Unit 6: Power and Energy Measurement: Single-Phase and Polyphase Electrodynamic Wattmeters

Unit 6 — ADCs, references and data acquisition

Why this unit matters. Connect converter architecture, voltage reference, sampling and digital processing to a reported number.

Application setting. Design a synthetic acquisition system and document what the digital numbers mean. The examples are synthetic, deliberately low-energy, and do not establish safe procedures for handling energized high-power systems.

Concept map

Converting input voltage to a code involves a reference, sampling capacitor, comparator or integrator, clock and digital processing. The ideal LSB quantifies nominal granularity only. A high-bit count can coexist with offset, reference, settling, clock jitter and INL errors. A performance claim must name the analysis bandwidth, input frequency and operating temperature.

Distributed observations require a consistent timebase. Channels sampled sequentially carry physically different observation instants even when displayed as a simultaneous table row. For an oscillatory signal, a microsecond-level skew may map into a large apparent phase shift at kilohertz frequencies. Preserve hardware sample times or characterize skew before computing cross-channel products.

Lessons and reading sequence

• Lesson 6.1 — [ADC quantization and transfer functions](../lessons/U06-L01-adc-quantization-and-transfer-functions.md)
• Lesson 6.2 — [Sample-and-hold settling and reference integrity](../lessons/U06-L02-sample-and-hold-settling-and-reference-integrity.md)
• Lesson 6.3 — [SNR, SINAD, ENOB and oversampling](../lessons/U06-L03-snr-sinad-enob-and-oversampling.md)
• Lesson 6.4 — [Timestamping, multiplexing and synchronized acquisition](../lessons/U06-L04-timestamping-multiplexing-and-synchronized-acquisition.md)

After reading the lessons, complete the [unit assessment](../assessments/unit-06.html), [research-style project](../projects/unit-06.html), and [interactive software model](../simulations/model-06.html). Follow the sequence for conceptual continuity, but revisit earlier examples when error analysis changes the meaning of a later result.

Five essential terms

• LSB: Nominal least significant code step.
• ENOB: Equivalent resolution estimated from SINAD.
• Settling: Convergence toward a stable value after a change.
• Skew: Difference between channel sampling instants.
• INL: Deviation of ADC transfer from an ideal reference line.

Cross-lesson example: competing explanations

Assume two engineers report conflicting results for the same notional specimen. One engineer repeats a measurement ten times and obtains a very narrow cluster. The second uses another range, reference, sample timing or wiring topology and obtains a systematic difference. The first engineer's repeatability does not prove that the second engineer is wrong. In this unit, list the physical differences between instrument chains and predict which could create a reproducible offset or scale change. Identify one controlled experiment that would separate the causes without relying on a higher number of decimal places.

Unit laboratory notebook template

Use fields for measurand, connection topology, symbol definitions, numeric input sources, model assumptions, instrument settings, raw observations, rejection criteria, derived values, uncertainty budget, limitations and conclusion. Include a citation for each manufacturer specification if a real instrument is discussed. Reserve a separate field for software version and simulator parameters; a teaching simulator is not a calibrated artifact.

Reader checkpoint

1. Explain the difference between an instrument's reported indication and the intended measurand.
2. Give one case in which making the display resolution finer does not improve the measurement result.
3. Identify one quantity in the unit simulator that must be experimentally calibrated before using the equation for a real system.
4. Define the most important limitation of the idealized model and a practical observation that could falsify it.

Answer guidance. A sound response links the observed quantity to the physical chain and names a bias or uncertainty source instead of vaguely saying 'human error.' It distinguishes a model prediction from independent evidence. Complete safety qualification is outside the scope of this book.

Safety note: The examples are synthetic and intended for analysis or software-only simulation. Do not connect measuring instruments to mains, high-energy equipment, CT secondaries or exposed conductors without authorized training, appropriately rated equipment and site-specific procedures. Instrument-category markings, isolation ratings and current limits must be verified in the manufacturer documentation.

6.1 — ADC quantization and transfer functions

Unit 6: ADCs, references and data acquisition Level: undergraduate, with prerequisite circuit theory, calculus and introductory probability. Estimated study: 70–100 minutes, including the numeric investigation. Tools: calculator, spreadsheet or the original offline simulator in this book. Evidence: examples are synthetic unless explicitly identified otherwise.

A problem that a measurement must answer

A uniform ideal N-bit ADC maps an analog range onto discrete codes. Code width is resolution, while integral and differential nonlinearities characterize how actual thresholds deviate from the ideal. Input overrange and missing codes are distinct from additive noise.

The application for this unit is: Design a synthetic acquisition system and document what the digital numbers mean. The meaningful output is not necessarily the number displayed by a meter. First decide whether it is the unperturbed circuit voltage, a time average, a reference-scale resistance, a physical temperature or a confidence-qualified estimate. That decision determines the topology, calibration approach and acceptable model. In an instrumentation report it is better to leave a result undecided than to replace the unmeasured quantity with a convenient proxy without disclosure.

