Sensors and Actuators

Chapter 2
Measurement Science and Sensor Characteristics

Bachelor Program in Mechatronics Engineering
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Measurement Science and Sensor Characteristics

Introduction

Sensors form the measurement interface between a physical process and a mechatronic control system. A temperature sensor, encoder, accelerometer, pressure transducer, or force cell converts a physical measurand into a usable signal. The quality of the resulting decision depends not only on the sensor itself, but also on calibration, signal conditioning, installation, sampling, and interpretation.

Measurement science, or metrology, provides the concepts needed to judge whether a measurement is trustworthy. In engineering, a sensor is never described adequately by its nominal range alone. Its accuracy, precision, sensitivity, resolution, dynamic response, environmental dependence, and interaction with the measured system must also be understood.

A useful measurement chain is:

physical measurand → sensing element → transduction → signal conditioning → data acquisition → computation → action

Each stage can introduce error. The engineer’s task is to characterize these effects, select suitable sensors, and design a measurement system whose uncertainty is acceptable for the application.

Learning Objectives

After studying this chapter, the reader should be able to:

  1. Distinguish accuracy, precision, sensitivity, resolution, repeatability, and related terms.
  2. Quantify static characteristics such as range, linearity, hysteresis, drift, dead band, saturation, loading, and cross-sensitivity.
  3. Explain calibration and uncertainty in practical sensor systems.
  4. Model zero-, first-, and second-order sensor dynamics.
  5. Calculate step responses, time constants, natural frequency, and damping effects.
  6. Select and evaluate sensors for a mechatronic application.

1. Measurement, Measurand, and Error

The measurand is the quantity intended to be measured, such as shaft angle, pressure, displacement, or temperature. The sensor output is ideally related to the measurand by a known transfer relationship. For a linear sensor,

y = Kx + b

where x is the measurand, y is the output, K is sensitivity or gain, and b is the zero offset. Units must be consistent: if x is measured in °C and y in volts, K has units V/°C.

The measured value differs from the true or reference value because of systematic and random effects. A practical error model is

e = ym − yr

where ym is the measured value and yr is the accepted reference value. Relative error is

er = (ym − yr) / yr

and percentage error is 100er percent, provided yr ≠ 0. For values near zero, absolute error or error relative to full scale is often more meaningful.

Systematic error tends to have a recognizable direction or pattern, such as zero offset or gain error. Random error varies unpredictably between observations. Calibration can reduce systematic error, whereas averaging may reduce random noise but cannot correct a persistent bias.

Accuracy and Precision

Accuracy describes closeness to an accepted reference value. Precision describes the closeness of repeated readings to one another. A sensor can be precise but inaccurate if it consistently reports a value with a large bias. It can also be approximately accurate on average but imprecise if readings fluctuate substantially.

For repeated readings x1, x2, …, xn, the sample mean is

x̄ = (1/n) ∑ᵢ₌₁ⁿ xᵢ

and the sample standard deviation is

s = √[∑ᵢ₌₁ⁿ (xᵢ − x̄)² / (n − 1)]

The standard deviation characterizes random spread under approximately independent, identically distributed observations. It does not by itself measure accuracy, because a systematic offset may remain.

Repeatability is agreement under the same conditions: same instrument, operator, location, procedure, and short time interval. Reproducibility concerns agreement when conditions change, such as operator, instrument, laboratory, or time.

2. Static Sensor Characteristics

Static characteristics describe the relationship between a steady input and the steady output after transients have disappeared.

Range and Span

The range is the interval over which a sensor is specified to operate, for example 0–10 bar or −40–125 °C. The span is the numerical difference between upper and lower range limits. A 0–10 bar sensor has a span of 10 bar; a −40–125 °C sensor has a span of 165 °C.

Operation outside the specified range can cause inaccurate readings, permanent damage, or unsafe behavior. The range should therefore include foreseeable transients, not merely the normal operating point.

Sensitivity and Gain

Sensitivity is the change in output per unit change in input:

S = Δy / Δx

For a differentiable nonlinear characteristic, local sensitivity is

S(x) = dy/dx

Sensitivity is not the same as accuracy. A highly sensitive sensor may still have large offset, hysteresis, or drift. Sensitivity may also vary with temperature, supply voltage, or operating point.

Resolution

Resolution is the smallest input change that can be distinguished reliably. In a digital measurement system with an ideal N-bit analog-to-digital converter and input span VFS, the ideal quantization step is

q = VFS / 2ᴺ

This is an ideal limit. Electrical noise, nonlinearities, reference instability, and sampling conditions may make practical resolution worse. A display with many digits does not necessarily provide equivalent measurement resolution.

