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.

free-space-path-loss-wavefronts

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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