Math Bridge: Two Meanings of Gain

See the relationship before changing it

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 measured calibration span, so the numeric fixture does not switch without explanation.

Derive the baseline in four named moves

1
Name the input. The chapter baseline is 96 units.
2
Name the relationship. corrected span = measured span x 100 / 96
3
Substitute with units. 96 x 100 / 96 = 100.00 units
4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change measured calibration span

Try Predict the direction of corrected span = measured span x 100 / 96. Test another measured calibration span, then compare corrected true span.

Observe The calibration factor corrects span; antenna dBi stays a separate fixture. Reset measured calibration span to 96 and compare corrected true span.

Explain The calibration correction expands a measured span to compensate for its compressed response, so applying the same correction factor makes corrected span grow proportionally.

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 measured calibration span moves here. Field effects named in the technical boundary stay fixed.

1. Refuse the bare word gain

Calibration gain multiplies a measured span after sensing. Antenna gain describes how transmitted power is concentrated in space. The same word does not make the quantities interchangeable.

Phoebe: Write the formula and unit beside gain before doing arithmetic.

2. Correct the sensor span

gcal=true span/measured span=100/96=1.0417

This dimensionless scale factor stretches the measured span by about 4.17%. It acts on sensor readings, not on radio power.

3. Concentrate radio power

Glinear=10^(GdBi/10); Ω=4π/G; EIRPdBm=PtdBm+GdBi

A directional ratio raises power density in some directions by reducing idealised coverage in others.

4. Try antenna gain

gcal=T/M; G=10^(dBi/10); EIRP=Pt+GdBi; Ω=4π/G

TryMove antenna gain while the chapter's 0 dBm transmitter, 1.5 dBi reference antenna, and 100/96 sensor calibration stay fixed.

ObserveAt 5 dBi, the antenna ratio is 3.16×, EIRP is 5.00 dBm or 3.16 mW, ideal sphere coverage is 31.6%, and ideal range is about 1.50× the 1.5 dBi case. Sensor gain remains 1.0417.

ExplainThe antenna control changes direction, EIRP, and the ideal range comparison. It never enters the 100/96 calibration correction because that belongs to another measurement chain.

Technical boundaries.

The sensor correction keeps the 100/96 calibration factor separate from the antenna comparison, which assumes lossless power redistribution and free-space range scaling. Antenna nulls, polarisation, cable loss, matching, surroundings, receiver sensitivity, regulations, and link margin are not included.

pattern nulls: Needs separate evidence
polarisation: Needs separate evidence
cable loss: Needs separate evidence
matching: Needs separate evidence
environment: Needs separate evidence
receiver sensitivity: Needs separate evidence
regulations: Needs separate evidence
link margin: Needs separate evidence

Use field evidence or a deeper model before release.

5. Check the 1.5 dBi reference

10^(1.5/10)=1.41×; EIRP=1.50 dBm=1.41 mW

The 5 dBi whip has 3.16/1.41 times the power-density ratio in its favored direction, so ideal range scales by the square root to about 1.50×.

6. Write two ledger rows

Record calibration gain with true span, measured span, units, and residuals. Record antenna gain with dBi, pattern, frequency, cable loss, and EIRP. Never add the two gains.

7. Check yourself

Why is 100/96 not 0.18 dB?

Answer: This chapter defines a linear calibration multiplier applied to sensor readings, not an RF power ratio.

Why does 5 dBi mean 3.16×?

Answer: Power ratios use 10^(dB/10), so 10^0.5 = 3.16.

Does higher antenna gain create free energy?

Answer: No. The ideal model concentrates radiation into less solid angle; real systems also have losses.

Honesty boundary.

The sensor span 100/96: True-to-measured sensor-span correction ratio
0 dBm transmitter: Fixed radio transmit power
1.5 dBi chip antenna: Reference chip-antenna directional gain
5 dBi whip reproduce the chapter's teaching case: Comparison whip-antenna directional gain

The range result is an ideal comparison, not a deployment promise or compliance calculation.