2. Static and Dynamic Characteristics of Instruments

Introduction and Context

An electrical measuring instrument is an information-processing system. It senses a physical quantity, converts that quantity into a usable signal, and presents an indication or transfers data to another system. The quality of measurement therefore depends not only on the sensing element, but also on the instrument’s calibration, mechanical or electronic response, environment, loading, and signal-processing characteristics.

Instrument characteristics are commonly divided into static characteristics and dynamic characteristics. Static characteristics describe performance when the measured quantity is constant or changes so slowly that transient effects can be neglected. Dynamic characteristics describe the relationship between an input that varies with time and the corresponding instrument output. This distinction is fundamental in selecting instruments, interpreting readings, designing control systems, and estimating measurement uncertainty.

A digital data-acquisition system, for example, may have excellent static accuracy but fail to reproduce a rapidly changing waveform because of limited bandwidth. Conversely, a fast sensor may track transients well but exhibit offset, drift, or poor linearity. Instrument evaluation must therefore consider both domains.

Learning Objectives

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

1. distinguish static and dynamic characteristics of measuring instruments;

2. formulate zero-order, first-order, and second-order instrument models;

3. evaluate accuracy, precision, sensitivity, resolution, threshold, hysteresis, linearity, drift, and dead zone;

4. derive and interpret step, ramp, impulse, and sinusoidal responses;

5. relate time constant, damping ratio, natural frequency, bandwidth, and fidelity;

6. identify environmental and loading effects on measurement quality;

7. select suitable instruments for steady-state and transient measurements;

8. diagnose common errors and recommend appropriate calibration and compensation methods.

Static Characteristics: What a Steady Reading Reveals

When the input quantity is constant, an instrument eventually reaches a steady indication. The static characteristic is the relationship between the steady output and the applied input. Ideally,

\[ y = Kx \]

where x is the measurand, y is the output, and K is the sensitivity or scale factor. Real instruments depart from this ideal relationship because of component tolerances, friction, temperature, aging, loading, and imperfect calibration.

Accuracy and Error

Accuracy expresses the closeness of an indicated value to the accepted reference value. It is not normally a single property independent of conditions; it is specified over a range, under stated environmental and operating conditions.

If the true or reference value is x_t and the measured value is x_m, the absolute error is

\[ e = x_m - x_t \]

The magnitude of error is often expressed as a percentage of the true value:

\[ \%\,error = \frac{|x_m-x_t|}{|x_t|}\times100 \]

Instrument specifications may instead use percentage of full-scale deflection. This distinction is important. An instrument with an error of ±1% of full scale has a relatively large percentage error when measuring a small value near zero.

Systematic errors have a reproducible pattern and may result from calibration, loading, or environmental effects. Random errors vary unpredictably and are reduced, but not eliminated, by repeated observations and statistical treatment. Accuracy should not be confused with precision: precision describes the closeness of repeated readings to one another, whereas accuracy describes closeness to the reference value.

Sensitivity and Scale Factor

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

\[ S = \frac{dy}{dx} \]

For a linear instrument, S is constant. For a nonlinear instrument, sensitivity depends on the operating point. The reciprocal quantity, input divided by output, is sometimes called the scale factor. High sensitivity is useful when small changes must be detected, but it may increase susceptibility to noise, loading, and environmental disturbances.

Linearity

An instrument is linear if its calibration curve follows a specified straight-line relationship. Since different reference lines can be used, the stated method matters. Common definitions use the ideal line through end points, the best-fit straight line, or a zero-based full-scale line.

If the maximum deviation of the calibration curve from the selected reference line is e_max and the full-scale output is y_FS, nonlinearity may be expressed as

\[ \%\,nonlinearity = \frac{e_{max}}{y_{FS}}\times100 \]

Linearity is especially significant in computer-based systems because a nonlinear sensor requires correction, calibration tables, or polynomial compensation. A high-order correction may improve accuracy within the calibration range but can produce unreliable extrapolation outside it.

Repeatability, Reproducibility, and Precision

Repeatability is the closeness of results obtained under the same conditions over a short period. Reproducibility concerns agreement when one or more conditions change, such as operator, instrument, laboratory, or time. Precision is a broader description of dispersion and may be quantified using standard deviation or variance.

