Math Bridge: RFID Field Boundaries

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Math BridgeRFID bandsField boundary

When does antenna gain become the right model?

Compare LF and HF near-field depth, then screen the UHF trade between solid angle and forward range.

Eddie, the electronics guideEddie guides
The one targetConnect frequency to the model that fits the read zone.
The chapter case125 kHz LF, 13.56 MHz HF, and a 6-to-9 dBi UHF portal comparison.
What it buys youA disciplined choice between coil-coupling and far-field gain language.

A field team faces an unresolved physical question: When does antenna gain become the right model? They must answer it before changing uhf 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 uhf gain. The middle card applies this page's relationship. The green card is lf wavelength. 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.

UHF gain changes lf wavelength An input card leads through the page relationship to the lf wavelength result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Holding EIRP fixed trades angular coverage against forward reach; it does not create radiated power.

Derive the baseline in four named moves

  1. 1

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

  2. 2

    Name the relationship. dnf,LF = 2400/(2π) = 382 m dnf,HF = 22.12/(2π) = 3.52 m G9 = 10^0.9 = 7.94; ohm9 = 4π/7.94 = 1.58 sr range9/range6 = √(7.94/3.98) = 1.41

  3. 3

    Substitute the chapter fixture. Set uhf gain to 9. The page ledger gives lf wavelength as 2400 m.

  4. 4

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

Predict, then change uhf gain

Try Predict the direction of lf wavelength. Move one control, calculate, then check your prediction.

9
Chapter baseline
LF wavelength

Observe Holding EIRP fixed trades angular coverage against forward reach; it does not create radiated power. Reset the control to 9 and compare lf wavelength.

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

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

1. Start with the physical story

LF and HF reads sit deep inside reactive near fields, while a UHF portal operates where directional gain and EIRP are useful abstractions.

Eddie: Frequency does not merely label a band; it changes wavelength and therefore the physical scale of the field model.

2. Name every algebra move

1

Find wavelengthUse λ = c/f for LF and HF.

2

Mark depthUse dnf = λ/(2π), then compare it with the practical read.

3

Linearise gainUse G = 10^(dBi/10).

4

Screen coverageUse Ω = 4π/G and range ratio √(G/Gref).

3. Reproduce the chapter case

dnf,LF = 2400/(2π) = 382 m
dnf,HF = 22.12/(2π) = 3.52 m
G9 = 10^0.9 = 7.94; Ω9 = 4π/7.94 = 1.58 sr
range9/range6 = √(7.94/3.98) = 1.41

The near-field ratios explain the coil-coupled reads; the gain ratio only screens the UHF portal direction.

4. Try one real input

TryMove UHF portal gain while the LF/HF cases and the 36 dBm EIRP ceiling stay fixed.

UHF gain
LF wavelength
LF near-field boundary
LF boundary/read ratio
HF wavelength
HF near-field boundary
HF boundary/read ratio
Linear gain
Ideal solid angle
Sphere fraction
Forward range / 6 dBi
Conducted power
Conducted watts

ObserveAt 9 dBi the ideal sphere fraction is 12.6%, forward range is 1.41 times the 6 dBi screen, and conducted power falls to 27 dBm.

ExplainHolding EIRP fixed trades angular coverage against forward reach; it does not create radiated power.

Technical boundaries.

The λ/(2π) boundary and Ω = 4π/G relation are first-order model-selection screens.

Near field
Real loop size, turns, Q, coupling coefficient, resonance, load, orientation, and conductive material set LF/HF reads.
Far field
Published E/H patterns, efficiency, polarisation, cable loss, multipath, tag threshold, and regulation set UHF reads.
Geometry
Solid angle does not directly predict a rectangular doorway width.

Correct, not complete: this ledger does not choose a band or antenna.

5. Use the result in the lab

Choose the appropriate field model, then test the real coil or portal geometry with intended and excluded objects at its claimed boundary.

6. Record the evidence state

Keep band, lawful region, antenna or coil, matching state, power, geometry, object material, orientation, reads, and exclusions.

7. Check yourself

Is a 0.3 m LF read near the 125 kHz far field?
Answer: No. The λ/(2π) screen is about 382 m away.
Does 9 dBi permit more EIRP under the same ceiling?
Answer: No. Conducted power must fall as antenna gain rises.
Does 1.58 sr define the exact portal footprint?
Answer: No. It is an ideal solid-angle screen, not a measured antenna pattern.
Honesty boundary.

The ledger distinguishes model regimes and exposes one ideal UHF gain trade.

Computed
Wavelengths, boundary depths, ideal gain, solid angle, and forward ratio are reproducible.
Specified
Coil and antenna data, regulation, and installation geometry replace the examples.
Observed
Coupling, read-zone traces, misses, duplicates, and spillover decide the system.

Correct, not complete: use the right physical model before optimising the read zone.

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