Math Bridge: NFC Loop Coupling

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Math BridgeNFC tapLoop coupling

Why does a few centimetres change an NFC tap?

Use a simple coaxial-loop model to expose the geometry that protocol simulation leaves out.

Eddie, the electronics guideEddie guides
The one targetConnect loop separation to mutual inductance and induced-voltage ratio.
The chapter case13.56 MHz, 2.5 cm reader loop, 1.5 cm tag loop, 1-to-4 cm separation.
What it buys youA geometry hypothesis for physical tap testing beyond software logic.

A field team faces an unresolved physical question: Why does a few centimetres change an NFC tap? They must answer it before changing separation 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 separation. The middle card applies this page's relationship. The green card is 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.

Separation changes wavelength An input card leads through the page relationship to the wavelength result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Being inside the near field does not guarantee enough coupling; loop geometry still drives mutual inductance.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for separation is 4.

  2. 2

    Name the relationship. λ = 3x10⁸ / 13.56x10⁶ = 22.12 m rnf = 22.12/(2π) = 3.52 m M1cm = 14.20 nH; M4cm = 2.64 nH M4/M1 = 0.186

  3. 3

    Substitute the chapter fixture. Set separation to 4. The page ledger gives wavelength as 22.12 m.

  4. 4

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

Predict, then change separation

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

4
Chapter baseline
Wavelength

Observe Being inside the near field does not guarantee enough coupling; loop geometry still drives mutual inductance. Reset the control to 4 and compare wavelength.

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

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

1. Start with the physical story

NFC coils behave like loosely coupled transformer windings, so separation and alignment can change induced voltage steeply without any far-field beam.

Eddie: The simulation can replay protocol states; only the physical coils can prove the installed coupling envelope.

2. Name every algebra move

1

Mark the regimeUse λ = c/f and rnf = λ/(2π).

2

Set geometryConvert loop radii and separation to metres.

3

Estimate couplingEvaluate the coaxial-loop mutual-inductance approximation.

4

Compare voltageWith frequency and current fixed, induced voltage follows M.

3. Reproduce the chapter case

λ = 3×10⁸ / 13.56×10⁶ = 22.12 m
rnf = 22.12/(2π) = 3.52 m
M1cm = 14.20 nH; M4cm = 2.64 nH
M4/M1 = 0.186

The 4 cm state retains about 18.6% of the 1 cm coupling in this aligned, single-turn approximation.

4. Try one real input

TryMove aligned loop separation while frequency, radii, turns, and reader current stay fixed.

Separation
Wavelength
Near-field boundary
Boundary/separation
Mutual inductance
1 cm reference M
Coupling drop
Coupling retained
Relative induced voltage

ObserveAt 4 cm the tap remains about 88 near-field separations inside the λ/(2π) boundary but retains only 18.6% of the aligned 1 cm coupling.

ExplainBeing inside the near field does not guarantee enough coupling; loop geometry still drives mutual inductance.

Technical boundaries.

This is a coaxial, single-turn, small-loop approximation.

Geometry
Tilt, lateral offset, rectangular traces, turns, ferrite, shielding, and nearby metal change mutual inductance.
Circuits
Q, resonance, tuning, load modulation, drive current, rectifier threshold, and phone case remain separate.
Protocol
Mutual inductance alone does not predict transaction success or timing.

Correct, not complete: this ledger does not certify NFC range or antenna shape.

5. Use the result in the lab

Build a separation, tilt, and lateral-offset matrix with the real reader, tag, mounting stack, and transaction trace.

6. Record the evidence state

Keep coil drawings, turns, tuning, drive, load, case and metal stack, poses, success trace, errors, and retest trigger.

7. Check yourself

Is a 4 cm NFC tap in the far field?
Answer: No. The λ/(2π) screen is about 3.52 m.
Does four times separation mean one quarter of the coupling?
Answer: No. This geometry gives about 0.186 of the 1 cm value.
Can this model approve an installed read range?
Answer: No. Alignment, tuning, materials, circuits, and protocol evidence remain.
Honesty boundary.

The ledger exposes the geometry that a protocol-only NFC simulation cannot reproduce.

Computed
Wavelength, boundary depth, ideal mutual inductance, and relative voltage are reproducible.
Specified
Measured coil, tuning, load, and pose data replace the aligned-loop assumptions.
Observed
Field voltage, transaction success, errors, and repeatability decide the tap envelope.

Correct, not complete: carry the simulation result into a physical coupling trial.

Return to NFC Simulation Lab with the coupling maths unpacked →