Unit 6: Generation of High Alternating-Current Voltages: Cascaded and Series-Resonant Transformers

Unit 6: High-voltage measurement systems and uncertainty

Why this subject matters

A measurement chain contributes loading, stray coupling, frequency response, calibration uncertainty and data-acquisition errors. Simply dividing a scope trace by a nominal ratio cannot verify a test voltage. Rated voltage alone does not imply impulse bandwidth.

Common reference case and assumptions

Use low-energy equivalent circuits, uncertainty budgets and synthetic impulse samples to analyze measurement credibility. Physical dividers and energized probes are not assembled. The goal is competence in interpreting certificates and synthetic measurements, not a test-lab construction guide.

Outcomes and verification

Propagate uncertainty, distinguish random and systematic effects, assess sensor loading and determine when traceable calibration evidence is required.

A researcher's reading protocol

Distinguish an instrument input from a calibrated complete measurement system. Work through loading, transfer function, resolution, systematic error and uncertainty of the inferred measurand. An instrument display with six decimals need not carry six decimals of credibility. The model is intentionally limited to analytical and offline-computation exercises. A student should identify the quantity being predicted and ask what observational evidence would be needed to validate it. The location of the maximum physical field is especially significant: an average stress may conceal a sharp local concentration at a boundary, edge, void, accessory or transient discontinuity.

When reading Lessons 6.1–6.4, first reconstruct the derivation, then change one variable in the supplied worked example. Carry the physical units consistently and ask whether a plotted curve can be traced back to the same symbol definitions as the equation. Use the [interactive software model](../simulations/model-06.html) after the hand calculation so the plot serves as a diagnostic rather than an opaque answer generator.

What counts as evidence

The unit challenge is reporting what an independently calibrated chain can support, not what its screen is capable of rendering. Distinguish proof of code execution, numerical agreement with a mathematical model, calibration evidence, dielectric qualification, and safe operation. These are five separate accomplishments; none implies the next automatically. All practice in this book is software-only. Do not build test apparatus, charge capacitors or obtain energized equipment for these exercises.

Suggested learning route

Read all four lessons and reproduce the arithmetic in a notebook. Complete the project with a baseline case and three documented variations. Attempt the unit assessment before consulting its solution notes. Record a clear status for each conclusion: derived, numerically checked, simulated, empirically supported or unresolved. There are no fabricated laboratory observations in this book.

Lesson 6.1: Voltage-dividing ratios and loading

Unit: High-voltage measurement systems and uncertainty. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

An ideal resistive divider ratio is determined by two resistances, but finite instrument impedance loads the lower element. Frequency behavior adds stray capacitance and inductance, requiring more than a DC ratio for impulse measurement. This chapter analyzes low-energy math only.

Governing equations and interpretation

For R1 in series with R2 and measured output across R2, k=R2/(R1+R2). With instrument resistance Rin, replace R2 by R2∥Rin.

Fully stated worked example

For R1=9 MΩ and R2=100 kΩ, ideal k≈0.010989. With Rin=1 MΩ, the effective low side is 90.91 kΩ and k≈0.010000; ignoring loading would bias the inferred input.

Reconstructing the model

Define every boundary condition and symbol before calculating. Derive the supplied relation from a recognizable physical law: Gauss’s law, conservation of charge, electromagnetic-wave continuity, equivalent-circuit impedance or probability bookkeeping as appropriate. If a derivative or spatial gradient appears, write what is being held constant and what geometric symmetry is assumed. The derivation must connect to an experimentally meaningful quantity rather than stop at formal algebra.

Failure modes, model exclusions, and hazard boundary

A divider that is correct at DC can be grossly wrong for fast impulses; no physical assembly or energized test is described.

Lesson problems (answer guidance follows)

1. Reproduce the baseline. Start with the precise governing relation and numerical values above. Write every symbol, perform the substitutions and express the result in physically appropriate units. Explain the sign, direction or phase reference where relevant.

2. Parameter-sensitivity check. Increase the most sensitive input in the worked example by 10% without silently changing another parameter. Recompute, distinguish relative from absolute change, and identify whether the equation predicts linear, reciprocal, logarithmic, power-law or exponential sensitivity.

3. Failure of an assumption. Describe a physically specific modification to voltage-dividing ratios and loading that renders the chapter's simplified model unreliable. Identify at least two previously missing quantities needed to restore predictive value.

4. Evidence and uncertainty. A student claims the chapter's numerical result proves an insulation component is safe. Draft a technically precise rebuttal: which quantity has actually been calculated, which uncertainty remains, and what category of qualified evidence would be needed?

5. Alternative mechanism. Propose a competing explanation for an observed deviation in the relevant field, current, waveform or diagnostic trend. Give a discriminating independent observation rather than only repeating the original formula.

Solutions and evaluation guide

Baseline answer: For R1=9 MΩ and R2=100 kΩ, ideal k≈0.010989. With Rin=1 MΩ, the effective low side is 90.91 kΩ and k≈0.010000; ignoring loading would bias the inferred input. Reproduce the numeric substitution and state assumptions before reporting precision.

