International Journal of Advanced Computer Science, Vol. 2, No. 9, Pp. 348-351, Sep., 2012.

Manuscript Received:

10,Oct., 2011

Revised:

1,Apr., 2012

Accepted: 21,May,2012

Published: 15,Oct., 2012

Keywords

Azimuth,

bore sight,

Geosynchronous

satellite vehicle

(GSV),

satellite ground

control station,

Satellite look

angles,

Sub-satellite

point,

Elevation

Abstract Antenna look angles of

geostationary communications satellite

provide the information required to

ensure that control station antenna is

directed towards the satellite; more

specifically to ensure that the main lobe of

the antenna is aligned with the main lobe

of the satellite’s antenna, and to ensure

that the largest amount of energy is

captured from the satellite. To optimize

the performance of a satellite

communications system, the directions of

maximum gain of a satellite ground

control station antenna (referred to as

boresight) must be pointed directly at the

satellite. To ensure that the earth station

antenna is aligned, two angles must be

determined: the azimuth and the elevation

angle. Azimuth angle and elevation angle

are jointly referred to as the antenna look

angles.

This paper describes in detail, the

mathematical modelling of antenna look

angles of two models of satellite ground

control station. The mathematical models

developed are abstract models that use

mathematical equations to describe the

antenna look angles. The mathematical

representations presented takes into

consideration the redundancy of the

control stations. Two models are used in

order to pave way for redundancy so that

if one fails the other takes over.

Mathematical model of antenna look

angles is a mathematical representations

of the equations governing them.

1. Introduction

The orbital slot of a geostationary communications

satellite determines the look angles at which a ground

antenna needs to be positioned to see the satellite. Higher

This work was supported by the National Space Research and

Development Agency (NASRDA) organization.

Ogundele Daniel Ayansola (National Space Research and Development

Agency, NASRDA, Abuja, Nigeria, ayansolaodaniel@gmail.com) and

Adediran Yinusa A. (Electrical Department, University of Ilorin,

yinusade@yahoo.com).

look angles provide greater reliability by improving the

quality of the communication link. On the other hand, low

or shallow look angles usually face obstructions from trees,

nearby buildings, or other objects and are more subject to

interference, particularly in heavy rain. The antenna of a

satellite ground control station needs to be properly

positioned in order to be able to track geostationary satellite.

With geosynchronous satellites, the look angles of

control earth station antennas only need to be adjusted once

as the satellite will remain in a given position permanently,

except for occasional minor variations [Wayne, 2001]. To

communicate with a satellite, ground-based reflector (dish)

antennas are used. Reflector parabolic antennas can focus

the transmitted power from/to a narrow region of the sky.

This allows for establishment of communication links over

long distances, thus minimizing transmitted electromagnetic

power requirements. However, because the signal is

concentrated in a narrow region of the sky, the antenna must

be precisely pointed at the emitting/receiving source. The

problems in pointing an antenna can range from simple to

complex, depending on the motion of the satellite in its orbit

[Tomas and David, 1994].

Look angles are most commonly expressed as azimuth

(Az) and elevation (El), although other pairs exist. For

example right ascension and declination are standard for

radio astronomy antennas. Azimuth is measured eastward

(clockwise) from geographic north to the projection of the

satellite path on a (locally) horizontal plane at the earth

station. Elevation is measured upward from the local

horizontal plane at the earth station to the satellite path. In

all look angle determinations, the precise location of the

satellite is critical. A key location in many instances is the

subsatellite point [Timothy et al, 2003].

The location of a satellite is generally in terms of

latitude and longitude similar to the way the location of a

point on earth is described. However, because a satellite is

orbiting many miles above the earth’s surface, it has no

latitude or longitude. Therefore, it location is identified by a

point on the surface of earth directly below the satellite.

This point is called the sub satellite point (SSP).