A set of readings can be precise but inaccurate if a constant calibration error is present. It can also be accurate on average but imprecise if random noise is large. Good measurement practice seeks both low bias and low dispersion.

Resolution, Threshold, and Dead Zone

Resolution is the smallest input change that produces a discernible output change. In a digital instrument, it is often associated with the least significant count, although display resolution is not necessarily the same as measurement accuracy.

Threshold is the minimum input required to produce a detectable output. A dead zone is an interval over which input changes cause no output change. Dead zones may arise from friction, backlash, comparator hysteresis, quantization, or signal-conditioning circuits. They are particularly harmful when measuring small oscillations or reversing direction.

Hysteresis

Hysteresis occurs when the output for a given input depends on whether the input is increasing or decreasing. It is commonly caused by mechanical friction, magnetic remanence, elastic after-effect, or material properties.

For an input x, let y_u and y_d be outputs obtained during upscale and downscale measurements. The hysteresis difference is

\[ h(x)=y_u(x)-y_d(x) \]

The maximum value over the range, expressed as a percentage of full scale, is a common hysteresis specification. Hysteresis is not removed simply by averaging readings; the direction history must be controlled or modeled.

Drift and Stability

Drift is a gradual change in output when the input and operating conditions are nominally constant. Zero drift changes the output intercept, while sensitivity drift changes the slope of the calibration curve. Drift may be caused by temperature, aging, humidity, mechanical stress, battery voltage, or component migration.

Stability describes the ability to maintain characteristics over time. Warm-up periods, periodic recalibration, temperature compensation, reference standards, and redundant measurements are common methods of controlling drift.

Environmental Effects and Loading

Temperature affects resistance, semiconductor junction properties, dimensions, viscosity, and magnetic characteristics. Humidity can alter insulation resistance and cause leakage or corrosion. Vibration and shock can create spurious outputs or alter mechanical alignment. Electromagnetic interference may couple into signal leads, while changes in supply voltage can change sensitivity or offset.

The act of connecting an instrument can alter the measurand. A voltmeter with finite input resistance draws current and changes the voltage in a resistive network. An ammeter with nonzero burden voltage changes the circuit current. This is called loading error. A high-input-impedance voltage measurement, a low-resistance current shunt, buffering, isolation, and four-wire measurement are methods used to reduce loading.

Mathematical Models of Instrument Dynamics

Static specifications do not describe how quickly an instrument responds. Dynamic behavior is represented by a differential equation or transfer function. For a linear time-invariant instrument,

\[ H(s)=\frac{Y(s)}{X(s)} \]

where X(s) and Y(s) are Laplace transforms of input and output, and H(s) is the transfer function under zero initial conditions.

Zero-Order Instrument

A zero-order instrument has no energy storage and is modeled by

\[ y(t)=Kx(t) \]

or

\[ H(s)=K \]

Its output follows the input instantaneously, apart from the scale factor. An ideal resistive voltage divider or an appropriately modeled potentiometric sensor may approximate zero-order behavior over a limited frequency range. Real instruments can behave as zero-order systems only when the input variation is sufficiently slow compared with their internal dynamics.

First-Order Instrument

A first-order instrument is described by

\[ \tau\frac{dy(t)}{dt}+y(t)=Kx(t) \]

and has transfer function

\[ H(s)=\frac{K}{\tau s+1} \]

Here, τ is the time constant. For a step input of amplitude X_0,

\[ y(t)=KX_0\left(1-e^{-t/\tau}\right) \]

The output reaches approximately 63.2% of its final value after one time constant, 86.5% after two, 95.0% after three, 98.2% after four, and 99.3% after five. These values provide a practical settling-time interpretation.

For a sinusoidal input, the magnitude ratio and phase angle are

\[ |H(j\omega)|=\frac{K}{\sqrt{1+(\omega\tau)^2}} \]

\[ \phi=-\tan^{-1}(\omega\tau) \]

The corner frequency is ω_c=1/τ radians per second, at which the amplitude is 1/√2 of its low-frequency value and the phase lag is 45 degrees. Thus, a first-order instrument is a low-pass system: it attenuates high-frequency components and delays them.

