Ada Audits the Response-Time Limit

Ada re-derives the thermistor settling, bandwidth, and missed-peak numbers from this chapter’s own example

ADA · CALCULATION AUDIT

Ada Audits the Response-Time Limit

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.

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:

value(t) = final x (1 - e^(-t / tau))

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:

f_3dB = 1 / (2 x pi x tau) = 1 / (2 x 3.1416 x 2 s) = 0.0796 Hz

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:

event_rate = 1 / 0.5 s = 2 Hz; event_rate / f_3dB = 2 / 0.0796 = 25.1

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:

fraction_seen = 1 - e^(-0.5 / 2) = 1 - e^-0.25 = 0.221, or 22.1 percent

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.