27 Sensor Dynamics and Response Time

27.1 In 60 Seconds

Measure One Change Before Choosing the Sample Rate

Picture a pressure sensor that looks smooth but reports a short safety event too late. Sampling faster cannot recover motion that the sensor itself did not follow.

An analog-to-digital converter means the circuit that turns a measured voltage into a number; it is shortened to ADC. Bandwidth means the range of change rates a sensor or signal path can follow usefully.

Apply the same step several times and record the raw response before and after the ADC. Keep sensor identity, range, sample times, bandwidth estimate, rise time, final value, noise, and missed threshold time.

This test describes one sensor and setup, not every field condition. The deeper sections connect time constant, bandwidth, sampling, filtering, and a defensible measurement claim.

A sensor does not jump to a new value the instant the world changes. It eases toward the new value at a speed set by its own physics. That delay, called the sensor’s dynamic response, decides what a measurement can actually capture. Choosing a sample rate or a filter without knowing how fast the sensor itself can move is how teams end up sampling a slow curve very fast, or trusting a fast sensor reading that the sensor never truly reached.

27.2 Start With the Story

You will judge whether a sensor responds quickly enough for the events your system must detect. Start with its response to a sudden change before choosing a sample rate.

Follow one physical step across four beats to see why sampling must respect the sensor’s own response time.

27.3 Sensors Ease Toward the Truth

Drop a thermometer into a glass of cold water and it does not read the cold temperature immediately. It slides toward it over a few seconds, fast at first and then slowing as it gets close. That gradual approach is the sensor's response, and almost every physical sensor has one: thermal mass, chemical diffusion, mechanical inertia, or filtering all slow the reaction to a change.

The important idea is that the sensor, not only the converter, sets how fast you can measure. A reading taken before the sensor has settled is a snapshot of the sensor catching up, not of the world. Sampling faster cannot recover detail the sensor was too slow to follow.

The single most useful number is the response time, often summarized by a time constant: a slow sensor takes seconds to follow a step, while a fast one takes milliseconds. Knowing it tells you whether the sensor can see the events you care about at all.

For example, a refrigerated cabinet may use an NTC thermistor inside a stainless probe. When the door opens, warm air reaches the probe first, but the probe body and potting compound slow the bead. The logger can sample every 100 ms and still show a rounded climb. The design question is not whether the ADC can collect more points; it is whether the thermal assembly can follow the shortest door-open event the product must alarm on.

Compare the abrupt input step with the rising sensor output in Figure 27.1. The marked time constants show how the reading gradually approaches the final value.

Follow the output curve in Figure 27.1 from one time constant at about 63 percent to three at about 95 percent. The input has already changed; extra samples only add points along the same delayed response.

The One-Minute View

Beginner Examples

• A slow temperature probe averaged over seconds is ideal for room comfort but useless for catching a fast thermal spike.
• A fast pressure sensor can follow a sharp pulse, but only if the sampling keeps up with it.
• Reporting a temperature one second after a step change may report the sensor mid-climb, not the true temperature.

Sensor Response Knowledge Check

If this gives you the idea, you can stop here. Continue to Practitioner when you need to match sampling and filtering to a specific sensor.

The mathematical gist. A first-order sensor closes a fixed fraction of its remaining gap each second, so $dx/dt=(x_{\mathrm{final}}-x)/\tau$ produces the exponential response $x(t)=x_{\mathrm{final}}(1-e^{-t/\tau})$. The same time constant sets its $-3$ dB bandwidth, $f_{3\mathrm{dB}}=1/(2\pi\tau)$: for this chapter’s $\tau=2$ s thermistor that is $0.0796$ Hz, and no faster ADC setting can restore a half-second spike that the probe’s physics has already smoothed away.

Math Bridge · guided foundations Where do tau and the -3 dB bandwidth come from? Let Phoebe walk from derivatives and exponentials to every algebra step and this chapter's own 2-second thermistor. Learn the maths

27.4 Apply It: Match Sampling to the Sensor

The practical job is to compare how fast the sensor can move against how fast the thing you measure actually changes, then choose a sample rate and filter that respect both. The sensor's datasheet response time is the starting evidence.

