Unit 7: Corona Phenomena, Power Loss, Audible Noise, and Radio Interference

Unit 7: Diagnostics, interpretation and uncertainty

Engineering overview

Diagnostic signals are noisy evidence, not unambiguous labels. PD pulses, dielectric loss, thermal imaging, dissolved gases and acoustic events have different selectivity and uncertainty. The rate of false alarms matters particularly when a defect is rare.

Four separate studies are developed in this unit: Diagnostic sensitivity and specificity, Measurement uncertainty and calibration transfer, Competing faults and data fusion, and Trend detection, censoring and action thresholds. Their models have intentionally different input assumptions. They should not be combined simply because their numerical outputs share units. A source voltage, local electric field, risk probability and event count answer different questions and can have different uncertainty structures.

Learning outcomes

By the end of the unit, a reader should be able to state the physical source of the phenomenon; calculate a constrained model result with consistent units; identify its boundary conditions and initial data; conduct a controlled sensitivity test; and construct a short engineering review that explicitly identifies omitted mechanisms. The reader should also be able to separate a documented equipment rating from a quantity invented for an instructional example. A defensible result includes both a number and a statement of its limitations.

Evidence workflow

1. Source: identify what event or operating condition creates the stress or observation. Write down its time reference and appropriate signal representation.

2. Propagation or internal response: trace the signal through the relevant network, insulating material or sensor. Use the model only where its assumptions apply.

3. Measurement: name the quantities that would have to be observed for a real analysis, noting uncertainty, calibration and reference location.

4. Decision: state an actionable conclusion only when the acceptance criterion, equipment definition and hazard boundary are independently documented.

Integrated software case

Use Bayes theorem to calculate the positive predictive value for a hypothetical monitor. Distinguish model-based probability from measured calibration, and ask how the action threshold should account for failure consequence. Start with the default inputs and write a small table of three cases: baseline, one input increased, and a contrasting configuration. Give units for every column. Do not treat a “better-looking” curve as a demonstration that a real dielectric is more reliable; the curve was produced by the stipulated equations. Where the unit involves event probability or a failure model, compute exposure and population definitions before interpreting the percentage.

Reading order

• Lesson 7.1: Diagnostic sensitivity and specificity — A diagnostic alarm should trigger proportionate verification rather than immediate confidence in a fault location.
• Lesson 7.2: Measurement uncertainty and calibration transfer — Document sensor settings and calibration traceability before interpreting small changes as material degradation.
• Lesson 7.3: Competing faults and data fusion — A reviewer needs to know sensor physics, shared noise paths and conditional dependence before treating agreement as validation.
• Lesson 7.4: Trend detection, censoring and action thresholds — The alarm threshold must be justified by cost of missed events and false actions, not optimized only on historical accuracy.

Expected outputs

Submit a two-page technical note, the three-case numerical table, your answer to each lesson's three problems, and one diagram identifying missing evidence. The project and the unit assessment are separate; the project tests reasoning about a whole subsystem, while the assessment checks multiple independent concepts. A valid submission can conclude that physical validation is currently unavailable, provided the logical and numerical work is reproducible.

Safety boundary. This is entirely a software and analytical teaching resource. No live high-voltage test or energized hardware construction is included. Lightning, insulation switching and stored electrical energy can cause fatal electric shock or arc events even after isolation. Physical testing requires properly trained personnel, site-specific risk assessments, approved procedures, interlocked laboratories and calibrated equipment; numerical examples provide no permission to undertake those tests.

Lesson 7.1 — Diagnostic sensitivity and specificity

Unit: Diagnostics, interpretation and uncertainty Scope: mathematical explanation, traceable worked cases, no energized experiments. Prerequisites: electromagnetism, circuit theory, calculus, introductory probability and numerical reasoning. Status: instructional draft, expert review required.