Mathematical model and physical assumptions

Central relation: For ideal unipolar full scale V_FS, LSB≈V_FS/2^N; ideal quantization error magnitude ≤LSB/2 away from overload under round-to-nearest coding.

Derivation strategy

Start by writing the quantity and its reference state as a sentence. Identify every input that is actually measured and every value that is assumed. If a parameter came from a manufacturer specification, label it as a specification rather than calling it an independent observation. Use the equation to map those inputs onto the desired output. For a laboratory record, store instrument mode and environmental conditions alongside the number. In this case the key input evidence is the reading or physical parameter introduced above, not the final calculated answer. An experimenter should separate constants taken from a definition, instrument parameters taken from a datasheet and conditions measured on the day of use. If an assumed parameter is poorly characterized, a small model residual does not make the resulting prediction precise. The uncertainty of a quantity calculated from a product, quotient or nonlinear relation must follow its sensitivity to each contributing input, not a simple count of display digits.

Why the relation is useful

Architecture versus ideal resolution Converting input voltage to a code involves a reference, sampling capacitor, comparator or integrator, clock and digital processing. The ideal LSB quantifies nominal granularity only. A high-bit count can coexist with offset, reference, settling, clock jitter and INL errors. A performance claim must name the analysis bandwidth, input frequency and operating temperature. The formula above is one of the operational links between a physical system and a numerical record. It is not an endorsement of any particular manufacturer's ratings, which must be checked independently. For instance, ideal resistance or sampling calculations disregard temperature-dependent and frequency-dependent behavior unless those are included explicitly.

Fully worked numerical case

A 12-bit unipolar 3.3 V ADC has nominal LSB=3.3/4096=0.8057 mV. A 1.650 V ideal midscale input maps near code 2048, assuming the adopted transfer convention.

Uncertainty, nonidealities and instrumentation bias

Effective voltage range, reference error, input settling and nonlinear codes usually matter more than another printout digit. A 12-bit code does not guarantee 12 accurate bits in a noisy circuit. The first concern affects the credibility of the numerical parameter; the second suggests a rival explanation of the observed behavior. A useful uncertainty analysis therefore goes beyond listing a collection of percentages. It defines the output measurand, classifies major sources as additive or multiplicative, evaluates correlations when necessary, and identifies the sensitivity of the output to each one. An uncertainty budget should preserve documentary support for Type B inputs and raw observations for Type A estimates. If the model is an approximation, give a bounded validity statement in addition to any statistical interval.

A counterexample to overconfidence

Time integrity Distributed observations require a consistent timebase. Channels sampled sequentially carry physically different observation instants even when displayed as a simultaneous table row. For an oscillatory signal, a microsecond-level skew may map into a large apparent phase shift at kilohertz frequencies. Preserve hardware sample times or characterize skew before computing cross-channel products. This contextual distinction provides a way to design a discriminating check. For an instrument-chain problem, change input range, source impedance, frequency or wiring to see whether the measurement follows the predicted dependence. For a data-analysis problem, change sample rate, observation length or independent reference while holding the physical input constant. If a conclusion persists only because the same assumption is reused in every run, that persistence supplies little additional evidence.

Engineering use and what would be required to trust it

Document whether the top code represents Vref or Vref−1LSB for the selected conversion convention. Calibration records, environmental limits and independent checks belong in the method, not just in appendices. Choose an experiment that isolates the main influence while staying within authorized, properly rated equipment and low-energy pedagogical conditions. A formula computed with fictional values is evidence that arithmetic was performed, not that a real device passed a standard. When the output feeds a safety, certification or financial decision, the appropriate reviewer and agreed acceptance rule must be identified before collecting measurements.

Study exercise 1 — reconstruct the baseline

Derive or justify For ideal unipolar full scale V_FS, LSB≈V_FS/2^N; ideal quantization error magnitude ≤LSB/2 away from overload under round-to-nearest coding. from the relevant conservation law, circuit relation, transfer model or statistical definition. Recompute the worked example, label every unit and distinguish the physical value from what an actual instrument could have displayed. Identify one data item whose uncertainty is not quantified by the example. Explain what would happen if that datum were systematically too high by a small amount.

Study exercise 2 — sensitivity experiment

Use the same physical configuration but change a controlled parameter by 10% in the direction that would expose a plausible failure mechanism. Compute or predict the response using the stated equation, and state whether the output sensitivity is linear, inverse, nonlinear or conditional. Now specify one nonideal effect, grounded in this lesson, that could cause a real observation to differ. A technically sufficient answer must compare two predicted numerical or symbolic outputs; a verbal statement that the error “increases” is not enough where values are available.

