterdam)

feed: Philips corrugated horn (4 grooves)l

edge illumination = -12 dB (approx.)

lin accordance with the proposal the following expression for the radiation pattern of the feed is used: E = (all sin tp + a'l' cos tp)(cos O)N ",~::r

The radiation pattern of the Philips 4 grooves corrugated horn has been approximated by an appropriate choice of the value of "N". Furthermore we have chosen an edge illumination of approximately -12 dB, which gives us the value of "N".

74

85.0

, /: '';, DIMENSIONS: eM

I I ,

I • " , , , , " , , ..

/" I " I .... r.. I ,

I IflJ"'!' " I I I ........ ,

I I I '" II , 42.5 I. '... ",

/. I ........ "

I~~ ~3: _ ~ ____________ 3!~ l~~ ___ ::~~ 7.31 I .1 ,..

: 28.7 ~: I 14 I I

I 62.9 1 ~ ~ , I

Figure .37: Shaped reflector in the YZ-plane.

95cm 95cm

Figure .38: Frontview of the aperture plane (XY-plane).

Figure 39: Frontview of the mechanical surface.

75

X-AXIS EUTELSAT (13°)

Z-AXIS

ASTRA1 (19.20

)

Figure 40: Scan angles of the reflector in the XZ-plane.

87.8

31.8

dimensions: mm

Figure 41: Positions of the feeds for the shaped reflector.

76

91.1

45.6

I

II

DIMENSIONS: eM

67.4

Figure 42: Offset parabola in the YZ-plane.

Figure 43: F'rontview of the aperture plane (XY-plane).

77

96.2 em

91.1cm . - -- ~ .. -- _ .... - - --:..---r-_

· • · I ________ J ___________ _

I I

• I I

• I · I I I

--------- ----.:--~-~

Figure 44: F'rontview of the mechanical surface.

90.4

34.4

dimensions: mm

Figure 45: Positions of the feeds for the offset parabola.

shaped reflector scan-angle [deg.] I height [em] I width [em] I gain [dB] I G/T [dB/K]

3.385 I 85 I 95 I 38.96 I 16.99

offset parabola scan-angle [deg.] I diameter [em] I gain [dB] I G/T [dB/KJ

3.385 I 91.1 I 38.86 I 16.89

78

EUTELSAT (13°) / ASTRA 1 (19.2°)

dB 40 ..... : ...... : ...... : ....... : ....... : .... ·7 co-pol. , .. . .. ... " ... " . ...

cross-pol.

20 '" '" ... ... . .. .. .. .. ,. .. . ... .. . .. ,. .. ..... .. .. .. . . . . .

E.T.S. . - "'" . ./ \' /'

I' . ( t \ •

I . \' I 1'1 I ~

o~~~~~~~--~~~----~--~--~~~~~~~~~ \ , "

/: " \ ,: \ '/' " /', \ , I' \ , '\ , ' . , .... .... ,. ,,' \" /'\

'/ \ " \,' " . \ '\ ' - , \' • \1. I • /' ": / '\ I I • ',. . \, • \ 1'\ I" '\

I \ /' \.. \. '\ ( r-~----------------~ , . \1 •

",).1:,,··· ,:,"'" •. \., '.,. ", .... \' '/'"

SHAPED REFLECTOR ,," : \/ -20

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

00 o

EUTELSAT (13°) I ASTRA 1 (19.20)

dB 40 co-pol. · . , . ~ . II" • .. .... f' II .. .. .. .. .. .. .. .. .. .. .. ..... .. .. .. .. .. ...' .. .. .. .. .. .. t" .. • .. .. ,. "." .. .. ..

. .. .. It .. .. .. .. .. ..

cross-pol.

20 .. .. .. .. .. .. .. .. .. .. .. .. ".. .. .. .. .. .. .. '.. .. .. .. .. .. .... .. .. .. .. .. .. I.. .. .. .. .. .. .. ' .... ..

I , ~ • • •

E.T.S. ;,.. , I

, \ , I I • \' I

, ' I , \ ' \ .,

I i O~--~~~~r-~b-~~----~--~~~~--~~~~~~

\{ \ ,.., \

\ ' : \

-20

,( '\ ,.' I '). ,.. , "'" \ , '\ · /' ' .\ \ (. ~ \' '/ '" • \ I • \ , ,\ ,. _

· \ " \ " \ I : I I ,\ '\ ,--, Ii: ,I: ,,' .,...:.', ___________ --,f • '( :\ I , '\ ' I ~ , , '" t . ~ .. \ I • . . , , , .:' ',',' •.. : 'I' 1 .. ,

I ,: \,': \ OFFSET PARABOLA : \' ,'' \.

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

Design 2a: Reception Astral (19.2° E) and Astra2 (28.2° E).

dimensions shaped reflector:

dimensions parabola:

wavelength = 2.65cm

F/D = 0.74 (approx.)

scan-angles = 2 * 4.884°

offset-angle = 37°

height of the aperture = 57cm width = 75cm offset height of the aperture = 28.5cm

diameter of the aperture = 60cm offset height of the aperture = 30cm

(frequency = 11.3 GHz)

(with respect to Amsterdam)

elevation angle astral = 29.71° (with respect to Amsterdam)

elevation angle astra2 = 27.36° (with respect to Amsterdam)

feed: Philips corrugated horn (4 grooves)

edge illumination = -12 dB (approx.)

81

DIMENSIONS: eM

57.0

28.

I 19.2 I

~ .1 I

I 42.2 I

'4 .. , ,

Figure 46: Shaped reflector in the YZ-plane.