Linearity

A sensor is linear if its input-output characteristic follows a straight line. Since the choice of reference line matters, specifications should state whether linearity is based on an independent, best-fit, or terminal-based line.

A common full-scale linearity error is

EL = (maximum deviation from reference line / full-scale span) × 100%

The numerator and denominator must use the same output units. Linearity error is a specification of deviation, not a claim that the sensor is unusable outside a narrow straight-line approximation.

Hysteresis

Hysteresis exists when the output at a given input depends on whether the input is increasing or decreasing. It may result from friction, magnetic remanence, elastic deformation, or material properties. Hysteresis error is often expressed as the maximum difference between upscale and downscale outputs at the same input, divided by full-scale span.

Hysteresis is not random noise. Repeating the same direction of motion may produce consistent results while reversing direction produces a different curve.

Zero Offset, Span Error, and Drift

A zero offset is an output error when the input is zero or at a defined reference value. A span error is an error in sensitivity or gain. In a linear model,

y = (K + ΔK)x + (b + Δb)

where ΔK is gain error and Δb is offset error.

Drift is a gradual change in output or sensitivity with time while the measurand remains constant. Zero drift changes the intercept; sensitivity drift changes the slope. Drift can arise from aging, thermal cycling, mechanical stress, contamination, or electronic component changes. It requires recalibration, compensation, or reference checking.

Dead Band and Threshold

A dead band is an interval of input change over which the output does not change appreciably. Mechanical backlash in a gear train is a common example. A threshold is the minimum input needed to produce a detectable output change from a defined starting state. Dead band and threshold reduce the ability to detect small changes and can cause control error, limit cycles, or missed events.

Saturation

Saturation occurs when output cannot increase further despite increasing input. For a bounded output,

y = ymax, x ≥ xsat

on the upper side, with an analogous lower limit. Saturation is important in amplifiers, actuators, ADCs, and sensors. It destroys proportional information and may cause integrator windup in feedback controllers. A sensor should be selected so that normal and transient inputs remain below saturation limits.

Loading Effect

A measurement device loads the system when its connection changes the measurand. An ideal voltmeter has infinite input resistance, but a real voltmeter has finite resistance. If a source with resistance Rs is measured by an instrument with input resistance Ri, the measured voltage for an ideal source voltage Vs is

Vm = Vs Ri / (Rs + Ri)

This follows the voltage-divider model and assumes linear resistances and negligible input capacitance. Loading is reduced by making Ri much larger than Rs. Similar effects occur in mechanical systems: attaching an accelerometer adds mass, and inserting a flow meter adds pressure loss.

Cross-Sensitivity

Cross-sensitivity is an output response to an unwanted input. For a multivariable sensor, a local linear model is

Δy = Sx Δx + Sz Δz

where x is the desired measurand, z is an interfering quantity, and Sz is cross-sensitivity. A load cell may respond to lateral force as well as axial force; a thermocouple may respond to electrical noise or reference-junction temperature.

Cross-sensitivity can be reduced through mechanical design, shielding, differential measurement, compensation sensors, calibration matrices, and software correction. Compensation is reliable only when the interfering variable is measured or sufficiently predictable.

3. Calibration and Uncertainty

Calibration compares a sensor with a traceable reference over specified input points. A calibration procedure should define the reference standard, environmental conditions, input sequence, settling time, loading arrangement, uncertainty, and acceptance criteria. Calibration does not make an instrument perfect; it establishes its behavior and evidence of conformity.

For independent uncertainty contributions u1, u2, …, un, the combined standard uncertainty, when contributions are independent, is commonly estimated by root-sum-square combination:

uc = √(u₁² + u₂² + … + uₙ²)

This assumes the quantities are expressed in compatible units and are uncorrelated. Correlated contributions require covariance terms. An expanded uncertainty may be reported as U = kuc, where k is a coverage factor selected according to the reporting convention and statistical assumptions.

An uncertainty statement should not be confused with an error guarantee. Error is the difference from the true value, which is generally unknown; uncertainty expresses knowledge about the possible value.

4. Dynamic Characteristics

Static specifications do not describe how quickly a sensor responds. Dynamic behavior matters when inputs vary rapidly, as in vibration measurement, robotic motion, and closed-loop control.

A sensor can be modeled by a differential equation relating input x(t) to output y(t). Under zero initial conditions, the transfer function is

G(s) = Y(s) / X(s)

where s is the Laplace-transform variable. Transfer functions are linear-system models and are applicable when the sensor operates within a region where linear approximation is reasonable.

