Friis Transmission Equation

Friis Transmission Equation

Friis Transmission Formula can be used to study RF communication links. The formula can be used in situations where the distance between two antennas are known and a suitable antenna needs to be found. Using Friis transmission equation, one can solve for the antenna gains needed at either the transmitter or receiver in order to meet certain design specifications.

${\frac {P_{receiver}}{P_{transmitter}}}=G_{t}G_{r}\left({\frac {\lambda }{4\pi \mathbb {R} }}\right)^{2}$

Where P_receiver is the recieved power in Watts (W), P_transmitter is the transmitted power, G_t is the transmitting antenna's gain, G_r is the receiving antenna's gain, $\lambda$ is the wavelength of the transmitted and received signal in meters, and R is the distance between the antennas in meters. The gain of the antennas is not in decibels. To convert to the gain back to a power ratio use: $G=10^{\frac {G_{dB}}{10}}$.

Example 1

An engineer is designing a communications link at 3 GHz where the receiver sensitivity is such that $1\mu W$ of power is needed to overcome receiver noise. The receiving antenna gain is 8dB, the transmitter antenna gain is 10dB, the transmitting power level is 25 Watts, and the distance between the two antennas is 1km. Will the communications link work?

Solution

The problem is solved by first converting the frequency of 3 GHz to a wavelength $\lambda$, and converting the antenna gains from decibels to a power ratio.

$\lambda ={\frac {c}{f}}$ where c is the speed of light in meters per second, and f is the frequency in Hz. We will use a c of 300 million m/s. This means that lambda is 0.1 meters. Using the conversion formula given above, the transmitter antenna gain G_t is 10, and the receiver antenna gain is 6.3.

Using Friis Equation we have the following relation $P_{receiver}=P_{transmitter}G_{t}G_{r}\left({\frac {\lambda }{4\pi R}}\right)^{2}$ where the transmitted power is 25 Watts, and the distance R is 1000 meters. Using the given values P_receiver is $99.88\cdot 10^{-9}$ or 99.88 nanowatts (nW) which is much less than the needed 1 microwatts needed to overcome receiver noise.

Answer
The communication link will not work unless the antenna gains or the transmitter power levels are dramatically increased.

Example 2

If an engineer is designing a communications link with a receiver antenna gain of 10dB and transmitter antenna gain of 12dB at a frequency of 10GHz, how much power will be received at the receiver if the transmitter is transmitting with a power of 25 watts across a distance of 1 kilometer.

ans: -46.4623721 dBm

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