Earth stations

Station data are available from mancufacturers, if not ask them. You should get a databook and a measurement protocol for a professional station. Aperture and pointing accuracies are important parameters of a station. A big dish is of no use if it's not properly fixed or moved with inaccurate motors. Do not trust all consumer stuff data, like LNBs from the supermarket with a noise figure of 0.2dB, that's often not true.

Earth station related data

 A m2 Antenna physical aperture area c m/s speed of light = 2.997925 × 108 d m Parabolic antenna diameter f Hz frequency F[dB] dB Noise figure gx dBi Antenna gain (relative to gain of an isotropic radiator that is linearly polarized) λ m wavelength = c / f ηx 1 Efficiency, a value between 0 and 1, or in % Pr W Reflected power due to mismatched load Pd W Delivered power that reaches the load Pf W Forward power from the generator, also named incident power Tsys K System noise temperature, see propagation theory site Tsys,nom K Nominal system noise temperature used by component manufacturers

Parabolic dish types of construction

Antenna gain depends on dish size, dish profile accuracy, feed illumination and blockage from feed assembly. Whether the dish is circular or elliptical does not matter too much, assuming the feed is designed to match the dish shape by appropriately distributing the power across the dish surface. Typically the power density in the middle of the dish will be about 14dB higher than at the edges. 11dB edge taper gives a better gain, 16dB reduces the sidelobes.

In the uplink direction, the power spectral density you transmit into adjacent satellites must be limited by reducing sidelobe emissions. This may be achieved by reducing the power or under-illuminating the dish. Sidelobes are improved if there is little scattering from obstructions like the feed assembly or the dish edges. Also, having extended corners across a particular direction will help, so an elliptical or diamond shaped dish will have better sidelobe discrimination along its widest axis. If this axis is orientated along the orbit, interference to and from the adjacent satellites will be minimised (typically -28dB).

Poor cross-polarisation isolation causes interference to and from other services. It's very much a function of feed design and position. An offset feed has a bad cross-pol isolation and that is tried to compensate with a rather long and mechanically awkward focal length (f/d=0.8), or using a gregorian sub-reflector, or a special, expensive, mode-matched feed. Also adjusting the polarisation angle very carefully is essential. A cross-pol interference of -29dB is typical, perhaps -20dB in the worst case for the cheapest of receive only dishes.

Antenna half power beamwidth HPBW (3-dB beamwidth)

In a plane containing the direction of the maximum of a beam, the angle between the two directions in which the radiation intensity is one half the maximum value of the beam. HPBW depends on how the feed horn illuminates the dish. The cosine aperture illumination distribution is close to the average if the illumination method adopted is not known.
φ3dB[deg] = x / (f[GHz] × d[m])

 aperture illumination distribution: x: uniform 17.508 cosine 21.825 cosine2 25.243 pedestal 19.936

Pointing attenuation

In practice the pointing accuracy is normally kept within one third of the half power beamwidth. The bigger the parabolic dish, the smaller is it's beamwidth and the better it has to be pointed. But big dishes are more difficult to adjust and have a higher wind load. Dishes with diameters of about 5m and more are supplied with automatic tracking motors to correct pointing errors with the help of a tracking beacon.

φ1 pointing accuracy dish to the satellite (≈ 10-30% of the half power beamwidth).
φ2 pointing stability due to wind and ageing. (≈ 0.2-0.5 degrees)
φ3 station keeping accuracy of the satellite (≈ ±0.16 degrees)

Pointing attenuation in dB: apoint ≈ 12 × (φ12 + φ22 + φ32) / (φ3dB)2

Efficiency of an aperture-type antenna

Aperture A: A surface near or on an antenna, on which it is convenient to make assumptions regarding the field values for the purpose of computing fields at external points. The aperture is often taken as that portion of a plane surface near the antenna, perpendicular to the direction of maximum radiation, through which the major part of the radiation passes.