Zero-Order Model

An ideal zero-order sensor has no dynamic lag:

y(t) = Kx(t)

or G(s) = K. Its output follows the input instantaneously. This is an approximation, because every physical sensor has finite thermal, mechanical, electrical, or computational response time. The model is useful when the sensor dynamics are much faster than the process dynamics.

First-Order Model

A first-order sensor is modeled by

τ dy(t)/dt + y(t) = Kx(t)

where τ is the time constant and K is the steady-state sensitivity. Its transfer function is

G(s) = K / (τs + 1)

For a step input x(t) = X₀u(t), with zero initial output,

y(t) = KX₀[1 − e^(−t/τ)], t ≥ 0

At t = τ, the output reaches 63.2% of its final value. Approximately 95% is reached after 3τ and 99.3% after 5τ. A smaller time constant means faster response, but not necessarily better noise performance.

The frequency response has magnitude

|G(jω)| = K / √[1 + (ωτ)²]

where ω is angular frequency in rad/s. The −3 dB cutoff angular frequency is ωc = 1/τ.

Second-Order Model

Many mechanical sensors and inertial devices behave approximately as second-order systems:

d²y/dt² + 2ζωₙ dy/dt + ωₙ²y = Kωₙ²x(t)

where ωₙ is natural angular frequency in rad/s, ζ is damping ratio, and K is steady-state gain. The transfer function is

G(s) = Kωₙ² / (s² + 2ζωₙs + ωₙ²)

For a unit step, the response depends on damping:

  • ζ > 1: overdamped, slow and non-oscillatory.
  • ζ = 1: critically damped, fastest response without oscillation.
  • 0 < ζ < 1: underdamped, oscillatory with overshoot.
  • ζ = 0: undamped ideal oscillation.

For the underdamped case, define ωd = ωₙ√(1 − ζ²). The normalized unit-step response is

y(t)/K = 1 − e^(−ζωₙt)[cos(ωdt) + (ζ/√(1−ζ²))sin(ωdt)]

This expression assumes zero initial conditions and a stable linear model with 0 < ζ < 1. The percent overshoot is

Mp = 100e^[−ζπ/√(1−ζ²)] %

The formula applies to a standard underdamped second-order step response. Increasing damping reduces overshoot but may increase settling time depending on the design target.

Dynamic Error and Bandwidth

A sensor may have excellent static accuracy but large dynamic error if it cannot follow the input. For example, a temperature sensor exposed to a rapidly changing gas temperature reports a filtered version of the actual temperature. The engineer must compare sensor bandwidth with the highest significant frequency of the measurand. Sampling frequency must also be high enough to avoid aliasing; an anti-aliasing filter is normally required before digitization.

5. Worked Examples

Example 1: Accuracy, precision, and uncertainty

A calibrated displacement reference is 50.00 mm. Three readings are 50.12, 50.10, and 50.11 mm.

The mean is

x̄ = (50.12 + 50.10 + 50.11)/3 = 50.11 mm

The mean error is 0.11 mm, or 0.22% of the reference. The readings are closely grouped, so precision is high, but the group is offset from the reference; accuracy is limited by bias. The sample standard deviation is approximately 0.010 mm. This illustrates why precision cannot be substituted for accuracy.

Example 2: First-order temperature sensor

A temperature sensor has K = 0.010 V/°C and τ = 4 s. A step from 20 °C to 70 °C is applied. The final output change is

Δy_∞ = KΔx = 0.010 × 50 = 0.50 V

After 8 s, which equals 2τ,

Δy(8) = 0.50[1 − e^(−2)] ≈ 0.432 V

Thus the sensor has achieved approximately 86.5% of its final change. If the initial output at 20 °C is 1.00 V, the output after 8 s is approximately 1.432 V.

Example 3: Loading

A source has Vs = 5.00 V and Rs = 10 kΩ. An instrument has Ri = 90 kΩ. Therefore,

Vm = 5.00 × 90/(10 + 90) = 4.50 V

The 0.50 V reduction is a 10% loading error. Increasing input resistance to 1 MΩ would reduce the error substantially.

6. Applied Case: Selecting a Sensor for a Robotic Joint

Background

An industrial robotic joint must measure angular position over ±180°, with a control bandwidth of 20 Hz. The joint operates in a dusty environment and may experience vibration and temperature changes.

Problem

A low-cost potentiometer offers simple voltage output and adequate static resolution, but its wiper introduces wear, friction, dead band, and possible loading. An optical encoder provides high resolution and low mechanical loading, but requires digital electronics and protection from contamination. A magnetic encoder is more tolerant of dust but may exhibit temperature-dependent offset and magnetic cross-sensitivity.