Study exercise 3 — measurement design

Design a low-energy or software-only verification that could distinguish the main mechanism from this competing explanation: A 12-bit code does not guarantee 12 accurate bits in a noisy circuit. Specify the independent variable, the measured variable, the reference standard or alternate observation, the expected trend and a stop criterion for an invalid run. Describe what metadata would be saved for another engineer to reproduce the analysis without interviewing the experimenter.

Solution and grading guide

Exercise 2. The independent variable is changed while the model assumptions remain fixed. For a linear proportional dependency, +10% in the input produces +10% in the relevant output contribution; for a reciprocal dependence the output becomes 1/1.1=0.9091 of the original; for a squared dependence it becomes 1.1²=1.21. Apply only the form actually supported by For ideal unipolar full scale V_FS, LSB≈V_FS/2^N; ideal quantization error magnitude ≤LSB/2 away from overload under round-to-nearest coding.; do not choose a generic factor without checking the equation. Explain how effective voltage range, reference error, input settling and nonlinear codes usually matter more than another printout digit. could invalidate an extrapolation.

Exercise 3. The strongest design isolates one cause using a second evidence pathway. A 12-bit code does not guarantee 12 accurate bits in a noisy circuit. should lead to a different prediction from the central mechanism. For example, changing source impedance while fixing source voltage is an effective loading test; changing sample rate while holding a physical sine frequency constant reveals aliasing; testing increasing and decreasing excitation reveals hysteresis. The appropriate case must match the instrument and phenomenon studied here.

Simulator and extension

Use [interactive Unit 6 model](../simulations/model-06.html) to explore the unit's central instrumentation mechanism. The simulator may focus on a related quantity rather than exactly this lesson's formula. Save the default model inputs and at least two modified cases; attach a screenshot or numerical result table. Compare one manually calculated point with the model output, and state a physical mechanism that the simulated equations omit. The original example dataset is not field evidence and must not be cited as such.

Safety note: The examples are synthetic and intended for analysis or software-only simulation. Do not connect measuring instruments to mains, high-energy equipment, CT secondaries or exposed conductors without authorized training, appropriately rated equipment and site-specific procedures. Instrument-category markings, isolation ratings and current limits must be verified in the manufacturer documentation.

6.2 — Sample-and-hold settling and reference integrity

Unit 6: ADCs, references and data acquisition Level: undergraduate, with prerequisite circuit theory, calculus and introductory probability. Estimated study: 70–100 minutes, including the numeric investigation. Tools: calculator, spreadsheet or the original offline simulator in this book. Evidence: examples are synthetic unless explicitly identified otherwise.

A problem that a measurement must answer

A switched-capacitor ADC input draws dynamic charge while its sampling capacitor settles. Source resistance, switch resistance and acquisition time limit the fidelity of each code. A voltage reference can add correlated drift that digital averaging cannot correct.

The application for this unit is: Design a synthetic acquisition system and document what the digital numbers mean. The meaningful output is not necessarily the number displayed by a meter. First decide whether it is the unperturbed circuit voltage, a time average, a reference-scale resistance, a physical temperature or a confidence-qualified estimate. That decision determines the topology, calibration approach and acceptable model. In an instrumentation report it is better to leave a result undecided than to replace the unmeasured quantity with a convenient proxy without disclosure.

Mathematical model and physical assumptions

Central relation: For an RC first-order step, residual error after acquisition t is e^(−t/RC) times initial step.

Derivation strategy

Begin with the lowest-complexity circuit or signal model that explains the key effect, then state the approximations it makes. For a loaded circuit, preserve current paths and terminal definitions; for a sampled signal, preserve sample timing and analog bandwidth; for a statistical model, preserve independence assumptions. A circuit may obey Kirchhoff laws perfectly while the inferred physical quantity is still wrong because the sensor transfer function is uncharacterized. In this case the key input evidence is the reading or physical parameter introduced above, not the final calculated answer. An experimenter should separate constants taken from a definition, instrument parameters taken from a datasheet and conditions measured on the day of use. If an assumed parameter is poorly characterized, a small model residual does not make the resulting prediction precise. The uncertainty of a quantity calculated from a product, quotient or nonlinear relation must follow its sensitivity to each contributing input, not a simple count of display digits.

Why the relation is useful

Architecture versus ideal resolution Converting input voltage to a code involves a reference, sampling capacitor, comparator or integrator, clock and digital processing. The ideal LSB quantifies nominal granularity only. A high-bit count can coexist with offset, reference, settling, clock jitter and INL errors. A performance claim must name the analysis bandwidth, input frequency and operating temperature. The formula above is one of the operational links between a physical system and a numerical record. It is not an endorsement of any particular manufacturer's ratings, which must be checked independently. For instance, ideal resistance or sampling calculations disregard temperature-dependent and frequency-dependent behavior unless those are included explicitly.

Fully worked numerical case

With R=10 kΩ, C=20 pF and acquisition 1 μs, τ=200 ns and the residual fraction is exp(−5)=0.00674. A full-scale step may therefore retain a multi-LSB error on a high-resolution converter.

