Math Bridge: Build the Path-Loss and Margin Ledger

A field team has a real problem to settle: Build the Path-Loss and Margin Ledger They must decide what happens before they change modelled distance on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is modelled distance. The middle card uses this page's rule. The green card is received power. Follow the arrows: set the input, use the rule, then read the result and its unit.

The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.

Follow the arrows. Path loss comes from 40 + 10(3.2)log10(d/1); the received-power and reserve readouts then use that exact ledger result.

Derive the baseline in four moves

1
Name the input. The chapter baseline for modelled distance is 30.
2
Name the rule. PL(d) = PL(d₀) + 10n log10(d/d₀); M_available = P_rx - P_sens - M_reserved
3
Put in the chapter value. Set modelled distance to 30. The page rule gives received power as -74.3 dBm.
4
Read the result. Keep dBm next to the value. Use it only within the limits on this page.

Predict, then change modelled distance

Try Predict what happens to received power. Move one control, calculate, then check your idea.

Observe Path loss comes from 40 + 10(3.2)log10(d/1); the received-power and reserve readouts then use that exact ledger result. Reset to 30 and compare received power.

Explain Only modelled distance moves here. The other chapter values stay fixed.

Check yourself

What should you do before you trust the result?

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

What does this small model leave out?

Answer: Only modelled distance moves. Field effects named in the page limits stay fixed.

1. Begin with the physical story

A link budget is ordinary bookkeeping in decibels. Transmit power and antenna gains are deposits. Cable, path, installation, and miscellaneous losses are withdrawals. Receiver sensitivity is the minimum balance, and the reserve protects the link when the site changes.

Phoebe: A logarithm answers “what power must the base be raised to?” Because log(ab) = log(a)+log(b), multiplied physical ratios become added dB rows. Distance growth d/d₀ enters the log-distance model through log10(d/d₀).

2. Put names and units on the maths

Keep the units beside every number. They are an error detector: only like units can be added or subtracted.

Symbol	Meaning	Unit
PL(d₀)	known loss at a reference distance	dB
n	measured environment exponent	no unit
P_rx	power predicted at the receiver	dBm
M_available	margin after sensitivity and reserve	dB

3. Derive it with every move named

PL(d) = PL(d₀) + 10n log10(d/d₀); M_available = P_rx − P_sens − M_reserved

Choose a referenceUse the chapter's 2.4 GHz loss PL(1 m) ≈ 40 dB.

Form the distance ratioAt 30 m, d/d₀ = 30/1 = 30.

Apply the environment slopePL(d) = 40 + 10n log10(30).

Build received powerAdd transmit and antenna gains; subtract cable, path, and miscellaneous losses.

Form raw marginM_raw = P_rx − P_sens.

Reserve variationM_available = M_raw − M_reserved.

4. Reproduce the chapter's numbers

At 30 m, free space n = 2.0 gives 40 + 20 log10(30) = 69.5 dB. The chapter's light-indoor n = 3.2 gives 40 + 32 log10(30) = 87.3 dB, spending 17.8 dB more before a separate wall row.

Its reviewed ledger separately uses a 118 dB path estimate: 14 + 0 − 1 + 3 − 1 − 118 − 2 = −105 dBm. Against −126 dBm sensitivity, raw margin is 21.0 dB; after a 12 dB reserve, 9.00 dB remains.

5. Try the formula

TryMove the modelled distance from 1 m to 100 m; pause at the chapter's 30 m indoor example.

ObserveObserve that each equal distance addition does not add equal loss: the formula responds to the distance ratio on a logarithmic scale.

ExplainPath loss comes from 40 + 10(3.2)log10(d/1); the received-power and reserve readouts then use that exact ledger result.

Technical boundaries.

This small widget varies one named input and holds the chapter constants fixed.

The honesty boundary below names what it does not model: Needs separate evidence

Use field evidence or a deeper model before release.

6. What the result buys you

The slider is a model review, not a range promise. It makes the effect of distance and n visible, but the chapter's release ledger must substitute measured antenna, installation, environment, receiver mode, and fade evidence. A model can choose survey points; only the survey can validate the site.

7. Check yourself

Try each question before revealing the answer.

1. Why does doubling distance add about 6 dB in free space?

Answer: 20 log10(2) = 6.02 dB.

2. What is the chapter's indoor loss at 30 m?

Answer: 40 + 10(3.2)log10(30) = 87.3 dB.

3. What remains after the reviewed ledger's 21 dB raw margin reserves 12 dB?

Answer: 21 − 12 = 9.00 dB.

Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

40 dB reference: Gain, loss, margin, or level ratio
30 m distance: Distance, wavelength, or size
n = 2.0: Named physical or model constant
3.2 comparisons: Time, interval, or service-life value
ledger terms: Chapter input or worked result
−126 dBm sensitivity: Radio power level
12 dB reserve come from the chapter: Gain, loss, margin, or level ratio
The slider keeps n fixed and omits shadowing Xσ: Chapter input or worked result
walls: Chapter input or worked result
Fresnel obstruction: Chapter input or worked result
antenna pattern: Sensor scale, pressure, or digital result
interference: Chapter input or worked result
receiver-specific RSSI: Chapter input or worked result

Under the Hood explains why those assumptions must be measured.