Analysis

The required sensor must have a range exceeding the mechanical travel, resolution finer than the allowable position error, and bandwidth comfortably above 20 Hz. Static accuracy alone is insufficient: backlash, hysteresis, sampling delay, and vibration-induced errors can destabilize the position loop. The mounting must avoid shaft misalignment, which can produce cyclic error and bearing load.

A suitable design might use a sealed magnetic absolute encoder with a specified accuracy better than the allowable joint error, a sampling rate well above 40 samples/s, and a digital diagnostic channel. The exact sampling rate should be chosen with respect to control design and anti-aliasing requirements, not merely the Nyquist minimum.

Approach and Solution

The engineering team should:

  1. Define the measurand, allowable error, range, bandwidth, environmental limits, and safety requirements.
  2. Compare candidate data sheets using common units and test conditions.
  3. Calibrate zero and scale after installation.
  4. Characterize repeatability, hysteresis, temperature drift, and dynamic delay.
  5. Implement plausibility checks, such as rate limits and redundant position limits.
  6. Validate the complete sensor–controller–actuator loop under vibration and thermal cycling.

Outcome

The selected encoder meets the required position accuracy and bandwidth, while its sealed housing reduces contamination risk. Temperature compensation is implemented using an internal temperature measurement and a calibration map. A failure-detection routine places the robot in a safe state if the encoder signal becomes invalid or inconsistent with the commanded motion.

Learning Points

The best sensor is not necessarily the one with the greatest resolution. Selection must consider accuracy, dynamic response, loading, environmental robustness, maintainability, diagnostics, safety, interface compatibility, and total system cost.

7. Practical Design Guidance and Limitations

A professional measurement design should:

  • define the measurand and acceptable uncertainty before selecting hardware;
  • distinguish sensor error from signal-conditioning and software error;
  • avoid operating near saturation or range limits unless characterized;
  • assess loading in mechanical, electrical, thermal, and fluid systems;
  • calibrate across the actual operating range and environmental conditions;
  • record traceability, calibration date, uncertainty, and adjustment history;
  • use shielding, grounding, filtering, and differential interfaces appropriately;
  • verify dynamic response, not only a steady-state reading;
  • consider failure modes, diagnostics, and safe-state behavior;
  • avoid reporting excessive digits unsupported by resolution or uncertainty.

Models are approximations. A first-order model may fail near resonance, saturation, hysteresis, or strong temperature variation. A second-order model may omit nonlinear friction, structural modes, sensor electronics, and sampling delay. Therefore, model parameters should be identified experimentally and validated against the intended operating conditions.

8. Contemporary Developments and Professional Responsibility

Modern sensors increasingly include embedded calibration, digital communication, self-diagnostics, compensation tables, and sensor fusion. MEMS inertial sensors combine accelerometers and gyroscopes with algorithms that estimate orientation. Machine-learning methods can compensate nonlinearities or detect abnormal sensor behavior, but they must not conceal poor calibration or produce unexplained safety-critical decisions.

Measurement data used in automated systems should be traceable, time-stamped, and protected against unauthorized alteration. Engineers must understand applicable standards for calibration, functional safety, electromagnetic compatibility, and machinery safety. A sensor failure can cause physical harm; consequently, redundancy, diagnostic coverage, fail-safe design, and documented validation may be more important than nominal precision.

Chapter Summary

Measurement science provides the language and models needed to evaluate sensors in real engineering systems. Accuracy concerns closeness to an accepted reference, while precision concerns repeatability of readings. Sensitivity describes output change per input change, and resolution describes the smallest distinguishable input change. Range, span, linearity, hysteresis, zero offset, span error, drift, dead band, saturation, loading, and cross-sensitivity describe different limitations and must not be treated as interchangeable.

Calibration establishes sensor behavior relative to a traceable reference, while uncertainty expresses the quality of knowledge about the measured value. Dynamic performance is modeled using zero-, first-, and second-order systems. The time constant characterizes first-order speed; natural frequency and damping ratio characterize second-order response. Sensor selection therefore requires simultaneous consideration of static accuracy, dynamic bandwidth, environment, loading, interface, safety, maintainability, and system-level consequences.

Key Terms

Measurand: Quantity intended to be measured.

Accuracy: Closeness of a measurement to an accepted reference value.

Precision: Closeness of repeated measurements to one another.

Sensitivity: Change in output per unit change in input.

Resolution: Smallest input change that can be reliably distinguished.

Range: Interval of input values for which a sensor is specified.

Span: Difference between upper and lower range limits.

Linearity: Agreement of a sensor characteristic with a specified straight-line reference.

Hysteresis: Dependence of output on the direction of input change.

Repeatability: Agreement under unchanged measurement conditions.

