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MATHEMATICAL MODELS OFPHENOMENA IN ELECTRICAL CONTACTS

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The Russian Academy of Sciences
Siberian Branch
A.P. ERSHOV INSTITUTE OF INFORMATICS SYSTEMS
S.N. KHARIN
MATHEMATICAL MODELS OF PHENOMENA IN
ELECTRICAL CONTACTS
Novosibirsk 2017
UDС 517.58
LBC22.1
K 42
Kharin S.N. Mathematical models of phenomena in electrical contacts:
Мonograph . / A.P. Ershov Institute of Informatics system , Siberian Branch of
RAS, 2017. 193 p.
ISBN 978-601-06-4797-8
The different methods of solution of mathematical models describing
phenomena in electrical contact systems by arcing are presented in the book. All
phenomena at contact opening and closure are considered as consecutive stages
including the initial stage of Joule’s heating, appearance and explosion of a liquidmetallic bridge, ignition, evolution and extinction of the electrical arc in metallic
and gaseous phases. Step by step the mechanism of the arc erosion due to
vaporization, as well as the new mechanisms of erosion due to thermo-capillary
convection and erosion in solid state due to thermo- elastic stresses are introduced
and considered. Mathematical models of arc floating, bouncing and welding, arcto-glow transition have practical applications in design of vacuum circuit breakers
and other electrical contact systems.
Most of above models are based on the boundary-value problems for the
systems of partial differential equations of parabolic and elliptic types in domains
with free boundaries, and included the phase transformations such as melting and
evaporation. For their solution were elaborated, in particular, the method of
majoring functions, methods with using integral error functions, heat polynomials,
dual integral equations.
Results of the modeling can be used for development of new circuit breakers
and other electrical contact systems.
Researches, results of which are presented in the book, are supported by the
Grant 5133/ГФ4 – «Methods of special functions for the solution of free boundary
problems and their applications for modeling of heat and mass transfer at electrical
arc phenomena» awarded by the Science Committee of the Ministry of Education
and Sciences of Kazakhstan.
Established for publishing by Scientific Council
Of A.P. Ershov Institute of Informatics system SB RAS
ISBN 978-601-06-4797-8
UDС 517.58
© Kharin S.N., 2017
© A.P.Ershov Institute of Informatics
Systems SB RAS, 2017
Preface
5
Chapter 1 STATIONARY CONTACTS.
1 THE CONTACT SURFACE
1.1 Modeling of contact surface. Contact resistance.
1.2 Dynamics of contact spot when current passing
2 PHENOMENA IN CLOSED CONTACTS.
2.1 Joule heating
2.2 Thomson effect
2.3 Peltier effect
2.4 Kohler effect
2.5 Galvano-magnetic and thermo-magnetic phenomena
3 MODELING OF CONTACT HEATING
3.1 S Spherical contact model
3.2 Cylindrical model of ideal and non-ideal contacts
4 ELECTROMAGNETIC FIELD IN CONTACTS
4.1 Mathematical model.
4.2 Electric potential and constriction resistance.
4.3 The model of a contact with a short constriction
4.4 Influence of the skin effect.
5 STATIONARY CONJUGATED FIELDS OF THE
TEMPERATURE AND ELECTRICAL POTENTIAL
5.1 Ideal symmetric contact.
5.2 Bimetallic contacts
5.3 Heating of ideal contacts with Thomson effect
5.4 Special cases
5.5 The limit of maximal current
5.6 Non-ideal contact
6 NON-STATIONARY TEMPERATURE FIELD
6.1 Transient model of Joule contact heating.
6.2 Special cases.
6.3 The role of the arc flux and Joule heating in the erosion
of electrical contacts
7 BOUNCING AND WELDING OF ELECTRICAL CONTACTS
7.1 Static welding.
7.2 Dynamic bouncing and welding during contact closure
7.3 The model of blow-off phenomena
CHAPTER 2 MODELING OF THE BRIDGE TRANSFER
1 CYLINDRICAL MODEL OF A BRIDGE
1.1 Introduction
3
6
6
6
9
11
11
12
13
14
16
17
17
17
22
22
24
25
27
30
30
36
39
44
47
51
57
57
59
60
69
69
80
88
103
103
103
1.2 Pre-arcing stage.
2 GENERAL MODEL OF BRIDGE TRANSFER
2.1 Bridge phenomena.
2.2. Mathematical model
2.3 Results of solution.
2.4 The mechanism of bridge rupture.
2.5 Criteria of bridging.
2.6 Self-restrained bridge erosion.
CHAPTER 3 ELECTRICAL CONTACT ARC
1 GENERAL MODEL OF ELECTRICAL ARC IN THE FRAME
OF MHD-THEORY
1.1 Review of main theoretical results.
1.2 Phenomena in the arc column.
1.3 Phenomena in near-electrode zones.
1.4 Intra-electrode phenomena.
2 MODELS OF ELECTRICAL ARC EROSION
2.1 Introduction.
2.2 Vaporization.
2.3 Liquid droplets erosion
2.4 Thermo-capillary mechanism of arc erosion.
2.5 Solid state type of erosion
2.6 Arc-to-glow transition
2.7 Dependence of the arc erosion on the current frequency
3 ARC DURATION AND CONDITIONS OF ARC INSTABILITY
3.1 Mathematical model of arc temperature and conductivity
at metallic arc phase
3.2 Transition from metallic arc phase to gaseous arc phase
3.3 Phenomena in thermo-chemical cathodes
3.4 Arc to glow transition
3.5 Dependence of the arc erosion on the current frequency
REFERENCES
4
104
113
113
114
118
120
121
122
124
124
124
124
124
124
124
124
125
132
134
136
138
140
140
141
148
150
154
155
156
Preface
One of the main tendencies in the progress of modern low voltage apparatus is
the increase of their performance. The demand for creation of apparatus which
operate in micro and nanosecond ranges increases more and more. However
experimental investigations in this direction are faced with the problem of a
short duration of these processes. Besides it is possible often to obtain the
required information about the final result of a process but not about its
dynamics. In this case the mathematical modeling only is capable to get an idea
about evolution of a process in time which should be checked then
experimentally. The basic theory for the development of modeling phenomena
occurring in electrical contact is presented in various fundamental books The
stationary models of contact heating are considered and then generalized taking
into account the influence of Thomson and Kohler effects on contact heating.
However the demand to use extra-thin films in modern nanotechnology
generates the need to develop non-stationary models of contact heating where
the Kohler effect plays the main role.
Static welding of closed electrical contacts has been investigated in details It
was found that the probability of welding is proportional to the current pulse
duration. The explanation of this dependence can be given using the model of
the ring-shaped welding.
The theory of the bridge erosion presented early should be developed for the
high velocity of the contact opening. The investigation of transient bridge
phenomena and elaboration of non-stationary bridge models are extremely
important not only for the calculation of the fine bridge transfer but also for the
estimation of the initial contact gap of the arc ignition which defines the
duration of the arc metallic phase
Modeling of phenomena in the electrical arc is considered in many papers. It
should be noted that mathematical models are different for the short and for the
long arcs. A model for the long arc should be based on the equations of
magnetic gas dynamics describing phenomena in arc column, while the model
for a short arc describes processes in ionization zone, sheath and solid domains
of anode and cathode .
Mathematical modeling is very helpful to make clear dynamics of many arc
phenomena:
- transition from the metallic arc phase to the gaseous phase];
- various types of arc erosion such as evaporation of material from the
contact surface, the ejection of droplets of molten metal due to metallic
vapour or magnetic pressure, thermo-capillary mechanism of ejection of
liquid metal, ejection of solid particles of contact material when thermoelastic stresses overcome the breaking point;
- transformation arc-to-glow discharge];
5
- the influence of metallic vapours pressure on the blow-off repulsion of
closed contacts and on floating or bouncing with final welding at contact
closure;
- phenomenon of electrochemical electron emission
- non-monotonic dependence of arc erosion on the current frequency ].
The special analytical and numerical methods for the solution of these
problems are elaborated which are based on integral equations, special
functions, majorant functions etc. They can be used for practical calculations.
CHAPTER 1. STATIONARY CONTACTS
1.1 THE CONTACT SURFACE
1.1 Modeling of contact surface. Contact resistance.
Electrical contact between two current currying elements is formed as a
result of compressed mechanical force Pc called contact load.
One of main important characteristics of closed electrical contacts is the
contact area between them. It is well-known that whole geometrical surface of a
contact is so-called apparent (or nominal) contact area, while the true contact
area occurs only on separate spots, which total area are much less than apparent
area. Microscopic asperities, elevations and depressions on a surface with the
deep depressions remaining as voids in the contact stipulate it. Those usually
much spread voids have a negligible influence on the contact resistance. When
two surfaces are brought together, mechanical contact occurs only in those
regions where the surface asperities meet. Most metal surfaces are covered with
an oxide film, which must be ruptured before true metal-metal contact takes
place. As the force increases, the number and the area of these small metal-metal
contact spots will increase as a result of oxide film rupture and metal extrusion
through the ruptures. These spots, termed a -spots, are small cold welds
providing the only conducting paths for the transfer of electrical current. Current
passing across a contact interface is therefore constricted to flow through these
a -spots. Hence, electrical resistance of the contact due to this constricted flow
of current is called "constriction resistance" and is related to basic properties of
metal such as hardness and electrical resistivity.
The current, constricted to flow through narrow conducting path, will raise
the temperature of these paths Ts above that of bulk Tb . The contact spot
temperature (so-called "super temperature”) is related to the voltage drop across
the contact interface U as [1]
Ts 2  Tb 2 
6
U2
,
4L
(1.1.1)
where L is the Wiedemann - Franz - Lorenz constant with the value
2.45  108V  K 1 . Hence, a relatively small increasing in the contact voltage
drop U can considerably raise the contact spot temperature sufficiently to
produce basic metallurgical changes such as softening and melting of the
conducting areas.
The total contact resistance R of an interface where there are many
conducting spots is given by R. Holm [1] as
R

2na


,
2
(1.1.2)
where the first term refers to the constriction resistance of n spots with an
average radius a , while the second term is additional resistance due to
interaction between spots forming a cluster, calculated theoretically by J.A.
Greenwood [2] as
1
  aa 
    ai    i j  ,
2  i   i , j sij 

2
(1.1.3)
where ai and a j are the radii of individual spots and sij is the distance
between them.
In order to account for long-term degradation of stationary contacts,
R. Malucci [3] developed a model by introducing a third level of multi-spot
constrictions on the surface of each asperity. This model was used to simulate
contact degradation by fretting. It was shown that asperity deformation and film
growth are important variables in delaying fretting degradation. The model was
also used to provide statistical data on change in resistance as contact degrades.
However, the uncertainty of the parameter determining the fraction of the
oxidized contact region limits wider application of this model.
The constriction resistance and distribution of the asperities in the contact
zone are best described by the model of J.A. Greenwood and J.B.P. Williamson
[4]. This model relates the contact deformation to the topography of the surface
and establishes the criterion for distinguishing surfaces undergoing elastic or
plastic deformation. Based on this approach M.D. Bryant [5] used polynomial
and Gaussian probability density function to describe the degradation of copper
contact by oxidation.
Using fractal surface modeling and applying the analogy between thermal
and electrical constriction, M.T. Singer and K. Kshonze [6] developed a model
for the constriction resistance of random rough contacting surfaces. This model
correlates well with that of J.A. Greenwood and J.B.P. Williamson. The analogy
between thermal and electrical contact resistance was also used by R.A. Burton
and R.G. Burton [7] to describe the co-operative interaction of asperities in
sliding contacts.
It is worth mentioned that computer modeling, based on finite element
analysis, was also used to determine the contact resistance [8], [9]. However,
7
these methods, although very useful for solving particular practical problems,
are inadequate to describe the dynamic processes occurring in the contact zone.
Bearing in mind the above discussion, we shall keep the approach given in
[4] - [5], using, however a new modification. Hence, the total contact resistance
can be expressed as
(1.1.4)
R  Rc  Rm  R f ,
where Rc , Rm , R f are respectively the constriction resistance, the resistance of
asperities and the resistance of a film in a quasi-metallic part of a contact spot.
These resistances can be determined in terms of electrical conductivities  c ,
 m ,  f as
Rc 
1
1
1
, Rm 
, Rf 
,
c S
m S
fS
(1.1.5)
is the nominal contact area.
The total loud Pc acting on a unit area of the contact surface is calculated
in the Appendix 1:
where
S

z
 z2 
4 NE rm
3/ 2
Pc 
(
z

d
)
exp
  2 2  dz 
3   (1   2 ) d



3/ 2
2 rm 3/ 2
4 NE
(i erfc   i 3/2 erfc  * ) .
3
(1   ) 2
(1.1.6)
Hence, knowing the applied load Pc , the values for  ,  and d can be
derived from this transcendental equation.
The above-described model was used to calculate the initial contact
resistance of an aluminum-copper bolted connection with a nominal contact area
of S  33,33 106 m2 . The contact load was Pc  16 103 N . The surfaces of
aluminum and copper bus bars were first machined and then brushed with a
brush having a density of 1 106 wires  m2 [11]. Typical surface profile and height
distribution is shown in Fig. 1-1.
40
20
0
-20 0
-40
0,5
1
1,5
2
-60
-80
-100
Fig. 1.1. Typical surface profile of a brushed surface.Vert. scale = 25.786 A ,
Horiz. scale unit = 9.960 m .
8
The asperity heights are measured from an arbitrary reference plane. The
results of topographic analysis of several surface profiles yielded the following
values for
r0  5.8  105 m , N  16 ,   1.6  105 m , z  2 105 m .
By inserting these values in equation (1.1.18) and solving it with respect to
 , we obtain
  0.6  10-5 m , and d  1.4 105 m .
Using these values and equations (1.1.10), (1.1.13), (1.1.15) and (1.1.5),
the following values for the constriction resistance Rc , asperity resistance Rm
and film resistance R f are obtained
R f  4.5  106  .
Rc  2.3  106  ,
Rm  0.1  106  ,
The total contact resistance is then R  6.9 106  .
This value agrees well with the experimentally observed initial contact
resistance of a bolted aluminum-copper connection before being subjected to
current-cycling [11].
1.2 Dynamics of a contact spot when current passing
The expressions for the contact resistance obtained above correspond to the
elastic mechanism of deformations, and they can be used in a range of relatively
small contact loading, current and temperature. In the case of plastic or elasticplastic deformations the size of a contact spot is defined using the concept of the
contact hardness introduced by R. Holm [1]. It is equal numerically to the
average pressure in contact asperities (filaments), that is approximately three
times more then the pressure corresponding to the beginning of the plastic
deformation. It is necessary to take into account also decreasing of contact
hardness when the temperature of a contact surface  increases when current is
passing. In this case the formula (1.1.6) must be replaced by the expression
Pc ( n )  n  H B ( )dS ,
(1.1.7)
Sn
where Pc is the contact load applied to a single contact spot Sn , and  n is the
coefficient of compressibility in a neighborhood of a spot. This coefficient takes
into account the degree of a contact surface treatment , which is responsible for
its elastic-plastic properties. For the plastic deformation n  1 , while for elasticplastic deformation 0.3  n  1 . If the current is absent, then the summary value
of the elastic-plastic forces Pep , which is balanced by the total contact load Pc ,
can be found by summation of (1.1.20) with respect to n :
(n)
m
Pep  n  H B ( )dS ,
n 1
where
m
is the number of contact spots.
9
Sn
(1.1.8)
It will be shown below, that a contact spot S n is an equipotential and
isothermal surface for stationary phenomena, therefore one can derive from
(1.1.7)
H B ( )S ( )  const .
(1.1.9)
This relationship agrees well with experimental data.
Fig. 1-2 shows the temperature dependence of the hardness for copper,
silver and silver-nickel all
H B , 10 7 N / m 2
100
1
80
3
60
40
2
20
0
0
200
400
600
800
 , 0C
Fig. 1.2 Temperature dependency of hardness :
1  Cu,
2  Ag,
3  AgNi (0.1%Ni )
One can see that a hardness falls sharply just on reaching of the softening
temperature  s (about 200o C ). It decreases two-three times in the temperature
interval 200o C    500o C . Accordingly to the formula (1.1.9) the increasing of
the radius of the contact spot r at the temperature   500o C in comparison with
its value at the temperature   200o C is equal to the ratio

r (500)

r (200)
H B (200)
.
H B (500)
For copper   1.4 , for silver   1.65 , and for silver-nickel alloy   1.45 ,
that agrees well with the experimental data obtained by N.E. Lyssov [12], who
found that this ratio is equal to 1.5 for all these materials.
Increasing of the size of a contact spot in the process of heating is
accompanied by a rapprochement of contacts until elastic-plastic force is
compensated again with the contact load Pc . This motion is described by the
equation
d 2x
 Pep  Pc ,
dt 2
where m is the mass of moving contact, and x  x(t ) is the coordinate of a point
m
hard placed on it (Fig. 1.2).
The current area of the spot S (t )   r02 (t ) can be expressed in term of x(t )
by the formula
S (t )  S0 [1 
2 r0
( x  x0 )],
S0
S0  S (0)  2 r0 ( z  d ),
10
x0  x(0) .
(1.1.10)
I the range of a high current ( 103  104 A ) it is necessary to take into account
the electrodynamic force stipulated by interaction of electric and magnetic fields
in the current constriction region :
1
Ped  0  E  HdV ,
(1.1.11)

D
where  is the relative magnetic permeability, 0  1.257  106 H / m is the
magnetic permeability of vacuum,  is electric resistivity of the contact
material, E and H are vectors of electric and magnetic fields, D is the volume
of the contact. With due regard for this phenomena the equation of the contact
motion can be written in the form:
m
d 2x
 P  Pep  Pc ,
dt 2
(1.1.12)
where P is the component of electrodynamic force normal to the contact
surface.
Beside phenomena of contact material’s softening and electrodynamic
force other factors can influence on the change of a contact spot’s size, such as
fritting and additional repulsion force stipulated by explosive evaporation of
contact material at very high temperature on the contact spot. These phenomena
will be considered below.
1.2 PHENOMENA IN CLOSED CONTACTS
2.1 Joule heating
Heating of an electrical contact during passing current is a result of
interaction of multifarious phenomena. The most important factor of a contact
heating is undoubtedly the Joule sources in the constriction region. Specific
output q produced by these sources in the unit volume is defined by the formula
q
1

E2
(1.2.1)
The vector of electric field strength E can be expressed in terms of electric
potential  and current density j using expression
E   grad   j
(1.2.2)
Side by side with Joule sources thermoelectric effects of Thomson, Peltier
and Kohler stipulated by the relationship between electric and electromagnetic
fields can influence on the contact heating. It is concluded in numerous papers
devoted to these effects that their contribution in contact heating either
negligible [13] or can be important only for low current liquid contact bridges
[1], [14], [15]. It will be shown below that such conclusion made on the base of
rough estimations is not quite correct. The definition of the conditions when
11
either effect is essential can be obtained only in the frame of an adequate
mathematical model.
2.2 Thomson effect
Thomson effect becomes apparent in the case of uneven heated electrical
conductor. It consists in the additional heat transfer by electrical carriers from
more heated to less heated regions. This effect is positive if heat generation
occurs in a conductor, when the directions of current and heat flow are the same,
and negative when they are different. In other words the positive Thomson
effect acts as though heat transfer is proceed by the positive current carriers.
Otherwise the effect is considered as negative.
The estimation of Thomson effect is often based on the assumption that the
total quantity of the Thomson heat QT which was produced in a conductor with
the passing current I during the time t is proportional to the difference of the
temperatures at the ends of a conductor  , if this difference is not too large
[16]:
QT   T   I  t .
(1.2.3)
The sign of the Thomson coefficient  T coincides with the sign of the
effect. Experimental data show that the value of the Thomson coefficient  T
depends on the temperature, and this dependence is linear for most of metals.
The temperature dependence of  T is presented in Fig. 1.3 [1].
 T , 10 6 V / K
40
20
1
2
3
0
-20
4
-40
-60
5
6
-80
-100
7
0
2
1
3
 , 10 3 0 K
Fig. 1-3 The temperature dependence of the Thomson coefficient  T [1].
1 – Ag, 2 – Cu, 3 – Au, 4 – Mo, 5 – Pt, 6 – W, 7 – Pd
Applying this law to the element l of a conductor and dividing both sides
of (1.2.3) by the value of elementary volume S  l , where S is the area of the
cross-section of a conductor, one can obtain the formula
qT   T

j,
l
(1.2.4)
where qT is the specific power produced by the Thomson heat sources per unit
volume, and j is the current density.
12
Generalization of the formula (1.2.4) leads to the conclusion that qT is
proportional to the scalar product of the temperature gradient and electric
potential gradient
qT   T grad   grad   
T
E  grad  .

(1.2.5)
The difference between the formula (1.2.4), which is usually applied for an
estimation of the Thomson effect, and the exact formula (1.2.5) is evident.
Comparing the average value of the Thomson component of the electric field for
the whole constriction region ET   T  with the ohmic component E   j one
l
can find that ET E for all real range of considered parameters. Thus according
to the formula (1.2.4) the Thomson heating component should be negligible in
comparison with the Joule heating component. On the contrary the formula
(1.2.5) shows that small local zones may exist, where Thomson heating is
essential, if the temperature gradient into these zones is sufficiently large. This
situation is typical for non-stationary phenomena in the case, when a power of
high intensity is produced inside a small surface zone during very short time,
especially for materials with large Thomson coefficient. It will be shown in the
Chapter 4 that the Thomson heating may surpass the Joule heating at electrical
arcing phenomena between tungsten contacts for a current about 100 A.
2.3 Peltier effect
Peltier effect lies in the fact, that in the case of an electrical contact
between two different materials special heating or cooling takes place in
addition to the Joule heating. This thermo-electric effect can be explained by the
contact voltage at the interface between two different metals that decreases or
increases the average power of electrons crossing the contact.
Contrary to the Joule and Thomson heat sources, which act into a volume,
the Peltier sources are surface. Their specific power per unit area of the contact
spot  P is proportional to the current density j
 P  uP j .
(1.2.6)
The Peltier coefficient u P is positive for heating and negative for cooling
produced by Peltier sources. Similar to Thomson coefficient the Peltier
coefficient depends linearly on the temperature
u P  k S .
The coefficient of proportionality k S (Seebeck coefficient) is defined by the
expression
kS 
k n1
ln .
e n2
13
where k is the Boltzmann constant, e is the electron charge, n1 and n2 are the
number of free electrons per the unit volume of each contact metal.
The values of Seebeck coefficients for some contact pairs are given in the
Table 1.1.
Table 1.1 Seebeck coefficient k S ( 10 3V  K 1 ) for some contact pairs [17]
Metal
Contact
pair with
Cu
Fe
M
Mo
4.4
6.5
W
0.4
11.0
Zn Cu
0.1
0
11.0 11.4
Ir
1.1
12.0
Ta
2.6
14.5
Ni
Al
24.0 3.2
34.0
-
g
3.5
-
2.4 Kohler effect
The tunnel effect plays an important role in many fields of engineering
applications and modern technologies such as electric vacuum devices, semiconducting materials, super-conducting contacts, technology of thin films etc.
Various problems of tunnel conductivity, magnitude of tunnel current, tunnel
voltage across a contact film as well as properties of varied films have been
investigated and much progress made in the understanding of these
characteristics.
The problem of contact heating owing to tunnel effect is one of them, and it
was M. Kohler who discovered theoretically that an analogue to the well-known
Wiedemann - Franz' law holds for the electric and thermal currents through the
film [1]. It enables to estimate a possible temperature difference between
cathode and anode. Now this phenomenon of electrical contact overheating
owing to tunnel effect is known in the literature as Kohler effect.
Let us consider a circular contact spot between cathode and anode covered
with a thin (a few A ) chemisorbed or adhesive films that are penetrated by the
conduction electrons by means of tunnel effect. The electrons don't alter their
energy level when tunneling. Since they land in anode with a lower negative
potential than at the cathode, they have a surplus kinetic energy there. The
increment of kinetic energy is given off as heat creating the source on the
interface between anode and film with specific capacity
  j  u f  j 2  f ,
(1.2.7)
where j is current density in contact spot, u f is the tunnel voltage across the
film,  f is the electric tunnel resistivity per unit area of the film. A portion of
liberated heat with specific capacity
c 
f
,
W
(1.2.8)
where  f is the temperature difference and W is the specific thermal
resistance across the film, flows back to the cathode through the film whereas
the remainder
14
a   
f
W
(1.2.9)
flows into the anode (Fig. 1.4) .
 1 (r , z  0, )
r
 2 (r , z  0, )
d
Cathode
D1 :   z  0
Anode
D2 : 0  z  
f
a
c
z
0
f
r
Fig. 1.4. Cathode and anode heat flux ' across the contact film
Overheating of anode by tunnelling electrons is called Kohler effect. The
magnitude of tunnel resistance  f depends on the contact materials, particularly
on work function  , radius of contact spot f and thickness of the film d .
Specific thermal resistance of the film, W , is given by
W
d
f
,
(1.2.10)
where  f is the significant parameter of Kohler effect . However this formula is
of little use for practical calculations because of very poor information
concerning thermal conductivity of the film  f . But combining Kohler and
Wiedemann - Franz' laws  f  f    LT , where  f and  are electrical
resistivity of the film and the metal respectively,  is the thermal conductivity
of the contact metal, and using the relation  f   f d we can obtain instead of
(4) the expression
W 
f
,

(1.2.11)
that is much more efficient.
Further investigations carried out by R.Holm concerning the influence of
thin films on contact superconductivity [1] as well as results of other
investigators (I.Dietrich , P. Kisliuk, W. Meissner et al. ) led to the conclusion
that the contribution of tunnel effect to contact heating is visible only at low
current and small contact loading. It has to be noted that such estimations are
based on the integral balance of heat transfer between anode and cathode. So
they are rather rough and may be used successfully at the limited range of
current.
15
The modern tendency to use extra-low range of the current (about 10 6 A
and less) in many fields of engineering leads to the situation when investigators
are faced with new serious problems and phenomena that remain obscured at
ordinary current range but come to the forefront if the range is extra-low. The
Kohler effect is one of them. The overheating of anode in comparison with
cathode due to Kohler effect, which is not very important at ordinary conditions,
is essential at extra-low current, as it will be shown below. It leads to the
thermal asymmetry in the microscopic molten bridge that appears between two
electrodes during their opening just before arc ignition. If cathode and anode are
of the same material the point of the bridge with maximum temperature is
displaced toward anode, and when the temperature rises to the boiling point the
bridge breaks at this point, and molten metal transfers from anode to cathode.
After certain cycles of opening and closing operations one can see the formation
of very thin pips (or spires) on the cathode and craters on the anode. This
phenomenon, called bridge erosion, is very dangerous for micro-relays.
Reducing of the current diminishes the volumes of the spires but not their
height. Even the very thin spires are able to make sensitive micro-relays
unreliable.
The hypothesis that Kohler effect may be responsible for the bridge erosion
of electrical contacts was proposed by E. Justi and H. Schultz [18]. New aspects
of this theory and corresponding mathematical models are developed in the
papers [19] - [20]. and some methods for solution of the problem ( mainly
numerical ) are elaborated.
Mathematical modeling is very important for the understanding and
calculation of relative contributions of Joule and Kohler components of contact
heating and bridge erosion because experimental study of this phenomena is
very difficult, sometimes on account of microscopic size of a bridge (
10 6  10 8 m ) at extra-low current.
2.5 Galvano-magnetic and thermo-magnetic phenomena
If electrical contacts are placed in a sufficiently strong external magnetic
field H  , then following galvano-magnetic and thermo-magnetic phenomena
should be estimated.
Hall and Nernst effects.
These effects create additional transverse electrical fields
Ei H 
R0
 Ei  H  ,
i
N
Ei  Qi 0  grad i  H  ,
where R and Qi are Hall and Nernst constants.
Ettingshausen and Righi- Leduc effects.
16
(1.2.12)
(1.2.13)
These effects cause additional temperature gradients in transverse direction
grad E i 
Pi 0
 Ei  H  ,
i
R
grad i  S 0  H   grad i ,
(1.2.14)
(1.2.15)
where Pi and S are Ettingshausen and Righi-Leduc constants. In addition
electrical contact resistance may be changed also.
The problem appears now: is it possible that tangible galvano-magnetic and
thermo-magnetic phenomena can be emerged due to self-magnetic field creating
by current passing through contact?
This problem with regard to electrical arc phenomena has been discussed in
some papers, for example, in [21] and [13]. It was found that considered effects
should be taken into account for medium containing addition of bismuth or
antimony. However self-magnetic field of closed electrical contacts creating by
passing current is not sufficient to provoke a visible influence of considered
effects on the process of heat transfer. It can be explained by convincing
circumstance that for electric contact materials Hall constant R is multiple of
1010 m3 / A  sec , Nernst constant Qi is multiple of 108 m2 / sec K , Ettingshausen
constant Pi is multiple of 108 K  m2 / V  A  sec and Righi-Leduc constant is
multiple of 103 m2 / V  sec [22].
Electrical contact resistance can be appreciably changed in strong magnetic
fields only which have a strength of several tens or hundreds oersted [22].
Thus the influence of galvano-magnetic and thermo-magnetic phenomena
on the heating of closed electrical contacts can be neglected.
3. MODELING OF CONTACT HEATING
3.1 Spherical contact model
See Appendix 2.
3.2 Cylindrical model of ideal and non-ideal contacts
The mathematical equations describing non-stationary heat transfer
phenomena in electrical contacts have to be considered taking into account all
above mentioned effects. Owing to Joule and Thomson heat sources acting
inside the volume of electrodes, which are described by the formulas (1.2.1),
(1.2.5), the heat equation for each electrode can be written in the form
ci i
i

1
 div(i grad i )  Ti Ei  grad i  Ei 2 ,
t
i
i
(1.3.1)
where index i  1 is used for the cathode, which occupies the region D1 bounded
by the surface S1 , index i  2 corresponds to the anode occupying the region D2
17
with the boundary S 2 , ci is the thermal capacity,  i is the density, i is the
thermal conductivity,  i is the electrical resistivity,  Ti is the Thomson
coefficient,  i is the temperature, Ei is the electrical field, and t is the time.
If the Thomson effect is negligible, then simplified equation can be used
instead of (1.3.1):
ci i
i
1
 div(i grad i )  Ei 2 .
t
i
(1.3.2)
The system of Maxwell’s equations for electric field Ei and magnetic field
H i has to be added to the equation (1.3.1) or (1.3.2):
curl Ei   0 i
curl H i 
1
i
H i
,
t
Ei ,
(1.3.3)
(1.3.4)
div H i  0 .
(1.3.5)
The equations (1.3.1) – (1.3.5) are a bounded system because  i depends
on the temperature  i . The temperature dependence of magnetic permeability  i
is very small for diamagnetic and paramagnetic contact materials. The electrical
field Ei can be eliminated from the system (1.3.2) – (1.3.3), and it converts into
the form:
ci i
i
 div(i grad i )  i curl 2 H i ,
t
H i
 0 i
 curl( i curl H i ) .
t
Sometimes it is more convenient to introduce the electrical potential  i
instead of Ei using formula (1.2.2). Applying the divergence operator to the
equation (1.2.12) we obtain instead of (1.3.1) – (1.3.4) the following system of
the equations for variables  i and  i :
ci i
i