750m

Figure 47: Frontview of the aperture plane (XY-plane).

82

60.15 em

75em

Figure 48: Frontview of the mechanical surface.

X-AXIS ASTRA2 (28.20 )

Z-AXIS

Figure 49: Scan angles of the reflector in the XZ-pla.ne.

dimensions : mm

Figure 50: Positions of the feeds for the shaped reflector.

83

60

30

I I

I I

" II " I I '"

I I " I I "

I ""

'I:? I I ~;Ct) I

I II:?' .~! 1,(6 I', I I I

I'

" " " " " "

DIMENSIONS: eM

" " " " , , , "-

" " " ", /1: '.... " '1~ .3~ _ ~ _____________ 3:0 l~~ ~"::"~~~

5.1 I I 101 !"I

20.3 .: t

44.4

I I I I I

II

Figure 51: Offset parabola in the YZ-plane.

I

60cm

---7--------::..-- .... -­

63.3 em

I I

Figure 52: Frontview of the aperture plane (XY-plane).

Figure 53: Frontview of the mechanical surface.

84

108

. dimensions: mm

Figure 54: Positions of the feeds for the offset parabola.

shaped reflector scan-angle [deg.] I height [em] I width [em] I gain [dB] I G/T [dB/K]

4.884 I 57 I 75 I 35.66 I 13.69

offset parabola scan-angle [deg.] I diameter [em] I gain [dB] I G/T [dB/K]

4.884 I 60 I 35.86 I 13.89

85

dB 40

20

-20

ASTRA 1 (19.20 )/ ASTRA2(28.20 )

, It • • • •

.... ~ •• ,. ........................................ ~ .......... J .............. ,." .. co-pol . . .. .. .. -. .. .. .. .. .. .. I It • , •

I f1 • I ,

It .. .. I I

cross-pol. J , ••

t " t , I •

. .

.. .. • .. ... ... .. .. ... .. .. .. 4.. .. .. .. .. .. .. I.. .. .. .. .. • .. I.. .. .. • .. ... .o... .. .. .. .. .. .. .. .. .. .. .. .. ..'.. .. .. .. .. .. .. '.. .. .. .. .. .. .. ' .. .. '"' .. .. .. I .. • • 4 .. • .. , . . .

E.T.S.

.- I' - .... /, , . \ /, \

I. \ I . 1 . I' . . \ . \

. \1 . . \' . \ ' I .\ I . .J. • • • •• \

'\ I ' , \ I· ,".' .. ~ '\', .. '~'" shaped-reflector(75*57cm) "': ... " .,.: ..... '\

• \.0 ' • v • \

I : : reception: astra 1/2 :

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

dB 40

20

ASTRA 1 (19.2°)/ASTRA2(28.2°)

· . . . . '" ........ ,- ............. ,. ............................. ~ .... - .......... .. • • I f I

t • • f 1

• \I " • •

, . . . . • I • • "

• • f • •

• " , 1 •

, . . . , · . . . . ~ . . ~ . . . .

.. .. .. ., .. ~ ,. ........ '" " ............. '" ..

E.T.S. ",. "",- '" ; - ...............

" ". , . / .

co-pol. • .. .. .. .j .. .. .. .. .. ~ . . .

cross-pol. : . . . . . . .. .. ~ I.. .. .. .. .. .. .. '.. .. .. .. .. .. .. ' '" .. .. .. .. ..

...... " 04C~~--~~-/~·1--~--~--~--~--~~~~~

I: ' I t

I . I : .

-1 0 -8 -6 -4 -2 o 2 4 6 8 10 degrees

Design 2b: Reception Astral (19.2° E) and Astra2 (28.2° E).

dimensions shaped reflector:

dimensions parabola:

wavelength = 2.65cm

F/D = 0.74 (approx.)

scan-angles = 2 * 4.884°

offset-angle = 37°

height of the aperture = 75.6cm width = 80cm offset height of the aperture = 37.8cm

diameter of the aperture = 78.4cm offset height of the aperture = 39.2cm

(frequency = 11.3 GHz)

(with respect to Amsterdam)

elevation angle astral 29.71° (with respect to Amsterdam)

elevation angle astra2 = 27.36° (with respect to Amsterdam)

feed : Philips corrugated horn (4 grooves)

edge illumination = -12 dB (approx.)

88

75.6

" /I ...

, I "" DIMENSIONS: eM , I "

, I " I. "

" " ... " I " I ,

'q:, 1 " -'I\;, I "

I r....... I " I 'A..-J .... "'" " t I·'" 1 ......... " ,I I "

37. ,I " " ......... ", I,' ...."

I~~~:-l------------~!~~~~~~:~~ 6.51 I I

80em

",!'II I

:4 25.47 .: I

\I 55.94

.'

Figure 55: Shaped reflector in the YZ-plane.

8Dem

Figure 56: Frontview of the aperture plane (XY-plane).

Figure 57: Frontview of the mechanical surface.

89

X-AXIS ASTRA2 (28.20 )

Z-AXIS

ASTRA1 (19.2°)

Figure 58: Scan angles of the reflector in the XZ-plane.

142

86

dimensions: mm

Figure 59: Positions of the feeds for the shaped reflector.

, 90

DIMENSIONS: eM

78.

39.

58.02

Figure 60: Offset parabola in the YZ-plane.

Figure 61: Frontview of the aperture plane (XY-plane).

91

82. em

, -..... 1 ... --------, , , , , , , , , ,

78.4cm

Figure 62: Frontview of the mechanical surface.