Following efficiencies are defined very different in literature, but these definitions are the most logical ones:

Antenna efficiency ηa: is concerned with the effectiveness of an antenna in directing or collecting radiated power. It's the antenna system overall efficiency. It might be reduced to optimize other characteristics like low sidelobe level.
The effective area of a receiving antenna is the ratio of the delivered power at the antenna terminal to the power density over the whole aperture area. The antenna efficiency for a specified planar aperture is the effective area to the physical aperture area.

${\mathrm{\eta }}_{a}=\frac{\text{effective area}\phantom{\rule{0.5em}{0ex}}{A}_{e}}{\text{physical aperture area}\phantom{\rule{0.5em}{0ex}}A}\phantom{\rule{2em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}{A}_{e}=\frac{\text{terminal power}\phantom{\rule{0.5em}{0ex}}\left[W\right]}{\text{overall power density}\phantom{\rule{0.5em}{0ex}}\left[W/{m}^{2}\right]}$

Aperture efficiency (illumination factor) ηap = ηil × ηsp × ηbl × ηre
Includes all efficiencies that affect aperture and radiation pattern, co and cross-pol components are included:

ηil = Illumination efficiency describes the variation in the electric field produced by a feed across an antenna surface. An uniformly illuminated dish with equal radiation intensity and without energy spilled past the edges is not achievable in practice, but useful as a reference. A non-uniformly illuminated dish produces amplitude (taper) and phase errors which means loss of directivity.

${\mathrm{\eta }}_{\mathit{il}}=\frac{\text{directivity}}{\text{directivity of a uniformly illuminated antenna of the same aperture size}}$

Surface tolerance efficiency is included in taper and phase efficiency. In cassegrain antennas the shape of the subreflector is distorted to achieve uniform illumination across the main reflector. This results in a phase error that causes energy being radiated in undesired directions. That means an increase of the sidelobe level.

ηsp = Spillover efficiency represents the energy spilled over the edge of the main reflector (and subreflector of a cassegrain dish).

ηbl = Blocking efficiency. The feed or the subreflector and support structure like fixing material shades off the main reflector which results in a smaller aperture.

ηre = Reflector transparency. Energy may be radiated through the reflector, e.g. in mesh antennas.

Radiation efficiency ηrad: includes internal antenna losses like imperfect conductors, dielectrics and feed system dissipative energy which represents losses of waveguide components like feed horns, polarizers, orthomodal transducers...

$\phantom{\rule{0.5em}{0ex}}{\mathrm{\eta }}_{\mathit{rad}}=\frac{\text{total power radiated by antenna}}{\text{net power which antenna accepts from transmitter}}$

Polarisation efficiency ηpol: The ratio of the power received by an antenna from a given plane wave of arbitrary polarisation to the power that would be received by the same antenna from a plane wave of the same power flux density and direction of propagation, whose state of polarisation has been adjusted for a maximum received power.

Mismatch efficiency ηvswr: derived from reflection at the feed port and lossy waveguides due to impedance mismatch before the antenna. It can only be avoided under test conditions at a single frequency, not over a wide frequency band. Mismatch is therefor not part of antenna efficiency.

$\text{Complex reflection coefficient}\phantom{\rule{0.5em}{0ex}}\mathrm{\Gamma }=\frac{\text{reflected wave}\phantom{\rule{0.5em}{0ex}}{v}_{r}}{\text{forward wave}\phantom{\rule{0.5em}{0ex}}{v}_{f}}=\frac{{Z}_{\mathit{load}}-{Z}_{0}}{{Z}_{\mathit{load}}+{Z}_{0}}$

Γ describes both the magnitude (|Γ|) and the phase shift of the reflection:
|Γ| = -1: maximum negative reflection on a short-circuited line
|Γ| = 0: no reflection on a perfectly matched line
|Γ| = +1: maximum positive reflection on an open line

