Unit 2: Conduction and Dielectric Breakdown in Gases: Townsend and Streamer Mechanisms

Unit 2: Gaseous dielectrics and electrical discharge

Why this subject matters

An avalanche grows when electron impact ionization exceeds losses, but a visually smooth calculated voltage curve is not a deterministic ignition guarantee. Gas composition, temperature, pressure, electrode condition, waveform duration, and statistical time lag all affect observed breakdown.

Common reference case and assumptions

Use the Townsend avalanche and Paschen relations strictly as approximate interpretations of gas discharges in their stated validity regions. A Paschen plot can be instructive and still be wrong for highly nonuniform fields, extremely small separations, streamer-dominated gaps, and changing gas chemistry.

Outcomes and verification

Distinguish collision processes from streamer evolution, build a dimensionally consistent pressure-gap model, and interpret rather than overgeneralize a breakdown plot.

A researcher's reading protocol

For every claimed threshold, document pressure, temperature, gas, waveform and electrode geometry. Put an uncertainty interval around an empirical fit, and describe competing discharge mechanisms. Estimate whether a representative electron transport or avalanche calculation stays within the assumptions of the model. 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 2.1–2.4, 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-02.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 recognizing that a fit to one pressure-gap range may be misleading at a new scale. Label extrapolation rather than quietly presenting it as a prediction. 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 2.1: Electron impact ionization and the Townsend coefficient

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

Physical mechanism

In a gas, electrons gain energy between collisions and can create additional charge carriers by impact ionization. The first Townsend coefficient α describes electron multiplication per unit path in a local approximation. Attachment, diffusion and space charge can defeat an elementary avalanche model.

Governing equations and interpretation

In a uniform region with constant α and negligible attachment, n(x)=n0 exp(αx). With attachment coefficient η, a simplified effective model uses exp[(α−η)x], not exp(αx) independently of chemistry.

Fully stated worked example

For α=350 m⁻¹ over 5 mm, multiplication is exp(1.75)≈5.75. If η=100 m⁻¹ under the same illustrative assumptions, the effective factor is exp(1.25)≈3.49.

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

Avalanche multiplication alone is not a verified breakdown criterion; streamer and electrode processes can dominate.

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 electron impact ionization and the townsend coefficient 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 α=350 m⁻¹ over 5 mm, multiplication is exp(1.75)≈5.75. If η=100 m⁻¹ under the same illustrative assumptions, the effective factor is exp(1.25)≈3.49. 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: Avalanche multiplication alone is not a verified breakdown criterion; streamer and electrode processes can dominate. 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-02.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 2.2: Paschen law and the pressure–distance product

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

Physical mechanism

For an idealized gas and approximately uniform field, a Townsend-based breakdown model varies with the product p d. It predicts a minimum within a bounded regime, rather than monotonic improvement when gaps shrink. Constants must use consistent pressure and length units.

Governing equations and interpretation

Vb=[B p d]/[ln(A p d)−ln(ln(1+1/γ))] for compatible empirical A, B and secondary-emission γ. A physically meaningful positive denominator is required.

Fully stated worked example

Using illustrative air-fit parameters A=15 (torr·cm)⁻¹, B=365 V/(torr·cm), γ=0.01 and p d=1 torr·cm gives Vb≈365/[ln15−ln(ln101)]≈310 V. This is a model calculation, not certification.

Dimensional and limiting checks

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

Failure modes, model exclusions, and hazard boundary

The curve is inappropriate as a universal design rule for microgaps, humid air, rough electrodes, streamers or strongly nonuniform fields.

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 paschen law and the pressure–distance product 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: Using illustrative air-fit parameters A=15 (torr·cm)⁻¹, B=365 V/(torr·cm), γ=0.01 and p d=1 torr·cm gives Vb≈365/[ln15−ln(ln101)]≈310 V. This is a model calculation, not certification. Reproduce the numeric substitution and state assumptions before reporting precision.

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

Limitation answer: The curve is inappropriate as a universal design rule for microgaps, humid air, rough electrodes, streamers or strongly nonuniform fields. 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-02.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 2.3: Streamers, leaders and polarity effects

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

Physical mechanism

As charge density grows, the local space-charge field can significantly reshape the imposed field and an avalanche-based description loses predictive value. Streamer propagation and later hot-channel development depend on polarity, waveform, electrode shape and gas composition.

Governing equations and interpretation

A simple diffusion scale is ℓ≈√(2Dt) for constant diffusion coefficient D and elapsed time t. This is a transport estimate, not a full streamer velocity or ignition calculation.

Fully stated worked example

For an illustrative D=0.02 m²/s and t=1 µs, ℓ≈0.20 mm. This distance is far smaller than many macroscopic gaps, showing why a diffusion-only picture is incomplete.

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

Treating a simple electron multiplication estimate as a time-dependent streamer simulation can yield false safety assurance.