130

74

dimensions: mm

Figure 63: Positions of the feeds for the offset parabola.

shaped reflector scan-angle [deg.] I height [em] I width [em] I gain [dB] I G/T [dB/K]

4.884 I 75.6 I 80 I 37.9 I 15.93

offset parabola scan-angle [deg.] I diameter [em] I gain [dB] I G/T [dB/KJ

4.884 I 78.35 I 38.1 I 16.13

92

dB 40

20

-20

ASTRA 1 (19.2°)/ASTRA2(28.2°)

co-pol. .. .. II • "I ......... ..

• • .. I • •

.......... f' ............ ," ............ f· .......... "I" .......... "." .......... lOt" ... ..

......... '0< .. __ ........ " ............ '. ,. .... .. , , ,

,

E.T.S.

I ,/ ,: ,

/ '\' . ' ..... " ' oJ • I • ..... / •

........ 1 • / ., .

, , 1'--=_ ...... - ...... .

\ , .1

\ 1 ./ . .

1 ../ • \"/ .......... : ..... .

. shaped-reflector reception: ASTRA 1/2

-10 -8 -6 -4 -2 o 2 4

cross-pol. :

., ..... \. \

'/ '-\., \. 1 \

. . . l .;. . . . .\. ~ ,. . .. \ . \;' \ 7 \

. I i \ \ ,. \

6 8 10 degrees

dB 40

20

ASTRA 1 (19.2°)/ASTRA2(28.2°)

• • • ~ I •

.. .. .. - • t' .. .. .. .. .. .. ... .. .. .. .. ., ".. .. .. .. .. .. .. ~- .. .. • • .. ..,.. '" ... .. .. .. ..... .. ..

• t • , · . . , • • • I

, . , · . . .. ., ,. - '" .............. ' ........ ,. .. ,. ' ...... ..

, . . · . · . .

*"""" ....... .,- ...... t

/' , / " E.T.S. . /: ,,: / : " I. ,,/ • \

co-pol. ... '" .. "'," ..........

cross-pol.

. .

1 ' : \ 0~~~--~~-Y7-r-~--~--~--~~--~~-+-+~

-20

'/

· /. / -:. 1 ' _. I: \ / .

/ ;'\ / '

\ ,-, I.

\ I' \ . \ ;''' · \ \ I. ,

...,. \ ~" · \ I· \ • J' \ I

\1 offset-parabola reflector ...... : ...... :. --\ i' .. /. - -: ;. / .... ~ ...

reception: ASTRA 1/2

-10 -8 -6 -4 .. 2 o 2 4 6 8 10 degrees

Design 3a: Reception Eutelsat (13° E) Astral (19.2° E) and Astra2 (28.20 E).

dimensions shaped reflector:

dimensions parabola:

wavelength = 2.65cm

F/D = 0.74 (approx.)

height of the aperture = 57cm width = 75cm offset height of the aperture = 28.5cm

diameter of the aperture =60cm offset height of the aperture = 30cQ'l

(frequency = 11.3 GHz)

scan-angles = 1 * 1.505° and 2 * 8.2620 (with respect to Amsterdam)

offset-angle = 37°

elevation angle eutelsat = 30.680 (with respect to Amsterdam)

elevation angle astral = 29.71 ° (with respect to Amsterdam)

elevation angle astra2 = 27.36° (with respect to Amsterdam)

feed : Philips corrugated horn (4 grooves)

edge illumination = -12 dB (approx.)

95

57em

57.0

28.

I

I , I

I

" II "

J I "" I I "

I "

I 4t) I

',.'''''' I I • I '/& ',! I I : ............

I ' I' I

" " " " " " " ,

DIMENSIONS: eM

" " , " " " ,

" " , t. ........"" " t J ..... .... '"

11~ ~4~ _ ~ _____________ 3!~ l~~ ___ ::.~~ 4.8 I I

I I I I , , , ,

~I !ot I I 19.2 I

:~ -' I I

Figure 64: Shaped reflector in the YZ-plane.

75em

__ ... _____ L. ____ .... _ 60.15 em ,

75em

Figure 65: Frontview of the aperture plane (XY-plane).

Figure 66: Frontview of the mechanical surface.

96

X-AXIS

REFLECTOR

--...I ..... FEED2

EUTELSAT (130)

ASTRA1 (19.20 )

Z-AXIS

ASTRA2 (28.20 )

Figure 67: Scan angles of the reflector in the XZ-plane.

69 99

43

dimensions: mm

Figure 68: Positions of the feeds for the shaped reflector.

97

60

30

I I

I I

I I

, II .... "

I I ", I 1 ,

I I ",

I!:J 1 ',.. •• "! ' , I ~ ',.!. IICO I', 1 I , , I

/ I: ' '171.3° : I) ____ ~ ____ _

5.1..: :~ :

1 20.3 ~l ,f

, " " " " "

DIMENSIONS: eM

" " " " " " " " " " " " " " , "

------~!~~~~~~~~~

Figure 69: Offset parabola in the YZ-plane.

Figure 70: Frontview of the aperture plane (XY-plane).

98

63.3 em

60em I

---1----- --- -:..--.--I I

___________ J ___________ _

I I I I I I I I I I I

--- ------ ----::--~-~

Figure 71: Frontview of the mechanical surface.

73 109

53

dimensions: mm

Figure 72: Positions of the feeds for the offset parabola.

shaped reflector scan-angle [deg.] height [em] width [em] gain [dB]

1.505 8.262

scan-angle [deg.] 1.505 8.262

57 75 35.68 57 75 35.57

offset parabola diameter [em] gain [dB]

60 36.05 60 35.41

99

G/T [dB/K] 13.71 13.60

dB 40

20

EUTELSAT(13°)/ASTRA 1 (19.2°)/ASTRA2(28.2°)

, , , * I •

.. ~ ...... ,. ........ ,. .... - ........ "." ........ '" ~ ,- .......... ".'" ................ ..