Sensitivity answer: Apply the actual displayed governing equation to one 10% parameter change; do not assume a linear relationship when it contains a logarithm, a quadratic energy term or an exponential.

Limitation answer: A divider that is correct at DC can be grossly wrong for fast impulses; no physical assembly or energized test is described. These exclusions identify what evidence is still required. A model calculation cannot certify safety or material lifetime.

Evidence answer: Record the model result as analytically calculated, the simulator as software output, and an actual dielectric test as not performed. The fifth question requires an independent discriminating observation rather than a restatement of the model.

Software exercise

Use the [offline unit model](../simulations/model-06.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 6.2: Capacitive sensors, coupling and bandwidth

Unit: High-voltage measurement systems and uncertainty. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

Capacitive measurement chains are sensitive to parasitic capacitances, grounding arrangement and frequency. Their transfer functions often require calibrated dynamic response rather than a nominal capacitance ratio. The voltage delivered to an instrument can change with cable length.

Governing equations and interpretation

For two ideal series capacitances C1 and C2 with Vout across C2, Vout/Vin=C1/(C1+C2). This follows from equal series charge, not from the resistive-divider formula.

Fully stated worked example

With C1=10 pF and C2=1 nF, Vout/Vin≈0.009901. Adding 200 pF instrument capacitance in parallel with C2 reduces the ratio to ≈0.008264.

Dimensional and limiting checks

Check dimensions at each intermediate step. In a relationship containing logarithms or exponentials, ensure their arguments are dimensionless: empirical constants may carry units precisely to make this possible. Then push one variable toward a simple limiting case. Ask whether the outcome is consistent with the physical arrangement, and identify a parameter range where the equation would cease to describe the real world.

Failure modes, model exclusions, and hazard boundary

The calculation neglects distributed leads and stray-ground paths, and cannot be used to certify impulse measurement accuracy.

Lesson problems (answer guidance follows)

1. Reproduce the baseline. Start with the precise governing relation and numerical values above. Write every symbol, perform the substitutions and express the result in physically appropriate units. Explain the sign, direction or phase reference where relevant.

2. Parameter-sensitivity check. Increase the most sensitive input in the worked example by 10% without silently changing another parameter. Recompute, distinguish relative from absolute change, and identify whether the equation predicts linear, reciprocal, logarithmic, power-law or exponential sensitivity.

3. Failure of an assumption. Describe a physically specific modification to capacitive sensors, coupling and bandwidth that renders the chapter's simplified model unreliable. Identify at least two previously missing quantities needed to restore predictive value.

4. Evidence and uncertainty. A student claims the chapter's numerical result proves an insulation component is safe. Draft a technically precise rebuttal: which quantity has actually been calculated, which uncertainty remains, and what category of qualified evidence would be needed?

5. Alternative mechanism. Propose a competing explanation for an observed deviation in the relevant field, current, waveform or diagnostic trend. Give a discriminating independent observation rather than only repeating the original formula.

Solutions and evaluation guide

Baseline answer: With C1=10 pF and C2=1 nF, Vout/Vin≈0.009901. Adding 200 pF instrument capacitance in parallel with C2 reduces the ratio to ≈0.008264. Reproduce the numeric substitution and state assumptions before reporting precision.

Sensitivity answer: Apply the actual displayed governing equation to one 10% parameter change; do not assume a linear relationship when it contains a logarithm, a quadratic energy term or an exponential.

Limitation answer: The calculation neglects distributed leads and stray-ground paths, and cannot be used to certify impulse measurement accuracy. These exclusions identify what evidence is still required. A model calculation cannot certify safety or material lifetime.

Evidence answer: Record the model result as analytically calculated, the simulator as software output, and an actual dielectric test as not performed. The fifth question requires an independent discriminating observation rather than a restatement of the model.

Software exercise

Use the [offline unit model](../simulations/model-06.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 6.3: Uncertainty budgets and calibration chains

Unit: High-voltage measurement systems and uncertainty. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

Measurement uncertainty is a statement about the dispersion of values reasonably attributable to a measurand. Independent components combine differently from correlated systematic terms. A reported calibrated ratio without conditions, traceability and a coverage factor is not enough for high-voltage qualification.

Governing equations and interpretation

For independent standard uncertainties u1 and u2, uc=√(u1²+u2²). An expanded uncertainty may be U=k uc when the chosen coverage factor is justified; it is not automatically a universal tolerance.

Fully stated worked example

Independent relative standard uncertainties of 1% and 2% combine to 2.236%. Using k=2 illustratively yields 4.472% expanded uncertainty, conditional on the distribution and coverage assumptions.

Comparing alternative mechanisms

Write down at least two competing explanations for the same observation. One can be the intended low-order model, and the other can be an omitted physical process such as moisture, a floating conductor, spatial inhomogeneity, instrument loading, dielectric ageing, or waveform bandwidth. Explain what additional observation would discriminate the hypotheses. A falsifiable comparison is more useful than repeating a likely answer.

Failure modes, model exclusions, and hazard boundary

Correlated offsets cannot be combined as though independent; repeatability is not the same as calibration accuracy.

