Frii's equation models free space path loss and predicts the amount of power (Pr) that can be received by an antenna at a distance D from the transmitting antenna. The equation is: Pr = Pt * Gt * Gr * λ2 / (4πD)2 * L Where Pt is the transmitted power, Gt and Gr are the gains of the transmitting and receiving antennas, λ is the wavelength, D is the distance between antennas, and L accounts for transmission losses. According to the model, received power decays quadratically with distance, and factors like antenna gain and wavelength affect reception levels.

1. WIRELESS COMMUNICATION
SERIES
Free Space Path Loss Model –
Frii’s equation
[part – 1]

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3. Pt
Gt
Gr - Gain of the Receiving Antenna
Pr
D
Pt - Power applied to the transmitting antenna
Gt - Gain of the transmitting antenna
Pr - Amount of Power received at the receiving antenna
D – Distance Between the Tx and Rx Antenna (T- R Seperation)
Assumptions
#1. Antennas are LOSSLESS
#2. Antennas are ISOTROPIC
#3. NO obstacles are considered
Gr
Aim: To predict the amount of
Signal that can be effectively
received at the receiver end
(Pr)

4. The Power Density of the Transmitted Signal is given by
PD = Pt / 4π D2
Power density is the measure of
amount of power per unit volume.
PD = Power Received / Surface area
The above Eqn. is valid for the ISOTROPIC Antenna, But in practical situations most of the antennas will be DIRECTIONAL
Antenna. Which do not have the Unit Gain in all directions.( Gain will be Higher in a particular Direction)
Observing from the Transmitter Side:
In this case, We have to consider the Gain of the Transmitting antenna into account, Thus the Eqn.
Becomes
PD = (Pt / 4π D2) . Gt
Observing from the Receiver Side:
In electromagnetics and antenna theory,
antenna aperture, is a measure of how
effective an antenna is at receiving the
radiated power.
Ae = Power Received / Power Density
The amount of power effectively received by the receiving antenna is
decided by the Effective Aperture (Ae) of an Antenna.
Ae= Pr / PD Pr = PD . Ae Pr =
𝑷 𝒕 .𝑮 𝒕
𝟒𝝅𝑫 𝟐 . Ae
The Relation between the G and Ae is
given by
Ae =
λ 𝟐
𝟒π
. GPr =
𝑷 𝒕 .𝑮 𝒕
𝟒𝝅𝑫 𝟐 .
λ 𝟐
𝟒π
Gr Pr =
𝑷 𝒕 .𝑮 𝒕 .𝑮 𝒓.𝝀 𝟐
(𝟒𝝅𝑫) 𝟐.𝑳
Frii’s Free Space Equation

5. Pr (D) =
𝑷 𝒕 .𝑮 𝒕 .𝑮 𝒓.𝝀 𝟐
(𝟒𝝅𝑫) 𝟐.𝑳
Gr - Gain of the Receiving Antenna
Pt - Power applied to the transmitting antenna
Gt - Gain of the transmitting antenna
Pr - Amount of Power received at the receiving antenna
D – Distance Between the Tx and Rx Antenna (T- R Seperation)
L – Losses due to Transmission Line Attenuation, Filter Loss and Antenna Loss
INFERENCES:
1. The received power decays at the rate of square of the distance between the antenna
2. According to the Frii’s Free Space Eqn. Received Power is represented as the Function of Distance
Between the Tx and Rx. Pr (D)
3. Pt . Gt is referred to as the Effective Isotropic Radiated Power (EIRP) compared with isotropic
radiator. Unit (dBi – dB gain with respect to isotropic antenna)
4. In practice Effective Radiated Power (ERP) is considered instead of EIRP. Power radiated is
compared with the half wave Dipole antenna. Unit (dBd – dB gain with respect to Dipole antenna)

6. S U B S C R I B E
PD = Pt / 4π D2
PD = (Pt / 4π D2) . Gt
Pr = PD . Ae
Pr =
𝑷 𝒕 .𝑮 𝒕 .𝑮 𝒓.𝝀 𝟐
(𝟒𝝅𝑫) 𝟐.𝑳