Unit 7: Generation of Impulse Voltages: Single-Stage and Multi-Stage Marx Generators

Unit 7: Partial discharge and insulation diagnostics

Why this subject matters

Partial discharge is a localized event that does not completely bridge the electrode separation. The measurable terminal pulse depends on circuit coupling, instrument bandwidth and interference; apparent charge is not necessarily the actual charge transferred inside a void.

Common reference case and assumptions

Analyze a synthetic phase-resolved PD display with known injected pulse times. Use charge-based concepts to explain why calibration, noise rejection and a measurement window are necessary before interpreting a pattern.

Outcomes and verification

Interpret synthetic pulses and their limitations, distinguish diagnostic hypotheses, and design a defensible measurement-evidence report.

A researcher's reading protocol

Describe how a calibrated equivalent injection would be interpreted, without building the system. Separate the event detection rule, false-positive prevalence, phase reference and hypothesis about the source of the pulses. Beware of using an attractive phase-resolved image as proof of a unique defect. 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 7.1–7.4, 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-07.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 resisting overconfident condition diagnosis from a short synthetic sample. 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 7.1: Partial discharge versus complete breakdown

Unit: Partial discharge and insulation diagnostics. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

Localized discharge within or along insulation does not necessarily form a full electrode-to-electrode conductive path. Apparent charge is a circuit-dependent measured quantity, not directly the microscopic charge in every void. Inception and extinction may depend on previous cycles.

Governing equations and interpretation

Charge from a calibrated equivalent injection is q=Cinj ΔV in a simple capacitive calibration model. State that this is an injected reference charge in the calibration circuit.

Fully stated worked example

For Cinj=10 pF and ΔV=10 V, the mathematical injected charge is 100 pC. This does not establish the size or damage rate of an internal void discharge.

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

Instrument coupling, bandwidth, grounding, PD pulse overlap and electrical noise affect the displayed apparent charge.

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 partial discharge versus complete breakdown 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 Cinj=10 pF and ΔV=10 V, the mathematical injected charge is 100 pC. This does not establish the size or damage rate of an internal void discharge. 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: Instrument coupling, bandwidth, grounding, PD pulse overlap and electrical noise affect the displayed apparent charge. 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-07.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 7.2: Phase-resolved PD pattern interpretation

Unit: Partial discharge and insulation diagnostics. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

A phase-resolved view maps pulse amplitude or apparent charge against the instantaneous AC phase and repetition. Patterns can suggest possible discharge mechanisms but are not unique signatures; multiple physical faults and external interference can create similar images.

Governing equations and interpretation

If t is a pulse timestamp and excitation frequency f, the nominal electrical phase is φ=(360° f t) mod 360°, provided the phase reference and synchronization are correct.

Fully stated worked example

At 50 Hz, a pulse 5 ms after a reference crossing occurs at 90°, while one 15 ms after occurs at 270°. A 1 ms timing offset shifts phase by 18°.

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

Randomized synthetic pulses are not a trained diagnostic classifier; never report a defect type without corroborating evidence.

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 phase-resolved pd pattern interpretation 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: At 50 Hz, a pulse 5 ms after a reference crossing occurs at 90°, while one 15 ms after occurs at 270°. A 1 ms timing offset shifts phase by 18°. 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: Randomized synthetic pulses are not a trained diagnostic classifier; never report a defect type without corroborating evidence. 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-07.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 7.3: Noise, pulse discrimination, and frequency effects

Unit: Partial discharge and insulation diagnostics. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

PD sensing competes with switching transients, radio interference and sensor noise. A threshold can reject noise while also deleting small true pulses, creating a biased count. Choosing bandwidth or filtering requires a signal-and-noise tradeoff.

Governing equations and interpretation

If a threshold is T and observed amplitude is A, the simple decision rule |A|>T classifies a recorded event, but it does not prove the event is PD. False-positive and false-negative rates require labeled evidence.

Fully stated worked example

In a synthetic batch of 100 true pulses and 200 noise pulses, suppose a threshold captures 80 true pulses and 10 noise pulses: precision=80/90≈88.9%, recall=80/100=80%.

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

These rates describe only the specified synthetic dataset. Field performance can be very different due to site-specific interference.

