IEEE TRANSACTIONS ON MAGNETICS,VOL.42,NO.4,APRIL 2006 1007

Calculation of Electric and Magnetic Fields

in Simpliﬁed Chambers of Low-Voltage

Circuit Breakers

Yi Wu,Mingzhe Rong,Jian Li,and Jianyong Lou

State Key Laboratory of Electrical Insulation for Power Equipment,Xi’an Jiaotong University,Xi’an 710049,China

This paper is mainly devoted to the three-dimension calculation of the electric and magnetic ﬁelds in the simpliﬁed arc chamber of

low-voltage circuit breakers.Coupled with the plasma ﬂow ﬁeld,the electric potential and current density in arc plasma are calculated

according to the electric conductivity,which is mainly decided by the temperature.Based on the potential vector method,the magnetic

ﬂux density and the Lorentz force under the effect of the ferromagnetic splitter plates are obtained.Both the stationary and transient

results of the calculation are discussed in detail.

Index Terms—Arc plasma,circuit breaker,temperature.

I.I

NTRODUCTION

A

LOW-VOLTAGE circuit breaker is used to switch the

electric current on and off.Its voltage is up to 1 kV with

the current in the range of some kiloamperes.When a fault

current happens in the circuit,the contacts of the breaker are

separated with an electric arc established in the arc chamber.

During combustion of the arc,a lot of physical mechanisms,

such as the gas convection,heat conduction,radiation,and

the Lorentz force occur in the chamber.The investigation into

the arc behavior is helpful for the designers to improve the

performance of breakers.Except for the experimental method,

arc simulation has been used by many researchers to obtain the

arc behavior in the chamber.Literature [1]–[6] has contributed

to this study over these years.However,the special discussion

for the electric and magnetic ﬁelds in the arc chamber is absent.

The effect of ferromagnetic splitter plates on the arc plasma is

not discussed systematically.

In arc plasma,the ﬂow,electric,and magnetic ﬁelds interact

with each other.Fig.1 presents coupled relationships among

these ﬁelds.The arc transport coefﬁcients such as electric

conductivity,speciﬁc heat,dynamic viscosity,and thermal

conductivity are dependent on the arc temperature and pressure

of the ﬂow ﬁeld.The electric conductivity of the arc plasma

decides the distribution of the electric potential and current

density.Also,the arc current induces the magnetic ﬁeld and

Lorentz force.Joule heating serves as a heat source and the

Lorentz force serves as momentum source.Therefore,the

electric and magnetic ﬁelds play an important role in the arc

behavior.

This paper mainly contributes to the calculation of the elec-

tric and magnetic ﬁelds for the arc chamber in low-voltage cir-

cuit breakers.A three-dimensional (3-D) computational model

is built according to a simpliﬁed geometry.First,the mathemat-

ical formulation for the calculation is described.Second,one

stationary result without ferromagnetic plates is presented.A

Digital Object Identiﬁer 10.1109/TMAG.2006.871386

Fig.1.Coupled realtionship among the ﬂow,electric,and magnetic ﬁelds.

transient solution under the effect of ferromagnetic plates is per-

formed with the stationary result used as the initial state.Finally,

related transient results are also discussed in detail.In order to

reduce the complexity of the arc physics,the arc ignition,and

arc-electrode interaction are not modeled [3] in this paper.

II.C

ALCULATED

M

ODEL

A.Model Geometry

In this paper,we adopt a simpliﬁed arc chamber as the calcu-

lated model.As shown in Fig.2,the dimension of the geometry

is 40

10

8 mm in the

–

–

direction and the origin of the

coordinate is at the center of the chamber.Enclosed by elec-

trodes and sidewalls (not shown in Fig.2),the whole chamber

is ﬁlled with air.Both the electrodes and sidewalls have a thick-

ness of 3 mm.In Fig.2,

is the splitter plates made of ferro-

magnetic material,

is the anode,and

is the cathode.

B.Mathematical Formulation

The plasma ﬂowﬁeld is calculated by the following equations

[7]:

(1)

(2)

(3)

The ﬂow ﬁeld is described by (1)–(3),which represent mass,

momentum,and energy conservation equations.The ﬁrst and

0018-9464/$20.00 © 2006 IEEE

1008 IEEE TRANSACTIONS ON MAGNETICS,VOL.42,NO.4,APRIL 2006

Fig.2.Simpliﬁed geometry of arc chamber.

last parts of each equation are the transient termand source term,

respectively.The source term of the energy conservation equa-

tion includes the Joule heat,radiation loss,and viscous dissi-

pation.The Lorentz force is involved in the momentum source

term.For the stationary calculation,the transient term of each

equation is not considered.

