Parabolic Antenna Diameter

[Andy Bacon]

**These are my calculations, based upon the latest specification and
previous calculations given to me by Steve Maughan.**

The first method of calculating the needed diameter is from the
effective aperture required to produce the gain stated in the
specification.

Ae = (λ² * G) / (4*pi)

Where:
Ae = effecive aperture area
λ = Wavelength = 70cm
G = linear Gain

G = 20dB = 10 log 100
=> G (linear) = 100

Ae = (0.7² * 100) / 4*pi = 3.899m²

However due to inefficiencies in parabolic antenna design, the
effective area can realistically range from 30% to 70% of the physical
area of the disk (A):

Largest Estimate:
Ae = 0.3*A
A = 12.997m²
12.997 = pi*r²
r = 2.034m
=> D = Antenna Diameter = 2*r = 4.068m

Smallest Estimate:
Ae = 0.7*A
A = 5.556m²
r = 1.33m
=> D = 2.66m

There is an alternative which is mostly taken from the book
"Spacecraft Systems Engineering" (ISBN 0-471-61951-5). The book claims
to have an approximation (p 479) for the diameter of parabolic
antennas used as ground stations. Firstly though the slant range needs
to be calculated (p 117):

S = (Re + h) sin Ø / cos e

S = Slant range (Km)
Re = Radius of the Earth = 6,371Km (Mean value)
h = Satellite altitude = 700Km (Estimate, real orbit as yet unknown)
Ø = Geocentric semi-angle (degrees)
e = Minimum visible satellite elevation due to horizon problems = 0 -
5º

Ø = -e + cosˉ¹ ((Re / Re + h) cos e)

When e = 0º
Ø = 6371 / 7071 = 25.71º
S = (7071 sin 25.71) / cos 0
S = 3068

When e = 5º
Ø = -5 + cosˉ¹ (25.71 cos 5)
Ø = 21.16º
S = (7071 sin 21.16) / cos 5
S = 2562

The formula for the antenna diameter is given as follows:

D = (6x10³ * d / fc) * (sqrt(b / tp))

D = Antenna Diameter (m)
d = max satellite distance = slant range
fc = carrier frequency = 440MHz
b = bit rate = 9600 bits/s
tp = satellite transmission power = 0.3 - 0.5 W

Largest size estimate (e = 0 and tp = 0.3)

D = (6x10³ * 3068 / 440e6) * (sqrt(9600 / 0.3))
D = 0.0418 * 178.89
D = 7.477m

Smallest size estimate (e = 5 and tp = 0.5)

D = (6x10³ * 2562 / 440e6) * (sqrt(9600 / 0.5))
D = 0.0349 * 138.56
D = 4.836m

These equations give us antenna sizes from 2.66m to 7.477m and
everything in between (apart from a gap between 4m and 4.8m) so what
is needed is refinement of the effective aperture to make it more
realistic. What would also be helpful is to work out the derivation of
the second formula (from the book), if both of the equations can be
made to agree - then we should be in business.

Also, please check my working!

[Andy Bacon]

Update for this week

I realised that I forgot to point out a critical factor for the
formula stated in the above post:

D = (6x10³ * d / fc) * (sqrt(b / tp))

It is unknown what the value of bitrate (b) actually represents, it
could be:
1) The average desired bitrate
2) The maximum bitrate (when the satellite is directly overhead -
lowest noise)
3) The minimum bitrate (when the satellite is near the horizon -
greatest noise)

Depending on which of these is the correct, the results change wildly.
If it were an average then the results would be similar to those above
(as the decrease in noise when overhead should balance with the
increase near the horizon). If it is the maximum bitrate, then the
antenna would have to be even bigger, whereas if it were a minimum the
antenna diameter could be reduced.
Because of these uncertainties, I think it may be best to ignore the
results derived from this formula until all the variables can be
better established. It would be helpful if someone could find the
formula described elsewhere, preferably with a better description.

Steve had this to say:
We can always trade off antenna size for additional gain in the low-
noise amplifier stages. Of course, this is limited by the noise
temperature of the low-noise amplifier stage, so perhaps if we looked
to see what LNA gains/noise-temperatures are possible (ie. cutting
edge of research), and then look at what is generally commercially
available, then we might find that we have plenty of noise-headroom to
get another 8x or 16x gain out of the LNA's, thus decreasing the dish
size by the same amount - and suddenly you only need a 0.75m diameter
dish. And if that's the case, the antenna becomes pretty insignificant
and I guess it would be a lot easier for your department to justify it
to all the other people who might object (central university buildings
people/safety people/finance people etc...).