IEEE TRANSACTIONS ON MAGNETICS,VOL.42,NO.4,APRIL 2006 1007
Calculation of Electric and Magnetic Fields
in Simpliﬁed Chambers of LowVoltage
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 threedimension calculation of the electric and magnetic ﬁelds in the simpliﬁed arc chamber of
lowvoltage 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
LOWVOLTAGE 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 lowvoltage cir
cuit breakers.A threedimensional (3D) 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
arcelectrode 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
00189464/$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 thermoemission [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 lowcir
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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Manuscript received June 20,2005 (email:wuyic51@mailst.xjtu.edu.cn).
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