An electrical arc erosion model valid for high current:

Vaporization and Splash Erosion.

Frédéric Pons 1,2,3

and Mohammed Cherkaoui 1,2

(1) George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, USA (2) UMI 2958 Georgia Tech-CNRS, Georgia Tech Lorraine, Metz, France

(3) Laboratoire de Physique et Mécanique des Matériaux, Université Paul Verlaine, Metz, France

Abstract—Electrical arc erosion plays an important role in

limiting the life of power switching devices. Consequently, a

macroscopic arc erosion model, valid for high current, has been

developed. This is the first of its kind since it takes into account

two modes of contact material erosion: vaporization and splash

erosion. Specifically, this model is used to measure the amount of

contact material removed after one electrical discharge. It,

therefore, allows the comparison of electrical contact erosion

resistances between materials. Furthermore, this model is the

first step in determining the composition of a suitable AgCdO

substitute by identifying the most influential material properties

in the arc erosion process.

Keywords—Arc erosion, splash erosion, AgCdO

I. INTRODUCTION

AgCdO is one of the most widely used contact materials in

the world because of its outstanding performance.

Nevertheless, due to environmental considerations, it will soon

be forbidden by European environmental directives [1-3].

Therefore, finding a good substitute is of crucial importance.

Electrical arc erosion plays a crucial role in the reliability

and life of power switching devices. Depending on the contact

material behavior in response to an electrical arc, surface

damages can induce severe changes in contact material

properties that will impact the power switching device functioning.

Arc erosion process mainly results from two modes:

vaporization of contact material under heat energy coming

from the electrical arc and splash erosion induced by driving

forces such as electromagnetic force, and the Buoyancy and

Marangoni effects. Most of the models dealing with electrical

arc erosion only take into account the vaporization part [4-11].

But these can only be valid for low current since the splash

erosion contribution grows with current intensity [4-10].

Xixiu’s model [11], based on a kinetic energy approach, was

the first one to describe sputter erosion of electrical contact material.

The purpose of this paper is to present a complete

macroscopic arc erosion model that describes, for the first

time, the arc erosion process during a breaking operation

under DC conditions. This model, based on the coupling of

three different models, combines:

• an Arc energy transport model [4] to get the

energy brought by the electrical arc at time t

• a Thermal model to get the arc erosion due to

vaporization

• a Magneto-hydrodynamic model to get the arc

erosion due to splash erosion

Since this model we developed takes into account the two modes of arc erosion, it describes the erosion due to an

electrical arc for low and high currents.

II. ARC EROSION MODEL

The model that we propose in this study, is based on the

coupling (see fig.1) of an arc energy transport model, a

thermal model and a magneto-hydrodynamic model. It

calculates the amount of contact material removed after one

electrical arc corresponding to one breaking operation under

DC conditions assuming that all vaporized and ejected molten

materials have been removed. It does not take into account deposition mechanisms.

Figure 1. Diagram of the models outputs and their coupling

Fig. 1 describes the interactions between the three basic

models by showing outputs of one model used as inputs for

another one.

987-1-4244-1902-9/08/$25.00 IEEE. 9

A. Arc Energy Transport Model

The purpose of this model developed by Swingler and

McBride [4] is to give, function of input parameters related to

experimental conditions such as the current intensity, circuit

voltage and opening velocity, the amount of energy brought

by the electrical arc that will hit the electrode surface at time t

of the breaking process under DC conditions.

Figure 2. Power dissipation through plasma region

It considers that the arc is divided into three regions (fig.2):

the anode region, the plasma region and the cathode region.

The power dissipation within each of these three regions is

computed from the current through the region and the

potential drop across it. Then, power dissipation through the

arc, from region to region up to the contact surface, is

calculated considering two ways of energy transport processes that encompass all the mechanisms, at a microscopic scale,

involved in the energy transport process: radial and channel

transport processes. The power flux density out of the plasma

region transporting energy towards the cathode is given by:

( )1

22 2 22

p a

c

p

P P k rq k

r r rπ+

= ++

(1)

Where k1 is the proportion of energy radially dissipated

k2 = 50% related to the channel transport process

The power flux density out of the plasma region

transporting energy towards the anode is given by the same

equation as (eq.1) using k3 instead of k2.