1
 div(i grad i )  Ti grad i  grad i  grad 2 i ,
t
i
i
1

div  grad i   0 .
 i

(1.3.6)
(1.3.7)
This general system of equations can be concretized and simplified in
dependence on the geometry of a contact spot. If we assume that the contact spot
is a circle with the radius r0 , then the axial symmetry takes place, and use of
cylindrical coordinates is most convenient. In this case the equations (1.2.14)
and (1.2.15) take the form:
18
ci i
i 1   i    i   Ti  i i i  


 i r
   i


+
t
r r 
r  z  z  i  r r
z z 
2
1  i 
i 2 
(
) ,

+ 
i  r 
z 
1   1 i    1 i 
 r
 
 0.
r r  i r  z  i z 
(1.3.8)
(1.3.9)
In the case of along constriction [1] characterized by the inequality f R0 ,
where R0 is the radius of the contact cross-section, electrodes can be considered
as two half-spaces. The cathode region (i  1) is D1 (  z  0, 0  r  ) , the
anode region (i  2) is D2 (0  z  , 0  r  ) , and the contact spot is
S ( z  0, 0  r  r0 ) (Fig. 1.6). For a quasi-metallic contact with a film of very
small thickness d (d r0 ) we can apply the model of non-ideal heat contact
through the spot S with the temperature difference  f across the film like the
difference between left-hand limit for  2 (r, z, t ) and right-hand limit for 1 (r, z, t )
as z  0 and 0  r  r0 :
 f   2 (r,0, t )  1 (r,0, t ) .
 1 (r , z  0, t )
Cathod
D1 : e   z
r
 2 (r , z  0, t )
Anod
d
0
D2 e: 0  z  
r0
a
c
z
0
r0
r
Fig. 1.6 The axial section of a contact with a film
Let us formulate now the boundary conditions. If  i is the supertemperature of an electrode, that means the difference between the electrode
temperature and surroundings temperature, the at the initial time t  0 (the
moment of current switching) we get
 i t 0  0 .
(1.3.10)
More general initial condition
 i t 0  F ( M i ) ,
M i  Di
may be used for consecutive stages of contact heating during commutations.
19
Due to Kohler and Peltier effects the surface heat source appears on the
anode side of the film with specific power
   f  P
(1.3.11)
where components on the right side are defined by the expressions (1.2.6) and
(1.2.8).
A portion of this heat with specific power
c 
f
Wf
,
(1.3.12)
where W f is the specific thermal resistance, flows back to the cathode through
the film whereas the reminder
a   
f
(1.3.13)
Wf
enters the anode.
Thus the boundary conditions for the temperature on the contact spot are
z  0,
1
0  r  r0 ,
2
1 ( r, 0, t ) 2 ( r, 0, t )  1 ( r, 0, t )
,

z
Wf
2 ( r, 0, t )
 ( r, 0, t )  1( r, 0, t)
.
 2
z
Wf
(1.3.14)
(1.3.15)
The boundary conditions for the potential at the same place are
z  0,
2 ( r, 0)  1( r, 0)  u f  uP ,
1 1 ( r, 0)
1  2 ( r, 0)

.
1 (1 )
z
 2 (2 )
z
0  r  r0
(1.3.16)
(1.3.17)
The condition (1.2.22) describes the potential difference across the film,
while (1.2.23) corresponds to the continuity of the current density passing
through the contact spot.
There is no heat and current transfer outside the contact spot, hence the
conditions of thermal and electrical insulation must be written
z  0,
1 ( r, 0, t )
2 ( r, 0, t )
 0,
0
z
z
1 ( r, 0)
2 ( r, 0)
 0,
 0.
z
z
r0  r  
(1.3.18)
(1.3.19)
Finally it has to be assumed that far from the contact spot
z   or
1  0,
r 
20
2  0
(1.3.20)
1  
uc
,
2
2  
uc
.
2
(1.3.21)
As a rule u f  uP uc , therefore significance of Kohler and Peltier
thermoelectric effects as volumetric heat sources is negligible in comparison
with Joule heating, however, Kohler effect as a surface heat source may be very
important. Sometimes this effect is responsible for for tunnel overheating and
thermal asymmetry of the constriction zone.
Specific thermal resistance of the film
Wf 
d
f
,
(1.3.23)
where  f is the thermal conductivity of the film, is the significant parameter of
Kohler effect. However the formula (1.2.28) is of little use for practical
calculations because of very poor information concerning  f . But combining
Kohler and Wiedemann-Franz laws
 f  f    LT ,
where  f is electrical resistivity of the film,  and  are thermal conductivity
and electrical resistivity of the electrode material, on which surface film is
formed, and using the relation  f   f d we can obtain instead of (1.2.28) the
expression
Wf 
f

,
(1.3.24)
that is much more convenient for practical calculations.
If we want to take into account Thomson effect not only in the metallic
constriction region, but inside the film itself, considering the film as a conductor
with the length d and the cross-section S   f 2 , then we have to add the special
heat flux due to Thomson effect
T   T j (2  1 ) z 0,
0 r  r0
additionally to the main heat flux  c entering the cathode due to heat
conduction. It seems as though Thomson effect increases the heat conductivity
of the film up to the value  Ti j . Thus to take into account Thomson effect inside
the film we have to introduce the concept of equivalent heat resistivity
Wf 
1
 /  f   T j
(1.3.25)
instead of the previous expression (1.2.29). It has to be noted that this correction
is very small as a rule.
Al observed thermoelectric effects are significant mainly for low current
contacts ( I  10 A) , when the radius of a contact spot is small (r0  10 5 m) because
21
of a small contact load. Particularly this case is typical for liquid metal bridges
appearing during contact separation. The same thermoelectric effects occur on
the interface between liquid bridge and solid electrode as on the interface
between two different metals.
The boundary conditions (1.2.20), (1.2.21) show that anode temperature
and cathode temperature are different even material of both electrodes is the
same, if there exists a film exiting tunnel effect between contacts. Thus above
formulated heat conduction problem is always asymmetrical about contact
plane. It is very important for the substantiation thermal theory of bridge erosion
which will be considered in the Chapter 3.
4 ELECTROMAGNETIC FIELD IN CONTACT S
4.1 Mathematical model
Mathematical models describing electromagnetic field in electrical contacts
are based as a rule on the hypothesis of constant current density in a contact
spot. It enables to reduce the problem of determination of components of electric
field to the solution of the Neumann’s problem for the Laplace equation which
can be found easily [15]. Such a model is approximate and can be used for
determination of contact resistance or average temperature in the contact
constriction domain only. However essential local overheating of periphery of
contact spot experimentally observed at high current at welding can be
explained only by nonuniformity of current density in contact spot. It is
stipulated by three factors, namely: 1) due to nature of current constriction, 2)
due to influence of cross-section of radius of conductor, 3) due to skin effect in
the case of alternative current.
To estimate each of above factors we consider first the case when two
R,
semi-cylinders
of
the
radius
and
D1 (0  r  R,    z  0)
D2 (0  r  R, 0  z  ) contact each other by a current-carrying circle spot
2
r 2  z 2  r0 . The axisymmetric vector of electrical field E has two components,
radial Er and axial Ez , while the vector of magnetic field H has the only angular
component H . It follows from the Maxwell equations (1.2.11)- (1.2.13) that H
satisfies the equation
H 
t
(i )
(i )

1    H     i 
(i ) 

(rH  )   .
  i
 
0   z 
z  r  r r
 
(1.4.1)
Index i  1 relates to cathode and index i  2 relates to anode.
The boundary conditions can be written in the form
H (r , z,0)  0 ,
(i )
22
(1.4.2)
H
(i )
 0,
z
(1.4.3)
z 0
0 r  r0
H
(i )
H
(i )

z 0
r0  r  R
r
H
z
I
2 r
,
(1.4.4)
 0,
(1.4.5)
0.
(1.4.6)
r R
(i )
z 
The conditions (1.4.2) - (1.4.6) are evident. In particular the condition
(1.4.4) describes Ampere's circuital law. The electrical field can be derived from
magnetic field using Maxwell equations:
I 2 i
qi ( r, z, r0 )  i ( jz  jr ) 
[2(2 2  s)( r02   2 )]1
2
2
2
or
qi ( r, z, r0 ) 
I 2 i
q0 ( r, z, r0 ) ,
16 2 r0 2
(1.4.7)
where

1  z 2  ( r  r0 )2
z 2  ( r  r0 )2
(1.4.8)
q0 ( r, z, r0 )   2


2
.
r  z  ( r  r0 )2
z 2  ( r  r0 )2

Near by the axis r  0 another representation of q0 (r, z, r0 ) is more
convenient
q0 ( r, z, r0 ) 
16r02
1

z 2  ( r  r0 )2 z 2  ( r  r0 )2 [ z 2  ( r  r0 )2  z 2  ( r  r0 )2
. (1.4.9)
It follows from the above expressions that the maximal current density and
power of Joule sources are placed on the edge of a contact spot z  0 , r  r0 ,
where they have an integrable singularity. Such contact model enable to explain
the phenomena of ring-shaped welding at high currents, while a model based on
a priori given constant current density in a contact spot is useless in this case.
The components of current density on the contact plane z  0 are
determined by the expressions
jr ( r,0, r0 )  0 , jz ( r,0, r0 ) 
I
2 r0 r02  r 2
, if r  r0 .
(1.4.10)
They can be derived from the relations
2
2

 r  r0 ,
lim   
z 0
0,


23
if
r  r0
if
r  r0
(1.4.11)
0,

 
r
lim

,
z 0 r
 r2  r 2
0

if
r  r0
if
r  r0
r0

  2 2 ,
  r0  r
lim
z 0 z

0,

if
r  r0
if
r  r0
Electrical field in the constriction region corresponding to the cylindrical
model is presented in Fig.1.8.
Accordingly to well-known    relation [1] the electrical field and the
temperature field for symmetric contacts in the stationary regime are the same.
The circle contact spot of radius f is an infinitely thin disk conducting a
current and a heat flow into a half-space electrode. In contrast to the spherical
model equipotential and isothermal surfaces (1) are not spheres, but ellipsoids
defined by the equation (1.4.14), while electrical and temperature gradient lines
form a family of confocal paraboloids of revolution.
z
2
1
r
r0
0
r0
r
Fig. 1.8 Electrical (temperature) field in the contact constriction region.
1 – equipotential (isothermal) surfaces; 2 – electrical (thermal) current lines
4.2 Electric potential and constriction resistance
To get more detail information about phenomena in closed contacts it is
useful to find also the distribution of electrical potential  i ( r , z ) and constriction
resistance Rc . Taking into account the conditions (1.2.27) one can get
1 ( r, z )  
 2 ( r, z ) 
z
z
uc
u I
1 
u
I 1  

  Ez (1)dz   c  1  2
dz   c 
  arctan  , 1.4.12)
2
2 
2 2  r0   z
2 2 r0  2
r0 


uc
u I
1

u
I 2  

  Ez (2)dz  c  2  2
dz  c 
 arctan

2
f
2 z
2 2 z r0   z
2 2 f  2
The constriction resistance of a half-space contact is
formula [22] :

Rc ( i ) 
2
( Er ( i ) H ) rdr .
z 0
I 2 0
From the relations (1.4.16), (1.4.22) one can derive
24

.

(1.4.13)
defined by the
Er
(i )
z 0
0
if


I i

, if

 2 r r 2  r 2
0

I i

, if r  r0

,
  2 r0 r0 2  r 2

0,
if r  r0

r  r0
r  r0
, Ez
(i )
z 0
(1.4.14)
therefore using (1.4.4)
Rc
(i )
i 
dr


 i .

2
2
2 r r r  r0
4r0
0
Constriction resistance of whole contact is
Rc  Rc (1)  Rc (2) 
1   2
4r0
.
(1.4.15)
For homogeneous contacts 1  2   and

Rc 
2r0
.
(1.4.16)
This formula enables to get the relation between current and contact
voltage and to express a potential in any point in their terms. Using conditions
(1.2.22), (1.2.23) and expressions (1.4.23), (1.4.24) we get
uc  u f  uP 
 uc
i ( r, z )  ( 1)i 
2

I ( 1   2 )
,
4r0
(1.4.17)

 
(uc  u f  uP )   arctan   , i  1,2 .
 1   2
r0  
2
2
i
(1.4.18)
It has to be noted that if contacts are homogeneous ( 1   2 ) and
thermoelectric effects are negligible ( u f  uP  0 ), then formula (1.4.29) in this
special case coincides with well-known expression for the contact potential [1]
 i (r , z )  (1) i
uc

arctan

r0
.
(1.4.19)
Finally we note that the expressions (1.4.17), (1.4.18) enables to conclude
about identity of current density in each contact Di , while specific power of
Joule sources qi is different and it is greater into electrode with greater value of
 i . Kohler and Peltier phenomena don’t influence on the symmetrical
expression for qi about the contact plane z  0 .
25
4.3 The model of a contact with a short constriction
Let us consider now the case of a short constriction, when the cross-section
radius of cylindrical contacting members R0 can not be negligible, and a
constriction is limited to a distance comparable with the radius of a spot f
(Fig. 1.9).
R 2  r0
R0
r0
0
2
r0
R0
Fig. 1.9 The model of a contact with a short constriction.
We can accept as a first approach that the expressions (1.4.7), (1.4.8),
(1.4.13), (1.4.14) describe the electromagnetic field only inside the region
bounded by the contact plane z  0 , 0  r  R0 and based on it ellipsoid
r2
z2

 1 . Outside this constriction region the magnetic field is radial,
R0 2 R0 2  r0 2
and the electric field is axial.
To check how well such model is working, let us compare the expression
for the axial component of electrodynamic repulsion force P calculated by use
of this model with the exact expression presented in [1]:
P 
0 I 2 R0
ln
.
4
r0
(1.4.20)
Formula (1.1.24) applying to above model gives
that coincides with (1.4.31)
If it requires more detail investigation of the influence of R0 on the
distortion of electromagnetic field, then one has to consider the equation (1.4.9)
with the conditions (1.4.3) – (1.4.6) in a bounded cylinder. Herewith the
magnetic field can be presented in the form of the Fourier-Bessel series:
26
H ( r , z )  
I
2

C
n 1
n

z  r
exp  n J 1  n  ,
r0   r0 

(1.4.21)
where n are the roots of the equation
J1 (n )  0 ,
and
Cn

r0
,
R0
can be found from the solution of the dual series equations:

C J
n 1
n 1
 n r  1

 ,
 r0  r

C  J
n 1
n n 1
 n r 

0,
 r0 
if r0  r  R0 ,
if 0  r  r0 .
(1.4.22)
Dual series equations of such type is considered in the paper [23]. Using
these results one can conclude that the coefficients C n can be determined by the
formula
Cn 
2 2

 2 
J 3/ 2 (n )
n
1

1
4

(1.368  0.216 2 )
(1.4.23)
with the error not exceeding  .
If R0   , then the series (1.4.32) can be summarized and reduced to the
expression (1.4.13).
5
4.4 Influence of the skin effect
Finally
we consider the case of alternating periodic current
J (t )  J 0 exp(it ) . Let us assume for simplicity that contacts are homogeneous
and symmetrical,   const , R0   .The equation (1.4.1) is reduced in this case
to the form
  2 H   1 

  2  
rH  ,
t
r  r r

 z
H
where  2 

0
2
(1.4.24)
. If a periodical current passes through contacts over a
sufficiently long period of time, then the equation (1.4.1*) has to be considered
without the initial condition (1.4.2) (it is assumed that   t   ), only for the
boundary conditions (1.4.3) – (1.4.6) with R0   .
The solution of this problem can be found in the form

H ( r , z , t ) 
J0
exp(it )  exp(   2  k 2 J 1 ( r )  ( )d  ,
2
0
27
(1.4.25)
where k 2  
i
2
, and  ( ) is unknown function determined from the conditions
(1.4.3), (1.4.4) , that gives the dual integral equations


 2  k 2 J 1 ( r )  ( )d   0 ,
if 0  r  r0 ,
(1.4.26)
0

1
 J ( r )  ( )d   r ,
if r0  r   .
1
(1.4.27)
0
If we represent a solution of the equations (1.4.36), (1.4.37) in the form
 sin t

 cos  t dt ,
 t

f
 ( )  J 0 ( f )    (t ) 
0
(1.4.28)
then the equation (1.4.37) is satisfied for any  (t ) due to the well-known
property of discontinuity of Weber’s integral. Replacing  ( ) in the equation
(1.4.36) by the expression (1.4.38) and using the formula
 r 2 J 1 ( r ) 
d 2
[r J 2 ( r )]
dr
we get

d
2  k2
 sin t

r  ( r )   r 2 J 2 ( r )
d    (t ) 
 cos t dt ,
dr 0

 t

0
f
2
where

 (r)  
0
2  k2
J 1 (  r ) d  .

Integrating both sides with respect to r from 0 to r , then interchanging
the procedure of integration and calculating the inner integral, we obtain the
equation
r

t 2 (t )dt
r2  t2
0
f
r
  r  ( r )dr   K0 ( r, t ) (t )dt ,
2
0
(1.4.29)
0
where

2  k2
1


0 



K0 ( r, t )  r
2

 sin  t

 cos t  d  .
J 2 ( r ) 

 t


(1.4.30).
The left side of (1.4.39) is an integral operator of the Abel’s type.
Reversion of this operator leads to the equation
f
 ( r )   K ( r, t ) (t )dt  F ( r ) ,
0
where
28
(1.4.31)
2 d
K0 (u, t )
r2  u2
du
2

 r dr 0
u
r
K ( r, t ) 
(1.4.32)
and F (r ) after simplification is

F (r)  
2 1 d
2  k2
d
.
J 0 ( f )r 5/ 2 J 5/ 2 ( r )
2

 r dr 0


(1.4.33)
The equation (1.4.41) belongs to the type of Fredholm integral equations of
the second kind with the kernel K ( r , t ) permissible for the representation of the
solution in the form of the series:

 ( r )   n ( r ) ,
0 ( r )  F ( r ) ,
n 1
(1.4.34)
f
 n 1 ( r )  F ( r )   K ( r, t ) n (t )dt .
0
Thus, the solution of the equation (1.4.1*) for the periodic alternating
current with the boundary conditions (1.4.3) – (1.4.6) is determined by the
expressions (1.4.35), (1.4.38) – (1.4.44). In the case of a faint skin effect
kf
1,
(1.4.35)
which often corresponds to a real conditions for electrical contacts, it is more
convenient to represent the solution in another form:
 ( ) 
sin f 
 sin t

   (t ) 
 cos  t dt .
f
 t

0
f
Taking into account that in this case g ( ) 
k2
2 2
and calculating
corresponding integrals one can find the following description for the magnetic
field:
H ( r , z , t )  

I 0 exp(it )  1 
z  k 2z
1


(2 2  2 z  z 2  r 2 )  .
 

2
 r    6r

(1.4.36)
If   0 , then this expression transforms into the formula (1.4.13).
Replacing H in the formulas (1.4.7), (1.4.8) by the representation (1.4.46) one
can find the components of electrical field, hence, the current density.
Particularly, the current density on the contact spot due to expressions (1.4.22) is
1 
I 0 exp(i t )  k 2 f 2 
3r 2 
jz ( r,0, t ) 
[rH ( r,0, t )]  
1 
1 
 .
r r
3  2 f 2 
2 f f 2  r 2 
(1.4.37)
The second term in square brackets of the right side corresponds to the
component of additional increasing of the current density at the edge of a
contact spot due to skin effect.
Let us estimate now the range of the correctness of the condition (1.4.45).
For non-magnetic materials   1 . Taking into consideration that
29
 ~107  108   m ,
0  1.257 107 H  m1 we conclude that the thickness of skin-
layer is
1


c



k
0
20

where c~0.03  0.10 .
For the current frequency   50Hz and f ~108  104 m one can find that
kf ~ 7  107  3  102 , thus the condition (1.4.45) is accomplished. The same
conclusion is true for ferromagnetics with the temperature above the Curie point.
Thus, influence of skin effect on the electromagnetic field can be important
in exceptional cases only, for example, for high-frequency current (  103 Hz ) or
for extra-high contact loads ( Pc ~104 N ) providing considerable size of a contact
spot ( f ~102 m ) .
Herewith one can conclude that an unevenness of current density on a
contact spot is stipulated mainly by the constriction phenomena, and skin effect
is of no importance for real conditions of electrical contacts commutation.
5 STATIONARY CONJUGATED FIELDS OF THE TEMPERATURE
AND ELECTRICAL POTENTIAL.
5.1 Ideal symmetric contacts
Stationary temperature and electromagnetic fields in symmetric electrical
contacts can be described by the differential equations
div( grad  ) 
1

grad 2   0 ,
1

div  grad    0 ,


where   T  T0 is the difference between current contact temperature T and
initial contact temperature T0 before heating,  ,  ,  are electrical potential,
heat conductance and electrical resistivity respectively.
In cylindrical coordinates these equations can be written as
1         1
 r
  

r r  r  z  z  
2
   2    

 
  0 ,
 r   z  
1   1     1  
r

0.
r r   r  z   z 
30
(1.5.1)
(1.5.2)
The boundary conditions are

z
 z 0,  0
0 r  r0

z  or
r 
 0,
(1.5.3)
z 0

z
z  0,
r0  r 
 0,
 z  or  uc / 2 .
 0,
r 
(1.5.4)
(1.5.5)
It has to be mentioned that this problem is essentially non-linear due to
temperature dependence of thermal conductivity    ( ) and electrical
conductivity    ( ) . The method of the solution can be obtained from the
suggestion that the identity of equipotential and isothermal surfaces in contacts,
which is correct for stationary fields in linear case, keeps safe for non-linear case
as well. In linear case these surfaces are ellipsoids of revolution
r2
z2

1 .
 2  r0 2  2
(1.5.6)
Therefore if  (r, z)  const is required equipotential and isothermal surface,
then electromagnetic and temperature fields can be characterized by the only one
coordinate  instead of two independent variables r and z . That is the
motivation for an attempt to find the solution of the equations (1.5.1) and (1.5.2)
in the form
(1.5.7)
   ( ) ,    ( ) ,    (r, z) .
Putting this expression in the equations (1.5.1)- (1.5.2) we get
[ ( ) ( )  ( )   ( )2   ( ) ( ) ( )2 ]  ( )2   ( ) ( ) ( )2  0 ,
(1.5.8)
  ( ) ( )   ( ) ( ) ( ) ( )2   ( ) ( )2  0 ,
(1.5.9)
 2 1   2
     
( )2       , 2  2 

.
r
r r z 2
 r   z 
(1.5.10)
where
2
2
Hereby, if a solution in the form (1.5.7) exists, then the ratio 2 /( )2
must be not dependent on the variables r and z , it should depend on the only
variable  . Indeed, using the relationship (1.5.6) it is easy to check that
 2
2
 2
.
2
( )
r0   2
31
(1.5.11)
This result is expected because a temperature dependence of the potential at
any point of electromagnetic field may change from physical point of view a
magnitude only, but not the configuration of equipotential surface, which
remains the same for both linear and non-linear problems.
Using (1.5.11) one can rewrite the equations (1.5.8) and (1.5.9) in the form
2
1
  ( ) ( ) 2 
 ( ) 2  0 ,
2
 ( )
r0  
2
 ( )  ( )   ( )  ( )  ( )   ( )  ( ) 2 2  0 .
r0  
 ( )  ( )   ( )  ( )
2
(1.5.12)
(1.5.13)
To write the boundary conditions in term of variable  note that


   ( ) ,
z
z


  ( ) .
z
z
(1.5.14)
Taking into account that

z
=
z=0
r0
if
r0  r 2
2
0  r  r0
and

z
if
=0
r0  r  
z=0
we can conclude that all boundary conditions (1.5.3)-(1.5.5) are satisfied if we
put:
for   0
for   
 (0)  0 ,
 (0)  0 ,
 ()  0 ,
 ( ) 
uc
2
.
(1.5.15)
(1.5.16)
(1.5.17)
(1.5.18)
Thus, the primary problem is reduced to the system of equations (1.5.12) –
(1.5.13) with the boundary conditions (1.5.15) –(1.5.18). We begin its solution
from the equation (1.5.13) rewritten in the form



 ( )
2
 ( )  2
 0.
 ( )
r0   2
Integration of this equation gives the relationship between   and  :
32
ln[ ( )]  ln{[ ( )]}  ln( r0   2 )  ln C ,
2
thus
  ( ) 
C  ( )
.
2
r0   2
(1.5.19)
The constant C can be defined using Ohm’s Law. The equipotential surface
(5.6) approaches the sphere of the radius  as    . If the potentials of two
such spheres with the radii  and   d are equal correspondingly to
 and   d , then accordingly to the Ohm’s Law
d  I  ( )
thus
  ( ) 
I  ( )
2 2
d
2 2
 .
as
Comparing this expression with (1.5.19) and calculating the limit as   
, we can find
C
I
2
,
(1.5.20)
thus
  ( ) 
I  ( )
.
2
2 (r0   2 )
(1.5.21)
Using this expression, we can transform the equation (5.12) to the form
 ( )  ( ) 
2
2
  I  ( )  0

(

)

2
2
r0   2
4 2 (r0   2 )
to solve this equation we use the substitution
  arctan
T ( )   ( ) ,

r0
(1.5.22)
reducing the functions domain      to the domain  / 2     / 2 and the
heat equation to the form
 (T )T ( )   (T )T ( ) 2 
I 2  (T )
0.
2
4 2 r0
The derivative T ( ) can be expressed from this equation as
dT


[2   (T ) (T )dT ]1/ 2
d  (T ) T ( )
T (0)
where
33
,
(1.5.23)
2 
I2
.
2
4 2 r0
(1.5.24)
This equation can be solved with respect to the variable 
 (T ) 
T (0)
1
 2

T (0)
 (T1 )[
T

 (T2 ) (T2 )dT2 ]1/ 2 dT1 .
(1.5.25)
T1
The temperature T ( ) can be found from this equation as an inverse
function, if the functions    (T ) and    (T ) are given.
Equating the heat flux passing through isothermal surface and specific
power generating by Joule heating we get

dT
1 d
 j 
,
d
 d
where j is current density. Thus

d d
d

  
d dT
dT
and integration yields well-known Kohlrausch’s temperature-potential relation
1/ 2
 T0

   2   dT 


 T

,
T0  T (0) .
(1.5.26)
However, the formula (1.5.25) has an advantage over (1.5.26) because it
enables one to get the information about the space distribution of temperature
field.
Special cases.
1. Thermal conductivity and electrical resistivity are constant.
 (T )  0  const ,  (T )  0  const
In this case the expression (1.5.25) becomes the form
 (T ) 
1
2


[T (0)  T ]
and the maximum temperature on the contact spot T (0) can be found from the
condition
34
T  0 if  

2
,
that gives well-known expression [16]
T (0) 
I 2 0
.
2
32r0 0
The expressions for spatial distribution of temperature and potential are
4
I 2 
4



T  T0  1  2  2  
 1  2 arctan  ,
r0 
 
 32r0  
I

arctan .
   
2 r0
r0
(1.5.27)
(1.5.28)
2. Thermal conductivity is constant, electrical resistivity is a linear function
of the temperature
 (T )  0 (1  T ) ,
 (T )  0  const
After integration of (1.5.25) we get
 (T ) 
0 0
1  T
arccos
 
1  T0
1
and the condition (1.5.26) gives the expression
T ( ) 
1 cos( 0 / 0 )
[
 1] ,
 cos(  /  )
0
0
2
(1.5.29)
which has been presented in [17].
The maximum temperature in this case is
T (0) 
1


sec(

0 / 0 )  1 .


