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HIGH VOLTAGE Breakdown phenomena Assistant Professor Suna BOLAT KRÖGER

Eastern Mediterranean University

Department of Electric & Electronic Engineering

Determination of Townsend’s coefficients ,

Townsend’s second ionization coefficient,

is the number of ion pairs made by one positive ion per unit drift (1 cm) in the direction of the field.

Positive ions removes electron from the cathode when they strike to cathode.

Townsend’s ionization coefficient is about that phenomena, not about ionization by collision.

Townsend’s 2nd ionization coefficient,

: is the number of electrons removed from the cathode by one positive ion striking to cathode.

𝛾: 1

50,… ,

1

5000

If =1/50, one electron can be removed from the cathode when 50 positive ions strike to it.

x = 0 x = d x

E

= 0.02 0.00002

depends on the material of the cathode and the type os gaseous insulator.

Material

of cathode

Gas

Air Neon

Al 0.035 0.1

Cu 0.025 0.065

Fe 0.02 0.06

At a distance x = d,

𝑒𝛼𝑑 − 1 positive ions are produced by collison

𝛾 𝑒𝛼𝑑 − 1 electrons are produced

Number of electrons

removed from the cathode by one positive ion

Townsend’s criteria for spark breakdown

If this condition is fulfilled, at least one electron is removed from the cathode and discharge can continue.

This is also called as; condition of self-sustainability of discharge.

𝛾 𝑒𝛼𝑑 − 1 = 1

Breakdown voltage in uniform fields by Townsend’s theory

Self sustaining discharge: 𝛾 𝑒𝛼𝑑 − 1 = 1

𝑒𝛼𝑑 − 1 =1

𝛾

𝑒𝛼𝑑 = 1 +1

𝛾

𝛼𝑑 = ln(1 +1

𝛾)

• 𝛼 = 𝐴. 𝑝. 𝑒−𝐵.𝑝

𝐸 : Townsend’s 1st ionization coefficient

• For breakdown:

𝛼 = 𝐴. 𝑝. 𝑒

−𝐵.𝑝

(𝑈𝑏

𝑑 )

= 𝐴. 𝑝. 𝑒−𝐵.𝑝.𝑑

𝑈𝑏

𝛼𝑑 = 𝐴. 𝑝. 𝑑. 𝑒−𝐵.𝑝.𝑑

𝑈𝑏 = ln(1 +

1

𝛾)

𝑈 = 𝑈𝑏

𝐸 = 𝐸𝑏

𝑒−𝐵.𝑝.𝑑

𝑈𝑏

=ln(1 +

1𝛾)

𝐴. 𝑝. 𝑑

𝑒𝐵.𝑝.𝑑

𝑈𝑏

=𝐴. 𝑝. 𝑑

ln(1 +1𝛾)

𝐵. 𝑝. 𝑑

𝑈𝑏= ln

𝐴. 𝑝. 𝑑

ln(1 +1𝛾)

Equation for breakdown voltage in uniform fields by Townsend’s theory (Paschen Equation)

𝑈𝑏 = f(p. d)

Breakdown voltage depends on multiplication of pressure and electrode gap

𝑈𝑏 =𝐵. 𝑝. 𝑑

ln𝐴. 𝑝. 𝑑

ln(1 +1𝛾)

Breakdown voltage curve (Paschen Curve)

Torr: mm.Hg

Critical (p.d) value that makes the breakdown voltage minimum

•𝑑𝑈𝑏

𝑑(𝑝.𝑑)= 0 → (𝑝. 𝑑)𝑐𝑟= ? → 𝑈𝑏 (𝑝. 𝑑)𝑐𝑟

𝑑𝑈𝑏𝑑(𝑝. 𝑑)

=

𝐵. ln𝐴. 𝑝. 𝑑

ln(1 +1𝛾)− 𝐵. 𝑝. 𝑑.

𝐴

ln(1 +1𝛾)

𝐴. 𝑝. 𝑑

ln(1 +1𝛾)

ln𝐴. 𝑝. 𝑑

ln(1 +1𝛾)

2

= 0

𝐵. ln𝐴. 𝑝. 𝑑

ln(1 +1𝛾)− 𝐵 = 0

ln𝐴. 𝑝. 𝑑

ln(1 +1𝛾)− 1 = 0

ln𝐴. 𝑝. 𝑑

ln(1 +1𝛾)= 1 = ln 𝑒 →

𝐴. 𝑝. 𝑑

ln(1 +1𝛾)= 𝑒 = 2.718

(𝑝. 𝑑)𝑐𝑟=𝑒. ln(1 +

1𝛾)

