Learning Objectives

Introduction to Gas Breakdown

• Growing demand for electrical energy increases power system size and complexity

• Higher transmission voltages (e.g., 400 kV, 800 kV, 1200 kV planned globally) enhance power transfer and reduce losses

• Conventional insulating materials for high-voltage systems are bulky and costly

• New insulating materials are investigated to reduce cost and space

• Insulation media classification: gases, liquids, solids, and composite systems

Role of Insulation in Power Systems

• Live conductors supported by insulating materials

• Air clearances: Prevent flashovers or short circuits in overhead lines

• Insulating liquids: Reduce container size while ensuring insulation (transformers, capacitors)

• Solid insulation: Provides mechanical support and electrical isolation

• Study focuses on breakdown mechanisms of gases, liquids, and solids

Properties of Gases

• At normal temperature and pressure (NTP), gases are excellent insulators

• Natural current conduction: \(\sim 10^{-10}\) to \(10^{-12} \, \mathrm{A/cm^2}\), due to:

  • Cosmic radiation (primary and secondary)

  • Natural radioactive substances

  • Photoionization from UV radiation

• At high electric fields, charged particles gain energy, leading to ionization on impact

• Critical field strength: Typically 30 kV/cm for air at STP

Ionization by Electron Impact

• Electrons gain kinetic energy from electric fields: \(KE = eVd\) where \(d\) is mean free path

• Elastic collisions: Minimal energy loss (\(\sim 10^{-5}\) of electron energy), rapid energy buildup

• Inelastic collisions: Energy transforms into potential energy, ionizing molecules

• Primary process for gas breakdown under strong electric fields

Photoionization

• Occurs when photon energy exceeds ionization energy:

  \[A + h\nu \rightarrow A^+ + e^- \quad \text{(if } h\nu > W_i\text{)}\]

• Secondary ionization process, crucial for streamer breakdown and corona discharges

• Photoexcitation: absorbed energy raises atom to higher energy level if below ionization threshold

• Photon sources: Recombination radiation, excited atom de-excitation

Ionization by Metastables

• Metastable states: Long-lived excited electronic states (microseconds to seconds)

• Metastables have high potential energy, can ionize neutral particles:

  \[A + B^{\mathrm{m}} \rightarrow A^+ + B + e^-\]

• Responsible for long time lags in some gases (particularly noble gases)

• Lifetime examples: He metastables: \(\sim 10^{-4}\) s, Ne metastables: \(\sim 10^{-2}\) s

Thermal Ionization

• Involves molecular collisions, radiation, and electron collisions at high temperatures

• High-temperature gas molecules gain kinetic energy, ionizing neutral particles

• Electrons and high-velocity molecules release more electrons

• Recombination balances ionization at thermodynamic equilibrium

Saha’s Equation for Ionization

• Degree of ionization \(\beta\) derived by Saha:

  \[\frac{\beta^2}{1 - \beta^2} = \frac{2.4 \times 10^{-4}}{p} T^{5/2} e^{-W_i / kT}\]

• \(p\): pressure (Torr), \(W_i\): ionization energy (eV), \(k\): Boltzmann constant, \(T\): temperature (K)

• Significant ionization only above 1000 K

• At \(\beta = 0.5\) (50% ionization): \(T \approx 0.75 W_i/k\) for \(p = 760\) Torr

Experimental Setup for Townsend’s Study

• Parallel plate capacitor with gas as insulating medium, plates separated by distance \(d\)

• Without electric field: equilibrium between electron and ion generation due to natural decay processes

• Applying high electric field disturbs equilibrium, initiating ionization avalanche

Current Variation with Voltage

• Townsend’s observation of current vs. voltage characteristics:

  • Region I: Low voltage - current increases proportionally with voltage (Ohm’s law)

  • Region II: Intermediate voltage - current saturates at \(I_0\) (saturation current)

  • Region III: High voltage - current increases exponentially due to gas ionization