Walkthrough: From Response Time to Sample Plan

1. Find the response time. Read the sensor's time constant or its time to reach 90 percent of a step from the datasheet or a bench test.
2. Name the fastest real event. Decide the quickest change the application must capture, such as a thermal spike or a pressure pulse.
3. Compare the two. If the event is faster than the sensor can follow, no sample rate will recover it; you need a faster sensor.
4. Set the sample rate. If the sensor can follow the event, take several samples across the sensor's response so the curve is well described.
5. Account for lag and drift. Record the response delay, any hysteresis, and expected drift so validation does not mistake them for real change.

Worked Example: A Two-Second Thermistor

A thermistor has a time constant of about 2 seconds, so after a sudden temperature step it reaches roughly 95 percent of the change in about 6 seconds. Sampling it 100 times per second does not measure temperature faster; it records the same slow climb in fine detail. If the process being measured spikes and falls within half a second, this sensor physically cannot report the peak, and the fix is a faster sensor, not a faster ADC.

The same time constant also sets the sensor's usable bandwidth. A slow sensor is, in effect, a low-pass filter on the world.

A reliable Worked Example: A Two-Second Thermistor review needs the figure Figure 27.2. Its Sample what the sensor can actually follow and frequency markers reveal where Sample plans should cover what the sensor can actually follow; beyond the sensor limit, extra samples mostly record lag or noise enters the running decision.

Read along the frequency axis in Figure 27.2, comparing the event band with the sensor limit. The region beyond useful coverage is marked as waste and noise; a faster sample plan cannot remove physical lag.

Dynamic Effects to Record

Challenge Dynamic Effects to Record with the worked review in Figure 27.3. Event fixes the deadline before Response test measures the sensor; Bandwidth choice and Filter budget then make every added delay visible before Validate repeats the test in the production assembly.

Inspect Figure 27.3 at Event: the release must reach 95 percent of a temperature step within 7,000 ms. Response test turns the measured 2,000 ms time constant into a 6,000 ms sensor contribution; Bandwidth choice records both the 0.080 Hz sensor limit and a 500 ms worst-case sampling wait. Filter budget adds the smoother’s 250 ms group delay rather than hiding it. Finally, Validate repeats the mounted, enclosed, production-filtered step test: every one of 10 trials must reach at least 95 percent by 7,000 ms. The resulting 6,750 ms total leaves 250 ms of margin and turns the figure into a bounded Dynamic Effects to Record decision.

Incremental Practice

Oversampling Knowledge Check

If you can match a sample plan to a sensor's response, you can stop here. Continue to Under the Hood for the response math and the failure modes behind it.

27.5 Under the Hood: Response Math and Failure Modes

Most sensor responses are well described as first-order or second-order systems. The math turns a vague "it is slow" into specific numbers for delay, bandwidth, overshoot, and settling.

First-Order Response and Its Constants

A first-order sensor following a step approaches the final value exponentially. The fraction completed depends only on how many time constants have passed:

• After 1 time constant, about 63 percent of the change is complete.
• After 3 time constants, about 95 percent is complete.
• After 5 time constants, about 99 percent is complete.

This is why response is often quoted as a time to reach 90 percent, which is about 2.3 time constants, and why "settled" usually means several time constants have elapsed.

Bandwidth Is the Same Fact in Frequency

A slow time constant is a low-pass filter on the input. The half-power bandwidth follows directly:

A 2-second time constant corresponds to a -3 dB bandwidth below 0.1 Hz, so signals that change several times per second are strongly attenuated. Sampling much faster than this response scale still produces closely correlated points on the lagging response; anti-alias protection must nevertheless cover whatever attenuated high-frequency content reaches the ADC.

A second-order response has a physical mechanism behind its equation. Figure 27.4 connects proof-mass displacement to differential capacitance and marks resonance as the bandwidth boundary.

In Figure 27.4, spring k and damper b constrain the proof mass m, while C1 rises as C2 falls to turn displacement x into an electrical signal. The resonance boundary at ω0 is not a useful measurement boost: operation stays well below it so displacement remains a stable proxy for acceleration.

