The Friis Transmission Equation relates the power received to the power transmitted between two antennas separated by a distance $R \gt2D^2/λ$, where D is the largest dimension of either antenna. Referring to Figure let us assume that the transmitting antenna is initially isotropic. If the input power at the terminals of the transmitting antenna is Pt, then its isotropic power density W0 at distance R from the antenna is

$$W_0 = e_t\frac{P_t}{4{\pi}R^2}$$

Where $e_t$ is the radiation efficiency of the transmitting antenna. For a non isotropic transmitting antenna, the power density W0 in the direction θt, φt can be written as

$$W_t = \frac{P_t \ G_t \ (θ_t, \ Φ_t)}{4{\pi}R^2} = e_t\frac{P_t \ D_t \ (θ_t, \ Φ_t)}{4{\pi}R^2}$$

Where $G_t \ (θ_t, \ Φ_t)$ is the gain and $D_t \ (θ_t, \ Φ_t)$ is the directivity of the transmitting antenna in the direction $θ_t, \ Φ_t$. Since the effective area $A_r$ of the receiving antenna is related to its efficiency $e_r$ and directivity $D_r$ by

$$A_r = e_r \ D_r (θ_r, \ Φ_r)\left(\frac{λ^2}{4{\pi}}\right)$$

The amount of power $P_r$ collected ny the receiving antenna can be written, using and (2-115), as

$$P_r = e_r \ D_r \ (θ_r, \ Φ_r)\frac{λ^2}{4{\pi}} \ W_t = e_t \ e_r \ \frac {{\lambda}D_t (θ_r, \ Φ_r) D_r(θ_r, \ Φ_r) P_t}{(4{\pi}R)^2} \ |\hat{ρ}_t - \hat{ρ}_r|^2$$

or the ratio of the received to the input power as

$$\boxed{\frac{P_r}{P_t} =e_t e_r\frac{λ^2 D_t (θ_r, Φ_r) D_r(θ_r, Φ_r)}{(4{\pi}R)^2}}$$

The power received based on above expression assumes that the transmitting and receiving antennas are matched to their respective lines or loads (reflection efficiencies are unity) and the polarization of the receiving antenna is polarization-matched to the impinging wave (polarization loss factor and polarization efficiency are unity). If these two factors are also included, then the ratio of the received to the input power of above result is represented by

$$\boxed{\frac{P_r}{P_t} = e_{cdt} e_{cdr}(1 - |Γ_t|^2)(1-|Γ_r|^2)\left(\frac{λ}{4{\pi}R}^2\right)D_t (θ_r, Φ_r) D_r(θ_r, Φ_r)|\hat{ρ}_t - \hat{ρ}_r|^2}$$

For reflection and polarization-matched antennas aligned for maximum directional radiation and reception, above equation reduces to

$$\boxed{\frac{P_r}{P_t} = \left(\frac{λ}{4πR}^2\right) G_0t G_0r}$$ Equations are known as the Friis Transmission Equation, and it relates the power Pr (delivered to the receiver load) to the input power of the transmitting antenna Pt. The term $(λ/4πR)^2$ is called the free-space loss factor, and it takes into account the losses due to the spherical spreading of the energy by the antenna.