4.6 Friis transmission equation

Friis transmission equation

The Friis transmission equation describes how much power a receiving antenna captures from a transmitting antenna across a free-space link. It ties together transmitted power, antenna gains, wavelength, and separation distance into a single expression, making it the starting point for any link budget analysis in wireless system design.

Harald T. Friis, a Danish-American electrical engineer, published the equation in 1946. It applies to line-of-sight (LOS) radio links under idealized conditions and forms the backbone of design calculations for cellular networks, Wi-Fi, satellite links, radar, and radio astronomy.

Core equation and derivation

Linear (power) form

The Friis equation in its standard form is:

$P_r = P_t \, G_t \, G_r \left(\frac{\lambda}{4\pi d}\right)^2$

where:

• $P_t$ = transmitted power (W)
• $P_r$ = received power (W)
• $G_t$ = transmitting antenna gain (dimensionless ratio, not dB)
• $G_r$ = receiving antenna gain (dimensionless ratio)
• $\lambda$ = wavelength of the signal (m)
• $d$ = distance between antennas (m)

The factor $\left(\frac{\lambda}{4\pi d}\right)^2$ is the free-space path loss (FSPL) expressed as a power ratio. It captures the inverse-square-law spreading of power density over a sphere of radius $d$.

Where the equation comes from

The derivation follows three steps:

1. Power density at distance $d$. A transmitter radiating power $P_t$ through an antenna of gain $G_t$ produces a power density at distance $d$ of $S = \frac{P_t \, G_t}{4\pi d^2}$ This is just the total radiated power spread over the surface area of a sphere, scaled by the antenna's directional gain.

2. Effective aperture of the receiving antenna. Any antenna with gain $G_r$ has an effective aperture (capture area) given by $A_e = \frac{G_r \, \lambda^2}{4\pi}$ This relationship comes from antenna reciprocity and the definition of gain relative to an isotropic radiator.

3. Received power. Multiply the incident power density by the effective aperture: $P_r = S \cdot A_e = \frac{P_t \, G_t}{4\pi d^2} \cdot \frac{G_r \, \lambda^2}{4\pi} = P_t \, G_t \, G_r \left(\frac{\lambda}{4\pi d}\right)^2$

Key assumptions

• Free-space propagation with no obstacles, reflections, or multipath.
• Matched polarization between transmit and receive antennas.
• Matched impedance at both antenna terminals.
• Far-field separation: the antennas are many wavelengths apart so that the radiated field has settled into its $1/r$ dependence.
• No atmospheric absorption or scattering.

Decibel form

Taking $10\log_{10}$ of both sides converts every multiplication into addition:

$P_r\,(\text{dBm}) = P_t\,(\text{dBm}) + G_t\,(\text{dBi}) + G_r\,(\text{dBi}) - L_{\text{FSPL}}\,(\text{dB})$

where the free-space path loss in dB is:

$L_{\text{FSPL}}\,(\text{dB}) = 20\log_{10}\!\left(\frac{4\pi d}{\lambda}\right)$

This can also be written in terms of frequency $f$:

$L_{\text{FSPL}}\,(\text{dB}) = 20\log_{10}(d) + 20\log_{10}(f) + 20\log_{10}\!\left(\frac{4\pi}{c}\right)$

For $d$ in km and $f$ in MHz, the constant evaluates to about 32.45 dB, giving the handy form:

$L_{\text{FSPL}}\,(\text{dB}) \approx 32.45 + 20\log_{10}(d_{\text{km}}) + 20\log_{10}(f_{\text{MHz}})$

Why the dB form matters

• Gains and losses stack by simple addition and subtraction, which makes link budget spreadsheets straightforward.
• Component specs (amplifiers, cables, connectors) are almost always quoted in dB, so you can plug them straight in.
• Quick mental estimates become possible once you memorize a few reference values.

Key parameters in detail

Antenna gain and effective aperture

Gain quantifies how well an antenna focuses energy in its peak direction compared to an isotropic radiator. It is measured in dBi (decibels relative to isotropic). A half-wave dipole, for example, has a gain of about 2.15 dBi.

Effective aperture $A_e$ is the equivalent "capture area" of the receive antenna. The two are linked by:

$A_e = \frac{G \, \lambda^2}{4\pi}$

A larger aperture (physically bigger dish, or a higher-gain array) collects more of the incoming wave's power. At shorter wavelengths (higher frequencies), the same physical aperture yields a higher gain because $G = 4\pi A_e / \lambda^2$ grows as $\lambda$ shrinks.

Effective isotropic radiated power (EIRP)

$\text{EIRP} = P_t \, G_t$

EIRP is the power an isotropic antenna would have to radiate to produce the same peak power density as the actual transmitter-antenna combination. Regulatory bodies often specify maximum EIRP rather than maximum transmit power, because EIRP captures the combined effect of power and directivity.

Free space path loss (FSPL)

FSPL is not caused by absorption; it is purely geometric spreading. Two dependencies stand out:

• Distance: FSPL grows as $d^2$. Double the distance and you lose 6 dB of received power.
• Frequency: FSPL grows as $f^2$. Double the frequency and you lose another 6 dB. This does not mean the atmosphere absorbs more at higher frequencies. It means the effective aperture of a fixed-gain antenna shrinks with wavelength. If you instead keep the physical aperture constant (like a dish), the receive gain rises with frequency and compensates for this term.

