Vacuum Systems

• A vacuum system maintains pressure below atmospheric pressure, measured in mm of mercury (torr).

• Standard atmospheric pressure at \(0^{\circ}~\mathrm{C} = 760~\text{mm of Hg}\).

• \( 1 \, \text{torr} = 1 \, \text{mm Hg}; \, 1 \times 10^{-3} \, \text{torr} = 1 \, \text{micron} \)

• Modern systems achieve pressures as low as \( 1 \times 10^{-8} \, \text{torr} \).

• Applications: Vacuum interrupters, particle accelerators, and high-voltage insulation systems.

• Design challenges: Maintaining ultra-high vacuum against outgassing and leaks.

Vacuum Classification

• Different vacuum levels have distinct characteristics.

• High and ultra-high vacuum required for electrical applications.

• Roughing vacuum: Initial pump-down stage.

• Molecular flow regime: Pressure \(< 10^{-3}\) torr.

Kinetic Theory and Molecular Density

• Mean free path formula:

  \[\lambda = \frac{1}{\sqrt{2} \times n \times \sigma}\]
  where \(n\) is molecular density and \(\sigma\) is collision cross-section.

• Molecular density varies with vacuum level:

  Vacuum Level	Pressure (torr)	Molecules/cm³
  Atmospheric	760	\(2.7 \times 10^{19}\)
  High Vacuum	\(10^{-6}\)	\(3.5 \times 10^{10}\)
  Ultra-High Vacuum	\(10^{-9}\)	\(3.5 \times 10^{7}\)

• At \(10^{-6}\) torr, mean free path \(\approx\) 50 meters.

• Relationship: \(n = \frac{P}{k_B T}\) (ideal gas law).

• Knudsen number: \(Kn = \lambda/L\) determines flow regime.

Mean Free Path Comparison

Townsend Discharge

• In gas, short mean free path leads to electron avalanches via ionization:

  \[I = I_0 e^{\alpha d}, \quad \text{where } \alpha \text{ is ionization coefficient.}\]

• In vacuum (\( 1 \times 10^{-5} \, \text{torr} \)), mean free path is several meters.

• For small gaps (few mm), electrons cross without collisions, preventing avalanche formation.

• Gas liberation in vacuum (e.g., from electrode surfaces) enables Townsend-like discharge.

• Implication: Vacuum insulation design must account for residual gas effects.

• Critical parameter: \(\alpha d < 1\) for vacuum conditions.

Paschen’s Law in Vacuum Context

• Paschen’s law fails at \(pd < 10^{-2}\) torr·cm.

• Transition from gas discharge to field emission dominated breakdown.

• Critical pressure: \(P_c \approx \frac{1}{d \times 10^2}\) torr (for gap \(d\) in cm).

• Left branch of Paschen curve: Field emission regime.

Transition from Gas to Vacuum Breakdown

• Gas discharge regime (\(pd > 1\) torr·cm):

  • Townsend avalanche multiplication

  • Secondary emission coefficient \(\gamma\) important

  • Breakdown condition: \(\gamma(\exp(\alpha d) - 1) = 1\)

• Intermediate regime (\(10^{-2} < pd < 1\) torr·cm):

  • Mixed mechanisms: collision and field emission

  • Statistical time lag becomes significant

  • Voltage-dependent breakdown probability

• Vacuum regime (\(pd < 10^{-2}\) torr·cm):

  • Field emission dominated

  • Electrode surface conditions critical

  • Micro-projection enhancement effects

Vacuum Arc

• Neutral atoms, ions, and electrons originate from electrodes via evaporation.

• Large mean free path results in dielectric strength 1000x higher than gas due to minimal ionization.

• Breakdown strength depends on gap length and electrode surface condition.

• Polished, degassed electrodes increase breakdown strength.

• Roughened electrodes reduce strength; improved by high-voltage impulses.

• Optimal gap for metals (e.g., Ag, Bi-Cu) is \( < 3 \, \text{mm} \) at \( 1 \times 10^{-6} \, \text{torr} \).