Question that motivates the calculation

A detector sensitivity describes how often it finds a true fault, and specificity how often it correctly rejects a nonfault. A positive result is less predictive for rare faults than many dashboards imply. Consider what an engineer would need to know before computing a number. A source condition and an observation location must be named; the disturbance travels through a physical structure, and the inference must distinguish an equivalent network quantity from a local material stress. This lesson focuses on the specific observables relevant to diagnostic sensitivity and specificity, rather than treating every high-voltage problem as a breakdown-field calculation.

Governing relation and conditions

Working relation: PPV=Se·p/[Se·p+(1−Sp)(1−p)], where p is a stated defect prevalence.

Worked engineering illustration

For Se=0.90, Sp=0.95 and p=0.02, PPV=0.018/(0.018+0.049)≈26.9%; most positives are false positives under these assumptions.

First robustness check

Compare the same sensor at p=2% and p=20% to see how the base rate changes action value. To turn this into a sensitivity analysis, change one input while keeping all the others fixed. If the model predicts an output proportional to an input, a ten percent increase produces a ten percent change. If the relation is a square law, the same input change can produce a twenty-one percent change. If the relation contains logarithms, exponential decay or a nonlinear power, compute the actual new value; do not apply a linear rule automatically. State the chosen parameter, the updated input, and whether the original physical assumptions still hold.

Second robustness check

Real prevalence, label quality and correlated event windows are often unknown; classroom PPV is not a site-calibrated probability. Identify which omitted effect would most likely overturn the numerical conclusion. Ask whether the apparent physical change could instead be a measurement-location effect, a mode not included in the circuit, a transient initial condition or an unverified population parameter. Rank these omissions by their causal relevance to the particular conclusion; not all unspecified real-world complexities deserve equal emphasis. A non-converged software calculation should be rejected before arguing about possible dielectric ageing.

Competing mechanisms and decision boundary

A diagnostic alarm should trigger proportionate verification rather than immediate confidence in a fault location. Compare the same sensor at p=2% and p=20% to see how the base rate changes action value. Two physical explanations can yield similar single-point voltage or event counts, so a useful experiment must predict different observations. For a transient network, compare arrival time, reflection polarity or waveform shape at two locations. For insulation, compare temperature, condition history or discharge signatures with controlled confounders. For asset probabilities, compare rates normalized to the same exposure rather than absolute totals. Always identify who is authorized to collect actual high-voltage measurements; this document does not authorize them.

Guided study: three substantive problems

1. Baseline reconstruction. Derive or explain the working relation from its relevant conservation law or constitutive assumption, identify every input in the numerical example, reproduce the result with units, and explain what a physically observable quantity would be. Then perform a dimensional check. Your answer must include the geometric or time-domain reference under which the formula is meaningful.

2. Alternative scenario. Use the particular comparison: Compare the same sensor at p=2% and p=20% to see how the base rate changes action value. Predict whether the model value rises, falls or changes sign. Provide a numerical recalculation where the supplied inputs allow it, and explain which untouched assumptions make the comparison fair. Avoid claiming a physical outcome that is absent from the equation.

3. Engineering review. A report claims that the worked example proves a complete network or insulation design is suitable. Identify the missing evidence implied by this lesson: Real prevalence, label quality and correlated event windows are often unknown; classroom PPV is not a site-calibrated probability. Propose two observations or records that would independently test the interpretation, and write the strongest narrower claim justified by the model.

Fully reasoned answer guidance

Problem 1: Start with PPV=Se·p/[Se·p+(1−Sp)(1−p)], where p is a stated defect prevalence. The illustrative case is: For Se=0.90, Sp=0.95 and p=0.02, PPV=0.018/(0.018+0.049)≈26.9%; most positives are false positives under these assumptions. A correct answer must show the steps and resolve any negative sign, time origin, population condition or reference terminal used in the relation. Merely copying the result without a model domain and physically meaningful units is incomplete. If the formula contains an illustrative statistical parameter, identify its assumed status and do not mislabel it as a field-measured prevalence or strike distribution.

