Measurements, Sensors and Instrumentation

Introduction

Measurement connects electrical theory with physical evidence. Engineers use instruments not merely to obtain numbers, but to determine whether those numbers are sufficiently accurate, traceable, safe and relevant for a decision. A voltage reading may be repeatable yet biased; a sensor may be accurate in the laboratory but unreliable after installation; and a digital display may suggest more certainty than the measurement system actually possesses.

A complete measurement system usually converts a physical quantity into an electrical signal, conditions that signal, digitizes it and presents or stores the result. Each stage can introduce loading, noise, drift, nonlinearity and uncertainty. Sound instrumentation therefore requires both circuit knowledge and statistical judgment.

Learning Objectives

After studying this chapter, students should be able to:

• use analog and digital electrical instruments safely;
• distinguish accuracy, precision, resolution, sensitivity and calibration;
• identify systematic, random and gross errors;
• evaluate instrument loading and insertion effects;
• calculate standard and expanded measurement uncertainty;
• propagate uncertainty through measurement equations;
• apply Wheatstone bridges and related sensor interfaces;
• explain the operation and limitations of common sensors and transducers;
• design basic signal-conditioning and data-acquisition chains;
• summarize and interpret experimental data statistically; and
• document measurements in a traceable and professionally responsible manner.

1. Measurement Systems and Terminology

A measurand is the quantity intended to be measured, such as terminal voltage, conductor temperature or shaft torque. The measurand should be defined together with relevant conditions. For example, “resistance” is incomplete if temperature materially affects the result.

A generalized measurement chain is shown below.

Figure: Functional stages of a measurement and data-acquisition system.

A sensor responds to a physical quantity. A transducer converts energy or information from one form to another. The terms overlap, although a transducer may include both sensing and conversion elements.

Important static characteristics include:

• Accuracy: closeness of a result to an accepted reference value. It is commonly specified by maximum permissible error rather than as an absolute property.
• Precision: closeness among repeated results under stated conditions.
• Resolution: smallest input change that can be meaningfully distinguished.
• Sensitivity: change in output divided by the corresponding change in input.
• Linearity: closeness of the input–output relation to a specified straight line.
• Hysteresis: different outputs for the same input depending on whether the input was approached from above or below.
• Repeatability: agreement under the same method, instrument, operator, location and short time interval.
• Drift: gradual change in response while the input remains constant.
• Range: interval over which specified performance is maintained.

Accuracy and precision are not equivalent. Repeated readings may cluster closely around an incorrect value because of calibration bias. Conversely, scattered readings may have an average near the reference value while individual measurements remain imprecise.

2. Analog and Digital Instruments

2.1 Analog instruments

Analog indicating instruments produce a continuous pointer displacement. In a permanent-magnet moving-coil movement, current produces torque on a coil in a magnetic field. A spring supplies restoring torque, and damping limits oscillation. Such mechanisms directly respond to DC; rectification is required for AC measurement.

Analog meters reveal trends effectively, but their reading is affected by scale interpolation, parallax and mechanical friction. Parallax is minimized by viewing the pointer perpendicular to a mirrored scale. Analog voltmeters may have lower input resistance than modern digital multimeters and can therefore cause substantial loading.

2.2 Digital multimeters

A digital multimeter (DMM) commonly measures voltage through an input attenuator, current through a shunt resistor, and resistance by applying a known test stimulus. Its analog-to-digital converter displays a numerical result. Advantages include high input resistance, good resolution, automatic polarity and data communication.

A DMM display count specifies the maximum displayed value, while resolution describes the smallest displayed increment on a selected range. Neither guarantees accuracy. A specification such as ±(0.5% of reading + 2 counts) combines a proportional term and a range-dependent digit term.

Example—interpreting a DMM specification. A meter reads 8.00 V on a range having 0.01 V resolution. Its specification is ±(0.5% of reading + 2 counts). The maximum permissible error is

The indicated result may therefore be reported as 8.00 V with an instrument limit of ±0.06 V under the manufacturer’s specified environmental and calibration conditions.