Uncertainty, nonidealities and instrumentation bias

Input multiplexer switching, source buffers and reference distribution can introduce further dynamics; simple RC is a starting model. Improving initial reference accuracy does not solve settling errors from a high-impedance sensor. The first concern affects the credibility of the numerical parameter; the second suggests a rival explanation of the observed behavior. A useful uncertainty analysis therefore goes beyond listing a collection of percentages. It defines the output measurand, classifies major sources as additive or multiplicative, evaluates correlations when necessary, and identifies the sensitivity of the output to each one. An uncertainty budget should preserve documentary support for Type B inputs and raw observations for Type A estimates. If the model is an approximation, give a bounded validity statement in addition to any statistical interval.

A counterexample to overconfidence

Time integrity Distributed observations require a consistent timebase. Channels sampled sequentially carry physically different observation instants even when displayed as a simultaneous table row. For an oscillatory signal, a microsecond-level skew may map into a large apparent phase shift at kilohertz frequencies. Preserve hardware sample times or characterize skew before computing cross-channel products. This contextual distinction provides a way to design a discriminating check. For an instrument-chain problem, change input range, source impedance, frequency or wiring to see whether the measurement follows the predicted dependence. For a data-analysis problem, change sample rate, observation length or independent reference while holding the physical input constant. If a conclusion persists only because the same assumption is reused in every run, that persistence supplies little additional evidence.

Engineering use and what would be required to trust it

Check worst-case previous channel to current channel step, not only repeated readings of a single steady input. Calibration records, environmental limits and independent checks belong in the method, not just in appendices. Choose an experiment that isolates the main influence while staying within authorized, properly rated equipment and low-energy pedagogical conditions. A formula computed with fictional values is evidence that arithmetic was performed, not that a real device passed a standard. When the output feeds a safety, certification or financial decision, the appropriate reviewer and agreed acceptance rule must be identified before collecting measurements.

Study exercise 1 — reconstruct the baseline

Derive or justify For an RC first-order step, residual error after acquisition t is e^(−t/RC) times initial step. from the relevant conservation law, circuit relation, transfer model or statistical definition. Recompute the worked example, label every unit and distinguish the physical value from what an actual instrument could have displayed. Identify one data item whose uncertainty is not quantified by the example. Explain what would happen if that datum were systematically too high by a small amount.

Study exercise 2 — sensitivity experiment

Use the same physical configuration but change a controlled parameter by 10% in the direction that would expose a plausible failure mechanism. Compute or predict the response using the stated equation, and state whether the output sensitivity is linear, inverse, nonlinear or conditional. Now specify one nonideal effect, grounded in this lesson, that could cause a real observation to differ. A technically sufficient answer must compare two predicted numerical or symbolic outputs; a verbal statement that the error “increases” is not enough where values are available.

Study exercise 3 — measurement design

Design a low-energy or software-only verification that could distinguish the main mechanism from this competing explanation: Improving initial reference accuracy does not solve settling errors from a high-impedance sensor. Specify the independent variable, the measured variable, the reference standard or alternate observation, the expected trend and a stop criterion for an invalid run. Describe what metadata would be saved for another engineer to reproduce the analysis without interviewing the experimenter.

Solution and grading guide

Exercise 2. The independent variable is changed while the model assumptions remain fixed. For a linear proportional dependency, +10% in the input produces +10% in the relevant output contribution; for a reciprocal dependence the output becomes 1/1.1=0.9091 of the original; for a squared dependence it becomes 1.1²=1.21. Apply only the form actually supported by For an RC first-order step, residual error after acquisition t is e^(−t/RC) times initial step.; do not choose a generic factor without checking the equation. Explain how input multiplexer switching, source buffers and reference distribution can introduce further dynamics; simple rc is a starting model. could invalidate an extrapolation.

Exercise 3. The strongest design isolates one cause using a second evidence pathway. Improving initial reference accuracy does not solve settling errors from a high-impedance sensor. should lead to a different prediction from the central mechanism. For example, changing source impedance while fixing source voltage is an effective loading test; changing sample rate while holding a physical sine frequency constant reveals aliasing; testing increasing and decreasing excitation reveals hysteresis. The appropriate case must match the instrument and phenomenon studied here.

Simulator and extension

Use [interactive Unit 6 model](../simulations/model-06.html) to explore the unit's central instrumentation mechanism. The simulator may focus on a related quantity rather than exactly this lesson's formula. Save the default model inputs and at least two modified cases; attach a screenshot or numerical result table. Compare one manually calculated point with the model output, and state a physical mechanism that the simulated equations omit. The original example dataset is not field evidence and must not be cited as such.

Safety note: The examples are synthetic and intended for analysis or software-only simulation. Do not connect measuring instruments to mains, high-energy equipment, CT secondaries or exposed conductors without authorized training, appropriately rated equipment and site-specific procedures. Instrument-category markings, isolation ratings and current limits must be verified in the manufacturer documentation.