Problem 2: The key comparison is Compare the same sensor at p=2% and p=20% to see how the base rate changes action value. The perturbation should preserve all unmodified conditions, so a change in one parameter can be attributed to the mathematical model. Explain the equation's sensitivity and one reason the real system may depart from that dependence. A graph generated from the same equation is a re-expression of the prediction, not independent validation.

Problem 3: A sufficient critique must cite Real prevalence, label quality and correlated event windows are often unknown; classroom PPV is not a site-calibrated probability. It should also explain the required evidence pathway: appropriate physical representation, verified input data, uncertainty evaluation, instrument qualification and a specific acceptance reference relevant to the real equipment. If those inputs are absent, the decision is unresolved. Failure to prove suitability is not by itself proof of failure.

Connection to the unit simulator

Open the [Unit 7 interactive software model](../simulations/model-07.html). Its parameters isolate a small part of the unit; it is not a design tool and does not evaluate the specific example of every chapter. Record default inputs, plotted axes and the reported calculated result. Repeat with one controlled parameter change. Independently calculate at least one data point. If the graph appears smooth, ask what is missing from the model and identify a quantity the software could not measure. Use Bayes theorem to calculate the positive predictive value for a hypothetical monitor. Distinguish model-based probability from measured calibration, and ask how the action threshold should account for failure consequence.

Summary and transition

The core result in this lesson concerns diagnostic sensitivity and specificity under explicit ideal conditions. The most important limitation is not insufficient decimal places but real prevalence, label quality and correlated event windows are often unknown; classroom PPV is not a site-calibrated probability. Retain the distinction among mathematical output, simulated behavior and verified real-world evidence when connecting this lesson to the remaining cases in the unit.

Safety boundary. This is entirely a software and analytical teaching resource. No live high-voltage test or energized hardware construction is included. Lightning, insulation switching and stored electrical energy can cause fatal electric shock or arc events even after isolation. Physical testing requires properly trained personnel, site-specific risk assessments, approved procedures, interlocked laboratories and calibrated equipment; numerical examples provide no permission to undertake those tests.

Lesson 7.2 — Measurement uncertainty and calibration transfer

Unit: Diagnostics, interpretation and uncertainty Scope: mathematical explanation, traceable worked cases, no energized experiments. Prerequisites: electromagnetism, circuit theory, calculus, introductory probability and numerical reasoning. Status: instructional draft, expert review required.

Question that motivates the calculation

Uncertainty budgets combine sensor gain, calibration reference, bandwidth, digitization, repeatability and environmental variability. Independent random standard uncertainties may be combined quadratically, but correlated sources cannot. Consider what an engineer would need to know before computing a number. A source condition and an observation location must be named; the disturbance travels through a physical structure, and the inference must distinguish an equivalent network quantity from a local material stress. This lesson focuses on the specific observables relevant to measurement uncertainty and calibration transfer, rather than treating every high-voltage problem as a breakdown-field calculation.

Governing relation and conditions

Working relation: If independent relative standard uncertainties u1 and u2 are small, uc≈√(u1²+u2²); expanded uncertainty U=k·uc uses a stated coverage factor.

Worked engineering illustration

For u1=1% and u2=2%, uc≈2.24%; k=2 gives a nominal expanded uncertainty ≈4.47% if assumptions hold.

First robustness check

Compare uncertainties from a shared reference with truly independent repeat measurements. To turn this into a sensitivity analysis, change one input while keeping all the others fixed. If the model predicts an output proportional to an input, a ten percent increase produces a ten percent change. If the relation is a square law, the same input change can produce a twenty-one percent change. If the relation contains logarithms, exponential decay or a nonlinear power, compute the actual new value; do not apply a linear rule automatically. State the chosen parameter, the updated input, and whether the original physical assumptions still hold.

Second robustness check

A systematic common calibration bias shared by both channels should not be counted as independent noise. Identify which omitted effect would most likely overturn the numerical conclusion. Ask whether the apparent physical change could instead be a measurement-location effect, a mode not included in the circuit, a transient initial condition or an unverified population parameter. Rank these omissions by their causal relevance to the particular conclusion; not all unspecified real-world complexities deserve equal emphasis. A non-converged software calculation should be rejected before arguing about possible dielectric ageing.