. . . .. .. .. .. • 1. .. .. ., .. .. .. t.. .. .. .. .. .. .. ". .. ..

, . ,

E.T.S.

· . . · . . · .

, "" , ;'

co-pol . • .. .. .. "J .. • - • .. ..

, .

cross-pol ..

I

O ~T--r.---;~--~--~--_\~'~--~--~--~~--~--~ \ '

-20

, I \ , , , \' I " \ , I

.J- \' \ / ..... ;' • , , , 'I • , , .,. l, . \ '\

-,' I. ,. \ ' \ I . ;' , I' \ r " ./ .' \ ,"" ....

I ' \ '\ .\ ' I I. \ . I , . \ I • • , \

L • . • • • ' . " ' • , r • • '\ ' • ..' • , '\' • " • • • • • •

I : \ I shaped-re'flector(75*57cm) ~ ,': \ : \1 reception: astra 1 ,"

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

...... o ......

dB 40

20

EUTELSAT(13°)/ASTRA 1 (19.2°)/ASTRA2(28.2°)

. . '"' ..................... t ...... " ...... too .. ..

E.T.S.

.. ... .. I. <II ....... ,.1 .......... .. . · . · · :..----. --'~ "",. f ..... -!.. _ ,- , .......: · . · . .

· ...... ~' ............ .. ' .. ...... ,. .. · . · . · . · . · . · . · . · . . . O~~+---~--~--~--~--~--~~~~~~~

7 • •

( '. \ I . : \

':' I· t. t \ /:-'

. /. \. I : •. ,. \ " ...... I· \ . . \, . \ .

,I " ..... / • /. • • • • \. • " .... I' . . . . . . . \ . I \.

..... ~ ...... ~ ... shaped-reflector(75*57cm) ... ~ .... \ 'j" ... \ : : reception: eutelsat, astra2 : ~,': \

-20

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

dB 40

20

EUTELSAT(13°)1 ASTRA 1 (19.2°)1 ASTRA2(28.2°) • • • • • • • I •

• I • • • •

.......... t' ............ .- .......... .. I'" ............ ," .......... "." ................. ..

• • t I ,

• • t I . .

· . . .. .. .. .. .. \ .......... .. ,_ ............ toO .. · . .

E.T.S. ,

\ " /

co-pol.

cross-pol. · . · .

.. .. ,,' ............. ' ............. ' .......... . . . .

O . I ~~-+--~~--~--~--~\~'~/---7--~~~~--~~~

\.

-20

\ ·1 ,

d .\ : \ ,- ~ : \ 1 • \

· \ -" . \ , . , . · \ 1 . \

..... : .. \ .. , ... : ... \. .. · \, . \,

I:' I· \ .

.... " I' I. I . \ . \. . . . I . , . \

. ,. ... ~ "\ .( .. ~ ... offset-parabola(d=60cm) I • I •

I : \' : reception: ASTRA 1 I

" \ , \ I

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

EUTELSAT(13°)/ASTRA 1 (19.2°)/ASTRA2(28.2°)

dB 40 • • • • • I

.. .. .. ... .. t' ........................ .. ,- ............ ,- .......... "." ................. .. co-pol. ..... :- . -...

cross-pol. : · . .

,. , I. I 20 .... -, ...... ' ....... '. .................... ... . .. ' ............ - .. ' .. - .. .

-20

E.T.S.

./ ''/

? ,..: -. /' .

" I ..... "'" • ", , , .

",.. ""7 - "'" • "", -'-.

, \ . · ",. ""', · ".... .' , . , · ., ..... -" · -' , · . · .' .

...... .. -, ........... ·0" ....... ..

· . · . .... -: ..... ': ... offset-parabola( d=60cm) · , · , reception: eutelsat,astra2

-1 0 -8 -6 -4 -2 0 2 4 6 8 10 degrees

Design 3b: Reception Eutelsat (13° E) Astral (19.2° E) and Astra2 (28.2° E).

dimensions shaped reflector:

dimensions parabola:

wavelength = 2.65cm

F/D 0.74 (approx.)

height of the aperture = 75.6cm width = 80cm offset height of the aperture = 37.8cm

diameter of the aperture = 78.4cm offset height of the aperture = 39 .. 2cm

(frequency = 11.3 GHz)

scan-angles = 1 * 1.505° and 2 * 8.262° (with respect to Amsterdam)

offset-angle 3r

elevation angle eutelsat = 30.68° (with respect to Amsterdam)

elevation angle astral = 29.71° (with respect to Amsterdam)

elevation angle astra2 = 27.36° (with respect to Amsterdam)

feed : Philips corrugated horn (4 grooves)

edge illumination -12 dB (approx.)

104

DIMENSIONS: eM

75.6

37.

55.94

Figure 73: Shaped reflector in the YZ-plane.

Figure 74: Frontview of the aperture plane (XY-plane).

105

80cm

Figure 75: Frontview of the mechanical surface.

X-AXIS

REFLECTOR

-__�� __ FEED2

EUTELSAT (13°)

ASTRA1 (19.20 )

Z-AXIS

ASTRA2 (28.20 )

Figure 76: Scan angles of the reflector in the XZ-plane.

dimensions: mm

Figure 77: Positions of the feeds for the shaped reflector.

106

78.

" " , " " DIMENSIONS: eM I I ,

, I ' 1 t " ,

I , , " I ...

I ...

