Math Bridge: Piezo Tap Rate and Charge

A field team faces an unresolved physical question: Why does a piezo sensor feel a tap but forget a hold? They must answer it before changing piezo force rise time in milliseconds 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 piezo force rise time in milliseconds. The middle card applies this page's relationship. The green card is average source current. 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.

Walk the arrows. The total generated charge stays 200 pC as the press slows, but the current falls and the 100 ms leakage path drains more of it before the peak is reached.

Derive the baseline in four named moves

1
Name the input. The chapter baseline for piezo force rise time in milliseconds is 5.
2
Name the relationship. Q=dF; iavg=Q/Δt; τ=RfCf; Vpeak=(Q/Cf)(τ/Δt)(1-e^(-Δt/τ))
3
Substitute the chapter fixture. Set piezo force rise time in milliseconds to 5. The page ledger gives average source current as 40.0 nA.
4
Read the result. Keep nA beside the value. Use it only inside the technical boundary on this page.

Predict, then change piezo force rise time in milliseconds

Try Predict the direction of average source current. Move one control, calculate, then check your prediction.

Observe The total generated charge stays 200 pC as the press slows, but the current falls and the 100 ms leakage path drains more of it before the peak is reached. Reset the control to 5 and compare average source current.

Explain Only piezo force rise time in milliseconds 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 piezo force rise time in milliseconds moves. Field effects named in the page's technical boundary stay fixed.

1. Start with separated charge

A piezoelectric material shifts bound charge when force strains it. The generated charge follows force: Q=dF.

Phoebe: A held force can leave charge on the crystal, but it does not keep sending fresh charge down the wire.

2. Differentiate to see current

Charge lawQ(t)=dF(t).

Rate lawi=dQ/dt=d·dF/dt.

Average rampFor a force ramp, iavg=Q/Δt.

3. Let the amplifier count charge

A feedback capacitor integrates the input current, so the ideal fast-tap peak is Q/Cf. A feedback resistor gives the circuit a leakage time constant τ=RfCf.

Videal=Q/Cf; Vpeak=(Q/Cf)(τ/Δt)(1−e^(−Δt/τ))

4. Try the force rise time

Q=dF; iavg=Q/Δt; τ=RfCf; Vpeak=(Q/Cf)(τ/Δt)(1−e^(−Δt/τ))

TrySlow the chapter's same 10 N deformation while keeping its PVDF and amplifier values fixed.

ObserveA 5 ms tap creates 200 pC, averages 40.0 nA, and reaches about 0.195 V after leakage—97.5% of the simple 0.200 V peak.

ExplainThe total generated charge stays 200 pC as the press slows, but the current falls and the 100 ms leakage path drains more of it before the peak is reached.

Technical boundaries.

The leakage-aware formula assumes force rises linearly, d is constant, and the amplifier is an ideal first-order charge amplifier.

dielectric loss: Needs separate evidence
mechanical resonance: Needs separate evidence
temperature dependence: Needs separate evidence
cable capacitance: Needs separate evidence
amplifier bias current: Needs separate evidence
noise: Needs separate evidence
saturation: Needs separate evidence
a non-ideal force waveform: Needs separate evidence

Use field evidence or a deeper model before release.

5. Work the chapter's fast tap

Q=20 pC/N×10 N=200 pC

iavg=200 pC/5 ms=40.0 nA; Videal=200 pC/1 nF=0.200 V

6. Compare tap with leakage

τ=100 MΩ×1 nF=0.100 s=100 ms

Five milliseconds is only 5% of τ, so the simple peak is good to a few percent. A 200 ms press spans two time constants and reaches only about 0.0865 V even though Q=dF still gives 200 pC.

7. Check yourself

Why does a held force stop sourcing current?

Answer: Because i=d·dF/dt, and dF/dt becomes zero after the force stops changing.

Why is a charge amplifier better than a plain resistive divider?

Answer: It holds the input near virtual ground and integrates the tiny delivered charge instead of swamping it through a low resistance.

Does a slower 10 N press generate less total ideal charge?

Answer: No. Q=dF is still 200 pC; the measured peak falls because leakage competes with the slower delivery.

Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

d=20 pC/N: Chapter input or worked result
F=10 N: Capacitance value
Cf=1 nF: Capacitance value
Rf=100 MΩ: Resistance or impedance value
0.200 V ideal result: Voltage or voltage-step value
5 ms tap is explicitly catalog-typical: Named teaching assumption

The 0.195 V result adds a stated linear-ramp leakage model; it refines rather than silently replaces the chapter's fast-event approximation.