135 lines
4.6 KiB
Markdown
135 lines
4.6 KiB
Markdown
---
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title: Link budget
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tags: link-budget, eirp, g-t, fspl, margin, rain
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summary: The arithmetic of whether a signal arrives strong enough to be used, and every term that turns a comfortable design into an impossible one.
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---
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# Link budget
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A link budget is the addition of every gain and loss between two antennas,
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compared with the power needed to be heard. It is arithmetic, but it is
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arithmetic on quantities that are all uncertain, and the margin left over is
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the only thing standing between a design and an outage.
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## Definition
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The received power at the far end is
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$$
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P_r = \frac{\text{EIRP} \, G_t}{L}
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$$
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with `L` the total propagation and implementation loss. The link closes when
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`P_r` exceeds the sensitivity of the receiver, and the **link margin** is the
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difference in decibels.
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Every term is a ratio, so the budget is written in decibels throughout:
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$$
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\begin{aligned}
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\text{EIRP} &= P_t + G_t - L_{\text{feeder}} \\
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P_r &= \text{EIRP} + G_r - L_{\text{space}} - L_{\text{pol}} - L_{\text{point}} \\
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\text{margin} &= P_r - \text{sensitivity}
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\end{aligned}
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$$
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The two that decide the result are free space loss, which is fixed by
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geometry, and the losses that are not: rain, pointing, polarisation
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mismatch.
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## Mathematics
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### Free space loss
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The loss grows with the fourth power of distance and with the square of
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frequency:
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$$
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L_{\text{fs}} = \left( \frac{4 \pi d f}{c} \right)^2
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$$
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in linear units, or in decibels,
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$$
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L_{\text{fs,dB}} = 92.45 + 20 \log_{10} d_{\text{km}} + 20 \log_{10} f_{\text{GHz}}
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$$
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The fourth power is why doubling the frequency costs six decibels and
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doubling the distance costs six. A constellation at 550 km versus one at
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1100 km has twelve decibels less margin for the same equipment, which is more
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than a doubling of the transmit power recovers, because doubling power is only
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three decibels.
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### Polarisation loss
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An antenna receives a polarisation close to its own. A mismatch of an angle
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`psi` costs
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$$
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L_{\text{pol}} = -10 \log_{10} \left( \cos^2 \psi \right) \ \text{dB}
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$$
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which is small below a few degrees and large beyond thirty. A tilted
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heliostat or a rotating spacecraft body can put a link over this threshold
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during a pass, and this is a common cause of an outage that appears only in
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part of the orbit.
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### Rain attenuation
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Attenuation from rain at frequencies above 10 GHz is significant. It scales
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roughly with `f^0.75` up to about 30 GHz and is then given by the specific
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attenuation `gamma` times the path length through the rain cell. The design
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figure is not the average attenuation but the value exceeded for a stated
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fraction of the worst hour in a year, because the budget is closed against
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the rain, not against the median.
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### Pointing and pointing loss
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Pointing loss is the amount of beam that misses. It is small when the ground
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station tracks a narrow beam well and large when a wide beam is used to reach
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a mobile user, where the beam is pointed at the user rather than at a fixed
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station.
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### Margin
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Margin is what is left after every term. What it is spent on depends on what
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was left out: a modest margin covers pointing jitter and small fades; it does
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not cover a rain cell that was sized from an average.
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## Unit conventions
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- Power in decibels relative to one milliwatt, abbreviated dBm.
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- Gains and losses in decibels.
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- Distances in kilometres, frequencies in gigahertz, which is the form the
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free space formula is usually written in.
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- EIRP in dBW when the figure is compared with a spacecraft power, dBm when
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with a receiver.
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## Applicability
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A budget computed with average losses over a terrain and an average rain rate
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does not close a link. The two terms that have to be exceeded rather than
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averaged are rain and pointing, and they are the two that cause outages in
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systems whose budget was signed off.
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Above roughly 30 GHz the rain attenuation keeps rising and the fade margin
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required becomes large enough to dominate the hardware. This is the reason
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the geostationary bands fill the sky below about 10 GHz and the higher bands
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are used where rain is rare or where the link can afford a large antenna.
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A budget does not account for interference, which is not a loss but a
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corruption; it needs [[interference]] treated separately.
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## Standards
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- Free space loss and the link budget convention: ITU-R recommendations
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on radiowave propagation and on the satellite broadcasting and
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telecommunications services
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- Rain attenuation: ITU-R P.838 and P.618
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- Regional rain rate data: ITU-R P.837
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## See also
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- [[doppler-and-rate]] for what is measured once the link closes
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- [[atmospheric-propagation]] for the medium the budget travels through
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- [[constellations]] for how the budget sets the orbital shell |