2.3 Oscilloscopes and current probes

An oscilloscope displays voltage as a function of time. Important specifications include bandwidth, sample rate, input impedance, probe attenuation and maximum safe input. A passive ×10 probe typically increases effective input resistance and reduces capacitance relative to a ×1 connection. Probe compensation must be adjusted to avoid waveform distortion.

Current may be measured using a shunt, current transformer, Hall-effect probe or Rogowski coil. Current transformers must not be operated with an open secondary while primary current flows because hazardous voltage can develop.

3. Safe Measurement Practice

Electrical measurement can expose the operator to shock, burns, arc flash and equipment damage. Safe practice includes:

1. Identify the expected voltage, current, frequency and transient environment before connecting an instrument.
2. Use an instrument, leads and accessories with appropriate voltage, current and measurement-category ratings.
3. Inspect insulation, probe guards, connectors and fuses.
4. Place leads in the correct terminals before measurement.
5. De-energize, isolate and verify absence of voltage before resistance or continuity tests.
6. Connect voltage instruments in parallel and current instruments in series, unless a non-contact probe is used.
7. Never place a meter configured for current directly across a voltage source.
8. Use appropriately rated high-breaking-capacity fuses specified by the manufacturer.
9. Prefer one-handed probing, barriers and remote acquisition where risk assessment requires them.
10. Discharge capacitors and account for stored mechanical or magnetic energy.

IEC measurement categories distinguish transient environments. In simplified terms, CAT II applies to loads connected to receptacle circuits, CAT III to building distribution, and CAT IV to the origin of a low-voltage installation. A category rating is meaningful only with its associated voltage rating. Local regulations and institutional procedures always take precedence.

A non-contact voltage detector is useful for preliminary screening but does not, by itself, prove that a conductor is de-energized. Verification requires an approved method and, commonly, testing the voltage detector before and after use on a known source.

4. Loading and Insertion Effects

An instrument becomes part of the circuit it measures. An ideal voltmeter has infinite input resistance; an ideal ammeter has zero resistance. Real instruments depart from these ideals.

4.1 Voltmeter loading

Represent a source network by its Thévenin equivalent: voltage V_th in series with resistance R_th. Connecting a voltmeter of input resistance R_m gives

Here V_m is the meter reading in volts. The equation assumes a DC or purely resistive circuit. For AC, resistances must be replaced by complex impedances.

The relative loading error referenced to the open-circuit voltage is

Worked example—voltmeter loading. A 10.0 V Thévenin source has R_th=100 kΩ. A DMM with R_m=1.00 MΩ is connected.

The loading error is approximately −9.09%. A 10 MΩ meter would read 9.90 V, reducing but not eliminating loading.

4.2 Ammeter burden

A real ammeter introduces shunt resistance and internal voltage drop. If an ideal source V_s drives load resistance R_L, insertion of ammeter resistance R_A changes the current from I₀=V_s/R_L to

Electronic ammeters may specify burden voltage, the voltage developed across the measuring circuit at a stated current. Burden can seriously disturb low-voltage circuits.

5. Error, Calibration and Traceability

5.1 Error classifications

Measurement error is conceptually the measured value minus a reference quantity value. Because the true value is generally unknowable, uncertainty—not “exact error”—is normally evaluated.

• Systematic effects produce consistent shifts, such as gain error, lead resistance or uncorrected temperature dependence.
• Random effects cause unpredictable variation, such as electrical noise.
• Gross errors arise from mistakes, damaged equipment or incorrect procedures.

Known systematic effects should be corrected where practical. The uncertainty of the correction must still be included.

5.2 Calibration and adjustment

Calibration establishes the relation between instrument indications and reference values under stated conditions. It does not necessarily adjust the instrument. Adjustment changes the instrument to improve its indication. Calibration should be performed after adjustment if valid post-adjustment results are required.