Second-Order Instrument

A standard second-order model is

\[ \frac{d^2y}{dt^2}+2\zeta\omega_n\frac{dy}{dt}+\omega_n^2y=K\omega_n^2x(t) \]

with transfer function

\[ H(s)=\frac{K\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2} \]

where ω_n is the undamped natural frequency and ζ is the damping ratio. The damping ratio determines the character of the response:

• ζ=0: undamped oscillation;
• 0<ζ<1: underdamped response with overshoot and ringing;
• ζ=1: critically damped response without overshoot and with the fastest nonoscillatory behavior;
• ζ>1: overdamped response, slower and nonoscillatory.

For 0<ζ<1, the damped natural frequency is

\[ \omega_d=\omega_n\sqrt{1-\zeta^2} \]

For a unit step, the normalized underdamped response is

\[ y_n(t)=1-\frac{e^{-\zeta\omega_nt}}{\sqrt{1-\zeta^2}}\sin(\omega_dt+\theta) \]

where θ=cos⁻¹ζ. The percentage overshoot is

\[ M_p=100e^{-\zeta\pi/\sqrt{1-\zeta^2}} \]

A low damping ratio gives rapid initial movement but significant overshoot. A high damping ratio reduces overshoot but increases settling time. Instrument designers choose damping according to the measurement task: a vibration sensor may require controlled resonance and bandwidth, while a pointer instrument may prioritize a quick readable indication without oscillation.

Dynamic Performance: Fidelity, Speed, and Bandwidth

Fidelity

Fidelity is the ability of an instrument to reproduce the changes in the measured quantity without distortion. A system has high fidelity over a frequency range if its amplitude ratio is nearly constant and its phase shift is predictable or small throughout that range.

An instrument with insufficient bandwidth suppresses rapid changes and rounds sharp transitions. An instrument with excessive resonant peaking may exaggerate particular frequency components. Fidelity is therefore not identical to speed: a fast but resonant instrument may respond quickly while distorting the signal.

Rise Time, Settling Time, and Overshoot

Rise time is the time required for the output to move between specified percentages, commonly 10% and 90%, of its final value. Settling time is the time required for the output to remain within a specified error band, such as ±2% or ±5%. Overshoot is the amount by which the response exceeds its final steady value.

For a first-order system, the 10–90% rise time is approximately 2.2τ and the 2% settling time is approximately 4τ. For a second-order system, these quantities depend strongly on ζ and ω_n. Reporting dynamic performance without stating the tolerance band can lead to misleading comparisons.

Frequency Response and Bandwidth

Frequency response describes the steady-state output magnitude and phase for sinusoidal inputs of different frequencies. The bandwidth is the frequency range over which the instrument meets a specified response criterion. For a low-pass instrument, the −3 dB frequency is often used as the upper cutoff.

A measurement system must have adequate bandwidth relative to the highest significant frequency in the measurand. For a periodic waveform, the fundamental frequency alone is not sufficient: sharp edges and nonsinusoidal shapes contain harmonics. An oscilloscope or data-acquisition channel that passes the fundamental but attenuates higher harmonics may display the correct period but an incorrect waveform shape.

Response to Ramp and Impulse Inputs

Step tests are convenient, but many physical measurements are ramps, pulses, or stochastic signals. For a first-order system with ramp input x(t)=at, the output eventually follows the ramp with a constant lag approximately equal to aτ, apart from the initial transient. This shows why a temperature sensor can appear to be continuously behind a steadily changing process.

The impulse response is the output produced by an ideal impulse input. It is a fundamental system descriptor because convolution of the input with the impulse response gives the output of a linear time-invariant system. In practice, short calibrated pulses or broadband excitation are used to estimate dynamic behavior.

Measurement Errors as a Combined Model

A useful practical model separates the indicated value into several components:

\[ y_m=f(x)+b+d(t)+h(x,\dot{x})+n(t)+e_L \]

Here f(x) represents the nominal calibration relation, b is a constant bias, d(t) is drift, h represents hysteresis, n(t) is random noise, and e_L is loading or interaction error. Dynamic distortion must also be included when x changes rapidly.