2

(1.5.30)
The expression for electrical potential can be obtained by integration of the
equation (A5.21) using substitution (1.5.22). It gives
 ( ) 
00 sin( 0 / 0 )
.
 cos(   /  )
0
0
2
35
3.Thermal conductivity and electrical resistivity are linear functions of the
temperature.
 (T )  0 (1   T ) ,
 (T )  0 (1  T ) .
(1.5.31)
(1.5.32)
In this case to the second-order infinitesimal
 (T ) (T )  00[1  (   )T ]
and the expression (1.5.25) can be integrated
 (T ) 
0 (   )


0
1


1  (   )T
  arccos
  [1  (   )T0 ]2  [1  (   )T ]2 
1  (   )T0


(1.5.33)
The expression for the temperature T ( ) can be obtained by the solving of
the equation (1.5.33) using the condition  (0)   / 2 for determination of T0 .
5.2 Ideal asymmetric contacts
In the case of bimetallic contacts the equations (1.5.12) and (1.5.13) should
be considered for each contact separately. In term of variable  these equations
can be written as
i (T )Ti( )  i(T )Ti( )2   2i i  0 ,
i ( )  i ,
where i  1 if  / 2    0 and i  2 if 0     / 2 .
(1.5.34)
(1.5.35)
The boundary conditions on the contact spot (1.5.15)-(1.5.16) have to be
replaced by the conditions of continuity of heat flux and current density
(1.5.36)
T1 (0)  T2 (0) , 1T1 (0)  2T2 (0) ,
1 (0)  2 (0) ,
1
1
1 (0) 
1
2
 2 (0) .
(1.5.37)
The conditions (1.5.17)-(1.5.18) should be changed for the new conditions
T1 ( / 2)  0 ,
T2 ( / 2)  0 ,
1 ( / 2)  UC / 2 , 1 ( / 2)  UC / 2 .
(1.5.38)
In asymmetric contacts the temperature maximum Tm is displaced from the
contact surface having temperature T0 toward the contact which material has
36
greater value of i i . Let us assign the index
(Fig. 5.2)
i  2 for this contact material
T
Tm
T0
T1
T2


2
 
2
0 m
Fig. 5.2 Temperature distribution in asymmetric contacts
The solution of the problem (1.5.34)-(1.5.38) is:
For the domain  / 2    0
 
1

T0
T0
Tm
1/ 2
 1 (2  11dT  2  2 2 ) dT ;
T
T
(1.5.39)
T0
T0
T
0
U
1 ( )   C  (2  1 1dT )1/ 2 (2  1 1dT )1/ 2
2
0
T
for the domain 0     m

1
Tm
T
 (2    dT )

2
T0
2
1/ 2
2
dT ;
T
 2 ( ) 
(1.5.40)
T
T
m
m
UC
 (2  2  2 dT )1/ 2 (2  2  2 dT )1/ 2
2
0
T
and for the domain  m     / 2
 m 
1
Tm
Tm
 (2    dT )

2
T
T
2
2
1/ 2
dT ;
(1.5.41)
T
T
m
m
U
 2 ( )  C  (2  2  2 dT )1/ 2  (2  2  2 dT )1/ 2 .
2
0
T
The values T0 and Tm can be found from the system of equations

2
T0
T0
Tm
0
T
T0
  1 (2  1 1dT  2  2  2 dT ) 1/ 2 dT
37
(1.5.42)

Tm
Tm
T0
T
  2 (2  2  2 dT ) 1/ 2 dT 
2
Tm
Tm
0
0
(1.5.43)
  2 (2  2  2 dT ) 1/ 2 dT
The temperature maximum occurs at the point
m 
1
Tm
Tm
 (2    dT )

2
2
T0
1/ 2
2
(1.5.44)
dT
T
In terms of cylindrical coordinates ( r , z ) the axial displacement of
temperature maximum in axial direction at r  0 is
(1.5.45)
zm  r0 tan  m .
In the case of linear temperature dependence of i and i (1.5.31), (1.5.32)
integration of the expressions (1.5.39) - (1.5.41) yields for the domain
 / 2    0
  1 A1 (arcsin
10

 arcsin 1 ) 
s
s
(1.5.46)
 B1 ( s  10  s   )
2
2
2
2
1
for the domain 0     m
   2 [ A2 (arcsin

2
 arcsin 20 ) 
2 m
2 m
(1.5.47)
 B2 ( 2 m  2  2 m  20 )
2
2
2
2
and for the domain  m     / 2
   m   2 ( A2 arccos
2
2
2
 B2 2 m  2 )
2 m
(1.5.48)
where
i 
i  1  (i  i )T , i 0  1  (i  i )T0 , im  1  (i  i )Tm ,
s  10 2 
20 20 1  1
2
2
(2 m  20 ) ,
10 10  2   2
38
Ai 
i
i  i
,
1

Bi 
i 0
,
i 0 (i  i )
i
i  i
.
5.3 Heating of ideal contacts with Thomson effect
Let us consider the problem of stationary temperature and electromagnetic
fields in closed heterogeneous contacts taking into account the Thomson effect.
We assume that specific electrical resistivity  i and Thomson coefficient  Ti are
linearly dependent on the temperature  i :
i  i 0 (1  ii ) ,
 Ti   Ti 0 (1  ii )
(1.5.49)
but thermal conductivity i is constant, because its temperature dependence is
not so essential and can be taken as average for considered temperature interval.
The system of equations for temperature  i and potential  i can be written in
the form
i i 
 Ti
1
i  i  2i  0 ,
i
i
div(
1
i
i )  0
(1.5.50)
In the cylindrical coordinate system these equations take the form
(1  ii )(
 i 1 i  2i

 i i i

 2 )  Ti 0 (1  ii )( i

)
2
r
r r z
i i 0
r r
z z


[( i )2  ( i )2 ]  0
i 0 r
z
(1.5.51)
1
(1  ii )(
 2i 1 i  2i
   

 2 )  i ( i i  i i )  0 , i  1,2 .
2
r
r r z
r r z z
(1.5.52)
The index i  1 corresponds to the cathode region D1 (0  r  ,   z  0)
and the index i  2 corresponds to the anode region D2 (0  r  ,0  z  ) .
The boundary conditions for electrical potential and temperature can be
written as
1  
2 
(1.5.55)
uc
,
2
uc
,
2
if
if
r2  z2   ,
r2  z2   ,
z0
z0
(1.5.54)
1 (r, 0)  2 (r, 0)
1
1
2

10 [1  11 ( r, 0)] z  20 [1  22 ( r, 0)] z
1
1 ( r, 0)  2 ( r, 0)

z
z
i  0 ,
(1.5.53)
if
r2  z2   ,
39
(1.5.56)
(1.5.57)
i  1,2
(1.5.58)
1 (r,0)   2 (r,0) ,
1
r  r0
1 (r ,0)
 (r ,0)
 2 2
,
z
z
1 (r ,0)
 0,
z
(1.5.59)
r  r0
 2 (r ,0)
 0,
z
(1.5.60)
r  r0
(1.5.61)
It has to be mentioned that this problem is essentially nonlinear because of
equations (1.5.51), (1.5.52), (1.5.56).
In terms of variable  we get
(1  ii )i 
 Ti 0
1
2
(1  ii )ii 
(i )2  (1  ii )i 2
0
i i 0
i i 0
f 2
(1  ii )i  iii  (1  ii )i
2
0
f 2
(1.5.63)
2
To write the boundary conditions in term of variable
(1.5.62)

note that
i

z
f 2r 2

i 
 i ( )
 ( 1) i ( ) [1  2
]
z
z

( f   2 )2  f 2r 2
(1.5.64)
Taking into account the relations (1.4.22) we can conclude that the
conditions (1.5.13) and similarly (1.5.9) are satisfied automatically at such
choice of the function  (r, z ) .
The relation (1.5.22) at z  0 , r  f takes the form
i
z
i (0) 
z 0
f
f 2 r
,
thus the condition (1.5.12) and similarly the condition (1.5.8) are reduced to the
form
1
10 [1  11 (0)]
1 (0) 
1
 20 [1  22 (0)]
11 (0)  22(0)
2 (0)
(1.5.65)
(1.5.67)
Finally, instead of conditions (1.5.5), (1.5.6), (1.5.7), (1.5.10), and (1.5.11)
we may write the corresponding conditions in term of  :
40
1 ( )  
uc
2
(1.5.68)
uc
2
1 (0)  2 (0)
 2 ( ) 
(1.5.69)
(1.5.70)
1 ()  2 ()  0
(1.5.71)
1 (0)  2 (0)
(1.5.72)
Thus, the primary problem is reduced to the system of equations (1.5.20) –
(1.5.21) with the boundary conditions (1.5.23) – (1.5.29). We begin its solution
from the equation (1.5.21) rewritten in the form
i
i

ii
2
 2
0
1  ii f   2
Integration of this equation gives the relationship between i and i :
i ( ) 
Ci (1  ii )
f 2 2
(1.5.73)
The constants C i can be defined from the following consideration. The
equipotential surface (1.5.18) approaches the sphere of the radius  as    .
If the potentials of two such spheres with the radii  and   d are equal
correspondingly to i and i  di , then accordingly to the Ohm Law
di  I i 0 (1  ii )
thus
i ( ) 
d
2
I i 0 (1  ii )
2 2
as
  
Comparing this expression with (1.5.30) and calculating the limit as
   , we can find
Ci 
I i 0
2
thus
i ( ) 
(1.5.74)
I i 0 1  ii
2 f 2   2
It has to be noted that such definition of a constant
automatic satisfiability of the condition (1.5.23).
(1.5.75)
Ci
involves an
Now the equation (1.5.20) can be reduced using (1.5.32) to the form
41
i  
I Ti 0
I 2
1  ii
2

(1  ii ) 2 i 2  2 i 0
 2
 0
2
2 2
2 i
2i
f 
4 i ( f   )
f 
Let us try to find the solution of this equation in a form
i ( )  i ( ) ,
  arctan
1
i
(1.5.76)
f
 i (  ) are
Then the equations concerning
 i  i 2 (i 

)  2 i (1  i  i ) i  0
(1.5.77)
where
i 
I
2f
 i0 i
,
i
i 
I Ti0
4i f
(1.5.78)
It will be shown below that Thomson effect is not so important for heating
of closed contacts in comparison with Joule effect, thus the coefficient  i can be
considered as a small parameter. An approximate solution of the nonlinear
differential equation (1.5.34) can be found in this case using the method of
Krylov-Bogolubov [24] :
i ( )  
1
i
 exp(  i )( Ai cos i  Bi sin i )
where
 i   i (1 
The constants Ai and
(1.5.24), (1.5.28), (1.5.29):
Ai 
Bi
(1.5.79)
i
)
i
are determined from the boundary conditions
1 r1

r

1 1
[ exp(  1 )  2 exp(   2 )  ( 1)i (  ) 3i
 1
2
2
2
1 2
1

i
i
Bi  ( 1)i { exp[( 1)i  i ]cos ec
 Ai cot
}
i
2
2
2
(1.5.80)
where
i  ri [cos
  1   2 ,
ri  ii sin
3i
2
i

i
 ( 1)i i sin
],
2
i
2
The potentials i ( ) can be found now by integration of the equation
(1.5.32) with the boundary conditions (1.5,25), (1.5.26):
42

u I
1  11 ( )
u
I
1 ( )   c  01 
d    c  10
2
2
2
2  f  
2 2 f

(


[1  11 ( )]d  
 / 2


1
uc I 101 1

{( A1  1 B1 )[exp( 1 ) cos 1  exp( 1 ) cos

2
2
1
2
2 f   1
2
2
(1.5.81)
1

1
A1  B1 )[exp( 1 )sin 1  exp( 1 )sin
]}
1
2
2
Similarly
 2 ( ) 
uc I  20 2  2


2

{( A2  2 B2 )[exp(   2 ) cos

2
2
2
2 f    2
2
2
2
 exp(  2 ) cos 2 ) 
(
(1.5.82)
2

2
A2  B2 )[exp(   2 )sin
 exp(  2 )sin 2 ]}
2
2
2
where
  arctan

.
f
(1.5.83)
Using the expressions (1.5.34) and Ohm Low it is not difficult to check that
the functions (1.5.38) and (1.5.39) satisfy the condition (1.5.27).
Thus, an approximate solution of the problem (1.5.3) – (1.5.13) is
determined by the formulas (1.5.30) – (1.5.40), where  is given by the
expression (1.5.18).
It has to be noted that the temperature maximum is displaced from the
contact plane to anode or cathode due to heterogeneous properties of electrode
materials and due to thermoelectric effects. Such a displacement can have an
influence on a shape of contact liquid bridge, zones of plasticity and contact
welding, as well as on other phenomena accompanied with structural phase
changes, particularly, melting of contact materials which arises inside of an
electrode. If the position of isotherm  0 where the temperature reaches its
maximum is found, then one can analyze parameters determining this position.
Thereby, it enables to predict the course of contact heating phenomena and to
operate it in a certain degree by a corresponding choice of contact materials.
The value  0 can be found from the extremum test for the function
(1.5.36). It gives the formula
0 
1
 i i
arctan
 Ai   i Bi
,
 i Ai  Bi
(1.5.84)
where  i  1  i / i and the index i  1 or i  2 has to be chosen corresponding
to the material with the greater value  i .
43
The constriction resistance Rc is determined by the expression
10 0 1  11
 20  1   2 2
Rc 
 f 2   2 d  2 0 f 2   2 d 
2 
10


 20


[ A1 sin 1  B1 (cos 1  1)] 
[ A2 sin 2  B2 (cos 2  1)]
2 f 1
2
2
2 f 2
2
2
(1.5.85)
5.4 Special cases
Let us consider some special cases when it is available to get the exact
solution of the problem.
1. Thomson coefficient is constant.
(1.5.34) takes the form
 i  2 i i  i (i 
1
i
In this case i  0 , the equation
)0
(1.5.86)
and its solution is determined by the same expressions (1.5.33) – (1.5.34) for the
temperature and (1.5.38) – (1.5.39) for the potentials, in which  i and i must
be replaced with  i and i 1 
 Ti 0
4 i 0i i
respectively.
2. Thomson effect is negligible.
In this case  Ti 0  0 , i  0 and
expressions (1.5.36) – (1.5.39) converts into the formulas
1
i ( )  
where
Ai 
Bi 
i
 Ai cos i  Bi sin i ,
i  1,2
(1.5.87)
1 11



1
1
[
sin 2  2 2 sin 1  ( 1)i (  )  3i ,
 1
2
2
2
1  2
3i3i

[(
1
1


1

1

1  cos 2  cos 2  cos 1 ],
2
2
2
1
2
2
2


  1   2 ,
 i  ii cos i  sin 3i
1
) cos

2
and
1 ( )  
2 ( ) 
2
uc


 1011 [ A1 (sin 1  sin 1 )  B1(cos 1  cos 1)]
2
2
2
uc


  2022 [ A2 (sin 2  sin 2 )  B2 (cos 2  cos 2 )]
2
2
2
These formulas become more convenient for calculation, if we introduce
into consideration the temperature and potential in the contact spot:
44
11
1 22
2
tan

tan

4
2
4
0  1 (0)   2 (0)  1
1
2
11 cot
 22 cot
2
(1.5.88)
0  1 (0)   2 (0)  
2
1
uc
2
 1011 (  0 ) tan

1
2
4
2
u
2
 c   20 22 (  0 ) tan
2
2
4
In this case
 i ( )  
1
i

(1  i0 )sin {i [

2
 ( 1)i  ]}  sin [i ( 1)i  ]
i sin
i
(1.5.89)
(1.5.90)
2
i  1,2
i ( )  0  i 0ii [ Ai sin 
i  Bi (1  cos 
i )]
(1.5.91)
The constriction resistance of the region bounded by the contact surface
and the isotherm  is determined from the expression
( 1)i
Ri ( ) 
[i ( )  0 ] ,
I
i  1,2
(1.5.92)
3.Homogeneous contacts. Thomson effect is negligible.
In this case 10  20  0 , 1  2   , 1  2   , 1  2   ,  Ti  0 and the
expressions (1.5.44) have the simplest form
( ) 
1 cos 
(
 1)
 sin 
uc
 0
1


(sin   sin
)
2
 cos 
2
2
u

1

 2 ( )  c  0
(sin
 sin  )
2
 cos 
2
2
1 ( )  
(1.5.93)
(1.5.94)
(1.5.95)
These expressions coincide with well-known formulas presented in the
paper [25]. The constriction resistance bounded by the contact surface and the
equipotential surface  is determined in this case as
45
R( ) 
 0 sin 
.
2 f  cos 
(1.5.96)
2
The constriction resistance of one contact member (    / 2 ) is
R( / 2) 
0

tan
,
2 f 
2
(1.5.97)
and the total resistance is double.
4. Heterogeneous contacts.
resistivity is constant.
In this case
 Ti  0 ,
i ( )  
i  0 .
Thomson effect is negligible.
 20   10
If
I 2 i 0
 2  Ai  Bi ,
2
2
8 i f
Electrical
, then
R
1   2
,
2
(1.5.98)
where
I 2  20 10
Ai 
(

),
 1  2 32 f 2 2 1
2
i
0 
I 2 (  201  102 )
B1  B2 
32 f 2 (1  2 )
 (  201  102 )
4  20 (1  2 )
The greatest displacement of the temperature maximum isotherm  0 does
not exceed the radius f of the contact spot, because  0 

4
If the temperature interval is relatively small (or
constriction resistance R can be written in the form
R
10  20
2f
1
where i 0  Ai 
i

.
 i is
small), then the
2 10110  20220 1012  202 1

(20  10 )
3
2f
4 f (1  2 )
(1.5.99)
is the value of the temperature on the contact spot.
5. Heterogeneous contacts with Thomson effect. Electrical resistivity is
constant.
In this case i  0 , and the equation (1.5.36) transforms to
i  2i (1  i i )i   i  0
where
46
(1.5.100)
i 
I 2 i 0
4 2 f 2i
The substitution
i ( )  yi ( xi )
(1.5.101)
where
xi
yi ( xi )  exp{ i  [1  i  i ( )]d } ,
xi  Pi ,
Pi  ( i i i )1/ 3
(1.5.102)
0
reduces this equation to the form
d
1 d2
{
[ yi ( xi )]}  1  0 ,
dxi yi ( xi ) dxi 2


2
P1  x1  0 ,
0  x2 

2
P2
The general solution of the last equation can be expressed in terms of the
Bessel functions:
2
2 3/ 2
yi ( xi )  vi [Ci J 1/ 3 ( vi 3/ 2 )  DY
)] ,
i 1/ 3 ( vi
3
3
vi  xi  Ei
(1.5.103)
The constants Ci , Di , Ei are determined using conditions (1.5.24), (1.5.28),
(1.5.29) which can be written due to equations (1.5.30), (1.5.58), (1.5.59) in the
form
1P12 
2 P2 2

[ y (0)  y1 (0)] 
,
11 1
 2 2
1 P1 
1 P
[ y1 (0)  1]  [ 2 y2 (0)  1]
1 1
2  2
(1.5.104)
y1( 

2

y2 ( P2 )   2 P2
2
P1 )  1P1 ,
with two additional conditions
yi (0)  1 ,
i  1,2
(1.5.105)
arising from the expression (1.5.59).
The potentials  i are determined by the previous expressions
u
I
i ( )  {( 1) c  i 0
2 2 f

i

[1  i  i ( )]d
(1.5.106)
( 1)i  / 2
5.5 The limit of maximal current
It has to be noted that all above derived formulas make sense only if i  1 ,
in other words
I
i
 0  
2f
i 0i
(1.5.107)
47
Such a rather unusual constraint can be explained by a temperature
dependence of electrical resistivity , which can stipulate the increasing of the
contact voltage uc even to infinity, but it leads to unlimited increasing of the
temperature and the constriction resistance while the value of the current
remains limited by the inequality (1.5.64). Let us prove this inequality. We
consider for simplicity the case of symmetrical homogeneous contacts without
Thomson effect. The equation (1.5.20) for the temperature  (  ) both for the
cathode as well as for the anode can be written in this case as
(1   )  
1
0
 2  (1   ) 
2
0
f 2
(1.5.108)
2
The solution of this equation is given by the expressions (1.5.50), (1.5.40):
 ( ) 
1

cos ( arctan
[

cos

f
)
 1]
(1.5.109)
2
where

 0

I
2f
The expression (1.5.67) for
it
 ( ) 

(1.5.110)
was found using the equation (1.5.30), that
C (1   )
f 2 2
(1.5.111)
and the constant C was found from the physical consideration using the formula
(1.5.34). Let us find it now directly from the boundary conditions for the
potential, which can be written in this case due to the contact symmetry as
 (0)  0 ,
 ( ) 
uc
.
2
(1.5.112)
Putting the expression (1.5.68) in the equation (1.5.65) and solving this
equation we obtain the same formula (1.5.66), where

C
f

0 
Integration of the equation (1.5.68) using formulas (1.5.66) and (1.5.69)
gives the expression for the potential
 ( ) 
C
f
sin ( arctan
cos

2

f
)

 0

sin ( arctan
cos

Thus
 ( ) 
therefore
48
uc
 0


tan
,
2

2
2

f
.

2

arctan(

) ,
 0
uc
2
(1.5.113)
and
C f
 0 2
u
arctan ( c
 
2

)
 0
(1.5.114)
It can be derived directly from the expression (1.5.70) that   1 , therefore
the inequality (1.5.64) is proved. In that way there exists the maximum value of
the current for every given radius of the contact spot, which is defined by the
properties of the contact material. However this value never reaches because it
corresponds to infinite temperature, while increase of the radius of the contact
spot occurs already at the softening temperature and the more so at the melting
temperature.
The ratio of the current and diameter of the contact spot corresponding to
the maximum value
 I 

 2 f   0    

 max
0
as well as corresponding to the melting temperature  m
 I

2f


1
   m  2
arccos
 0
1   m
m
is given for some materials in the Table 1-1.
The values  0 and  m could be really rather greater because of a heat
emission from the contact surface.
It has to be noted that if the temperature dependence of electrical resistance
is ignored the current limitation will not occur. This conclusion is confirmed by
the formula
 ( ) 
I 2 0
4

(1  arctan 2 )
2
32 f 

f
(1.5.115)
that can be derived from 1.5.66) as limit for   0 . This fact testifies to a
limited applicability of the formula (1.5.72). It means also that the correct
mathematical statement of the contact heat problem with temperature
dependence of contact resistance involves the voltage as input instead of the
current.
TABLE 1-1. Limit of maximal current

0
Material
Al
Cr
Fe
Ni
10
8
m
2.9
20.0
10.0
8.0
10
3
K
1

1
10 W  m  K
2
4.0
2.0
6.5
5.0
2.10
0.67
0.60
0.70
49
1
0
m
106 A  m 1
106 A  m 1
4.23
1.29
0.96
1.32
3.42
1.09
0.90
1.21
Cu
Zn
Mo
Ag
Cd
Sn
Ta
Au
W
Pt
Pd
Pb
1.75
6.1
5.8
1.65
7.5
12.0
14.0
2.3
5.5
11.0
10.8
21.0
4.0
3.7
4.5
4.0
4.0
4.5
3.0
4.0
5.0
3.8
3.3
4.0
3.80
1.10
1.40
4.18
0.90
0.64
0.54
3.10
1.90
0.7
0.7
0.35
7.33
2.19
2.30
7.92
1.72
1.08
1.13
5.78
2.62
1.29
1.39
0.64
6.45
1.63
1.59
6.87
1.23
0.73
1.05
5.07
2.53
1.18
1.25
0.46
It is appropriate to mention here that the solution of the problem (1.5.3) (1.5.13) could be obtained using a formal change of the cylindrical co-ordinates
(r , z ) for the ellipsoidal co-ordinates ( , ) , which are bounded up by the
relationship
Fig. Ellipsoidal coordinates
r2
z2

1
2  f 2 2
r2
2

(1.5.116)
z2
1
f 2  2
(1.5.117)
50
The ellipsoidal co-ordinates are determined by the family of isothermal
ellipsoids of revolution and orthogonal family of hyperboloids of one sheet that
are given by the equations (1.5.73) and (1.5.74). From these equations one can
derive the inverse relationship
r

f 2  2 ,
f
-     ,
z

f 2  2 ,
f
.
0   f
The contact spot given in cylindrical co-ordinates by z  0, 0  r  f is
replaced in ellipsoidal co-ordinates by   0,   f , while the remaining part of
the contact plane z  0, r  f is replaced by   f ,   0 . The solution of the
problem can be obtained after such change of co-ordinate systems using method
of separation of variables.
One of the most important theoretical results of this section is the
conclusion that the temperature of an ideal contact spot (even nonhomogeneous) is the same at stationary regime for the all area of the contact
spot z  0, 0  r  f . It can be derived from the expressions (1.5.29)-(1.5.36) at
  0 . It will be shown below that this conclusion is not valid for a non-stationary
regime of contact heating, when temperature maximum is displaced to the edge
of the contact spot, where current density is maximal.
The temperature distribution along contact spot is also not uniform in the
case of a non-ideal contact.
5.6 Non-ideal contact
The temperature distribution into cathode 1 (r, z ) and into anode  2 (r, z )
occupying the cylindrical half-spaces D1 (  z  0,0  r  ) and
D2 (0  z  ,0  r  ) respectively are described by the heat equations
 2 i 1  i  2 i j 2



0

r 2 r r z 2
(1.5.118)
with i  1 for D1 and i  2 for D2 .
The current density j is determined by the expression [5]
j 2  jr  j z 
2

G 
2
I2
1
,
 G G (G   G  ) 2
z  (r  f )
2

2

(1.5.119)
2
Because of very small thickness of contact film we apply the model of nonideal contact through the circular spot S 0 ( z  0,0  r  f ) with the temperature
difference  f across the film like the difference between left-hand limit for  2
and right-hand limit for  1 when z  0 and 0  r  f :
 f   2 (r,0)  1 (r,0)
(1.5.120)
51
The boundary conditions for such non-ideal contact may be obtained from
(2), (3), (6) as z  0,0  r  f :
 1 (r ,0)  2 (r ,0)

z
W
 2 (r ,0)
 (r ,0)  1 (r ,0)
 2
 2
z
W
1
(1.5.121)
(1.5.122)
There is no heat exchange outside the contact spot, hence
z  0, f  r   :
 1 ( r,0)  2 (r ,0)

0
z
z
(1.5.123)
 i (r , z ) denotes the difference between contact temperature and
If
surroundings temperature , then far from contact spot
z  , or z  , or r   :
(1.5.124)
1   2  0
Solution. To solve the equations (6) with the conditions (8) - (12) we
introduce new unknown functions u(r, z ) and v(r , z ) instead of previous
1 (r , z ) and  2 (r, z ) by substitution
(1.5.125)
u ( r , z )   2 ( r , z )   1 ( r , z ) ,
v ( r , z )   2 ( r , z )   1 ( r , z )
where z  0 .
If we change z for  z in the equation (6) for i  1 , afterwards add and
subtract left and right parts of these equations for i  1 and i  2 and take into
account the equality
j 2 (r , z )  j 2 (r , z ) , then for u (r , z ) and v(r , z ) we obtain equations :
 2 u 1 u  2 u 2 j 2



 0 , r  0, z  0

r 2 r r z 2
 2 v 1 v  2 v


0 ,
r  0, z  0
r 2 r r z 2
After addition and subtraction the boundary conditions
reduced to the conditions for u(r, z ) and v(r , z ) :

u (r ,0)  if

z
0 if
(1.5.126)
(1.5.127)
(9) - (12) are
r f
r f
(1.5.128)
2

v(r ,0)   v(r,0) if r  f


W
z

if r  f
 0
uv0 ,
if r  
or
52
z
(1.5.129)
Hence we obtain two independent boundary problems (14) , (16) , (18)
for u(r, z ) and (15) , (17) , (18) for v(r , z ) . The solution of first of them is
already known [5] :
u (r , z ) 
f


 exp( xz ) J
0
( xr )  J 1 ( xf )
0
dx

x
I 
4

(1  2 arctan 2 )
2
f
16 f 

2

where
1
2
(1.5.130)
s  s2  4 f 2z2 ,
s  z2  r2  f 2
The solution of the second problem may be represented in the form

v(r , z )   A( x)  exp(  xz )  J 0 ( xr )dx
(1.5.131)
0
To satisfy the boundary condition (17) we get the system of dual integral
equations for unknown function A(x) :

2
 ( x  W )  A( x)  J
0
( xr )dx 
0

, 0r  f
2
(1.5.132)

 A( x)  J
0
f r 
( xr ) xdx  0 ,
0
After substitution
f
A( x)    (t ) cos( xt )dt
0
system (21) can be reduced to the Fredholm integral equation for  (t ) :
2
 (t ) 
f 2 t2 

f
4
ln
 W 0
f 2 t2 
f 2  t1
t 2  t1
2
2
 (t1 )dt1
(1.5.133)
The kernel of this integral equation has a week singularity at t  t1 ,
therefore it may be solved by the standard Picard's method.
It is available at the frame of this approach to get the analytical expression
for the Kohler overheating in the center of contact spot. From (20) at z  0 and



0
0
v(r ,0)   A( x) J 0 ( xr )dx   J 0 ( xr )dx   (t ) cos( xt )dt 
from (22) we get r

0
hence
0
 (t )
1
r2 t2
v(0,0)   (0) 

2
dt  
0
 (rx)
1 x2
dx
.
53
To find  (0) we rewrite the equation (22) in the form
 (t ) 
2
f 2 t2 

(1.5.134)

4f 1
 2 1   2  x2
 (tx )dx
  ln
 W  0
1 x2
Since
lim
1
   ( t 0 )


 ln
 2 1   2  x2
1 x
0
t  0 we obtain
 (0) 
2f


2
v(0,0) 
k 

2
hence from (23) at
1
1  2 f /( W )
and
where
  (tx )dx   (0) 
f

k
1 
,

(1.5.135)
2f
W
(1.5.136)
Then from (19) we find
I 2
u (0,0) 

  k  2 J
 16 f 2 
f
(1.5.137)
where
I 2
J 
32 f 2 
(1.5.138)
Finally from (13) we can obtain the temperature in the center of contact
spot on the cathode and on the anode
k 
,
2 1 
 2
 2 (0,0)   J  k
2 1 
 1 (0,0)   J 
(1.5.139)
Thus temperature difference across the contact film in the center of the
contact spot is
 f (0)   2 (0,0)  1 (0,0) 
k
1 
(1.5.140)
It has to be noted that the magnitude  k is equal to the temperature that the
cathode should be heated by tunnel mechanism only without Joule sources and
heat transfer to the cathode. In contrary the magnitude  J is the maximum
temperature at the center of contact spot at Joule mechanism of heating only
without tunnel effect.
54
Criterion of tunnel heating. To estimate the influence of Kohler effect on
the contact heating we introduce the new criterion Ko , called Kohler criterion,
that is the ratio of film resistance R f   f /(f 2 ) and contact resistance without
film ( so called constriction resistance ) Rc that is determined by the wellknown formula [2] Rc 

.
2f
Hence we introduce the Kohler criterion as
Ko 
Rf
Rc

2 f
(1.5.141)
 f
It defines the level of additional heating of anode contact surface owing to
tunnel effect in comparison with the Joule heating of constriction zone. In terms
of criterion Ko the formula (1) can be written as  
where I is the current, hence
k 
Therefore
Ko 
f


16

I 2 f
2f 4
Ko   J
 k
16  J
Further from (5) , (30) , (25)
W  Ko 
,
(1.5.142)
(1.5.143)
f
, 4 1
2
 Ko
(1.5.144)
and using the formulas (28) we get
1 (0,0)   J [1  1 ( Ko)]
 2 (0,0)   J [1  2 ( Ko)]
(1.5.145)
where
1 ( Ko) 
32
1
 1  4 /(  Ko)
16
1  2 /(  Ko)
 2 ( Ko)  Ko 

