Static and dynamic characteristics of sensors

Why it matters

Every datasheet you will ever read for a sensor is a list of static and dynamic characteristics. Choosing a sensor, sizing its signal conditioning and estimating the error in a reading all depend on knowing what "±0.5 % FS non-linearity" or "τ = 2 s" actually means for your measurement.

Key ideas

Static characteristics describe the input–output relation when the measurand is constant or changes slowly enough that the sensor is always in equilibrium. They are found by static calibration: apply known inputs, record outputs, going up and then down the range.

• Range and span: range is the lowest-to-highest input the sensor accepts (say 0–100 kPa); span is the difference (100 kPa). Full-scale output (FSO) is the output at the top of the range.
• Accuracy: closeness of the reading to the true value; usually quoted as ± % of full scale (% FS) or ± % of reading. A ±1 % FS error on a 0–100 kPa sensor is ±1 kPa at every point, so it is a 10 % error at a 10 kPa reading.
• Precision / repeatability: closeness of repeated readings of the same input under the same conditions. A sensor can be precise but inaccurate (consistent bias), which calibration can remove; random scatter cannot be calibrated out.
• Sensitivity (static gain): slope of the calibration curve, output change per unit input change. For a linear sensor it is one number; for a non-linear sensor it varies with the operating point.
• Linearity: maximum deviation of the calibration curve from a reference straight line, as % FSO. The reference can be the terminal line (through end points), the best-fit (least-squares) line, or the independent line — always state which, because the number changes.
• Hysteresis: maximum difference between the increasing-input and decreasing-input outputs at the same input, as % FSO. Caused by friction, magnetic effects, elastic after-effects.
• Resolution and threshold: resolution is the smallest input change that produces a detectable output change; threshold is the smallest input from zero that produces a detectable output.
• Drift: change of output with time or ambient temperature at constant input. Zero drift shifts the whole curve; sensitivity drift changes the slope.
• Dead zone: the largest input change to which the sensor does not respond (often due to backlash).

Dynamic characteristics describe how faithfully the output follows a time-varying input. Most sensors are modelled as zero-, first- or second-order linear systems.

• Zero order: output follows input instantly (an ideal potentiometer). Only static gain K.
• First order: one energy-storing element, e.g. a thermometer bulb storing heat. Characterised by K and the time constant τ. After a step the output reaches 63.2 % of its change in τ, 95 % in about 3τ and 98.2 % in 4τ. For a sinusoidal input it behaves as a low-pass filter with corner frequency 1/(2πτ).
• Second order: two energy stores, e.g. a spring–mass accelerometer or a galvanometer. Characterised by K, natural frequency ωn and damping ratio ζ. With ζ < 1 the step response overshoots and rings; ζ ≈ 0.6–0.7 gives the widest flat frequency response with little overshoot, which is why many instruments are designed near this value.
• Dynamic error and fidelity: the difference between true and indicated value while the input varies, and how faithfully the waveform shape is reproduced. Speed of response is measured by rise time, settling time or time constant. Measuring lag is the delay between input change and output response.
• Bandwidth: the frequency range over which the amplitude ratio stays within a stated tolerance (commonly −3 dB, i.e. 0.707).

Formulas

S = Δq_o / Δq_i S = static sensitivity (output unit per input unit, e.g. V/kPa); Δq_o = change of output; Δq_i = change of input. For a linear region only.

Non-linearity (% FSO) = (maximum deviation from reference line / full-scale output) × 100

Hysteresis (% FSO) = (maximum up–down output difference / full-scale output) × 100

τ·dq_o/dt + q_o = K·q_i (first-order model) τ = time constant (s); K = static gain.

q_o(t) = q_o(0) + K·Δq_i·(1 − e^(−t/τ)) (step response, first order)

M = 1 / √(1 + (ωτ)²), φ = −tan⁻¹(ωτ) (first order, sinusoidal input) M = amplitude ratio (dimensionless); ω = input angular frequency (rad/s); φ = phase lag.

d²q_o/dt² + 2ζωn·dq_o/dt + ωn²·q_o = K·ωn²·q_i (second-order model) ωn = undamped natural frequency (rad/s); ζ = damping ratio (dimensionless).

