Friis Transmission Formula explained

 

The Free Space Path Loss (FSPL) equation originates from the Friis Transmission Formula:

FSPL = (4πd / λ)²

Since wavelength is related to frequency by:

λ = c / f

Substituting λ into the FSPL equation:

FSPL = (4πdf / c)²

Converting to decibels:

FSPL(dB) = 20 log₁₀(d) + 20 log₁₀(f) + 20 log₁₀(4π / c)

Using distance in meters and frequency in hertz:

FSPL(dB) = 20 log₁₀(d_m) + 20 log₁₀(f_Hz) − 147.55

The practical Free Space Path Loss (FSPL) formula commonly uses:

• Distance in kilometers (km)
• Frequency in megahertz (MHz)

Therefore, the unit conversions are:

d_m = 1000 × d_km

f_Hz = 10⁶ × f_MHz

Substituting into the FSPL Equation

FSPL = 20 log₁₀(1000 d_km) + 20 log₁₀(10⁶ f_MHz) − 147.55

Expanding the Logarithms

FSPL = 20 log₁₀(d_km) + 20 log₁₀(f_MHz) + 20 log₁₀(1000) + 20 log₁₀(10⁶) − 147.55

Evaluating the Constant Terms

20 log₁₀(1000) = 60

20 log₁₀(10⁶) = 120

Combining the Constants

60 + 120 − 147.55 = 32.45

After rounding:

32.44

This is the origin of the constant used in the practical FSPL equation:

FSPL(dB) = 20 log₁₀(d_km) + 20 log₁₀(f_MHz) + 32.44

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