Angle of elevation (sometimes called elevation angle)

is the vertical angle formed between the direction of travels

of an electromagnetic wave radiated from an earth station

antenna pointing directly toward a satellite and the

horizontal angle. The smaller the angle of elevation, the

greater the distance a propagated wave must pass through

earth’s atmosphere. As with any wave propagated through

Mathematical Model of Antenna Look Angle of

Geostationary Communications Satellite Using Two

Models of Control Stations Ogundele Daniel Ayansola & Adediran A. Yinusa

Ogundele Daniel Ayansola et al.: Mathematical Modeling of Antenna Look Angle of Geostationary Communications Satellite Using Two Models of Control Stations.

International Journal Publishers Group (IJPG) ©

349

earth’s atmosphere, it suffers absorption and may also be

severely contaminated by noise. Azimuth angle is the

horizontal angular distance from a reference direction,

either the solution or northern most point of the horizon.

Azimuth angle is defined as the horizontal pointing angle of

an earth station antenna. For navigation proposes, azimuth

angle is usually measured in a clockwise direction in

degrees from true north.

In the geometry of the range and elevation angle

calculation shown in Figure 1, two models of satellite

ground control stations are presented for the determination

of antenna look angles. One satellite ground control station

can be used as main station while the other can be used as

the back-up or redundant system, or vice-versa. The

redundant system will take over the control of the satellite

whenever the main control system failed or its link is being

obstructed by the rain attenuation. The models will aid

availability of the system at all time. The two ground

control stations are linked by a geostationary satellite for

continuous communications. Model 1 is for Satellite

Ground Control Station X while model 2 is for Satellite

Ground Control Station Y.

The antenna look angles of the satellite ground control

stations X and Y are modelled using Fig. 1.

re

cos Ɣ1

Center of earth

Subsatellite point

Satellite Ground Control

Station

rs

El1

Satellite

re

El2

90 – Ɣ1

θ2 =

90 – Ɣ2

90 +

Ɣ2

β1 =

90 –

E

l 1 -

Ɣ1

β2

= 9

0 –

El 2

– Ɣ

2

re

cos γrs -

Ψ1

Z

re

re

X Y

rs = distance from the center of the earth to the

satellite

re = distancer from the center of the earth to the

earth station

d = distance from the earth station to the satellite

Ɣ1 and Ɣ2 are angles between re and rs

Ψ1 and Ψ2 are angles between re and d1 and re and

d2

d1 d2

Ɣ1 Ɣ2

Satellite Ground Control

StationΨ2

θ1 =

90 +

Ɣ1

D

Equator

O

V

Fig. 1 Geometry of the range and elevation angle calculation

2. Mathematical Modeling of the

Antenna Look Angle

The mathematical representations of the antenna look

angles of geostationary communications satellite are

developed for two models of satellite ground control station

designed. The look angles are separately developed for

each of the models.

A. Model 1: For Satellite Ground Control Station X

From ∆ZXO: Using cosine formula, we have

𝑑12 = 𝑟𝑠

2 + 𝑟𝑒2 − 2𝑟𝑒𝑟𝑠𝑐𝑜𝑠𝛾1 (Equ. 1)

𝑑1 = [𝑟𝑠2 + 𝑟𝑒

2 − 2𝑟𝑒𝑟𝑠𝑐𝑜𝑠𝛾1]1

2 (Equ. 2)

𝑑1 = 𝑟𝑠 [1 + (𝑟𝑒

𝑟𝑠)2

− 2(𝑟𝑒

𝑟𝑠) 𝑐𝑜𝑠𝛾1]

1/2

(Equ. 3)

Similarly, from ∆ZXO

𝑟𝑒2 = 𝑑1

2 + 𝑟𝑠2 − 2𝑑1𝑟𝑠𝑐𝑜𝑠(90 − 𝛾1 − 𝐸𝑙1) (Equ. 4)

= 𝑑12 + 𝑟𝑠

2 − 2𝑑1𝑟𝑠sin (𝛾1 + 𝐸𝑙1) (Equ. 5)