6.3 — SNR, SINAD, ENOB and oversampling

Unit 6: ADCs, references and data acquisition Level: undergraduate, with prerequisite circuit theory, calculus and introductory probability. Estimated study: 70–100 minutes, including the numeric investigation. Tools: calculator, spreadsheet or the original offline simulator in this book. Evidence: examples are synthetic unless explicitly identified otherwise.

A problem that a measurement must answer

Effective number of bits is inferred from converter performance including noise and distortion, rather than the number of digital output bits. In an ideal sine-wave ADC, the theoretical quantization-limited SNR rises about 6.02 dB per bit. Additional averaging helps only when noise assumptions permit it.

The application for this unit is: Design a synthetic acquisition system and document what the digital numbers mean. The meaningful output is not necessarily the number displayed by a meter. First decide whether it is the unperturbed circuit voltage, a time average, a reference-scale resistance, a physical temperature or a confidence-qualified estimate. That decision determines the topology, calibration approach and acceptable model. In an instrumentation report it is better to leave a result undecided than to replace the unmeasured quantity with a convenient proxy without disclosure.

Mathematical model and physical assumptions

Central relation: Ideal SNR≈6.02N+1.76 dB; ENOB=(SINAD−1.76)/6.02 for the conventional sinusoidal test.

Derivation strategy

Classify each influence by its pathway to the output. Consider bias corrections separately from dispersion. A standard uncertainty is not necessarily equal to a specified maximum tolerance; a rectangular bounded-error model, for example, converts half-width a into u=a/√3 only if that probability model is justified. In a multivariable equation, derivatives and covariance decide how input uncertainty propagates. In this case the key input evidence is the reading or physical parameter introduced above, not the final calculated answer. An experimenter should separate constants taken from a definition, instrument parameters taken from a datasheet and conditions measured on the day of use. If an assumed parameter is poorly characterized, a small model residual does not make the resulting prediction precise. The uncertainty of a quantity calculated from a product, quotient or nonlinear relation must follow its sensitivity to each contributing input, not a simple count of display digits.

Why the relation is useful

Architecture versus ideal resolution Converting input voltage to a code involves a reference, sampling capacitor, comparator or integrator, clock and digital processing. The ideal LSB quantifies nominal granularity only. A high-bit count can coexist with offset, reference, settling, clock jitter and INL errors. A performance claim must name the analysis bandwidth, input frequency and operating temperature. The formula above is one of the operational links between a physical system and a numerical record. It is not an endorsement of any particular manufacturer's ratings, which must be checked independently. For instance, ideal resistance or sampling calculations disregard temperature-dependent and frequency-dependent behavior unless those are included explicitly.

Fully worked numerical case

If measured SINAD is 62 dB for a sinusoidal test, ENOB=(62−1.76)/6.02≈10.0 bits, even if the converter outputs 12-bit words.

Uncertainty, nonidealities and instrumentation bias

Tone amplitude, windowing, harmonics, clock jitter and measurement bandwidth change SINAD; oversampling does not automatically fix systematic nonlinearity. Random dither can decorrelate quantization patterns but necessarily adds noise energy. The first concern affects the credibility of the numerical parameter; the second suggests a rival explanation of the observed behavior. A useful uncertainty analysis therefore goes beyond listing a collection of percentages. It defines the output measurand, classifies major sources as additive or multiplicative, evaluates correlations when necessary, and identifies the sensitivity of the output to each one. An uncertainty budget should preserve documentary support for Type B inputs and raw observations for Type A estimates. If the model is an approximation, give a bounded validity statement in addition to any statistical interval.

A counterexample to overconfidence

Time integrity Distributed observations require a consistent timebase. Channels sampled sequentially carry physically different observation instants even when displayed as a simultaneous table row. For an oscillatory signal, a microsecond-level skew may map into a large apparent phase shift at kilohertz frequencies. Preserve hardware sample times or characterize skew before computing cross-channel products. This contextual distinction provides a way to design a discriminating check. For an instrument-chain problem, change input range, source impedance, frequency or wiring to see whether the measurement follows the predicted dependence. For a data-analysis problem, change sample rate, observation length or independent reference while holding the physical input constant. If a conclusion persists only because the same assumption is reused in every run, that persistence supplies little additional evidence.

Engineering use and what would be required to trust it

Specify input frequency, amplitude, sample rate, analysis bandwidth and exclusion rules with every ENOB number. Calibration records, environmental limits and independent checks belong in the method, not just in appendices. Choose an experiment that isolates the main influence while staying within authorized, properly rated equipment and low-energy pedagogical conditions. A formula computed with fictional values is evidence that arithmetic was performed, not that a real device passed a standard. When the output feeds a safety, certification or financial decision, the appropriate reviewer and agreed acceptance rule must be identified before collecting measurements.