,'J!! I ", , ..... ,..., l " I , N· ... I ... I'cti ,...... "

I' r "

I' ... 39. I I .........',

" I ....."" I If......... "

_ _ _ _ ~~ .30

_ ~ _____________ 3!~ l ~ ~ ______ :.~~ 6.6 I I I

1>1 r4 I

: 26.5 .: ~ -I

1 58.02 ~ ~ , I

Figure 78: Offset parabola in the YZ-plane.

82.8 em

, ---{ .. - ....... ---, , , , , , , , , ,

78.4cm

Figure 79: Frontview of the aperture plane (XY-plane).

Figure 80: Frontview of the mechanical surface.

107

sea

I

I

dimensions: mm

Figure 81: Positions of the feeds for the offset parabola.

1.505 8.262

1.505 8.262

[deg.]

gain [dB] G/T [dB/K] 38.1 16.13 37.6 15.63

offset parabola diameter [em] gain [dB] G/T [dB/K]

78.4 38.4 16.43 78.4 37.6 15.63

108

EUTELSAT(13°) / ASTRA 1 (19.20 ) / ASTRA2(28.5°) J • • • • ,

dB 40 · .... ~ ...... ~ ...... : ....... : ....... : ....... : .... co-pol. /

• • lOt • ,. ••• ,.

. cross-pol.

20 • .. ... .. .. "" ... .. .. .. .. .. too .. ~ .. .. .. • I.. .. .. ... .. . . .

E.T.S. ~....:..-- -..: '\. I",

\ 1 \ . I

0~~-4~~~-4~--~~--_\~:~/--~--~~~~~~~~~ \.

-20

l' I \ .1 I \. I \ 1

I ~, , ;.

. ':, I , ,: " \ ,_ 1-";' I 1.\1 . , . I· t' \.1 \

I '\ ., I \ . '/ \ · i 0 •• ~ ,0 .,. • 0 0 ~ i i 0 0 0 0 shaped-offset reflector .. 0 r ': .... , . i: ... 0 \ 0

" " I • II I" , " \ \ I : 1/ : 1/ reception: ASTRA 1 I~: ',,: \ I

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

I-' I-' o

ASTRA 1 (19.2°)/ASTRA2(28.2°)/EUTELSAT(13°)

dB 40 . . ....... f' ........ .......... "." ... .... .. 'I, • ........ ... ,_ ....... 'I," .. . co-pol. ..... : ..... .

,

cross-pol. : . .

, , . , 20 ..... , ...... ' ....... '. . . .. .' ....... ' .... . .. • • ~... • .. .. .. .. sf.. • .. .. R .. ..* • .. ,. .. • ..

E.T.S. .",.,.'----. "'" ' ......

, ,. " , " " '/ . \

I: : \ I ,

I : o~~~~~--~~~--~--~~~~~~~~

-20

• I --.:. ,. - -" , ." ", . . ,. .

_ -....,. - t •

· . · , . ·

\ . \ ' ,.. ....

/ , : \: , \ .

\' -.

~--~--~--~--~ ,.,I' \ .'

\'/ ' , \' I .... , .

..... : ...... :' .... shaped-reflector reception: astra2/eutelsat

. .. ',' ..... ' ..... \ .. , \.1 \ : \ ~ \ , ,

-1 0 -8 -6 -4 -2 o 2 4 6 8 10 degrees

EUTELSAT(13°) / ASTRA 1 (19.20 ) / ASTRA2(28.5°)

dB 40 ..... ~ ...... : ...... : ....... : ....... : ....... : .... co-pol. .. '.' .... . : : : : : /: . . . . . . . . cross-pol.

20 ....................... .. -............. ' ............. " . ................... .. * .. • .. !

E.T.S.

\ :/ \

• , WI \ :,. .. "

-20

r \. \ /. \ / I,' . , . I ) \ I . \ .'" "

/ \ .\ , . \ /. . " / • I .1 I. "\ /'

I' • .\ .\ • • • I. \ ' \ . \ I . \ I , / I ' I .

\ .. ,. .. ~ t . I ... ~ \ '" .. offset-parabola reflector" 1/ .: ... \,' i ': .. \ .. ;' \1 '11 ., I' /.

I , . l reception: ASTRA 1 \1: \ " \I

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

dB 40

20

ASTRA 1 (19.2°)/ASTRA2(28.2°)/EUTELSAT(13°)

..... ~ .. , .. , : ..... " ":" " " . " , ":" " " " " " ":" " " . " " ':" ." co-pol. , , , , , /' , . . . " " • I t I I I I •

... . -, ..... .

cross-pol. :

E.T.S. .r

/,

/ ' / '

O~~~~~-H~/~:--~--~~~--~--~~

\ I' , 1 •• '" \

1 : \ /-, '. /..,. ...... _I '. .... , . , .. \ ....... " ....... -_/. . " ,.I'. ,

/ , '.----'-' -----'-----'----'---------'-----, ..... .,;, , . . ..... /

..... : ..... ':' ... , offset-parabola ... ':' ..... ':' ... , . -20

: : reception: astra2/eutelsat : :

-1 0 -8 -6 -4 -2 0 2 4 6 8 10 degrees

Design 3c: Reception Eutelsat (130 E) Astral (19.20 E) and Astra2 (28.20 E).

dimensions shaped reflector:

dimensions parabola:

wavelength = 2.65cm

F/D = 0.74 (approx.)

height of the aperture = 85cm width = 110cm offset height of the aperture = 42.5cm

diameter of the aperture = 99.7cm offset height of the aperture = 49.8cm

(frequency = 11.3 GHz)

scan-angles = 1 * 1.505° and 2 * 8.262° (with respect to Amsterdam)

offset-angle = 37°

elevation angle eutelsat = 30.68° (with respect to Amsterdam)

elevation angle astral = 29.71° (with respect to Amsterdam)

elevation angle astra2 = 27.36° (with respect to Amsterdam)

feed : Philips corrugated horn (4 grooves)

edge illumination = -12 dB (approx.)