The electric ﬁeld is described by following equations:

(4)

(5)

The current density is deﬁned by

(6)

Some published literatures [1],[2] study the magnetic ﬁeld

by Biot–Savart law.However,coupled with the ﬂowﬁeld of the

compressible gas,this method consumes too much time to it-

erate during the arc calculation course.Additionally,it is not

suitable for the case of considering the ferromagnetic plates.

This paper adopts the potential vector way to calculate the mag-

netic ﬁeld

and the shortcomings described above are avoided.

According to Maxwell’s equations,we can obtain the following

equations:

(7)

(8)

(9)

Deduced from (6)–(9),we can obtain

(10)

(11)

The potential vector

is computed by (11).At the inter-

face between the ferromagnetic plates and plasma,the tangential

component of

remains,while the tangential component of

jumps.

In the equations,

is the density,

is the speciﬁc heat,

is the

dynamic viscosity,

is the thermal conductivity,

is the electric

conductivity,

is the time,

is the pressure,

is the temperature,

is the electric potential,

is the velocity vector

is the enthalpy,

is the viscous dissipation,

is the current

density

is the electric ﬁeld,

is the magnetization

ﬁeld strength,and

is the potential vector.Dependent on the

temperature and pressure solved in the plasma ﬂowﬁeld,the arc

transport coefﬁcients in above equations are obtained from the

literature [8].

C.Electric and Magnetic Boundary Condition

According to (5) and (6),the current density is used to deﬁne

the electric potential boundary condition in this paper.On all

the chamber walls,zero current density is applied.Due to the

current emission contribution of the cathode,a current density

condition should also be imposed on the interface between the

cathode and arc plasma.However,it is difﬁcult to decide the real

distribution of the current density at this interface.In this paper,

the Richardson’s law is used to deﬁne the current density at the

cathode/arc interface,i.e.,we mainly consider the mechanism

of thermo-emission [9] at the cathode/plasma interface.Thus,

the current density at this boundary is mainly dependent on the

temperature of interface elements and the total arc current.As

for the anode/plasma interface,we take it as a collector for neg-

ative particles.Therefore,Dirichet condition is applied to deﬁne

the potential boundary condition,i.e.,zero electric potential is

imposed on the anode/plasma interface.

As shown in (9)–(11),the potential vector is used to calcu-

late magnetic ﬁeld in this paper.The potential vector decreases

to zero in inﬁnite points.However,it is not feasible to build

the calculated region without limits due to conﬁned memory

of computer.According to the fact that the magnetic ﬁeld de-

creases with the reciprocal of

(

is the distance fromthe cur-

rent source),the potential vector at some distance away fromthe

arc chamber is set to zero in our calculation.

III.S

OLUTION AND

R

ESULTS

Based on the equations and boundary conditions described

above,the calculation work is performed by a modiﬁed com-

putational ﬂuid dynamics (CFD) code (Fluent 6.1).The whole

calculation includes the stationary and transient course.First,

the stationary calculation without ferromagnetic plates is carried

out.Second,with the stationary result used as the initial state,

a transient solution under the effect of ferromagnetic plates is

performed.

Due to the fact that the arc behavior includes an electromag-

netic process which is combined with aerodynamic action,we

adopt the couple solution method to solve above equations.

Compared with the segregated solution method,it has higher

accuracy especially for the compressible ﬂows and coupled

problem.The description of the solution course is presented

by Fig.3.It starts with the initialization and ends till the

convergence of the calculation.With the physical properties,

source term,and boundary condition updated,the discrete

equations are formed and solved.During this course,the mass,

momentum,and energy conservation equation are solved syn-

chronously ﬁrst.After that,the electric and magnetic ﬁelds are

computed according to the temperature distribution obtained

before.

The calculation is carried out with the arc current equal to

200 A.Due to the symmetry of the geometry,a half model is

built to reduce the data memories.Hexahedral cell conﬁgura-

tion is used to mesh the calculated domains.According to the

WU et al.:CALCULATION OF ELECTRIC AND MAGNETIC FIELDS 1009

Fig.3.Program of solution course.

Fig.4.Temperature distribution of the middle

–

plane with

mm.

maximal ﬂow velocity and the mesh size,the time step size for

the transient solution is set to 1

s in this paper.

A.Stationary Result

As the initial state of the transient solution,the stationary re-

sult is calculated without considering the ferromagnetic plates.

The arc column is located in the center of the arc chamber.In

this case,the sumof the Lorentz force applied to the arc column

is equal to zero and the plasma ﬂow keeps a balance.

1) Temperature Distribution:As the primary parameter of

the arc plasma,the temperature is used to decide the physical

properties of the arc plasma including the electric conductivity.

Thus,the temperature distribution is signiﬁcant for the calcu-

lation of electric and magnetic ﬁelds.Fig.4 shows the temper-

ature distribution in the middle

–

plane (

mm) of the

arc chamber.The temperature in the arc core is much higher

than other areas and the maximal value is up to 19.2 kK,which

means the arc core provides a current path due to high electric

conductivity.