The total power dissipated in the arc is used as an input for

the thermal model. Model results about the total power

dissipated in the arc for I=9A, shown on fig.3, show good

agreement between experimental data [4] and the model

results.

Figure 3. Arc power output for I=9A

B. Thermal Model

This model, using the same approach as Swingler and

McBride in [4], consists on a transient axi-symmetric thermal

conduction model (see eq.2) within the electrode based on

Fourier’s law whose heat energy input is the power flux

density output from the arc energy transport model acting at the center of the electrode, as it can be seen on fig.4.

Figure 4. Meshing scheme

Meshing scheme and the boundary conditions can be

seen on fig.4. Eq.2 has been solved using finite differences.

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1

'''r z

T T Tk r k g c

r r r z z tρ∂ ∂ ∂ ∂ ∂+ + =

∂ ∂ ∂ ∂ ∂ (2)

Where rk and zk are the directional conductivities

'''g is the generation rate per unit volume and is only

nonzero for the element hit by the electrical arc

The thermal model gives the temperature distribution (see

fig.5) at each time t, and so, it allows us to deduce the molten

pool size at each time step and the total amount of material

vaporized after one breaking operation. We will consider that all vaporized material is part of the arc erosion.

Figure 5. Temperature distribution at the arc extinction (I = 9 A, V = 64

VDC, S = 1 m/s)

Temperatures of all the elements which have been vaporized

during the breaking process are set to 0 K (see fig.5).

C. Magneto-hydrodynamic Model

This model considers the formation of a molten metal jet, at

the contact surface center of the molten pool and along the z-

axis, under the influence of driving forces induced by the

electrical arc during the breaking operation. Whenever this

molten metal jet reaches critical conditions of stability, a

droplet comes off the molten metal jet tip and is ejected. All

the ejected droplets after one breaking operation are assumed

to constitute the splash erosion part of the arc erosion.

To develop our model, we use the same approach as J. N. Anno used in the case of a free, viscous inertial jet [12].

• Molten pool driving forces Molten metal is driven by gravity, buoyancy, surface

tension and electromagnetic forces [13-26]. We consider to be

in the “worst erosion case” where only surface tension

prevents the molten metal jet from growing and the three other

driving forces are working together to make it grow.

The buoyancy force, induced by the density change with

the spatial temperature distribution, is given by:

( )Buoy mF g T Tρ β= − (3)

Where g is the acceleration of gravity

is the density

is the thermal expansion coefficient

mT is the melting temperature

The buoyancy force at time t depends on the value of the jet

tip temperature at time t, output of the thermal model.

The electromagnetic force, resulting from the interaction

between the current flow and its self-induced magnetic field,

is given by:

ElecF J B= × (4)

Based on the assumptions of Kumar and DebRoy [27], the electromagnetic force at the center of the contact material

surface is:

2

2 20 2

m

Elecr

arc

I dF

r c

µπ→

= (5)

Where mµ is the magnetic permeability of the material

arcr is the arc radius

c is the thickness

d is the current density distribution factor

The electromagnetic force at time t depends on the value of

the intensity at time t, output of the energy transport model.

• Axial velocity distribution The momentum equation for the molten metal cylindrical jet

can be developed by evaluating the forces acting on a differential element of the jet (see fig.6)[12]. Considering this

element as a control volume and the case of steady flow,

Newton’s second law projected on the z-axis gives:

.z z

controlvolume

F v v dAρ= ∆ (6)

This gives the following differential equation:

( )

3 / 2

0 0

2

2 2

3 1 1

2

11 ( )

2

z z z

z z

m

m

zarc

u u u

z z u z zut w

I dg T T

ur c

µ γρ ρ

µβρπ

∂ ∂ ∂∂= +∂ ∂ ∂ ∂

+ + − +

(7)

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Where zu is the axial velocity

µ is the molten metal viscosity (=0.01kg.m-1.s-1)

γ is the surface tension (=1N.m-1)

0t is the molten pool radius

0w is the velocity at the contact surface

On fig.6, where we can see the molten liquid jet at time t,

we assume that the molten pool surface (at z = 0) corresponds

to the contact surface of the electrode under study.

