Path-Loss and Margin Numbers: Checked Calculations

Path-Loss and Margin Numbers: Checked Calculations

Turn logarithmic signal loss into a link-margin decision you can audit

foundations
math-foundations
wireless-propagation
path-loss
intermediate
Ada Ada · calculation audit
Start with the unresolved radio link

The spreadsheet says 9 dB remains. Why do field packets still disappear?

A sensor transmits at +14 dBm toward a gateway whose receiver can detect down to −126 dBm. After antenna gains, enclosure and cable loss, a 118 dB path, implementation loss, and a 12 dB reserve, the chapter’s ledger leaves 9 dB. That looks healthy—but only if every signed term and every propagation assumption is honest.

Ada will first show what signal spreading means, then unpack the logarithms, and finally walk every gain and loss from transmitter to receiver. The calculation does not replace a field survey; it tells the survey exactly which assumptions to challenge.

Companion to Path Loss and Link Budgets. Every constant and worked result comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is link distance at 915 mhz. The middle card applies this page's rule. The green card is free-space loss baseline. 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 link distance at 915 mhz, so the numeric fixture does not switch without explanation.

Link distance at 915 MHz changes free-space loss baseline An input card leads through the rule FSPL = 20 log10(distance in m) + 31.6801 dB to the free-space loss baseline result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Distance changes spreading while frequency and installed losses remain fixed.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 100 m.

  2. 2

    Name the relationship. FSPL = 20 log10(distance in m) + 31.6801 dB

  3. 3

    Substitute with units. 20 log10(100) + 31.6801 = 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 link distance at 915 mhz

Try Predict the direction of FSPL = 20 log10(distance in m) + 31.6801 dB. Test another link distance at 915 mhz, then compare free-space loss baseline.

100 m
Chapter baseline
Free-space loss baseline

Observe Distance changes spreading while frequency and installed losses remain fixed. Reset link distance at 915 mhz to 100 and compare free-space loss baseline.

Explain Distance changes spreading while frequency and installed losses remain fixed.

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 link distance at 915 mhz moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only link distance at 915 mhz. It is not a field prediction.

Try

Follow the signed ledger, then move distance around the reviewed 1 km case.

Observe

Doubling distance adds about 9.6 dB of loss when the exponent is 3.2.

Explain

Decibels turn multiplication into addition, so gains add and losses subtract in one readable chain.

Picture the ledger as signal passing through a sequence

Read the figure left to right. The transmitter and gateway antenna add useful terms. The enclosure, path, cable, and implementation subtract terms. The final −105 dBm receive level is 21 dB above the −126 dBm sensitivity; reserving 12 dB for fading and shadowing leaves 9 dB available.

A signed radio link-budget ledger from sensor to gateway Seven connected blocks show plus 14 dBm transmit power, zero dBi transmit gain, minus 1 dB enclosure loss, minus 118 dB path loss, plus 3 dBi receive gain, minus 1 dB cable loss, and minus 2 dB implementation loss, giving minus 105 dBm at the receiver. A margin bar compares this with minus 126 dBm sensitivity and a 12 dB reserve. Add gains. Subtract losses. Keep every sign. TX power+14 dBm TX gain+0 dBi Enclosure−1 dB Path−118 dB RX gain+3 dBi Cable−1 dB Other−2 dB 14 + 0 − 1 − 118 + 3 − 1 − 2 = −105 dBm Sensitivity−126 dBm 9 dB available 12 dB reserved −105 dBm received
Walk the sign of every box. The 21 dB raw gap between −105 and −126 dBm is split into 12 dB reserved for fading and shadowing plus 9 dB still available.

Derive the 900 MHz free-space reference

The chapter uses distance in kilometres and frequency in megahertz:

FSPLdB = 20 log10(dkm) + 20 log10(fMHz) + 32.45
  1. 1

    Put values in the formula’s named units. The worked reference is already d = 1 km and f = 900 MHz.

  2. 2

    Evaluate the distance logarithm. log10(1) = 0 because 100 = 1, so 20 log10(1) = 0 dB.

  3. 3

    Evaluate the frequency logarithm. log10(900) = 2.9542, so 20 × 2.9542 = 59.084 dB.

  4. 4

    Add the three decibel terms. FSPL = 0 + 59.084 + 32.45 = 91.534 dB ≈ 91.5 dB

The unitless logarithms do not mean the physical units vanished. They were fixed by the formula’s kilometre/megahertz convention and its 32.45 constant.

Use an anchor to see what distance changes

At 2.4 GHz and 1 m, the same free-space equation gives about 40.05 dB. The chapter then compares environments with the log-distance model:

PL(d) = PL(d0) + 10n log10(d/d0)

Name the ratio: 30 m / 1 m = 30, with metres cancelling. Choose the exponent: n = 2.0 for the free-space example or n = 3.2 for the indoor example. Calculate the added loss:

Model Step shown Worked result
Free space, n = 2.0 40.0 + 10(2.0)log10(30) = 40.0 + 29.54 69.54 dB, about 69.5 dB
Indoor example, n = 3.2 40.0 + 10(3.2)log10(30) = 40.0 + 47.27 87.27 dB, about 87.3 dB

A larger n makes distance more expensive because the model represents stronger obstruction and scattering. It is an environment model to validate, not a universal property of every building.

Reproduce the chapter’s 9 dB margin

  1. 1

    Start at the transmitter and preserve every sign. Prx = 14 + 0 − 1 − 118 + 3 − 1 − 2 = −105 dBm

  2. 2

    Compare receive power with sensitivity. Subtracting a negative adds the gap: Raw margin = −105 − (−126) = 21 dB

  3. 3

    Reserve the stated uncertainty allowance. Available margin = 21 dB − 12 dB = 9 dB

Move distance around the reviewed 1 km case

Try The widget anchors the chapter’s 118 dB path at 1 km and uses its n = 3.2 distance example. Predict whether doubling distance consumes less than, equal to, or more than the 9 dB available margin.

1.00 km
Distance
Modelled path loss
Receive power
Available margin
Model verdict

Ready: predict the remaining margin, then calculate.

Observe At 1 km, the readouts return 118 dB path loss, −105 dBm receive power, and 9 dB available margin. At 2 km, the model adds 32log10(2) = 9.63 dB, just exhausting that reserve.

Explain Distance is inside a logarithm, but the 10n multiplier matters. With n = 3.2, one doubling costs slightly more than the entire 9 dB buffer.

Check yourself

Why does doubling distance add about 6 dB in free space when n = 2?
Answer: 10n log10(2) = 20 × 0.3010 = 6.02 dB.
Why is raw margin −105 − (−126) = 21 dB rather than −231 dB?
Answer: Margin is the distance between two levels. Subtracting the negative sensitivity becomes addition: −105 + 126 = 21.
Does the 9 dB calculation prove reliable packet delivery?
Answer: No. It exposes the assumptions to test. Terrain, walls, fading distribution, interference, noise floor, antenna orientation, weather, diffraction, polarisation, and retries can change the field result.
Technical boundaries. The distance widget extends the chapter’s 118 dB path anchor with a deterministic n = 3.2 median model. It omits terrain and wall geometry, fading distribution, interference, noise-floor variation, antenna mismatch and orientation, weather, diffraction, polarisation, and packet-level retries.

What the audit buys you: the spreadsheet becomes a signed, testable story. If field RSSI, signal-to-noise ratio, retries, or delivery disagree, you know which gain, loss, reserve, or environment assumption to revisit.

Every number above is taken from the companion chapter’s link ledger and re-derived step by step.