Pr
Pt
= Gt Gr
λ2
(4πR)2
Study notes for undergraduate electrical engineering students on the fundamental equation for wireless communication link budget calculation
The Friis Transmission Equation (also known as the Friis transmission formula) is a fundamental equation in antenna theory and wireless communications. It is used to calculate the power received by one antenna from another antenna under ideal conditions.
Developed by Harald T. Friis in 1946, this equation is essential for designing wireless communication systems, calculating link budgets, and understanding the factors that affect signal strength in free space.
Key Application: The equation is primarily used to determine if a radio communication link is feasible given specific transmitter power, antenna gains, distance, and frequency.
The Friis transmission equation in its basic form is:
Where:
The ratio of received power to transmitted power (also known as the path loss in linear terms).
The power available at the receiving antenna terminals.
The power supplied to the transmitting antenna.
The gain of the transmitting antenna relative to an isotropic radiator (dimensionless ratio, often expressed in dBi).
The gain of the receiving antenna relative to an isotropic radiator.
The wavelength of the transmitted signal. λ = c/f, where c is the speed of light (3×108 m/s) and f is the frequency.
The separation between the transmitting and receiving antennas.
Note: The equation assumes the antennas are in far-field of each other and are polarization-matched.
Adjust the parameters below to see how they affect the received power:
Received power decreases with the square of the distance (1/R² factor in the equation).
Higher frequency signals have shorter wavelengths, which reduces received power for the same antenna gains.
The Friis transmission equation makes several important assumptions:
In real-world applications, additional factors must be considered: atmospheric absorption, rain attenuation, obstructions, reflections, and system losses. These are often accounted for by adding margin to the link budget calculation.
Calculate the received power using the Friis transmission equation:
Problem: A wireless communication link operates at 5.8 GHz. The transmitter has an output power of 20 W and uses an antenna with a gain of 15 dBi. The receiver antenna has a gain of 12 dBi. If the distance between antennas is 2 km, calculate the received power.
Step 1: Convert gains from dBi to linear scale
Gt (linear) = 1015/10 = 101.5 ≈ 31.62
Gr (linear) = 1012/10 = 101.2 ≈ 15.85
Step 2: Calculate wavelength
λ = c/f = (3×108 m/s) / (5.8×109 Hz) ≈ 0.0517 m
Step 3: Apply Friis equation
Pr = Pt Gt Gr (λ/(4πR))2
Pr = 20 × 31.62 × 15.85 × (0.0517/(4π×2000))2
Pr ≈ 20 × 31.62 × 15.85 × (0.0517/25132.7)2
Pr ≈ 20 × 31.62 × 15.85 × (2.057×10-6)2
Pr ≈ 20 × 31.62 × 15.85 × 4.23×10-12
Pr ≈ 4.24×10-8 W = 42.4 nW
Step 4: Convert to dBm for practical interpretation
Pr (dBm) = 10×log10(42.4×10-9 / 0.001) ≈ -43.7 dBm
Interpretation: The received power is approximately -43.7 dBm, which is a typical value for many wireless communication systems. This power level would need to be compared with the receiver sensitivity to determine if the link is feasible.