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Link Budget Equations

Complete derivation of communications link budget calculations.

Overview

A link budget accounts for all gains and losses in a communications link, from transmitter to receiver.

The fundamental equation:

\[ P_{rx} = P_{tx} + G_{tx} - L_{tx} - L_{path} + G_{rx} - L_{rx} \]

All quantities in dB.

EIRP (Effective Isotropic Radiated Power)

EIRP is the power that would be radiated by an isotropic antenna to produce the same field intensity:

\[ EIRP = P_{tx} + G_{tx} - L_{tx} \]

Where: - \(P_{tx}\) = Transmitter power output (dBW) - \(G_{tx}\) = Transmit antenna gain (dBi) - \(L_{tx}\) = Transmit losses (feed, radome, etc.) (dB)

For Phased Arrays

\[ P_{tx} = 10\log_{10}(N \cdot P_{elem}) \]

Where \(N\) is element count and \(P_{elem}\) is power per element in watts.

Path Loss

Free Space Path Loss (FSPL)

\[ L_{FSPL} = 20\log_{10}\left(\frac{4\pi d f}{c}\right) \]

Or equivalently:

\[ L_{FSPL} = 32.45 + 20\log_{10}(f_{MHz}) + 20\log_{10}(d_{km}) \]

Or:

\[ L_{FSPL} = 92.45 + 20\log_{10}(f_{GHz}) + 20\log_{10}(d_{km}) \]

Example: 10 GHz, 100 km

\[ L_{FSPL} = 92.45 + 20\log_{10}(10) + 20\log_{10}(100) = 92.45 + 20 + 40 = 152.45 \text{ dB} \]

Additional Losses

Total path loss:

\[ L_{path} = L_{FSPL} + L_{atm} + L_{rain} + L_{pol} + L_{misc} \]
Loss Type Typical Range
Atmospheric 0.1-2 dB (depends on f, elevation)
Rain fade 0-20 dB (depends on f, availability)
Polarization 0-3 dB (mismatch)

Received Power

\[ P_{rx} = EIRP - L_{path} + G_{rx} \]

Noise Power

Thermal Noise

\[ N = kTB \]

Where: - \(k\) = Boltzmann constant = 1.38×10⁻²³ J/K - \(T\) = System noise temperature (K) - \(B\) = Bandwidth (Hz)

In dB:

\[ N_{dBW} = 10\log_{10}(kTB) = -228.6 + 10\log_{10}(T) + 10\log_{10}(B) \]

System Noise Temperature

\[ T_{sys} = T_{ant} + T_{rx} \]

Where:

\[ T_{rx} = T_0(F - 1) \]
  • \(T_0\) = Reference temperature (290 K)
  • \(F\) = Noise figure (linear)

Noise Figure

\[ F_{dB} = 10\log_{10}(F) = 10\log_{10}\left(1 + \frac{T_{rx}}{T_0}\right) \]

Total noise power including noise figure:

\[ N_{dBW} = 10\log_{10}(kT_0B) + NF_{dB} \]

Signal-to-Noise Ratio

\[ SNR = P_{rx} - N \]

Expanding:

\[ SNR = EIRP - L_{path} + G_{rx} - 10\log_{10}(kT_0B) - NF \]
\[ M = SNR - SNR_{required} \]

Positive margin indicates the link closes with room to spare.

G/T Figure of Merit

For receive systems, G/T characterizes sensitivity:

\[ \frac{G}{T} = G_{rx} - 10\log_{10}(T_{sys}) \]

Units: dB/K

Given: - Frequency: 10 GHz - Range: 100 km - TX array: 8×8, 1 W/element, 65% efficiency - TX losses: 1.5 dB - RX antenna gain: 30 dBi - RX noise figure: 3 dB - RX noise temp: 290 K - Bandwidth: 10 MHz - Required SNR: 10 dB - Atmospheric loss: 0.5 dB

Calculation:

  1. TX Power: $$ P_{tx} = 10\log_{10}(64 \times 1) = 18.1 \text{ dBW} $$

  2. TX Gain: $$ G_{tx} = 10\log_{10}(0.65 \times \pi \times 64) = 21.2 \text{ dB} $$

  3. EIRP: $$ EIRP = 18.1 + 21.2 - 1.5 = 37.8 \text{ dBW} $$

  4. Path Loss: $$ L_{path} = 152.4 + 0.5 = 152.9 \text{ dB} $$

  5. Received Power: $$ P_{rx} = 37.8 - 152.9 + 30 = -85.1 \text{ dBW} $$

  6. Noise Power: $$ N = 10\log_{10}(1.38 \times 10^{-23} \times 290 \times 10^7) + 3 = -131.0 \text{ dBW} $$

  7. SNR: $$ SNR = -85.1 - (-131.0) = 45.9 \text{ dB} $$

  8. Link Margin: $$ M = 45.9 - 10 = 35.9 \text{ dB} $$

Trade-offs

EIRP Improvements

Change EIRP Gain
Double TX power +3 dB
Double elements (2×) +6 dB (gain + power)
Double aperture +6 dB

Reducing Path Loss

  • Lower frequency (but larger antenna for same gain)
  • Shorter range (\(L \propto d^2\))

Improving Receiver

Change Effect
Higher G/T Better sensitivity
Lower NF +1 dB NF reduction = +1 dB SNR
Narrower BW Lower noise (but also signal BW)

See Also