Mp = e^(−πζ / √(1 − ζ²)), ωd = ωn·√(1 − ζ²) (second order, ζ < 1) Mp = peak overshoot as a fraction of the step; ωd = damped natural frequency (rad/s).

Worked examples

Example 1 — static calibration. A 0–100 kPa pressure sensor gives the following outputs (V). Input (kPa): 0, 25, 50, 75, 100. Increasing: 0.00, 1.27, 2.53, 3.76, 5.00. Decreasing: 0.02, 1.31, 2.58, 3.80, 5.00. Find the terminal-line sensitivity, non-linearity (increasing run) and hysteresis, both in % FSO.

1. Terminal line through (0, 0) and (100, 5.00): S = (5.00 − 0)/(100 − 0) = 0.05 V/kPa.
2. Ideal outputs: 0, 1.25, 2.50, 3.75, 5.00 V. Deviations of increasing run: 0, 0.02, 0.03, 0.01, 0 V. Maximum = 0.03 V.
3. Non-linearity = 0.03/5.00 × 100 = 0.6 % FSO.
4. Up–down differences: 0.02, 0.04, 0.05, 0.04, 0 V. Maximum = 0.05 V at 50 kPa.
5. Hysteresis = 0.05/5.00 × 100 = 1.0 % FSO.

Answer: S = 0.05 V/kPa, non-linearity 0.6 % FSO, hysteresis 1.0 % FSO.

Example 2 — first-order dynamics (GATE level). A thermometer with τ = 2 s, initially at 25 °C, is plunged into a bath at 125 °C. (a) What does it read at t = 3 s? (b) It is then used to track a sinusoidal temperature. Up to what frequency is the amplitude ratio at least 0.9, and what is the phase lag at that frequency?

1. Step response: T(t) = 25 + 100·(1 − e^(−t/τ)).
2. T(3) = 25 + 100·(1 − e^(−1.5)) = 25 + 100 × 0.7769 = 102.7 °C.
3. Amplitude condition: 1/√(1 + (ωτ)²) = 0.9 → (ωτ)² = 1/0.81 − 1 = 0.2346 → ωτ = 0.4843.
4. ω = 0.4843 / 2 = 0.2422 rad/s; f = ω/(2π) = 0.0385 Hz.
5. φ = −tan⁻¹(0.4843) = −25.8°.

Answer: (a) 102.7 °C; (b) f ≤ 0.0385 Hz, phase lag 25.8° at that frequency.

Example 3 — second-order overshoot. An accelerometer has fn = 1 kHz and ζ = 0.6. Find the peak overshoot to a step and the damped natural frequency.

1. Mp = e^(−π × 0.6/√(1 − 0.36)) = e^(−2.356) = 0.0948, i.e. 9.5 %.
2. fd = fn·√(1 − ζ²) = 1000 × 0.8 = 800 Hz.

Common mistakes

• Treating accuracy and precision as the same thing: a precise sensor with an offset is consistently wrong.
• Reading a % FS error as % of reading — at low readings the relative error is much larger.
• Quoting non-linearity without saying which reference line was used.
• Thinking the output reaches its final value in one time constant; it reaches only 63.2 %. Use about 4τ–5τ for "settled".
• Forgetting that a first-order sensor also has a phase lag: for a sine input the reading is attenuated and delayed.
• Using ωn where the formula needs ωd, or assuming any damping gives no overshoot (only ζ ≥ 1 does).

For GATE IN

Expect NAT questions on first-order step and ramp response (reading after time t, time to reach a given percentage), amplitude ratio and phase of first-order sensors, overshoot and damped frequency of second-order sensors, and % FS error versus % of reading. MCQs test definitions — hysteresis, resolution, threshold, drift, dead zone. Practise converting between ζ, overshoot and settling quantities quickly.

Quick check

1. A ±0.5 % FS sensor of range 0–200 °C reads 40 °C. What is the possible error as a percentage of the reading?
2. A first-order sensor has τ = 4 s. How long does it take to reach 95 % of a step change?
3. What is the amplitude ratio of a first-order sensor when ωτ = 1?
4. Which damping ratio gives zero overshoot at the boundary between oscillatory and non-oscillatory response? Answers: 1. ±1 °C, i.e. ±2.5 % of reading; 2. about 12 s (3τ, exactly 11.98 s); 3. 0.707 (−3 dB), phase −45°; 4. ζ = 1 (critical damping).