• Higher \(V/d\) (electric field) accelerates electrons, increasing kinetic energy and electron production

Definition of Townsend’s First Ionization Coefficient

• Defined as \(\alpha\): number of ionizing collisions made by an electron per unit length in field direction

• Physical meaning: probability of ionization per unit path length

• Let \(n_0\) be electrons leaving cathode; at distance \(x\), number of electrons is \(n\)

• In distance \(dx\), \(n\) electrons produce \(dn\) additional electrons:

  \[dn = \alpha n \, dx\]

Derivation of Electron Avalanche

• Differential equation for electron multiplication:

  \[\frac{dn}{dx} = \alpha n \quad \Rightarrow \quad \frac{dn}{n} = \alpha \, dx\]

• Integrate from \(x=0\) (where \(n=n_0\)) to distance \(x\):

  \[\int_{n_0}^{n} \frac{dn'}{n'} = \int_0^x \alpha \, dx' \quad \Rightarrow \quad \ln n - \ln n_0 = \alpha x\]

• Solution for electron avalanche:

  \[n(x) = n_0 e^{\alpha x}\]

• At anode \((x=d)\): \(n = n_0 e^{\alpha d}\), hence current: \(I = I_0 e^{\alpha d}\)

• Term \(e^{\alpha d}\) is the avalanche multiplication factor

\(\alpha/p\) vs \(E/p\) Relationship

• Empirical relationship for most gases:

  \[\frac{\alpha}{p} = A e^{-Bp/(E/p)} = A e^{-B p d / V}\]

• \(A\) and \(B\): gas-dependent constants

• \(E/p\): reduced electric field ( \(V \cdot \mathrm{cm}^{-1}\cdot \mathrm{Torr}^{-1}\))

Role of Cathode in Gas Discharge

• Cathode supplies electrons for initiation, sustenance, and completion of gas discharge

• Electrons in metal are bound by electrostatic forces; energy required to release them is the work function \(W\) (typically 2-6 eV)

• Electron emission mechanisms:

  • Thermionic emission (high temperature)

  • Field emission (high electric field)

  • Secondary emission (ion/photon bombardment)

Thermionic Emission

• At high temperatures (\(T \geq \mathrm{1500}{K}\)), electrons gain thermal energy to overcome surface barrier

• Richardson’s equation for saturation current density:

  \[J_s = A_0 T^2 e^{-W / kT}, \quad A_0 = \frac{4\pi m_e k^2}{h^3} = 120 \times 10^4 \, \mathrm{A·m^{-2}·K^{-2}}\]

• Practical value: \(A \approx 60 \times 10^4 \, \mathrm{A·m^{-2}·K^{-2}}\) due to surface imperfections

• Low work function metals preferred (tungsten: 4.5 eV, barium oxide: 1.0 eV)

Field Emission (Schottky Effect)

• Strong electric field reduces effective work function:

  \[W' = W - \sqrt{\frac{e^3 E}{4\pi\epsilon_0}} \approx W - 3.79 \times 10^{-5} \sqrt{E} \text{ eV (E in V/cm)}\]

• Modified Richardson equation:

  \[J_s = A T^2 e^{-W' / kT} = A T^2 e^{-(W - 3.79 \times 10^{-5} \sqrt{E}) / kT}\]

• At very high fields (\(E > 10^7 \, \mathrm{V/cm}\)), quantum tunneling dominates

• Fowler-Nordheim equation for field emission:

  \[J = 1.54 \times 10^{-6} \frac{E^2}{\phi} \exp\left(-6.83 \times 10^7 \frac{\phi^{3/2}}{E}\right)\]

Secondary Emission by Ion Bombardment

• Secondary emission: Positive ions with sufficient energy strike cathode, releasing electrons

• Energy condition: \(W_k + W_p \geq 2W\) where:

