Deriving the Chapter’s FSPL Numbers

Deriving the Chapter’s FSPL Numbers

Ada reproduces the chapter’s free-space path-loss figures from one formula, across both unit conventions

ADA · CALCULATION AUDIT

Deriving the Chapter's FSPL Numbers

The physics story is spherical spreading; the mathematics is a logarithmic unit check. The chapter’s distance, frequency, and comparison numbers should all reproduce from one formula — 100 m at 915 MHz costs about 71.7 dB, the same span at 2.4 GHz costs about 80 dB, and doubling distance always adds a fixed 6.02 dB. This audit rebuilds each figure from one equation across both unit conventions.

Companion to the chapter Free-Space Path Loss — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is radio path distance. The middle card applies this page's rule. The green card is free-space path loss at 915 mhz. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only radio path distance, so the numeric fixture does not switch without explanation.

Walk the arrows. Doubling distance adds 6.02 dB when frequency stays at 915 MHz.

Derive the baseline in four named moves

1. 1

   Name the input. The chapter baseline is 100 m.

2. 2

   Name the relationship. loss = 20 log10(distance in m) + 31.68 dB

3. 3

   Substitute with units. 20 log10(100) + 31.68 = 71.68 dB

4. 4

   Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change radio path distance

Try Predict the direction of loss = 20 log10(distance in m) + 31.68 dB. Test another radio path distance, then compare free-space path loss at 915 mhz.

Radio path distance

100 m

Chapter baseline

Free-space path loss at 915 MHz

Observe Doubling distance adds 6.02 dB when frequency stays at 915 MHz. Reset radio path distance to 100 and compare free-space path loss at 915 mhz.

Explain Doubling distance adds 6.02 dB when frequency stays at 915 MHz.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only radio path distance moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only radio path distance. It is not a field prediction.

TryCompare the two fixed rows—100 m at 915 MHz and 100 m at 2.4 GHz—then press Calculate to follow the arithmetic.

ObserveFree-space loss moves from about 71.7 to 80 dB, and the unit-converted formula reproduces the same values.

ExplainInverse-square spreading becomes a 20 log distance term, so every distance doubling adds the same 6.02 dB penalty.

Ready: use the stated baseline inputs, then compare each displayed result.

Convert the Distance Once

1. Convert the distance once. The chapter uses the km/MHz form of FSPL, so the 100 m comparison becomes 0.1 km before any logarithms are taken:

FSPL(dB) = 20 log₁₀(d_km) + 20 log₁₀(f_MHz) + 32.45

Chapter value	Calculation	Checked result
100 m at 915 MHz	20 log10(0.1) + 20 log10(915) + 32.45 = -20.00 + 59.23 + 32.45	71.68 dB, so about 71.7 dB
100 m at 2.4 GHz	20 log10(0.1) + 20 log10(2400) + 32.45 = -20.00 + 67.60 + 32.45	80.05 dB, so about 80 dB
Extra 2.4 GHz loss	80.05 - 71.68, or 20 log10(2400 / 915)	8.37 dB, so about 8.4 dB less headroom

Verify the Same Answer With the Meters/Hz Convention

2. Verify the same answer with the meters/Hz convention. The alternate constant is useful only if the units change too:

20 log₁₀(100) + 20 log₁₀(915000000) - 147.55 = 40.00 + 179.23 - 147.55 = 71.68 dB

Check the Distance-Doubling Rule From the Physics

3. Check the distance-doubling rule from the physics. Free-space power thins with distance squared, so the decibel penalty for doubling distance is independent of the starting range:

20 log₁₀(2d) - 20 log₁₀(d) = 20 log₁₀(2) = 6.02 dB

What the calculation buys you: these numbers are not range promises. They prove the open-air baseline before the review adds walls, antenna mismatch, Fresnel blockage, fade margin, receiver mode, and field evidence.

Every number above is taken from the chapter’s own FSPL example and re-derived step by step.

Technical boundaries. Shadowing, multipath, rain, cable loss, antenna gain, Fresnel blockage, and receiver behavior are excluded from this free-space-only equation.