Math Bridge: How can one exponent collapse 315 m to 27 m?

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How can one exponent collapse 315 m to 27 m?

Connect wavelength, reference path loss, path exponent, link budget, battery power loss, and ideal radio range for the ad hoc lab.

Packet Pete, the guidePacket Pete guides
The one targetSolve a link budget for ideal range and expose two different causes of shrinkage.
The chapter case2.4 GHz, 0 dBm TX, −90 dBm sensitivity, n=2 versus 3.5, then −3 dB.
What it buys youA simulator setting with a named physical cause.

A technician must decide whether free-space range is safe before changing fade-margin deduction on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is fade-margin deduction. The middle card applies this page's rule. The green card is free-space range. 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 fade-margin deduction, so the numeric fixture does not switch without explanation.

Fade-margin deduction changes free-space range An input card leads through the rule range = 10^((49.95 dB - fade margin) / 20) to the free-space range result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Every added fade-margin decibel reduces the range allowed by the same budget.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 0 dB.

  2. 2

    Name the relationship. range = 10^((49.95 dB - fade margin) / 20)

  3. 3

    Substitute with units. 10^(49.95 / 20) = 314.4 m

  4. 4

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

Predict, then change fade-margin deduction

Try Predict the direction of range = 10^((49.95 dB - fade margin) / 20). Test another fade-margin deduction, then compare free-space range.

0 dB
Chapter baseline
Free-space range

Observe Every added fade-margin decibel reduces the range allowed by the same budget. Reset fade-margin deduction to 0 and compare free-space range.

Explain Every added fade-margin decibel reduces the range allowed by the same budget.

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 fade-margin deduction moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only fade-margin deduction. It is not a field prediction.

1. Start with the physical question

Solve a link budget for ideal range and expose two different causes of shrinkage. A simulator setting with a named physical cause.

Packet Pete: Keep the units and the model boundary visible from the first line.

2. Name every algebra move

1

Build the reference lossPL0=20log10(4πd0/λ).

2

Find usable budgetB=Pt−sensitivity−margin.

3

Isolate the distance logarithmlog10(d/d0)=(B−PL0)/(10n).

4

Undo log base tend=d0×10^((B−PL0)/(10n)).

5

Spend 3 dB separatelyRepeat with B−3 for the battery-sag power cut.

3. Reproduce the chapter case

B=90 dB; d=10^((B−PL0)/(10n)); PL0=40.05 dB
n=3.5: d=26.7 m; n=2: d=315 m; collapse=11.8×; 3 dB-cut open range=223 m

The arithmetic reproduces the chapter case while keeping its assumptions explicit.

4. Try the controlling input

TryMove the control and watch every displayed result come from the shown formula.

Exponent n
1 m reference
Link budget
Range at chosen n
Open-field range
Obstruction collapse
Sagged open range
Sagged share

ObserveAt n=3.50, the ideal range is 26.7 m versus 314.5 m in free space, while a separate 3 dB cut leaves 222.6 m in free space.

ExplainObstruction changes the distance exponent; battery sag spends transmit-power budget. Both shrink range, but they are different fault causes.

Technical boundaries.

This compact engine isolates one relationship; it is not a deployment certificate.

Propagation
A single exponent omits shadowing, fading, antenna orientation, and interference
Radio
Voltage sag is represented as a specified 3 dB RF power cut
Range
Sensitivity crossing is not a packet-delivery or latency guarantee

Measure the real system and reopen the decision when its inputs change.

5. Name the injected fault

When the lab shrinks range, state whether it represents obstruction, transmit-power loss, antenna damage, interference, or a deliberate policy. Capture the corresponding evidence.

6. Keep the fault record

Record frequency, transmit power, sensitivity, exponent, margin, battery state, antenna state, range setting, packet results, intended cause, owner, and retest trigger.

7. Check yourself

Why is the free-space estimate about 315 m?
Answer: 10^((90−40.05)/20) is about 314.5 m.
Why does n=3.5 collapse it to about 26.7 m?
Answer: The same 49.95 dB distance budget is divided by 35 instead of 20 before undoing the logarithm.
Does the 222.6 m sagged result predict a real deployment?
Answer: No. It isolates a 3 dB power cut in ideal free space; measured propagation and packet evidence still govern.
Honesty boundary.

The worked values are traceable chapter examples or explicitly labelled teaching assumptions.

0 and −90 dBm
Catalog-typical teaching radio values
n=3.5
Explicit obstructed example
3 dB
Illustrative battery-driven RF power cut

Correct, not complete: field evidence still decides acceptance.

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