  • \(W_k\): kinetic energy of ion

  • \(W_p\): potential energy of ion

  • \(W\): work function of cathode

• One electron neutralizes the ion, another is emitted

• Key mechanism in Townsend discharge

Photoemission and Metastable Bombardment

• Photoemission: UV photons with \(h\nu > W\) release electrons from cathode

  \[h\nu + \text{cathode} \rightarrow e^- + \text{cathode}^+\]

• Quantum efficiency: typically 0.01-0.1 electrons/photon for metals

• Metastable bombardment: Excited atoms transfer energy to cathode

  \[A^m + \text{cathode} \rightarrow A + e^- + \text{cathode}^+\]

• Important in noble gas discharges due to long metastable lifetimes

Need for Second Ionization Coefficient

• First coefficient equation: \(I = I_0 e^{\alpha d}\) predicted slower current growth

• Townsend observed current increases faster than predicted

• Additional ionization sources identified:

  • Positive ions bombarding cathode

  • Photons causing photoemission

  • Metastable atoms at cathode

  • Volume recombination radiation

• Need for secondary ionization coefficient \(\gamma\)

Defining Townsend’s Second Ionization Coefficient

• \(\gamma\): Number of secondary electrons released from cathode per incident positive ion

• Physical processes contributing to \(\gamma\):

  • Ion impact: \(\gamma_i\) (dominant at low pressures)

  • Photon impact: \(\gamma_{ph}\) (important in streamer formation)

  • Metastable impact: \(\gamma_m\) (significant in noble gases)

• Total: \(\gamma = \gamma_i + \gamma_{ph} + \gamma_m\)

• Typical values: \(\gamma = 10^{-4}\) to \(10^{-1}\) for most gas-cathode combinations

Derivation of Current with Secondary Effects

• Electron balance at cathode:

  • Primary electrons: \(n_0\) (from external sources)

  • Secondary electrons: \(n_+\) (from positive ion bombardment)

• Electrons reaching anode: \(n = (n_0 + n_+) e^{\alpha d}\)

• Positive ions produced in gap: \(n - (n_0 + n_+) = (n_0 + n_+)(e^{\alpha d} - 1)\)

• Each positive ion releases \(\gamma\) electrons:

  \[n_+ = \gamma (n_0 + n_+)(e^{\alpha d} - 1)\]

Final Current Expression with Secondary Effects

• Solving for \(n_+\):

  \[n_+ = \frac{\gamma n_0 (e^{\alpha d} - 1)}{1 - \gamma (e^{\alpha d} - 1)}\]

• Total electrons at anode:

  \[n = n_0 \frac{e^{\alpha d}}{1 - \gamma (e^{\alpha d} - 1)}\]

• Final current expression:

  \[I = I_0 \frac{e^{\alpha d}}{1 - \gamma (e^{\alpha d} - 1)}\]

Factors Influencing \(\gamma\)

• Cathode material: Lower work function \(W\) increases emission

  • Clean tungsten: \(\gamma \sim 10^{-3}\)

  • Oxidized surfaces: \(\gamma \sim 10^{-2}\)

  • Alkali metals: \(\gamma \sim 10^{-1}\)

• Gas type: Affects ion energy and photon production

• Pressure: Influences ion energy distribution

• \(E/p\) ratio: Higher fields produce more energetic ions and photons

• Surface conditions: Contamination, roughness, temperature

Townsend Breakdown Criterion

• Current with secondary effects:

  \[I = I_0 \frac{e^{\alpha d}}{1 - \gamma (e^{\alpha d} - 1)}\]

• Current becomes infinite (breakdown) when denominator equals zero:

  \[1 - \gamma (e^{\alpha d} - 1) = 0\]

• Townsend criterion for breakdown:

  \[\gamma (e^{\alpha d} - 1) = 1 \quad \text{or} \quad \gamma e^{\alpha d} = \gamma + 1\]

• For \(\gamma \ll 1\): \(\boxed{\gamma e^{\alpha d} = 1}\) (simplified form)