Figure 6. Differential element of the jet

By putting (eq. 7) into dimensionless form, it gives:

'

'' ' 1W

W W WW W

λ= + − − (8)

with ( )1/ 3

2

2 2

31 ( )

2

mz m

arc

I du g T T W

r c

µµ βρ ρπ

= + − +

( )1/ 32 / 3 2

2 2

31 ( )

2

mm

arc

I dz g T T

r c

µµ β ξρ ρπ

−

= + − +

( )2 / 32 / 3 2

2 2

0 0

1 11 ( )

2 3 2

mm

arc

I dg T T

t r c W

µγ ρλ βρ µ ρπ

−

= + − +

In the dimensionless non linear second order differential

equation (Eq.8), parameter depends on the current intensity I

at time t, output of the arc energy transport model and on the

jet tip temperature and the molten pool radius at time t,

outputs of the thermal model. Eq.8 has been plotted on fig.7

for two significant values of lambda.

Figure 7. Plot of the dimensionless form of the velocity differential equation

As it can be seen in Eq.8, W is related to the axial

velocity of the molten metal jet and ksi is related to the axial

coordinate z.

• Stability conditions During the arcing process, the molten liquid jet keeps

growing up to reach a critical length after which a droplet will

form. These stability conditions are derived from a mechanical

energy balance for a closed, stationary control volume of

volume V and surface area S [12]:

2

2

1

2

10

2

i i ij j i j j

V S S

V V

X u dV n u dS u n q dS

q dV RdVt

ρ τ ρ

ρ

+ −

∂− − =∂

(9)

Where iX is the generalized body force

ijτ is the stress tensor

iu is the velocity component in the ith direction

jn is the projection of the outward unit normal to the

surface in the jth direction

i iq u u= R is the viscous energy dissipation, given by

( ) ( ), , , ,2

i j j i i j j iR u u u uµ= + + (10)

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This leads to the following stability condition,

corresponding to the jet breakup time:

072

crit

tT

µγ

= (11)

Where 0

t is the molten pool radius

µ is the molten metal viscosity

γ is the surface tension

• Droplet size From [12], we assume that the droplets are spherical and

their radii R0 are:

0

2 cR t= (12)

Where ct is the jet radius at breakup

Consequently, the amount of material removed by splash

erosion is given by:

3

0

4

3spl

Droplets

m Rπρ= (13)

D. Model results

Total masses of material removed by arc erosion during the

breaking process for different intensities from the model are

plotted on fig.8 and compared with experimental data [4].

Figure 8. Experimental data [4] and model results of AgCdO cathode arc

erosion

Fig.8 shows good agreement between model results and

experimental data (Polynomial (Model results) corresponding

to the curve of best fit polynomial of the model results).

III. CONCLUSION

A complete macroscopic arc erosion model which takes into

account the two arc erosion modes has been developed. This

model will be used to compare the arc erosion ability of

different contact material in order to find the best substitute to AgCdO. It will also be used to evaluate the most influential

contact material properties on the arc erosion process.

Other experiments will be conducted to finish validating our

model. White light interferometry will be used to get the total

amount of material removed due to the electrical arc from

each sample (see fig.9 & 10).

In our analysis of composite contact material such as

AgCdO, we assumed homogeneous material properties

(Surface tension, viscosity, etc..) reflecting the global

behaviour of the contact material under study.

Effects related to particle size, their distribution and percentage of CdO on the material properties will be the

object of future studies, where Molecular Dynamics will be

used to develop a model.

Figure 9. White light interferometry picture of a crater after one breaking

operation (400 A, 28 VDC)

Figure 10. White light interferometry picture of a crater after one breaking

operation (400 A, 28 VDC)

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ACKNOWLEDGMENT

The authors would like to acknowledge Idriss Ilali (Project Manager at Leach Int.), Adeline Aubouin and Stephane

L’Attention (Research Engineers at Leach Int.), Alain Boyer

(Leach Int. Research Manager), Jean Michel Stocklouser

(Leach Int. R&D Manager), Dominique Bauthian (Leach Int.

General Manager), Jean-Michel Sigaud (Leach Int. President)

for their support.

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