This model helps organize calibration. A static calibration can estimate f(x), bias, nonlinearity, and hysteresis. A time-dependent test can estimate τ, ω_n, ζ, delay, and bandwidth. Repeated measurements can characterize noise. Environmental tests reveal temperature coefficients, humidity effects, and vibration sensitivity.

Practical Selection and Characterization of Instruments

Instrument selection begins with the measurand and the decision that the measurement must support. The engineer should identify the expected range, rate of change, required uncertainty, allowable loading, environmental conditions, safety category, and interface requirements.

A static calibration usually applies known reference inputs at several points during increasing and decreasing cycles. The resulting data can be used to determine sensitivity, offset, nonlinearity, hysteresis, repeatability, and uncertainty. A dynamic characterization applies a known step, ramp, pulse, or sinusoidal sweep and estimates the relevant model parameters.

For a sensor connected to a digital acquisition system, the sensor bandwidth, signal-conditioning bandwidth, sampling rate, anti-aliasing filter, quantization, and computational delay must be considered together. Sampling faster than the signal bandwidth is necessary but does not by itself prevent aliasing; an analog anti-aliasing filter is normally required before sampling.

Digital calibration may use a lookup table, piecewise-linear interpolation, polynomial correction, or model-based compensation. Such corrections must be traceable to reference standards and validated with independent test points. Excessive polynomial order can fit calibration noise rather than the true characteristic.

Illustrative Case Study: Selecting a Current Sensor for a Motor Drive

Background

A postgraduate engineering laboratory is instrumenting a 15 kW variable-speed induction motor drive. The current waveform contains a fundamental component whose frequency changes with speed, switching ripple from the power converter, and short-duration transients during acceleration.

Situation and Problem

A conventional moving-coil meter gives stable readings but cannot display the transient current. A current transformer provides isolation and good high-frequency performance, but its response near very low frequency and during saturation is unsuitable for all operating conditions. A Hall-effect transducer offers isolation and a broad usable range but has offset drift and temperature dependence.

The team must select a measurement chain for control validation and a separate chain for routine energy monitoring.

Analysis

For routine monitoring, the principal requirement is accurate average or RMS current over a relatively slow time scale. Static accuracy, thermal stability, burden, and isolation are more important than microsecond response. For control validation, the waveform and transient current are required. The sensor and acquisition system must therefore have adequate bandwidth, controlled phase response, and sufficient sampling rate.

The team performs a static calibration at zero, 25%, 50%, 75%, and 100% of rated current, both increasing and decreasing. It records offset, sensitivity, nonlinearity, hysteresis, and temperature coefficient. A step-current test is then applied to estimate rise time and settling time. A frequency sweep identifies attenuation and phase shift.

Approach and Solution

A Hall-effect sensor with closed-loop compensation is selected for the control-validation channel because it provides isolation and acceptable dynamic response. The signal-conditioning amplifier is checked for saturation and bandwidth. An anti-aliasing filter is placed before the data-acquisition input, and the sampling rate is selected with the highest relevant harmonic in mind.

For routine monitoring, a calibrated RMS transducer with lower bandwidth but improved long-term stability is used. The two channels are not treated as interchangeable: their calibration records, uncertainty budgets, and intended uses are separately documented.

Outcome

The laboratory avoids the common error of choosing one instrument solely on the basis of full-scale accuracy. The dynamic channel reproduces acceleration transients with acceptable fidelity, while the monitoring channel provides stable operational readings. Temperature tests reveal a measurable zero drift, so an automatic zero check is included during scheduled shutdowns.

Learning Points

The case illustrates that instrument choice is a system-level decision. Static accuracy cannot compensate for inadequate bandwidth, and bandwidth cannot compensate for excessive drift or poor calibration. Requirements should be written in terms of the measurement task rather than the sensor name alone.

Challenges, Limitations, and Common Mistakes

Several errors recur in the evaluation of instrument characteristics. Treating displayed resolution as accuracy produces unjustified confidence. Comparing percentage full-scale specifications with percentage-of-reading specifications without conversion leads to incorrect conclusions. Measuring a dynamic quantity with a slow instrument can create a plausible but distorted waveform. Ignoring phase shift is particularly dangerous in power, vibration, and control measurements.