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 streamers, leaders and polarity 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: For an illustrative D=0.02 m²/s and t=1 µs, ℓ≈0.20 mm. This distance is far smaller than many macroscopic gaps, showing why a diffusion-only picture is incomplete. 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: Treating a simple electron multiplication estimate as a time-dependent streamer simulation can yield false safety assurance. 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-02.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 2.4: Surface flashover, contamination and atmospheric effects

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

Physical mechanism

A surface path follows a boundary rather than a uniform bulk gas gap; contaminants and water films alter field distribution and leakage current. Pollution severity, wetting, material hydrophobicity and creepage geometry interact. Altitude and humidity influence discharge behavior.

Governing equations and interpretation

Surface resistance approximation R=ρs L/w is valid only for an ideal uniform sheet with sheet resistivity ρs measured in Ω per square, track length L and effective width w. This is not a flashover-voltage formula.

Fully stated worked example

For ρs=100 MΩ/square, a simple track of length 60 mm and width 10 mm has R≈600 MΩ. At 1 kV this model gives 1.67 µA leakage, with no assertion of flashover threshold.

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

Actual contaminated surfaces have spatially varying moisture, dry bands, heating and arcs; a single sheet resistance cannot replace an insulator test.

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 surface flashover, contamination and atmospheric 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: For ρs=100 MΩ/square, a simple track of length 60 mm and width 10 mm has R≈600 MΩ. At 1 kV this model gives 1.67 µA leakage, with no assertion of flashover threshold. 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: Actual contaminated surfaces have spatially varying moisture, dry bands, heating and arcs; a single sheet resistance cannot replace an insulator test. 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-02.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Unit 2: Gaseous dielectrics and electrical discharge

Why this subject matters

An avalanche grows when electron impact ionization exceeds losses, but a visually smooth calculated voltage curve is not a deterministic ignition guarantee. Gas composition, temperature, pressure, electrode condition, waveform duration, and statistical time lag all affect observed breakdown.

Common reference case and assumptions

Use the Townsend avalanche and Paschen relations strictly as approximate interpretations of gas discharges in their stated validity regions. A Paschen plot can be instructive and still be wrong for highly nonuniform fields, extremely small separations, streamer-dominated gaps, and changing gas chemistry.

Outcomes and verification

Distinguish collision processes from streamer evolution, build a dimensionally consistent pressure-gap model, and interpret rather than overgeneralize a breakdown plot.

A researcher's reading protocol

For every claimed threshold, document pressure, temperature, gas, waveform and electrode geometry. Put an uncertainty interval around an empirical fit, and describe competing discharge mechanisms. Estimate whether a representative electron transport or avalanche calculation stays within the assumptions of the model. 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 2.1–2.4, 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-02.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 recognizing that a fit to one pressure-gap range may be misleading at a new scale. Label extrapolation rather than quietly presenting it as a prediction. 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 2.1: Electron impact ionization and the Townsend coefficient

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

Physical mechanism

In a gas, electrons gain energy between collisions and can create additional charge carriers by impact ionization. The first Townsend coefficient α describes electron multiplication per unit path in a local approximation. Attachment, diffusion and space charge can defeat an elementary avalanche model.

Governing equations and interpretation

In a uniform region with constant α and negligible attachment, n(x)=n0 exp(αx). With attachment coefficient η, a simplified effective model uses exp[(α−η)x], not exp(αx) independently of chemistry.

Fully stated worked example

For α=350 m⁻¹ over 5 mm, multiplication is exp(1.75)≈5.75. If η=100 m⁻¹ under the same illustrative assumptions, the effective factor is exp(1.25)≈3.49.

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

Avalanche multiplication alone is not a verified breakdown criterion; streamer and electrode processes can dominate.

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 electron impact ionization and the townsend coefficient 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 α=350 m⁻¹ over 5 mm, multiplication is exp(1.75)≈5.75. If η=100 m⁻¹ under the same illustrative assumptions, the effective factor is exp(1.25)≈3.49. 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: Avalanche multiplication alone is not a verified breakdown criterion; streamer and electrode processes can dominate. 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-02.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 2.2: Paschen law and the pressure–distance product

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

Physical mechanism

For an idealized gas and approximately uniform field, a Townsend-based breakdown model varies with the product p d. It predicts a minimum within a bounded regime, rather than monotonic improvement when gaps shrink. Constants must use consistent pressure and length units.

Governing equations and interpretation

Vb=[B p d]/[ln(A p d)−ln(ln(1+1/γ))] for compatible empirical A, B and secondary-emission γ. A physically meaningful positive denominator is required.

Fully stated worked example

Using illustrative air-fit parameters A=15 (torr·cm)⁻¹, B=365 V/(torr·cm), γ=0.01 and p d=1 torr·cm gives Vb≈365/[ln15−ln(ln101)]≈310 V. This is a model calculation, not certification.

Dimensional and limiting checks

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

Failure modes, model exclusions, and hazard boundary

The curve is inappropriate as a universal design rule for microgaps, humid air, rough electrodes, streamers or strongly nonuniform fields.

