Measurement system characteristics: static and dynamic

Why it matters

Every control loop is only as good as the sensor feeding it. Static characteristics tell you how close a steady reading is to the truth; dynamic characteristics tell you how faithfully the sensor follows a changing input. Choosing a sensor, setting a sampling rate or tuning a controller all start from these numbers on the data sheet.

Key ideas

A measurement system is usually drawn as sensor → signal conditioning → display or controller. Its behaviour is split into two parts.

Static characteristics (input held constant or changing very slowly):

• Range and span: range is the lowest to highest measurable input (e.g. 0–200 °C); span is the difference (200 °C).
• Accuracy and error: error = measured value − true value. Accuracy is usually quoted as a maximum error, often as a percentage of full-scale (FS) output or of reading. A ±1 % FS sensor of 0–200 °C may be wrong by ±2 °C anywhere in its range, so small readings carry a large percentage error.
• Precision (repeatability): closeness of repeated readings of the same input under the same conditions. A sensor can be precise but inaccurate (consistent bias) — bias can be removed by calibration, scatter cannot.
• Sensitivity: slope of the calibration curve, ΔOutput/ΔInput (e.g. mV/°C). For a nonlinear sensor it varies along the range.
• Linearity (non-linearity): maximum deviation of the calibration curve from a reference straight line, expressed as % FS.
• Resolution: smallest input change that produces a detectable output change. In a digital system it is often set by the ADC step or encoder count, not the sensing element.
• Threshold and dead zone: the smallest input from zero that gives an output, and a band of input with no output response.
• Hysteresis: different outputs for the same input depending on whether the input is rising or falling; caused by friction, magnetic effects, elastic after-effect.
• Drift: change in output with time or temperature at constant input — zero drift shifts the whole curve, sensitivity drift changes its slope.

Dynamic characteristics (input changing with time). The sensor is modelled by an ordinary differential equation:

• Zero order: output follows input instantly (an ideal potentiometer).
• First order: one energy storage, e.g. a thermometer bulb (thermal capacitance and resistance) or an RC filter. Characterised by the static sensitivity K and the time constant τ. After a step, the output reaches 63.2 % of the change at t = τ, 95 % at 3τ and 98.2 % at 4τ.
• Second order: two energy stores, e.g. accelerometers, pressure diaphragms, galvanometers. Characterised by K, undamped natural frequency ωn and damping ratio ζ. ζ < 1 gives overshoot and ringing; ζ ≈ 0.6–0.7 is the usual design choice because it gives the flattest amplitude response over the widest band.
• Response time / settling time: time to enter and stay within a band (e.g. 2 % or 5 %) of the final value.
• Bandwidth: frequency range over which amplitude ratio stays within a stated tolerance (often −3 dB).
• Dynamic error and lag: for a ramp input a first-order sensor lags by τ in time and by (rate × τ) in reading; for a sine input it attenuates amplitude and adds phase lag.

Connections: the time constant links to RC filters in signal conditioning; the bandwidth sets the minimum sampling rate in data acquisition; second-order models return for accelerometers and pressure sensors.

Formulas

e = x_m − x_t ; % error of reading = 100·(x_m − x_t)/x_t ; % error of FS = 100·(x_m − x_t)/span

• x_m: measured value, x_t: true value (unit of the measurand). Applies to static readings.

K = Δq_o / Δq_i

• K: static sensitivity (output unit per input unit, e.g. V/°C); q_o: output; q_i: input. Valid on the linear part of the curve.

τ·dq_o/dt + q_o = K·q_i

• τ: time constant (s). First-order sensor model.

q_o(t) = q_f + (q_0 − q_f)·e^(−t/τ)

• q_0: initial reading, q_f: final reading, t: time since step (s). Step response of a first-order sensor.

|G(jω)| = K / √(1 + (ωτ)²) ; φ = −tan⁻¹(ωτ)

• ω: input angular frequency (rad/s) = 2πf. Sinusoidal steady state of a first-order sensor.

e_ss(ramp) = r·τ

• r: ramp rate of the input (unit/s). Steady-state lag error of a first-order sensor.