27.5.1 Second-Order Behavior: Damping

A moving sensor element does not jump instantly to its final position. A proof mass contributes inertia, a suspension contributes a restoring force, and motion through the suspension or surrounding gas dissipates energy. Write each force separately before combining them:

$$F_m=m\ddot{x},\qquad F_b=b\dot{x},\qquad F_k=kx.$$

Here $m$ is proof mass, $b$ is viscous damping coefficient, $k$ is spring stiffness, $x$ is displacement, $\dot{x}$ is velocity, and $\ddot{x}$ is acceleration. Newton’s law gives the driven equation of motion

$$m\ddot{x}+b\dot{x}+kx=F(t).$$

Dividing by $m$ exposes the two parameters that shape the transient:

$$\ddot{x}+2\zeta\omega_n\dot{x}+\omega_n^2x=\frac{F(t)}{m}, \qquad \omega_n=\sqrt{\frac{k}{m}}, \qquad \zeta=\frac{b}{2\sqrt{km}}.$$

The natural frequency in hertz is

$$f_n=\frac{\omega_n}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}.$$

This is a resonance boundary, not a claim that measurements remain flat all the way to $f_n$. As excitation approaches resonance, amplitude and phase error grow; a measurement bandwidth is normally chosen comfortably below it and verified from the actual transfer function. When $\zeta<1$, the sensor is underdamped and rings. At $\zeta=1$ it is critically damped. When $\zeta>1$, it is overdamped and approaches the final value without overshoot but more slowly.

An accelerometer turns base acceleration into relative proof-mass displacement. At frequencies well below resonance, dynamic terms are small enough that the spring force balances the inertial force in magnitude:

$$k\,\Delta z \approx m a \quad\Longrightarrow\quad a\approx\frac{k}{m}\Delta z.$$

The sign depends on the chosen axis: the mass deflects opposite the base acceleration, so firmware must preserve the mechanical and electrical sign conventions. The derivation also shows the trade-off. A softer spring or larger mass produces more displacement per unit acceleration, improving displacement sensitivity, but lowers $f_n$ and therefore narrows the usable dynamic range.

The readout completes the instrument. Differential capacitive plates are well suited to micrometre and sub-micrometre travel because a small gap change produces a measurable capacitance imbalance, and the opposing capacitors reject some common motion and parasitics. Inductive displacement sensing is often a better fit for larger travel and rugged conductive targets, but needs coils, excitation, and magnetic-field management. Neither method changes the mechanics: it only determines how faithfully $\Delta z$ becomes an electrical estimate.

An early reading from an underdamped sensor can be higher or lower than the true value. That is why a complete sensor specification records natural frequency, damping or step-response evidence, measurement bandwidth, overshoot, and settling tolerance rather than reporting rise time alone.

The transfer function makes the response families explicit. With zero initial conditions, the Laplace transform gives

$$(ms^2+bs+k)X(s)=F(s), \qquad \frac{X(s)}{F(s)}=\frac{1}{ms^2+bs+k}.$$

The characteristic roots are

$$s_{1,2}=-\zeta\omega_n\pm\omega_n\sqrt{\zeta^2-1}.$$

For $0<\zeta<1$, the square root is imaginary, so the output rings at $\omega_d=\omega_n\sqrt{1-\zeta^2}$ inside a decaying envelope $e^{-\zeta\omega_n t}$. At $\zeta=1$, the two real poles coincide at $-\omega_n$. For $\zeta>1$, two different negative real poles remain and the slower one controls the long tail. The familiar step-response estimates now have a cause:

$$M_p=\exp\!\left(\frac{-\pi\zeta}{\sqrt{1-\zeta^2}}\right), \qquad t_s\approx\frac{4}{\zeta\omega_n}\quad\text{for a 2\% band}.$$

Turn those symbols into observable evidence in the interactive below. Start with the Proof mass: underdamped preset, then walk the four stages from applied acceleration through displacement, poles, and the exact fixed-grid 2% settling check. Watch the final displacement alongside overshoot: the first is the static sensitivity result, while the second belongs to the transient.

