Battery Bruno guidesA field team faces an unresolved physical question: Why does occupancy sensing need a settling window? They must answer it before changing cutoff 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 cutoff. The middle card applies this page's relationship. The green card is time constant. 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.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline for cutoff is 10.
- 2
Name the relationship. dT/dt=4/0.1=40 K/s i=200 uC/m²Kx2 mm²x40 K/s=16.0 nA τ=1/(2πx10)=15.9 ms; 3τ=47.7 ms 8 mAx0.030 s+2 mAx0.050 s=0.34 mA·s
- 3
Substitute the chapter fixture. Set cutoff to 10. The page ledger gives time constant as 15.9 ms.
- 4
Read the result. Keep ms beside the value. Use it only inside the technical boundary on this page.
Predict, then change cutoff
Try Predict the direction of time constant. Move one control, calculate, then check your prediction.
Observe A wider first-order filter responds faster, but it also admits more noise; the slider does not invent that missing noise model. Reset the control to 10 and compare time constant.
Explain Only cutoff moves here. The other chapter fixtures remain fixed.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
1. Start with the physical story
A pyroelectric crystal produces charge when its temperature changes. A moving warm body crosses Fresnel zones and creates a tiny alternating current; a stationary warm body becomes quiet.
2. Name every algebra move
Find temperature rateDivide 4 K by 0.1 s.
Find currentMultiply pA by dT/dt.
Find time constantUse τ=1/(2πfc).
SettleUse three time constants as the review window.
Price sensingAdd MCU current-time and sensor current-time.
3. Reproduce the chapter case
i=200 µC/m²K×2 mm²×40 K/s=16.0 nA
τ=1/(2π×10)=15.9 ms; 3τ=47.7 ms
8 mA×0.030 s+2 mA×0.050 s=0.34 mA·s
The independent filter calculation lands close to the chapter's 50 ms value without forcing the match.
4. Try one real input
TryChange filter cutoff and predict how the three-time-constant window moves.
ObserveHigher cutoff shortens settling, while the fixed crossing and chapter charge ledger remain unchanged.
ExplainA wider first-order filter responds faster, but it also admits more noise; the slider does not invent that missing noise model.
This is a signal-and-settling ledger, not an occupancy classifier.
- Motion
- PIR current supports motion evidence, not a person count.
- Filter
- Three time constants is a settling convention.
- Noise
- Changing cutoff would also change noise and threshold behavior.
Correct, not complete: the calculation explains the window, not field detection reliability.
5. Use the result in the design
Measure the actual module's filter, warm-up, threshold, false-alarm rate, and missed-motion rate before letting ACE reuse or replace sensing.
6. Record the evidence state
Keep cutoff, settle rule, lens, crossing speed, temperature contrast, current trace, threshold, occupancy claim, and cache policy together.
7. Check yourself
Why does a stationary warm object become quiet?
Why is 10 Hz close to 50 ms settling?
Does 16 nA prove occupancy?
The arithmetic uses illustrative PIR material and geometry plus the chapter's sensing currents.
- Motion
- PIR current supports motion evidence, not a person count.
- Filter
- Three time constants is a settling convention.
- Noise
- Changing cutoff would also change noise and threshold behavior.
Correct, not complete: the calculation explains the window, not field detection reliability.