Another mistake is estimating a first-order time constant from an output that is actually limited by the acquisition filter or display update rate. Mechanical hysteresis may be mistaken for random noise if the direction of input change is not recorded. Drift may be hidden by recalibrating before its pattern is understood. Finally, a calibration curve obtained under laboratory conditions may not represent field performance if temperature, vibration, electromagnetic interference, or installation loading differs substantially.

Every specification has a domain of validity. Linear models are approximations, and a transfer function identified at one amplitude or operating point may not apply near saturation, magnetic core nonlinearity, friction breakaway, or thermal limits.

Professional, Safety, and Governance Considerations

Measurement results often support protection decisions, compliance reports, equipment acceptance, and financial or operational claims. Calibration records should therefore identify the reference standard, method, date, environmental conditions, uncertainty, and responsible personnel. Traceability means that the result can be related through an unbroken documented chain of calibrations to an appropriate reference.

Safety-related measurements require more than nominal accuracy. Isolation rating, creepage and clearance, overvoltage category, fault energy, insulation condition, and fail-safe behavior must be considered. A sensor or instrument should never be connected to a hazardous circuit solely because its voltage or current range appears adequate.

Professional integrity requires reporting unfavorable dynamic behavior, drift, saturation, and uncertainty rather than selectively presenting only stable readings. Automated systems should retain raw data, time stamps, calibration status, and alarm records where the measurement influences safety or regulatory decisions.

Digital Instrumentation and Emerging Practice

Modern instruments combine sensors, analog front ends, embedded processors, communication interfaces, and software compensation. Digital filtering can improve noise performance, but it also introduces delay and may conceal transients. Engineers should distinguish analog bandwidth from digital bandwidth and document filter type, cutoff frequency, phase response, and latency.

Model-based diagnostics can compare measured response with an expected zero-, first-, or second-order model to identify sensor aging, mounting faults, or abnormal damping. Machine-learning methods may classify complex faults, but they do not remove the need for traceable calibration or physical validation. A data-driven correction is trustworthy only within the range represented by its training and validation data.

Networked instruments also introduce clock synchronization, packet loss, cybersecurity, and data-integrity concerns. In a distributed measurement system, a numerically accurate value with an incorrect time stamp may be less useful than a slightly less accurate value that is correctly synchronized with other channels.

Best Practices for Engineering Use

A reliable characterization program should:

• define the measurand, range, uncertainty, response time, and environmental envelope before selecting hardware;
• separate static calibration from dynamic testing;
• state the reference line and error definition used for linearity;
• test both increasing and decreasing inputs to reveal hysteresis;
• record warm-up time, temperature, supply conditions, and installation details;
• evaluate the complete measurement chain, including sensor, cable, amplifier, filter, converter, and software;
• verify bandwidth, phase shift, delay, and sampling adequacy for time-varying signals;
• maintain traceable calibration records and uncertainty estimates;
• validate compensation using test data not used to create the correction;
• report limitations and operating boundaries with the result.

Chapter Summary

Static characteristics describe an instrument under constant or slowly changing input conditions. Accuracy, precision, sensitivity, resolution, linearity, hysteresis, drift, threshold, dead zone, repeatability, and environmental influence determine the quality of a steady measurement. Loading must be considered because the instrument can alter the circuit or process being measured.

Dynamic characteristics describe time-dependent behavior. Zero-order instruments reproduce the input through a constant scale factor. First-order instruments are governed by a time constant and exhibit exponential step response, attenuation, and phase lag. Second-order instruments are characterized by natural frequency and damping ratio; their response may be underdamped, critically damped, or overdamped. Rise time, settling time, overshoot, fidelity, and bandwidth are essential in assessing transient performance.

Instrument selection is a system-level activity. Sensor behavior, signal conditioning, filtering, sampling, software, environment, safety, and calibration must be evaluated together. Digital compensation and diagnostic methods are valuable, but they require traceability, validation, and awareness of latency and model limitations.

Key Terms

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

Bandwidth: Frequency range over which an instrument satisfies specified amplitude or phase requirements.

Dead zone: Input interval over which the output does not change appreciably.

Drift: Gradual output change with constant input and nominally constant conditions.

Damping ratio: Dimensionless parameter describing energy dissipation in a second-order system.