Competing mechanisms and decision boundary

Document sensor settings and calibration traceability before interpreting small changes as material degradation. Compare uncertainties from a shared reference with truly independent repeat measurements. Two physical explanations can yield similar single-point voltage or event counts, so a useful experiment must predict different observations. For a transient network, compare arrival time, reflection polarity or waveform shape at two locations. For insulation, compare temperature, condition history or discharge signatures with controlled confounders. For asset probabilities, compare rates normalized to the same exposure rather than absolute totals. Always identify who is authorized to collect actual high-voltage measurements; this document does not authorize them.

Guided study: three substantive problems

1. Baseline reconstruction. Derive or explain the working relation from its relevant conservation law or constitutive assumption, identify every input in the numerical example, reproduce the result with units, and explain what a physically observable quantity would be. Then perform a dimensional check. Your answer must include the geometric or time-domain reference under which the formula is meaningful.

2. Alternative scenario. Use the particular comparison: Compare uncertainties from a shared reference with truly independent repeat measurements. Predict whether the model value rises, falls or changes sign. Provide a numerical recalculation where the supplied inputs allow it, and explain which untouched assumptions make the comparison fair. Avoid claiming a physical outcome that is absent from the equation.

3. Engineering review. A report claims that the worked example proves a complete network or insulation design is suitable. Identify the missing evidence implied by this lesson: A systematic common calibration bias shared by both channels should not be counted as independent noise. Propose two observations or records that would independently test the interpretation, and write the strongest narrower claim justified by the model.

Fully reasoned answer guidance

Problem 1: Start with If independent relative standard uncertainties u1 and u2 are small, uc≈√(u1²+u2²); expanded uncertainty U=k·uc uses a stated coverage factor. The illustrative case is: For u1=1% and u2=2%, uc≈2.24%; k=2 gives a nominal expanded uncertainty ≈4.47% if assumptions hold. A correct answer must show the steps and resolve any negative sign, time origin, population condition or reference terminal used in the relation. Merely copying the result without a model domain and physically meaningful units is incomplete. If the formula contains an illustrative statistical parameter, identify its assumed status and do not mislabel it as a field-measured prevalence or strike distribution.

Problem 2: The key comparison is Compare uncertainties from a shared reference with truly independent repeat measurements. The perturbation should preserve all unmodified conditions, so a change in one parameter can be attributed to the mathematical model. Explain the equation's sensitivity and one reason the real system may depart from that dependence. A graph generated from the same equation is a re-expression of the prediction, not independent validation.

Problem 3: A sufficient critique must cite A systematic common calibration bias shared by both channels should not be counted as independent noise. It should also explain the required evidence pathway: appropriate physical representation, verified input data, uncertainty evaluation, instrument qualification and a specific acceptance reference relevant to the real equipment. If those inputs are absent, the decision is unresolved. Failure to prove suitability is not by itself proof of failure.

Connection to the unit simulator

Open the [Unit 7 interactive software model](../simulations/model-07.html). Its parameters isolate a small part of the unit; it is not a design tool and does not evaluate the specific example of every chapter. Record default inputs, plotted axes and the reported calculated result. Repeat with one controlled parameter change. Independently calculate at least one data point. If the graph appears smooth, ask what is missing from the model and identify a quantity the software could not measure. Use Bayes theorem to calculate the positive predictive value for a hypothetical monitor. Distinguish model-based probability from measured calibration, and ask how the action threshold should account for failure consequence.

Summary and transition

The core result in this lesson concerns measurement uncertainty and calibration transfer under explicit ideal conditions. The most important limitation is not insufficient decimal places but a systematic common calibration bias shared by both channels should not be counted as independent noise. Retain the distinction among mathematical output, simulated behavior and verified real-world evidence when connecting this lesson to the remaining cases in the unit.

