What Response Time Does to a Reading You Took Too Soon

Why this matters

An unsettled reading is not "a little off." It is a point on a curve whose shape you know, whose direction you know, and whose remaining error you can calculate to within a few percent while you are standing there. Techs treat it as noise, which throws away the most useful thing about it. The number is wrong by a predictable amount in a predictable direction, and that means an early reading is sometimes completely conclusive and sometimes worthless, on the same instrument, in the same minute, depending on which side of the accept line you are approaching from. Knowing which case you are in is the difference between confidently calling a system out of range in thirty seconds and condemning a system that was fine the whole time.

What a first-order response actually is

Change the thing an instrument is measuring in one step and the reading does not jump. It closes the gap between where it was and where it is going at a rate proportional to how much gap is left, which produces the exponential approach every field instrument shows. The single number that describes it is the time constant, written as tau: the time to cover 63.2 percent of a step.

The remaining error is always the same fraction of the original step, whatever the step was. That is why this table travels: it is the same for a temperature probe, a pressure transducer settling through a length of tubing, and a moisture meter coming up to reading.

Two conditions on it. It describes a single dominant lag responding to a step, so a probe inside a thermowell, which has two lags in series, starts more slowly than the table predicts and then catches up, and a quantity that is ramping rather than stepping never settles at all, it just trails behind by a fixed offset. And it is an approach, not an arrival: mathematically the reading never gets there, which is why the practical rule below is stated in multiples of tau rather than as "when it stops moving."

The gate

Wait three time constants after the last change before the reading counts, per step change, per probe in its actual installation. Where the margin between your expected reading and the accept/reject line is under 5 percent of the step you just made, wait five.

Three tau leaves 5 percent of the step as error. Five tau leaves 0.7 percent. Choosing between them is not caution, it is arithmetic: if the remaining error at three tau is larger than the margin you are judging against, three tau cannot settle the question and you have not waited long enough regardless of how still the display looks.

The display looking still is not the test. Damping and averaging inside the instrument make a slowly-closing reading look stationary well before it has arrived, and a sibling article owns what that processing hides (see References). Use the clock, not the digits.

Before you stand there waiting

Waiting is the instruction, so the hazard belongs to the waiting.

• Beside a firing appliance. A settling flue or supply-air reading can put you next to a running burner for minutes. Wear a personal carbon monoxide monitor in the occupied space before the appliance fires, and if ambient carbon monoxide climbs toward the 50 ppm eight-hour time-weighted average in 29 CFR 1910.1000 Table Z-1, everyone leaves immediately, no switch is touched on the way out, and the space is ventilated from outside before re-entry. Flue probes come out hot enough to burn; handle by the grip.
• In hot water or hot fluid. A probe held in a stream for a full settling period is a scald exposure for that whole period, not a moment. Clamp or fixture the probe rather than holding it, wear liquid-resistant gloves and eye protection, and throttle the flow.
• On an energized circuit. A settling electrical reading means the leads stay on live parts for the duration, which sits behind the gate at 29 CFR 1910.333(a)(1) and requires the electrical protective equipment 29 CFR 1910.335(a) specifies for the exposure. Clip the leads on rather than hand-holding probes for a long dwell, keep the free hand clear, and confirm instrument, leads and tips are all rated at or above the circuit's measurement category under IEC 61010-1.
• Do not shorten the wait by wiggling the probe. Moving it changes the coupling, which restarts the clock at a new value and hides the fact that you restarted it.

What sets the time constant, and why the published one is rarely yours

A manufacturer's response-time figure is measured under stated conditions, and those conditions are usually far more favourable than a field installation. A bare bead in fast-moving water settles in a small fraction of the time the same bead takes in still air, because the number is set by how quickly heat crosses from the fluid to the element, and still air is a poor conductor.

What lengthens tau in the field, in the order these normally matter:

• The medium and its velocity. Still air is the worst case by a wide margin, moving air is better, liquid better again. A published figure quoted without its medium and velocity is not usable.
• Mass around the element. A thermowell, a sheath, a heavy stem, a lump of thermal compound. Every bit of it has to change temperature before the element does.
• Poor contact. A surface probe not seated adds a slow leg to the path.
• Volume and tubing in a pressure or gas measurement. Filling a hose is a real time constant even though nothing thermal is happening.

So measure your own. Make a step, start a timer, and note how long the reading takes to cover 63 percent of the total change. Do that once for each probe-and-installation combination you use routinely and write it on the case. It is a one-time cost and it converts an argument into a number.