Use the Critical damping counterexample preset for the one-click comparison. It changes only damping, removes the 72.9% overshoot, and leaves the final 10.0 mm displacement unchanged; that is the direct test of the misconception that more damping changes final sensitivity.

Walk the coefficients. Let $m=0.020\ \mathrm{kg}$ and $k=50\ \mathrm{N/m}$. Then $\omega_n=50\ \mathrm{rad/s}$ and critical damping requires $b_c=2\sqrt{mk}=2.0\ \mathrm{N\,s/m}$. If $b=0.40\ \mathrm{N\,s/m}$, then $\zeta=0.20$, predicted overshoot is about $52.7\%$, and the 2% settling estimate is $0.40\ \mathrm{s}$. Raising $b$ to $2.0\ \mathrm{N\,s/m}$ removes overshoot at critical damping. At $b=4.0\ \mathrm{N\,s/m}$, $\zeta=2$ and the poles are approximately $-13.4$ and $-186.6\ \mathrm{s^{-1}}$; that first pole explains the visibly slower final approach.

Use Figure 27.5 to connect those roots to observable traces. Start at the ideal step, then compare the one-exponential first-order response with the second-order response, where rise, overshoot, oscillation, and settling all need separate measurements.

Follow the second-order trace in Figure 27.5 from its first peak through the shrinking oscillations until it remains inside the settling band. The vertical gap above the final value is $M_p$; the time between peaks corresponds to $2\pi/\omega_d$; and the shrinking amplitude follows $e^{-\zeta\omega_n t}$. The chart is therefore a visual reading of the same $m$, $b$, and $k$ terms derived above, not a separate classification to memorise.

Hysteresis and Drift Are Separate Errors

Hysteresis means the reading depends on whether the input is rising or falling, so a value approached from below differs from the same value approached from above. Drift means the output slowly changes with constant input, due to aging or temperature, which is why calibration has a finite useful life. Neither is captured by response time, and both must be measured separately.

Symptom

Likely Cause

Evidence to Collect

First Fix to Try

Peaks look too low

Sensor too slow for the event (dynamic lag).

Compare event duration with the time constant.

Use a faster sensor or accept averaged values.

Early reads unstable

Reading taken before settling.

Settling time at the required tolerance.

Wait several time constants before reading.

Up and down disagree

Hysteresis in the sensor.

Rising versus falling sweep at the same input.

Record direction and characterize the offset.

Slow baseline shift

Drift over time or temperature.

Long capture at constant input.

Schedule recalibration or compensate the drift.

Common Pitfalls

1. Oversampling a slow sensor. A high sample rate cannot add detail the sensor was too slow to capture.
2. Reading before settling. Early samples include transient or overshoot error, not the true value.
3. Confusing resolution with speed. More bits do not make a slow sensor follow a fast event.
4. Ignoring hysteresis and drift. Both produce errors that response time alone never reveals.

Sensor Bandwidth Knowledge Check

At this depth, sensor dynamics are the real bandwidth limit of a measurement. Characterize the response time, convert it to a bandwidth, respect settling and damping, and measure hysteresis and drift separately, so the sample plan fits the sensor's physics rather than an idealized signal.

Move from forces to poles before interpreting a step trace with Figure 27.6.

In the diagram Figure 27.6, sum the forces yields F = m·x” + b·x’ + k·x, and Move into the s-domain exposes natural frequency ωn and damping ratio ζ. Read poles as behavior then aligns underdamped overshoot, critical damping, and slower overdamped settling.

27.6 Summary

• Every physical sensor eases toward a new value over time; that response time, not the ADC, sets the fastest event it can capture.
• A first-order sensor completes about 63 percent of a step in one time constant, 95 percent in three, and 99 percent in five.
• The time constant sets the $-3$ dB bandwidth: $f_{3\mathrm{dB}}=1/(2\pi\tau)$, so a slow sensor is a low-pass filter on the world.
• Sampling faster than the sensor’s bandwidth adds correlated, lagging points rather than new information.
• Second-order sensors can overshoot, so settling time, not only rise time, decides when a reading is trustworthy.
• Hysteresis and drift are separate errors that response time does not reveal and must be measured on their own.

27.7 See Also