Lesson problems (answer guidance follows)

1. Reproduce the baseline. Start with the precise governing relation and numerical values above. Write every symbol, perform the substitutions and express the result in physically appropriate units. Explain the sign, direction or phase reference where relevant.

2. Parameter-sensitivity check. Increase the most sensitive input in the worked example by 10% without silently changing another parameter. Recompute, distinguish relative from absolute change, and identify whether the equation predicts linear, reciprocal, logarithmic, power-law or exponential sensitivity.

3. Failure of an assumption. Describe a physically specific modification to uncertainty budgets and calibration chains that renders the chapter's simplified model unreliable. Identify at least two previously missing quantities needed to restore predictive value.

4. Evidence and uncertainty. A student claims the chapter's numerical result proves an insulation component is safe. Draft a technically precise rebuttal: which quantity has actually been calculated, which uncertainty remains, and what category of qualified evidence would be needed?

5. Alternative mechanism. Propose a competing explanation for an observed deviation in the relevant field, current, waveform or diagnostic trend. Give a discriminating independent observation rather than only repeating the original formula.

Solutions and evaluation guide

Baseline answer: Independent relative standard uncertainties of 1% and 2% combine to 2.236%. Using k=2 illustratively yields 4.472% expanded uncertainty, conditional on the distribution and coverage assumptions. Reproduce the numeric substitution and state assumptions before reporting precision.

Sensitivity answer: Apply the actual displayed governing equation to one 10% parameter change; do not assume a linear relationship when it contains a logarithm, a quadratic energy term or an exponential.

Limitation answer: Correlated offsets cannot be combined as though independent; repeatability is not the same as calibration accuracy. These exclusions identify what evidence is still required. A model calculation cannot certify safety or material lifetime.

Evidence answer: Record the model result as analytically calculated, the simulator as software output, and an actual dielectric test as not performed. The fifth question requires an independent discriminating observation rather than a restatement of the model.

Software exercise

Use the [offline unit model](../simulations/model-06.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 6.4: Impulse parameter extraction and reporting

Unit: High-voltage measurement systems and uncertainty. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

A peak number from a digitizer requires a declared processing method. Filtering can smooth noise but also shift peak value and front time. Test reports should distinguish raw captured samples from corrected parameters and state method and uncertainty.

Governing equations and interpretation

For a sampled signal vi, the raw sampled maximum is max(vi). It is not necessarily the continuous-waveform peak. A local quadratic interpolation may improve some estimates, provided the sampling and noise assumptions are disclosed.

Fully stated worked example

A 100 kV synthetic pulse sampled at only 1 µs intervals can miss a narrow true peak between samples. Two traces with identical reported sample maxima need not have the same front-time dynamics.

Making a controlled decision

Set a baseline, change only one controlled variable, and recalculate the metric using the stated formula. Report the change numerically, including units. Then ask whether the change also introduces an adverse effect: higher stored energy, field enhancement, leakage current, uncertainty, ageing stress, or loss of a valid approximation. End with an evidence request that a qualified technical reviewer could act on.

Failure modes, model exclusions, and hazard boundary

Use IEC 60060-1:2025 and IEC 60060-2:2025 for the applicable formal definitions and measurement-system requirements; this lesson is not a substitute.

Lesson problems (answer guidance follows)

1. Reproduce the baseline. Start with the precise governing relation and numerical values above. Write every symbol, perform the substitutions and express the result in physically appropriate units. Explain the sign, direction or phase reference where relevant.

2. Parameter-sensitivity check. Increase the most sensitive input in the worked example by 10% without silently changing another parameter. Recompute, distinguish relative from absolute change, and identify whether the equation predicts linear, reciprocal, logarithmic, power-law or exponential sensitivity.

3. Failure of an assumption. Describe a physically specific modification to impulse parameter extraction and reporting that renders the chapter's simplified model unreliable. Identify at least two previously missing quantities needed to restore predictive value.

4. Evidence and uncertainty. A student claims the chapter's numerical result proves an insulation component is safe. Draft a technically precise rebuttal: which quantity has actually been calculated, which uncertainty remains, and what category of qualified evidence would be needed?

5. Alternative mechanism. Propose a competing explanation for an observed deviation in the relevant field, current, waveform or diagnostic trend. Give a discriminating independent observation rather than only repeating the original formula.

Solutions and evaluation guide

Baseline answer: A 100 kV synthetic pulse sampled at only 1 µs intervals can miss a narrow true peak between samples. Two traces with identical reported sample maxima need not have the same front-time dynamics. Reproduce the numeric substitution and state assumptions before reporting precision.

Sensitivity answer: Apply the actual displayed governing equation to one 10% parameter change; do not assume a linear relationship when it contains a logarithm, a quadratic energy term or an exponential.

Limitation answer: Use IEC 60060-1:2025 and IEC 60060-2:2025 for the applicable formal definitions and measurement-system requirements; this lesson is not a substitute. These exclusions identify what evidence is still required. A model calculation cannot certify safety or material lifetime.

Evidence answer: Record the model result as analytically calculated, the simulator as software output, and an actual dielectric test as not performed. The fifth question requires an independent discriminating observation rather than a restatement of the model.