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 noise, pulse discrimination, and frequency effects 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: In a synthetic batch of 100 true pulses and 200 noise pulses, suppose a threshold captures 80 true pulses and 10 noise pulses: precision=80/90≈88.9%, recall=80/100=80%. 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: These rates describe only the specified synthetic dataset. Field performance can be very different due to site-specific interference. 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-07.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 7.4: Condition monitoring and trends without false certainty

Unit: Partial discharge and insulation diagnostics. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

A diagnostic trend is meaningful only if the measurement process, operating conditions and normalization are sufficiently comparable. One apparently lower PD amplitude does not prove self-healing; the stimulus might have changed or the sensor might have drifted.

Governing equations and interpretation

For repeated apparent-charge readings qi, a sample mean qbar=Σqi/n and sample deviation s=√[Σ(qi−qbar)²/(n−1)] summarize a dataset but do not quantify failure probability.

Fully stated worked example

Synthetic readings 10, 12 and 14 pC have mean 12 pC and sample standard deviation 2 pC. A later single reading of 9 pC is inadequate evidence of an improving asset.

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

Service-life forecasting needs validated material-specific models, operating history, confounder control and appropriate physical inspections.

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 condition monitoring and trends without false certainty 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: Synthetic readings 10, 12 and 14 pC have mean 12 pC and sample standard deviation 2 pC. A later single reading of 9 pC is inadequate evidence of an improving asset. 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: Service-life forecasting needs validated material-specific models, operating history, confounder control and appropriate physical inspections. 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-07.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Unit 7: Partial discharge and insulation diagnostics

Why this subject matters

Partial discharge is a localized event that does not completely bridge the electrode separation. The measurable terminal pulse depends on circuit coupling, instrument bandwidth and interference; apparent charge is not necessarily the actual charge transferred inside a void.

Common reference case and assumptions

Analyze a synthetic phase-resolved PD display with known injected pulse times. Use charge-based concepts to explain why calibration, noise rejection and a measurement window are necessary before interpreting a pattern.

Outcomes and verification

Interpret synthetic pulses and their limitations, distinguish diagnostic hypotheses, and design a defensible measurement-evidence report.

A researcher's reading protocol

Describe how a calibrated equivalent injection would be interpreted, without building the system. Separate the event detection rule, false-positive prevalence, phase reference and hypothesis about the source of the pulses. Beware of using an attractive phase-resolved image as proof of a unique defect. 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 7.1–7.4, 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-07.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 resisting overconfident condition diagnosis from a short synthetic sample. 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 7.1: Partial discharge versus complete breakdown

Unit: Partial discharge and insulation diagnostics. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

Localized discharge within or along insulation does not necessarily form a full electrode-to-electrode conductive path. Apparent charge is a circuit-dependent measured quantity, not directly the microscopic charge in every void. Inception and extinction may depend on previous cycles.

Governing equations and interpretation

Charge from a calibrated equivalent injection is q=Cinj ΔV in a simple capacitive calibration model. State that this is an injected reference charge in the calibration circuit.

Fully stated worked example

For Cinj=10 pF and ΔV=10 V, the mathematical injected charge is 100 pC. This does not establish the size or damage rate of an internal void discharge.

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

Instrument coupling, bandwidth, grounding, PD pulse overlap and electrical noise affect the displayed apparent charge.

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 partial discharge versus complete breakdown 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 Cinj=10 pF and ΔV=10 V, the mathematical injected charge is 100 pC. This does not establish the size or damage rate of an internal void discharge. 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: Instrument coupling, bandwidth, grounding, PD pulse overlap and electrical noise affect the displayed apparent charge. 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-07.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 7.2: Phase-resolved PD pattern interpretation

Unit: Partial discharge and insulation diagnostics. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

A phase-resolved view maps pulse amplitude or apparent charge against the instantaneous AC phase and repetition. Patterns can suggest possible discharge mechanisms but are not unique signatures; multiple physical faults and external interference can create similar images.

Governing equations and interpretation

If t is a pulse timestamp and excitation frequency f, the nominal electrical phase is φ=(360° f t) mod 360°, provided the phase reference and synchronization are correct.

Fully stated worked example

At 50 Hz, a pulse 5 ms after a reference crossing occurs at 90°, while one 15 ms after occurs at 270°. A 1 ms timing offset shifts phase by 18°.

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

Randomized synthetic pulses are not a trained diagnostic classifier; never report a defect type without corroborating evidence.