Metrological traceability is a documented, unbroken chain of calibrations linking a result to a recognized reference, with each link contributing uncertainty. Calibration records should identify the instrument, serial number, method, environmental conditions, standards used, results, uncertainty and validity date.

Two-point calibration of an approximately linear sensor uses reference pairs (x₁,y₁) and (x₂,y₂). The fitted relation is

This model is appropriate only when linearity over the interval has been justified. More calibration points are needed to assess nonlinearity and hysteresis.

6. Measurement Uncertainty and Statistical Analysis

6.1 Descriptive statistics

For n repeated observations x_i, the arithmetic mean is

The sample standard deviation is

The denominator n-1 provides an unbiased estimate of population variance under independent sampling from a stable population. The estimated standard uncertainty of the mean is

This reduction with √n applies to independent random observations; it does not remove drift or systematic bias.

6.2 Type A and Type B evaluation

A Type A uncertainty component is evaluated statistically from repeated observations. A Type B component is evaluated using other information, such as calibration certificates, instrument specifications, resolution or prior data. “Type B” does not mean inferior.

If an error is assumed uniformly distributed between -a and +a, its standard uncertainty is

For rounding to the nearest display increment q, the rounding error lies from -q/2 to +q/2. Under a uniform model,

The assumed probability distribution must be defensible; a manufacturer’s maximum limit is not automatically a standard deviation.

6.3 Combining and propagating uncertainty

For a measurement model y=f(x₁,x₂,\ldots,x_m), first-order uncertainty propagation gives

Here u_c(y) is combined standard uncertainty, u(x_i) is the standard uncertainty of input x_i, and u(x_i,x_j) is covariance. The approximation assumes uncertainties are sufficiently small for local linearization. If inputs are independent, covariance terms are zero.

For independent uncertainty components already expressed in output units,

Worked example—combined uncertainty. Independent standard uncertainty components of 0.30 V and 0.40 V produce

Expanded uncertainty is

where k is the coverage factor. Under conditions approximating a normal distribution with adequate effective degrees of freedom, k≈2 corresponds to approximately 95% coverage. The value of k and the coverage interpretation must be stated.

Worked example—power uncertainty. Power is calculated from independently measured DC voltage and current: P=VI. Let V=12.00 V, u(V)=0.06 V, I=2.000 A and u(I)=0.010 A. Then

The measured power is 24.00 W with combined standard uncertainty 0.17 W. For k=2, U=0.34 W. This calculation assumes negligible voltage–current covariance and a valid DC product model.

Outliers should not be deleted merely because they are inconvenient. Their physical cause should be investigated, and any exclusion criterion should be defined and documented. Regression residuals, time plots and uncertainty analysis often reveal drift or model failure that a mean alone conceals.

7. Bridge Measurements

7.1 Wheatstone bridge

A Wheatstone bridge measures resistance by comparison. Define four arms as follows: R₁ is upper-left, R₂ lower-left, R₃ upper-right and R_x lower-right. A supply V_s is connected between the top and bottom nodes, and a null detector connects the two midpoint nodes.

At balance, detector current is zero and the midpoint potentials are equal:

Cross-multiplication gives

and therefore

This result depends on the stated arm definitions. Lead and contact resistances are assumed negligible, and the detector must not significantly load the bridge at null.

Worked example—bridge balance. If R₁=100 Ω, R₂=200 Ω and R₃=150 Ω, then

Null methods can achieve high sensitivity because the result depends mainly on resistance ratios rather than absolute detector calibration.

7.2 Sensor bridges

Strain gauges and resistive sensors are often arranged in quarter-, half- or full-bridge circuits. For a balanced quarter bridge with three fixed resistors equal to R and one active gauge R+∆R, the small-signal output magnitude is approximately

provided |∆R|≪R. Output polarity depends on the active arm and output-node definition. Half and full bridges can increase sensitivity and provide temperature compensation.

For AC impedance measurement, bridges compare complex impedances. Balance requires equality of both real and imaginary components. Stray capacitance, grounding and frequency stability then become important.