Software exercise

Use the [offline unit model](../simulations/model-06.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Unit 6: High-voltage measurement systems and uncertainty

Why this subject matters

A measurement chain contributes loading, stray coupling, frequency response, calibration uncertainty and data-acquisition errors. Simply dividing a scope trace by a nominal ratio cannot verify a test voltage. Rated voltage alone does not imply impulse bandwidth.

Common reference case and assumptions

Use low-energy equivalent circuits, uncertainty budgets and synthetic impulse samples to analyze measurement credibility. Physical dividers and energized probes are not assembled. The goal is competence in interpreting certificates and synthetic measurements, not a test-lab construction guide.

Outcomes and verification

Propagate uncertainty, distinguish random and systematic effects, assess sensor loading and determine when traceable calibration evidence is required.

A researcher's reading protocol

Distinguish an instrument input from a calibrated complete measurement system. Work through loading, transfer function, resolution, systematic error and uncertainty of the inferred measurand. An instrument display with six decimals need not carry six decimals of credibility. The model is intentionally limited to analytical and offline-computation exercises. A student should identify the quantity being predicted and ask what observational evidence would be needed to validate it. The location of the maximum physical field is especially significant: an average stress may conceal a sharp local concentration at a boundary, edge, void, accessory or transient discontinuity.

When reading Lessons 6.1–6.4, first reconstruct the derivation, then change one variable in the supplied worked example. Carry the physical units consistently and ask whether a plotted curve can be traced back to the same symbol definitions as the equation. Use the [interactive software model](../simulations/model-06.html) after the hand calculation so the plot serves as a diagnostic rather than an opaque answer generator.

What counts as evidence

The unit challenge is reporting what an independently calibrated chain can support, not what its screen is capable of rendering. Distinguish proof of code execution, numerical agreement with a mathematical model, calibration evidence, dielectric qualification, and safe operation. These are five separate accomplishments; none implies the next automatically. All practice in this book is software-only. Do not build test apparatus, charge capacitors or obtain energized equipment for these exercises.

Suggested learning route

Read all four lessons and reproduce the arithmetic in a notebook. Complete the project with a baseline case and three documented variations. Attempt the unit assessment before consulting its solution notes. Record a clear status for each conclusion: derived, numerically checked, simulated, empirically supported or unresolved. There are no fabricated laboratory observations in this book.

Lesson 6.1: Voltage-dividing ratios and loading

Unit: High-voltage measurement systems and uncertainty. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

An ideal resistive divider ratio is determined by two resistances, but finite instrument impedance loads the lower element. Frequency behavior adds stray capacitance and inductance, requiring more than a DC ratio for impulse measurement. This chapter analyzes low-energy math only.

Governing equations and interpretation

For R1 in series with R2 and measured output across R2, k=R2/(R1+R2). With instrument resistance Rin, replace R2 by R2∥Rin.

Fully stated worked example

For R1=9 MΩ and R2=100 kΩ, ideal k≈0.010989. With Rin=1 MΩ, the effective low side is 90.91 kΩ and k≈0.010000; ignoring loading would bias the inferred input.

Reconstructing the model

Define every boundary condition and symbol before calculating. Derive the supplied relation from a recognizable physical law: Gauss’s law, conservation of charge, electromagnetic-wave continuity, equivalent-circuit impedance or probability bookkeeping as appropriate. If a derivative or spatial gradient appears, write what is being held constant and what geometric symmetry is assumed. The derivation must connect to an experimentally meaningful quantity rather than stop at formal algebra.

Failure modes, model exclusions, and hazard boundary

A divider that is correct at DC can be grossly wrong for fast impulses; no physical assembly or energized test is described.

Lesson problems (answer guidance follows)

1. Reproduce the baseline. Start with the precise governing relation and numerical values above. Write every symbol, perform the substitutions and express the result in physically appropriate units. Explain the sign, direction or phase reference where relevant.

2. Parameter-sensitivity check. Increase the most sensitive input in the worked example by 10% without silently changing another parameter. Recompute, distinguish relative from absolute change, and identify whether the equation predicts linear, reciprocal, logarithmic, power-law or exponential sensitivity.

3. Failure of an assumption. Describe a physically specific modification to voltage-dividing ratios and loading that renders the chapter's simplified model unreliable. Identify at least two previously missing quantities needed to restore predictive value.

4. Evidence and uncertainty. A student claims the chapter's numerical result proves an insulation component is safe. Draft a technically precise rebuttal: which quantity has actually been calculated, which uncertainty remains, and what category of qualified evidence would be needed?

5. Alternative mechanism. Propose a competing explanation for an observed deviation in the relevant field, current, waveform or diagnostic trend. Give a discriminating independent observation rather than only repeating the original formula.

Solutions and evaluation guide

Baseline answer: For R1=9 MΩ and R2=100 kΩ, ideal k≈0.010989. With Rin=1 MΩ, the effective low side is 90.91 kΩ and k≈0.010000; ignoring loading would bias the inferred input. Reproduce the numeric substitution and state assumptions before reporting precision.