2) Electric Potential:Fig.5 presents the electric potential

distribution for the

–

plane (

mm).With the zero po-

tential deﬁned on the anode,a maximal value of potential drop

(about 19.5 V) near the cathode is visible fromthe contour.As a

contrast to the result of literature [3],some resemblance can be

found,although different electric boundary conditions are used

in this paper.It also should be noted that Fig.5 only presents the

potential in the arc column and the voltage drop of the anode and

cathode sheath is not included.

3) Electric Conductivity and Current Density:According to

the relationship between electric conductivity and the temper-

ature,the electric conductivity corresponding to the tempera-

ture shown in Fig.4 is decided.The current density of the arc

plasma is computed by (6).Combined with the vector of current

density,the electric conductivity distribution at the

–

plane

(

mm) for the stationary result is presented in Fig.6.It is

Fig.5.Electric potential ﬁeld of the middle

–

plane (

mm).

Fig.6.Electric conductivity and current density vector of the middle

–

plane (

mm).

Fig.7.Current density distribution at the

–

plane (

mm).

clear to see that the main current path is located in the arc core

area and the maximal value of current density in this plane is

about 1.2e8A/m

.In addition,a shrinkage phenomenon of elec-

tric conductivity near the electrode is visible.Thus,the value of

the current density in this region is much higher than other areas.

Such condition causes higher Lorentz force in the vicinity of

the electrodes,which leads to a stronger pinch effect on the arc

column.

Fig.7 shows the distribution of the current density at the

–

plane (

mm).The maximal value in the ﬁgure is about

1.16 e7A/m

.

B.Transient Result

1) Magnetic Field:For the stationary state,the arc column

is located at the center of the chamber,and the plasma ﬂowﬁeld

holds a balance.In the case of the transient calculation,we adopt

the stationary result as the initial state and ferromagnetic splitter

plates are taken into account.Such condition lead to the change

of the magnetic ﬁeld.Consequently,the arc plasma moves for-

ward under the effect of the magnetic force.The corresponding

magnetic ﬂux density at

ms on the

–

plane is shown

in Fig.8,which is calculated by (7)–(11).Due to the symmetry

of the geometry,only half of the plane is shown,and a max-

imal value about 65 mT is obtained in the ferromagnetic areas.

1010 IEEE TRANSACTIONS ON MAGNETICS,VOL.42,NO.4,APRIL 2006

Fig.8.Magnetic ﬂux density on the

–

plane at

ms with a maximal

value about 65 mT.

Fig.9.Temperature distribution of the middle

–

plane (a)

ms,

(b)

ms,and (c)

ms.

Fig.10.Electric potential of the middle

–

plane (a)

ms,(b)

ms,and (c)

ms.

Based on the current density and magnetic ﬂux density in the

arc plasma,the Lorentz force can also be obtained.

2) Temperature and Electric Potential During the Arc

Motion:Under the effect of the ferromagnetic plates,the

arc column moves to the right direction of the chamber.The

distributions of temperature and electric potential are changed

correspondingly.As the primary parameter which decides other

ﬁelds,the temperature distribution on the middle

–

plane

at different time is shown in Fig.9.It is clear to see that the

shape of the arc column is changed greatly.With the distance

between the arc column and ferromagnetic plates reduced,the

arc moves more and more fast due to the raised Lorentz force.

Fig.10 shows the distribution of the electric potential for the

middle

–

plane at time equal to 0.8,1.0,and1.05 ms.Fromthe

ﬁgure,the electric potential is different at different time.Such

a fact is caused by different temperature distributions,shown in

Fig.9.

IV.C

ONCLUSION

Combined with the plasma ﬂow ﬁeld,this paper calculates

the electric and magnetic ﬁelds for arc chambers in low-cir-

cuit breakers with ferromagnetic splitter plates taken into ac-

count.According to the electric conductivity decided by the arc

temperature,the distribution of the electric potential and cur-

rent density of the arc column are presented and discussed.The

shrinkage of the arc column near the electrode is visible through

the current density distribution and a maximal value of the elec-

tric potential drop occurs near the cathodes/plasma interface.In

addition,the Lorentz force has a pinch effect on the arc column

and its value is higher near the shrinkage region.Also,the mag-

netic ﬂux density is obtained by the magnetic potential vector

method,which is advantageous to improving the efﬁciency of

the arc plasma simulation.Under the effect of the ferromag-

netic plates,the arc column moves to the right direction of the

chamber,and electric potential is changed greatly with the va-

riety of temperature correspondingly.

A

CKNOWLEDGMENT

This work was supported by the National Natural Science

Foundation of China under Grant 50 477 025.

R

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