𝑟𝑒 = [𝑑12 + 𝑟𝑠

2 − 2𝑑1𝑟𝑠sin (𝛾1 + 𝐸𝑙1)]1

2 (Equ. 6)

= 𝑑1 [1 + (𝑟𝑠

𝑑1)2

− 2(𝑟𝑠

𝑑1) sin (𝛾1 + 𝐸𝑙1)]

1/2

(Equ. 7)

Also, from ∆ZXO

𝑟𝑠 = 𝑟𝑒 [1 + (𝑑1

𝑟𝑒)2

+ 2(𝑑1

𝑟𝑒) sin 𝐸𝑙1]

1/2

(Equ. 8)

Applying sine formula to ∆ZXO, we have

𝑠𝑖𝑛𝐸𝑙1 =(𝑟𝑠−

𝑟𝑒𝑐𝑜𝑠𝛾1

)sin (90+𝛾1)

𝑑1 (Equ. 9)

Therefore,

𝐸𝑙1 = sin−1 [(𝑟𝑠 −

𝑟𝑒

𝑐𝑜𝑠(𝛾1)) sin(90 + 𝛾1)

𝑑1]

= sin−1 [(𝑟𝑠𝑐𝑜𝑠(𝛾1) − 𝑟𝑒)

𝑑1] (Equ. 10)

Figure 2 shows the position of a hypothetical

geosynchronous satellite vehicle (GSV), subsatellite point

(SSP), and an earth station (ES) all relative to Earth’s

geocenter. The SSP has 300E longitude and 0

0 latitude. The

earth station has a location of 300W and 20

0N latitude. Ls

and ls and are respectively the latitude and longitude of the

subsatellite point, while Le and le are the latitude and

longitude of the Earth station respectively.

80

70

60

50

40

30

20

10

0

-10

-20

-30

-40

0 10 2030

4050

102030

4050

N

ESgeocenter

SSP

EW

GSV

Greenwich

“prime” meridian

(00 longitude)

A B(le, Le)

(ls, Ls)

ls - le

C

90

Fig. 2. Position of a hypothetical geosynchronous satellite vehicle (GSV),

International Journal of Advanced Computer Science, Vol. 2, No. 9, Pp. 348-351, Sep., 2012.

International Journal Publishers Group (IJPG) ©

350

its respective subsatellite point (SSP), and an arbitrary selected earth

station (ES)

From ΔABC in Fig. 2, using Napier’s rule for a

spherical right-angled triangle which states that the sine of

an angle is equal to the product of tangents of the two

adjacent angles; then

sin(𝐿𝑒) = tan𝐴𝑡𝑎𝑛|𝐵| = tan𝐴 tan|𝑙𝑠 − 𝑙𝑒| (Equ. 11) Therefore,

tanA =tan|𝑙𝑠−𝑙𝑒|

sin(Le), then 𝐴 = 𝑡𝑎𝑛−1 [

tan|𝑙𝑠−𝑙𝑒|

sin(Le)] (Equ. 12)

Once angle A is determined, the azimuth angle 𝐴𝑧 can

be found. Four situations must be considered, the results for

which can be summarized as follows:

1. 𝐿𝑒 < 0; 𝐵 < 0: 𝐴𝑧 = 𝐴

2. 𝐿𝑒 < 0; 𝐵 > 0: 𝐴𝑧 = 3600 − 𝐴

3. 𝐿𝑒 > 0; 𝐵 < 0: 𝐴𝑧 = 1800 + 𝐴

4. 𝐿𝑒 > 0; 𝐵 > 0: 𝐴𝑧 = 1800 − 𝐴

B. Model 2: Satellite Ground Control Station Y

From ∆ : Using cosine formula, we have

𝑑22 = 𝑟𝑠

2 + 𝑟𝑒2 − 2𝑟𝑒𝑟𝑠𝑐𝑜𝑠𝛾2 (Equ. 13)