8. Sensors and Transducers

8.1 Resistive sensors

A metal resistance temperature detector (RTD), commonly platinum, offers good stability. Over a limited interval near a reference temperature T₀, a linear approximation is

where R₀ is resistance at T₀, and α is the temperature coefficient in °C⁻¹. Wider ranges require an appropriate standardized nonlinear model. Two-wire RTDs include lead resistance; three- and four-wire arrangements reduce or eliminate its first-order influence.

A strain gauge obeys

where GF is gauge factor, ∆R/R is fractional resistance change and ε=∆L/L is mechanical strain. Temperature compensation, bonding quality and transverse sensitivity affect practical accuracy.

Thermistors provide high temperature sensitivity but are nonlinear. Self-heating occurs when measurement current dissipates power in the sensor.

8.2 Thermoelectric sensors

A thermocouple produces a voltage related to the temperature difference between its measuring and reference junctions. Over a small interval,

where S is the approximate Seebeck coefficient in V/°C, T_h is hot-junction temperature and T_c is reference-junction temperature. In general, S varies with temperature, so standardized polynomial tables are used. Cold-junction compensation measures the terminal temperature and applies the corresponding correction.

8.3 Capacitive sensors and leakage

For parallel conductive plates with uniform field and negligible fringing,

where C is capacitance in farads, ε is permittivity in F/m, A is overlap area in m² and d is separation in metres. Displacement, pressure, humidity or material composition may alter one of these quantities.

Capacitive sensors are especially susceptible to cable capacitance, contamination and leakage. A charged capacitive sensor connected to a finite leakage resistance R_leak forms a discharge path. With no replenishing source, its voltage decays as

Thus the time constant is τ=R_leakC. Even a very large leakage resistance eventually discharges the capacitor, so a simple charge-based capacitive sensor cannot preserve a static DC indication indefinitely. Practical interfaces use AC excitation, capacitance-to-digital conversion, guarding, clean insulation and very high-input-impedance amplifiers. Leakage is therefore not merely an abstract limitation: it causes measurable droop and limits low-frequency or static performance.

8.4 Inductive, magnetic and optical sensors

An LVDT uses transformer coupling to measure displacement. Its differential AC output is approximately proportional to displacement near the central null, while phase indicates direction. It requires AC excitation and demodulation.

Hall-effect sensors produce an electrical response to magnetic flux density and can measure current without direct electrical contact. Current transformers measure AC but cannot reproduce steady DC. Rogowski coils respond to the rate of change of current and require integration.

Photodiodes convert optical power to current. Reverse-bias operation improves speed but increases dark-current effects. Optical encoders provide digital position or speed information but require alignment and contamination control.

9. Signal Conditioning

Sensor outputs are often too small, noisy or incompatible with an analog-to-digital converter. Signal conditioning may provide excitation, amplification, filtering, isolation, linearization and protection.

An instrumentation amplifier offers high input impedance and high common-mode rejection, making it suitable for bridge outputs. Common-mode rejection ratio is

where A_d is differential gain and A_cm is common-mode gain. High CMRR requires both amplifier performance and balanced source impedances.

Low-pass filtering removes out-of-band noise and helps prevent aliasing. Shielding reduces electric-field coupling, twisted pairs reduce magnetic pickup, and differential measurement rejects shared interference. Incorrect grounding can create loops in which unintended current produces measurement error.

Industrial 4–20 mA loops are robust over long cables. The nonzero 4 mA “live zero” distinguishes a valid zero reading from some open-circuit faults and can power two-wire transmitters. Isolation may be required between field equipment and acquisition hardware.

10. Data Acquisition and Digital Conversion

10.1 Sampling and aliasing

For a signal strictly band-limited to maximum frequency f_max, perfect reconstruction theoretically requires

where f_s is sampling frequency. Equality is not a practical design target. Real filters have finite transition bands, so engineers sample faster and apply an analog anti-alias filter before conversion.