Voltage standing wave ratio (VSWR): e.g. VSWR 1.3:1 denotes a maximum standing wave magnitude that is 1.3 times greater than the minimum standing wave value. VSWR should not exceed 1.3:1, that is a return loss of 18dB and a mismatch efficiency of 0.983 (mismatch loss 0.075dB).
VSWR = (1 + |Γ|) / (1 - |Γ|) a value between 1 and infinity

$\text{Return loss}=10×\mathrm{log}\left(\frac{\text{forward power}\phantom{\rule{0.5em}{0ex}}{P}_{f}}{\text{reflected power}\phantom{\rule{0.5em}{0ex}}{P}_{r}}\right)=-20×\mathrm{log}|\mathrm{\Gamma }|=-20×\mathrm{log}\left(\frac{\mathit{VSWR}-1}{\mathit{VSWR}+1}\right)$

$\text{Missmatch efficiency}\phantom{\rule{0.5em}{0ex}}{\mathrm{\eta }}_{\mathit{VSWR}}=\frac{\text{delivered power}\phantom{\rule{0.5em}{0ex}}{P}_{d}}{\text{forward power}\phantom{\rule{0.5em}{0ex}}{P}_{f}}=1-{|\mathrm{\Gamma }|}^{2}=1-{\left(\frac{\mathit{VSWR}-1}{\mathit{VSWR}+1}\right)}^{2}$

Missmatch loss = -10 × log ηvswr (minus sign because a loss is positive)

Power Standing Wave Ratio PSWR = VSWR2

Cable attenuation between LNB and IRD is neglected. As a rule of thumb, this attenuation should be 30dB lower than the gain of LNB, otherwise there is significant reduction of G/T.

Antenna gains

Directive gain gd: is independent of actual power output and the distance of measurement. An isotropic radiator has an uniform spherical pattern with a directive gain of 0dB by definition. Radiation intensity in a given direction is the power radiated from an antenna per unit solid angle.

${g}_{d}=10×\mathrm{log}\left(\frac{4×\mathrm{\pi }×\left(\text{radiation intensity in a given direction}\right)}{\text{total power radiated by antenna}}\right)$

Directive gain is proportional to the aperture area and to the square of frequency! It neglects any losses, compared to the following gains.
Directivity = directive gain in the direction of maximum radiation.

Parabolic antenna directivity: gd = 10 × log(4 × Pi × A / λ2) = 10 × log(4 × Pi × A × (f / c)2)

Power gain gp: In a physical media the ratio of the power flux per unit area from an antenna to the power flux per unit area from an isotropic radiator with the same power input. When the direction is not stated, usually the power gain in direction of maximum.

${g}_{p}=10×\mathrm{log}\left(\frac{4×\mathrm{\pi }×\left(\text{radiation intensity in a given direction}\right)}{\text{net power which antenna accepts from transmitter}}\right)$

gp = gd + 10 × log(ηa) = gd + 10 × log(ηap) + 10 × log(ηrad) + 10 × log(ηpol)

Realized gain gr: Power gain reduced by the loss due to mismatch of antenna input impedance to a specified impedance.
gr = gp + 10 × log(ηvswr)

Emitted Isotropic Radiated Power EIRP[dBW] = gr[dB] + P[dBW]

Figure of merit G/T

Nominal G/T describes the system quality in receiving direction under "nominal" conditions θ0 and Tsys,nom. This value can be found in manufacturers' datasheets. The antenna efficiency of the receiving path ηa,rx might be slightly different to the transmitting path efficiency due to other waveguide components, therefore we use index rx. For calculation of T see the propagation site.
(G/T)nom = 10 × log(Gd × ηa,rx / Tsys,nom) = gp,rx - 10 × log(Tsys,nom)

The usable G/T is declined by environmental influences on T and pointing attenuation:
(G/T)sys = gp,rx - 10 × log(Tsys) - apoint

Literature

1. Digital Satellite Communications, Tri T. Ha, McGraw Hill
2. Antenna Introduction / Basics
3. Gain of Directional Antennas, John E. Hill, WJ Communications
4. Efficiency and sensitivity definitions for reflector antennas in radio-astronomy, Wim van Cappellen