𝑑2 = [𝑟𝑠2 + 𝑟𝑒

2 − 2𝑟𝑒𝑟𝑠𝑐𝑜𝑠𝛾2]1

2 (Equ. 14)

= 𝑟𝑠 [1 + (𝑟𝑒

𝑟𝑠)2

− 2(𝑟𝑒

𝑟𝑠) 𝑐𝑜𝑠𝛾2]

1/2

(Equ. 15)

Similarly, from ∆ZYO

𝑟𝑒 = [𝑑22 + 𝑟𝑠

2 − 2𝑑2𝑟𝑠sin (𝛾2 + 𝐸𝑙2)]1

2 (Equ. 16)

= 𝑑2 [1 + (𝑟𝑠𝑑2)2

− 2(𝑟𝑠𝑑2) sin (𝛾2 + 𝐸𝑙2)]

1/2

(Equ. 17)

Also, from ∆ZYO

𝑟𝑠 = 𝑟𝑒 [1 + (𝑑2

𝑟𝑒)2

+ 2(𝑑2

𝑟𝑒) sin 𝐸𝑙2]

1/2

(Equ. 18)

Applying sine formula to ∆ZYO, we have

𝑠𝑖𝑛𝐸𝑙2 =(𝑟𝑠−

𝑟𝑒𝑐𝑜𝑠𝛾2

)sin (90+𝛾2)

𝑑2 (Equ. 19)

Therefore,

𝐸𝑙2 = sin−1 [(𝑟𝑠 −

𝑟𝑒

𝑐𝑜𝑠𝛾2) sin(90 + 𝛾2)

𝑑2]

= sin−1 [(𝑟𝑠𝑐𝑜𝑠(𝛾2)−𝑟𝑒)

𝑑2] (Equ. 20)

Similarly, from ΔABC in Figure 2, using Napier’s rule for a

spherical right triangle

tanA =tan|𝑙𝑠−𝑙𝑒|

sin(Le), and 𝐴 = 𝑡𝑎𝑛−1 [

tan|𝑙𝑠−𝑙𝑒|

sin(Le)] (Equ. 21)

Once angle A is determined, the azimuth angle 𝐴𝑧 can be

found in the same manner as in Model 1.

For a satellite to be visible from a satellite ground

control station, 𝛾1and 𝛾2 must satisfy the inequalites

[Timothy et al, 2003]:

1) 0 ≤ 𝛾1 ≤ 81.30 𝑖. 𝑒. (0 ≤ 𝛾1 ≤ 1.4191 𝑟𝑎𝑑) 2) 278.700 ≤ 𝛾2 ≤ 3600 𝑖. 𝑒. (4.8649 ≤ 𝛾2 ≤

6.2840 𝑟𝑎𝑑)

From Figure 1, using < 𝑋𝐷 = 𝜃1 = 900 − 𝛾1and < 𝐷 = 𝜃2 = 900 − 𝛾2, the following are obtained:

a) For 0 ≤ 𝛾1 ≤ 81.30, we have (8.700 ≤ 𝜃1 ≤900) i.e. (0.1591 ≤ 𝜃1 ≤ 1.571 𝑟𝑎𝑑).

b) For 278.700 ≤ 𝛾2 ≤ 3600, we have (−2700 ≤𝜃2 ≤ −188.70) i.e. (−4.7130 ≤ 𝜃2 ≤−3.2939 𝑟𝑎𝑑).

Considering the values obtained in (a) and (b),

additional assumptions to the ones in (1) and (2) of Timothy

et al (2003) for a satellite to be visible from a Satellite

Ground Control Station are that 𝜃1 and 𝜃2 must satisfy the

inequalities:

1) (8.700 ≤ 𝜃1 ≤ 900) i. e. (0.1591 ≤ 𝜃1 ≤1.571 𝑟𝑎𝑑) and

2) (−2700 ≤ 𝜃2 ≤ −188.70) i.e. (−4.7130 ≤𝜃2 ≤ −3.2939 𝑟𝑎𝑑).