1  4 /(  Ko)
2
(1.5.146)
The tunnel overheating is
1
2
 f (0)   2 (0,0)   1 (0,0)   J  Ko  1 ( Ko)
(1.5.147)
The functions 1 ( Ko) and 2 ( Ko) are the ratio of tunnel and Joule
components for cathode and anode heating respectively. It is easy to calculate
that if Ko  0.02 , then tunnel heating of the anode is less 10% compared with
Joule heating ; if Ko  0.28 they are equal ; if Ko  1 , then tunnel component
of anode temperature is 10 times more than Joule component.
55
It is interesting to note that this ratio is limited for the cathode by the value
32 /  2  3.03 even if Ko   . It equals to 0.1 and 1 at Ko  0.04 and Ko  0.57
respectively .
Comparison with experimental data . Experimental data were obtained by
R.Holm and I.Dietrich [2] for Pt contacts at the conditions :
I  10 A , f  0.5  10 5 m ,   60W  m 1  K 1 ,
  9.5  10 7 Ohm m ,  f  1.3 10 12 Ohm  m 2
The measured temperature difference across the film was  f  176 o C .
Formulas (30) - (35) give
 J  2000 o C ,  k  1770 o C ,   7.34 , Ko  0.177
1 (0,0)  2780o C ,  2 (0,0)  2970o C ,  f  190 o C
One can see that coincidence of theoretical and experimental data is
satisfactory.
As it may be seen the tunnel overheating at this current is not so much.
Unfortunately other experimental data concerning temperature measurements
are still absent in the literature. But information about measured values of film
resistance enables us to conclude that at low current ( all the more at extra-low
current ) with small contact load the Kohler effect may plays main part in
contact heating.
Experimental data for the measured values of contact resistance for Au
contacts [2] , given in the Table 1 (columns 1 - 4 ) , and corresponding
calculated values of Ko criterion ( 5-th column ) lead to the conclusion that at
low current ( corresponding to low contact load ) tunnel heating is more much
than Joule one.
Table 1. Dependence of constriction resistance Rc , film resistance R f ,
and Ko criterion on the contact load N .
f
N
10 3 kg
10
400
35
11
1.15
8.6
5.0
6.1
4.6
13
Ohm  m
Rc
2
4
Rf
Ko
4
10 Ohm
10 Ohm
1.16
3.94
7.0
22.0
0.28
1.86
7.2
52.0
56
0.24
0.47
1.08
2.36
6 NON-STATIONARY TEMPERATURE FIELD
6.1 Transient model of Joule contact heating
For the temperature averaged values of  and  the equations (9) and (10)
take the form
T
1
 a 2 ( T 
grad 2  )
t

  0
(1.6.1)
(1..6.2)
where a 2   / C is the thermal diffusivity of contact material, C is the
thermal capacity, and  is the density. In cylindrical coordinates r, z
 2T 1 T  2T


r 2 r r z 2 ,


grad 2  ( )2  ( )2
T 
r
z
If a contact junction is symmetrical, then the domain occupied by one
contact member is cylindrical semi-space 0  r  , 0  z   . Since the boundary
conditions at this stage remain the same like (A.1.3) - (A.1.5) in the Appendix 1,
the expression for potential  (r, z) remains the same as well. Thus it can be
defined by the expression (A.1.28), where the current should be now timedependent:
 ( r, z ) 
I (t ) 

arctan
2 r0
r0
(1.6.3)
where    (r, z, r0 ) is the expression obtained from the equation (1.6.4)
 ( r, z, r0 ) 
1
s  s 2  4r0 z , s 2  r 2  z 2  r02
2
(1.6.4)
In principle substitution the expression (A..3.3) in the equation (A.3.1)
enables one to get a standard solution of linear heat equation with the boundary
conditions (A.1.3)-(A.1.5) and zero initial condition in the form
t
T ( r, z, t ) 


2
dt1  dz1  G( r, r1, z  z1, t  t1 )  grad 2  ( r1, z1 )rdr
1 1
C 0 
0
where G(r, r1 , z, t ) is the Green’s function defined by the expression
57
(1.6.5)
G( r, r1 , z, t ) 
1
4a 3
r 2  r12  z 2
rr
exp( 
) I 0 ( 12 )
2
3
4a t
2a t
t
(1.6.6)
2
However integrand grad  in (A.3.5) has a singularity at the point
z  0, r  r0 , and it creates a problem at the numerical calculation of the triple
integral (A.3.5). From physical point of view this singularity means that current
j (r , z ) 
1
grad 2
on the contact spot z  0 approaches infinity as r  r0 .
density
That is follows from the expression

j ( r,0) 
I
2 r0 r0 2  r 2
, r  r0
(1.6.7)
which can be easily derived from (A.1.3) and (A.1.4).
To avoid this singularity we use the relation
grad 2  
1
 2
2
which is valid for any harmonic function, in particular for  . Then the
substitution
u T 
1
2 
2
(1.6.8)
enables one to reduce the heat equation (A.3.1) to the form
u
1  2
 a 2 u 
t
2C t
(1.6.9)
which has no longer singularity.
The equation (A.3.9) at corresponding boundary conditions can be solved
using Laplace transform.
Finally we can write for T (r, z, t ) the expression
T ( r, z, t )   ( r, z, t ) 



0
  dz1  G ( r, r1 , z  z1 , t ) ( r1 , z1 , t ) r1dr1 
t


0

0
2  dt1  dz1  G ( r, r1 , z  z1 , t  t1 )
(1.6.10)

 ( r1 , z1 , t1 )r1dr
t1
where  (r, z, t ) is the corresponding quasi-stationary solution (stationary solution
with t as parameter) defined by the expression
58
 ( r, z, t ) 
I 2 (t ) 
4
 ( r, z, r0 )
[1  2 arctan 2
].
2
32r0 

r0
(1.6.11)
 ( r, z, r0 ) and G( r, r1, z, t ) are defined by the expressions (A.3.4), (A.3.6).
The another equivalent form of this solution
T ( r, z, t )  
 ( r, z, r0 )
I 2 (t ) 
arctan 2

2 2
8 r0 
r0




I 2 (t ) 
2  (r , z , r )
 dz1 0 G(r, r1, z  z1, t ) arctan 1 r01 0 r1dr1 
8 2 r0 

t
16r0 
2



0
[ I 2 (t )  I 2 (0)]  
0
(1.6.12)
I (t1 ) I (t1 ) 
dt1 
16r0 2
  dz1  G ( r, r1 , z  z1 , t  t1 ) arctan 2
 ( r1 , z1 , r0 )
r0
r1dr
can be derived from (A.3.1) by integration using the formula


 dz  G(r, r , z  z , t )rdr  1
1
1

1
1
1
0
In particular the temperature at the centre of the contact spot is


I 2 (t ) 
T (0,0, t )  2 2  dz  exp(  r 2  z 2 ) 
4 r0   0
 arctan 2
t

0
 (2a tr, 2a t z, r0 )
r0

rdr  

16r0 2
[ I 2 (t )  I 2 (0)]
(1.6.13)

I (t1 ) I (t1 ) 
dt1  dz  exp(  r 2  z 2 ) 
16r0 2

0
 arctan 2
 (2a t  t1 r, 2a t  t1 z, r0 )
r0
rdr
The integrals in (A.3.10)-(A.3.13) have no singularity now and can be
calculated easily using MathCad.
6.2 Special cases.
1. I (t )  I  const . In this case the formula (A.3.11) gives
T ( r, z, t )  

1
2

1
2
 2 ( ) 
(1.6.14)

 dz  G(r, r , z  z , t )
1

1
1
0
59
2
[ ( r1 , z1 , r0 )]r1dr1
and from (A.3.13)

T (0,0, t ) 
 arctan

I 2 (t ) 
dz exp(  r 2  z 2 ) 
2 2 r0 2 0 0
2  (2a tr , 2a t z , r0 )
r0
(1.6.15)
rdr 
If I (0)  0 , then

 arctan 2


I (t1 ) I (t1 )dt1  dz  exp( r 2  z 2 ) 
2
r0  5/ 2 0
0
0
t
T (0,0, t ) 
 (2a t  t1 r, 2a t  t1 z, r0 )
r0
rdr
(1.6.16)
If I (t )  I 0 sin t , then

 arctan

I 0 2 
sin 2t1dt1  dz  exp(  r 2  z 2 ) 
2r0 2 5/ 2 0
0
0
t
T (0,0, t ) 
2  (2a t  t1 r , 2a t  t1 z , r0 )
r0
(1.6.17)
rdr
6.3 The role of the arc flux and Joule heating in the erosion of electrical
contacts
Let us consider now the dynamics of heating of opening contacts by
volume and surface thermal sources. In this case power losses consumed for the
phase transformations (melting and evaporation) will be neglected. The
temperature and electromagnetic fields will be described by the system of
equations

1
 a 2 (  
grad 2  )
t

  0
(1.6.18)
(1.6.19)
with the initial and boundary conditions

 (r, z,0)  0
 (r , 0, t )
r
 P(r , t )
r
2
 z 2 
0
(1.6.20)
1.6.21)
(1.6.22)
(r , 0)  0,
0  r  r0
(r , 0)
 0,
z
r0  r  
(1.6.23)
Here we use the same notation as the previous part, with the averaged
values of  ,  and a2   /(c ) , c is the heat capacity,  is the density of the
contact material.
60
We represent the temperature in the form of two components
 (r, z, t )  1 (r, z, t )  2 (r, z, t )
(1.6.24)
where 1 (r, z, t ) is the temperature component corresponding to the volume Joule
heating, and 2 (r, z, t ) is the surface temperature component.
The function 1 (r, z, t ) is the solution of the problem (23)-(28) for P(r, t )  0 ,
and 2 (r, z, t ) is the solution of the same problem (23)-(28) for grad  0 .
Since (r, z) does not depend on the temperature in this case it can be
defined by the formula (14), i.e.
(r , z )   
I 2

arctan
2 r0
r
(1.6.25)
This function satisfies the Laplace equation (24), i.e. it is a harmonic
function for which
1
  2 
2
1 (r , z, t ) introduce the new unknown
grad 2 
Thus if we instead
u (r , z, t )  1 (r , z, t ) 
1
2
function u(r , z, t )
 2 (r , z )
(1.6.26)
Then we get for u(r , z, t ) the problem
  2u 1 u  2u 
u
 a2  2 


t
r r z 2 
 r
u (r , z,0) 
1
2
(1.6.27)
 2 (r, z)
(1.6.28)
u (r , 0, z )
0
z
I 2
u r 2  z 2   u 
2
32ro 
(1.6.29)
(1.6.30)
The last formula coincides with (12).
The boundary conditions (34)-(35) can be obtained using the formulas
r0

  2 2 ,
lim
  r0  r
r  r0 z 0 z 
0,
u( r,0, t )
1
 ( r,0)

 ( r,0)

z
z
 r 2  r 2 ,
0
lim   
z 0
0,
r  r0
0,

 ( r,0)   I 
r 2  r0 2
arctan
,

r0
 2 r0
 I
 ( r,0)  2 r

0
z
0,

1
r0  r 2
2
,
r  r0
61
r  r0
r  r0
r  r0
r  r0
r  r0
The solution of the problem (32) – (35) is
u( r, z, t )  u 

1
4a 3  t 3/ 2

 1

2
 dz1 0  2  (r1, z1 )  u  
 r 2  r12  ( z  z1 )2 
 rr 
I 0  12  exp  
 r1dr1
4a 2 t
 2a t 


Taking into account that

 r 2  r12  ( z  z1 )2 
 rr1 
3
3/ 2
dz
I
exp
 r1dr1 4a  t
 1 0 0  2a2t  
4a 2 t


we obtain
u( r, z, t ) 
1
8a 3  t 3/ 2



0
2
 dz1   (r1, z1 ) 
 r  r  ( z  z1 )2 
 rr 
I 0  12  exp  
 r1dr1
4a 2 t
 2a t 


2
2
1
Thus finally
1 (r , z, t ) 
1
2

 2 (r , z ) 
8a 3

1
2
 dz1 0  (r1, z1 ) 
 t 3/ 2 
(1.6.31)
 r  r  ( z  z1 ) 
 rr 
I 0  12  exp  
 r1dr1
4a 2 t
 2a t 


2
2
1
2
For small values of t this formula can be rewritten in more convenient
form
1 (r , z, t ) 

8a 3

1
2
2
 dz1 0  (r1, z1 )   (r, z)  
 t 3/ 2  
 r 2  r12  ( z  z1 ) 2 
 rr 
I 0  12  exp  
 r1dr1
4a 2t
 2a t 


(1.6.32)
It is not difficult to check that
1 (r , z, )  u 
1
2
 2 (r , z ) 
I 2 
4
2  
1  2 arctan 
2
r0 
32r0   
(1.6.33)
that coincides with the obtained above formula (13).
Thus the solution for large t can be represented as the small deviation from
the stationary solution
62
1 (r , z, t )  1 (r , z, )  u 

8a 3
1
 t 3/ 2 




dz1   2 (r1 , z1 ) 
0
 r 2  r12  ( z  z1 ) 2 
 rr1 
I 0  2  exp  
 r1dr1
4a 2t
 2a t 


u 
(1.6.34)
I 2
 1 (0, 0, )
2
32r0 
As is well known, at the initial stage of Joule heating the temperature
maximum occurs at the edge of the contact spot r  r0 , where the current density
is maximal, then it moves toward the centre of the spot, and finally at the
stationary regime whole contact spot becomes isothermal. Let us investigate this
motion of the isotherm r  rm (t ) on the contact spot z  0, 0  r  r0 where the
temperature reaches its maximum:
1 (rm (t ), 0, t )  max  (r, 0, t )
0 r  r0
We have
1 (r , 0, t ) 

1
8a 3  t 3/ 2

2
 dz1   (r1, z1 ) 

(1.6.35)
0
 r r z 
 rr 
I 0  12  exp  
 r1dr1
4a t
 2a t 


2
2
1
2
2
1
Thus the temperature at the centre and at the edge of the spot are
I 2
1 (0, 0, t ) 
2
32 5/ 2 a3 r0 t 3/ 2


 dz  arctan
2
1

0
1
r0

 r12  z12 
 exp  
 r1dr1
2
 4a t 
1 (r0 , 0, t ) 
I 2
2
32 5/ 2 a 3 r0 t 3/ 2




dz1  arctan 2
0
1
r0

 r r z 
 rr 
I 0  0 21   exp   0
 r1dr1
4a t
 2a t 


2
2
1
2
2
1
where
1 
1
2
2 2
S1  S1  4r0 z1 ,
2
S1  z1  r1  r0
2
2
2
If we use the substitution of variables in the integrals
r1  r0 x,
z1  r0 y
then they can be represented in the form
1 (0, 0, t ) 
I 2
 ( )
2 1
32 r0
1 (r0 , 0, t ) 
I 2
 1( )
2
32 r0
63
(1.6.36)
(1.6.37)
where
1 (  ) 
8

5.2


2
( x
 dy  arctan   x, y,    e

1( ) 
2
 y2 )
xdx,
0
8


 5/ 2 

dy  arctan 2   x, y,    
0
e 2  x I 0 (2  x ) exp  ( x   ) 2  y 2  xdx,
1
S ( x, y ,  )  S 2 ( x , y ,  )  y 2
2
x2 y 2
S ( x, y ,  )  2  2  1
 ( x, y ,  ) 


The graph f the functions 1 ( ) and 1 ( ) is represented in Fig. 3 and Fig. 4
Fig.3 Graph of the function 1 ( )
Fig.4 Graph of the function  1 ( )
The desired isotherm r  rm (t ) is determined from the maximum condition
64


 r
 r ( t ) r  r (t )  r (t ) r  
2
 dz1 0  (r1, z1 )   2a12t I1  m2a 2t 1   2ma 2t I 0  m2a 2t 1   
 r 2 (t )  r12  z12 
 exp   m
 r1dr1  0
4a 2t


After substitution
x
r1
,
2a t
y
z1
2a t
we get
I ( ) 




dy   2 ( x, y )   xI1 (2vx )  vI 0 (2vx )  
(1.6.38)
0
 exp  v ( )  x  y  xdx  0
2
2
2
where
  2a t ,
v( ) 
rm (t )
.
2a t
Expanding the left side of (41) with respect to  and equating the
coefficients at the same degree we get for small values of 
v( ) 
r0



2a
 o(1)
(1.6.39)
where  can be found from the transcendent equation
lim
 0
dI
0
d
(1.6.40)
Taking into account the asymptotic formulas
I 0 (2vx )  exp( 2vx ), I1 (2vx)  exp( 2vx) 
1
2  vx
, if   0
we get
lim I ( )  lim
 0

 0


 dy   ( (v  x ), y ) 
2
1

0
x1 (v  x1 )
 exp   x12  y 2  dx1  0
2  v(v  x1 )
One can derived from (42)
rm (t )  r0   t  o( t )
(1.6.41)
From the formula (40) we can obtain the expressions for the temperature at
the edge ( r  r0 ) and in the centre) of the contact spot ( r  0 ) if the time if
sufficiently small:
1 (0,0, t ) 
2 u
,
2 2
4 u
1 (r0 , 0, t )  3/ 2  ,
 
r
I 2
  0 , u 
2
32r0 
2a t
65
(1.6.42)
(1.6.43)
(1.6.44)
These formulas can be applied for   1, or in the terms of Fourier criteria
Fo 
a 2t
 1 .
2
r0
The comparison the graph for 1 ( ) (Fig. 3) with the asymptotic formula
(45) enables us to conclude that they both give the same value for   2 ,
1 (2)  0.05 . For the values   2 the formula (45) gives more exact result than
the graph in Fig. 3 due to an inaccuracy of calculation of the integral. Thus it is
recommended to use the graph only for 0    2 , and the asymptotic formula
(45) for   2 .
The magnified graph of 1 ( ) from Fig. 3 is represented in Fig. 5.
The same conclusion remains correct for the asymptotic formula (46)
which should be applied for   2 < while for the range 0    2 one can use the
graph.
Fig. 5 The graph of 1 ( ) for 0    2
It should be noted that in the range of very high current, opening velocity
and contact forces the temperature maximum at the edge of a contact spot may
reach the melting point with further formation of a hollow liquid bridge and
specific arc [9] – [10]. The asymptotic formula (46) may be very useful to
estimate this phenomenon.
Let us consider now the second temperature component 2 (r, z, t )
conditioned by the arc surface flux. It satisfied the heat equation (23), where
grad  0 , and the conditions (25) – (27). The solution of this problem can be
written in the form [11] – [12]
 2 (r , z, t ) 
t
1
2a 

e

P(r1 , t1 )

(t  t1 )3/ 2
0
 dt1 
0
r 2  r12  z 2
4a
2


rr
( t t1 )
I0  2 1
  r1dr1
 2a (t  t1 ) 
(1.6.45)
We consider two most typical radial distribution of the arc heat flux:
66
a) the constant flux along the contact spot
 P0  const ,
P(r , t )  
0,
0  r  r0
(1.6.46)
r  r0
b) the flux with the normal Gaussian distribution
P(r , t )  Pa (t )  e

r2
ra 2 ( t )
(1.6.47)
where ra (t ) is the effective radius of he entering heat flux. Sometimes it can be
identified with the arc radius.
For the constant heat flux (49) we have the solution
r
dt1 0 
 2 (r , z, t ) 
 3/ 2  e
2a  0 t1 0
t
P0
r 2  r12  z 2
4 a 2t1
 rr 
 I 0  21   r1dr1
 2a t1 
In particular,
2 (0,0, t ) 
P0r0

2 (  ),
2 (  ) 
1
 
1 
 i erfc 

(1.6.48)
Fig.5 Graph of the function 2 ( )
Similarly, using the formula
1
 xe
0
 x 2
I 0 (2 x)dx 
1 
e  e I 0 (2 ) 
4 
we get
2 (r0 , 0, t ) 
P0 r0
 2 (  ),
(1.6.49)
2
 2

 2 
x
1

e
I 0  2  dx

 x 

(1.6.50)
 
where
 2 ( ) 
1
2
1/ 

0
The graph of the function  2 ( ) is presented in Fig. 6
67
Fig.6 Graph of the function  2 ( )
For   1 we get
2 (1/  ) 
1
1 
1 

  2  
and
 2 (r0 , 0, t ) 
 1 
 o 2 
2  
 
P0 r0 1
(1.6.51)
Similarly for the heat flux (50) we get
 2 (r , z, t ) 
1
2a
t
Pa (t1 )dt1
  (t  t )
3/ 2

1
0
2
 r

r  r1  z 2  
rr1
 exp  

I 0  2
  r1dr1
2
0
 ra (t1 ) 4a (t  t1 )   2a (t  t1 ) 

2
1
2
2
(1.6.52)
Integrating and using formula

e
0
 r12
2
1  4
I 0 ( r )rdr 
e
2
we obtain
 2 ( r, z, t ) 
a

t
 
0
Pa (t1 )  ra 2 (t1 )

(t  t1 )1/ 2


z
r2
exp   2
 2

2
 4a (t  t1 ) ra (t1 )  4a (t  t1 )  dt

1
 ra 2 (t1 )  4a 2 (t  t1 ) 
2
68
(1.6.53)
In particular,
2 (0,0, t ) 
a
 
t
 (t  t )
P (t1 )  ra 2 (t1 )dt1
 ra 2 (t1 )  4a 2 (t  t1 ) 
a
1/ 2
0
1
and if
Pa (t )  Pa  const ,
ra (t )  r0  const
then
2 (0,0, t ) 
P0 ra
arctan
1
(1.6.54)

 
It is interesting to note that for   1 we get here the same asymptotic like
(54).
7 BOUNCING AND WELDING OF ELECTRICAL CONTACTS
7.1 Static welding
Stationary model of static welding. Symmetric contacts. Welding of
electrical contacts is a reason of failure in switchers, connectors, relays and other
electrical equipment. The problem of optimal choice of contact parameters with
low weldability is very important to enhance endurance and reliability of contact
systems. Analysis of investigations in this field shows that the formation of a
weld is a complicated function of current, voltage, contact force, parameters of
contact material, arc duration etc. The variation of the weld strength may be of a
wide range even at the same current. That is a reason why many experimental
data are presented in a form of cumulative probability [1]-[3]. As a rule
experimental investigation of welding phenomena is focused on the resulting
values of welded area and welding force because of very short duration of this
phenomenon, which embarrasses measurements in dynamics. Nevertheless
evolution of contact surface melting and its further welding is very important for
understanding and explanation of welding characteristics. Mathematical
modelling is very helpful in this situation. There are two types of welding in
electrical contacts. The first type is static welding, which occurs in
closed stationary contacts with a short circuit current. The second one is
dynamic welding stipulating by electric arc in closing contacts with bounce.
Simple expressions for approximate estimation of welding forces are given in
[4]. Some models and theoretical criteria of static weldability are presented in
[5] for the steady heat state and in [6] for the transient heat state. Later this
approach was developed for asymmetric contact pairs [7]-[8]. However
simplifying assumptions concerning adiabatic conditions of heating, neglect of
phase transformation, and dynamical increase of contact radius during melting
restrict significantly the range of application of these models. The radial
temperature distribution on the contact area is also important for specificity of
welding, especially in the range of high current and high contact loading force.
69
If that’s the case, then the welding area on a surface of closed contacts has ringshaped form due to displacement of current density and temperature maximum
hardly from the centre of contact spot to its edge .This phenomenon was
observed experimentally in [9], [10] and [29]. It occurs at the initial stage of
non-stationary contact heating. That is the reason why for a low range of current
and contact force such displacement can be observed because of very short
duration of transition to stationary temperature field in contacts, therefore
welding area in low current contacts is observed usually in the form of a circle.
Thus mathematical models for static welding in closed contacts should be
chosen different for low and high current.
Investigation of dynamic welding occurring at contact closure with arcing
is more complicated due to difficulties in mathematical description of arc
evolution. If welding occurs at the first contact touch after melting of a microasperity, then the welding force is weak as a rule due to very short arc duration.
However the problem becomes more serious with welding at contact bouncing.
It is interesting to note that strong contact welds may be formed in case of single
very short bounce arcs as opposed to long and multi-bounce arcs [11]. The
influence of asymmetry of contact material and polarity on anti-welding
properties and erosion is considered in [12]. Publications concerning
mathematical modelling of dynamic welding are very rare. The model presented
in [8], [13] is based on the thermionic electron emission at arcing and heat
equation. However dynamic parameters of bounce length and duration, time
dependence and space distribution of welding characteristics, dynamics of phase
transformation remain out of deliberation. An attempt to involve these aspects
into consideration is made in [14]-[15]. This paper presents a generalization of
static welding model for non-linear case of contact characteristics and for ac
current. Further development of the model of dynamic welding in dependence
on bouncing parameters is considered as well.
The fundamental potential-temperature relation discovered by Kohlrausch
(Appendix 1, formula (A.1.26)) and proved mathematically in [18] and [19] is a
theoretical base for explanation of many stationary phenomena occurring in
closed electrical contacts such as contact resistance, softening, melting and
boiling potentials, thermoelectric effects in contacts, fritting phenomena etc.
[16], [4]. This relation is very important for study of welding in closed
stationary contacts. The threshold welding current IW related to attainment of
melting temperature on the contact spot and the criterion of weldability W in
symmetric stationary contacts for different contact materials are presented in [5]
in the form
70
IW  r0 2 2W
W 
TW
TW
(1.7.1)
  (T )[   (T ) (T )dT ]
1/ 2
1
0
2
2
2
dT1
(1.7.2)
T1
The radius of the contact spot r0 in this formula was calculated taking into
account temperature dependence of contact hardness H  H (T )
r0 
F
 H (TW )
(1.7.3)
where F is the contact force and TW is melting temperature of contact material.
It has to be noted that the expression (3) should be corrected due to
electrodynamic repulsion force, thus r0 should be found as the solution of the
equation
F  107 IW ln
2
r0 
 H (TW )
r0
R
(1.7.4)
In the range of high current this correction is very important [4]. Results
presents in [5] are in a good agreement with experimental data.
However the problem of spatial temperature distribution, which is
important as well for welding characteristics, especially for welding area and
force in asymmetric contacts, was kept a mystery. Such information can be
obtained by the direct solution of the differential equations describing
temperature and potential distribution inside contacts rather than by use of
Kohlrausch’s law. This approach is presented in the Appendix 1. The threshold
welding current IW can be found from the equations (A.1.24) - (A.1.25) at the
values T (0)  TW ,  (0)   / 2 exactly at the same form (1). In the case of linear
temperature dependence of electrical resistivity and thermal conductivity the
formula for welding current can be derived from the expression (A.1.33) at
T  0, T0  TW ,    / 2
IW  4r0
0 (   )
1
{ arccos

0
1  (   )TW
  [1  (   )TW ]2  1]}
71
(1.7.5)
It is interesting to note that if we put here   0 , then calculating the limit as
  0 , using for r0 the expression (7.1.3) and replacing  0 by the averaged value
2
3
0 (1  TW ) [16] we get the well-known Hilgarth’s formula [20]
0TW F
IW  4 2
(1.7.6)
2
 H 0 (1  TW )
3
The relation between the radius of welding area r0 and current can be found
by the solving of the system of equation (7.1.1), (7.1.4). Fig. 1 depicts results of
calculation of the diameter d of welding area as a function of current [21]. The
curve 1 is calculated using Hilgarth’s formula (7.1.6), which gives redundant
temperature, the curve 2 relates to the formula (7.1.3), and the curve 3
corresponds to the expression (7.1.4), which is mostly inside the dashed area of
experimental data.
d , 103 m
2
1
3
3
2
1
0
8
1
6
2
4
3
2
I , kA
Fig.1 Diameter of welding zone as a function of
current [20]
The weld force FW can be estimated approximately as
FW   r0 ,
2
where  is the tensile strength of contact material.
The dependence of weld force on current is shown in Fig.2.
72
(1.7.7)
FW , N
3
10
1
100
3
2
10
1
4
5
6
7
8
9
10
I , kA
Fig. 2 Weld force as a function of current
The area between curves 1 and 2 corresponds
to experimental data [21], the curve 3 is
calculated
One can see a satisfactorily agreement of calculated values and
experimental data.
Asymmetric contacts. Welding phenomenon in asymmetric contacts is
characterized by a displacement of temperature maximum inside the contact
member with greater value of  . The values of this maximum Tm and
corresponding voltage  m can be obtained by modelling on the base of
temperature-potential relation at the assumption that m  UC / 2 [22]-[24].
However this assumption can be accepted as an approximation only. Sometimes
it fails, especially if materials used for contact junction have quite different
values of  and  . Furthermore, the information about the value of maximum
displacement, which is very important for welding parameters, can not be
achieved using this model. However it is available if we apply the description of
temperature and electrical fields by differential equations similar like in previous
section. Such model is presented in the end of Appendix I. The criterion of
weldability should be ascribed now not to temperature maximum Tm like in the
expression (3) but to the temperature on the contact surface T0 where welding
occurs (Fig. 3).
73
r
Tm ,  m
T0 , 0


2
0

2
m


T1 , 1
T2 ,  2
r
Fig.3 Distribution of temperature and potential in
asymmetric contacts
It can be defined from the equation (1.7.6) as
T0
T0
Tm
W   1 (2  1 1dT  2  2  2 dT ) 1/ 2 dT .
0
T
(1.7.8)
T0
The values of T0 , Tm and displacement  m can be calculated by the
numerical solving of the system of equations (A.2.9)-(A.2.11). Then the
threshold welding current and the weld force can be found from the same
expressions (7.1.1), ( 7.1.4) and (7.1.7).
Fig. 4 depicts the axial temperature distribution between Ag and W
contacts with radius of contact area r0  3.17 104 m at the current 2 kA calculated
using the expressions of Appendix II.
T ( z ), 0C
1500
1400
1300
1200
1100
1000
900
800
700
600
500
-0.4 -0.3 -0.2 -0.1
0
0.1
zm
0.3
0.4
z, mm
74
Fig. 4 Axial temperature distribution
between Ag and W contacts
One can see that the temperature on the contact spot is T0  963 0C , thus
welding should occur. The calculation for symmetric Ag contacts using
expressions of Appendix I yields at the same conditions the value T0  233 0C ,
while for symmetric W contacts the temperature should be T0  2788 0C . The
temperature maximum T0  1262 0C is displaced from the contact plane z  0 to
the point zm  1.79 104 m .
The potential distribution is shown in Fig. 5.
i , V
0.1
2
0
0.1
1
0.2
0.3
-0.4
-0.3
-0.2
-0.1
0
0.1
zm
0.3
0.4
z, mm
Fig.5 Potential distribution in Ag against W contact
junction
The total contact voltage in this case is UC  0.513 V , the potential of
contact spot is 0  0.168 V . The potential at the point zm  1.79 104 m , where
temperature maximum occurs, is m  0.029 V . Thus the assumption that m  0
[23] is sometimes not correct.
Transient model of static welding. In the case of ac current or dc pulse
current of a short duration the above considered stationary model ceases to be
valid because of time dependence of all parameters responsible for heating and
welding of closed contacts including temperature, voltage, current, contact area
etc.
The stationary mathematical model describing contact heating at the stage
up to attainment of melting temperature should be replaced by the nonstationary heat equation
C

1
 div( grad  )  grad 2 
t

while the equation for electric potential remains the same
75
(1.7.9)
1
div( grad  )  0
(1.7.10)