Breakdown Voltage vs Gap Distance

Field Emission Theory

• Fowler-Nordheim equation describes field emission current density:

  \[J = \frac{A \times E^2}{\phi} \times \exp\left(-\frac{B \times \phi^{3/2}}{E}\right)\]
  where \(A = 1.54 \times 10^{-6}\) A·eV·V\(^{-2}\), \(B = 6.83 \times 10^7\) eV\(^{-3/2}\)·V·cm\(^{-1}\)

• Work function values for electrode materials:

  Material	Work Function (eV)
  Copper (Cu)	4.7
  Silver (Ag)	4.3
  Tungsten (W)	4.5
  Bismuth (Bi)	4.2

• Temperature dependence: \(J \propto T^2 \exp(-\phi/k_B T)\) for thermionic emission.

• Field enhancement factor \(\beta\) at micro-projections: \(E_{local} = \beta \times E_{average}\).

Surface Electric Field and Micro-projections

• Field enhancement at micro-projections:

  \[\beta = \frac{h}{r} \quad \text{(height-to-radius ratio)}\]
  where \(h\) is projection height and \(r\) is tip radius.

• Critical field for emission: \(E_c \approx 10^7\) V/cm for most metals.

• Surface roughness effects:

  • Polished surface: \( R_a < 0.1 \, \mu\text{m} \)

  • Rough surface: \( R_a > 1 \, \mu\text{m} \)

  • Field enhancement: \(\beta = 10-100\) for rough surfaces

• Schottky effect: Image force reduces effective work function:

  \[\phi_{eff} = \phi - \sqrt{\frac{eE}{4\pi\epsilon_0}}\]

• Surface preparation crucial for reliable operation.

Vacuum Arc Visualization

Current Density and Temperature at Cathode Spots

• Cathode spot diameter: \( 1 \, \mu\text{m} \) to \( 100 \, \mu\text{m} \).

• Erosion rate: \( 50 \, \mu\text{g} \, \text{C}^{-1} \) for copper.

• Plasma density: \(10^{18} - 10^{19}\) particles/cm³.

• Current density: \(10^8 - 10^9\) A/cm² at cathode spots.

• Temperature: 3000-4000 K at active spots.

Electric Discharge Mechanisms

• Cathode spots form based on current:

  Current Level	Cathode Spot Behavior
  Low Current (\(< 100 ~\mathrm{A}\))	Single mobile spot
  High Current (\(> 1000 ~\mathrm{A}\))	Multiple stationary spots

• Spots are primary vapor sources for the arc.

• Discharge triggered by high electric fields or resistive heating at micro-projections.

• Cold cathode discharge: emission from spots, not entire cathode surface.

• Typical currents in vacuum interrupters: 100 A to 10 kA.

Cathode Spot Dynamics

Anode Phenomena in Vacuum Arc

• Anode modes depend on current density:

  Current Density	Anode Mode	Characteristics
  < \(10^3\) A/cm²	Diffuse	Low voltage drop
  \(10^3 - 10^5\) A/cm²	Intermediate	Transition mode
  > \(10^5\) A/cm²	Constricted	High voltage drop

• Anode heating: Joule heating + ion bombardment + radiation.

• Anode fall voltage: 10-20 V in diffuse mode, 50-100 V in constricted mode.

• Metal vapor generation: Both cathode and anode contribute.

• Retrograde motion: Cathode spots move against \(\vec{J} \times \vec{B}\) force.

Arc Voltage and Energy Balance

• Typical arc voltage: 20-50 V (nearly constant).

• Number of cathode spots: \(N \approx I/I_0\) where \(I_0 \approx\) 100 A per spot.

• Energy balance: Joule heating = evaporation + radiation + conduction.

• Power density at cathode: \(10^8 - 10^9~\mathrm{W/cm^2}\) .

Electron Emission Mechanisms

• Multiple emission mechanisms operate simultaneously.

• Field emission dominates at high electric fields.

• Secondary emission sustains the discharge.

• Explosive electron emission at very high current densities.

Discharge Stability

• Depends on contact material (vapor pressure) and circuit parameters (voltage, current, inductance, capacitance).

• High vapor pressure at low temperature (e.g., Zn, Bi) enhances stability.

• Thermal conductivity affects current chopping:

  • Good conductors cool faster, reducing vapor, leading to arc chopping.

  • Poor conductors maintain vaporization, stabilizing the arc.

Current Chopping Phenomenon

• Current chopping levels for different materials:

  Material	Chopping Level (A)
  Copper (Cu)	2-5
  Silver (Ag)	3-8
  Bismuth (Bi)	0.5-2
  Cu-Bi alloy	1-3

• Overvoltage: \(V = L \frac{di}{dt}\) can reach several kV.

