22 Free-Space Path Loss
22.1 Start With Distance Taking Energy Away
A wireless signal spreads as it travels, so the receiver gets less energy than the transmitter sent. Free-space path loss is the clean baseline: what the path costs before walls, trees, antennas, fading, or interference make it harder.
Use the formula as a first promise, not a final guarantee. Once you know the open-air loss, every real deployment adds evidence about materials, height, frequency, antennas, and margin.
Phoebe’s Why
This page states the FSPL formula and its two constants – 32.45 for km and MHz, \(-147.55\) for m and Hz – as facts to plug numbers into. Both constants, and the whole \(d^2\)-and-\(f^2\) shape of the formula, fall out of two separate physical ideas: how power spreads outward from a point source, and how a receiving antenna’s ability to “catch” that power depends on wavelength. Neither idea involves air soaking up more energy at higher frequency – free space does not absorb radio waves at all. The frequency penalty this page quotes belongs entirely to the antenna doing the catching, not to the space the wave crossed.
The Derivation
An isotropic transmitter spreads power \(P_t\) evenly over a sphere of radius \(d\):
\[S = \frac{P_t}{4\pi d^2}\]
An isotropic receive antenna collects that density over an effective aperture set purely by wavelength:
\[A_e = \frac{\lambda^2}{4\pi}\]
Received power \(P_r = S\,A_e\) gives the path-loss ratio:
\[\mathrm{FSPL} = \frac{P_t}{P_r} = \left(\frac{4\pi d}{\lambda}\right)^2 = \left(\frac{4\pi d f}{c}\right)^2\]
Converting to dB with \(d\) in km and \(f\) in MHz:
\[\mathrm{FSPL}_{dB} = 20\log_{10}d + 20\log_{10}f + \underbrace{20\log_{10}(4\pi) + 180 - 20\log_{10}c}_{K}\]
Worked Numbers: This Page’s Own 915 MHz / 2.4 GHz Comparison
- With \(c = 3.00\times10^8\) m/s: \(K = 20\log_{10}(4\pi) + 180 - 20\log_{10}(3.00\times10^8) = 32.4\) – matching this page’s own quoted 32.45 to the precision of \(c\) used.
- This page’s own 100 m at 915 MHz: \(20\log_{10}(0.1) + 20\log_{10}(915) + 32.4 = -20.0 + 59.2 + 32.4 = 71.7\) dB, matching this page’s own quoted 71.7 dB.
- This page’s own 100 m at 2.4 GHz: \(20\log_{10}(0.1) + 20\log_{10}(2400) + 32.4 = -20.0 + 67.6 + 32.4 = 80.0\) dB, matching this page’s own quoted 80.0 dB exactly.
- The delta is \(20\log_{10}(2400/915) = 8.38\) dB, matching this page’s own “8.4 dB” – entirely from the aperture term \(f^2\). The \(d^2\) spherical-spreading term (this page’s own “6 dB per doubling,” \(= 20\log_{10}2 = 6.02\) dB) is completely unchanged by frequency, because sphere geometry does not know what frequency the wave is.
Overview: FSPL Is The Open-Air Baseline
Free-space path loss estimates how much radio power is lost as a signal spreads through unobstructed space. It is the clean starting point for IoT range work: distance consumes margin, frequency changes the received power, and the result becomes one line in a larger link-budget record.
FSPL is not an indoor or city-range promise. It assumes line of sight, clear Fresnel space, matched antennas, and no walls, bodies, shelves, vehicles, rain fade, or multipath. Use it to set the baseline, then add deployment losses before deciding gateway spacing.
The practical value of FSPL is that it separates a radio design into two questions. First, what would the path cost under the cleanest possible geometry? Second, which real losses must be added for this deployment? That separation keeps teams from arguing about range as a single number. A 100 m path at 915 MHz is about 71.7 dB in free space, while the same 100 m path at 2.4 GHz is about 80.0 dB. The higher-frequency link starts with roughly 8.4 dB less headroom before walls, antenna orientation, interference, and receiver mode are considered.
Use those numbers as a baseline record, not as a guarantee. For an IoT gateway plan, the baseline should state the distance, frequency, formula convention, and receiver sensitivity before it states the conclusion. If a designer cannot point to those assumptions, the range estimate cannot be reviewed or repaired when the site changes. If the assumptions are visible, the team can decide whether to raise an antenna, move a gateway, change a band, accept a lower data rate, or schedule a field measurement.
Distance
Every distance doubling adds about 6 dB of free-space loss because the same power spreads over a larger wavefront.
Frequency
At equal distance, 2.4 GHz has about 8.4 dB more FSPL than 915 MHz because the frequency term is higher.