Aliasing makes frequencies above half the sampling rate appear as false lower frequencies. Once aliased components enter sampled data, digital filtering generally cannot identify and remove them uniquely.

10.2 ADC resolution and quantization

For an ideal N-bit ADC spanning an input range V_FS, one least significant bit has width

For ideal rounding, quantization error lies approximately between -q/2 and +q/2, except at range endpoints. Resolution does not include offset, gain error, nonlinearity, noise or reference uncertainty.

Worked example—ADC selection. A 12-bit ADC has a unipolar 0–5.000 V range:

The ideal rounding uncertainty under a uniform quantization model is q/√12=0.3524 mV. If the conditioned sensor sensitivity is 20 mV/°C, one code corresponds to approximately 0.0610°C. This is resolution, not guaranteed temperature accuracy.

Practical data acquisition also requires attention to aperture timing, channel multiplexing, settling time, clock accuracy, reference stability and effective number of bits. Averaging can reduce uncorrelated noise but cannot correct aliasing or systematic bias.

11. Applied Case: Motor-Bearing Temperature Monitoring

Background

This illustrative case concerns a manufacturing plant seeking early detection of bearing overheating in a 30 kW induction motor. The bearing housing normally operates between 45°C and 70°C, and an alarm is required at 85°C.

Problem

A long cable run passes near variable-frequency-drive conductors. An initial two-wire Pt100 installation shows unstable readings and a persistent temperature offset. Maintenance personnel have occasionally interpreted individual high readings as genuine alarms.

Analysis

At 0°C a Pt100 has nominal resistance 100 Ω. Near ambient temperature its sensitivity is approximately 0.385 Ω/°C. Two leads each having 1.2 Ω resistance add 2.4 Ω in a two-wire circuit, corresponding approximately to

This approximation is adequate for diagnosing the lead-resistance effect but does not replace the standardized RTD equation. Electromagnetic interference from the drive explains part of the observed variation, while the lead resistance explains the positive offset.

Approach and Solution

The engineering team implements:

• a three-wire Pt100 transmitter mounted near the motor;
• a low measurement current to limit self-heating;
• a shielded twisted cable routed separately from drive conductors;
• an isolated 4–20 mA output scaled from 0°C to 100°C;
• an anti-alias filter and 10 samples/s acquisition rate because bearing temperature varies slowly;
• calibration checks at two traceable reference temperatures;
• software alarms requiring both an 85°C threshold and a short persistence interval; and
• recorded uncertainty, calibration date and sensor identifier with each maintenance report.

Suppose the transmitter is scaled linearly from 4 mA at 0°C to 20 mA at 100°C. At 85°C the expected current is

Outcome

The redesign removes most lead-resistance bias, reduces noise and prevents isolated spikes from triggering unnecessary shutdowns. The persistence condition is selected using thermal dynamics and safety requirements—not merely to suppress inconvenient alarms.

Learning Points

A better sensor alone would not have solved the problem. Wiring, conditioning, installation, calibration, acquisition and decision logic all influenced the result. Alarm design must balance false alarms against the consequences of missed overheating.

12. Limitations, Governance and Contemporary Practice

Every measurement model is an approximation. Environmental changes, aging, cross-sensitivity, mounting stress, electromagnetic compatibility and software configuration can dominate laboratory sensor specifications. Good practice includes uncertainty budgeting, periodic verification, version-controlled software, time synchronization and retention of raw data.

Automated diagnostics and machine-learning systems can identify drift, anomalies and sensor disagreement. However, an algorithm cannot recover information destroyed by saturation, poor sampling or faulty installation. Training data may also encode prior sensor bias. AI-derived alarms should therefore be validated against physical knowledge and traceable references, with human oversight for safety-critical decisions.

Professional ethics require engineers to report uncertainty and limitations honestly, preserve unfavorable data, avoid unjustified precision and disclose calibration failures. Instruments used in regulated energy, medical, transportation or workplace-safety applications may be subject to legal metrology, functional-safety, electromagnetic-compatibility and cybersecurity requirements. Networked instruments should use authenticated access, controlled firmware updates and auditable configuration records.