The simplified adiabatic model of transient contact heating is presented in
[6] based on the assumption that div( grad )  0 and that current density is
averaged over contact cross-section. A criterion of weldability as a function of
melting temperature, heat capacity, resistivity and hardness was introduced.
Later this approach was extended for asymmetric contact junctions [8]. However
man-made assumptions, restrict significantly incidence of application of this
model. Neglect of thermal conductivity is correct only for a heating due to
current pulse of very short duration. From the other hand averaging-out of
current density for phenomena of a short duration is a problematical operation,
especially in the range of high current and contact load. For example, as it will
be shown below, the phenomenon of ring-shaped welding occurs due to radial
nonuniformity of current density on the contact spot, which maximum is
displaced to the edge of contact area.
The model based on the general heat equation (7.1.9) is presented in the
Appendix 3. The fulltime of contact Joule heating is divided in two stages, premelting and melting. At the first stage no phase transformation occurs, thus the
boundary conditions are described now as before by the equations (A.1.3)(A.1.5). To get an analytical solution for examination of heat transfer dynamics,
electrical conductivity  and thermal conductivity  are averaged over
temperature accordingly to recommendations advised in [16]. It is important at
this stage to take into account increase of contact area at the attainment of
softening temperature and the action of electromagnetic force using the
expression (7.1.4).
Results of calculation using the model presented in the Appendix 3 are
shown in Fig. 6 and 7
T , 0C
1000
900
800
700
600
500
400
300
200
100
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
t , ms
Fig. 6 Dynamics of contact temperature at
pre-melting stage
76
They depict dynamics of the temperature at the centre of the contact spot
and dynamics of contact radius during pre-melting stage for the following data:
contact material is Ag-Ni (20%), current is half-wave pulse of 1.4 kA in a peak
and 15 ms of duration, contact force is 1.4 N . These parameters were used in
experiments described in [25].
r0 (t ),  m
.
40
35
30
25
20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t , ms
Fig. 7 Dynamics of contact radius
at pre-melting stage
One can see increase of contact radius when contact temperature reaches
softening temperature and corresponding deceleration of temperature rate in
spite of current rise.
The second stage of the Joule heating begins when the temperature on the
contact spot riches the melting value. The main problem in modelling of
transient welding is estimation of melted area. It should be noted that due to
nonuniformity of the current density on the contact spot, which is expressed by
the formula (3.7) , the temperature maximum is displaced from the centre of a
spot to its edge. That is a reason why melting always starts from the
circumference of contact circle and expands toward its centre. After some time
the whole contact spot may be heated up to approximately the same temperature
and stationary regime comes. The time required for the transition from nonstationary to stationary temperature field in contact area should be estimated
using Fourier criterion (dimensionless time) Fom  a 2tm / r02 , where tm is the time
from the initial current passing to attainment of melting temperature Tm at the
hottest place z  0, r  r0 . This time can be calculated from the equation
T (r0 ,0, tm )  Tm
(1.7.11)
If Fom is much greater than 1, then stationary regime becomes very quickly
and simple stationary model presented in Appendix 1 can be successfully
applied. However if Fom is much less than 1, then the melted zone has the shape
of a anchor ring on the periphery of the contact spot. In any case this
phenomenon of the liquid phase formation away from the centre of spot should
77
be taken into consideration at the modelling of the initial stage of welding. In
this case in addition to the above mentioned boundary conditions the new
conditions on the interface z   (r, t ) between melted and solid zones should be
considered. These conditions, so-called Stefan conditions, can be written in the
form [26]
T (r,  (r, t ), t )  Tm
[1  (
 2 T
) ]
r
z
 ( r ,t )

z  ( r ,t )

(1.7.12)
 j 2dz  Lm
0

t
(1.7.13)
where Lm is the latent heat of fusion.
The solution of this problem is based on the theory of heat potentials with
following reduction of free boundary problem to non-linear integral equations.
To obtain approximate solution the new integral method is proposed. As
provided by this method the temperature rise is suggested being non-zero in the
most heated initially region 0  z  Z1 (r, t ) with a moving boundary S (t ) and then
it expands in both directions according to the equation
The temperature T (r, z, t ) and surface of phase transformation z   (r, t ) are
represented in the form
z 2   (r , t ) 2
T (r , z, t )  Tm (1 
)2 ,
2
2
Z1 (r , t )   (r , t )
 (r , t )   (t ){1 
[r  r0 (t )]2 1/ 2
}
 (t )2
The functions S (t ),  (t ), r0 (t ) are defined from the system of ordinary
differential equations.
Results of numerical calculation for Cu contacts (the amplitude of ac
current is 70 kA, the contact force is 3500 N) are represented in Fig. 8. They
enable to explain the experimentally observed [10], [27] a ring-shape zone of
most intensive welding 1 (t )  r  2 (t ) corresponding to this maximum and two
zones of partial welding: internal 0  r  1 (t ) and external 2 (t )  r  3 (t ) .
78
 (t ), 103 m
3 (t )
3
 2 (t )
2
1
1 (t )
0
Iˆ
0
2
4
6
8
1
t , 103 s
0
Fig.8 Evolution of welding zones
Fig. 9 depicts expansion of the melted zones from the edge of the contact
spot ( r  r / r0  1 ) to its centre ( r  0 ). Isothermal surfaces of melting temperature
Ti , i  1, 2,...,5 correspond to increasing Fourier criteria Foi 
a 2ti
from the start
r0 (ti )2
of Joule heating ( i  1, Fo1  0.01 ) to reaching of stationary temperature (
i  5, Fo5  2.5 ), when the temperature on the contact spot is constant along its
radius. One can see that ring-shape contact heating and welding occurs only if
Fo  2 , i.e. for sufficiently large values of the radius of a contact spot r0 , in other
words for large contact forces used in high current equipment. Otherwise this
non- stationary duration of the ring-shaped heating passes very quickly and can
not be observed at the ordinary conditions. For example it is equal about 2 µs for
the Cu-Cr contacts at current 20 A and contact force 200 N.
z
1.0
0.8
T5
0.6
T4
0.4
0.2
T3
T2
T1
0
0
0.25
0.50 0.75
1.00
1.25
1.50
1.75
r
Fig. 9 Dynamics of temperature zones
79
2.00
7.2 Dynamic bouncing and welding during contact closure
Introduction. Phenomena occurring at closure of electrical contacts in
vacuum circuit breakers are very important for understanding of arc evolution,
mechanism of contact erosion and welding. Dynamics of contact closure should
be considered as a consecutive chain of interrelated stages, and parameters at the
end of each preceding stage should be assigned as the initial conditions for the
following stage. Electrical breakdown, micro-arcing, melting and evaporation of
micro-asperities, compression (penetration) have to be considered at contact
approaching, and then stages at contact opening (restitution, bridging and
arcing) should be investigated as well.
Electrical arc igniting at make and break operations due to thermal
ionization or electrical breakdown is the main source of failure in vacuum circuit
breakers [1]. Formation, evolution and explosive rupture of molten contact
bridge at the pre-arcing stage of contact opening are very important for further
dynamics of electrical arc [2], [3].
Contact bouncing accompanying by welding is another cause of failure in
low voltage electrical apparatus [6]-[8]. It may be reduced by decrease of
contact force and deceleration of closure process; however another problem
appears in this case. Gas pressure in contact gap during arcing at closure may
produce contact quakes and phenomena of contact floating which increases arc
duration and contact erosion [4]. In the case of short-circuit current it may
provoke even contact repulsion [5].
The main goal of this paper is to interlink all above mentioned phenomena
at contact closure and opening in dynamics on the base of a general model.
Experimental set up. A standard vacuum circuit breaker used for
experiments. The contact system was mounted inside vacuum bottle on a
massive support. Two laser sensors and special equipment were used to measure
displacement of each contact member and to calculate the distance between
contact surfaces by Difference Path Method (DPM), which is very convenient
for analysis of contact gap dynamics, compression, restitution and fluctuations
of contact force. [4].
The current variation from 20 A to 2 kA (dc or half-wave ac) was measured
by non-inductive resistance. For voltage measurement, which variation is very
large, 3 voltage dividers were applied. An ordinary RC–divider was used to
measure high-range voltage up to 400 V, isolation amplifier (optical coupling)
was applied for voltage in the range from 4 V to 40 V, and a clamp diode device
with non-linear characteristic was inserted inside to measure voltage across
closed contact, which range varies from 0 to 4 V. All other parameters are the
same like described in the paper [4]. Experimental results of contact closure
dynamics are presented in Fig. 1.
80
Fig. 1 Dynamics of contact closure
Arc ignition and current start at
t0  0 ,
when the contact gap is
S0
 63 m ,
and the first contact touch in molten metal occurs at FW   rm (t ) 2 (the point A1).
Thereby both contact pieces, fixed and movable, get an impulse and bounce in
opposite directions due to the pressure of expanding vapour and molten metal
until the next touch in solid ground (the point A2) at t2  480  s . Then the last
bounce occurs, however without contact opening, and the final touch (the point
A3) is accompanied by contact welding at t3  860  s .
All consecutive interrelated stages of contact closure should be considered
step by step.
Initial stage of contact closure. Electrical contact at closure can be
established due to breakdown by strong electrical field or due to touch and
explosion of micro-asperities with thermal ionization.
Breakdown of the gap between approaching contacts in vacuum circuit
breakers occurs at electrical field [1]. It means that the critical contact gap
required for breakdown in low voltage vacuum circuit breakers ( U  400 V )
81
should be a few microns. This situation can be observed sometimes for a slow
contact speed (not greater than 0.1 m / s ). On the contrary, if closure speed is
0.5 m / s or greater, then the duration of micro-arc is insufficient to melt and
evaporate a micro-asperity, thus the most apparent mechanism of final electrical
contact in this case is the mechanical touch of micro-asperities on contact
surface.
Fig. 2 depicts the initial stage of contact closure before the first touch (A1)
with a greater time resolution.
Fig. 2 Contact gap, voltage and current between movable
contact piece (mcp) and fixed contact piece (fcp).
Cu (60%) – Cr (40%) , V  0.6 m / s
One can see that the arc indeed ignites at the contact gap 63  m , that is an
evidence of the ignition mechanism by the explosion of micro-asperities. The
micro-arc of 35  s duration (stage A in Fig. 2) is the cause of explosive
evaporation of contact material accompanying by rise of the pressure in contact
gap and backward contact motion. However this pressure decreases very quickly
because of vacuum condition, contacts reverse and approach again until the
touch of a new micro-asperity with new backward and forward steps of motion
(stage B). This phenomenon called contact floating is very dangerous for
vacuum circuit breakers [4]. The next touch occurs in a molten area on the
contact surface. The current density is not sufficient now for explosion of
contact material and arc ignition therefore only a small floating jump arises due
82
to extension of liquid metal and evaporation (stage C). Further expansion of
contact area leads finally to the full closure with contact solidification,
compression, restitution and following mechanical bounce if the repulsion force
is greater than the force of micro-welding.
Let us consider the model of contact dynamics at the initial stage before
mechanical bounce.
Three zones in contact area should be considered to find temperature
dynamics and pressure of metal vapours in contact gap, including the zone of
evaporated micro-asperity D [ l  z  0] with adjoining evaporated area inside
0
contact
D1 [0  z   b ( r , t )] ,melted
D3 [ m (r , t )  z  ]
D2 [ b (r , t )  z   m ( r , t )]
zone
and solid zone
(Fig. 3).
r
rm (t )
S ( z)
D1
rb (t )
D2
D3
h
 (t )
l
T0
T2
T1
D0
0
zb
T3
z
zm
z   m (r , t )
z   b (r , t )
r
Fig. 3 Temperature contact zones.
The process of Joule heating and evaporation of a micro-asperity and
adjoining zone of contact material should be considered in dynamics step by
step beginning from the time of contact touch. The initial stage of heating and
melting of a micro-asperity up to boiling temperature can be described by the
bridge model, which is presented in [2]. This liquid bridge can be considered as
a bar with variable cross-section S ( z ) . The heat equation for the temperature
field T ( z, t ) in such domain should be written in the form
0
c0 0
T0
t

1

S ( z ) z
[ S ( z )0
T0
z
I 0
2
]
2
S ( z)
(1.7.14)
where c0 ,  0 , 0 ,  0 are heat capacity, density, heat conduction, electrical
resistivity of molten contact material respectively, and I is the current. The
appropriate shape of the cross-section S ( z ) should be chosen from the analysis
83
of a Talyrond trace (profilogram) of the contact surface. In this paper it was
identified with a paraboloid having the altitude l and the radius of the base h .
If z   (t ) corresponds to the cross-section of boiling temperature T , then the
boundary condition on this surface after arc ignition is
b
T0
 0
z  ( t )
T0
z
 Tb
(1.7.15)
d
 P0  Lb 0
(1.7.16)
dt
z  ( t )
where P0 is the arc power density, and Lb is the latent heat of evaporation.
Heat equations for the domains D1 , D2 and
D3
can be written in standard
form
ci  i
Ti
t
 Ti  j i ,
2
i  1, 2, 3
(1.7.17)
where j is the current density.
The boundary conditions should be written separately for each stage.
Initially, when molten area is the micro-asperity only, the condition on the
boundary T ( r ,  b ( r , t ), t )  Tb is
T0
When the molten area expands into zone
z   m (r , t )
0
(7.2.5)
 T1
D1
T0
z
 1
T1
z
(1.7.18)
the conditions on the boundary
are
2
T2
n
 3
T3
n
 Lm m
T0
 T1
 m
t
(1.7.19)
(1.7.20)
Finally for the last stage when evaporation boundary goes inside contact
the condition on the boundary z   b (r , t ) is

T
T

 
 Lb 2 b
(1.7.21)
1
1
n
2
2
t
n
84
the last boundary condition away from the contact area ( z   or r   ) is
T3  0
(1.7.22)
The problem (1) – (10) is solved using method of majorant functions [9].
The interface of evaporation z   b (r , t ) is found from the equation
T ( r ,  b ( r , t ), t )  Tb
Thus the volume of exploded vaporized zone
V1 
2
3
D1
(1.7.23)
is
 rb 2 zb
(1.7.24)
where rb and zb are roots of the equations
 b ( rb , t )  0
and  b (0, t )  zb
Dynamics of the arc temperature TA ( r , z , t ) is calculated using model
presented in [10]. The pressure of metallic vapour p (t ) in the arc can be
estimated now by gas law
p (t ) 
kTA (t )
(1.7.25)
V1 (t )
and corresponding force is
F (t )   rb (t ) p(t )
2
(1.7.26)
Fig. 4 depicts dynamics of the vapour force during first and second touches
of micro-asperities.
F (t ), N
200
160
120
80
40
0
0
20
40
60
80
100
Fig. 4 Forces of metallic vapour pressure
85
To verify model experimental data of contact gap measurement
compared with values X c (t ) calculated from the motion equation
mX c (t )  F (t )  Fs
where
Fs
X e (t )
are
(1.7.27)
is the force of contact spring
Fs  kX c (t )
The results of comparison are shown in Fig. 5
X e (t ), X c (t )  m

. 150
120
90
1
60
2
30
0
0
25
50
75
t,  s
100
Fig. 5 Dynamics of contact gap
1 – measured, 2 – calculated
One can see a satisfactory agreement of calculated data with measured
values.
IV. BOUNCING
The next contact touch at the point A1 occurs in molten metal. Current
density
j (t ) 
I (t )
 rb (t )
2
(1.7.28)
at this point decreases very quickly in spite of current rise due to rise of contact
spot as well generating micro-welding and compression stage of contact closure
begins. This stage is described in details in the paper [4]. Contact motion can be
described by the equation
mX (t )  Fs (t )  Fep  Fed  Fw
86
(1.7.29)
where Fep , Fed and FW
are the forces of elastic-plastic deformation,
electromagnetic repulsion and welding respectively. Calculation show that
elastic-plastic force Fep is much greater than other terms in the right side of the
equation (7.2.15), therefore contact opening occurs at the restitution, which
accompanies by the arc ignition. Again explosive evaporation of contact
material provokes the contact repulsion, which can be modeled and described by
the same equations like at the initial stage. After the second bounce at the point
A2 the contact voltage decreases up to the value U c  1.5 V (Fig. 6) which is
sufficient for boiling and melting of contact material, but not sufficient to
maintain arcing.
Fig. 6 Dynamics of contact voltage and resistance
at bouncing
Actually it is a contact backlash rather then opening. Calculating the radius
of molten zone rm (t3 ) at the point of the last touch A3 ( t3  860  s ) and then the
welding force
87
FW   rm (t3 )
where
FW
8
  2  10 N / m
 Fep
2
2
is the tensile strength for Cu one can conclude that
, thus welding occurs in considered case.
7.3 The model of blow-off phenomena
Introduction. Dynamics of contact blow-open forces is very important to provide desired
opening conditions during a short circuit current for reduction of arc duration and welding
probability. It is suggested as a rule [1] that two forces appearing in closed contacts at inrush
of short circuit current are responsible mainly for a character of contact separation at blow-off
process. The first of them is electromagnetic force and the second one is arc plasma pressure
force. Influence of magnetic force on contact repulsion is investigated in details
experimentally [2] and by numerical simulation [3]-[5]. Dynamics of blow-open contact
repulsion is explained usually in the following way [6], [7], [8]. Increasing electrical current
produces magnetic force, which reduces the load of contact spring and the radius of contact
spot. It results in rise of magnetic force. When magnetic force becomes equal to contact
spring force contact separation starts, which accompanied by increasing of current density and
temperature up to the melting point with following liquid bridge formation. The arc igniting
after bridge rupture produces intensive evaporation of contact material and gas pressure in the
contact gap; this creates a repulsion impulse and blow-open process. The main problem here
is the evaluation of plasma pressure that can be done using some indirect methods. The
method employed for this aim in the paper [6] is based on the dependence of arc electrical
conductivity on the arc temperature, which is different for different pressure. It enables to
estimate average values of blow-open force using their excellent experimental data and Yos’s
theory for arc conductivity-temperature relationship [7]. However, as it was mentioned by
authors, a more accurate method for calculating the arc dynamics pressure needs to be
developed taking into account anode and cathode phenomena, and time dependence of all
parameters. The method used in [8] deals with the relationship between plasma emission
coefficient and plasma pressure. The analytical model, which is based on the force balance
equation, was elaborated to calculate the dynamics of contact gap. Some noticeable
discrepancy between results of calculation and experimental data due to simplification of this
model indicate the necessity of more detailed time-dependent description of all phenomena at
blow-open process. In particular the basic assumption in both papers that 70% of arc power is
consumed for arc radiation should be replaced by the more detailed arc power balance. All
parameters of blow-open repulsion including arc and contact temperature, phase
transformation, forces interaction have to be considered and modelled as time-dependent
characteristics.
2 Experimental set up and measurements methods
This investigation was concentrated on pre-breaking phenomena occurring
at short-circuit conditions in the double-break contact system of low-voltage
three-pole circuit breaker for motor protection. Circuit breakers of high breaking
capacity are able to clear off short-circuit within the first AC half-period of
power frequency. Therefore also the experimental equipment was set up in order
to provide facilities for simulation of such conditions. Attention was focused on
the pure blow-open phenomena of contacts, therefore no operating means for
switching of contacts have been involved.
88
Blow-open experiments were conducted on double-break asymmetric pairs
of AgC-AgNi contacts and arc chutes of commercial switching device. The
radius of contact cross-section was R0  1.75 mm . Movable contact piece was
firmly cemented into contact holder on whom a helical spring of determined
characteristics was attached in order to provide the required contact force
directly on contacts. No other driving means for switching or loading of contact
was attached. A lightweight screen was mounted on contact holder to provide
feasible measurement of contact displacement. Test circuit breaker was
connected into test circuit by screw terminals, which permitted to connect the
stranded conductor of effective cross-section 6 mm2.
Test circuit includes the current source in series with make switch, multitap air core reactor, precision shunt resistor, and tested device as shown in Fig.
1.
Fig. 1 Shows test circuit diagram
A capacitor bank of 10 electrolytic capacitors 2200 F (-10/+30%) 350 V
with charging accessory was used as the supply of test current. The discharge
was initiated by closing the make switch and by this way an aperiodic current
waveform of exponential time rise and fall characteristics was obtained. Air core
reactor is applied in order to adjust time-to-peak of test current with waveform
in its rising part and around its maximum essentially similar to the AC halfwave of power supply 50 Hz. The maximal achieved peak value was 2 kA.
All time-variable quantities are measured as voltage drops by 100 MHz
digital CRT oscilloscope over the period of the duration of test current wave.
Current trace was picking up by CRT probe on precision shunt resistor 120
mV/250 mA, ± 0,5 %. The terminals of CRT probe for registration of contact
voltage drop (and arc voltage) are connected onto power connection terminals of
fixed contact pieces. The measuring error due to the additional voltage drop on
89
contact pieces can be disregarded, as the probe terminals were placed as close to
the contact sites as possible and the rest of current-carrying parts are short and of
considerable cross-section. Two-probe method was used to eliminate other
parasitic voltage drops. By this way the measured voltage represents the sum of
voltage drops of double-break contact pairs.
The position of movable contact bridge was measured by optical sensor. A
screen firmly attached on the holder of movable contact bridge was placed in a
gap between source of laser beam, which emits 1 mm thick and 10 mm wide
beam, and laser-light sensor with orifice of the same cross-section as a receiver
for laser beam. The sensor's orifice was partially shadowed by movable screen
regarding the instantaneous position of movable contact piece. The output of
sensor unit is directly proportional to the unscreened area of orifice with the rate
1,02 V/mm of contact travel. The frequency response and linearity of laser
emitter-sensor system was confirmed by measurements as adequate to follow
movements at velocity of more than 8 m/s without distortion of its response.
It should be noted, that the total voltage drop measured as the sum of both
contact voltage drops in series does not give us any information of asymmetry of
double break. By measurement of contact movement only translation was
recorded although in the case of double-break contacts the presence of rotational
component is reported [2], [6]. The rotational component is remarkably reduced
but not eliminated by guiding of contact holder as per actual design of regular
CB type. The actual asymmetry was indicated at the end of experiment by
dismantling of contact system.
3 Experimental results
Measurements were carried out in test circuit with circuit parameters
maintained at constant values. In fact ohmic resistance, which was actually less
than 0, 3, could vary to a certain extent due to variable contact resistance of
test device. When the capacitor battery loaded on voltage UC was discharged, a
waveform of transient current was obtained, which simulates AC current halfwave. The magnitude of test current wave was adjusted for experiment by
charge voltage in the range up to 350 V. Contact system was loaded by tensile
force of helical spring. It is the only external force determining the dynamics of
movable contact due to negligible gravitation force. The initial value of this
force in closed contacts is F0  1.4 N with spring constant k  0.15 N / mm .
Fig. 2 depicts typical oscillograms of current, voltage and contact gap at
blow-open repulsion during half-wave current with 1.5 kA peak.
90
U (t ), V
60
50
40
I (t ), kA
1.8
1.5
1.2
I (t )
30
20
10
0
0.9
0.6
0.3
0
U (t )
x(t )
x(t ), mm
1.2
1.0
0.8
0.6
0.4
0.2
0
-0.2
0
2
4
6
8
1
0
t , ms
D
A BC
12
1
4
1
6
-0.4
-0.6
E F G
Fig. 2 Dynamics of voltage U (t ) , current I (t ) , and contact displacement x(t )
Trace of contact displacement starts from the beginning of blow-open to its
maximum and returns back into the position of reclose. As it can be seen from
graphs, the reclose position does not coincide with the initial close position due
to certain burnout of contact tips during blow-open cycle.
Analysis of repulsion dynamics enables to conclude that blow-open process
should be divided into several consecutive stages. At the first stage of contact
separation AB, which duration is t1  1.1 ms , voltage increases from 0 to its
boiling value 0.75 V. It is caused due to increase of current and contact
temperature. The magnetic force, which should reduce contact load force and
increase current density, contributes to this voltage increase as well, however its
role is less significant due to relatively small current at this time. Transition to
the next stage at the point B (Step 1) lasts from t1  1.1 ms to t2  1.2 ms . It is
accompanied by abrupt voltage rise to the value 13.5 V required for arc ignition
on the contact M 1 (Fig.1), while the second contact M 2 remains still in a good
electric conductance. This voltage is approximately constant for whole arc
duration BC in the contact M 1 from the time of arc ignition t2  1.2 ms to the
time t3  1.7 ms . The power of this transition is consumed for melting and
vaporisation of a zone in the constriction region and for cathode fall formation
required providing minimum arc voltage.
It will be shown below that this phenomenon occurs as explosion due to
Joule heating, however a pressure impulse of generated vapours is not
sufficiently high yet to initiate a contact motion.
Next voltage jump from 13.5 V to 24 V (Step 2) occurring at the point C
from t3  1.7 ms to t4  1.8 ms indicates the similar phenomena of Joule explosion
and arc formation on the second contact of bridge pair M 2 . Now the resulting
force of vapour pressure and magnetic repulsion in sum on both contacts plus
magnetic force becomes to exceed spring force, and contacts start to move.
91
Arc temperature and heat fluxes into contacts become greater provoking
increase of evaporation rate, rise of gas pressure and acceleration of contact
motion at the beginning of the next stage of separation. However after some
time gas pressure decreases due to increasing of contact gap and decreasing of
current, and when it becomes equal to spring pressure at the point D ( t5  9.1 ms )
the direction of contact motion change to reverse. The voltage also decreases to
the point E, at which it steps down from t6  13.92 ms to t7  14.1 ms indicating arc
extinguishing and contact closure of one contact, while the second contact is still
arcing. It closes later at the point F stepping down from t8  14.8 ms to
t9  14.88 ms . A bounce may be observed sometimes at the final stage of contact
closure DE from t9  14.88 ms to t10  16 ms .
The object of this paper is the clarification of the mechanism of open-blow
process, especially the dynamics of forces at contact opening. Therefore the
information about dynamics of current, voltage and displacement will be used
directly from the experimental data, rather than derived from the equations to
estimate each force component during separation. The mathematical model
should describe dynamics of blow-open phenomena as a chain of consecutive
stages before and after arc ignition, including pre-arcing stage of contact
separation, which continues from the start of opening to arc ignition at the first
contact pair M 1 , transition stage up to arc ignition at the second contact pair M 2 ,
and arcing stage. Each stage should be considered separately.
4. Pre-arcing stage of contact separation
The duration of the pre-arcing stage of contact separation can be obtained
from the oscillogram (the zone AB in Fig. 2). This stage should be divided into
two periods. The first period lasts from the start of current passing to attainment
of the melting temperature on the contact surface. The second period continues
from melting temperature to arc ignition at M 1 .
The total force F (t ) acting the contacts at the pre-arcing stage can be
represented as the sum of components
F (t )   Fc (t )  Fep (t )  Fed (t )  Fp (t )  Fg (t )
(1.7.30)
The force of contact spring Fc (t ) is defined by the expression
Fc (t )  F0  kx(t )
(1.7.31)
where F0 is the initial value of this force in closed contacts, and k is the spring
constant.
In the case of double-break bridge contacts the force pressing on each
contact is
Pc (t )  0.5  Fc (t ) ;
92
(1.7.32)
The force of elastic-plastic deformation of contact zones Fep (t ) acting in
both contacts, M 1 and M 2 , which compensates the spring force in state of
closed contact (after melting it becomes to equal zero), is defined by the formula
Fep (t )  2 rC (t ) H B
2
(1.7.33)
where rC (t ) is the radius of contact spot,  is the coefficient of surface treatment
(in considered case   1 ), and H B is the contact hardness [1] depending on the
temperature.
The electrodynamic magnetic force Fed (t ) acting initially in both two
contacts is defined as [1]
Fed (t )  2 I 2 (t ) ln
R0
rC (t )
(1.7.34)
where   107 N / A2 , I (t ) is the current, R0 is radius of contact cross-section.
The force Fp (t ) appearing due to pinch pressure from the electrode jet can
be estimated from the expression [15]
Fp (t )  107
I 2 (t )
 rC 2 (t )
(1.7.35)
The gravitation force Fg (t ) is equal to mg for vertical orientation of the
contact system and to 0 for horizontal orientation. In considered case the force
components Fp (t ) and Fg (t ) are negligible in comparison with the other force
components in the equation (1).
Electrodynamic force Fed (t ) , which appears in both contact constriction
zones at the first period simultaneously with rising current, reduces the contact
force Fc (t ) and develops a micro-motion of contacts. Strictly speaking it is
decompression of elastic-plastic deformation in constriction zone rather than
motion of contacts. This displacement occurs only away from the contact zone,
and laser sensors mounted on contact member record it, while the contact plane
remains unmoved in axial direction. The reduced contact load Fc (t )  Fed (t ) is
counterbalanced by elastic-plastic force Fep (t ) . The dependence of Fc (t ) on
displacement x(t ) at this micro-motion is negligible ( Fc (t )  kx(t ) ), thus one can
derive from (2), (4) and (5) the equation
rC (t ) 
Pc (t )   I 2 (t ) ln[ R0 / rC (t )]
 H B
(1.7.36)
which should be solved with respect to rC (t ) taking into account the dependence
of hardness H B on temperature.
93
The temperature field due to volumetric Joule heating in closed contacts
TJ (r , z, t , rC ) is given in the Appendix 1 by the expressions (A.6) or (A.9). At the
conditions corresponding to Fig.1 ( a2  104 m2 / s, t  103 s, rC 2  1010 m2 ) one can
conclude that Fourier criterion Fo 
a 2t
 103 is very large, thus quasi-stationary
2
rC
approach (A.9) is correct. Attainment of the melting temperature Tm at the centre
of the contact spot z  0, r  0 occurs at the time t  tm1 that can be calculated from
the equation
(1.7.37)
TJ (0,0, tm1 , rC (tm1 ))  Tm
giving the value tm1  0.686 ms
The second period of pre-arcing stage consists of two steps. The first one is
melting of contact spot, which starts from t  tm1 and continues up to the time of
boiling attainment tb1  1.1 ms that can be found from Fig. 2. The second step is
boiling with duration from tb1 to the time of arc ignition ta1  1.2 ms at the contact
M 1 , which can be easily found as well from oscillograms. It is interesting to note
that the solution of the equation
TJ (0,0, ta1 , rC (ta1 ))  Tb
(1.7.38)
gives the same value of ta1 , that confirms correctness of applied model.
The temperature distribution in liquid and solid zones at melting step can
be represented by the expressions (A.16) and (A.19). The contact radius rC (t )
and the melting isotherm  m (t ) can be found from the conditions (A.7) and
(A.8).
All three zones Ds , Dm and Db should be considered at the boiling step. The
temperature fields in liquid and solid zones are described by the equations
(A.22) and (A.19). The boiling isotherm can be found by the solution of the
equation (A.5). The main problem at modelling is to find the temperature
distribution in vaporised zone Db because of insufficient information about
electrical and thermal conductivities of contact material in vapour state.
Therefore it is reasonable to suggest that the spatial temperature  -profile is a
parabola with a top maximum at the contact spot   0 , which increases in time
from boiling temperature Tb to the temperature of metallic vapour ionisation Tvi ,
while the temperature at the boundary    b is equal to boiling value:
T0 ( , t )  T0 (0, t )   2 /  b (t )[T0 (0, t )  Tb ]
2
(1.7.39)
The duration of boiling stage is very short (a few hundred microsecond),
thus we may assume the linear rise of temperature maximum in time
94
T0 (0, t )  Tb 
t  tb
[Tvi  Tb ]
tvi  tb
(1.7.40)
Dynamics of the contact radius rC (t ) at the pre-arcing stage at the
conditions of Fig.2 is presented in Fig. 3. Corresponding temperature at the
centre of the contact spot M 1 is shown in Fig. 4
rC (t ),  m
140
120
100
80
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
t , ms
Fig. 3 Dynamics of contact radius rC (t ) at the pre-arcing stage
TJ (0,0, t ) 0C
4500
4000
3500
3000
2500
2000
1500
1000
500
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1.1 1.2
t , ms
Fig.4 Dynamics of contact spot temperature
due to Joule heating at pre-arcing stage
The oscillogram shows that no motion of the contact M 1 occurs up to the
time ta 2  1.8 ms because the contact force is counterbalanced this time by the
elastic force in the contact pair M 2 . The radius of the contact spot decreases at
the pre-softening stage from t  0 to tS  0.35 ms due to electrodynamic force
Fed (t ) . However this decreasing is very small (from 23  m to 22.8  m ) and can
be neglected. At the next stages one can see increasing rate of contact radius due
95
to reduction of contact hardness H B accompanying by corresponding decreasing
rate of contact temperature at the softening point tS  0.35 ms and melting point
tm1  0.66 ms . The axial distribution of temperature just before arc ignition is
shown in Fig. 5.
TJ (0, z, ta1 ), 0C
4500
4000
3500
3000
Db
2500
2000
1500
1000
500
0
Dm
Ds
20
40
60
80
100 120 140 160 180 200
z,  m
Fig. 5
Axial temperature distribution at the time of arc ignition
in vaporised zone Db , melted zone Dm and solid zone Ds
The high temperature gradient in the vaporised zone indicates importance
of heat transfer inside contact that should be taken into account in the power
balance at arcing.
5. Transition stage.
The transition stage of contact separation continues from the time ta1  1.2
of arc ignition at the first contact pair M 1 to the time ta 2  1.8 ms of arc ignition at
the second contact pair M 2 . The temperature of vaporised zone Db increases at
this stage from the boiling point Tb  2193 0C to the threshold temperature of
metallic vapour ionisation Tvi  4300 0C . Dynamics of metallic vapour pressure
and force due to explosive Joule heating is very important to clear up the
mechanism of contact repulsion. The partial metallic vapour pressure can be
estimated using formula
pm (t ) 
RT0 a (t ) b (t )

