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Friis Transmission Equation Basics With easy examples.

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The Friis transmission equation gives received power in free space: P_r = P_t x G_t x G_r x (lambda/(4 x pi x R))^2. Received power equals transmitted power times both antenna gains times the square of wavelength over the sphere of radius R across which the energy has spread.

Friis is the starting point of every link budget: Wi-Fi planners, satellite engineers and 5G designers begin here, then add corrections for buildings, rain and multipath. It shows instantly how range, frequency and gain trade off, making it the most-used formula in radio engineering.

Key Concepts

  • Free space: empty and unobstructed; real environments add loss on top of the Friis value.
  • Far field: valid only beyond ~2D^2/lambda from the antenna, where the wave is cleanly spherical.
  • Effective aperture: A_e = G_r x lambda^2/(4 x pi); the antenna's capture "net" grows with gain and wavelength.
  • Free-space path loss (FSPL): the (4 x pi x R/lambda)^2 factor; grows with the square of distance and frequency.
  • Matched conditions: assumes matched polarization and impedance; mismatches add extra loss.

Formulas

  • Friis: P_r = P_t x G_t x G_r x (lambda/(4 x pi x R))^2 (linear gains, consistent units).
  • FSPL: (4 x pi x R/lambda)^2 = (4 x pi x R x f/c)^2.
  • dB form: P_r(dBm) = P_t(dBm) + G_t(dBi) + G_r(dBi) - 20 log_10(4 x pi x R/lambda).
  • Assumptions: far field, free space, matched polarization/impedance, gains in the link direction.

Worked Example

2.4 GHz link: P_t = 1 W, G_t = G_r = 10 dBi, R = 100 m:

  1. lambda = 3x10^8/2.4x10^9 = 0.125 m.
  2. 4 x pi x 100/0.125 = 10053; FSPL = 20 log_10(10053) = 80.05 dB.
  3. P_t = 30 dBm, so P_r = 30 + 10 + 10 - 80.05 = -30.05 dBm (~1 microwatt).
  4. Doubling to 200 m adds exactly 6 dB of loss (P_r ~ -36 dBm): the inverse-square law in action.

Common Misconceptions

  • "Doubling distance halves power." Inverse-square quarters it: 6 dB lost, not 3 dB.
  • "Friis works at any distance." Inside the near field the wavefront is not spherical; check R against 2D^2/lambda first.
  • "Higher frequency always means shorter range." FSPL rises with frequency at fixed gain, but a fixed-size dish's gain rises with frequency squared and can more than compensate.

Real-World Applications

  • Wi-Fi/cell planning: first-cut coverage and access-point spacing.
  • Satellite budgets: Friis plus rain, atmospheric and pointing losses sizes transponder power.
  • Radar: the two-way Friis variant predicts detection range.
  • 5G mmWave: high 28 GHz FSPL countered with high-gain beamforming arrays.

Friis says received power falls with the square of distance and frequency, scaled by both antenna gains; in decibels it becomes simple addition and subtraction. Respect its limits: far field, free space, matched polarization, and always extra loss in the real world.

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