(1/ωn²)·d²q_o/dt² + (2ζ/ωn)·dq_o/dt + q_o = K·q_i

• ωn: natural frequency (rad/s), ζ: damping ratio (dimensionless). Second-order sensor.

M_p = e^(−ζπ/√(1 − ζ²)) (for 0 < ζ < 1)

• M_p: peak overshoot as a fraction of the step for a second-order sensor.

Worked examples

Example 1 (standard). A temperature probe with τ = 5 s, initially at 20 °C, is plunged into a bath at 60 °C. How long until it reads 50 °C?

1. Step response: T(t) = T_f + (T_0 − T_f)·e^(−t/τ).
2. Substitute: 50 = 60 + (20 − 60)·e^(−t/5) → −10 = −40·e^(−t/5) → e^(−t/5) = 0.25.
3. t = −5·ln(0.25) = 5 × 1.386 = 6.93 s.
4. Answer: t ≈ 6.93 s (that is 75 % of the step, between τ and 2τ, which is sensible).

Example 2 (GATE level). A first-order sensor has K = 1 and τ = 0.2 s. (a) Find the amplitude ratio and phase lag for a 1 Hz sinusoidal input. (b) What maximum τ would keep the amplitude error within 2 % at 5 Hz? (c) The input ramps at 10 °C/s. What is the steady-state reading error?

1. (a) ω = 2π × 1 = 6.283 rad/s; ωτ = 6.283 × 0.2 = 1.257.
2. Amplitude ratio = 1/√(1 + 1.257²) = 1/√2.579 = 0.623, so the sensor shows only 62 % of the true amplitude.
3. φ = −tan⁻¹(1.257) = −51.5°.
4. (b) Need 1/√(1 + (ωτ)²) ≥ 0.98 → (ωτ)² ≤ 1/0.98² − 1 = 0.0412 → ωτ ≤ 0.203.
5. ω = 2π × 5 = 31.42 rad/s → τ ≤ 0.203/31.42 = 0.00646 s.
6. (c) e_ss = r·τ = 10 × 0.2 = 2 °C (the reading lags 2 °C behind).
7. Answers: (a) 0.623, −51.5°; (b) τ ≤ 6.5 ms; (c) 2 °C lag.

Common mistakes

• Treating accuracy and precision as the same thing; a tight cluster of readings can still be biased.
• Quoting % FS error as % of reading — at 10 % of range a ±1 % FS error is ±10 % of reading.
• Confusing resolution (smallest detectable change) with sensitivity (slope).
• Using the 63.2 % rule for a second-order sensor; τ is defined only for first-order systems.
• Writing the step response with the wrong sign: q_o(t) = q_f + (q_0 − q_f)e^(−t/τ), not q_0·e^(−t/τ) when q_0 ≠ 0.
• Using f in Hz where ω in rad/s is needed (factor 2π).
• Assuming a low damping ratio is "faster" in a useful way — it overshoots and rings.

For GATE ME

Expect NAT questions on first-order step response (time to reach a reading, reading after time t), the amplitude ratio and phase lag of a first-order sensor under a sine input, ramp lag error, and second-order overshoot from ζ. MCQs test definitions: hysteresis vs drift, resolution vs sensitivity, accuracy vs precision, which systems are first or second order. Practise rearranging the exponential response quickly and keeping ω in rad/s.

Quick check

1. A sensor's readings are 50.4, 50.5, 50.4 °C for a true 48.0 °C. Is it precise, accurate, both or neither?
2. What fraction of a step change does a first-order sensor show after 3τ?
3. Name the static characteristic that differs between rising and falling input.
4. For a first-order sensor, what is the phase lag when ωτ = 1?
5. Why is ζ ≈ 0.7 preferred for second-order instruments?

Answers: 1. Precise but not accurate (bias ≈ 2.4 °C). 2. About 95 %. 3. Hysteresis. 4. 45°. 5. It gives the flattest amplitude response over the widest frequency band with small overshoot.