𝐴

Minimum breakdown voltage

𝑈𝑏 (𝑝. 𝑑)𝑐𝑟 = 𝑈𝑏,𝑚𝑖𝑛 =𝐵. (𝑝. 𝑑)𝑐𝑟

ln𝐴. (𝑝. 𝑑)𝑐𝑟

ln(1 +1𝛾)

𝑈𝑏,𝑚𝑖𝑛 =𝐵.𝑒. ln(1 +

1𝛾)

𝐴

ln𝐴.𝑒. ln(1 +

1𝛾)

𝐴

ln(1 +1𝛾)

= 𝐵.𝑒. ln(1 +

1𝛾)

𝐴

(𝑝. 𝑑)𝑐𝑟

Minimum breakdown voltage

𝑈𝑏,𝑚𝑖𝑛 = 𝐵. (𝑝. 𝑑)𝑐𝑟

Breakdown strength

𝐸 = 𝑈 𝑑

𝐸𝑏 =𝑈𝑏𝑑

𝐸𝑏 =𝐵. 𝑝

ln𝐴. 𝑝. 𝑑

ln(1 +1𝛾)

Generalized Paschen Equation

• 𝑇0 = 273 + 20℃ = 293°𝐾 • 𝑇 = 273 + 𝜃℃: gas temperature in °𝐾 • 𝑝: gas pressure in mmHg • A, B: gas constants • 𝛾: number of electrons removed from the cathode by positive ions

𝑈𝑏 =𝐵.𝑇𝑇0. 𝑝. 𝑑

ln𝐴.𝑇𝑇0. 𝑝. 𝑑

ln(1 +1𝛾)

𝐴 → 𝐴. 𝑇 𝑇0

𝐵 → 𝐵. 𝑇 𝑇0

Streamer Theory

• This theory explains electrical discharges in big gaps and high pressure

• 𝑝. 𝑑 > 500 𝑚𝑚𝐻𝑔. 𝑐𝑚

• The Streamer theory predicts the development of a spark discharge directly from a single avalanche

1. Streamer Theory for uniform electric field

+ +

+

+

+ +

+

- -

-

-

-

Electric field between the avalanche and the anode is big

E

- - -

- - - - - -

- -

-

- - - -

- - -

- + + + + + + +

E

U

electrons Positive ions

- - - - -

- -

- - - - - + + + +

the electrons advance rapidly, the positive ions are left behind in a relatively slow-moving tail

Streamer process

• The space charge produced in the avalanche causes sufficient distortion of the electric field that those free electrons move towards the avalanche head, and in so doing generate further avalanches in a process that rapidly becomes cumulative.

• As the electrons advance rapidly, the positive ions are left behind in a relatively slow-moving tail

• The field will be enhanced in front of the head. Just behind the head the field between the electrons and the positive ions is in the opposite direction to the applied field and hence the resultant field strength is less.

• Again between the tail and the cathode the field is enhanced. • Due to the enhanced field between the head and the anode, the space

charge increases, causing a further enhancement of the field around the anode.

• The process is very fast and the positive space charge extends to the cathode very rapidly resulting in the formation of a streamer.

𝑈𝑏 = 5.27 × 10−7𝛼. 𝑑. 𝑒𝛼𝑑

𝑑𝑝

Breakdown voltage by streamer theory

d: electrode gap [cm]

(electrode seperation)

p: gas pressure [mmHg]

: ionization coefficient of the eletron [1/cm]

Ub: breakdown voltage [V]

cm

V

px

eE

x

r

71027.5

1/cm

cm cm

mmHg

V

𝐸𝑏 =𝑈𝑏𝑑= 5.27 × 10−7

𝛼. 𝑒𝛼𝑑

𝑑𝑝

2. Streamer Theory for non-uniform electric field

r: curvature radius

d: electrode gap

Non-uniform field: 𝑑

𝑟> 40 → 𝑑 > 40 𝑟

r

d

U

Different non-uniform electric field cases

U (+)

E

- - - - - -

- -

- -

- -

+ + + +

U (-)

E

- - - - - -

- -

- -

- -

+ + + +

(+)

- E1

+

+ E1

-

U (-)

U (+)

E

Electrode gap – breakdown voltage

d

Ub

- +

- +

Discharge time

𝑡𝑏 = 𝑡𝜎 + 𝑡𝑝

Creepage time: time required for an electron to appear after the moment which voltage applied

Progress time: time required for such an electron appeared in the gap to lead a breakdown after the moment it appeared

Discharge time

creepage time

progress time