Angles 𝛾1 and 𝛾2 are related to the earth station north

latitude Le and west longitude le and the subsatellite point at

north latitude Ls and west longitude ls by [Timothy et al,

2003]

cos(𝛾1) = cos(𝐿𝑒) cos(𝐿𝑠) cos(𝑙𝑠 − 𝑙𝑒) + sin(𝐿𝑒) sin(𝐿𝑠) (Equ. 22) cos(𝛾2) = cos(𝐿𝑒) cos(𝐿𝑠) cos(𝑙𝑠 − 𝑙𝑒) + sin(𝐿𝑒) sin(𝐿𝑠) (Equ. 23)

For most geostationary satellites, the subsatellite point

is on the equator at longitude ls, while latitude Ls is 0.

Equ. (22) and (23) therefore simplify to

cos(𝛾1) = cos(𝐿𝑒) cos(𝑙𝑠 − 𝑙𝑒) (Equ. 24) cos(𝛾2) = cos(𝐿𝑒) cos(𝑙𝑠 − 𝑙𝑒) (Equ. 25)

3. Comparison of Results

The comparison is done using the real parameters of

Abuja Satellite Ground Control Station and Nigerian

Communications Satellite (Nigcomsat-1) and the results

obtained using the mathematical models developed. The

parameters of Abuja Satellite Ground Control Station and

Nigcomsat-1, given by Chai (2005) are as follows: satellite

longitude (sub-satellite point), 𝑙𝑠 = 42.50𝐸; satellite

latitude, 𝐿𝑠 = 00; satellite ground control station longitude,

𝑙𝑒 = 7.38910 and satellite ground control station latitude

𝐿𝑒 = 8.99160𝑁; Azimuth angle, Az = 101.5810; Elevation

angle, El = 48.6210 and Range, R = 37, 168 km.

Substituting the values above into

Eqns. (33), (3), (28), (30) and using radius of the Earth

𝑟𝑒 = 6,378.14 km, and orbital radius 𝑟𝑠 = 42,164.17 km

then,

A. tℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 is

𝛾1 = 𝑐𝑜𝑠−1(cos(𝐿𝑒) cos(𝑙𝑠 − 𝑙𝑒)) = 36.10020

= 0.6301𝑟𝑎𝑑

B. range is

𝑑1 = 𝑟𝑠 [1 + (𝑟𝑒𝑟𝑠)2

− 2(𝑟𝑒𝑟𝑠) 𝑐𝑜𝑠𝛾1]

1

2

= 37,201.0110 𝑘𝑚

C. elevation angle is

𝐸𝑙1 = 𝑠𝑖𝑛−1 [(𝑟𝑠𝑐𝑜𝑠𝛾1−𝑟𝑒)

𝑑1] = 48.10200 = 0.8396 𝑟𝑎𝑑

Ogundele Daniel Ayansola et al.: Mathematical Modeling of Antenna Look Angle of Geostationary Communications Satellite Using Two Models of Control Stations.

International Journal Publishers Group (IJPG) ©

351

D. Azimuth angle is

Az = 𝐴 = 𝑡𝑎𝑛−1 [tan|𝑙𝑠−𝑙𝑒|

sin(Le)] = 77.46760 = 1.3522 𝑟𝑎𝑑

𝐵 = |𝑙𝑠 − 𝑙𝑒| = |42.50 − 7.38910| = 35.11090 𝑖. 𝑒. 𝐵 > 0, Since 𝐿𝑒 > 0 𝑎𝑛𝑑 𝐵 > 0 , then

𝐴𝑧 = 1800 − 77.46760 = 102.53240 = 1.7898𝑟𝑎𝑑 The real values of Abuja Satellite Ground Control

Station and Nigcomsat-1 given by Chai (2005) are

compared with the using Table 1.