That frequency dependence trips up a lot of students. The "extra loss" at higher frequency is really a consequence of the receive antenna becoming electrically smaller relative to the wavefront, not of the wave itself losing energy.

Worked example

Step 1 — Compute FSPL.

$\lambda = \frac{c}{f} = \frac{3 \times 10^8}{2.4 \times 10^9} = 0.125\;\text{m}$

$L_{\text{FSPL}} = 20\log_{10}\!\left(\frac{4\pi \times 100}{0.125}\right) = 20\log_{10}(10{,}053) \approx 80.0\;\text{dB}$

Step 2 — Apply the dB-form Friis equation.

$P_r = 30 + 6 + 3 - 80.0 = -41.0\;\text{dBm}$

At $-41$ dBm the link has plenty of margin; typical Wi-Fi receivers need roughly $-70$ to $-80$ dBm for a good connection.

Noise, SNR, and link budgets

Signal-to-noise ratio (SNR)

The Friis equation tells you $P_r$, but whether communication succeeds depends on how $P_r$ compares to the noise floor. SNR is defined as:

$\text{SNR (dB)} = P_r\,(\text{dBm}) - N\,(\text{dBm})$

where $N$ is the total noise power at the receiver input. Higher SNR means a cleaner, more reliable link.

Noise figure and noise temperature

• Noise figure $F$ (in dB) describes how much a receiver component degrades the SNR. An ideal noiseless component has $F = 0$ dB.
• Noise temperature $T_e$ (in Kelvin) is an equivalent way to express the same thing: $T_e = T_0(F_{\text{linear}} - 1)$, where $T_0 = 290$ K.

These quantities feed into the system noise power $N = k_B \, T_{\text{sys}} \, B$, where $k_B$ is Boltzmann's constant and $B$ is the receiver bandwidth.

Link budget

A full link budget extends the Friis equation by adding every real gain and loss in the chain:

$P_r = P_t + G_t - L_{\text{cable,tx}} - L_{\text{FSPL}} - L_{\text{atm}} - L_{\text{misc}} + G_r - L_{\text{cable,rx}}$

You then compare $P_r$ to the receiver sensitivity (minimum detectable power) to find the link margin. A positive margin means the link closes; a negative margin means it doesn't.

Extensions to real-world conditions

The basic Friis equation is a best-case starting point. Real links introduce additional loss factors.

Polarization mismatch

If the transmit and receive antennas have different polarizations, a polarization loss factor $p$ (between 0 and 1) multiplies the received power. For example, a vertically polarized transmitter and a horizontally polarized receiver give $p = 0$ (complete mismatch). A factor of $\cos^2\theta$ applies when the polarization axes are offset by angle $\theta$.

Atmospheric absorption and scattering

• Oxygen and water vapor absorb energy at specific bands (notably around 22 GHz and 60 GHz).
• Rain, snow, and fog scatter millimeter-wave signals, adding several dB/km of extra attenuation depending on precipitation rate.
• These losses are added as extra terms in the link budget.

Reflection and multipath

When signals bounce off buildings, terrain, or other surfaces, multiple copies arrive at the receiver with different delays and phases. This causes constructive or destructive interference (fading), which the basic Friis equation does not predict. Statistical fading models (Rayleigh, Rician) are used alongside Friis to account for these effects.

Applications

Wireless communication systems

The Friis equation is used to size cells in cellular networks (determining how far apart base stations can be for a given transmit power and frequency band), plan indoor Wi-Fi coverage, and set power levels for short-range protocols like Bluetooth Low Energy and Zigbee.

Satellite communication

Satellite links involve very large distances (hundreds to tens of thousands of km), so FSPL values are enormous. High-gain dish antennas on both ends compensate. The Friis equation, combined with noise temperature analysis, determines the required dish size, transmit power, and data rate for systems like GPS, Iridium, and Starlink.

Radar

In radar the signal travels to the target and back, so the one-way Friis equation is applied twice (with the target's radar cross-section replacing the receive antenna gain on the outbound leg). This leads to the radar range equation, where received power falls as $1/d^4$ instead of $1/d^2$.

Radio astronomy

Radio telescopes use the Friis framework in reverse: given a known (very large) dish aperture and a measured received power, astronomers infer the radiated power of distant cosmic sources like pulsars, quasars, and galaxies.

Common mistakes to avoid

• Mixing linear and dB quantities. Gains must be dimensionless ratios in the linear equation and dBi values in the dB equation. Mixing them is the single most common error in link budget problems.
• Forgetting that $\lambda$ and $f$ are inversely related. If a problem gives frequency, convert to wavelength ($\lambda = c/f$) before plugging into the linear FSPL expression.
• Interpreting the frequency dependence as atmospheric loss. The $f^2$ term in FSPL is geometric, not absorptive.
• Applying Friis in non-LOS or near-field scenarios. The equation assumes far-field, unobstructed propagation. Using it indoors or in cluttered environments without additional loss terms will overestimate received power.