Study exercise 1 — reconstruct the baseline

Derive or justify Ideal SNR≈6.02N+1.76 dB; ENOB=(SINAD−1.76)/6.02 for the conventional sinusoidal test. from the relevant conservation law, circuit relation, transfer model or statistical definition. Recompute the worked example, label every unit and distinguish the physical value from what an actual instrument could have displayed. Identify one data item whose uncertainty is not quantified by the example. Explain what would happen if that datum were systematically too high by a small amount.

Study exercise 2 — sensitivity experiment

Use the same physical configuration but change a controlled parameter by 10% in the direction that would expose a plausible failure mechanism. Compute or predict the response using the stated equation, and state whether the output sensitivity is linear, inverse, nonlinear or conditional. Now specify one nonideal effect, grounded in this lesson, that could cause a real observation to differ. A technically sufficient answer must compare two predicted numerical or symbolic outputs; a verbal statement that the error “increases” is not enough where values are available.

Study exercise 3 — measurement design

Design a low-energy or software-only verification that could distinguish the main mechanism from this competing explanation: Random dither can decorrelate quantization patterns but necessarily adds noise energy. Specify the independent variable, the measured variable, the reference standard or alternate observation, the expected trend and a stop criterion for an invalid run. Describe what metadata would be saved for another engineer to reproduce the analysis without interviewing the experimenter.

Solution and grading guide

Exercise 2. The independent variable is changed while the model assumptions remain fixed. For a linear proportional dependency, +10% in the input produces +10% in the relevant output contribution; for a reciprocal dependence the output becomes 1/1.1=0.9091 of the original; for a squared dependence it becomes 1.1²=1.21. Apply only the form actually supported by Ideal SNR≈6.02N+1.76 dB; ENOB=(SINAD−1.76)/6.02 for the conventional sinusoidal test.; do not choose a generic factor without checking the equation. Explain how tone amplitude, windowing, harmonics, clock jitter and measurement bandwidth change sinad; oversampling does not automatically fix systematic nonlinearity. could invalidate an extrapolation.

Exercise 3. The strongest design isolates one cause using a second evidence pathway. Random dither can decorrelate quantization patterns but necessarily adds noise energy. should lead to a different prediction from the central mechanism. For example, changing source impedance while fixing source voltage is an effective loading test; changing sample rate while holding a physical sine frequency constant reveals aliasing; testing increasing and decreasing excitation reveals hysteresis. The appropriate case must match the instrument and phenomenon studied here.

Simulator and extension

Use [interactive Unit 6 model](../simulations/model-06.html) to explore the unit's central instrumentation mechanism. The simulator may focus on a related quantity rather than exactly this lesson's formula. Save the default model inputs and at least two modified cases; attach a screenshot or numerical result table. Compare one manually calculated point with the model output, and state a physical mechanism that the simulated equations omit. The original example dataset is not field evidence and must not be cited as such.

Safety note: The examples are synthetic and intended for analysis or software-only simulation. Do not connect measuring instruments to mains, high-energy equipment, CT secondaries or exposed conductors without authorized training, appropriately rated equipment and site-specific procedures. Instrument-category markings, isolation ratings and current limits must be verified in the manufacturer documentation.

6.4 — Timestamping, multiplexing and synchronized acquisition

Unit 6: ADCs, references and data acquisition Level: undergraduate, with prerequisite circuit theory, calculus and introductory probability. Estimated study: 70–100 minutes, including the numeric investigation. Tools: calculator, spreadsheet or the original offline simulator in this book. Evidence: examples are synthetic unless explicitly identified otherwise.

A problem that a measurement must answer

Separate acquisition channels may be sampled sequentially rather than simultaneously. A timestamp should be associated with the actual sampling instant, not simply the time when software processed the buffer. Clock skew introduces phase error in power or vibration measurements.

The application for this unit is: Design a synthetic acquisition system and document what the digital numbers mean. The meaningful output is not necessarily the number displayed by a meter. First decide whether it is the unperturbed circuit voltage, a time average, a reference-scale resistance, a physical temperature or a confidence-qualified estimate. That decision determines the topology, calibration approach and acceptable model. In an instrumentation report it is better to leave a result undecided than to replace the unmeasured quantity with a convenient proxy without disclosure.

Mathematical model and physical assumptions

Central relation: For a sinusoid, phase error from sampling skew Δt is Δφ=2πfΔt radians.

Derivation strategy

Use both a numerical consistency check and a counterexample. Change one input to challenge sensitivity; change the topology or data-generating assumption to challenge validity. A result is only as robust as the least defensible assumption. When discussing acceptance, state a decision rule, uncertainty and consequences for an incorrect classification. In this case the key input evidence is the reading or physical parameter introduced above, not the final calculated answer. An experimenter should separate constants taken from a definition, instrument parameters taken from a datasheet and conditions measured on the day of use. If an assumed parameter is poorly characterized, a small model residual does not make the resulting prediction precise. The uncertainty of a quantity calculated from a product, quotient or nonlinear relation must follow its sensitivity to each contributing input, not a simple count of display digits.

