Math Bridge: TPMS diaphragm and cell

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Math BridgeDesign MethodologyStruggle-friendly runway

How does tyre pressure become a voltage and a ten-year claim?

Follow pressure through silicon strain and bridge voltage, then keep cell energy and self-discharge in their own ledger.

Blueprint Bina, the design guideBlueprint Bina guides
The one targetKeep the sensing and battery chains traceable.
The chapter case3 bar; 25 µm diaphragm; 220 mAh cell; ten years.
What it buys youA TPMS datasheet claim with visible assumptions.

See the relationship before changing it

The figure reads from left to right. The blue card is diaphragm gauge factor. The middle card applies this page's rule. The green card is bridge resistance change. 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 model keeps those stated values fixed and changes only diaphragm gauge factor, so the numeric fixture does not switch without explanation.

Diaphragm gauge factor changes bridge resistance change An input card leads through the rule resistance change = gauge factor x 488 microstrain x 100 to the bridge resistance change result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Gauge factor converts fixed diaphragm strain into resistance change.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 100.

  2. 2

    Name the relationship. resistance change = gauge factor x 488 microstrain x 100

  3. 3

    Substitute with units. 100 x 488e-6 x 100 = 4.88%

  4. 4

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

Predict, then change diaphragm gauge factor

Try Predict the direction of resistance change = gauge factor x 488 microstrain x 100. Test another diaphragm gauge factor, then compare bridge resistance change.

100
Chapter baseline
Bridge resistance change

Observe Gauge factor converts fixed diaphragm strain into resistance change. Reset diaphragm gauge factor to 100 and compare bridge resistance change.

Explain Gauge factor converts fixed diaphragm strain into resistance change.

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 diaphragm gauge factor moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only diaphragm gauge factor. It is not a field prediction.

1. Pressure cannot enter software directly

Pressure bends a silicon diaphragm. Bending strains piezoresistors, resistance changes unbalance a bridge, and the bridge produces a voltage. The battery is a separate physical chain and must not be used to hide sensor assumptions.

Blueprint Bina: Draw two ledgers—measurement and energy—then join them only in the release record.

2. Name the algebra moves

1

Convert pressure3 bar=300,000 Pa.

2

Calculate strainε=3Pa²(1−ν²)/(4Et²).

3

Calculate deflectionw₀=3Pa⁴(1−ν²)/(16Et³).

4

Reach voltageΔR/R=GFε and Vout≈VexcΔR/R.

5

Derate the cellE=CV/1000 and retained=(1−k)^years.

3. Reproduce the 3 bar case

εedge=488 µε; w₀=1.22 µm=4.88% of thickness
ΔR/R=100×488×10⁻⁶=4.88%
Vout≈3.0×0.0488=146 mV
Ecell=220×3.0/1000=0.660 Wh; retained=0.985¹⁰=86.0%

The pressure control changes strain and voltage. The cell outputs stay fixed because tyre pressure does not change nameplate energy or annual self-discharge.

4. Try the pressure swing

TryMove the teaching pressure from 3 bar toward 1 bar while die geometry stays fixed.

Pressure
Edge strain
Deflection
Thickness ratio
Resistance change
Bridge output
Nameplate energy
Ten-year retained

ObserveIn this small-deflection model, strain, deflection, resistance change, and voltage scale with pressure. Cell energy does not.

ExplainPressure appears once in both plate equations, so changing P changes those outputs linearly while geometry and material stay fixed.

Technical boundaries.

This is a catalog-typical teaching model, not a TPMS qualification model.

Diaphragm
Real die geometry, residual stress, temperature, packaging, and nonlinear calibration vary
Bridge
The simplified voltage relation omits bridge topology, amplification, offsets, and ADC behaviour
Cell
Load pulses, temperature, passivation, cutoff, and aging alter usable energy

Use the named part's datasheet, calibration, environmental tests, and current traces.

5. Test both chains

Pressure-test calibrated units across temperature while separately logging radio pulses, sleep current, cell voltage, and end-of-life behaviour.

6. Record the qualification state

Store die and package, pressure range, calibration, accuracy conditions, excitation, ADC chain, cell lot, duty cycle, temperature, years, and derating rule.

7. Check yourself

What physical quantity changes first?
Answer: Pressure bends the diaphragm and creates strain.
Why keep cell arithmetic separate?
Answer: It answers a different physical question and has different assumptions.
Does 86.0% retained prove ten-year service?
Answer: No. It omits load, temperature, cutoff, passivation, and variability.
Honesty boundary.

The dimensions, materials, gauge factor, and cell values reproduce the chapter's catalog-typical example.

488 µε
Small-deflection circular-plate teaching result
146 mV
Simplified full-swing bridge estimate
86.0%
Self-discharge-only retained fraction

Correct, not complete: these two ledgers do not qualify a TPMS design.

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