TABLE 1 COMPARISON OF THE REAL VALUES AND VALUES OBTAINED

THROUGH MODELLING FONT

S/N Parameters Real

Values

Values from

the model

Percentage

Difference

1 Range (km) 37,168.00 37,201.011 -0.089

2 Elevation

angle

(degree)

48.6210 48.6210

(0.8396 rad)

1.067

3 Azimuth

angle

(degree)

101.5810 102.5320

(1.7898 rad)

-0.936

As seen in the table, the values of range, elevation

angle and azimuth angle obtained through modelling are

very close to the real values given by Chai (2005) indicating

that the results obtained using the mathematical models

developed for the antenna look angles of geostationary

communications satellite are in comformity with the real

values provided by Chai (2005).

The mathematical modelling presented in this paper is

a good tool that can be used to determine look angles for

pointing satellite ground control station antenna to true

geostationary satellites. The real values of Abuja Satellite

Ground Control Station and Nigeria Communication

Satellite (Nigcomsat-1) provided by Chai (2005) were

compared with the values obtained through modelling. The

real values and those obtained through modelling are very

close, indicating that, the modelling can be used to

determine look angles of satellites moving in orbits.

Acknowledgment

The authors acknowledge financial support from the

National Space Research and Development Agency

(NASRDA), Nigeria and the assistance rendered by the

Federal University of Technology Minna, Nigeria in the

course of writing this paper.

References

[1] S. Tomas, & W. David, “Determination of Look Angles to

Geostationary Communication Satellites” (1994), National

Geodetic Survey, Silver Spring, MD 20910, pp. 115-126.

[2] Chai J., Ground Control Station (GCS) System Design,

Beijing Institute of Telemetry, Tracking and Telecommand

(BITTT), Beijing: 2005, pp. 1-48.

[3] Evans B. G., Satellite Communication Systems, The

Institution of Electrical Engineers, London: 1999, pp. 68-

260.

[4] Evans B. G., Satellite Communication Systems, The

Institution of Electrical Engineers, London: 1999, pp. 68-

260.

[5] Timothy P., Charles B., & Jeremy A., Satellite

Communications, John Wiley & Sons, Inc., New York, 2003,

pp. 1-43.

[6] Wayne T., Electronic Communications Systems:

Fundamentals Through Advanced, Fourth Edition: Pearson

Education, Inc., 2001; pp. 790-800.

[7] Ippolito L. J., Satellite Communications Systems

Engineering, ITT Advanced Engineering & Sciences, USA,

and The George Washington University, Washington, DC,

USA, John Wiley and Sons, Ltd, pp. 30-36.

Ogundele Daniel Ayansola received B.TECH in Electronics

and Electrical Engineerig from

Ladoke Akintola University of

Technology, Nigeria in 2000.

Presently, he is undertaking his

Masters Degree of

Communication Engineering at

Federal University of

Technology, Minna, Nigeria and it is near completion. He is one

of the Nigerian Engineers sent to China Academy of Space

Technology, China and Beijing Institute of Tracking and

Telecommunication Technology (BITTT), Beijing, China for the

Know How Technology Transfer on the design, control and

operation of Nigeria Communication Satellite (Nigcomsat-1), and

design, control and installation of Abuja and Kashi Satellite

Ground Control Station. His research interests include spacecraft

dynamics and control, Telemetry, Tracking and Command (TT &

C), design of Satellite Ground Control Station and orbital

mechanics and astrodynamics.

Yinusa A. Adediran attended

Budapest Technical University,

Hungary where he obtained an

M.Sc degree in

Telecommunications in 1980. He

also obtained an M.Sc degree in

Industrial Engineering from the

University of Ibadan in 1987 and

Ph.D in Industrial and Production

Engineering in 1999 from Federal

University of Technology, Minna.

He has contributed to knowledge

through various articles in journals and proceedings. He is

currently an Associate Professor of Electrical and Computer

Engineering at Federal University of Technology, Minna.