Friis transmission

Linear power ratios

$P_t$ is transmitted power and $P_r$ is received available power, both in watts. Lowercase $g_t$ and $g_r$ are the transmit and receive antenna power gains relative to isotropic; $\ell$ denotes a linear loss factor. These ratios have no units: a gain or loss factor of 1 leaves the power unchanged.

The same equation in dB

Apply $10\log_{10}$ to each power ratio: $G=10\log_{10}(g)$ and $L=10\log_{10}(\ell)$. Multiplication becomes addition; division by a loss factor becomes subtraction. Power in dBm uses a 1 mW reference: $P[\mathrm{dBm}]=10\log_{10}\!\left(\frac{P}{1\,\mathrm{mW}}\right)$.

For example: a 10 dB loss means a linear loss factor of 10. Divide the power in watts by 10, or subtract 10 from the power in dBm. A 0 dB loss means a factor of 1; 0 dBi likewise means antenna gain 1.

What the loss terms mean

All three uppercase $L$ terms are in dB. The calculator derives free-space path loss; you enter polarization mismatch and other losses as non-negative dB values.

$L_{\mathrm{FS}}$Free-space path loss

The loss from geometric spreading, referenced to isotropic antennas. It depends on separation $d$ and wavelength $\lambda=c/f$:$$\ell_{\mathrm{FS}}=\left(\frac{4\pi d}{\lambda}\right)^2$$$$\begin{aligned}L_{\mathrm{FS}}&=10\log_{10}(\ell_{\mathrm{FS}})\\&=20\log_{10}\!\left(\frac{4\pi d}{\lambda}\right)\end{aligned}$$Here $c$ is the speed of light and $f$ is frequency. Use the same length unit for $d$ and $\lambda$. Antenna gains are accounted for separately.

$L_{\mathrm{pol}}$Polarization mismatch

The power penalty when the arriving wave’s polarization does not match the receiving antenna. Use 0 dB for matched polarization. This is separate from antenna pointing and directional gain.

$L_{\mathrm{other}}$Other losses

Additional losses you supply, such as feeder or connector loss, or a separately accounted impedance mismatch loss. Include a loss only if it is absent from the entered powers and gains at your chosen reference planes; count it once.

When this model applies

• Unobstructed free-space far-field reference, with available received power into a matched load. Antenna dimensions and wavelength determine far-field suitability; separation in wavelengths alone does not.
• Use gains toward the other antenna and compatible power reference planes. Enter polarization and other losses once; include mismatch or feeder loss only if absent from the other inputs.
• The entered gains stay fixed when frequency changes. Obstruction, atmosphere, multipath and fading are not calculated.