Physical Interpretation of Townsend Criterion

• At \(\gamma e^{\alpha d} = 1\): Self-sustaining condition reached

• Each primary electron creates \((e^{\alpha d} - 1)\) positive ions in the gap

• These ions produce \(\gamma(e^{\alpha d} - 1) = 1\) secondary electron at cathode

• This electron creates another avalanche → continuous process

• Result: Discharge continues even if initial source \(I_0\) is removed

Derivation of Paschen’s Law

• From Townsend criterion: \(\gamma e^{\alpha d} = 1\)

• Taking natural logarithm: \(\ln \gamma + \alpha d = 0\)

• Substituting \(\alpha/p = A e^{-Bp/(E/p)}\):

  \[\ln \gamma + A p d e^{-Bp d / V} = 0\]

• Rearranging: \(A p d e^{-Bp d / V} = -\ln \gamma\)

• Let \(K = -\ln \gamma\) (positive constant), then:

  \[A p d e^{-Bp d / V} = K\]

Paschen’s Law - Final Form

• From \(A p d e^{-Bp d / V} = K\), solving for \(V\):

  \[e^{-Bp d / V} = \frac{K}{A p d}\]

• Taking natural logarithm:

  \[-\frac{B p d}{V} = \ln\left(\frac{K}{A p d}\right) = \ln K - \ln A - \ln(p d)\]

• Paschen’s Law:

  \[\boxed{V_b = \frac{B p d}{\ln(A p d) - \ln K} = \frac{B p d}{\ln\left(\frac{A p d}{K}\right)}}\]

Paschen Curve Characteristics

• Paschen curve shows \(V_b\) vs \(pd\) relationship

• Key features:

  • Minimum breakdown voltage at optimal \(pd\)

  • Left branch: low pressure, long mean free path

  • Right branch: high pressure, short mean free path

• Minimum occurs when: \(\frac{d}{d(pd)}[V_b] = 0\)

Paschen’s Law - Practical Applications

• Minimum breakdown voltage for air: \(\sim 327\) V at \(pd = 1\) Torr·cm

• Applications:

  • Design of gas-insulated equipment

  • Vacuum system breakdown prediction

  • High-altitude equipment insulation

  • Gas discharge tube design

• Deviations from Paschen’s Law:

  • Very low pressures (mean free path > gap)

  • Very high pressures (attachment effects)

  • Non-uniform fields

  • Electrode surface effects

Limitations of Townsend Mechanism

• Townsend theory assumes uniform field and steady-state conditions

• Observed discrepancies:

  • Faster breakdown than predicted electron drift time

  • Lower breakdown voltages in non-uniform fields

  • Independence of breakdown time on gap distance

  • Luminous channels (streamers) observed

• Typical conditions where Townsend fails:

  • \(pd > 1000\) Torr·cm

  • Non-uniform electric fields

  • Large gap distances (\(d > 1\) cm at atmospheric pressure)

Streamer Formation Mechanism

• Initial phase: Townsend avalanche creates space charge

• Critical avalanche size: \(\sim 10^8\) electrons (Raether criterion)

• Space charge effects: Distort electric field

  • Enhanced field at avalanche head

  • Reduced field at avalanche tail

• Streamer initiation: When space charge field comparable to applied field

• Photoionization: Provides seed electrons ahead of avalanche

Raether Criterion

• Critical condition for streamer formation:

  \[\boxed{\alpha d \geq 18-20}\]

• At this point, avalanche contains \(\sim 10^8\) electrons

• Space charge field becomes significant compared to applied field

• Physical reasoning:

  • Electron cloud radius: \(\sim 10^{-4}\) m

  • Space charge field: \(E_{sc} \sim 10^6\) V/m

  • Comparable to applied breakdown fields

Streamer Propagation

• Positive streamer: Propagates toward cathode

  • High field at streamer tip ionizes gas

  • Electrons drift toward anode, leaving positive channel

  • Propagation velocity: \(\sim 10^7\) m/s

• Negative streamer: Propagates toward anode (less common)