(1.7.41)
where R is absolute gas constant,  is atomic weight of contact metal,  b (t ) is
density of metallic vapour and T0 a (t ) is volume-averaged vapour temperature
(10), i.e.
96
2
1
T0 a (t )  T0 (0, t )  Tb
3
3
(1.7.42)
To find  b (t ) we use the law of power dependence of vapour electrical
conductivity  b on vapour density  b at isochoric heating of non-ionised
metallic vapour [14]
 b  Ab b
(1.7.43)
where Ab  8.3 ohm1m1 and   0.56 for silver if  b is measured in kg / m3 .
Therefore, if  1 and  2 are the values of vapour density,  1 and  2 are the values
of electrical conductivity of vaporised zone Db at respective values of
temperature T1 and T2 , then
1   2(
 1 1/ 
)
2
(1.7.44)
The law (15) remains to be correct for very high values of temperature T2
and pressure, when the values  2 and  2 for metallic vapour approach to
corresponding values for solid metal, therefore the expression (15) can be
written in the form
1   s (
 1 1/ 
)
s
(1.7.45)
where  s and  s are density and electrical conductivity of the solid metal. In
terms of electrical resistance of vaporised zone Db this expression can be
represented in the form
 b (tb )   s [
R1 (tb ) 1/ 
]
R1 (ta1 )
(1.7.46)
where R1 (tb ) and R1 (ta1 ) are the values of electrical resistance of the zone Db
before and after vaporisation. These values are determined from experimental
data for power and resistance, which can be calculated as product and ratio
between measured voltage and current. Results are presented in Fig. 5.
97
W1 (t ), 103W
R1 (t ), m
40
35
30
R1 (t )
25
20
15
10
W1 (t )
5
0
0
0.5
1
1.5
2
2.5
t , ms
3
Fig. 5 Resistance R1 (t ) and power W1 (t ) at the transition steps
At both transition steps, step 1 ( 1.1 ms  t  1.2 ms ) and step 2 (
1.7 ms  t  1.8 ms ) power and resistance can be considered linearly dependent on
the temperature. Then metallic vapour pressure pm (t ) can be evaluated using
formula (12). Corresponding force Fm (t ) can be represented in the form
Fm (t )  pm (t )Sb (t )
(1.7.47)
where Sb (t ) is the surface area of vaporised zone which is the semi-ellipsoid of
revolution about z-axis (See the equation (A3) in Appendix)
r2
z2

1
b 2  rC 2 b 2
This area can be calculated by the standard formula for a surface area of
revolution of the curve
r ( z )  b  rC
2
2
1
z2
2
rC
about z-axes:

2
 dr 
Sb (t )  2  r ( z ) 1    dz ,
 dz 
0
Evaluating the integral we get
where  

rC

1  1  2
2
Sb (t )   rC 1  2  2 1  2 ln




 .


(1.7.48)
 tan  b .
The calculation of vapour force at the arc ignition using expressions (12),
(19) gives the value Fb (ta1 )  0.653 N . The electrodynamic force at this time is
Fed (ta1 )  107 I 2 (t ) ln
R0
 0.105 N
rC (ta1 )
98
(1.7.49)
Since the sum Fb (ta1 )  Fed (ta1 )  0.758 N is less than the spring force
Fc (ta1 )  1.4 N , no contact motion occurs still at this time.
The two next steps of the transition stage are the melting of the second
contact pair M 2 from ta1  1.2 s to tm2  1.7 s and its boiling from tm2  1.7 s to
tb 2  1.8 s . All characteristics for these steps can be calculated similarly like for
the contact pair M 1 above. The contact M 1 is in the arcing state this time and its
temperature should be calculated as the sum of two components (A.23) taking
into account both causes of heating, to volumetric Joule sources and surface
source by arc heat flux. The Fourier criterion at this time ceases to be small due
to increasing of contact radius at arcing. Therefore the quasi-stationary solution
of heat equation should be replaced by non-stationary solution with components
defined by the expressions (A.10) and (A.26).
The main peculiarity of this model is the expansion of the contact radius
rC (t ) , i.e. arc root radius, during arcing. It is reasonable to suggest [9] that the
arc root is attached to the expanding vaporised zone Db in the course of further
arc evolution. Thus it is identified with the radius of boiling isotherm rb (t ) , which
can be found as above from the conditions (A.4) – (A.5). Calculation shows that
when arc ignites, the contact radius increases abruptly from 122  m to 570  m ,
thus the current density at the contact M 1 decreases significantly (Fig. 6).
j (t ), A / m2
12
10
11
10
10
10
9
10
8
10
0
0.2 0.4 0.6 0.8 1
1.2 1.4 1.6 1.8 2
t , ms
Fig. 6 Dynamic of current density at transition stage
It explains the appearance of high vapour pressure at explosion in contact
zone due to Joule heating, which is however is not sufficient still to initiate
contact motion. But estimation shows that for a current density greater than
1011 A / m2 contact repulsion may start even at this first step.
The heat flux Pc (t ) entering contact from arc should be calculated taking
into account positive components due to arc radiation, electron (or ion)
bombardment of anode (cathode) contact surface, inverse electrons from the arc
column, and negative components due to power losses for evaporation,
radiation, electron emission cooling and heat conduction inside the contact body.
The expressions for all these components can be found in the paper [11] and
99
[12]. However the model in considered case can be simplified because the
information about current, voltage and displacement is available from
experiment. Therefore it is more convenient to use the arc power balance
equation
CA
dTA
dp
 WA  Wc  A
dt
dt
(1.7.50)
Here WA is the total power generated by arc in a unit volume, which can be
calculated directly from the measured values of arc voltage U A (t ) , arc current
I A (t ) and contact displacement x(t ) as
WA 
I A (t ) U A (t )
 rA2 (t ) x(t )
(1.7.51)
The last term in the right side of (21) is small and can be neglect. The force
due to arc plasma pressure FA (t ) consists of two components during arcing
FA (t )  Fm (t )  Fa (t )
(1.7.52)
The first component on the right side is caused by partial pressure of
metallic vapours in plasma, while the second one appears due to partial pressure
of heated gas (air). The relation (14) is not valid for plasma, thus Clapeyron
equation should be used to calculate both components
Fm (t ) 
M c (t ) RTA (t )
M (t ) RTA (t )
 rC 2 (t )  c
mcVA (t )
mc x(t )
Fa (t ) 
M a (t ) RTA (t )
M (t ) RTA (t )
 rA2 (t )  a
maVA (t )
ma x(t )
(1.7.53)
(1.7.54)
Here R is the gas constant, TA (t ) is the volume averaged gas temperature,
mc , ma and M c (t ) , M a (t ) are the atomic weights and mass of evaporated metal
(index c) and gas (index a) in the arc column, and VA (t )   rA2 (t )  x(t ) is the arc
volume. It should be noted that the force component due to gas pressure is
absent for metallic arc phase, when the arc temperature is less than temperature
of gas ionisation (for air it is  5000 0C ) or for a vacuum arc. The mass of
evaporated metal M c (t ) in the expressions (25) should be calculated from its
volume, which can be identified with the region of evaporated zone between
contact surface z  0 and isothermal surface of boiling temperature z   b (r, t )
(Fig. A.1).
The calculation of all parameters for consecutive phenomena, which occur
on the contact pair M 2 at the second step from t  1.7 ms to t  1.8 ms , can be
100
performed similarly like for the contact pair M 1 above. It should be noted that in
considered case no liquid metal bridge exists between contacts at the start of
contact separation because of formed vaporised zones. However in the range of
more high current the contact repulsion may be initiated by electromagnetic
force. In this case liquid bridge appears at the initial stage of repulsion [8] and
its final length should be added to the length of contact gap. This bridge length
lbr can be found as the product of bridge duration and opening velocity. The
bridge duration may be calculated as the time corresponding to the boiling
temperature. Alternatively it may be found directly from an oscillogram. Above
described mathematical model can be applied in this case as well if the contact
gap x(t ) in the expressions (25) and (26) is replaced by the sum x(t )  lbr .
One can see that the metallic plasma force Fm (t ) appearing due to Joule
heating at the transition stage is much greater than the magnetic force Fed (t ) . It
becomes equal to the spring force due to Joule explosion in the second contact
pair M 2 at t  1.8 ms , and then contacts begin to move. However the magnetic
force being relatively small contributes as well into start of contact repulsion
because the plasma force alone is not sufficient to overcome the spring force.
6. Arcing stage. Gas pressure due to arc heating
Further extension of contact gap and current rise increase the arc heat flux
entering contact, rate of evaporation and gas pressure. Calculation of contact
temperature using expression (23) shows that now arc component TA (r , z, t )
becomes much greater than Joule component TJ (r, z, t ) and motive power is
caused already by the gas plasma force, which is predominant in comparison
with magnetic force (Fig. 8 A). It is interesting to note that in contrast to results
published in [6] and [7] heat flux component due to electron and ion
bombardment amounts to 65%, while flux component due to arc radiation is
35%. It seems suggestion concerning 70% portion of arc heat flux for radiation
may be correct only for high current which was used in above referred papers.
At the time when current ceases to increase, gas pressure becomes to decrease
and at critical time tcr  3.1 ms it is equalised with spring force. However reverse
motion begins later at tr  9.1 ms (the point D in Fig. 3) due to inertia of contact
motion.
101
F (t ), N
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
F (t ), N
1.6
Fc (t )
1.55
1.5
FA (t )
FA (t )  Fed (t )
1.45
Fm (t )
1.4
Fc (t )
1.35
FA (t )
1.3
Fed (t )
1.25
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
t , ms
t , ms
Fig. 8 Dynamics of forces for the whole duration of blow-off repulsion
5. Experimental verification of forces dynamics modelling.
Verification of above considered modelling of force dynamics could be
achieved by comparison of calculated contact motion due to resulting force with
measured values of contact displacement. The resulting motive force Fres (t ) can
be obtained from the expression
Fres (t )  Fg (t )  Fed (t )  Fc (t )
Corresponding acceleration W (t ) 
(1.7.55)
1
Fres (t ) is presented in Fig. 9.
m
W (t ), mm / s 2
0.1
0.05
0
0.05
0.1
0.15
0 1
2
3
4
5
6
7 8
9 10 11 12 13 14 15
t , ms
Fig. 9 The acceleration W (t ), mm / ms 2 , versus time t , ms
The contact displacement x(t ) can be calculated by the formula
t
x(t )   (t   )W ( )d
(1.7.56)
0
Results of calculation using MathCad and data in Fig. 9 in comparison with
experimental data from oscillogram in Fig. 2 are presented in Fig. 10
102
x(t ), mm
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
t , ms
Fig. 10 Measured (solid) and calculated (dashed) contact displacement
One can see a good agreement of measured and calculated values of contact
displacement.
It should be noted that presented above mathematical model may be
simplified significantly if we use the experimental information about contact
displacement for direct calculation of acceleration and total force rather than for
verification of the model.
CHAPTER 2. MODELING OF THE BRIDGE TRANSFER
1 CYLINDRICAL MODEL OF A BRIDGE
1.1 Introduction
Phenomena at the initial stage of electrical arc depend on preceding conditions
of separation and heating of electrical contacts. The sequences of voltage
evolution before and after appearance of melting liquid bridge are described in
the paper [1]. According to this observed voltage waveform one can distinguish
4 sequences during opening: heating of solid material (I), fast adiabatic melting
(II), quasi-stationary extended melting (III), quasi-adiabatic pinch and heating
(I\/). Experimental voltage-time characteristics enable authors to establish
duration of each sequence, change in slope and magnitude of voltage and to get
some information about bridge phenomena.
However more detailed estimations [2] show that it is necessary to
ground the hypothesis about adiabatic melting and quasi-adiabatic heating. It
seems such an assumption may be correct only for limited range of opening
extent. It is well known those heat conduction phenomena's influences on the
temperature increasing in contacts as well as on the bridge and arc duration. This
conclusion will corroborate with the observed solidification of bridges just after
it's melting.
The experiments T. Takagi with co-authors [3]--[4] show considerable
decreasing of bridge and arc duration (in metallic phase) when electrodes are
heated preliminary.
103
Mathematical model mentioned below is an attempt to take into account
thorough kinds of heat transfer in power balance of solid contacts, bridges and
electrical arc that is more simple comparing with previous models [5] -- [8].
1.2 Pre-arcing stage
Models of contact heating. This stage lasts from the start of contact
separation up to ignition of electrical arc. It includes phenomena in solid
electrodes, contact melting, appearance, evolution and rupture of liquid contact
bridge.
There exist two models describing voltage and temperature waveforms
during contact separation.
One of them is based on the classical Holm model with constriction
resistance Rc   / 2ro , where  is electric resistivity and r0 is radius of contact
spot. Well-known Holm relations [9] between voltage and temperature, current
and radius of spot when temperature rises to melting point describe satisfactorily
quasi-stationary contact phenomena for a slow separation velocity. This model
was extended concerning the non-stationary phenomena and non-ideal electrical
contact [5], [7], when contact opening is so rapid that Holm relationship will no
longer be valid.
Another approach [1], [10] is associated with a filament or roughness
between the electrodes, connecting the contact with the resistance R f  2  / ro
,where 2 is the length of filament, that has to be added to the constriction
resistance Rc . Further heating of this filament leads to its melting, creating of a
molten bridge and arc ignition after bridge boiling. However the assumptions
about adiabatic heating and melting of filament [1] or about constant
temperature of the ends of filament [10] don't enable to take into account real
quantity of the power dissipation caused by heat conduction from filament to
electrodes that is very important as it is noted above. Clearly model [1] yields
overstated temperature in filament while model [10] understates it.
To correct both these models we consider the axial contact of thin semifilament (  z  0) with electrode (0  r, z  ) through the circle z  0,0  r  r0 .
It is the symmetrical half of whole filament  2  z  0 .
HEATING OF SOLID CONTACTS. At the first period 0  t  t m of nonstationary heating the temperature fields of a solid filament 1 ( z, t ) and solid
electrode  2 (r, z, t ) can be found from the solution of the heat contact problem
[11] in the form:
104
 1 ( z, t ) 
1

(e kt  1) 
 (t ) z
[3(  1) 2  1] 
6

(2.1.1)


e k (t  )  ( )d
 0
2 t
 2 (r, z, t )   21(r, z, t )   22 (r, z, t ),
(2.1.2)
where  (t )  0 (1  e t ), is the heat flux entering electrode from the filament, the
first term in the right side of (2) corresponds to the heating of the electrode by
the heat flux  (t ) from the filament, while the second one is responsible for the
Joule heating inside electrode.
The constants are defined by formulas:
 0
 0 j 2
 [cot()  r0 ]
k
, 0 
,j
r 

c
  cos 0
4
where  0 , , c,  ,  are electrical resistivity, its temperature coefficient, thermal
capacity, mass density and thermal conductivity respectively, j 
I
r0 2
is the
current density in the filament.
Expressions for components  21 (r, z, t ) and  22 (r, z, t ) are presented in the
paper [11].
The model considered in [1] may be obtained from the expression (1) if the
heat conduction from filament to electrodes is neglected ( (t )  0) . When t is
small we get adiabatic heating
1 ( z, t ) 
kt


 0 j12
 t.
c
(2.1.3)
Let us compare the results of calculation corresponding to three models.
1) Adiabatic model [1], formula (3);
2) Model with the constant ambient temperature  0 on the interface of
filament with electrodes [10];
3) General model given by the expression (8).
These results for the data given in Table 1 are presented in Fig.1.
Table 1. Data for the calculation of the temperature in filament
c  3.9 10 3 J  kg 1 K 1
  8.9  103 kg  m 3
  3.8  10 2 W  m 3 K 1
 0  1.55  10 8 Ohm  m
  4  10 3 K 1
J  25 A , t  t m  1.4  10 5 s
105
1 ( z , t m ) , o C
1
1000
3
800
600
2
400
200
l
0
z
Fig.1 Temperature distribution in the Cu half-filament,
25 A, r0  0.78  10 5 m,   1.44  10 5 m [1] ;
1- equation (3), 2 – paper [10], 3 – equation [1].
The time t m required for the melting of filament according to adiabatic
model given in [1] (curve 1) is equal to 1.4  10 5 s . At this time the temperature
corresponding to the model given in [10] is 1 (, t )  342o C in the center of the
filament and ambient temperature  0  20o C on its edge. According to the
general model the real temperature is 1 (, t )  7400 C , 1 (0, t )  3360 C .
It is not so difficult to calculate that only 54% of dissipated energy
W  J U  t m is spent for the heating of filament, while 46% is passed through
interface of electrode owing to heat conduction.
Therefore the radius r0 and the length  of the filament calculated in [1]
using adiabatic energy balance has to be corrected.
MELTING OF CONTACT CONSTRICTION REGION.
This period
6
5
(t m  t  t1 ) is of very short duration (10 s  10 s). It is identified in [1] with
the time required for the melting of a cylindrical filament or conducting
roughness that is transformed into molten bridge . From measured values of
voltage amplitude U and duration t 0 (Fig.2) it is possible to calculate
geometrical extent of bridge and current density.
106
U,V
1.0
0.8
2
25A
0.6
2
0.4
12A
0.2
Fig. 2 Fast bridge melting , Cu -electrodes. Melting
voltage 0.41 V. Molten (1) and solidified (2) bridges [1]
Let  b (t ) be the boundary between liquid part of filament    z   b (t ) and
its solid part  b (t )  z  0 , so  b (tm )   ,  b (t1 )  0 . It may be found from the
equation
1 ( b (t ), t )   m ,
(2.1.4)
where 1 ( z, t ) is given by the formula (1) and  m is melting temperature. The
energy balance of the liquid and solid parts of filament at the time t is given by
the expression
W  Wd  WL  Wc
(2.1.5)
where
Wd  r0
t
2
  ( )d
(2.1.6)
tm
is the energy dissipated to the electrode through interface z  0 ,
2
WL  L  r0 [ b (t )  ] ,
(2.1.7)
( L is latent heat melting ) is the energy required for melting of the liquid region
   z   b (t ) , and
Wc  r0 c
2
0
 ( x, t )dx
(2.1.8)
1
 b (t )
is the energy for the heating of the solid region  b (t )  z  0 .
Taking into account
W  0.5U  J  t , U  J 0 (1  1 )

ro 2
(2.1.9)
we can calculate from (23) - (28) , r0 ,WL ,Wd ,Wc .The calculations show that
Wd (t1 )  0.23  10 5 J ,
when t  t1  4.7 106 s , then WL (t1 )  0.21  105 J ,
107
Wc (t1 )  0.06  105 J , so the radius and the length of the bridge are more small
comparing with adiabatic model :
r0  4.6  106 m ,   8.2 10 6 m
The temperature and potential  b, b
described by more simple equations
c b
for the bridge
  z  0
 b 

J 2
2

(b b )  T  b  2 b4
t
z
z
rb
 r0
are
(2.1.10)
d 1 d b
(
)  0,
dz  b dz
(2.1.11)
where  T is total heat emission owing to both radiation and convection [12] .
The boundary conditions take into account continuity of temperature and heat
fluxes with phase transformation on the interfaces.
The results of calculation using model (10) - (13) are given in the Table 2.
Table 2 Comparison of models ( I = 25 A )
W 10 5 J
Wc
V
5
15
10
1
2
3
0.5
0.5
-
J
0
0.28
-
10
m
3
2.74
0.84
0.39 - 1.54
r

6
6
j
10 m
10 m
1011 A  m 2
7.8
6.1
-7
14.4
7.2
5- 10
1.3
2.6
2 - 3.5
1 - calculations using model [1]
2 - calculations using model (10) - (13)
3 - experimental data [1]
One may resume that model (10) - (13) is much closer to experimental data
in comparison with the adiabatic model.
QUASI – STATIONARY EXTENSION AND HEATING OF MOLTEN
BRIDGE. Experimental data [1] concerning first period ( extension of molten
bridge ) are given in Table 3.
If extension velocity of molten bridge is low, quasi - stationary model may
be used successfully, but it has to be corrected taking into account the change of
bridge shape during contact separating. Under the action of the forces of surface
tension, gravitation, pinch - effect the bridge takes shape of revolution surface
round z – axis.
108
TABLE 3 Characteristics of bridge’s extension
v  0.1m  s 1
Duration , 10 6 s
Contact initial
gap, 10 6 s final
Voltage, initial
V final
Voltage velocity,
v  0.01m  s 1
10
1
2
0.5 - 0.8
1.0
100
4
5
0.5
1.0
5- 8
0.4-0.5
5
5
dU
10 4 V  s 1
dt
Voltage gradient,
dU
, 105V  m 1
dz
Mathematical model describing temperature and electromagnetic fields is
based on the one dimensional heat equation for the bridge with variable crosssection [7]
d 2 b 2 y  b
J 2 b




0
dz 2
y z z  2 b y 4
1 d b
J

 2
 b dz y
(2.1.12)
(2.1.13)
Other equations for 1 , 2 , 1 ,  2 are the same as at preceding stage.
The solutions of the equations (14) , (15) are found in the form :
(16)
where
z
  2 y (  vt, t ) 
0
dz
,
y ( z, t )
2

r0
2
and constants A and B are chosen to satisfy the boundary conditions [7],
[11].
The free boundary (the shape of the bridge) r  y( z, t ) is determined from
the variation principle declaring that free energy of bridge must be minimum for
the really observed bridge shape, i.e. it is necessary to find minimum of
functional
F ( y)  Fk ( y)  FH ( y)  Fg ( y)
where Fk ( y), FH ( y), Fg ( y) are components of surface tension, electromagnetic
and gravitation energy respectively described in details in [7], [11].
109
When temperature in narrow cross-section of bridge rises to the point
corresponding to the abrupt decreasing of surface tension coefficient, quasi stationary period of this stage finishes and next heating period bridge starts.
Experimental data for the calculations at this period with J  50 A and
v  0.1m  s 1 are given in the Table 4.
Table 4 Voltage in the rupturing bridge
Duration ,
6
0.05
0.10
10 s
-
Initial gap
,
6
2
10 m
Voltage ,
initial
1
10
U
final
- 20
Voltage
velocity ,
1
- 10
dU
,
dt
108 V  s 1
The results of calculation according to above mathematical model are given
on Fig. 5.
 , 10 3 0 C
U, V
2
2
1
1
5
t , 10 3 s
0
Fig. 5 Voltage U (1) and temperature  b (0, t ) (2) in
Cu - contacts. I  50A, v  10 2 m / s
110
The comparison of calculations with experimental curves [1] shows a good
coincidence.
INFLUENCE OF BRIDGING ON THE ARC DURATION. Transition from
bridging to arcing is accompanied by a voltage drop. A typical oscillogram of
voltage and contact gap corresponding to consecutive periods of contact opening
is shown in Fig. 6
l(t)
u
uA
2
lcr
u
ign
lmin
umin
lb
1
tb
t
cr
t
Fig. 6 Voltage and contact gap of opening
contacts. 1- contact gap,
2- voltage, t a - bridge lifetime, lb - bridge length, t cr - critical time, lcr -critical gap,
uig -voltage of arc ignition , umin - min voltage, uarc - voltage of cathode arc
One can see following typical stages :
1). The stage of a liquid molten bridge with the length lb and the lifetime tb
;
2). The transition stage of an anode dominated arc phase ( tb  t  tcr ) with
the linear decreasing voltage umin  u  uig in the region lb  r  lmin
and
increasing voltage umin  u  uarc in the region lmin  r  lcr ; voltage umin appearing
on contacts just after the rupture of a bridge bears a direct relation to the cathode
fall of the material. ;
3). The stage of a diffusive cathode dominated arc ( t  tcr ) with constant
voltage uarc .
At the beginning of the second stage, just after bridge rupture and arc
ignition, the distance between the cathode with single cathode spots on the
surface and the anode is very small , therefore the power capacity from the arc
to the anode is large. As a result the anode undergoes powerful local overheating
and becomes a source of vapors influencing on the next arc stage. Owing to the
plasma from the cathode region being formed as truncated cone with the greater
111
base on the anode, there is a certain critical gap lcr , which becomes too large to
support the power capacity required for the further melting and evaporation of
the anode. The another reason for the appearance of the critical gap is a cooling
effect due to increasing of evaporation surface on the anode .
The time of the beginning of the third stage depends on the duration of two
previous stages, particularly, on the bridge time life and bridge length. If the
length of a bridge lb exceeds the critical gap distance lcr , then the second
transition stage could be not appearing at all. On the contrary, if the bridge
length is negligible, or if bridge does not appear at all , then duration of the
transition anode dominated arc stage ta  tcr  tb is maximal.
The dependence of t a on the bridge time-life tb for the Co and Mo
electrodes [13] is given in Fig. 7 .
ta ,10-4 sec
16
1
12
3
8
4
4
0
2
4
2
6
8
-4
t b ,10 sec
Fig.7 The dependence of the anode arc duration t a on the bridge lifetime tb
;
1- Cu, I  650 A ; 2 - Cu, I  270 A
3- Mo , I  650 A; 4- Mo , I  450 A
Experimental observed values of anode arc duration have considerable
deviations ( Fig. 8 ). It is a result of deviations of bridge lifetimes. For cases
when bridges don't appear at all (hatched column in Fig. 8 the anode arc
duration is maximal.
112
W
0.25
0.20
Mo
0.15
0.1
0.05
0
10
20
30
40
50
60 70
80 l -6
b ,10 m
Fig. 8 The histogram of statistical distribution of frequency of a bridge
appearance W in dependence on the bridge length lb
If the velocity of contact opening V0  0. 05m  sec1 , then the transition anode
dominated arc stage is not observed in the case of Cu -electrodes for the bridge
length lb  40 106 m . For the velocity range V0  0. 2m  sec1 this threshold of the
bridge length is lb  85106 m . This phenomena can be explained by suggestion,
that the mechanism of bridge generation for such materials as Cu and Ni is
stipulated by stretching of a liquid metal from the electrode rather than by
melting of a surface filament.
Bridges on the Mo -electrodes are stable and generated accordingly to the
normal length distribution low with the maximum magnitude lb,max  25 106 m .
The analysis of experimental data [13] enables to conclude that for the Cu electrodes tb  ta , thus in this case the bridge is the main factor predicting the
following transition arc stage or its absence . On the contrary , for the Mo electrodes tb  ta , and one can observe relative self-dependence of the transition
arc phenomena.
2 GENERAL MODEL OF BRIDGE TRANSFER
2.1Bridge phenomena
A molten metal bridge which is formed between electrical c2ontacts in the last
stage of contact opening is a cause of unreliable performance of low current
relays and control systems working with milli- and micro amperes. Duration and
other characteristics of an electrical arc ignited in vapours of ruptured bridge
depend also on the parameters of a bridge [1]. Problems of experimental and
theoretical investigations of bridge phenomena have been considered in many
papers. One can find the detailed review concerning this matter in the book [2].
However the problem of mathematical modeling of bridging remains still one of
the main problems in the modern theory of electrical contacts.
113
The mathematical model describing the dynamics of the temperature field
at the consecutive stages of contact opening [3] enables to conclude , that the
hottest section of the bridge is displaced from the centre because of
thermoelectric effects. It was shown further in the papers [4] , [5], that if
contacts are homogeneous, then the tunnel effect is most important, and it is
responsible for the rupture of a bridge at its anode side. This rupture is
accompanied by metal transfer from the anode to the cathode. During next
commutation contact surfaces consist of the same material as the anode,
therefore the material transfer occurs again from the anode to the cathode due to
the same tunnel overheating. As a result of such transfer a thin pip of the anode
material, called spire, builds up on the cathode, while a crater appears on the
anode .
The process of material transfer is quite different if one could succeed in
choosing of a contact pairs composition to provide the negative materials
transfer from the cathode to the anode at the first opening operation. In this
case the materials transfer is positive , it comes from the anode to the cathode.
This transfer continues as long as all transferred during first opening operation
material is replaced again to the cathode, and after that negative transfer occurs
again. Very scanty thickness of the cathode material has a share in the transfer at
this case, thus the reliability of the contacts could be sufficiently improved.
Empirical choice of such contact pairs composition , that provides above
self-restrained bridge transfer, was fulfilled by W.G. Pfann [6] , who has
discovered the contact pairs:Au - Pt , Ag - Pt , Au - Pt Ir (20%), Au - Pd , Pd Pt with the first component as anode . M.A.Razumikhin [7] has confirmed the
principle of choice of contact pairs with initial negative transfer by using alloys
Pt Ni ( 5%) as anode and Pd Ag (40%) as cathode . He found that this
composition provides very small layer of cathode material , which has more
resistivity, performing in the transfer even during long working .
2.2 Mathematical model
The regions of bridging . A mathematical model of contact bridging
transfer must describe the dynamics of formation, evolution and rupture of a
molten liquid bridge . Many investigators proceed from the assumption that
visible part of a bridge has the shape of a cylinder with the axis parallel to the
contact plane. Of course, such assumption is true in the case of high current
bridge at high velocity of contact opening. It enables to simplify the theoretical
investigation of bridging phenomena in a large measure. All the same time it is
necessary to take into account the shape of the bridge discrepant from
cylindrical if contact opening duration is commensurable with the time of bridge
formation . However no investigations have been carried to determine the
bridges geometry and its influence on contact opening phenomena , except
114
classical results obtained by P.Davidson as early as 1954 , who has considered
bridges as nodoids and cathenoids formed due to surface tension force .
We assume in general case, that under the action of forces of surface
tension, gravitation, and pinch effect the bridge takes the shape of a surface of
revolution around z - axis :
r  y( z, t )
r1 (t )  r  (t ) ,
for
 ( t )  z  0 ,
and
z  10 (r , t )
for
where (t ) is the contact gap at the time t .
Geometry of such bridge is presented on Fig. 1 . It occupies the region
D2  D3 where D2 {0  r  y( z, t ); (t )  z  0} is the visible part of the bridge ,
and D3 {10 (r , t )  z  1 (r , t ) , if r1 (t )  r  (t ) ; 0  z  1 (r , t ) , if 0  r  r1 (t )} is
the part of the bridge embedded inside solid electrode occupying the
D4 {1 (r , t )  z   , if 0  r  (t ) ; 0  z   if (t )  r  } .
region
The bridge is generated from the electrode, which material is more fusible,
if contact materials are heterogeneous , or from the anode in the case of
homogeneous material due to tunnel extra heating at the previous stage . The
region D1 in Fig. 1 is more refractory and fuses later than D3 .
z  2 (r , t )
D1
(t )
2
r0
 2 (t )
D2
r
r
0
3
D3
r1 ( t )
(t )
z  1 ( r , t )
z  10 ( r , t )
D4
z
Fig.1 Geometry of a bridge and adjoining regions
Differential equations and boundary conditions.
The general
mathematical model describing electromagnetic and temperature fields in the
bridge and adjoining regions by the system of MHD - equations is presented in
the papers [ 8 ] - [ 9 ] . However this model is very complicated and requires
special computer programmes . There is shown in the paper [ 9 ] , that if velocity
of contact opening is not high , precisely d / dt  4m  sec1 , then bridging process
can be considered as quasi-steady and described by equations, where t is
parameter only.
115
For this case we present rather simple model which enables to analyze an
influence of different factors on the bridging process. In accordance with this
model equations
i i 2i  Ti i i  (i ) 2  0 ( 1 ),
curl(
1
i
i )  0 ,
i=1,3,4
(2.1.15)
where  i is thermal conductivity , i  i 0 (1  i i ) is specific electrical
resistance ,  i is its temperature coefficient , and  Ti is Thomson coefficient ,
describe the temperature and electromagnetic fields i and  i in each region
Di ( i = 1, 3, 4 ) .
To choose a simple equation for the bridge we use the result of F.L. Jones [10] who showed that temperature gradient in a bridge cross-section is
negligible . In this case a bridge can be observed as a bar with variable crosssection y 2 ( z, t ) carrying only the axial component of heat flux . The equations
of quasi-steady heat transfer and potential distribution in the visible part D2 of a
bridge in this model are :
d 2 2 2 yz  2 I T 2 d 2 I 2 20 (1   2 2 )