113

85cm

85

42.5

, /

I I

" II " I I '"

I , " I , "

I "

I

'''' ' I .... r . I

, I J?!'" J.. I I"'" I ........ "'" , I ,

,I

l

" " " " " " "

DIMENSIONS: eM

" " " , " " " " " " " .... "

" I ' I .... '"

'1~~40 7.2, , i------------!!~t:~~~~~~

110cm

, , , , , , . ,

", rill : '28.5 ,4 , 14

62.9 .'

Figure 82: Shaped reflector in the YZ-plane.

--------t------.

110cm

Figure 83: Frontview of the aperture plane (XY-plane).

Figure 84: Frontview of the mechanical surface.

114

X-AXIS

REFLECTOR

---- FEED2

EUTELSAT (130)

ASTRA1 (19.20 )

Z-AXIS

ASTRA2 (28.20 )

Figure 85: Scan angles of the reflector in the XZ-plane.

92 132

dimensions: mm

Figure 86: Positions of the feeds for the shaped reflector.

115

Figure 87: Offset parabola in the YZ-plane.

Figure 88: Frontview of the aperture plane (XY-plane).

116

105. em

99.7<:m

Figure 89: Frontview of the mechanical surface.

138

82

dimensions: mm

Figure 90: Positions of the feeds for the offset parabola.

shaped reflector scan-angle [deg.] height [em] width [em] gain [dB] G/T [dB/K]

1.505 85 110 39.1 17.13 8.262 85 110 38.5 16.53

offset parabola scan-angle [deg.] diameter [em] gain [dB] G/T [dB/K]

1.505 99.7 40.3 18.33 8.262 99.7 38.1 16.13

117

...... ...... 00

EUTELSAT(13°) / ASTRA 1 (19.2°) / ASTRA2(28.5°) • • I I I I

dB 40 ..... ~ ...... ~ ...... : ....... : ....... : ....... : .. " co-pol. ~/

. . ", ..... .

. cross-pol.

20 ................... .. 'I' ... ' .. .. " .............. ' ..... D .. .. I' ........... " ... . . . . . . • 0 0 . / - '" . ,. - , .

...... -;' ., : , E.T.S.

o~~-+-+~~--~~~----~--~~~~--~~~~~

-20

o I 1' ...... {'..... I 0 ,

o \ , \ I 0

I 0 \ 0 \ •

I \ I 0 I I I 0 \ I -0,

': \ I : \. "': I I : \ / ....: - I • 0 • \ .,

I' J • 1/ . \!. I .\ /' I . \ I 0 "0 0 0 0 0 • \ I . ,

\ '0 I 0 0 \. 0 \

•• '\' T ~ ••• '\i . ~ ... ,. .. shaped-offset reflector ..... : ....... :. \/ .. . \ I . • • .

.. 0 reception: ASTRA 1

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

EUTELSAT(13°) / ASTRA 1 (19.20 ) / ASTRA2(28.5°) .. .

dB 40 , . , , .. I

, .... ~ ...... : .. , .. , .: ....... :. , ..... : ..... '/':, . ,. co-po. , l I 9 •

, , . I. . .. , f , • • • • •

. • .. "'t· • '" •••

cross-pol. · .

20 .............. 'lit ....... I ... '" ...... .. " ..... ,. .... " .............. t,. ...... ..

• I • • •

• • t ,

• • I •

E.T.S. , ,-...... f

. " ..... . " ? ~ /

I. I .

I

I \

-20

'f • • I I .

• "" - ".' I I '.' ./ . \ I I I • : /-, i. : I J' \

/-,./ '"" • • I I , .

, ' J. • --!.. •• ----L..-' -----''----_..!--_-'--_---!........, ,/I , " '\: .-- ,. " '" \, \ .\, . , ... ~ ...... ~ .... shaped-offset reflector ..... : .. '( I. •• ~, •. ,. •• . . \ ,

reception :eutelsat/astra2 . : 'I I,

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

EUTELSAT(13°) / ASTRA 1 (19.20 ) / ASTRA2(28.5°)

dB 40 ..... ~ ...... ~ ...... : ....... : ...... : ..... : .... co-pol. ,L---,--"\ /: •

. .. • •• .. .. \I .....

. . cross-pol.

20 ......................... ' ................ '" ................................... .. , t ~ • •

, "-,' ...... " I . / . / '\ \ . I .

E T S \.., • • • . I

\. I I: 1

~I

~20

: I \ I ;" \

. 1\ :, \ 1 :,,' \: I' / \' I \'1 \ I : \ I \: I \

'l' I :, v . I , :' \,' \ I. , I '\' \ 1 : \' I" \ I: \ I

\ f: " :" r . . . ': ;' : . ",':' .. : ':. . .. offset-parabola reflecto r ....• ~ .... ~: '; .. \, ~/' .

,I : ,,' reception: ASTRA 1 .. \I i I

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

dB 40

20

EUTELSAT(13°) / ASTRA 1 (19.20 ) / ASTRA2(28.5°)

. , · . . , . . .. .. .. ., • #' .. .. • .. .. .. .., .. .. .. " .. .. ,. .. .. .. .. '" .. t'" .. .. • .. .. "." ... .. .. .. .. '"'," .. .. · . . .