Margin
The deployment decision uses received power minus receiver sensitivity, with reserve for fading, walls, antenna mismatch, and aging.
Baseline formula: with distance in km and frequency in MHz, FSPL(dB) = 20 log10(d) + 20 log10(f) + 32.45. With distance in meters and frequency in Hz, use 20 log10(d) + 20 log10(f) - 147.55.
Practitioner: Turn FSPL Into A Link-Budget Record
A field-ready link budget starts with transmit power, antenna gains, cable losses, FSPL, extra deployment losses, receiver sensitivity, and required margin. The decision is not whether a theoretical signal exists. The decision is whether enough margin remains after the site-specific losses are added.
Write the record so another engineer can reproduce the decision without re-running the whole survey. Keep the radio configuration beside the path estimate: band, channel width or spreading mode, transmit power limit, antenna gain, cable length, receiver sensitivity mode, and the packet success target. Then put the FSPL line in the middle of the ledger, not at the end. The path baseline should be followed by explicit allowances for walls, shelving, foliage, vehicle blockage, antenna mismatch, Fresnel clearance, weather exposure, and the fade margin required by the service.
This record also prevents overfitting to a single optimistic test. A field RSSI reading from a clear doorway might validate the arithmetic for that moment, but it does not prove the link during inventory movement, seasonal leaf growth, rain, or a changed enclosure. Treat the first calculation as a release candidate: approve it only when the remaining margin survives reasonable site losses and when the measurement plan checks the riskiest paths, not just the easiest path.
A smart-building estimate that passes in free space can still fail after wall loss, metal shelving, antenna orientation, and people moving through the path. The useful output is a margin with assumptions, not a single range number.
Under The Hood: Model Boundaries Matter
FSPL follows from spreading loss and antenna aperture assumptions. The formula is powerful because it gives a repeatable open-air baseline, but it stops being sufficient when obstacles, reflections, antenna placement, terrain, or fading dominate the received signal.
The logarithmic shape is the key engineering idea. Doubling distance adds about 6 dB of free-space loss because the wavefront area grows with the square of distance. Increasing frequency also raises the loss term in the usual link-budget form, so a radio comparison at the same range still needs the band written down. These terms are deterministic; the uncertainty enters when the path stops looking like the model. That is why the formula should be paired with a model-boundary note rather than copied into a range claim.
A useful boundary note names which losses are inside the calculation and which are still external assumptions. Free-space spreading is inside. Wall attenuation, enclosure detuning, polarization mismatch, partial Fresnel blockage, body shadowing, rain fade, interference, and receiver implementation margin are outside unless the record adds them explicitly. Once those assumptions are named, the design review can ask a precise question: is the remaining margin large enough for this service, or must the team collect measurements before release?
The log-distance model adds an environment exponent: L(d) = L0 + 10 n log10(d / d0) + X_sigma. Free space uses about n = 2. Indoor and obstructed environments often need a larger exponent plus explicit wall, body, foliage, or fade-margin allowances.
The exponent is not a decoration on the formula; it changes how quickly distance consumes margin. An exponent near 2 describes clear open space. Larger values describe environments where reflections, absorption, diffraction, and clutter make each distance increase more expensive. The shadowing term X_sigma is the reminder that two paths with the same distance can still behave differently. For IoT deployments, the under-the-hood review should therefore check model choice, unit consistency, and validation evidence together.
Unit Discipline
The constant changes with units. Use km/MHz with 32.45, or meters/Hz with -147.55, and record the convention.
Environment Drift
Warehouses, hospitals, farms, and city streets do not share one path-loss exponent. Measure or choose conservative assumptions for the actual site.
Verification Trigger
Retest when frequency, antenna, gateway height, enclosure, wall layout, seasonal foliage, or packet reliability requirements change.
22.2 Summary
Free-space path loss gives the clean baseline for wireless range estimates: distance and frequency consume signal margin before any real-world losses are added. A deployment-ready link budget then adds antenna gains and losses, receiver sensitivity, wall or foliage loss, fading allowance, and a required margin. FSPL is a starting point, not a field guarantee.
22.3 Key Takeaway
Use FSPL to create the baseline, then record units, site losses, receiver sensitivity, required margin, and field-verification evidence before approving IoT gateway spacing.
22.4 See Also
Material Attenuation and RSSI
Extend the baseline with wall, body, glass, metal, and measured RSSI effects.
Link Budget and Coverage Planning
Turn transmit power, receiver sensitivity, losses, and margin into a coverage decision.
Fresnel Zones and Practical Deployment
Check antenna height and clearance before trusting a long wireless path.
Wireless Access: Wi-Fi
Apply path-loss thinking to indoor Wi-Fi planning, channel choices, and coverage margins.