Sensitivity answer: Apply the actual displayed governing equation to one 10% parameter change; do not assume a linear relationship when it contains a logarithm, a quadratic energy term or an exponential.

Limitation answer: A divider that is correct at DC can be grossly wrong for fast impulses; no physical assembly or energized test is described. These exclusions identify what evidence is still required. A model calculation cannot certify safety or material lifetime.

Evidence answer: Record the model result as analytically calculated, the simulator as software output, and an actual dielectric test as not performed. The fifth question requires an independent discriminating observation rather than a restatement of the model.

Software exercise

Use the [offline unit model](../simulations/model-06.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 6.2: Capacitive sensors, coupling and bandwidth

Unit: High-voltage measurement systems and uncertainty. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

Capacitive measurement chains are sensitive to parasitic capacitances, grounding arrangement and frequency. Their transfer functions often require calibrated dynamic response rather than a nominal capacitance ratio. The voltage delivered to an instrument can change with cable length.

Governing equations and interpretation

For two ideal series capacitances C1 and C2 with Vout across C2, Vout/Vin=C1/(C1+C2). This follows from equal series charge, not from the resistive-divider formula.

Fully stated worked example

With C1=10 pF and C2=1 nF, Vout/Vin≈0.009901. Adding 200 pF instrument capacitance in parallel with C2 reduces the ratio to ≈0.008264.

Dimensional and limiting checks

Check dimensions at each intermediate step. In a relationship containing logarithms or exponentials, ensure their arguments are dimensionless: empirical constants may carry units precisely to make this possible. Then push one variable toward a simple limiting case. Ask whether the outcome is consistent with the physical arrangement, and identify a parameter range where the equation would cease to describe the real world.

Failure modes, model exclusions, and hazard boundary

The calculation neglects distributed leads and stray-ground paths, and cannot be used to certify impulse measurement accuracy.

Lesson problems (answer guidance follows)

1. Reproduce the baseline. Start with the precise governing relation and numerical values above. Write every symbol, perform the substitutions and express the result in physically appropriate units. Explain the sign, direction or phase reference where relevant.

2. Parameter-sensitivity check. Increase the most sensitive input in the worked example by 10% without silently changing another parameter. Recompute, distinguish relative from absolute change, and identify whether the equation predicts linear, reciprocal, logarithmic, power-law or exponential sensitivity.

3. Failure of an assumption. Describe a physically specific modification to capacitive sensors, coupling and bandwidth that renders the chapter's simplified model unreliable. Identify at least two previously missing quantities needed to restore predictive value.

4. Evidence and uncertainty. A student claims the chapter's numerical result proves an insulation component is safe. Draft a technically precise rebuttal: which quantity has actually been calculated, which uncertainty remains, and what category of qualified evidence would be needed?

5. Alternative mechanism. Propose a competing explanation for an observed deviation in the relevant field, current, waveform or diagnostic trend. Give a discriminating independent observation rather than only repeating the original formula.

Solutions and evaluation guide

Baseline answer: With C1=10 pF and C2=1 nF, Vout/Vin≈0.009901. Adding 200 pF instrument capacitance in parallel with C2 reduces the ratio to ≈0.008264. Reproduce the numeric substitution and state assumptions before reporting precision.

Sensitivity answer: Apply the actual displayed governing equation to one 10% parameter change; do not assume a linear relationship when it contains a logarithm, a quadratic energy term or an exponential.

Limitation answer: The calculation neglects distributed leads and stray-ground paths, and cannot be used to certify impulse measurement accuracy. These exclusions identify what evidence is still required. A model calculation cannot certify safety or material lifetime.

Evidence answer: Record the model result as analytically calculated, the simulator as software output, and an actual dielectric test as not performed. The fifth question requires an independent discriminating observation rather than a restatement of the model.

Software exercise

Use the [offline unit model](../simulations/model-06.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 6.3: Uncertainty budgets and calibration chains

Unit: High-voltage measurement systems and uncertainty. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

Measurement uncertainty is a statement about the dispersion of values reasonably attributable to a measurand. Independent components combine differently from correlated systematic terms. A reported calibrated ratio without conditions, traceability and a coverage factor is not enough for high-voltage qualification.

Governing equations and interpretation

For independent standard uncertainties u1 and u2, uc=√(u1²+u2²). An expanded uncertainty may be U=k uc when the chosen coverage factor is justified; it is not automatically a universal tolerance.

Fully stated worked example

Independent relative standard uncertainties of 1% and 2% combine to 2.236%. Using k=2 illustratively yields 4.472% expanded uncertainty, conditional on the distribution and coverage assumptions.

Comparing alternative mechanisms

Write down at least two competing explanations for the same observation. One can be the intended low-order model, and the other can be an omitted physical process such as moisture, a floating conductor, spatial inhomogeneity, instrument loading, dielectric ageing, or waveform bandwidth. Explain what additional observation would discriminate the hypotheses. A falsifiable comparison is more useful than repeating a likely answer.

Failure modes, model exclusions, and hazard boundary

Correlated offsets cannot be combined as though independent; repeatability is not the same as calibration accuracy.