Safety boundary. This is entirely a software and analytical teaching resource. No live high-voltage test or energized hardware construction is included. Lightning, insulation switching and stored electrical energy can cause fatal electric shock or arc events even after isolation. Physical testing requires properly trained personnel, site-specific risk assessments, approved procedures, interlocked laboratories and calibrated equipment; numerical examples provide no permission to undertake those tests.

Lesson 7.3 — Competing faults and data fusion

Unit: Diagnostics, interpretation and uncertainty Scope: mathematical explanation, traceable worked cases, no energized experiments. Prerequisites: electromagnetism, circuit theory, calculus, introductory probability and numerical reasoning. Status: instructional draft, expert review required.

Question that motivates the calculation

Partial-discharge pulse trains, temperature anomalies and switching artifacts can overlap. Evidence may be dependent; naively multiplying correlated likelihoods exaggerates certainty. Consider what an engineer would need to know before computing a number. A source condition and an observation location must be named; the disturbance travels through a physical structure, and the inference must distinguish an equivalent network quantity from a local material stress. This lesson focuses on the specific observables relevant to competing faults and data fusion, rather than treating every high-voltage problem as a breakdown-field calculation.

Governing relation and conditions

Working relation: If evidence tests A,B are conditionally independent given H, posterior odds equal prior odds times LR_A×LR_B. Independence must be checked.

Worked engineering illustration

If prior odds are 1:19 and two independent signals each have LR=4, posterior odds become 16:19, or probability ≈45.7%.

First robustness check

Compare corroborating acoustic and electrical measurements with two electrical channels coupled through the same interference source. To turn this into a sensitivity analysis, change one input while keeping all the others fixed. If the model predicts an output proportional to an input, a ten percent increase produces a ten percent change. If the relation is a square law, the same input change can produce a twenty-one percent change. If the relation contains logarithms, exponential decay or a nonlinear power, compute the actual new value; do not apply a linear rule automatically. State the chosen parameter, the updated input, and whether the original physical assumptions still hold.

Second robustness check

With highly correlated sensors the product LR=16 may be unjustified, leading to an overconfident posterior. Identify which omitted effect would most likely overturn the numerical conclusion. Ask whether the apparent physical change could instead be a measurement-location effect, a mode not included in the circuit, a transient initial condition or an unverified population parameter. Rank these omissions by their causal relevance to the particular conclusion; not all unspecified real-world complexities deserve equal emphasis. A non-converged software calculation should be rejected before arguing about possible dielectric ageing.

Competing mechanisms and decision boundary

A reviewer needs to know sensor physics, shared noise paths and conditional dependence before treating agreement as validation. Compare corroborating acoustic and electrical measurements with two electrical channels coupled through the same interference source. Two physical explanations can yield similar single-point voltage or event counts, so a useful experiment must predict different observations. For a transient network, compare arrival time, reflection polarity or waveform shape at two locations. For insulation, compare temperature, condition history or discharge signatures with controlled confounders. For asset probabilities, compare rates normalized to the same exposure rather than absolute totals. Always identify who is authorized to collect actual high-voltage measurements; this document does not authorize them.

Guided study: three substantive problems

1. Baseline reconstruction. Derive or explain the working relation from its relevant conservation law or constitutive assumption, identify every input in the numerical example, reproduce the result with units, and explain what a physically observable quantity would be. Then perform a dimensional check. Your answer must include the geometric or time-domain reference under which the formula is meaningful.

2. Alternative scenario. Use the particular comparison: Compare corroborating acoustic and electrical measurements with two electrical channels coupled through the same interference source. Predict whether the model value rises, falls or changes sign. Provide a numerical recalculation where the supplied inputs allow it, and explain which untouched assumptions make the comparison fair. Avoid claiming a physical outcome that is absent from the equation.

3. Engineering review. A report claims that the worked example proves a complete network or insulation design is suitable. Identify the missing evidence implied by this lesson: With highly correlated sensors the product LR=16 may be unjustified, leading to an overconfident posterior. Propose two observations or records that would independently test the interpretation, and write the strongest narrower claim justified by the model.