Why the relation is useful

Architecture versus ideal resolution Converting input voltage to a code involves a reference, sampling capacitor, comparator or integrator, clock and digital processing. The ideal LSB quantifies nominal granularity only. A high-bit count can coexist with offset, reference, settling, clock jitter and INL errors. A performance claim must name the analysis bandwidth, input frequency and operating temperature. The formula above is one of the operational links between a physical system and a numerical record. It is not an endorsement of any particular manufacturer's ratings, which must be checked independently. For instance, ideal resistance or sampling calculations disregard temperature-dependent and frequency-dependent behavior unless those are included explicitly.

Fully worked numerical case

At 1 kHz, channel skew 100 μs corresponds to 2π×1000×100e−6=0.6283 rad, or 36°. This is large enough to corrupt a simple phase-based power calculation.

Uncertainty, nonidealities and instrumentation bias

Independent clocks drift; packet arrival time is not a trustworthy replacement for hardware acquisition time. An apparent causal delay may be a channel ordering artifact in a multiplexed data logger. The first concern affects the credibility of the numerical parameter; the second suggests a rival explanation of the observed behavior. A useful uncertainty analysis therefore goes beyond listing a collection of percentages. It defines the output measurand, classifies major sources as additive or multiplicative, evaluates correlations when necessary, and identifies the sensitivity of the output to each one. An uncertainty budget should preserve documentary support for Type B inputs and raw observations for Type A estimates. If the model is an approximation, give a bounded validity statement in addition to any statistical interval.

A counterexample to overconfidence

Time integrity Distributed observations require a consistent timebase. Channels sampled sequentially carry physically different observation instants even when displayed as a simultaneous table row. For an oscillatory signal, a microsecond-level skew may map into a large apparent phase shift at kilohertz frequencies. Preserve hardware sample times or characterize skew before computing cross-channel products. This contextual distinction provides a way to design a discriminating check. For an instrument-chain problem, change input range, source impedance, frequency or wiring to see whether the measurement follows the predicted dependence. For a data-analysis problem, change sample rate, observation length or independent reference while holding the physical input constant. If a conclusion persists only because the same assumption is reused in every run, that persistence supplies little additional evidence.

Engineering use and what would be required to trust it

Use common clocks, trigger distribution or timestamp calibration according to the required simultaneity. Calibration records, environmental limits and independent checks belong in the method, not just in appendices. Choose an experiment that isolates the main influence while staying within authorized, properly rated equipment and low-energy pedagogical conditions. A formula computed with fictional values is evidence that arithmetic was performed, not that a real device passed a standard. When the output feeds a safety, certification or financial decision, the appropriate reviewer and agreed acceptance rule must be identified before collecting measurements.

Study exercise 1 — reconstruct the baseline

Derive or justify For a sinusoid, phase error from sampling skew Δt is Δφ=2πfΔt radians. from the relevant conservation law, circuit relation, transfer model or statistical definition. Recompute the worked example, label every unit and distinguish the physical value from what an actual instrument could have displayed. Identify one data item whose uncertainty is not quantified by the example. Explain what would happen if that datum were systematically too high by a small amount.

Study exercise 2 — sensitivity experiment

Use the same physical configuration but change a controlled parameter by 10% in the direction that would expose a plausible failure mechanism. Compute or predict the response using the stated equation, and state whether the output sensitivity is linear, inverse, nonlinear or conditional. Now specify one nonideal effect, grounded in this lesson, that could cause a real observation to differ. A technically sufficient answer must compare two predicted numerical or symbolic outputs; a verbal statement that the error “increases” is not enough where values are available.

Study exercise 3 — measurement design

Design a low-energy or software-only verification that could distinguish the main mechanism from this competing explanation: An apparent causal delay may be a channel ordering artifact in a multiplexed data logger. Specify the independent variable, the measured variable, the reference standard or alternate observation, the expected trend and a stop criterion for an invalid run. Describe what metadata would be saved for another engineer to reproduce the analysis without interviewing the experimenter.

Solution and grading guide

Exercise 2. The independent variable is changed while the model assumptions remain fixed. For a linear proportional dependency, +10% in the input produces +10% in the relevant output contribution; for a reciprocal dependence the output becomes 1/1.1=0.9091 of the original; for a squared dependence it becomes 1.1²=1.21. Apply only the form actually supported by For a sinusoid, phase error from sampling skew Δt is Δφ=2πfΔt radians.; do not choose a generic factor without checking the equation. Explain how independent clocks drift; packet arrival time is not a trustworthy replacement for hardware acquisition time. could invalidate an extrapolation.

Exercise 3. The strongest design isolates one cause using a second evidence pathway. An apparent causal delay may be a channel ordering artifact in a multiplexed data logger. should lead to a different prediction from the central mechanism. For example, changing source impedance while fixing source voltage is an effective loading test; changing sample rate while holding a physical sine frequency constant reveals aliasing; testing increasing and decreasing excitation reveals hysteresis. The appropriate case must match the instrument and phenomenon studied here.