Lesson problems (answer guidance follows)

1. Reproduce the baseline. Start with the precise governing relation and numerical values above. Write every symbol, perform the substitutions and express the result in physically appropriate units. Explain the sign, direction or phase reference where relevant.

2. Parameter-sensitivity check. Increase the most sensitive input in the worked example by 10% without silently changing another parameter. Recompute, distinguish relative from absolute change, and identify whether the equation predicts linear, reciprocal, logarithmic, power-law or exponential sensitivity.

3. Failure of an assumption. Describe a physically specific modification to uncertainty budgets and calibration chains that renders the chapter's simplified model unreliable. Identify at least two previously missing quantities needed to restore predictive value.

4. Evidence and uncertainty. A student claims the chapter's numerical result proves an insulation component is safe. Draft a technically precise rebuttal: which quantity has actually been calculated, which uncertainty remains, and what category of qualified evidence would be needed?

5. Alternative mechanism. Propose a competing explanation for an observed deviation in the relevant field, current, waveform or diagnostic trend. Give a discriminating independent observation rather than only repeating the original formula.

Solutions and evaluation guide

Baseline answer: Independent relative standard uncertainties of 1% and 2% combine to 2.236%. Using k=2 illustratively yields 4.472% expanded uncertainty, conditional on the distribution and coverage assumptions. Reproduce the numeric substitution and state assumptions before reporting precision.

Sensitivity answer: Apply the actual displayed governing equation to one 10% parameter change; do not assume a linear relationship when it contains a logarithm, a quadratic energy term or an exponential.

Limitation answer: Correlated offsets cannot be combined as though independent; repeatability is not the same as calibration accuracy. These exclusions identify what evidence is still required. A model calculation cannot certify safety or material lifetime.

Evidence answer: Record the model result as analytically calculated, the simulator as software output, and an actual dielectric test as not performed. The fifth question requires an independent discriminating observation rather than a restatement of the model.

Software exercise

Use the [offline unit model](../simulations/model-06.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 6.4: Impulse parameter extraction and reporting

Unit: High-voltage measurement systems and uncertainty. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

A peak number from a digitizer requires a declared processing method. Filtering can smooth noise but also shift peak value and front time. Test reports should distinguish raw captured samples from corrected parameters and state method and uncertainty.

Governing equations and interpretation

For a sampled signal vi, the raw sampled maximum is max(vi). It is not necessarily the continuous-waveform peak. A local quadratic interpolation may improve some estimates, provided the sampling and noise assumptions are disclosed.

Fully stated worked example

A 100 kV synthetic pulse sampled at only 1 µs intervals can miss a narrow true peak between samples. Two traces with identical reported sample maxima need not have the same front-time dynamics.

Making a controlled decision

Set a baseline, change only one controlled variable, and recalculate the metric using the stated formula. Report the change numerically, including units. Then ask whether the change also introduces an adverse effect: higher stored energy, field enhancement, leakage current, uncertainty, ageing stress, or loss of a valid approximation. End with an evidence request that a qualified technical reviewer could act on.

Failure modes, model exclusions, and hazard boundary

Use IEC 60060-1:2025 and IEC 60060-2:2025 for the applicable formal definitions and measurement-system requirements; this lesson is not a substitute.

Lesson problems (answer guidance follows)

1. Reproduce the baseline. Start with the precise governing relation and numerical values above. Write every symbol, perform the substitutions and express the result in physically appropriate units. Explain the sign, direction or phase reference where relevant.

2. Parameter-sensitivity check. Increase the most sensitive input in the worked example by 10% without silently changing another parameter. Recompute, distinguish relative from absolute change, and identify whether the equation predicts linear, reciprocal, logarithmic, power-law or exponential sensitivity.

3. Failure of an assumption. Describe a physically specific modification to impulse parameter extraction and reporting that renders the chapter's simplified model unreliable. Identify at least two previously missing quantities needed to restore predictive value.

4. Evidence and uncertainty. A student claims the chapter's numerical result proves an insulation component is safe. Draft a technically precise rebuttal: which quantity has actually been calculated, which uncertainty remains, and what category of qualified evidence would be needed?

5. Alternative mechanism. Propose a competing explanation for an observed deviation in the relevant field, current, waveform or diagnostic trend. Give a discriminating independent observation rather than only repeating the original formula.

Solutions and evaluation guide

Baseline answer: A 100 kV synthetic pulse sampled at only 1 µs intervals can miss a narrow true peak between samples. Two traces with identical reported sample maxima need not have the same front-time dynamics. Reproduce the numeric substitution and state assumptions before reporting precision.

Sensitivity answer: Apply the actual displayed governing equation to one 10% parameter change; do not assume a linear relationship when it contains a logarithm, a quadratic energy term or an exponential.

Limitation answer: Use IEC 60060-1:2025 and IEC 60060-2:2025 for the applicable formal definitions and measurement-system requirements; this lesson is not a substitute. These exclusions identify what evidence is still required. A model calculation cannot certify safety or material lifetime.

Evidence answer: Record the model result as analytically calculated, the simulator as software output, and an actual dielectric test as not performed. The fifth question requires an independent discriminating observation rather than a restatement of the model.