Fully reasoned answer guidance

Problem 1: Start with If evidence tests A,B are conditionally independent given H, posterior odds equal prior odds times LR_A×LR_B. Independence must be checked. The illustrative case is: If prior odds are 1:19 and two independent signals each have LR=4, posterior odds become 16:19, or probability ≈45.7%. A correct answer must show the steps and resolve any negative sign, time origin, population condition or reference terminal used in the relation. Merely copying the result without a model domain and physically meaningful units is incomplete. If the formula contains an illustrative statistical parameter, identify its assumed status and do not mislabel it as a field-measured prevalence or strike distribution.

Problem 2: The key comparison is Compare corroborating acoustic and electrical measurements with two electrical channels coupled through the same interference source. The perturbation should preserve all unmodified conditions, so a change in one parameter can be attributed to the mathematical model. Explain the equation's sensitivity and one reason the real system may depart from that dependence. A graph generated from the same equation is a re-expression of the prediction, not independent validation.

Problem 3: A sufficient critique must cite With highly correlated sensors the product LR=16 may be unjustified, leading to an overconfident posterior. It should also explain the required evidence pathway: appropriate physical representation, verified input data, uncertainty evaluation, instrument qualification and a specific acceptance reference relevant to the real equipment. If those inputs are absent, the decision is unresolved. Failure to prove suitability is not by itself proof of failure.

Connection to the unit simulator

Open the [Unit 7 interactive software model](../simulations/model-07.html). Its parameters isolate a small part of the unit; it is not a design tool and does not evaluate the specific example of every chapter. Record default inputs, plotted axes and the reported calculated result. Repeat with one controlled parameter change. Independently calculate at least one data point. If the graph appears smooth, ask what is missing from the model and identify a quantity the software could not measure. Use Bayes theorem to calculate the positive predictive value for a hypothetical monitor. Distinguish model-based probability from measured calibration, and ask how the action threshold should account for failure consequence.

Summary and transition

The core result in this lesson concerns competing faults and data fusion under explicit ideal conditions. The most important limitation is not insufficient decimal places but with highly correlated sensors the product LR=16 may be unjustified, leading to an overconfident posterior. Retain the distinction among mathematical output, simulated behavior and verified real-world evidence when connecting this lesson to the remaining cases in the unit.

Safety boundary. This is entirely a software and analytical teaching resource. No live high-voltage test or energized hardware construction is included. Lightning, insulation switching and stored electrical energy can cause fatal electric shock or arc events even after isolation. Physical testing requires properly trained personnel, site-specific risk assessments, approved procedures, interlocked laboratories and calibrated equipment; numerical examples provide no permission to undertake those tests.

Lesson 7.4 — Trend detection, censoring and action thresholds

Unit: Diagnostics, interpretation and uncertainty Scope: mathematical explanation, traceable worked cases, no energized experiments. Prerequisites: electromagnetism, circuit theory, calculus, introductory probability and numerical reasoning. Status: instructional draft, expert review required.

Question that motivates the calculation

A rising diagnostic signal may reflect ageing, changed load, altered calibration or selection bias. Multiple checks and documented operating conditions are essential when event counts are sparse. Consider what an engineer would need to know before computing a number. A source condition and an observation location must be named; the disturbance travels through a physical structure, and the inference must distinguish an equivalent network quantity from a local material stress. This lesson focuses on the specific observables relevant to trend detection, censoring and action thresholds, rather than treating every high-voltage problem as a breakdown-field calculation.

Governing relation and conditions

Working relation: For Poisson counts over an exposure interval T, an event-rate estimate is λhat=N/T and approximate standard error √N/T when N is not extremely small.

Worked engineering illustration

For N=100 events in 10 h, λhat=10/h and approximate standard error=1/h; a subsequent 11/h estimate is weak evidence without comparable exposure and uncertainty.

