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.

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Introduction to the Radiating field regions of an antenna

The space surrounding an antenna is usually subdivided into three regions:

• reactive near-field
• radiating near-field (Fresnel) and
• far-field (Fraunhofer) regions 

Although no abrupt changes in the field configurations are noted as the boundaries are crossed, there are distinct differences among them. The boundaries separating these regions are not unique, although various criteria have been established and are commonly used to identify the regions.

Reactive near-field region: 

Reactive near-field region is defined as “that portion of the near-field region immediately surrounding the antenna wherein the reactive field predominates.” For most antennas, the outer boundary of this region is commonly taken to exist at a distance R < 0.62(D³/ƛ)^1/2 from the antenna surface, where λ is the wavelength and D is the largest dimension of the antenna. “For a very short dipole, or equivalent radiator, the outer boundary is commonly taken to exist at a distance λ/2π from the antenna surface.”

Radiating near-field (Fresnel) region: 

Radiating near-field (Fresnel) region is defined as “that region of the field of an antenna between the reactive near-field region and the far-field region wherein radiation fields predominate and wherein the angular field distribution is dependent upon the distance from the antenna. If the antenna has a maximum dimension that is not large compared to the wavelength, this region may not exist. For an antenna focused at infinity, the radiating near-field region is sometimes referred to as the Fresnel region on the basis of analogy to optical terminology. If the antenna has a maximum overall dimension which is very small compared to the wavelength, this field region may not exist.” The inner boundary is taken to be the distance R < 0.62(D³/ƛ)^1/2 and the outer boundary the distance R < 2D²/λ  where D is the largest∗ dimension of the antenna. This criterion is based on a maximum phase error of π/8. In this region the field patterns, in general, a function of the radial distance and the radial field component may be appreciable.

Far-field (Fraunhofer) region: 

Far-field (Fraunhofer) region is defined as “that region of the field of an antenna where the angular field distribution is essentially independent of the distance from the antenna. If the antenna has a maximum∗ overall dimension D, the far-field regions commonly taken to exist at distances greater than 2D2/λ from the antenna, λ being the wavelength. For an antenna focused at infinity, the far-field region is sometimes referred to as the Fraunhofer region on the basis of analogy to optical terminology.” In this region, the field components are essentially transverse and the angular distribution is independent of the radial distance where the measurements are made.

It is observed that the patterns are almost identical, except for some differences in the pattern structure around the first null and at a level below 25 dB. Because infinite distances are not realizable in practice, the most commonly used criterion for minimum distance of far-field observations is 2D2/λ.

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