13. Best-Practice Measurement Workflow

A defensible experimental workflow is to:

1. define the measurand and required decision uncertainty;
2. estimate range, bandwidth and environmental conditions;
3. select a suitable sensor and safe instrument category;
4. model loading and other interactions;
5. plan calibration and traceability;
6. design excitation, conditioning, shielding and anti-alias filtering;
7. choose sampling rate, ADC range and resolution;
8. collect raw data with units, timestamps and metadata;
9. inspect trends, distributions and residuals;
10. correct justified systematic effects and propagate uncertainty;
11. report the result with coverage information and limitations; and
12. retain sufficient records for independent review.

Chapter Summary

Electrical measurement is an interaction between a measurand, an instrument and its environment. Accuracy concerns closeness to a reference, whereas precision concerns agreement among repeated observations. Resolution, sensitivity, drift, hysteresis and bandwidth describe different aspects of instrument performance and must not be confused.

Real voltmeters and ammeters load circuits. Calibration establishes relationships to reference values, while metrological traceability links results through documented calibrations. Measurement uncertainty combines statistical Type A evaluations and knowledge-based Type B evaluations. Sensitivity coefficients and covariance provide a general framework for propagating input uncertainties to calculated results.

Wheatstone bridges enable sensitive resistance comparison and interface naturally with strain gauges and other resistive sensors. RTDs, thermocouples, capacitive devices, LVDTs, Hall sensors and photodiodes each require suitable excitation and conditioning. Capacitive sensors are limited at very low frequencies by leakage, which causes charge and voltage to decay with time constant R_leakC.

Signal conditioning provides gain, filtering, isolation and interference rejection. Data-acquisition design must prevent aliasing and distinguish ADC resolution from system accuracy. Safe procedures, trustworthy records, cybersecurity and honest communication of uncertainty are integral—not optional—parts of professional instrumentation.

Key Terms

Measurand: Quantity intended to be measured.

Accuracy: Closeness of a measurement result to a reference quantity value.

Precision: Closeness among repeated measured values under stated conditions.

Resolution: Smallest input change that can be meaningfully distinguished.

Sensitivity: Ratio of output change to corresponding input change.

Loading effect: Change in a circuit or system caused by connecting an instrument.

Burden voltage: Voltage developed across a current-measuring instrument.

Calibration: Establishment of the relationship between indications and reference values.

Metrological traceability: Documented calibration chain linking a result to a reference.

Measurement uncertainty: Non-negative parameter characterizing dispersion of values attributed to a measurand.

Type A evaluation: Statistical evaluation of an uncertainty component from observations.

Type B evaluation: Uncertainty evaluation based on non-statistical information.

Coverage factor: Multiplier used to obtain expanded uncertainty from standard uncertainty.

Wheatstone bridge: Four-arm resistance network used for comparison or sensor measurement.

Transducer: Device that converts energy or information from one form to another.

Signal conditioning: Processing that makes a sensor signal suitable for measurement or conversion.

Aliasing: False frequency representation caused by inadequate sampling.

Quantization: Mapping of a continuous input amplitude to discrete digital levels.

Common-mode rejection: Ability of a differential system to reject signals shared by both inputs.

Cold-junction compensation: Correction for thermocouple reference-junction temperature.

Review Questions

Short Answer Questions

1. Distinguish accuracy, precision and resolution.
2. Why can a voltmeter alter the voltage it is intended to measure?
3. What is the difference between calibration and instrument adjustment?
4. Why is an anti-alias filter placed before an ADC?
5. Explain why leakage limits static charge-based capacitive measurements.

Long Answer Questions

1. Explain the complete measurement chain and identify potential errors introduced at each stage.
2. Discuss safe procedures for voltage, current and resistance measurement in industrial circuits.
3. Derive the Wheatstone bridge balance equation using the arm definitions given in this chapter.
4. Compare RTDs, thermocouples, thermistors and capacitive temperature or humidity sensors in terms of interface requirements and limitations.
5. Explain Type A and Type B uncertainty evaluations, uncertainty propagation, covariance and expanded uncertainty.