• • • I · . . , · . . , " , , . · . · . · , , , · .

, ,

-- - ...... :.". ". -" : ETS

'", . \ . . . .( , .\ /'

/ ' . .

co-pol. . .. .. .. .. . .. ... .. .. ..

.

cross-pol.

o~--~--~~--~--~--~--~~~--~~+-+-~ • '" • • , I '\

/ .-; \ I .\ ",' , 'I •

/ , ,'" \' /' ,/ , , '\ I " , ;, ,. I· . I \ .,. ,

, . ," ... \ /' ,/ ',/ - I. • • ..... \

/... • ''/', ,'. • "I ,/.

-20 I • I • •

, ... ': ' .... ': ... offset-parabola reflector ...... : ...... : ..... .

: : reception: eutelsat/astra2 : :

-10 -8 -6 -4 -2 o 2 4 6 8' 10 degrees

Design 4: Reception Eutelsat (13° E) Astral (19.2° E) and Telenor (lO W).

dimensions shaped reflector:

dimensions parabola:

wavelength = 2.65cm

F/D = 0.74 (approx.)

height of the aperture = 85cm width HOem offset height of the aperture = 42.5cm

diameter of the aperture = 99.7cm offset height of the aperture = 49.8cm

(frequency = 11.3 GHz)

scan-angles = 1 * 4.17° and 2 * 10.81° (with respect to Amsterdam)

offset-angle = 32°

elevation angle eutelsat 30.68° (with respect to Amsterdam)

elevation angle astral = 29.71° (with respect to Amsterdam)

elevation angle telenor = 30.79° (with respect to Amsterdam)

feed: Philips corrugated horn (4 grooves)

edge illumination = -12 dB (approx.)

122

85

" /, " I , ", DIMENSIONS: eM

I I , I , '

I 1 "" I " I ,

I "

I " I "

I I " ,I('() I " , r~. "

, Iq;; ',.!. ", ~ I I ..... '" " ,I' "

42.5 ,I' ", /.: .............', ,I 1 ................""