Simulator and extension

Use [interactive Unit 6 model](../simulations/model-06.html) to explore the unit's central instrumentation mechanism. The simulator may focus on a related quantity rather than exactly this lesson's formula. Save the default model inputs and at least two modified cases; attach a screenshot or numerical result table. Compare one manually calculated point with the model output, and state a physical mechanism that the simulated equations omit. The original example dataset is not field evidence and must not be cited as such.

Safety note: The examples are synthetic and intended for analysis or software-only simulation. Do not connect measuring instruments to mains, high-energy equipment, CT secondaries or exposed conductors without authorized training, appropriately rated equipment and site-specific procedures. Instrument-category markings, isolation ratings and current limits must be verified in the manufacturer documentation.

Project 6 — ADCs, references and data acquisition: reproducible measurement review

Research question. Can two plausible measurement chains give meaningfully different estimates of the same physical measurand when one ideal-model assumption is violated?

Starting case. Design a synthetic acquisition system and document what the digital numbers mean. This is a software-only project. It must not be implemented by handling energized installations or using the simulation as a substitute for calibration.

Background and model selection

Converting input voltage to a code involves a reference, sampling capacitor, comparator or integrator, clock and digital processing. The ideal LSB quantifies nominal granularity only. A high-bit count can coexist with offset, reference, settling, clock jitter and INL errors. A performance claim must name the analysis bandwidth, input frequency and operating temperature.

Distributed observations require a consistent timebase. Channels sampled sequentially carry physically different observation instants even when displayed as a simultaneous table row. For an oscillatory signal, a microsecond-level skew may map into a large apparent phase shift at kilohertz frequencies. Preserve hardware sample times or characterize skew before computing cross-channel products.

Select one formula from each of Lessons 6.1–6.4, but model one of them in depth. For the chosen case, clearly define independent input, measured indication, desired output, units, reference conditions and primary sensitivity. Explain why changing a software parameter is a model intervention, not a laboratory measurement.

Task A — baseline and synthetic evidence

Open [Unit 6 simulator](../simulations/model-06.html). Save the four slider values, the computed numerical output, the axes and the model description. Change one input to three different values and record the response. Independently calculate one point. Create a small table including percent deviations from the baseline and at least one limiting-case check. Where a plotted variable is not the same as an instrument's recorded output, document that distinction.

Task B — deliberately introduce error

Choose one missing mechanism from the lesson limitations. The candidates specific to this unit include: Effective voltage range, reference error, input settling and nonlinear codes usually matter more than another printout digit. Another candidate is: Input multiplexer switching, source buffers and reference distribution can introduce further dynamics; simple RC is a starting model. Write a small correction term, state its assumed magnitude and investigate how the inferred result changes. Do not assert that an invented error magnitude is representative of a commercial instrument or a deployed installation.

Task C — contrast two explanations

Two technical reports disagree. One report blames calibration; the other blames a physical effect: Random dither can decorrelate quantization patterns but necessarily adds noise energy. Design a low-energy or observational check that would produce different predictions under those explanations. State the independent evidence source and explain how correlated calibration references could defeat the check. If both explanations predict the same observable under your selected settings, your proposed test is not discriminating and must be redesigned.

Task D — quantitative uncertainty and decision

Assign three clearly labeled synthetic standard uncertainties and propagate them through the selected measurement equation. Where appropriate, include a common-reference correlation. Present a result estimate, combined standard uncertainty and an optional expanded uncertainty with specified k. Introduce a fictional tolerance limit and a predeclared decision rule. Show a near-boundary example where the displayed reading by itself would give a different decision from a guard-banded approach. Do not conflate a rejected acceptance decision with proof of nonconformity.

Deliverables

• Reproducible spreadsheet or notebook with units, formulas and a synthetic-data flag.
• At least one parameter-sensitivity plot and one tabulated calculation.
• A short model-assumption statement and a distinguishable rival hypothesis.
• A traceable uncertainty budget with corrections separated from uncertainty contributions.
• A final one-page technical memo explaining what is supported and what remains unmeasured.

Evaluation and extension

A report is complete only when a reader can reproduce the model calculation, recognize synthetic inputs and identify the decisive missing evidence for a real-world conclusion. Additional complexity earns credit only if it addresses the dominant uncertainty, not simply because the diagram becomes more elaborate. For an extension, estimate how much improved reference accuracy or increased acquisition time would actually reduce the dominant combined uncertainty under the stated model.

Safety note: The examples are synthetic and intended for analysis or software-only simulation. Do not connect measuring instruments to mains, high-energy equipment, CT secondaries or exposed conductors without authorized training, appropriately rated equipment and site-specific procedures. Instrument-category markings, isolation ratings and current limits must be verified in the manufacturer documentation.