Math Bridge: RFID Beam Footprint

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Math BridgeRFID labBeam screen

What does antenna gain reshape at a read-zone boundary?

Turn a panel's gain into an ideal square-beam width, solid angle, conducted power, and footprint before testing real boundary tags.

Eddie, the electronics guideEddie guides
The one targetConnect gain to an ideal beam-footprint and EIRP-split screen.
The chapter case6 dBi versus 9 dBi panels at a 36 dBm EIRP ceiling and 1 m zone distance.
What it buys youA placement hypothesis for controlled boundary-tag trials.

A field team faces an unresolved physical question: What does antenna gain reshape at a read-zone boundary? They must answer it before changing antenna gain 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 antenna gain. The middle card applies this page's relationship. The green card is linear gain. 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.

Antenna gain changes linear gain An input card leads through the page relationship to the linear gain result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Higher ideal directivity concentrates the same ceiling into less angle. The geometric estimate only tells the lab where to test.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for antenna gain is 9.

  2. 2

    Name the relationship. D_6 = 10^0.6 = 3.98 → θ_6 = sqrt(41253/3.98) = 101.8° D_9 = 10^0.9 = 7.94 → θ_9 = 72.1° ohm_9 / ohm_6 = 3.98 / 7.94 = 0.501 P_tx,9 = 36 - 9 = 27 dBm

  3. 3

    Substitute the chapter fixture. Set antenna gain to 9. The page ledger gives linear gain as 7.94 times.

  4. 4

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

Predict, then change antenna gain

Try Predict the direction of linear gain. Move one control, calculate, then check your prediction.

9
Chapter baseline
Linear gain

Observe Higher ideal directivity concentrates the same ceiling into less angle. The geometric estimate only tells the lab where to test. Reset the control to 9 and compare linear gain.

Explain Only antenna gain 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 antenna gain moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only antenna gain. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

Antenna gain does not create energy. It concentrates radiation into a smaller solid angle, changing which boundary tags sit inside the favoured region.

Eddie: A calculated beam edge is a test hypothesis; real patterns, reflections, tags, and mounting decide the observed zone.

2. Name every algebra move

1

Linearise gainConvert dBi with 10^(G/10).

2

Estimate beamUse sqrt(41253/D) for the ideal square-beam width.

3

Find solid angleUse Ω = 4π/D.

4

Hold EIRPSubtract antenna gain from the ceiling to find conducted power.

5

Screen geometryProject the half-power width at the zone and compare a placement shift.

3. Reproduce the chapter case

D_6 = 10^0.6 = 3.98 → θ_6 = sqrt(41253/3.98) = 101.8°
D_9 = 10^0.9 = 7.94 → θ_9 = 72.1°
Ω_9 / Ω_6 = 3.98 / 7.94 = 0.501
P_tx,9 = 36 − 9 = 27 dBm

At one metre, the square-beam footprint screen narrows from about 2.46 m to 1.46 m. A 10 cm shift changes the look angle by 5.71°, enough to motivate a measured boundary retest.

4. Try one real input

TryMove panel gain while the EIRP ceiling stays fixed. Beam, footprint, solid angle, and conducted power recompute together.

Antenna gain
Linear gain
Square-beam width
Ideal solid angle
Solid angle / 6 dBi
Conducted power
Conducted watts
1 m beam footprint
Footprint / 6 dBi
10 cm placement angle
Shift / half-beam

ObserveAt 9 dBi, the ideal solid angle is half the 6 dBi value, the 1 m footprint is about 1.46 m, and conducted power is 27 dBm.

ExplainHigher ideal directivity concentrates the same ceiling into less angle. The geometric estimate only tells the lab where to test.

Technical boundaries.

This is an ideal directivity and geometry screen, not an antenna pattern.

Antenna
Gain is treated as directivity; efficiency, polarisation, sidelobes, cable loss, and published E/H patterns are separate.
Geometry
The square-beam approximation does not predict the exact −3 dB contour or near-field behavior.
Zone
Multipath, materials, tags, power limits, collisions, and middleware filtering must be tested.

Correct, not complete: this ledger does not select an antenna or release a read zone.

5. Use the result in the lab

Place controlled intended and excluded tags around the predicted edge, then move or tilt the antenna one variable at a time and retain the trace.

6. Record the evidence state

Keep antenna model, E/H beamwidths, gain, cable loss, power, mounting pose, tag set and orientation, geometry, trace, filters, and retest trigger.

7. Check yourself

Does 3 dB more gain double the regulatory EIRP?
Answer: Not when the ceiling is held fixed; conducted power is reduced by the same 3 dB.
Is the 72° value the panel's guaranteed beamwidth?
Answer: No. It is an ideal square-beam estimate; use the antenna data and measured pattern.
Can the footprint alone release the zone?
Answer: No. Intended, missed, duplicate, and stray reads must pass controlled trials.
Honesty boundary.

The ledger turns gain into a testable placement hypothesis.

Computed
Ideal directivity, square beam, solid angle, EIRP split, and footprint are reproducible.
Specified
Published antenna patterns and lawful power settings replace the ideal assumptions.
Observed
Boundary reads, misses, duplicates, spillover, and repeatability decide the lab outcome.

Correct, not complete: measure the zone before accepting placement.

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