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 paschen law and the pressure–distance product 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: Using illustrative air-fit parameters A=15 (torr·cm)⁻¹, B=365 V/(torr·cm), γ=0.01 and p d=1 torr·cm gives Vb≈365/[ln15−ln(ln101)]≈310 V. This is a model calculation, not certification. Reproduce the numeric substitution and state assumptions before reporting precision.

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

Limitation answer: The curve is inappropriate as a universal design rule for microgaps, humid air, rough electrodes, streamers or strongly nonuniform fields. 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-02.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 2.3: Streamers, leaders and polarity effects

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

Physical mechanism

As charge density grows, the local space-charge field can significantly reshape the imposed field and an avalanche-based description loses predictive value. Streamer propagation and later hot-channel development depend on polarity, waveform, electrode shape and gas composition.

Governing equations and interpretation

A simple diffusion scale is ℓ≈√(2Dt) for constant diffusion coefficient D and elapsed time t. This is a transport estimate, not a full streamer velocity or ignition calculation.

Fully stated worked example

For an illustrative D=0.02 m²/s and t=1 µs, ℓ≈0.20 mm. This distance is far smaller than many macroscopic gaps, showing why a diffusion-only picture is incomplete.

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

Treating a simple electron multiplication estimate as a time-dependent streamer simulation can yield false safety assurance.

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 streamers, leaders and polarity 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: For an illustrative D=0.02 m²/s and t=1 µs, ℓ≈0.20 mm. This distance is far smaller than many macroscopic gaps, showing why a diffusion-only picture is incomplete. 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: Treating a simple electron multiplication estimate as a time-dependent streamer simulation can yield false safety assurance. 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-02.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Lesson 2.4: Surface flashover, contamination and atmospheric effects

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

Physical mechanism

A surface path follows a boundary rather than a uniform bulk gas gap; contaminants and water films alter field distribution and leakage current. Pollution severity, wetting, material hydrophobicity and creepage geometry interact. Altitude and humidity influence discharge behavior.

Governing equations and interpretation

Surface resistance approximation R=ρs L/w is valid only for an ideal uniform sheet with sheet resistivity ρs measured in Ω per square, track length L and effective width w. This is not a flashover-voltage formula.

Fully stated worked example

For ρs=100 MΩ/square, a simple track of length 60 mm and width 10 mm has R≈600 MΩ. At 1 kV this model gives 1.67 µA leakage, with no assertion of flashover threshold.

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

Actual contaminated surfaces have spatially varying moisture, dry bands, heating and arcs; a single sheet resistance cannot replace an insulator test.

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 surface flashover, contamination and atmospheric 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: For ρs=100 MΩ/square, a simple track of length 60 mm and width 10 mm has R≈600 MΩ. At 1 kV this model gives 1.67 µA leakage, with no assertion of flashover threshold. 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: Actual contaminated surfaces have spatially varying moisture, dry bands, heating and arcs; a single sheet resistance cannot replace an insulator test. 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-02.html) to change one variable and interpret the graph using the stated equation. This is not a dielectric-withstand test.

Unit 2 project: Gas breakdown model audit

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

Problem statement

Create a pressure–distance sensitivity plot from a declared Paschen relation, and separately prepare an avalanche multiplication table.

An avalanche grows when electron impact ionization exceeds losses, but a visually smooth calculated voltage curve is not a deterministic ignition guarantee. Gas composition, temperature, pressure, electrode condition, waveform duration, and statistical time lag all affect observed breakdown.

Use the underlying quantities from Lessons 2.1–2.4 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

Change γ in the declared empirical model while keeping A and B fixed; flag points with nonpositive mathematical denominators. Compare any apparent minimum with the model’s own validity range.

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-02.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

Deliver the equation, units on empirical constants, valid and invalid sampled points, and a narrative about streamer or surface processes.

Reference check: At pd=1 torr·cm with the illustrative model, Vb≈310 V. No real breakdown guarantee follows.

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 2: Gaseous dielectrics and electrical discharge

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 Electron impact ionization and the Townsend coefficient, 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 Paschen law and the pressure–distance product, 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 Streamers, leaders and polarity 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 Surface flashover, contamination and atmospheric effects, 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 2.1 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 2.2'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 2.4 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 2 project: Gas breakdown model audit

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

Problem statement

Create a pressure–distance sensitivity plot from a declared Paschen relation, and separately prepare an avalanche multiplication table.

An avalanche grows when electron impact ionization exceeds losses, but a visually smooth calculated voltage curve is not a deterministic ignition guarantee. Gas composition, temperature, pressure, electrode condition, waveform duration, and statistical time lag all affect observed breakdown.

Use the underlying quantities from Lessons 2.1–2.4 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

Change γ in the declared empirical model while keeping A and B fixed; flag points with nonpositive mathematical denominators. Compare any apparent minimum with the model’s own validity range.

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-02.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

Deliver the equation, units on empirical constants, valid and invalid sampled points, and a narrative about streamer or surface processes.

Reference check: At pd=1 torr·cm with the illustrative model, Vb≈310 V. No real breakdown guarantee follows.

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.