Software exercise

Use the [offline unit model](../simulations/model-06.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Unit 6 project: Measurement credibility notebook

Type: offline investigation only. Time: suggested three study sessions. Dataset: hypothetical values and deterministic simulator output; no real equipment observations are claimed.

Problem statement

Compare nominal and instrument-loaded divider ratios and build a two-component standard uncertainty budget.

A measurement chain contributes loading, stray coupling, frequency response, calibration uncertainty and data-acquisition errors. Simply dividing a scope trace by a nominal ratio cannot verify a test voltage. Rated voltage alone does not imply impulse bandwidth.

Use the underlying quantities from Lessons 6.1–6.4 rather than a single unexamined software result. Define the components of the system, model assumptions, time scale and spatial location of quantities. Make a table with input names, units, baseline values, variations and equations used. The notebook should be reproducible by another student without their guessing what your graph labels or unit conventions mean.

Phase 1 — derive and reproduce

Write a system diagram with all reference directions. Re-derive at least two of the unit's governing relations, including a limiting-case check. Compute the baseline numerical examples independently. Verify every logarithm argument, SI conversion and phase reference; do not copy values from a graph without stating their coordinates. In the model, account for the source of any numerical peak or numerical minimum: it may arise from a mathematical function's domain, a numerical sampling grid, or a physical boundary. Those distinctions are essential to interpreting the result.

Phase 2 — controlled variations

Vary instrument resistance independently of nominal R2. Decide which observed changes are sensor loading versus uncertainty estimation.

Create a separate table for each variation, changing only one parameter at a time. Note how far the new point lies from the model's stated assumptions. When an assumption becomes invalid, stop and mark the result as an extrapolation. Explain whether the outcome changes because of a physical law, geometry, empirical coefficient, or measurement convention. Include one deliberately invalid point to demonstrate the limits of the simplified model, rather than silently removing difficult samples.

Phase 3 — reproducible software study

Open [the unit's model](../simulations/model-06.html) from the offline HTML package, document its input settings, and compare its result with your independent calculation. Preserve the chart as an SVG or screenshot only as supporting evidence; include the numerical parameter table in plain text so it can be checked without image-reading software. Examine two scales for each variable if useful: a full-range overview and a zoomed view near the reference case.

Phase 4 — failure-first review

Describe one failure mechanism hidden by the chosen model, one instrument error that could masquerade as a physical effect, and one environmental change that might shift the answer. Propose a separate independent observation that could falsify the model in a qualified laboratory. Do not design or conduct energized physical testing. Include a hypothetical hazard register naming shock, arc, stored-energy and unexpected re-energization risks; record that these are outside the software activity.

Deliverables and answer check

Provide a calibration-evidence checklist, clear units, and distinction between corrected signal and raw samples.

Reference check: With 9 MΩ, 100 kΩ, and instrument Rin=1 MΩ, the loaded ratio is approximately 0.010000.

A complete project has a properly labeled baseline table; at least three calculations derived from written equations; at least two changed-parameter traces; numerical and dimensional verification; model exclusions; and an evidence-status conclusion. The graph alone is insufficient. The written conclusion must distinguish mathematical verification, simulation, real-world measurement and regulatory compliance. For each physical acceptance claim that cannot be established, use the status not established and give a reason.

Assessment rubric

Award 25% for correct equations and consistent units, 20% for independent numerical reproduction, 20% for parameter-sensitivity logic, 20% for explicit limitations and hazard boundaries, and 15% for evidence traceability. No part of the rubric rewards inventing measured values or pretending that software has completed a dielectric test. The grading criterion is reasoning quality, not the number of graphs or pages.

Assessment for Unit 6: High-voltage measurement systems and uncertainty

Instructions: offline, closed-hardware assessment. Show assumptions and conversions. Complete all parts. These questions check whether the student can distinguish a mathematical model from a physical approval.

Questions

1. Derivation and geometry. In the context of Voltage-dividing ratios and loading, reconstruct the stated governing relation, identify the boundary conditions, and produce a dimensional check. Give an example of a geometry or environment where the derivation fails.

2. Quantitative comparison. In the context of Capacitive sensors, coupling and bandwidth, reproduce the numeric example, then change one stated input by 10%. Give both the new quantitative result and the reason the scaling law applies. Do not substitute an unrelated formula.

3. Competing explanations. In the context of Uncertainty budgets and calibration chains, describe an observed numerical change with two possible physical explanations. Name an independent observation capable of separating them.

4. Engineering consequence. In the context of Impulse parameter extraction and reporting, evaluate what a student may and may not conclude from the supplied worked result. Name one omitted loss or degradation mechanism, one calibration concern, and one hazard category.

5. Evidence table. Create a four-row table for the unit separating a derived fact, a software-simulated result, a proposed qualified laboratory observation, and a standard-specific acceptance claim. Mark the last two as not performed unless verified evidence exists.

6. Software sensitivity. Run the offline simulation for this unit with two parameter settings. Describe the axes, the mathematical relation displayed, the numerical change and whether extrapolation occurs.

7. Professional decision. An organization wants to qualify an insulation system solely from your notebook. Write a concise release decision describing at least three missing kinds of evidence and which kind of specialist would oversee testing.

