Math Bridge: Contact Lens Near-Field Storage

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Math BridgeApplicationsStruggle-friendly runway

How does a battery-free lens bridge time away from its reader?

Use wavelength to classify the link, then turn sensing and sleep energy into a storage-capacitor requirement.

Light Lucy, the optical sensing guideLight Lucy guides
The one targetSize a harvested-energy buffer for a reader gap.
The chapter case13.56 MHz, 5 cm, 30 minutes, 2 microwatts, and 1.5 V.
What it buys youA lens constraint expressed in coupling and capacitance, not mAh.

A field team faces an unresolved physical question: How does a battery-free lens bridge time away from its reader? They must answer it before changing reader gap 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 reader gap. 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.

Reader gap 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. At constant load, stored energy grows with time; at constant voltage, capacitance must grow with that energy.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for reader gap is 30.

  2. 2

    Name the relationship. λ=3.00x10^8/13.56x10^6=22.1 m dnf=λ/(2π)=3.52 m; 5 cm is 1.42% of dnf Esense=6x2 uWx0.150 s=1.80 uJ Esleep=0.2 uWx1800 s=360 uJ C=2x361.8 uJ/(1.5 V)²=322 uF I≤2 uW/0.6 V=3.33 uA

  3. 3

    Substitute the chapter fixture. Set reader gap to 30. 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 reader gap

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

30
Chapter baseline
Wavelength

Observe At constant load, stored energy grows with time; at constant voltage, capacitance must grow with that energy. Reset the control to 30 and compare wavelength.

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

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

1. Start with the physical story

The reader and lens behave more like loosely coupled coils than distant antennas. When the reader is absent, a capacitor must supply sensing and sleep energy. Its voltage stores energy quadratically.

Light Lucy: First decide which physical regime applies; only then choose the energy equation.

2. Name every algebra move

1

Find wavelengthDivide light speed by 13.56 MHz.

2

Mark near fieldDivide wavelength by 2π and compare with 5 cm.

3

Count readingsDivide the reader gap by the five-minute reading interval.

4

Add energyUse power times time for sensing and sleep.

5

Solve for capacitanceRearrange E=CV²/2 into C=2E/V².

6

Bound electrode currentDivide 2 microwatts by 0.6 V.

3. Reproduce the chapter case

λ=3.00×10^8/13.56×10^6=22.1 m
dnf=λ/(2π)=3.52 m; 5 cm is 1.42% of dnf
Esense=6×2 µW×0.150 s=1.80 µJ
Esleep=0.2 µW×1800 s=360 µJ
C=2×361.8 µJ/(1.5 V)²=322 µF
I≤2 µW/0.6 V=3.33 µA

The storage result is large for a lens-scale device, so the gap, sleep load, voltage window, and leakage all matter.

4. Try one real input

TryChange the time away from the reader and predict how storage capacitance scales.

Reader gap
Wavelength
Near-field boundary
5 cm share
Readings in gap
Sensing energy
Sleep energy
Total energy
Storage capacitance
Electrode current bound

ObserveDoubling the gap nearly doubles required capacitance because sleep energy dominates and voltage stays fixed.

ExplainAt constant load, stored energy grows with time; at constant voltage, capacitance must grow with that energy.

Technical boundaries.

This is an ideal energy-balance screen.

Coupling
The near-field boundary classifies regime; it does not calculate mutual inductance or delivered power.
Capacitor
Leakage, rectifier loss, ESR, allowed voltage drop, and geometry are omitted.
Biosensor
Power divided by bias is only an upper current bound, not analyte calibration.

Correct, not complete: a sufficient ideal capacitor does not prove a safe or clinically useful lens.

5. Use the result in the design

Measure harvested power versus alignment and separation, then integrate the real load and choose an allowed capacitor voltage window.

6. Record the evidence state

Keep coil geometry, distance, alignment, frequency, rectifier efficiency, storage capacitance, leakage, voltage limits, sensing cadence, electrode bias, temperature, and exposure limits.

7. Check yourself

Why is a 5 cm lens-reader path not a far-field link?
Answer: It is only about 1.42% of the 3.52 m near-field boundary at 13.56 MHz.
Why does a longer reader gap need more capacitance?
Answer: The load consumes more energy, and C=2E/V² at fixed storage voltage.
Does 3.33 microamps predict glucose concentration?
Answer: No. It only bounds current from circuit power and bias voltage.
Honesty boundary.

The arithmetic extends the chapter's illustrative frequency, gap, power, and voltage values.

Coupling
The near-field boundary classifies regime; it does not calculate mutual inductance or delivered power.
Capacitor
Leakage, rectifier loss, ESR, allowed voltage drop, and geometry are omitted.
Biosensor
Power divided by bias is only an upper current bound, not analyte calibration.

Correct, not complete: a sufficient ideal capacitor does not prove a safe or clinically useful lens.

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