Dead time is not a time constant

There is a second kind of delay and the exponential does not describe it at all. Dead time is transport delay: the interval before anything happens, while the sample physically travels. A flue-gas sample line, a long capillary, a duct run between the source and the sensor, a slow scanning cycle in a datalogger.

The distinction matters because the two require different responses. During dead time the reading is not partially correct, it is completely stale, and no fraction of it is usable. The rule is additive: wait out the dead time first, then start counting the three or five time constants from the moment the reading begins to move. Techs who treat a long sample line as a slow time constant start the clock at the wrong moment and then wonder why a 3 tau wait was not enough.

One gate, two outcomes on the same early reading

A probe sitting in a van at 70 F goes into a delivered water stream. Nothing else about the setup differs between the two cases below: same probe, same installation, same stream.

Measure tau first. Step the probe into the stream, and it covers 63 percent of the change in 20 seconds. So tau is 20 seconds, 3 tau is 60 seconds, and 5 tau is 100 seconds.

The stream is genuinely at 140 F, which makes the step 70 F. Read the probe at 30 seconds, which is 1.5 tau. The remaining error fraction is e to the minus 1.5, which is 0.223, so 0.223 times 70 F is 15.6 F still to go and the display reads 124.4 F. For reference, at 60 seconds the remaining error is 0.050 times 70, or 3.5 F, giving 136.5 F, and at 100 seconds it is 0.007 times 70, or 0.5 F, giving 139.5 F.

Case A: is delivered temperature above an upper limit of 120 F? The early reading is 124.4 F. The probe is approaching from below, so every remaining second moves it further above 120, never back toward it. The answer is already settled at 30 seconds and waiting the full 60 only makes it more so. Report it as over the limit, note the reading was taken at 1.5 tau and rising, and move on.

Case B: does delivered temperature meet a minimum of 130 F? Same 124.4 F, same instant. Now the reading is on the wrong side of the line and the known error points across it. At 3 tau the reading is 136.5 F, a pass, and the truth is 140 F, a comfortable pass. Acting on the 30-second number writes up a failure on a system that meets the requirement, and the tech has evidence in hand, the probe was still climbing, that they did not use.

What decides it. Not the reading, not the instrument, not the wait. The relationship between the direction of approach and the side of the boundary. An unsettled reading that has already crossed the line it is moving away from is conclusive; an unsettled reading on the near side of a line it is moving toward is not evidence of anything yet.

Reverse the step and the sign reverses with it. Take the same probe out of the 140 F stream into 70 F room air and read it at 1.5 tau, and it reads 70 plus 0.223 times 70, which is 85.6 F, now 15.6 F high rather than 15.6 F low. Same magnitude, opposite direction. Note that tau itself is not the same in still room air as it was in the moving stream, so the 20-second figure applies to the immersion and the return leg will be slower; the sign and the shape are what carry over, not the constant.

The failure mode. A tech takes a fast reading, sees it fail a minimum, and writes it up. On the callback the second tech waits and it passes. Now the shop has two readings that disagree, no timing on either one, and a customer who has stopped believing both. The fix is one extra field on the record: seconds after the step. Without it, an early reading and a settled reading look identical on paper.

How to verify you got this right

• Write down when the step happened and when you read it. Two clock values, and the difference is the only thing that lets anyone judge the number later.
• Take one confirming reading a full tau after the first. If the value has moved more than the remaining-error fraction predicted, either your tau is wrong or the quantity itself is changing, and those need different responses.
• Check the direction against the mechanism. Coming up from cooler, the reading is low. Coming down from warmer, it is high. If your explanation of a stubborn reading has the sign the other way, the mechanism you named is not the one operating.
• Confirm you counted the dead time separately. If the reading sat flat for a while before it began to move, your three time constants start at the moving, not at the step.

References

• 29 CFR 1910.1000 Table Z-1 - the carbon monoxide exposure limit used as the evacuation trigger during a long settling period beside a firing appliance
• 29 CFR 1910.333(a)(1) and 1910.335(a) - the energized-work gate and required electrical protective equipment where leads stay on live parts for a full settling dwell
• IEC 61010-1 - measurement category and voltage ratings for instrument, leads and tips, binding through the product's listing
• Instrument manufacturer documentation - the medium, velocity and mounting a published response-time figure was measured under
• See related: Averaging and Spikes: What Your Meter Hides; Why Where You Put the Probe Decides the Answer; How to Take a Reading Another Tech Can Reproduce