Reproducibility: Agreement when specified measurement conditions change.

Drift: Gradual output or sensitivity change with time at constant input.

Dead band: Input interval producing negligible output change.

Saturation: Limiting condition in which output cannot increase further.

Loading effect: Measurement-induced alteration of the measurand.

Cross-sensitivity: Response to an unwanted or interfering input.

Time constant: First-order time parameter controlling transient response.

Natural frequency: Characteristic oscillation frequency of a second-order model.

Damping ratio: Parameter describing decay and oscillation in a second-order system.

Calibration: Comparison with a traceable reference under stated conditions.

Review Questions

Short Answer Questions

  1. Distinguish accuracy from precision.
  2. What is the difference between range and span?
  3. Explain hysteresis and give one physical cause.
  4. Why can a sensor with high resolution still be inaccurate?
  5. What does the time constant represent in a first-order sensor?

Long Answer Questions

  1. Explain the principal static characteristics of a sensor and discuss how they affect sensor selection.
  2. Derive the first-order step response from the first-order transfer function and interpret the result at one time constant.
  3. Compare overdamped, critically damped, and underdamped second-order responses.
  4. Discuss loading and cross-sensitivity in electrical and mechanical measurement systems.
  5. Describe a complete calibration and validation procedure for a sensor used in industrial automation.

Analytical / Application Questions

  1. A sensor has a time constant of 2 s and a steady-state gain of 0.02 V/°C. For a 30 °C step, calculate its output change after 4 s and explain whether the sensor is suitable for a process changing significantly within 1 s.
  2. A voltage source has internal resistance 4 kΩ and open-circuit voltage 3.3 V. Calculate the reading of a 36 kΩ instrument and identify the loading error relative to the open-circuit value.
  3. A robotic actuator requires angular measurement with low hysteresis, high bandwidth, and operation in a dusty environment. Compare a potentiometer, optical encoder, and sealed magnetic encoder, and justify a selection using at least four sensor characteristics.

Multiple Choice Questions with Answers

Question 1

Which statement best distinguishes accuracy from precision?

A. Accuracy is closeness to a reference; precision is closeness among repeated readings.

B. Accuracy is output range; precision is sensor sensitivity.

C. Accuracy concerns response speed; precision concerns hysteresis.

D. Accuracy and precision are interchangeable terms.

Correct Answer: A. Accuracy is closeness to a reference; precision is closeness among repeated readings.

Question 2

A sensor output changes by 2 V when its input changes by 10 units. What is its average sensitivity?

A. 0.02 V/unit

B. 0.2 V/unit

C. 5 V/unit

D. 20 V/unit

Correct Answer: B. 0.2 V/unit

Question 3

For an ideal N-bit converter with full-scale input span V_FS, the quantization step is:

A. V_FS × 2ᴺ

B. N/V_FS

C. V_FS/2ᴺ

D. 2ᴺ/V_FS

Correct Answer: C. V_FS/2ᴺ

Question 4

At one time constant, the step response of a first-order sensor reaches approximately:

A. 25.0% of its final value

B. 50.0% of its final value

C. 63.2% of its final value

D. 99.3% of its final value

Correct Answer: C. 63.2% of its final value

Question 5

Which condition describes sensor saturation?

A. Output changes randomly around its mean

B. Output cannot increase further despite increasing input

C. Output differs on increasing and decreasing input

D. Output changes proportionally with input

Correct Answer: B. Output cannot increase further despite increasing input

Question 6

A source has 5 V, source resistance 10 kΩ, and a measuring instrument resistance 90 kΩ. What voltage does the instrument read?

A. 0.50 V

B. 4.00 V

C. 4.50 V

D. 5.50 V

Correct Answer: C. 4.50 V

Question 7

Which characteristic occurs when output depends on whether the input is increasing or decreasing?

A. Resolution

B. Hysteresis

C. Bandwidth

D. Quantization

Correct Answer: B. Hysteresis

Question 8

For a standard underdamped second-order system, increasing damping ratio generally:

A. Increases overshoot

B. Reduces overshoot

C. Makes natural frequency infinite

D. Creates saturation

Correct Answer: B. Reduces overshoot

Question 9

Cross-sensitivity means that a sensor:

A. Has no response below its range

B. Responds to an unwanted input in addition to the desired measurand

C. Always has zero offset

D. Has a perfectly linear response

Correct Answer: B. Responds to an unwanted input in addition to the desired measurand

Question 10

Which action most directly reduces systematic zero offset?

A. Increasing display digits

B. Averaging many biased readings

C. Calibration or zero adjustment

D. Increasing sampling noise

Correct Answer: C. Calibration or zero adjustment
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