I ~ I ..... "

~~~4:_~ ____________ !!~~~~~:~~ 7.2, , , , ~~ , I

I '28.3 I I I ~ , ,4 I , I 62.9 I ~ ~ I I

Figure 91: Shaped reflector in the YZ-plane.

~10~m. · . · . : : ~~ : :b.;~. · . --------~------ --------· . · .

Figure 92: Frontview of the aperture plane (XY-plane).

123

Figure 93: Frontview of the mechanical surface.

X-AXIS

REFLECTOR ASTRA1 (19.2° E)

EUTELSAT (130 E)

Z-AXIS

TELENOR (1 0 W)

Figure 94: Scan angles of the reflector in the XZ-plane.

85 136

131

dimensions: mm

Figure 95: Positions of the feeds for the shaped reflector.

124

99.7

49.8

I

rill 33.68 I :. .:

I ~

DIMENSIONS: eM

73.78

Figure 96: Offset parabola in the YZ-plane .

Figure 97: Frontview of the aperture plane (XY-plane).

125

105.2 em

. 99.7em ,

...... 1-- ..... -----, , ,

· , · · , ----- .. -----~- .... -~-------, ,

Figure 98: Frontview of the mechanical surface.

89 200

144

dimensions: mm

Figure 99: Positions of the feeds for the offset parabola.

shaped reflector scan-angle [deg.] height [em] width [em] gain [dB] G/T [dB/K]

4.17 85 110 39.0 17.03 10.8 85 110 38.3 16.33

offset parabola scan-angle [deg.] diameter [em] gain [dB] G/T [dB/K]

4.17 99.7 39.9 17.93 10.8 99.7 36.8 14.83

126

40~--~----~--~----~--~--~----~--~----~--~

dB 30

t • • I

,. .. .. .. .. .. "' .. .. .. .. .. ,'" .. . . .. .. ... - . .. .. .. .. .. ... :/ co-pol. : . . .. .. .. .. .. "." .. .. .. .. .. .... . .. .. .. .. ". .. .. ... .. ... ..

cross-pol. 20 · . . ...................... __ ............... .

10 E.T.S.

\ O-r--+-~-r~~~----~----------------~\---,~--~~-+--'~~

:...... \ ,: \ · • '. \ I. . • I,. \ . · ,,,,. \ I' . . . \ . · ./ . . . . I , . /., . . . . . • . . . . . . r .. \ .. I .•.. t f •••••••••••••••••••••••• f .••••• \ •••••••••••.••••••• .,.... . \,. . . . I . I . \ ....

" . \ I. ,11 ',. \ I' \ 1\ I \ 1 \, : \': 11 1,: ,,: \ f : \ / "

I \:: ": \':". . . t: " : \ I : \ I ••• \ ••••.• J J.~...... . ........ \t .. .•. , . . 1 •••

, :: \ I shaped-offset reflector :.: \ I I' 'I . . \ I I,' : 4.17 degrees scannin ~

-30~--~---+--~==~====~==~==~--~---+--~

-10

-20

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

ASTRA 1 (19820 E)/EUTESAT(13° E)/TELENOR(1° W) 40~--~----~--~--~--~----~--~--~----~--~

, • •• '" I

: : : : : : / co-pol. : dB 30 ..... : ....... :. _ ..... : .. _ .... : ..... _ .: ....... '~. _ .. , .. _ ... _, .... __ .: ___ .. .

: cross-pol.:

20

10 E.T.S. :

-10

O~~~~~~--~-+----------~~--~~~~~~~

I : ' ..... -:, , , !" ..... . \ /. \ . '/;'" \ /. . . , .

, . . . . . ,. ~ .. .. .. .. ~ -,,- --: .-. "'1".. '"' .. .. .. .. .. •• .. .. .. .. .. .. I_ .. .. .. .. .. .. t.. .. co .. .. ~ .. ,.. .. .. .. .. .I _... D ..

. . .. ................. -......... .. , '

',:, -, ',', '\ ,-\ , I' - .. - -. - - -. - -~ -. - -.------------~- -. ~/-·-1-· - - -.\ - -,- --

shaped-offset reflector \ : ,I :' I , " "I

1 0.81 degrees scanning . ': : ~ I -30~--~---+--~~~F=~~~~~~---+-L-+~~

-20

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

ASTRA1 (19.20 E)/EUTELSAT(13° E)ITELENOR(1 0 W) 40~--~----~--~----~--~~--~--~----~--~--~

. , .

dB 30 • t .. f .. .. .. w .. .. w ~ .. .. .. .. •• .. .. .. • .. .. I. .. .. .. .. .. .....

:,: co-pol. , : / .. ~ -.. ... -~ .... ~ ,. -r ....... ," .. . , . cross-pol.

20 • I" , ' , I ,'I

E. T. S .: , i ,: I •. _ ••• ,." •• - . - 7'- •• -"';"",,_. -10

, I" \.

o~----~~~~~--~------------~~--~~~~~~ . \ .\ ,I , \ I ' " : \ 1,,\:

1", I I . I . · • I \' I' , • . \ ,. / \ .

• • • • . . • . . ;..~ • . . . . . ,. • • • . • ... . . . • • .•• . . . . . .,. . . • • . .,. \ I. . . . \., .,. . .,. • ~ . • • . • • • I" I ~ I, , , 'J ' ./ , , " . I . I \' I \ r , I" I I' \ ,\ II .

'-, r \ . \1 '\ I ~ \ /' \ I I • \,. , \' I

I \', I' '\ , , I, \ I" 'ff:" .\ ... ~ - _. _.. . ... -: .. - -. - -:-. -'f/--

., , offset-parabola

-10

-20

4.17 degrees scanning -30~--~--~--~P===~==~===F==~~~~--+-~

-10 -8 -6 -4 -2 o 2 4 6 8 10 degrees

ASTRA 1 (19.2° E)/EUTELSAT(13° E)fTELENOR(1 ° W) 40~--~--~--~--~--~--~--~--~--~--~

dB 30

20

10

. : : : : : : co-pol. : · .... :./ '

-- ...... : ........ :-- ..... : ........ ' .......... : ..... -~.- ... - ..... - .......... - ... :.-- ... .. • " I

I • I •

I • I • · .. . · .. , I • • • • ........ .. ,- ............. - ............ ,- ............ ,- .......... "." .. '" ...... .. • G ." I ' • I · . . . . .. .

E.T.S. : ~-~-- ....... . ,.

, \ .. .. -to. ........ . \

\ \

cross-pol. :

04-------~~~------------~----~~-4-+~ ~ . / \

/ . . \ .\ / . . / . , ... . \ / \. . . . \ \. /, · . /,. , . . . /. .

.....•...... ~ I ............. , .......•.......•....... , ....... \. J •• \ •••• - •.•• • • • , • • , 'V \ '/ \

.I : : : : : : "": \/ ....... : "' .......... /. .., "...... I I I I , • • I

-10

• I I • , • ,

I • • • t • I ............. ~ .... ,.-------------r ................. . :: offset-parabola ::

-20

: : 10.81 degrees scanning : : -30~--~--~--~~~~=F~=F~=F~~--~--~

-1 0 -8 -6 -4 -2 o 2 4 6 8 10 degrees

10 Conclusions

• The method of calculation is very flexible and is usable for both the symmetrical and offset-shaped reflector. In addition the calculation can also be used for the design of parabolic reflectors (symmetrical and offset).

• The shaped reflector can be used for scan angles up to 2 * 30°.

• For all the four designs presented in this report apply that with the shaped reflector one has less trouble with interference from nearby satellites than the offset parabola.

• The gain reduction as a function of the scan angle is smaller for the shaped reflector than for the offset parabola.

• A smaller reflector surface will result in a lower gain. However, a smaller reflector surface means also smaller phase errors in the aperture plane. So, there is less gain reduction as a result of the increased scan angle for smaller reflector surfaces.

• To compensate for the difference in elevation angle between the different satellites, we have to tilt the entire dish antenna a little bit.

• The positions of the feeds (Philips 4 grooves corrugated horns) do not need to be extra adjusted for different countries inside Europe.

• For all the four designs presented in this report apply that there is enough space to place the different feeds.

• To obtain a circular aperture, we have to choose the mechanical height of the reflector surface a little bit bigger than the diameter of the aperture.

131

132

References

[1} C.M. Rappaport and W.P. Graig: High Aperture Efficiency Symmetric Reflector Anten­nas with up to 60° Field of view, IEEE Transactions on Antennas and Propagation, Vol. 39, Nf!... 3 , March 1991.

[2] C.M. Rappaport and W.P. Graig: A high Aperture Efficiency, Wide-Angle Scanning Offset Reflector Antennas, IEEE Transactions on Antennas and Propagation, VoL 41, Nf!... 11 , November 1993.

[3] A Roederer: Private communications.

[4} C.A. Balanis: Antenna Theory, Arizona State University, John Wiley & Sons, 1982.

[5] R.M.M. Mattheij. Inleiding Numeriek Methoden, Lecture notes Eindhoven University of Technology, faculty Mathematics and Infomation science.

133