• Impact: Affects TRV (Transient Recovery Voltage) in circuit breakers.

Surface Contamination Effects

Non-metallic Emission

• Pre-breakdown current originates from non-metallic surfaces (e.g., oxides, organic residues).

• Micro-inclusions (e.g., trapped gas pockets) reduce breakdown strength.

• Contamination (e.g., Na, K, B, Al, Si) from glass sealing, fingerprints, or pump oil deposits on electrodes.

• Contaminants enhance:

  • Field emission

  • Secondary electron emission

  • Stimulated desorption of molecules/ions

• Reduces electric strength to \( 10 \, \text{kV} \, \text{cm}^{-1} \) (vs. \( 1 \times 10^4 \, \text{kV} \, \text{cm}^{-1} \) for pure field emission).

Gas Desorption and Virtual Leaks

• Electron stimulated desorption (ESD):

  • Electrons impact surface → gas molecules released

  • Desorption yield: \(\eta = 10^{-4} - 10^{-2}\) molecules/electron

  • Common desorbed gases: \(H_2\), \(H_2O\), \(CO\), \(CO_2\)

• Ion stimulated desorption (ISD):

  • Higher desorption yield than ESD

  • Physical sputtering + chemical desorption

• Virtual leaks:

  • Trapped volumes in threads, welds, porous materials

  • Slow outgassing affects vacuum level

  • Solution: Proper venting and design

• Permeation: Hydrogen through metal walls at elevated temperatures.

Clump Mechanism Process

Clump Mechanism

• Assumes a loosely bound particle (clump, \( 1 \, \mu\text{m} \) to \( 100 \, \mu\text{m} \)) on electrode surface.

• Steps:

  1. Clump charges under high voltage.

  2. Detaches and is attracted to opposite electrode.

  3. Impact releases vapor/gas, triggering discharge.

• Frequent breakdowns increase withstand voltage via electrode conditioning.

• Practical implication: Critical for reliable operation of vacuum interrupters in circuit breakers.

Alternative Breakdown Theories

• Micro-particle theory: Conducting particles in vacuum gap initiate breakdown.

• Triple point theory: Breakdown at electrode-insulator-vacuum junction due to field enhancement.

• Statistical nature: Breakdown follows Weibull distribution:

  \[P(V) = 1 - \exp\left[-\left(\frac{V}{V_0}\right)^n\right]\]
  where \(n\) is shape parameter (typically 5-15) and \(V_0\) is scale parameter.

Breakdown Statistics and Time Lags

• Statistical time lag (\(t_s\)):

  • Time required for initiating electron to appear

  • Follows Poisson distribution: \(P(t) = 1 - \exp(-t/t_s)\)

  • Typically: \(t_s = 10^{-6} - 10^{-3}\) seconds

• Formative time lag (\(t_f\)):

  • Time for discharge development after initiation

  • Generally: \(t_f = 10^{-8} - 10^{-6}\) seconds

  • Depends on gap length and applied voltage

• Area effect: Breakdown voltage decreases with electrode area:

  \[V_b = V_0 \times A^{-n}, \quad n = 0.1-0.3\]

• Electrode material influence: Work function, thermal properties, surface roughness.

Conditioning Process Comparison

Conditioning Methods

• Destroys micro-emission sites, increasing withstand voltage.

• Methods:

  • Hydrogen glow discharge (consistent results, typically hours).

  • Pre-breakdown current flow or high-temperature heating in vacuum (days for large systems).

  • Repeated spark breakdowns (time-consuming, weeks for unpolished electrodes).

• Larger electrode area reduces breakdown voltage: \(V_b \propto A^{-0.5}\), where \(A\) is electrode area.

Quantitative Conditioning Effects

• Typical improvement factors: 2-10x increase in breakdown voltage.

• Surface roughness specifications: \( R_a < 0.1 \, \mu\text{m} \) for optimal performance.

• Outgassing rates:

  • Before conditioning: \(10^{-6}\) torr·L/(s·cm²)

  • After conditioning: \(10^{-9}\) torr·L/(s·cm²)

• Conditioning effectiveness depends on electrode material and surface preparation.

