Friis-Transmission Equation of Antenna: Derivation

The Friis-Transmission equation relates the power received to the power transmitted between two antennas separated by a distance  R > 2 (D²/ƛ) , where D is the largest dimension of either antennas.

Let us assume that the transmitting antenna is initially isotropic ( having equal radiation in all directions ). If the input power at the input power at the terminals of the transmitting antenna is  P_t  then it’s isotropic power density W₀ at distance R from the antenna is 

                                                        W₀ = e_t ( P_t / 4ПR² )

Where e_t = radiation efficiency of the transmitting antenna.

If the transmitting antenna be non-isotropic , then the power density in the direction ϴ_t , Ф_t can be  written

                                                        W_t = P_t G_{t(ϴt ,Фt)} / 4ПR²

                                                                         =  e_t ( P_t D_{t(ϴt ,Фt)} / 4ПR²) ---------(1)

                                                  So, P_t =  ( W_t 4ПR²) / e_t D_{t(ϴt ,Фt)}

Where G_{t(ϴt ,Фt)} = gain

             D_{t(ϴt ,Фt)} = directivity

    and  gain = radiation efficiency х directivity

Now whole surface of the receiving antenna don’t receive the radiating wave simetrically. Major part of the radiating wave incident in a particular area rather than all the area. This particular area is known as effective area.  This effective area A_r  of the receiving antenna is related to its efficiency e_r and the directivity D_r by –

            A_r   =   e_r  D_{r(ϴr ,Фr)} (ƛ² / 4П)   ---------------(2)

As the transmitting power is collected by the effective area that is why total amount of power P_r collected by the receiving antenna can be written – 

           Received power = effective area х transmitting power density

                               P_r  = A_r  W_t

                                           = e_r  D_{r(ϴr ,Фr)} (ƛ² / 4П) e_t ( P_t D_{t(ϴt ,Фt)} / 4ПR²)

[ using 1 and 2 we get the above relation ]

              Now, P_r / P_t  =  ( e_t e_r  D_{t(ϴt ,Фt)} D_{r(ϴr ,Фr)} ƛ² ) / (4ПR)²  -----------(3)

If the antenna’s have reflection and radiation losses due to matching and polarization factor then the efficiency of the antennas become-

    e_t  = e_cdt  ( 1 + |П|² )

    e_r  = e_cdr  ( 1 - |П|² )

where e_cdt = radiation efficiency of transmitting antenna

                  = e_c e_d  = conduction efficiency x dielectric efficiency

            ( 1 - |П|² ) = e_r = reflection efficiency of transmitting antenna

            Similar to receiving antenna

Including the two factor matching and polarization we get from (3)-

 P_r / P_t  =  e_cdt  e_cdr  ( 1 - |П|² ) ( 1 - |П|² )D_{t(ϴt ,Фt)} D_{r(ϴr ,Фr)}    ( ƛ/4ПR)²  ---------(4)

When the antennas are reflection and polarization matched then

    e_t  = e_cdt  ( 1 - |П|² )  = 1

    e_r  = e_cdr  ( 1 - |П|² )  =  1

so gain = radiation efficiency х directivity       

  reduce to gain = directivity  

That represents the isotropic gain. 

     

Considering this fact equation (4) reduce to 

    P_r / P_t  =  G_0t G_0r ( ƛ/4ПR)²  ---------(5)

Where  G_0t  = isotropic gain of the transmitting antenna  

Equations (3), (4), or (5) are known as the Friis-Transmission Equation considering different circumstances and it relates the power P_r (delivered to the receiver load) to the input power of the transmitting antenna P_t . 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.