Analytical / Application Questions

1. A 15.0 V Thévenin source has an output resistance of 220 kΩ. Calculate the readings and loading errors for voltmeters having input resistances of 1 MΩ and 10 MΩ. Recommend the more appropriate instrument.
2. Design a data-acquisition chain for a strain-gauge bridge whose useful bandwidth is 100 Hz and whose full-scale differential output is 12 mV. Justify amplification, filtering, ADC range and sampling rate.
3. A calibrated temperature result is calculated from a sensor voltage, amplifier gain and reference-junction correction. Construct a measurement model and explain how you would estimate and combine its principal uncertainty components.

Multiple Choice Questions with Answers

Question 1

Repeated voltage readings are tightly clustered but consistently 0.8 V above a traceable reference. Which description is most appropriate?

A. High precision but poor accuracy

B. High accuracy but poor precision

C. High resolution and necessarily high accuracy

D. Low sensitivity but zero systematic effect

Correct Answer: A. High precision but poor accuracy

Question 2

A 10.0 V Thévenin source with R_th=100 kΩ is measured by a voltmeter with R_m=1.00 MΩ. What voltage is indicated?

A. 5.00 V

B. 9.09 V

C. 10.0 V

D. 11.0 V

Correct Answer: B. 9.09 V

Question 3

In the chapter's Wheatstone bridge, R₁ is upper-left, R₂ lower-left, R₃ upper-right and R_x lower-right. Which equation gives R_x at balance?

A. R_x=R₁R₂/R₃

B. R_x=R₁R₃/R₂

C. R_x=R₂R₃/R₁

D. R_x=(R₁+R₂)/R₃

Correct Answer: C. R_x=R₂R₃/R₁

Question 4

Two independent standard uncertainty components are 0.30 V and 0.40 V. What is their combined standard uncertainty?

A. 0.10 V

B. 0.35 V

C. 0.50 V

D. 0.70 V

Correct Answer: C. 0.50 V

Question 5

What is the ideal code width of a 12-bit ADC with a 0–5.000 V input range?

A. 0.305 mV

B. 1.221 mV

C. 2.441 mV

D. 4.883 mV

Correct Answer: B. 1.221 mV

Question 6

A strain gauge has GF=2.0 and experiences ε=500 με. What is ∆R/R?

A. 0.00025

B. 0.00050

C. 0.00100

D. 0.00200

Correct Answer: C. 0.00100

Question 7

Why is cold-junction compensation required in thermocouple measurement?

A. A thermocouple responds to the temperature difference between measuring and reference junctions

B. It eliminates all thermocouple nonlinearity

C. It converts the thermocouple into a resistive sensor

D. It prevents ADC quantization

Correct Answer: A. A thermocouple responds to the temperature difference between measuring and reference junctions

Question 8

A charged capacitive sensor is connected to a finite leakage resistance. What directly causes its static output voltage to decrease with time?

A. Charge flows through the leakage path with time constant R_leakC

B. The plate area necessarily decreases after charging

C. The dielectric permittivity always becomes zero at DC

D. The ADC increases the sensor capacitance without limit

Correct Answer: A. Charge flows through the leakage path with time constant R_leakC

Question 9

A signal is strictly band-limited to 1.8 kHz. Which sampling rate satisfies the theoretical strict Nyquist inequality?

A. 1.8 ksample/s

B. 3.0 ksample/s

C. 3.6 ksample/s

D. 5.0 ksample/s

Correct Answer: D. 5.0 ksample/s

Question 10

Which action is correct when measuring resistance with a conventional DMM?

A. Measure the resistance while the circuit is energized

B. Connect the meter in current mode across the component

C. De-energize the circuit and discharge stored energy first

D. Open-circuit a current-transformer secondary during operation

Correct Answer: C. De-energize the circuit and discharge stored energy first