0,
 22 y 4
dz 2
y z 2 y 2 dz
d 2 20 (1  2 )

dz
y 2
(2.1.16)
(2.1.17)
where I is the current .
The boundary conditions are :
a). For the boundaries of the visible part of the bridge
z  0 , if 0  r  r1 (t ) ,
and z  (t ) , if 0  r  r0
i  3 for z  (t ) )
( i  1 for z  0 ,
and
we can write the conditions of continuity of temperature, heat fluxes, electrical
potentials, and current densities :
i  i 1 ,  i
 i

  i 1 i ,
z
z
i  i 1 ,
1  i
1  i 1

i z i 1 z
(2.1.18)
b). On the lateral surface of the bridge as well as on the contact surfaces
outside the bridge z  0 and z  (t ) the heat and electrical transfer can be
neglected :
 i
 0,
n
 i
0
n
(2.1.19)
c). In regions faraway from the contact zone electrical potentials are given
, and the temperature is constant ( this constant we identify as 0 ) :
1  0 ,
4  0 ,
1
2
1   uc ,
1
2
 4  uc
z  0 , r 2  z2   ,
if
, if
116
z  0 , r 2  z2  
(2.1.20)
(2.1.21)
d). On the surface z  1 (r , t ) between liquid and solid zones ( D1 and D4
) the temperature is equal to the melting point :
3  4  m ,
if z  (r , t )
(2.1.22)
Equations for free boundaries . The boundaries z  1 (r , t ) , r  y( z, t ) ,
z  10 (r , t ) are free and have to be determined from the additional conditions.
The interface z  1 (r , t ) between solid and liquid zones can be found from the
Stefan condition, that transforms in the stationary case into simple equation
3
 3

 4 4
n
n
(2.1.23)
The free unknown boundaries r  y( z, t ) and z  10 (r , t ) for the bridge
shape can be derived from the Euler equation, which can be reduced to the form
( see Appendix 1 ) ;
 yy zz  yz  1  0 I 2
 20

 k 2 ,  ( t )  z  0 ,
2
2 y (1  yz ) 3/ 2 8 2 y 2
2
r ( 10 ) rr  ( 10 ) r [1  ( 10 ) r ]  0 I 2 1
1
 20

 2  k2 , r1 (t )  r  (t ) ,
2 3/ 2
2
2
8 r
r[1  ( 10 ) r ]
rm
2
where
k2 
0 I 2
 k3 ,
2
8 2 r1 ( t )
k3  ( 20  2 3 ) z 0 ,
(2.1.24)
(2.1.25)
(2.1.26)
 20 is the surface tension between the bridges molten metal and surroundings
(air),  0  1. 257 106 H  m1 is magnetic permeability of vacuum, rm  rm (t ) is the
point of maximum of the function z  10 (r , t ) in the domain r1 (t )  r  (t ) .
It is necessary to put boundary conditions to obtain the unique solutions of
the equations ( 9 ) and ( 10 ). The first of them for the equation ( 9 ) is evident :
y( (t ), t )  r0
(2.1.27)
and the second can be obtained from Young formula for a boundary wetting
angle  2 :
cos  2  [1  yz (  ( t ), t )]1/ 2 
2
 10   21
 20
(2.1.28)
where  ij is the coefficient of the surface tension between the domains Di and
D j ( index 0 corresponds to the surroundings ).
The boundary conditions for the equation ( 10 ) are :
10 ( (t ), t )  0, ( 14 )
cos  3 
1  [( 1 ) r  ( 10 ) r ]r  (t )
[ 1  ( 1 ) r  1  ( 10 ) r ]r   (t )
2
2
117

 40   34
 30
(2.1.29)
2.3 Results of solution.
The equations ( 1 ) and ( 2 ) can be reduced to the ordinary differential
equations by appropriate substitutions, and their solutions , as well as the
solutions of the equations ( 3 ) , ( 4 ) , and the surface of phase transformation
z  1 (r , t ) , are found in analytical form ( Appendix 2 ). Similar solutions are
presented also in the paper [5].
To find the free surfaces y ( z, t ) and 1 (r , t ) we shall use the Ritz method.
In addition to the boundary conditions ( 12 ) - ( 15 ) we must take into account
r1 (t )  r  (t ) ,
the balance of volumes of the bridge meniscus Dm :
0  z  10 (r , t ) and its visible part D2 , thus
0
2

dz
 ( t )
y ( z ,t )
 (t )
0
r1 ( t )
 (1   2 )rdr  2

10
(r , t )rdr ,
(2.1.30)
where  is the thermal expansion coefficient .
Besides that the conditions of smooth conjugation on the interface z  0 ,
r  r1 have to be provided :
y (0, t )  10 (r1 , t )

 1.
z
r
10 (r1 , t )  0 ,
y (0, t )  r1 ,
(2.1.31)
In accordance with the Ritz method the boundary problem for equations
(9) - ( 10 ) to a variational problem for minimum of certain functional F ,
constructed in [8], [9] , which is equal to the free energy of a bridge :
F  Fg  FH  F
(2.1.32)
where the terms on the right side are equal to gravitational component,
electromagnetic component, and surface tension component of the bridge energy
respectively. The term Fg is negligible, and for a low current ( less than 500 A )
FH  0 . In this case
F  F  2
0
 (t )
r0
 ( t )
0
0
2
2
  20 y 1  y z dz  2   34 1  ( 1 ) r rdr  2   12 rdr
(2.1.33)
Unknown y ( z, t ) , and 10 (r , t ) are represented in the form
y( z, t ) 
z( z  )
2
C
z
2
( z   )( 2
10 ( r , t ) 
r1  r0

 cot  2 ) 
z2
2
( r1  r0 )  r1 ,
( r  r1 )(   r ) ( r  r1 ) 2 (   r )

cot  3
C (   r1 ) 2
(  r )2
(2.1.34)
(2.1.35)
It is not difficult to verify that the conditions ( 12 ) - ( 15 ) , ( 17 ) are
satisfied. Parameter C is determined from ( 16 ). Minimizing the functional F
by parameter r1 , we obtain the last equation
F
0
r1
(2.1.36)
Thus the bridging problem is reduced to the system of equations ( 8 ), ( 16
), ( 22 ) with the additional equation
118
0
 0  2r0

 ( t )
dz
y 2 ( z, t )
(2.1.37)
for parameters  0 ,  m , C , r1 .
This system was solved using method of half-dividing intervals. The results
of calculations of the bridge rupture voltage Vr are given in the Table 1 in
comparison with experimental data which review is adduced in [10].
Apparently, formula Vr  V3 (m )  V2 (0) used for calculation gives satisfactory
coincidence for all metals except nickel, apparently because of oxidizing
reactions , that can not be described in the frame of presented model.
Table 1. The voltage Vr of bridge rupture ( volts )
M
Voltage, experiment (V)
etal
Voltage,
calculated
(V)
0.8
1
Cu
0.64-0.68
(1,5,f)
1.0-1.5
(1,4,8,d)
0.44
(1,6,f)
1.0-1.5
(1,4,f)
0.8
(2,4,b)
0.8
(1,4,7,d)
Au
0.84
(1,5,g)
0.85
(1,4,d)
0.88
(1,4,g)
1.0
(2,4,b)
2
1.14
(1,6,g)
1.21
(1,5,g)
1.44
(1,4,g)
1.44
(1,4,a)
2
1.0
(1,6,f)
1.22
(1,5,f)
1.26
(1,6,f)
1.06
(1,6,g)
1.2
(1,4,f)
1.45
(1,5,f)
0.9
(1,6,f)
1.02
(1,5,f)
1.52
(1,4,d)
1.05
(1,5,h)
0.96
(1,4,d)
1.4
(2,4,b)
1.06
(1,6,f)
1.29
(1,4,a)
1.34
1.0
(1,6,f)
1.0
(1,5,f)
1.1
(1,4,f)
1.08
(1,6,g)
1.32
(1,4,g)
1.6
1.12
(1,5,g)
1.32
(1,4,d)
1.6
1.16
(1,5,f)
1.32
(1,4,e)
1.35
Ir
Fe
M
o
Ni
Pd
Pt
1.1
1,4,d)
1.20
(1,5,h)
(
1.18
(1,5,g)
1.4
(1,4,h)
0.9
1.1
1.0
3
1.1
3
119
0.4
1
1.3
1.4
1
Ag
W
(1,4,f)
(2,4,b)
(2,4,c)
(1,4,b)
0.35
(1,4,d)
0.35
(1,4,d)
1.5
(1,6,f)
0.35
(1,4,d)
0.35
(1,4,d)
1.5
(1,6,f)
0.35
(1,4,d)
0.35
(1,4,d)
1.5
(1,6,f)
0.35
(1,4,d)
0.35
(1,4,d)
1.5
(1,6,f)
0.5
8
1.4
2
1 - quasi static method, 2 - oscillographic method, 3 - method is unknown,
4 - in air, 5 - in vacuum, 6 - in vacuum with cleaned electrodes, 7,8 - different
bridges are given a - A.Fairweather, b - J.Lander and L.Germer, c - J,Warham,
d - L.Jones, e - R.Holm and E.Holm, f - C.Jones, G - R.James, h - F.Llevellin
Jones and M.Price
2.4 The mechanism of bridge rupture.
The process of a bridge rupture may be caused by mechanical or thermal
forces dependently on criteria considered hereafter. Mechanical rupture occurs
when surface tension force is less than electromagnetic and gravitation forces.
The bridge rupture begins at its narrowest cross-section z  zr which can be
determined from the equation
y ( zr , tr )
0
z
(2.1.38)
Thermal mechanism of a bridge rupture can be consisted of one or two
stages. It is necessary to consider two different cases separately.
Case 1. The melting temperature m1 of the electrode, occupied the
region D1 is more than the boiling temperature of the bridge : m1  b . In this
case the bridge rupture occurs when the temperature reaches the boiling point in
any cross-section z  zb at the time t  tb . Many experiments show that the
bridge rupture is accompanied by its explosion. It confirms the fact, that the
surface tension
force can not be a cause of the bridge rupture and can influence only on the
location of the cross-section with maximal temperature where explosion occurs
during boiling. Thus, unknown values zb and tb can be found from the
equations
 2
 0,
z
2  b
for
z  zb , t  tb
that transforms in this case to the expressions
 2b 
1
22  22
 2 A2   2   2  B2
2
arctan
2
 2 2   2 2  A2   2 B2
120
(2.1.39)
b  
1
 exp (  2 2b )[ A2 cos( 2b  2   2 )  B2 sin( 2b  2   2 )] ,
2
2
2
2
2
where
z

2b
 2r
0
b
dz

2
  (t ) y ( z, t b )
b
.
Thus
rb  y ( zb , tb ) .
The metal transfer due to bridging is directed then to the region D1 , on
which side the portion

zb
y
2
( z, t b )dz
of the bridge is displaced. The portion
 ( tb )
0
of the bridge   y 2 ( z, t b )dz remains on the other electrode.
zb
If m1  b , the melting of the second electrode begins at some
time t  tm2 before the bridge rupture. Three new zones instead of one D1
appear during further opening process :
  z  (t ) , if
D11 :   z  2 (r , t ) , if 0  r   2 (t ) ,
and
Case 2.
 2 (t )  r  
2 (r , t )  z  (t ) D12 : , 0  r   2 (t ) ;
D13 : (t )  z  1 (t ) ,
0  r  y( z, t )
The function 1 (t ) is determined by viscosity and thermal parameters of
liquid metal. It can be taken as 1 (t )  0.5[(t )  (tm2 )] for a first approximation.
The equation of heat transfer in the forms ( 1 ) or ( 3 ) remains correct for
each of these zones. The functions i (r1 , z, tm2 ) are the initial temperature
distribution in this system ( i = 1, 2, 3, 4 ), but additional boundary conditions
are the same. The cross-section z  zb and the time of explosion tb are
determined as usually from ( 25 ), however, in this case cross-section z  zb
may be placed in zone D13 or in D2 because of essential temperature gradient
along the bridge.
The mechanism of the rupture for quasi-static bridges is thermal as a rule.
It was established, that the diameter of a bridge increases with current as linear
function for the current range 10 A - 100 A ( both for the minimal diameter
d min  2rb and for the maximal diameter d max  2(t ) ).
2.5 Criteria of bridging
The analysis obtained results enables us to estimate the influence of
different factors on the bridging process before the solution of the problem.
They are determined by following criteria :
121
a). Surface tension criterion ( Weber criterion )
We  103 , then
We 
0
. If
g 0 2 ( t )
r1
 2 , the curvature of a bridge surface is large , and surface
rb
tension is very important during rupture process.
b). The inverse value of We is Bond criterion Bo  We 1 , which determines
the value of gravitation force with respect to the surface tension force. It is not
essential in the current range up to 100 A .
c). Pinch effect criterion Pi 
 0 I 2(t )
is very important if its value is 102
2
4  2 r0  0
and more. If I  103 A , then Pi  103 , and a bridge may be squeezed and crushed
by the pinch effect only.
d). Thermal criteria. Pomerantsev criterion Po 
criterion Th 
I 2 2  ( t )
,
 2 rb 2  2
Thomson
2 f
I T 2
, Kohler criterion Ko 
, where  f is tunnel resistivity,
 2 rb
 2 r1
are responsible for the thermal mechanism of bridge rupture due to the Joule
heating and thermoelectric effects.
2.6 Self-restrained bridge erosion.
The analysis of the expressions describing the temperature field in a molten
bridge enables us to obtain the theoretical conditions for self-restrained erosion.
They are :
1). The anode begins to melt first, while the cathode melts later, i.e.
m1  m2 .
(2.1.40)
2).The temperature along the bridge , generating mainly from the anode
material, increases monotonously and reaches the maximum on the contact
surface with the anode. The inequalities
1  4 ,
1   4 ,
2

 1
c2  2 c1 1
(2.1.41)
enable to satisfy this suggestion.
3).The melting point of the cathode material is less than the boiling point of
the anode material :
m2  b1,
(2.1.42)
i.e. the cathode melts before the bridge rupture .
4). The anode is more fusible than the cathode :
b2  b1 ,
(2.1.43)
but the maximum of the bridge temperature is placed on the cathode part of the
bridge , and the boiling temperature at this place reaches early than the boiling
temperature of the anode part.
Thus , the inequalities ( 26 ) - ( 29 ) provide the conditions for optimal
self-restrained bridge transfer.
122
These conditions can be verified for the found by W.G.Pfann [ 6 ] selfrestrained contact pairs given in the Table 2.
Table 2. Contact pairs with self-restrained erosion
No
1
2
3
4
5
Material
Ag – anode, Pdcathode
Ag – anode,
Pt - cathode
Au – anode,
Pt - cathode
Au – anode,
Pd - cathode
Ag – anode,
Pt-Ir20% - cathode
i / ci  i
i
i
8
10   m
1.65
10.5
1.65
11.0
2.30
11.0
2.30
10.5
1.65
32.0
watt / m K
418
71
418
70
310
70
310
71
418
8.92
10
14
4
m K  j
0.676
4.00
0.676
3.93
0.92
3.93
0.92
4.00
0.676
8.92
bi
 mi
3
10 C
0
0.960
1.554
0.960
1.773
1.063
1.773
1.063
1.554
0.960
1.830
103 C 0
2.000
4.000
2.000
4.400
2.970
4.400
2.970
4.000
2.000
4.400
It has to be noted, that alloys give more ample opportunity for a choice of selfrestrained contact compositions in comparison with pure metals. The best result
from all proposed by W.G. Pfann pairs gives the pair No 5.
The obtained above criteria enables us to find new contact compositions
with self-restrained bridge erosion. They are :
1). Ag  Au (10%) - anode
2). Ag  Pd (5%) - anode
Pd  Ag ( 40%) - cathode
Pt  Ir ( 20%) cathode
This contact composition were tested in the Lab. of Electrical Apparatus of
Kharkov Polytechnic Institute, and it was found, that bridge erosion after
350,000 operations, for the current I  1 A and the voltage uc  8volts , is very small
( 2 1013  51014 cm3 / A2 ), especially for the second contact composition because of
the self-restrained mechanism of the process.
123
CHAPTER 3. ELECTRICAL CONTACT ARC
1. GENERAL MODEL OF ELECTRICAL ARC IN THE FRAME
OF MHD-THEORY
1.1 Review of main theoretical results.
1.2 Phenomena in the arc column.
1.3 Phenomena in near-electrode zones.
1.4 Intra-electrode phenomena
(See Appendix 7)
2. MODELS OF ELECTRICAL ARC EROSION
2.1 Introduction
The arc erosion in opening electrical contacts depend on many factors such
as the range of current and voltage, opening velocity, properties of contact
materials and surroundings, parameters of electrical circuit etc. Erosion of
electrical contacts at low current due to vaporization of material was considered
in many papers [1] – [4]. It was found that direction of transfer of contact
material depends on the mechanism of electrical conductivity in the arc column
and changes from anode to cathode during the transition from the metallic arc
phase to the gaseous arc phase [5] – [9].
It is very important to know information about duration of each phase to
provide minimum of arc erosion due to vaporization. Experimental investigation
of vaporization in dynamics is rather difficult because of very short of
phenomena life time. Therefore mathematical models describing this
mechanisms of erosion seems to be very important. Such models of phenomena
accompanying transition from metallic to gaseous arc are presented in [10] –
[11]. However they are complicated for engineering application and need to be
simplified.
The mechanism of erosion stipulated by gaseous and electromagnetic
pressure with the effect of these and other forces in a liquid melt on the surface
has been considered in [12], but no detailed analysis or method of calculation
were provided. Incomplete estimates of surface tension, as well as convective
and electrodynamic forces were given in [13] – [14]. A rigorous substantiation
of the hypothesis of the thermo-capillary mechanism of contact erosion during
arcing was given in [13].
Mechanism of contact erosion in the solid state was treated for zirconium
carbide in [16] and that for tungsten in [17]. It was confirmed that eroded
particles have distinct crystalline cleavage faces. Some approach for modelling
of this phenomena was given in [18] – [19].
124
This paper is an approach of further development of previous models describing
different types of contact erosion (vapors, droplets, solid particles) as well as
removal mechanism of contact material.
2.2 Vaporization
Mathematical model. Mathematical description of electrical contact
erosion in dynamics due to vaporization has to take into account such
phenomena as ion and electron bombardment of electrode surfaces, electron
emission from the cathode, inverse electron flux and radiation from the arc
column, time-dependent electromagnetic and temperature fields with heat
conduction, melting and vaporization. Dynamics of these phenomena has to be
investigated in dependence on given current and voltage, opening velocity,
inductance of circuit, properties of contact material and surroundings.
A general theoretical model describing dynamics of electric arc was
presented in [10]. However it is rather difficult to apply this model for practice,
therefore simplified approach has to be retrieved from the general model without
losses of main properties of arc dynamics. It can be obtained by replacing of
differential equations for the arc column, sheath , ionization zone, near-anode
region with more simple balance energy equations, while differential equations
remain only for electrodes with given heat fluxes from the arc column.
As a result of arc heat flux a pool of liquid metal forms on the contact
surface.
Fig.1
depicts
the
axial
symmetric
region
D1 ( h (r, t )  z  hm (r, t ), 0  r   (t ) ) occupied by melted metal and the region
D2 ( ( z  0, r  0) \ D2 ) occupied by solid zone of electrode. The heat equation for
both these regions can be represented in the form
Ci  i
Ti
2
 div (i Ti )   i ji
t
(3.2.1)
where Ci ,  i , i ,  i , ji , Ti , and t
are thermal capacity, density, thermal
conductivity, electrical resistivity, current density, temperature, and time
respectively, h (r , t ) and h (r , t ) are the isotherms of evaporation and melting, 
and h0 are the radius and the depth of the melt, f is the radius of arc root, index
i  1 is related to the melted zone, while i  2 - to the solid zone.
At the isothermal surface of vaporization z  h (r, t ) the equilibrium of
heat fluxes given as
125
 1
h
T1
 Q f  L  1 
n
t
(3.2.2)
rA (t )
QA
z  h (r , t )
r
 t 
 t 
0
r
D1
z  hm r , t 
D2
z
Fig. 1 Melted ( D1 ) and solid ( D2 ) regions of electrode
with heat flux from the arc column
Q A  q 0 (t ) exp( 
r2
rA (t ) 2
)
(3.2.3)
where q0 (t ), L , rA (t ), and n are magnitude of heat flux at the center of contact
spot, heat of vaporization, effective radius of normal distribution, and surface
normal respectively.
The rate of vaporization can be described by the Langmuir law
1
h

B

exp( A  ) ,
t
T1
T1
(3.2.4)
where   (2RT / M )1/ 2 ,
RT is the gas constant, M is the vapour molecular
weight, A and B are the constants of vaporization. If we suggest that portion of
entering the electrode heat flux Q (r , t ) consumed for vaporization has normal
distribution along radius
Q  q (t ) exp( 
r2
r (t ) 2
)
(3.2.5)
where q (t ) and r (t ) are magnitude and effective radius of vaporization, then
instead of Langmuir Law (4) one can consider only boundary condition (4)
corrected by the subtraction of the expression (5). Effective radius of
vaporization in this case can be found from the equation
126
T1 ( r (t ), 0, t )  Tb
(3.2.6)
where Tb is the boiling temperature. The magnitudes q0 (t ) and q (t ) are
determined from the energy balance equation for arc column, anode and cathode
surfaces.
At the interface surface z  hm (r, t ) between melt and solid states the Stefan
conditions are applicable
 1
h
T1
T
 2 2  Lm 1 m
n
n
t
(3.2.7)
T1  T2  Tm
(3.2.8)
are the latent heat of melting and melting temperature
where Lm and Tm
respectively.
For closing of this system of equations it is necessary to put additionally
the condition of axial symmetry
T1
 0,
r
if
r 0
(3.2.9)
T1 (0,0,0)  Tm
(3.2.10)
and initial conditions for the temperature
T2 (r, z,0)  T0 (r, z),
where Tm is the melting temperature. The expression for T0 (r , z) can be found
from the solution of heat equation at the period preceding melting [11].
Results of calculations. Copper electrodes. The calculations are carried
out for the Cu electrodes with the current I  100 A, 300 A, 500 A and opening
velocity V0  0.5 m / sec, 5 m / sec, 20 m / sec . The results are presented
in Fig. 2 – 4.
127
Ta , Tc 10 3 0 C
6,00
1
2
3
6
5
5,00
4,00
4
3,00
2,00
1,00
0
0,2
0,4
0,6
0,8
1
1,2
t, msec
Fig. 2 The temperature Ta (0, t ) and Tc (0, t ) at the center of the anode
and the cathode spots respectively as functions of t . Cu - contacts.
1 - I  100 A, 2 - I  300 A, 3 - I  500 A .
For anode:
For cathode: 4 - I  100 A, 5 - I  300 A, 6 - I  500 A
Critical time: t cr  0.4 m sec (300 A), 0.65m sec (500 A), 0.8m sec (500 A)
Just after bridge rupture the temperature at the center of the anode spot
Ta (0, t ) ( Fig. 2 ) jumping up to the value 6 103 K decreases then during the time
less then 1103 sec down to the values comparable with the cathode group
temperature ( Tc  4 103 K ). Herewith the anode ceases active evaporation, and
the arc transforms from the metallic phase into the gaseous phase at the critical
points t cr of intersection of the anode and cathode temperatures. The current
range is very important because the lesser life time of the anode arc corresponds
to the lesser current . Decreasing of the temperature at the center of the anode
spot is a result both of the anode heat flux decreasing owing to increasing of a
contact gap and of evaporation cooling .
Therefore conditions of appearance and evolution of the gaseous arc stage
depend on properties of a contact material such as melting and boiling
temperature , specific heat of vaporization etc., that have to be taken into
account in a mathematical model of a short arc .
Vaporization of material from the arc roots is also very important for the
evolution of a short arc . The effective radius of evaporation from the anode spot
r (t ) increases with current increasing ( Fig. 3 ) , while rv (t ) for the cathode
changes very slowly .
128
r t , 10 6 m
3
3
2,5
2
2
1,5
1
6
5
4
1
0,5
0
t , m sec
0
0,2
0,4
0,6
0,8
1
1,2
Fig. 3 Dynamics of the effective radius of evaporation rev (t ) from the
anode spot
( 1 - I = 100 A , 2 - I = 300 A , 3 - I = 500 A)
and
from the cathode spot ( 4 - I - 100 A ,
5 - I = 300 A , 6 - I = 500 A ) . (
Co - contacts)
The anode rate of increasing
drev
changes from 0.1m / sec to 0. 2m / sec and
dt
approach 0 to the end of the anode arc stage .
The curves of the anode evaporation losses Pev (t ) presented on the Fig. 4
rises rapidly for the first time up to the maximum at t  0.1 - 0.5 msec and then
falls down to zero intersecting the cathode curves at the critical points .
Q a t , Q c t 
10 3 W
10
8
6
6
3
4
2
2
5
1
4
0
t , m sec
0
0,2
0,4
0,6
0,8
1
Pev ( t ) ( 1 - I = 100 A , 2 - I =
Fig. 4 Anode evaporation losses
300A, 3 - I = 500 A) and cathode evaporation losses Qev (t ) ( 4 - I = 100 A , 5
Co - contacts
- I = 300 A, 6 - I = 500 A ) , V0  0. 2m / sec .
129
The maximum point of the anode vaporization is displaced slightly when
the current increases to the greater time . This phenomena can be explained by
the changing of the relationship between the temperature rate and vaporization
intensity . At the initial stage vaporization energy rises owing to the sharp
increasing of the anode local temperature .This time rapidly increasing radius
rv (t ) stipulates an intensive vaporization . After maximum point the temperature
decreases as a result of both vaporization cooling effect and decreasing of heat
flux from the arc due to increasing of contact gap. The vaporization intensity is
very sensible to the change of the temperature because it of exponential
dependence ( 4 ) as well as to a current value . One can conclude from the
Fig.4 that the anode evaporation comes to an end at t = 5 msec for the
current
I = 100 A , while at the same time evaporation for the current I = 500 A is
maximum.
The comparison of the results of these calculations with experimental data
[20] showed satisfactory conformity with the error not more than 20%.
Erosion of AgMeO electrodes. Similar calculations were carried out for
silver metal oxide AgCdO and AgCdO at the conditions : DC I  20 A and
voltage U  14 V , opening velocity V0  0.2 m / sec , inductance
I  40 A ,
L  0 and L  50 mH . These conditions are typical for interruption of current
flow in automobile area as well as in telecommunication.
The results of calculations are presented in Fig. 5 – 7.
130
Mass variation (ng/op)
400
1
2
200
0
-200
1
10
3
4
100
1000
10000
-400
Mass variation (ng/op)
Arc length ( m)
100
1
2
0
1
10 3
-100
100
1000
10000
4
-200
Arc length
( m )
Mass variation (ng/op)
Fig. 5 AgCdO mass variation versus arc length. L  0
mass variation versus arc length. L  0
.
1 - I  40 A ( cathode ), 2 - I  20 A ( cathode ) .
( cathode ), 2 - I  20 A ( cathode )
3 - I  20 A ( anode ), 4 - I  40 A ( anode ).
( anode ), 4 - I  40 A ( anode ).
400
200
0
-200 1
-400
-600
-800
-1000
Fig. 6 AgSnO2
1 - I  40 A
3 - I  20 A
3
1
10
2
100
1000
10000
4
Arc length
( m )
Fig. 7 AgCdO and AgSnO2 mass variation
versus arc length.
L  50 mH , I  40 A
1 - AgCdO cathode ,
2 - AgCdO
3 - AgSnO2 anode ,
4 - AgSnO2
anode,
cathode
These results are in a good agreement with experimental data[9], [22]. Its
analysis enables to conclude that in metallic arc phase (short arc length) , which
131
is characterized by material transfer from the anode to the cathode, the erosion
of AgSnO2 contacts is considerably small than erosion of AgCdO contacts both
for resistive and inductive circuits, while in gaseous arc phase (long arc length)
with opposite material transfer the rate of erosion depends on the inductance. If
the inductance L  0 , then AgSnO2 contacts have smaller erosion in comparison
with AgCdO contacts, however for inductive circuits situation is quite different,
thus use of AgCdO contacts in the case of long arcs burning in gaseous phase is
more preferable.
2.3 Liquid droplets erosion
Side by side with vaporization the another mechanism of contact erosion in
the form of ejected liquid droplets occurs in the range of moderate and high
current. It is stipulated by interaction of various forces in the region of liquid
metal pool, such as electromagnetic and gas-kinetic pressure, surface tension,
explosion and spraying of gaseous inclusions inside molten metal etc.
Mathematical model (1) – (10) presented above has to be corrected to take
into account heat and mass transfer in a thin boundary layer on the molten
surface. The motion equation for the liquid metal in the region D1 can be written
as
V
1
 V  V  P  V  F
t
1
(3.2.11)
and the continuity equation is
 V  0
(3.2.12)
The heat equation (1) for i  1 has to replaced by the energy equation
C1 1 (
T1
2
 V  T1 )  div (1T1 )  1 j1
t
(3.2.13)
while for i  2 it remains the same. Here V (Vr , V z ) is the velocity of molten
metal, P is the gas-kinetic pressure, F  ( 0 /  1 ) j  H is the electromagnetic
force,  and  0 are viscosity and magnetic permeability.
Additionally to the boundary conditions for the temperature similar
conditions must be given for the velocity
Vr
0
z
on the surface
132
z  h (r , t )
(3.2.14)
Vr  V z  0
on the surface
z  hm ( r , t )
(3.2.15)
To simplify this problem we can use the radial distribution of pressure as
P  P0  (1  r 2 /  2 )
(3.2.16)
Using the theory of similarity one can derive that the characteristic time t h
for the heat transfer phenomena is much greater in comparison with the
t for the hydrodynamic phenomena. Hence the
characteristic time
hydrodynamic parts of the above considered equations can be solved separately
using quasi-stationary heat approximation, i.e. assuming that T1 , hm and h are
constants for a given time t .
Applying the law of energy conservation
1

