91Ó°ÊÓ

The Friis transmission formula is essential in understanding how power is transmitted between two antennas in free space. It is used to calculate the power density at a certain distance from the antenna.
The formula can be represented as:
\( S = \frac{P \times G}{4 \times \pi \times R^2} \)
where:
\(S\) is the power density (in \( W/m^2 \))
\(P\) is the input power (in Watts)
\(G\) is the antenna gain on a linear scale (unitless)
\(R\) is the distance from the antenna (in meters)
This formula essentially tells us how the transmitted power decreases as the distance from the antenna increases. The factor of \(4 \times \pi \times R^2 \) in the denominator represents the surface area of a sphere (in squared meters) with radius R, spreading the power evenly in all directions.
Understanding and using this formula helps in predicting the performance of communication systems, such as satellite links and wireless networks.

Antenna gain is usually given in dBi (decibels relative to an isotropic radiator), which is a logarithmic measure. For many calculations, we need to convert this gain to a linear scale.
The conversion formula is:
\( G_{linear} = 10^{(G_{dBi}/10)} \)
For example, if an antenna has a gain of 12 dBi:
\( G_{linear} = 10^{(12/10)} = 10^{1.2} \)
This can be calculated to approximately 15.85. By converting antenna gain from dBi to a linear scale, we make it easier to apply in various equations, such as the Friis transmission formula.
This step is crucial as it normalizes the gain in a way that aligns with the physical properties of power and energy distributions. Decibels are useful for expressing large ranges of values in a compact form, but linear gain is necessary for accurate multiplication and division in formulas.

Calculating linear gain is a straightforward process once you understand the conversion from dBi to a linear scale. The calculation for our example was:
\( G_{linear} = 10^{1.2} \)
This results in a linear gain of approximately 15.85.
With this value, you can now substitute it into equations that require a linear representation of antenna gain.
Here are a few key points to remember:

• Linear gain provides a direct, proportional relationship to the input power and the resulting power density or field strength
• Once in linear form, you avoid the complications of logarithmic addition or subtraction
• In systems analysis, knowing both the dBi and linear gain values can give a more complete understanding of system performance

This understanding helps in various practical applications, like designing and evaluating antenna systems for better performance.