First robustness check

Compare apparent rates from unequal observation windows and demonstrate exposure normalization. To turn this into a sensitivity analysis, change one input while keeping all the others fixed. If the model predicts an output proportional to an input, a ten percent increase produces a ten percent change. If the relation is a square law, the same input change can produce a twenty-one percent change. If the relation contains logarithms, exponential decay or a nonlinear power, compute the actual new value; do not apply a linear rule automatically. State the chosen parameter, the updated input, and whether the original physical assumptions still hold.

Second robustness check

For low counts use an appropriate exact or Bayesian interval rather than relying on the normal approximation. Identify which omitted effect would most likely overturn the numerical conclusion. Ask whether the apparent physical change could instead be a measurement-location effect, a mode not included in the circuit, a transient initial condition or an unverified population parameter. Rank these omissions by their causal relevance to the particular conclusion; not all unspecified real-world complexities deserve equal emphasis. A non-converged software calculation should be rejected before arguing about possible dielectric ageing.

Competing mechanisms and decision boundary

The alarm threshold must be justified by cost of missed events and false actions, not optimized only on historical accuracy. Compare apparent rates from unequal observation windows and demonstrate exposure normalization. Two physical explanations can yield similar single-point voltage or event counts, so a useful experiment must predict different observations. For a transient network, compare arrival time, reflection polarity or waveform shape at two locations. For insulation, compare temperature, condition history or discharge signatures with controlled confounders. For asset probabilities, compare rates normalized to the same exposure rather than absolute totals. Always identify who is authorized to collect actual high-voltage measurements; this document does not authorize them.

Guided study: three substantive problems

1. Baseline reconstruction. Derive or explain the working relation from its relevant conservation law or constitutive assumption, identify every input in the numerical example, reproduce the result with units, and explain what a physically observable quantity would be. Then perform a dimensional check. Your answer must include the geometric or time-domain reference under which the formula is meaningful.

2. Alternative scenario. Use the particular comparison: Compare apparent rates from unequal observation windows and demonstrate exposure normalization. Predict whether the model value rises, falls or changes sign. Provide a numerical recalculation where the supplied inputs allow it, and explain which untouched assumptions make the comparison fair. Avoid claiming a physical outcome that is absent from the equation.

3. Engineering review. A report claims that the worked example proves a complete network or insulation design is suitable. Identify the missing evidence implied by this lesson: For low counts use an appropriate exact or Bayesian interval rather than relying on the normal approximation. Propose two observations or records that would independently test the interpretation, and write the strongest narrower claim justified by the model.

Fully reasoned answer guidance

Problem 1: Start with For Poisson counts over an exposure interval T, an event-rate estimate is λhat=N/T and approximate standard error √N/T when N is not extremely small. The illustrative case is: For N=100 events in 10 h, λhat=10/h and approximate standard error=1/h; a subsequent 11/h estimate is weak evidence without comparable exposure and uncertainty. A correct answer must show the steps and resolve any negative sign, time origin, population condition or reference terminal used in the relation. Merely copying the result without a model domain and physically meaningful units is incomplete. If the formula contains an illustrative statistical parameter, identify its assumed status and do not mislabel it as a field-measured prevalence or strike distribution.

Problem 2: The key comparison is Compare apparent rates from unequal observation windows and demonstrate exposure normalization. The perturbation should preserve all unmodified conditions, so a change in one parameter can be attributed to the mathematical model. Explain the equation's sensitivity and one reason the real system may depart from that dependence. A graph generated from the same equation is a re-expression of the prediction, not independent validation.

Problem 3: A sufficient critique must cite For low counts use an appropriate exact or Bayesian interval rather than relying on the normal approximation. It should also explain the required evidence pathway: appropriate physical representation, verified input data, uncertainty evaluation, instrument qualification and a specific acceptance reference relevant to the real equipment. If those inputs are absent, the decision is unresolved. Failure to prove suitability is not by itself proof of failure.