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 phase-resolved pd pattern interpretation 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: At 50 Hz, a pulse 5 ms after a reference crossing occurs at 90°, while one 15 ms after occurs at 270°. A 1 ms timing offset shifts phase by 18°. 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: Randomized synthetic pulses are not a trained diagnostic classifier; never report a defect type without corroborating evidence. 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-07.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 7.3: Noise, pulse discrimination, and frequency effects

Unit: Partial discharge and insulation diagnostics. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

PD sensing competes with switching transients, radio interference and sensor noise. A threshold can reject noise while also deleting small true pulses, creating a biased count. Choosing bandwidth or filtering requires a signal-and-noise tradeoff.

Governing equations and interpretation

If a threshold is T and observed amplitude is A, the simple decision rule |A|>T classifies a recorded event, but it does not prove the event is PD. False-positive and false-negative rates require labeled evidence.

Fully stated worked example

In a synthetic batch of 100 true pulses and 200 noise pulses, suppose a threshold captures 80 true pulses and 10 noise pulses: precision=80/90≈88.9%, recall=80/100=80%.

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

These rates describe only the specified synthetic dataset. Field performance can be very different due to site-specific interference.

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 noise, pulse discrimination, and frequency effects 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: In a synthetic batch of 100 true pulses and 200 noise pulses, suppose a threshold captures 80 true pulses and 10 noise pulses: precision=80/90≈88.9%, recall=80/100=80%. 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: These rates describe only the specified synthetic dataset. Field performance can be very different due to site-specific interference. 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-07.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 7.4: Condition monitoring and trends without false certainty

Unit: Partial discharge and insulation diagnostics. Mode: theoretical analysis and offline simulation. Prerequisites: elementary electromagnetism, circuits, differential equations, logarithms and unit conversion. Review status: teaching draft.

Physical mechanism

A diagnostic trend is meaningful only if the measurement process, operating conditions and normalization are sufficiently comparable. One apparently lower PD amplitude does not prove self-healing; the stimulus might have changed or the sensor might have drifted.

Governing equations and interpretation

For repeated apparent-charge readings qi, a sample mean qbar=Σqi/n and sample deviation s=√[Σ(qi−qbar)²/(n−1)] summarize a dataset but do not quantify failure probability.

Fully stated worked example

Synthetic readings 10, 12 and 14 pC have mean 12 pC and sample standard deviation 2 pC. A later single reading of 9 pC is inadequate evidence of an improving asset.

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

Service-life forecasting needs validated material-specific models, operating history, confounder control and appropriate physical inspections.

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 condition monitoring and trends without false certainty 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: Synthetic readings 10, 12 and 14 pC have mean 12 pC and sample standard deviation 2 pC. A later single reading of 9 pC is inadequate evidence of an improving asset. 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: Service-life forecasting needs validated material-specific models, operating history, confounder control and appropriate physical inspections. 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-07.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Unit 7 project: Phase-resolved synthetic pulse review

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

Problem statement

Analyze generated PD-like pulses against a controlled noise background; do not infer a real insulation fault.

Partial discharge is a localized event that does not completely bridge the electrode separation. The measurable terminal pulse depends on circuit coupling, instrument bandwidth and interference; apparent charge is not necessarily the actual charge transferred inside a void.

Use the underlying quantities from Lessons 7.1–7.4 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 threshold and phase offset separately. Compute detected count, a false-positive example with known synthetic labels, and the resulting tradeoff.

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-07.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

Export a phase plot, threshold table, phase-reference statement and list of additional independent field evidence needed.

Reference check: At 50 Hz a 1 ms time-reference offset shifts the phase by 18 degrees.

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 7: Partial discharge and insulation diagnostics

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 Partial discharge versus complete breakdown, 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 Phase-resolved PD pattern interpretation, 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 Noise, pulse discrimination, and frequency effects, 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 Condition monitoring and trends without false certainty, 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 7.1 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 7.2'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 7.4 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 7 project: Phase-resolved synthetic pulse review

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

Problem statement

Analyze generated PD-like pulses against a controlled noise background; do not infer a real insulation fault.

Partial discharge is a localized event that does not completely bridge the electrode separation. The measurable terminal pulse depends on circuit coupling, instrument bandwidth and interference; apparent charge is not necessarily the actual charge transferred inside a void.

Use the underlying quantities from Lessons 7.1–7.4 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 threshold and phase offset separately. Compute detected count, a false-positive example with known synthetic labels, and the resulting tradeoff.

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-07.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

Export a phase plot, threshold table, phase-reference statement and list of additional independent field evidence needed.

Reference check: At 50 Hz a 1 ms time-reference offset shifts the phase by 18 degrees.

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.