Dynamic characteristic: Instrument behavior when the input varies with time.

Fidelity: Ability to reproduce input variations without unacceptable distortion.

Hysteresis: Dependence of output on the previous direction or history of input change.

Linearity: Degree to which the calibration relation follows a specified straight line.

Loading error: Error caused by the instrument altering the system being measured.

Natural frequency: Characteristic frequency of an undamped second-order system.

Precision: Closeness of repeated measurements to one another.

Resolution: Smallest input change that produces a discernible output change.

Sensitivity: Change in output per unit change in input.

Settling time: Time required for output to remain within a specified final-value band.

Static characteristic: Instrument behavior for constant or slowly varying input.

Time constant: First-order parameter determining the speed of exponential response.

Transfer function: Ratio of output transform to input transform under zero initial conditions.

Review Questions

Short Answer Questions

1. Distinguish accuracy from precision.

2. What is meant by sensitivity drift?

3. Define hysteresis and state one physical cause.

4. What does the time constant signify in a first-order instrument?

5. Distinguish bandwidth from fidelity.

Long Answer Questions

1. Explain the principal static characteristics of an instrument and show how they influence calibration quality.

2. Derive the step response and sinusoidal frequency response of a first-order instrument.

3. Explain the effect of damping ratio on the response of a second-order instrument.

4. Discuss environmental effects, loading error, and methods of compensation.

5. Describe a complete procedure for static and dynamic characterization of a sensor system.

Analytical / Application Questions

1. A first-order sensor has a time constant of 0.20 s and a sensitivity of 2 V per unit input. Determine its output at 0.20 s and 0.60 s after a unit step, and estimate its 10–90% rise time.

2. A measurement channel has a natural frequency of 100 rad/s and damping ratio 0.5. Determine whether its step response overshoots and calculate the approximate percentage overshoot.

3. An instrument is specified as ±1% of full scale on a 100 V range. Evaluate the worst-case percentage error when it measures 10 V, and explain whether a lower-range instrument would be preferable.

22. Multiple Choice Questions with Answers

Question 1

Which characteristic describes the closeness of repeated readings to one another?

A. Accuracy

B. Precision

C. Sensitivity

D. Linearity

Correct Answer: B. Precision

Question 2

A first-order instrument reaches approximately what percentage of its final step response after one time constant?

A. 37.0%

B. 50.0%

C. 63.2%

D. 86.5%

Correct Answer: C. 63.2%

Question 3

For a first-order instrument, increasing the time constant generally causes the instrument to

A. respond faster and increase bandwidth

B. respond more slowly and reduce bandwidth

C. eliminate phase lag

D. become a second-order system

Correct Answer: B. respond more slowly and reduce bandwidth

Question 4

A second-order instrument with damping ratio less than one is described as

A. underdamped

B. critically damped

C. overdamped

D. zero-order

Correct Answer: A. underdamped

Question 5

The output of an instrument differs for increasing and decreasing input because of

A. resolution

B. hysteresis

C. bandwidth

D. sampling

Correct Answer: B. hysteresis

Question 6

An ammeter changes the circuit current because of its internal resistance. This is an example of

A. random error

B. loading error

C. zero drift

D. quantization error

Correct Answer: B. loading error

Question 7

At the corner frequency of a first-order low-pass instrument, the magnitude is approximately

A. equal to zero

B. twice its low-frequency value

C. 1/square root of 2 of its low-frequency value

D. independent of frequency

Correct Answer: C. 1/square root of 2 of its low-frequency value

Question 8

Which specification is most directly associated with the ability to reproduce rapid waveform changes?

A. Bandwidth

B. Zero offset

C. Hysteresis width

D. Full-scale range

Correct Answer: A. Bandwidth

Question 9

A constant output change caused by temperature while the input remains fixed is best classified as

A. temperature-induced drift

B. resolution

C. natural frequency

D. fidelity

Correct Answer: A. temperature-induced drift

Question 10

Why is an anti-aliasing filter used before an analog-to-digital converter?

A. To increase sensor sensitivity

B. To remove all phase shift

C. To limit frequency components that could be aliased during sampling

D. To convert a first-order system into a zero-order system

Correct Answer: C. To limit frequency components that could be aliased during sampling