Connection to the unit simulator

Open the [Unit 7 interactive software model](../simulations/model-07.html). Its parameters isolate a small part of the unit; it is not a design tool and does not evaluate the specific example of every chapter. Record default inputs, plotted axes and the reported calculated result. Repeat with one controlled parameter change. Independently calculate at least one data point. If the graph appears smooth, ask what is missing from the model and identify a quantity the software could not measure. Use Bayes theorem to calculate the positive predictive value for a hypothetical monitor. Distinguish model-based probability from measured calibration, and ask how the action threshold should account for failure consequence.

Summary and transition

The core result in this lesson concerns trend detection, censoring and action thresholds under explicit ideal conditions. The most important limitation is not insufficient decimal places but for low counts use an appropriate exact or Bayesian interval rather than relying on the normal approximation. Retain the distinction among mathematical output, simulated behavior and verified real-world evidence when connecting this lesson to the remaining cases in the unit.

Safety boundary. This is entirely a software and analytical teaching resource. No live high-voltage test or energized hardware construction is included. Lightning, insulation switching and stored electrical energy can cause fatal electric shock or arc events even after isolation. Physical testing requires properly trained personnel, site-specific risk assessments, approved procedures, interlocked laboratories and calibrated equipment; numerical examples provide no permission to undertake those tests.

Unit 7 investigation — Diagnostics, interpretation and uncertainty

Design brief

A hypothetical network operator requests a desk-based technical comparison of two options affecting diagnostics, interpretation and uncertainty. Both options are abstractions; no physical components, voltages or laboratory procedures are supplied. You may use the teaching equations and the interactive model, but you must not turn synthetic assumptions into claimed equipment specifications.

Diagnostic signals are noisy evidence, not unambiguous labels. PD pulses, dielectric loss, thermal imaging, dissolved gases and acoustic events have different selectivity and uncertainty. The rate of false alarms matters particularly when a defect is rare.

Required deliverables

A. Analytical framework. Draft a block diagram connecting the initiating phenomenon, medium or circuit, local stress, sensor evidence and possible decision. Include at least two feedback or hidden variables. For the chosen operating condition, describe why diagnostic sensitivity and specificity and competing faults and data fusion may have competing impacts. Demonstrate with one of the unit's equations, with dimensional units shown.

B. Three-case calculation. Use Bayes theorem to calculate the positive predictive value for a hypothetical monitor. Distinguish model-based probability from measured calibration, and ask how the action threshold should account for failure consequence. Make the baseline and both variations reproducible. Label input values as invented, externally sourced, or assumptions. Where the change is nonlinear, explain the physical intuition and derive the relevant numerical scaling rather than claiming every input behaves linearly.

C. Sensitivity and model breakdown. Consider specifically: A systematic common calibration bias shared by both channels should not be counted as independent noise. and For low counts use an appropriate exact or Bayesian interval rather than relying on the normal approximation. These are different failure mechanisms; decide which one would most affect the proposed comparative result. Record at least one missing distribution, geometry, time profile, material constant or calibration measurement.

D. Verification plan. Propose a staged desk review: compare limit cases, check units, reproduce one point independently, vary numerical timestep if appropriate, compare an independent data source, and request qualified engineering review before any real installation recommendation. No energized experiment or switchgear operation is permitted as part of this project.

E. Decision memorandum. State a conservative conditional conclusion: what the toy model predicts, what remains unknown and which documented evidence could change the decision. Include at least two alternatives and explain why choosing the most attractive plot alone is insufficient.

Rubric

Model correctness 25%, units and reproduction 20%, sensitivity reasoning 20%, evidence and uncertainty 20%, decision clarity 15%. A polished figure without a correct scope declaration earns no credit for validation.

Safety boundary. This is entirely a software and analytical teaching resource. No live high-voltage test or energized hardware construction is included. Lightning, insulation switching and stored electrical energy can cause fatal electric shock or arc events even after isolation. Physical testing requires properly trained personnel, site-specific risk assessments, approved procedures, interlocked laboratories and calibrated equipment; numerical examples provide no permission to undertake those tests.