Suggested answer notes

Q1: Use the relation from Lesson 6.1 and verify symbols before calculating. A correct solution explicitly states relevant geometry and symmetry. A relation that is dimensionally correct may still be physically inapplicable.

Q2: Reproduce Lesson 6.2's stated reference example and change a named variable only. Show the original and changed substitutions. Scaling must follow the actual equation, which may be linear, inverse, exponential or logarithmic.

Q3: A defensible pair of explanations differs in a falsifiable prediction. Include material/environmental effects or sensor-system coupling as appropriate. A visually consistent model trace cannot identify a real defect by itself.

Q4: The result of Lesson 6.4 is a conditional estimate. No safety margin, equipment approval or time-to-failure is established without independently validated parameters, a specified acceptance rule and an authorized measurement program.

Q5: The table is graded on correct evidence labeling. Calculated and simulated answers are not fabricated measurements. Citing IEC standards establishes the existence of requirements, not the fact of compliance.

Q6: A complete software analysis names both input states and both output values, reproduces the output from the stated mathematical model, and explains what a real-world experiment would still need to show.

Q7: The appropriate decision is not approved on this evidence. Recommend evaluation by a qualified high-voltage laboratory and asset-specific subject-matter experts before any physical acceptance or energized testing.

Submission quality checks

Before submitting, verify all physical units, numerical substitutions, source-versus-measurement labels, limitations and standard references. Do not submit personal or invented measurement data. Models are not protective equipment, and the exercises do not authorize hands-on high-voltage activities.

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Unit 6 project: Measurement credibility notebook

Type: offline investigation only. Time: suggested three study sessions. Dataset: hypothetical values and deterministic simulator output; no real equipment observations are claimed.

Problem statement

Compare nominal and instrument-loaded divider ratios and build a two-component standard uncertainty budget.

A measurement chain contributes loading, stray coupling, frequency response, calibration uncertainty and data-acquisition errors. Simply dividing a scope trace by a nominal ratio cannot verify a test voltage. Rated voltage alone does not imply impulse bandwidth.

Use the underlying quantities from Lessons 6.1–6.4 rather than a single unexamined software result. Define the components of the system, model assumptions, time scale and spatial location of quantities. Make a table with input names, units, baseline values, variations and equations used. The notebook should be reproducible by another student without their guessing what your graph labels or unit conventions mean.

Phase 1 — derive and reproduce

Write a system diagram with all reference directions. Re-derive at least two of the unit's governing relations, including a limiting-case check. Compute the baseline numerical examples independently. Verify every logarithm argument, SI conversion and phase reference; do not copy values from a graph without stating their coordinates. In the model, account for the source of any numerical peak or numerical minimum: it may arise from a mathematical function's domain, a numerical sampling grid, or a physical boundary. Those distinctions are essential to interpreting the result.

Phase 2 — controlled variations

Vary instrument resistance independently of nominal R2. Decide which observed changes are sensor loading versus uncertainty estimation.

Create a separate table for each variation, changing only one parameter at a time. Note how far the new point lies from the model's stated assumptions. When an assumption becomes invalid, stop and mark the result as an extrapolation. Explain whether the outcome changes because of a physical law, geometry, empirical coefficient, or measurement convention. Include one deliberately invalid point to demonstrate the limits of the simplified model, rather than silently removing difficult samples.

Phase 3 — reproducible software study

Open [the unit's model](../simulations/model-06.html) from the offline HTML package, document its input settings, and compare its result with your independent calculation. Preserve the chart as an SVG or screenshot only as supporting evidence; include the numerical parameter table in plain text so it can be checked without image-reading software. Examine two scales for each variable if useful: a full-range overview and a zoomed view near the reference case.

Phase 4 — failure-first review

Describe one failure mechanism hidden by the chosen model, one instrument error that could masquerade as a physical effect, and one environmental change that might shift the answer. Propose a separate independent observation that could falsify the model in a qualified laboratory. Do not design or conduct energized physical testing. Include a hypothetical hazard register naming shock, arc, stored-energy and unexpected re-energization risks; record that these are outside the software activity.

Deliverables and answer check

Provide a calibration-evidence checklist, clear units, and distinction between corrected signal and raw samples.

Reference check: With 9 MΩ, 100 kΩ, and instrument Rin=1 MΩ, the loaded ratio is approximately 0.010000.

A complete project has a properly labeled baseline table; at least three calculations derived from written equations; at least two changed-parameter traces; numerical and dimensional verification; model exclusions; and an evidence-status conclusion. The graph alone is insufficient. The written conclusion must distinguish mathematical verification, simulation, real-world measurement and regulatory compliance. For each physical acceptance claim that cannot be established, use the status not established and give a reason.

Assessment rubric

Award 25% for correct equations and consistent units, 20% for independent numerical reproduction, 20% for parameter-sensitivity logic, 20% for explicit limitations and hazard boundaries, and 15% for evidence traceability. No part of the rubric rewards inventing measured values or pretending that software has completed a dielectric test. The grading criterion is reasoning quality, not the number of graphs or pages.