# FRIIS FREE SPACE EQUATION

Today we will discuss a very important relation related to wave propagation. The equation is known as Friis free space equation.

As only a small fraction of radiated power is received at the receiver from an isotropic radiator in free space, but the received signals, must be 10-20 dB above the receiver noise to complete the link between transmitter (T_{X}) and receiver (R_{X}) antenna. The amount of received power depends on transmitted power, gains of transmitter and receiver antennas and separation between them, operating frequency and path attenuation. Thus in order to describe the characteristics of wave propagation, it is necessary to derive equation relating to these parameters. The expression relating these parameters is known as Friis free space wave equation.

Let us take an isotropic radiator transmitting power in free space, so the medium surrounding is homogeneous and non absorbing with dielectric constant unity. The power density at distance d from the centre of the radiator will be

P_{D} = P_{t}/4πd^{2} …(1) (*a*)

where P_{D} = Power density (W/m^{2})

P_{t}= Transmitted power (W)

*d* = Distance between transmitter and receiver (km)

4Π*d*^{2} = Spherical surface area (m^{2})

If a directional antenna is used at receiver, the receiving density will increase by the multiple of gain of the transmitting antenna. *i.e.*,

P_{D} = P_{t} G_{t} /4πd^{2}

where G* _{t}* = maximum directive gain of the T

_{X}antenna and

= 6(R/λ)^{2} in the case of microwave dish antenna

in which R is larger aperture of antenna and l is operating wavelength.

If a is attenuation of the medium the power density is modified to

P_{D} = P_{t}G_{t}/4πd^{2}a

As the transmitted power spreads over a spherical area of many kilometers, the receiving antenna picks up only a small fraction of the radiated power. The amount of power at the R_{X} antenna will be area of the receiving antenna (A) times the power density at the T_{X} antenna.

P_{D}= P_{t}G_{t}A/4πd^{2}a …(1) (*b*)

As the gain of receiving antenna is

G* _{r}* = 4πA/λ

^{2}

then A = G* _{r}* λ

^{2}/4π …(2)

The power received at the receiver will be

P* _{r}* = P

_{D}A

Put equation (1.b) and (2)

P* _{r}* = P

_{t}G

_{t}G

_{r}a( λ

^{2}/4πd

^{2}) …(3)

or P_{r} = P* _{t}* G

*G*

_{t}*a 1/L*

_{r}_{p}

Where L_{P} = L* _{p}* = 4πd

^{2}/ λ

^{2}= Free space path loss. This may be defined as the ratio of antenna area one wavelength square to area over which the transmitted power has been spread.

In equation (3) all the parameters can be determined easily except attenuation a. The a depends upon atmospheric conditions that vary with time and local weather.

Equation 3 is known as fundamental equation of free space propagation or Friis free space wave equation.