Response-Time Limit Calculation Audit
Response-Time Limit Calculation Audit
Ada re-derives the thermistor settling, bandwidth, and missed-peak numbers from this chapter’s own example
ADA · CALCULATION AUDIT
Response-Time Limit Calculation Audit
A response-time number is a physics contract: it tells you how much of a real event the sensor can actually turn into evidence before firmware ever samples it.
A thermistor with a time constant of about 2 seconds is sampled 100 times per second during a slow temperature change, reaching roughly 95 percent of a step in about 6 seconds — but the chapter warns that if the real process spikes and falls within half a second, this sensor physically cannot report the peak. This audit asks the question that warning invites: does the physics of a 2-second time constant really block a half-second event, and is the honest fix a faster sensor rather than a faster ADC?
Companion to the chapter Sensor Dynamics and Response Time — every number here comes from that chapter.
See the relationship before changing it
The figure reads from left to right. The blue card is sensor time constant. The middle card applies this page's rule. The green card is approximate 95 percent settling time. 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 sensor time constant, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 2 s.
- 2
Name the relationship. settling time = 3 x time constant
- 3
Substitute with units. 3 x 2 s = 6.0 s
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change sensor time constant
Try Predict the direction of settling time = 3 x time constant. Test another sensor time constant, then compare approximate 95 percent settling time.
Observe A slower physical time constant lengthens settling even if firmware samples more often. Reset sensor time constant to 2 and compare approximate 95 percent settling time.
Explain A slower physical time constant lengthens settling even if firmware samples more often.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Use tau = 2 s in the first-order response and evaluate both the 95% settling time and the fraction reached during a 0.5 s spike.
The model needs about 6 s for 95% and 10 s for 99%; after only 0.5 s it reports roughly 22.1% of the true step.
A first-order sensor stores energy and therefore behaves as a low-pass element: faster sampling adds points along the same physical rise but cannot accelerate that rise.
1. The two-second thermistor does not settle in two seconds
This chapter uses a thermistor with a time constant of about 2 seconds and the first-order step equation:
At three time constants the response is nearly complete enough for many release checks, and at five time constants it is essentially settled:
| Check | Arithmetic shown | Meaning |
| 95 percent point | 3 x tau = 3 x 2 s = 6 s | The chapter's 6-second estimate follows directly from the 2-second time constant. |
| 99 percent point | 5 x tau = 5 x 2 s = 10 s | A strict settling gate waits closer to 10 seconds before trusting the final value. |
| 100 samples/s at 95 percent | 100 samples/s x 6 s = 600 samples | Oversampling creates many points on the same slow climb; it does not make the probe reach the endpoint faster. |
2. The same time constant is a bandwidth limit
The chapter gives the half-power bandwidth formula:
That is below 0.1 Hz, matching the text. The chapter's half-second spike is equivalent to a short event-rate check around 2 Hz, far above this sensor's useful bandwidth:
3. A half-second spike is physically under-reported
The practitioner section warns that a process spike lasting 0.5 seconds cannot be captured by this slow thermistor. The first-order response shows why:
What the audit buys you: a sample-rate decision can be checked against settling time and bandwidth before deployment. If the event is faster than the sensor physics, the honest fix is a faster sensor or a different requirement, not a faster ADC setting.
Every number above is taken from this chapter's own worked example and re-derived step by step.
Technical boundaries: This single-pole step model omits transport delay, nonlinear thermal coefficients, airflow and mounting effects, noise, asymmetric heating and cooling, ADC limits, and multi-time-constant packaging.