Advanced Conditioning Techniques

• Laser conditioning:

  • Pulsed laser ablation removes surface contaminants

  • Selective removal of micro-projections

  • Precise control over surface modification

• Ion bombardment conditioning:

  • Ar\(^+\) or Ne\(^+\) ion sputtering

  • Removes surface layers with contaminants

  • Creates atomically clean surfaces

• Chemical conditioning:

  • Controlled oxidation followed by reduction

  • Reactive gas cleaning (e.g., \(H_2\) at elevated temperature)

• Electropolishing: Electrochemical surface smoothing.

• In-situ monitoring: Real-time current measurement during conditioning.

Pressure vs. Breakdown Voltage

Pressure Effects

• Small gaps (\( < 1 \, \text{mm} \), \( 1 \times 10^{-2} \, \text{torr} \) to \( 1 \times 10^2 \, \text{torr} \)): No change in breakdown voltage due to collision-free electron paths.

• Large gaps (20 cm): Pressure variation significantly lowers withstand voltage due to increased collision probability.

• Higher pressures introduce gas molecules, reducing mean free path and enabling ionization.

• Approximate relation: \(V_b \propto P^{-0.1}\) for large gaps, where \(P\) is pressure.

Effect of Gas Species on Breakdown

• Residual gas composition affects breakdown characteristics:

  Gas	Ionization Potential (eV)	Relative Effect
  \(H_2\)	15.4	High
  \(H_2O\)	12.6	Medium
  \(N_2\)	15.6	High
  \(O_2\)	12.1	Medium
  \(CO\)	14.0	Medium-High
  \(CO_2\)	13.8	Medium-High

• Electronegative gases (\(O_2\), \(H_2O\)): Electron attachment reduces avalanche.

• Noble gases: Higher breakdown voltages due to high ionization potentials.

• Partial pressure effects: Most harmful gas determines breakdown characteristics.

• Gas evolution: Electrode heating → increased gas pressure → reduced breakdown strength.

Practical Applications and Limitations

• Vacuum interrupter specifications:

  • Operating pressure: \(< 10^{-4}\) torr

  • Contact gap: 8-20 mm

  • Withstand voltage: 50-200 kV

• Maintenance requirements: Vacuum monitoring, leak detection.

• Economic advantages: No gas handling, environmental friendliness.

• Life cycle: 20-30 years with proper maintenance.

Vacuum System Design Principles

• Key design parameters:

  • Pumping speed: S (L/s)

  • Conductance: C (L/s)

  • Leak rate: Q (torr·L/s)

• Ultimate pressure: \(P = \frac{Q_{leak} + Q_{outgas}}{S_{eff}}\)

• Seal technology: Metal gaskets, elastomer O-rings, glass-to-metal seals.

Getter Materials and Vacuum Measurement

• Getter materials maintain vacuum by absorbing residual gases:

  Getter Type	Operating Temp. (\(^{\circ}\mathrm{C}\))	Pumping Speed
  Titanium	400-600	High for \(O_2\), \(N_2\)
  Barium	300-500	Excellent for \(H_2O\)
  Zirconium	500-700	Good for \(H_2\)

• Vacuum measurement techniques:

  • Pirani gauge: \(10^{-4}\) to 1 torr (thermal conductivity)

  • Ion gauge: \(10^{-10}\) to \(10^{-3}\) torr (ionization)

  • Capacitance manometer: \(10^{-5}\) to 1000 torr (absolute)

• Leak detection: Helium mass spectrometry, sensitivity \(< 10^{-10}\) torr·L/s.

• Outgassing rate target: \(< 10^{-9}\) torr·L/(s·cm²) for UHV applications.

Summary

• Vacuum breakdown differs fundamentally from gas discharge due to large mean free path and electrode-derived particle emission.

• Cold cathode discharge with cathode spots serves as the primary mechanism driving vacuum arcs.

• Non-metallic emissions, surface contamination, and clump mechanisms significantly reduce breakdown strength.

• Electrode conditioning through various methods (glow discharge, heating, spark conditioning) substantially enhances breakdown performance.

• Pressure effects on breakdown voltage vary significantly with electrode gap dimensions and residual gas composition.

• Vacuum technology offers superior dielectric strength and environmental advantages for high-voltage applications.

• Future developments focus on advanced materials, improved conditioning techniques, and enhanced reliability for next-generation vacuum interrupters.