• Photoionization role: Creates seed electrons ahead of streamer tip

• Channel conductivity: Partially ionized, conductive path

Streamer-to-Arc Transition

• Leader formation: Streamer channel heats up, becomes highly conductive

• Return stroke: High current flows through leader channel

• Arc formation: Thermal ionization sustains discharge

• Current levels:

  • Streamer: \(\mu\)A to mA

  • Leader: mA to A

  • Arc: A to kA

Corona Discharge in Non-Uniform Fields

• Occurs in highly non-uniform electric fields

• Common geometries:

  • Wire-to-plane (transmission lines)

  • Point-to-plane (lightning rods)

  • Wire-to-cylinder (coaxial cables)

• Characteristics:

  • Localized ionization near high-field electrode

  • Self-limiting current due to space charge

  • Continuous discharge below breakdown voltage

  • Visible glow, audible noise, ozone production

Peek’s Formula for Corona Inception

• For wire-to-plane geometry, corona inception field:

  \[E_0 = E_{0,air} \left(1 + \frac{k}{\sqrt{r}}\right)\]

• Where:

  • \(E_{0,air} = 30\) kV/cm (critical field for air at STP)

  • \(k = 0.03\) cm\(^{1/2}\) (surface roughness factor)

  • \(r\): wire radius (cm)

• Corona inception voltage:

  \[V_c = E_0 r \ln\left(\frac{D}{r}\right)\]

• \(D\): distance to ground plane

Types of Corona

• Positive corona: Wire at positive potential

  • Streamer formation from wire surface

  • Pulsed, irregular discharge

  • Higher inception voltage

  • Greater power loss and radio interference

• Negative corona: Wire at negative potential

  • Glow discharge near wire

  • Steady, uniform discharge

  • Lower inception voltage

  • Less radio interference

Corona Effects and Mitigation

• Detrimental effects:

  • Power loss (especially in wet conditions)

  • Radio and TV interference

  • Audible noise

  • Ozone and NOx production

  • Conductor surface erosion

• Mitigation strategies:

  • Larger conductor radius (bundled conductors)

  • Higher conductor height

  • Corona rings on insulators

  • Smooth conductor surfaces

Gas-Insulated Systems

• Sulfur hexafluoride (\(SF_6\)): Superior insulating gas

  • Breakdown strength: 2-3 times air at same pressure

  • High electronegativity (electron attachment)

  • Chemically inert, non-toxic at normal concentrations

  • Self-healing after breakdown

• Applications:

  • Gas-insulated switchgear (GIS)

  • Gas-insulated transmission lines (GIL)

  • High-voltage circuit breakers

  • Transformers (as alternative to oil)

Design Guidelines for Gas Insulation

• Field uniformity: Avoid sharp edges and points

• Surface finish: Smooth, clean surfaces essential

• Particle control: Metallic particles drastically reduce breakdown strength

• Pressure optimization: Balance between insulation and cost

• Temperature effects: Account for gas density variations

• Moisture control: Water vapor reduces breakdown strength

Vacuum Insulation

• Principles: Eliminate gas molecules to prevent avalanche

• Pressure requirements: \(< 10^{-4}\) Torr for effective insulation

• Breakdown mechanisms in vacuum:

  • Field emission from cathode (primary)

  • Microdischarges from surface impurities

  • Electrode vapor formation

• Applications: Vacuum circuit breakers, vacuum capacitors, electron tubes

Summary

• Gas breakdown mechanisms:

  • Townsend avalanche for uniform fields and low \(pd\)

  • Streamer mechanism for non-uniform fields and high \(pd\)

  • Corona discharge in highly non-uniform fields

• Key parameters:

  • First ionization coefficient \(\alpha\)

  • Secondary emission coefficient \(\gamma\)

  • Townsend criterion: \(\gamma e^{\alpha d} = 1\)

  • Paschen’s Law: \(V_b = f(pd)\)

• Engineering applications: GIS, vacuum systems, transmission lines

Key Takeaways

References