Free-space Path Loss

Friss Equation

Friss Transmission Equation gives an estimate of power received by an antenna from another antenna radiating power at some distance away.


If the antenna radiates power uniformly in all directions, it is called isotropic antenna. Power(P_t) flows out from the transmit antenna in spherical wavefronts. Thus the power density at a distance R from the transmit antenna is \frac{P_t}{4pi R^{\tiny 2}}(watt/m^2) .

If an antenna radiates power in a specific direction, it is called directional antenna. It is characterized by gain(G).

\mbox{Gain of antenna(G)} = {\mbox{Power density of the antenna}\over\mbox{Power density of isotropic antenna}}

The units of antenna gain are expressed in dBi.

If the gain of transmit antenna is G_t, then power density at the receive antenna is \frac{P_t G_t}{4pi R^{2}}. If the receive antenna has an aperture area of A_e, then the total receive power is \frac{P_t G_t}{4\pi R^2}A_e (watts)

The effective antenna aperture area is related to wavelength lambda and receive antenna gain G_r as A_e = \frac{\lambda^2 G_r}{4\pi}

Therefore received power of antenna is

    \[P_r = \frac{P_t G_t G_r \lambda^2}{(4\pi R)^2}\]

Path loss is the power loss in the channel and is given by

(1)   \begin{equation*} L_p = \frac{P_t}{P_r}=\frac{(4\pi R)^2}{G_t G_r \lambda^2},,,watts  \end{equation*}

(2)   \begin{equation*} L_{p(dB)} = 20 \log(\frac{4\pi R}{\lambda}) - G_t(dB) - G_r(dB) \end{equation*}

where, L_p → path loss in free space
r → distance from the transmit antenna(m).
\lambda → wavelength of propagating wave in free space(m^{-1}).
f → frequency in Hz.
c=f\lambda → speed of light in free space, (m/s).

It can alternatively be expressed as

(3)   \begin{equation*} L_{p(dB)} = 32.4 + 20 log(F) + 20 log(D) - G_{t(dB)} - G_{r(dB)}  \end{equation*}

where,F → frequency in MHz, and
D → distance in km

The pathloss increases with frequency of transmission and distance. For example signals transmitting at 2.4GHz incur much higher path loss than the signals transmitting at 900MHz.

References :

RF Link Budget Calculator

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