Math Bridge: Path-Loss Exponents in a Packet Simulator

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Math BridgeCore NetworkingStruggle-friendly runway

How does one path-loss exponent remove 13 dB?

One thread from inverse-square spreading to realistic margin and packet energy.

Pete, the packet guidePete guides
The one targetRead the exponent as an environmental loss rate.
The chapter case2.4 GHz, 20 m, +10 dBm, −85 dBm; n=2 versus n=3.
What it buys youStop a simulator from promising free-space reliability indoors.

A field team faces an unresolved physical question: How does one path-loss exponent remove 13 dB? They must answer it before changing path-loss exponent on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is path-loss exponent. The middle card applies this page's relationship. The green card is free-space loss. 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 added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Path-loss exponent changes free-space loss An input card leads through the page relationship to the free-space loss result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The exponent changes the modeled path, not the bytes. A weaker path creates retries, and retries repeat the per-packet energy cost.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for path-loss exponent is 3.

  2. 2

    Name the relationship. PL(n)=FSPL+10(n-2)log10(20/1); M=10-PL-(-85)

  3. 3

    Substitute the chapter fixture. Set path-loss exponent to 3. The page ledger gives free-space loss as 66.07 dB.

  4. 4

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

Predict, then change path-loss exponent

Try Predict the direction of free-space loss. Move one control, calculate, then check your prediction.

3
Chapter baseline
Free-space loss

Observe The exponent changes the modeled path, not the bytes. A weaker path creates retries, and retries repeat the per-packet energy cost. Reset the control to 3 and compare free-space loss.

Explain Only path-loss exponent moves here. The other chapter fixtures 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 path-loss exponent moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only path-loss exponent. It replays the chapter fixture; it is not a field prediction.

1. Free space is the clean baseline

A wave spreads over a growing sphere. At twice the distance, the same power covers four times the area. In dB, that inverse-square law becomes a 20 log10 distance term.

Pete: Free space is useful because it is a best case, not because a factory floor behaves like empty sky.

2. Let the exponent describe clutter

1

Build wavelengthλ=c/f.

2

Find free-space lossFSPL=20log10(4πd/λ).

3

Charge for clutterExtra loss over free space is 10(n−2)log10(d/d0).

3. Turn loss into margin

PL=FSPL+10(n−2)log10(d/d0); M=Pt−PL−Srx

Path loss is subtracted from transmit power. Sensitivity is a negative threshold, so margin subtracts that negative number.

4. Try the path-loss exponent

PL(n)=FSPL+10(n−2)log10(20/1); M=10−PL−(−85)

TryMove n from open free space toward an obstructed industrial path. Packet sizes and energy stay fixed so path and protocol costs remain separate.

Wavelength
Free-space loss
Extra environmental loss
Total path loss
Link margin
14-byte packet
16-byte packet
18-byte packet
No-checksum annual
Checksum annual
CRC annual

ObserveAt n=3, 20 m free-space loss is 66.1 dB and clutter adds 13.0 dB, leaving 15.9 dB rather than 28.9 dB margin. Annual packet energy remains 39.2, 44.9, and 50.5 mWh before retries.

ExplainThe exponent changes the modeled path, not the bytes. A weaker path creates retries, and retries repeat the per-packet energy cost.

Technical boundaries.

One log-distance exponent compresses walls, shelves, reflections, and placement into an average slope.

It does not model fast fading, interference, antenna orientation, shadow maps, collision probability, retry policy, or confidence intervals
Needs separate evidence

Use field evidence or a deeper model before release.

5. Reproduce the chapter values

At 2.4 GHz, λ=0.125 m. At 20 m, FSPL=66.0666… dB, reported as 66.1 dB. Moving from n=2 to n=3 adds 10log10(20)=13.0 dB, so margin falls from 28.9 to 15.9 dB. The 14-, 16-, and 18-byte packets cost about 0.00124, 0.00142, and 0.00160 µWh.

6. Prove the model rather than trust it

Record simulator model and seed, measured or justified n, reference loss, distance, antenna placement, interference, packet success, retries, latency distribution, energy accounting, warm-up, and confidence interval across independent runs.

7. Check yourself

Why is free space a lower-loss bound?
Answer: It includes ideal geometric spreading but none of the extra obstruction, absorption, scattering, or interference of a real indoor site.
Does n=3 mean every packet loses exactly 13 dB extra?
Answer: No. It describes the model's mean distance slope; individual locations and times still vary.
Why connect margin to energy?
Answer: A weak link often retries, and each retry pays the packet transmission energy again.
Honesty boundary.

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

10 mW/+10 dBm
Radio power level
250 kbps
Chapter input or worked result
86,400 packets/day
Frequency, sample rate, or event rate
2.4 GHz
Frequency, sample rate, or event rate
20 m
Distance, wavelength, or size
−85 dBm
Radio power level
n=3
Named physical or model constant
66.1 dB
Gain, loss, margin, or level ratio
13.0 dB
Gain, loss, margin, or level ratio
28.9 dB
Gain, loss, margin, or level ratio
15.9 dB
Gain, loss, margin, or level ratio
packet-energy
Sensor scale, pressure, or digital result

They do not validate a particular industrial site.

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