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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  Pt  then it’s isotropic power density W0 at distance R from the antenna is 
                                                        W0 = et ( Pt / 4ПR2 )
Where et = radiation efficiency of the transmitting antenna.

Friis-Transmission equation about antenna

If the transmitting antenna be non-isotropic , then the power density in the direction ϴt , Фt can be  written
                                                        Wt = Pt Gt(ϴt ,Фt) / 4ПR2
                                                                         =  et ( Pt Dt(ϴt ,Фt) / 4ПR2) ---------(1)
                                                  So, Pt =  ( Wt 4ПR2) / et Dt(ϴt ,Фt)
Where Gt(ϴt ,Фt) = gain
             Dt(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 Ar  of the receiving antenna is related to its efficiency er and the directivity Dr by –
            Ar   =   er  Dr(ϴr ,Фr) 2 / 4П)   ---------------(2)
As the transmitting power is collected by the effective area that is why total amount of power Pr collected by the receiving antenna can be written – 
           Received power = effective area х transmitting power density
                               Pr  = Ar  Wt
                                           = er  Dr(ϴr ,Фr) 2 / 4П) et ( Pt Dt(ϴt ,Фt) / 4ПR2)
[ using 1 and 2 we get the above relation ]

              Now, Pr / Pt  =  ( et er  Dt(ϴt ,Фt) Dr(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-
    et  = ecdt  ( 1 + |П|2 )
    er  = ecdr  ( 1 - |П|2 )
where ecdt = radiation efficiency of transmitting antenna
                  = ec ed  = conduction efficiency x dielectric efficiency
            ( 1 - |П|2 ) = er = reflection efficiency of transmitting antenna
            Similar to receiving antenna

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

 Pr / Pt  =  ecdt  ecdr  ( 1 - |П|2 ) ( 1 - |П|2 )Dt(ϴt ,Фt) Dr(ϴr ,Фr)    ( ƛ/4ПR)2  ---------(4)

When the antennas are reflection and polarization matched then
    et  = ecdt  ( 1 - |П|2 )  = 1
    er  = ecdr  ( 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 

    Pr / Pt  =  G0t G0r ( ƛ/4ПR)2  ---------(5)
Where  G0t  = 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 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.
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