1
V
2 t
2
   2V  E H  E M
(3.2.17)
where the terms on the left hand side correspond to kinetic and potential energy,
while terms on the right-hand side represent the action of hydrodynamic and
electromagnetic forces. Using functional analysis methods the following
approach can be obtained
tA
 1 V 2      V 2 dt  WM  WV
(3.2.18)
0
where t A is the arcing time, and the expressions for E M , E H , WM , WV are given
in the paper [15].
The relationship (18) allows the hydrodynamic and electromagnetic forces
to be estimated without solving the differential equations but comparing each
term on the right side of (18).
This equation becomes more effective by replacing region D1 with the
boundary layer near the vaporization surface z  h (r, t ) . Here the temperature
and velocity fields at vaporization can be described by the system of equations
C1 1
T1
T

2
 (1 1 )   1 j r ,
z
t
z
2
Vr
 Vr

,
t
z 2
Vr Vr V z


 0,
r
r
z
V z
 2V r  0
1 F


jr H 
 1 z
1
t
z 2
(3.2.19)
that can be solved by numerical methods.
The melting isotherms T of a copper cathode under the action of arc heat
flux Q A and Joule sources with current density j are shown in the Fig. 5 for the
parameters
133
Q A  Q0 exp(  i r 2 ) ,
Q0  2.2 10 7 W / m 2 ,
 i  3.18 10 4 m 2
j 0  1.9 10 6 A / m 2 ,
 j  1.3 10 2 m 2
j  j 0 exp(  j r 2 ) ,
The dynamics of melting isotherms during arcing is shown in Fig. 8
0
1
2
3
1
4
2
5
3
6
7
8
r , 10 3 m
4
1
2
3
4
z , 10 4 m
Fig. 8 Dynamics of melting isotherms of convective heat transfer during arcing.
I  100 A , 1  t  0.2 m sec , 2  t  0.4 m sec , 3  t  0.6 m sec , 4  t  0.8 m sec
It enables to estimate erosion in the form of liquid droplets which kinetic
energy is greater than the energy of surface tension.
2.4 Thermo-capillary mechanism of erosion
Some times it is impossible to explain measured values of erosion in the
forms of droplets due to motion of liquid metal from the center to periphery by
the action of electromagnetic and gas-kinetic pressure, especially for such
refractory materials as tungsten, molybdenum, zirconium etc. In this case
erosion phenomena can be explained by the influence of thermo-capillary
Marangoni effect which provokes an intensive convective flow in a narrow
surface layer of melting zone owing to temperature dependence of surface
tension of liquid metal.
To take into account this effect it is necessary to correct the above
presented model of convective heat and mass transfer by replacing of the
condition (14) with the special boundary condition for thermo-capillary forces
causing radial stresses on the melted surface:

Vr
d T1

,
z
dT1 r
z  h (r , t )
(3.2.20)
where  is the dynamic viscosity and  is the surface tension. Beside that,
additional terms E and W responsible for the thermo-capillary effect must be
added in the right side of the equations (17) and (18).
134
In this case, the analog of the thermo-capillary Reynolds number
Re  W10 h0  1 /  ,
W10 
d Tm  T0
, the Prandtl number
dT1 
Pr 

1
and the
Marangoni number Ma  Re Pr play key role.
The melting isotherms of melting tungsten for the same above given
conditions are shown in Fig. 9 taking into account thermo-capillary convection
0
1
2
3
1
4
2
5
3
6
7
8
r , 10 3 m
4
1
2
3
4

z , 10 4 m
Fig. 9
I  100 A ,
Melting
Ma  2 10
isotherms of thermo-capillary heat transfer.
3
It was shown in [15] that for tungsten with current density
j  6.45 10 7 A / m 2 and heat flux Q0  3.2 10 8 W / m 2 the Marangoni number is
Ma  1.13 10 2 , while the rate of thermo-capillary convection V r at the molten
surface reaches 13 m / sec , thus causing the ejection of metal from the molten
pool by the thermo-capillary forces.
Fig. 10 depicts a typical picture of thermo-capillary waves moving from the
center of molten pool to its periphery.
 c is the cathode material density .
Fig.10 Thermo-capillary waves [21]
135
2.5 Solid state type of erosion
Brittle fraction of metallic surface layer caused by thermal stresses
generated by high power current pulses of short duration and concentrated heat
sources from the arc discharge (  i  10 7 m 2 ) can also significantly contribute
to the overall erosion of contacts as the electric arc takes the effect.
The nature of thermal stresses is strongly dependent on the current density
in the area of the arc. At the current density 10 8 A / m 2 and higher, the
temperature rise is due to localized heating in the region under the arc spot,
while at the density 10 7 A / m 2 and lower, the surface heat sources prevail.
The occurrence of thermo-elastic stresses due to Joule heat sources can be
described by the model based on the same heat equation like equation (1):
c
T
 div (T )    j 2 ,
t
r  0,
z0
(3.2.21)
The current density in the case of a pulse can be calculated by solving of
the equation for electric potential distribution in the form [11]:
j (r , z, t ) 
t
1
(  )
4fr

(3.2.22)
where  is the rate of the current growth, f  f (t ) is the arc spot radius growing
over time, and
 [
z 2  (r  f ) 2
z  (r  f )
2
2
]1 / 4
The solution of the equation (21) with the boundary conditions in the form
of heat flux as P0 t entering the contact from the arc is well-known [11]. Using
this solution and thermo-elastic potential, it is possible to find stress
components. Numerical calculation show that thermo-elastic stresses according
to this model become important only when the rate of the current growth is very
high (   10 7 A / sec for tungsten and   10 9 A / sec for copper ). In this case the
stress is highest along the outer rim of the arc spot ( z  0, r  0 ), where
densities of current and heat sources are also highest, whereas in the arc spot
center the stresses are considerably lower as seen from the equation (22).
When the surface sources of heat are considerable, the stress distribution is
opposite. In this case, it is essential to use a simple spherical Holm’s model
describing the temperature field in the electrode rather than a cylindrical one.
Hence, a heat-receiving spot with radius f has to be replaced with a hemisphere
136
having an ideal conductivity with radius b  f 2 , for which entering heat flux is
equal to the value

T (b, t )
 P0 t
r
(3.2.23)
The principal components of the stress tensor can be determined using the
quasi-static approximation from the expressions
 rr 
2 E 0 1
1  r 3
r
     
2
 x T ( x, t )dx ,
b
E 0 1
[
1  r 3
r
x
2
T ( x, t )dx  T (r , t )] (3.2.24)
b
Solution of the equation (21) at the condition (23) is
T (r , t ) 
bP0 t

[erf (   )  2(   )  ierfc (   )] 
(3.2.25)
1
4  exp{(   ) [(   )  (   
(]}
2
2
2
where

b
,
2a t

r
,
( z )  exp( z 2 )  erfc z
2a t
By inserting formula (25) in (24), we find the stress tensor components. If
 y is the yield strength of contact material, the condition excluding the fracture
by temperature in its simplest form is  rr      y , i.e.
E 0 P0 t 0 b 4 
 1  4  2 [1  (1 / 2  )]   y (3.2.26)
1  

where t 0 is the pulse duration.
Let us calculate as an example the thermo-elastic stresses generated in
tungsten contacts as described in [17]. Assuming
I  t 0  1000 A ,
then
b  2.55 10 4 m ,
   2.2 10 9 kg / m 2
,
that
t 0  2 10 6 sec ,
(1 / 2 )  0.94
exceeds
the
f 2  5 10 4 m ,
,
yield
and
stress
consequently
of tungsten
 y  1.3 10 9 kg / m 2 . The experimentally observed fracture of contact material
confirmed the validity of the inequality (26).
Using the expression  y   rr    we can calculate the characteristic
size l 0 of the region exposed to the action of attenuating thermo-elastic wave
that causes the ejection of contact material.
137
 rr    , 10 8 kg / m 2
6
4
 y  1.3  10 8 kg / m 2
2
0
0.5
0.7
0.9
1.1
1.3

Fig. 12 Variation of stress tensor  rr   
As seen from Fig. 12 the inequality
    0.65
with   
 y   rr   
is valid if
r  b  1.3a t . Hence the fracture of electrode occurs at the
i.e.
depth
l 0  1.3a t 0  1.67 10 5 m
It appears that the inequality (26) can be very useful for calculating the
thermal stresses in contacts operating in a pulsing regime.
2.6 Arc to glow transition
For some contact materials at certain circuit parameters (low current and
high inductance) the arc instability may lead to the transformation of arc to glow
discharge rather than to the arc extinction. Fig. 14 depicts such transformation
for all stages of arc root immobility and arc running.
t t=43ms
tG
tA
i(t)
10ms
u(t)
1A
UG
5 ms, d0,6mm
13 ms, d1,6mm
IG
29 ms, d3,5mm
250V
30 ms, d3,6mm
32 ms, d3,8mm
Fig. 14 The unstable arc to glow transition when nickel was used.
This phenomenon was described in detail in [13]. It was found that such
transition appears in low current inductive circuits and accompanied by a step of
138
spasmodic voltage increase and current decrease with duration 107 106 s. (Fig.
15).
UA, V
IA, A
0.25
500
400
UA
0.2
0.15
300
IA
200
0.1
0.05
100
0
0
1
2
3
4
t,  s
Fig. 15 Transition voltage and current. AgCdO contacts,
I  0.22 A Dashed- calculation, solid- experiment
cr
At certain conditions arc stage duration becomes much smaller than glow
stage duration. The problem is to find criteria and optimal choice of
interdependent parameters (material properties, current, voltage, resistance,
inductance, pressure, opening velocity etc.) providing arc instability and
controlled arc-to-glow transformation.
Such information is very important, because new resources for diminution
of failure and for enhancement of time life and reliability of electrical contacts
may be found due to reduction of arc duration at the expense of enlarging of
glow duration, which burns practically without erosion. The conditions of arc
instability from electrical point of view in terms of circuit parameters are
discussed in [13].
2.7 Dependence of the arc erosion on the current frequency
The rate of the arc erosion in opening electrical contacts at ac current has
non-monotonic dependence on the frequency f . It increases in the range
50Hz  f  500Hz and then decreases for f  500Hz . This phenomenon can be
explained by the redistribution of the components of the arc heat flux between
anode and cathode during the transition from the metallic arc phase to the
gaseous arc phase [34], [48]. Shifting of the arc temperature maximum in time
entails the change of the time duration t when the arc temperature is greater
139
than the temperature of gas ionization (  50000 K ) at which evaporation is more
intensive (Table 2).
Table 2. Dependence of t on the current frequency
f , Hz
t , ms
50
49
250
67
500
78
750
62
1000
36
The table 3 gives the information about relationship between current
frequency and evaporated domain.
Table 3. Dependence of erosion on the current frequency
Frequency
Hz
Mass, mg
50
0.54
250
0.73
500
0.75
750
0.48
1000
0.23
The comparison of these values with the experimental data [34] confirms
the conclusion about the non-monotonic dependence of arc erosion on the
current frequency.
3 ARC DURATION AND CONDITIONS OF ARC INSTABILITY
Investigation of dynamical arc phenomena in opening electrical contacts is
very important for performance build-up of circuit breakers by means of
decrease of arc duration and erosion. Mayr’s and Cassie’s models [1] and their
generalization [2] based on the power balance method are not applicable to
describe arc temperature field at the initial arc stage just after arc ignition.
Elenbaas-Heller equation gives information about radial distribution of the arc
temperature however it is correct for stationary arcs only [3]. Arc dynamics
should be described by transient heat equation taking into account nonlinear arc
characteristic. It is the first intent of this paper. The second one is to device a
method for calculation of arc erosion in dynamics.
\
140
3.1 Mathematical model of arc temperature
and conductivity at metallic arc phase
Equation for the temperature. The arc temperature  (r , t ) in opening
contacts just after ignition is less than the threshold value required for gas
ionization, ig , however it is sufficient to ionize metallic vapours in the contact
gap, which takes place at the temperature im :
im    ig
This initial stage, called metallic arc phase, has very short duration and
occurs in a small contact gap. Therefore the arc takes the form of a disk, which
thickness is much less than radius, and the axial temperature component can be
neglected in comparison with radial component. In this case the heat equation
for the arc should be written in the form
C
 1 


( r )   E 2  Wr
t r r
r
(3.3.1)
where C and  are thermal capacity and density,  and  are heat and
electrical conductivities of the arc plasma, E is electrical field and Wr is power
loss due to arc radiation and heat conduction from arc column to electrodes. The
initial temperature distribution along radius
 (r,0)  f (r )
(3.3.2)
can be found from the solution of the heat equation for metallic vapours at the
pre-arcing stage [4]-[5]. We can approximate the function f (r )  0 J 0 (1r / rA ) by
parabola
f (r )  0 (1 
r2
)
rA2
or by the Bessel function
f (r )  0 J 0 (1r / rA )
(3.3.3)
where   2.405 is the first root of the Bessel function and  0 is the temperature
maximum at the centre of arc disc.
The temperature on the interface r  rA between ambient air and arc plasma
should be equal to threshold of metal ionization
1
 (rA , t )  mi
(3.3.4)
It should be noted that thermal and electrical plasma conductivities,  and  ,
depend essentially on the temperature and this dependence can not be averaged.
141
In contrast the arc radiation Wr can be neglected for metallic arc phase, which
temperature is relatively low : im < ig  5000 0C (See Fig. 1).
 , C ,Wr
6
5
1
4
2
3
2
1
0
3
5
6
7
8
9
10
T , 103 K
Fig,1 Temperature dependence of  ,  and Wr
1-  , Wm1K 1 ; 2- C, 102 Jm3 K 1
3- Wr , 1011 Wm3 [6]
Equation for electrical conductivity  . To solve the heat equation (1) we
use the Kirchhoff’s substitution

  ( )d
S ( ) 
(3.3.5)
mi
Then the equation (1) transforms to
C S 1  S

(r )   E 2  Wr
 t r r r
(3.3.6)
Solving the equation (5) with respect to  we get
  mi  g (S )
(3.3.7)
Since the function    ( ) is given (See Fig. 1), we can write this function
in term of  using (6), i.e.    (S ) . Linearization of this function gives the
expression (See Fig. 2)
g
(3.3.8)
  bS ,
b  tan  
S gi
where  g is given electrical conductivity at the transition from metallic arc
phase to gaseous arc phase, when    gi , and

S gi 
gi

 ( )d
mi
142
(3.3.9)
 (S )
g

S gi
0
S
Fig. 2 Linear approximation of  (S )
Substituting (8) in (7) and using notation
r
x
E b
, 
C
 E 2b
(3.3.10)
we can write the equation with respect to 

  2 1 
 2 

t
x
x x
(3.3.11)
It should be noted that  can be considered as constant because the thermal
diffusivity a2  C /  is approximately constant (See Fig. 1). The domain for this
equation is 0  x  x0 , where x0  rA E b .
The boundary conditions (2)-(4) transform to the type
(3.3.12)
 ( x,0)  F ( x) ,
with
f (x/ E b)
F ( x)  b


 ( )d
mi
 ( x0 , t )  0
(3.3.13)
The solution of the problem (11)-(13) can be found in the form of FourierBessel series

 ( x, t )   Cn exp[(kn 2  1)t /  ]J 0 (k0 x)
n 1
where
143
(3.3.14)
Cn 
x0
2
x0 2 J12 ( n ) 0
F ( x) J 0 (kn x) xdx ,
kn  n / x0 ,
and n are roots of the Bessel function:
J 0 (n )  0,
n  1, 2,3,...
For approximation (3)
 ( x,0)   0 J 0 (1 x / x0 )
and the solution (14) takes the simple form
 ( x, t )   0 exp[(
12
x0 2
 1)t /  ]J 0 ( 1 x / x0 )
Taking into account (10) we get finally the expression for arc electrical
conductivity in the form
a 2t
 (r , t )   0 exp[(   E brA ) 2 ]J 0 ( 1r / rA )
rA
2
1
2
2
(3.3.15)
The arc temperature can be found now from the expressions (5) and (8).
Let us introduce the criterion of arcing
  E 2brA2  12
(3.3.16)
We should distinguish three cases (Fig. 3):
 (0, t )
 0
0
 0
 0
t
Fig. 3 Evolution of arc conductivity
1).   0 . Rise of arc conductivity, power and
temperature due to Joule heating.
2).   0 . Maximum value of arc conductivity and power
3).   0 . Arc conductivity, power and temperature
decrease, thus the arc should extinguish.
144
Interaction between arc and contact surface. At the first stage of contact
opening   0 and then changes the sign. To find the critical point   0 we need
to know the dynamics of arc radius rA , which expands during arcing. Then using
formula
E
I
 rA2
(3.3.17)
and the expression (16) we can find the critical time t  tcr at which   0 . For
this purpose we consider the region DA occupied by arc interacting with contact
surface (Fig. 4). This interaction results into phase transformations of contact
material and formation of three zones:
1)
The zone of evaporated material Db : 0  r  rb (t ), 0  z   b (r, t ) ,
Dm :  b (r , t )  z   m (r , t ),
2)
The zone of melted material if 0  r  rb (t ), and
0  z   m (r , t ) if rb (t )  r  rm (t )
3)
The solid zone
Ds :  m (r , t )  z  , if 0  r  rm (t ),
and 0  z   if rm (t )  r  
x(t)
rm (t )
z   m (r , t )
Ds
Dm
Db
rb (t )
rA (t )
0
zb (t )
zm (t ) z
z   b (r , t )
Fixed contact
DA
Movable contact
r
Fig. 4 The arc and contacts geometry: arc region
DA , evaporated zone Db , melted zone Dm and
solid zone Ds
The contact temperature TC (r, z, t ) can be presented as the sum
TC (r, z, t )  TJ (r, z, t )  TS (r, z, t )
145
(3.3.18)
where TJ (r, z, t ) and TS (r , z, t ) are the temperature components due to volumetric
Joule heating and due to surface arc flux heating respectively. The expression
for calculating of the first component is given above. It can be shown that the
Joule component TJ (r, z, t ) is important at the pre-arcing stage only, and it can be
neglected after arc ignition. The expression for the second component can be
found similarly in the form
t

0
0
TS (r , z, t )   dt1  [ Pc (r1 ,t1 )  Pb (r1 , t1 )
(3.3.19)
 Pm (r1 , t1 )]G (r , r1 , z, t  t1 )r1dr1
Here Pc (r , t ) is the total heat flux (power per unit area) entering the contact
surface during arcing, Pb (r , t ) and Pm (r , t ) are portions of this flux consumed for
evaporation and melting of contact material, which can be found by the
expressions
Pb (r , t )  Lb
 b (r , t )
 (r , t )
, Pm (r , t )  Lm m
t
t
(3.3.20)
where Lb and Lm are specific heat for evaporation and melting,  is density of
contact material,
It reasonable to assume that the isothermal surfaces z   b (r, t ) and
z   m (r , t ) are ellipsoids of revolution that can be found from the equations
r2
z2

1
rb (t )2 zb (t ) 2
r2
z2

 1,
rm (t )2 zm (t )2
in other words
 b (r , t )  zb (t ) 1  r 2 / rb (t )2
 m (r , t )  zm (t ) 1  r 2 / rm (t )2
(3.3.21),
The functions rb (t ) , zb (t ) , rm (t ) , and zm (t ) should be found from the
equations
TC (rb (t ),0, t )  Tb ,
TC (0, zb (t ), t )  Tb
TC (rm (t ),0, t )  Tm , TC (0, zm (t ), t )  Tm (3.3.22)
where Tm is the melting temperature of the contact material.
If the heat fluxes Pc (r , t ) Pb (r , t ) , Pm (r , t ) obeys the normal Gauss’s radial
distribution
Pc (r , t )  Pc (t ) exp(
146
r2
)
rA (t )2
Pb (r , t )  Pb (t ) exp(
r2
)
rA (t )2
Pm (r , t )  Pm (t ) exp(
(3.3.23)
r2
)
rA (t )2
then the integral with respect to r in the formula (19) can be calculated and the
expression for the contact temperature becomes more simple form
TS (r , z , t ) 
[ Pc (t1 )  Pb (t1 )  Pm (t1 )]rA (t1 ) 2
  0 [rA (t1 )2  4a 2 (t  t1 )] t  t1
t
a
z2
r2

exp[ 2
]d
4a (t  t1 ) rA (t1 ) 2  4a 2 (t  t1 )
(3.3.24)
The heat flux Pc (t ) should be calculated taking into account positive
components due to arc radiation, electron (or ion) bombardment of anode
(cathode) contact surface, inverse electrons from the arc column, and negative
components due to power losses for evaporation, radiation, electron emission
cooling and heat conduction inside the contact body. The expressions for all
these components can be found in the paper [7]. However the model in
considered case can be simplified because the information about current, voltage
and displacement is available from experiment. Therefore it is more convenient
to calculate power generated by arc WA directly from the measured values of arc
voltage U A (t ) , arc current I A (t ) and then arc heat flux entering contact is
Pc (t ) 
I A (t ) U A (t )
P (t )
 A2
2
2 rA (t )
2 rA (t )
(3.3.25)
This expression is the final equation, which enables in the aggregate with
other cited above equations to calculate dynamics of contact melting,
evaporation, arc radius rA (t ) and arc power PA (t ) .
Fig.5 and Fig. 6 depict dynamics of arc power and temperature for AgCdO
contacts calculated using above considered model at the conditions: supplied
voltage U 0  14 V , current I 0  20 A , inductance L  47.5 mH , opening velocity
V  0.2 m / s [2]
PA (t ), W
400
350
300
250
200
150
100
50
0
Fig. 5 Dynamics of arc power PA (t )
0
5
10
15
20
25
30
35
40
t , ms
147
TA , 103 K
8
7
6
5
2
4
1
3
1
0.1
10
t , ms
100
Fig. 6 Arc temperature. 1 – Experimental data [2], 2 – calculation
One can see that critical time in this case is tcr  10 ms , however the
maximum of arc temperature occurs a little bit later, at t  15 ms due to thermal
inertia.
3.2 Transition from metallic arc phase to gaseous arc phase
Temperature field and erosion. The duration of metallic phase is very
short, therefore the arc thickness is still small and above considered model can
be applied to describe the transition from metallic to gaseous phase if we replace
all parameters of metallic vapours by parameters of gaseous vapours. Dynamics
of this transitions is represented in Fig. 7. One can see that at the first stage of
arcing, when the contact gap does not exceed 20 m , anode temperature rises
very sharp in comparison with cathode temperature.
Ta , Tc , 103 K
4
Ta
3
Tc
2
1
0
0
0.1
0.2
0.3
0.4
0.5
t , ms
Fig. 7 Dynamics of anode and cathode temperature at the
centre of arc root.
It can be explained by the fact that in a short arc, which length is
comparable with the length of ionization zone, electron temperature Te is much
greater than ion temperature Ti , therefore kinetic energy of electrons
148
bombarding anode,
3 kTe
je , exceeds significantly kinetic energy of ions entering
2 e
3 kTi
ji . Moreover, calculation shows that in this range of contact gap
2 e
electron component of current density je is much greater than ion component ji ,
cathode,
that is an additional reason for anode overheating and material transfer from
anode to cathode. However intensive evaporation from anode and increasing of
anode arc spot radius, that entails decreasing of current density, cause anode
cooling and decreasing of its temperature, while cathode temperature continues
to increase. The point of intersection of anode and cathode temperature
occurring at tac  0.15 ms corresponds to change the direction of material transfer
for inverse and to beginning of compensation arc stage, which continues up to
t1  1.8 ms and transforms then into cathodic stage (Fig. 8).
Calculation enables to conclude that cathodic arc stage begins in metallic
phase with temperature about 4700 K, that is less than threshold ionisation,
however transition to gaseous phase occurs just at t1  2 ms . Results of calculated
erosion given in Fig. 8 are evidence of the fact, that the main portion of erosion
in inductive circuits occurs in gaseous phase. Calculated values for metallic
phase are slightly greater than experimental data. It can be explained by
recycling phenomenon, i.e. re-deposition of evaporated material, which is
ignored in the mathematical model.
M ea , M ec 107 g
4
5
3
0
2
-5
1
-10
-15
-20
-25
-30
0.001
0.0
1
0.
1
1
10
t, ms
100
Fig. 8 Anode and cathode mass transfer.
From bottom to top:
1 - M ec (calculated),
2 - M ec (experiment [2]),
3 - M ea (experiment [2]), 4 - M ea (calculated)
Influence of inductance on arc duration. Similar calculations were carried
out for different values of inductance in the range from 1 mH to 400 mH. It was
found that arc duration increases proportionally inductance and depends on
current at relatively small values of inductance (Fig. 9). However for inductance
149
greater than 10 mH this dependence becomes negligible. This result correlates
with experimental data observed in [2].
tA , s
100000
10000
100
100
1
3
1
2
1
0.
0.001
0.01
0.
1
1
100
L, mH
Fig. 9 Arc duration versus inductance
1 - I 0  0.6 A , 2 - I 0  1 A , 3 - I 0  20 A
Increasing of arc duration with inductance occurs on account of enlarging
of gaseous arc phase, while variation of metallic phase is relatively small. The
same conclusion may be proposed for increasing of erosion. However further
increasing of inductance up to a few hundred millihenry in the range of low
current leads to decrease arc duration and erosion due to arc-to-glow
transformation, which is considered below.
3.3 Phenomena in thermo-chemical cathodes
The current density at the cathode is one of important characteristics
responsible for the evolution of electrical arc between opening contacts. In the
case of refractory materials such as tungsten and carbon, the normal cathode
spot reaches temperatures of about 3000 o K – 4000 o K, which is sufficient for
thermionic emission of observed current density, maintaining of evaporation and
arcing. The problem appears when this model is applied for explanation of the
rate of erosion for Cu , Zr , Hf and other materials used for thermo-chemical
cathodes of plasma generators. In this case it is not possible to explain
mechanism of erosion, because the basic Richardson-Dushman formula even
with Schottky correction gives too small values for the current density of
electron emission in comparison with experimental observed values. The reason
of such discrepancy is stipulated by the too large value of  eff which varies from
3 eV to 4 eV . Thus, some other mechanism has to be involved to explain required
level of current density or decreasing of the value of effective work function  eff
. Many attempts to ground the probable diminution of the work function were
undertaken, such like electron affinity of the semiconductor [1], electrical field
enhancing factor due to asperity peaks on the cathode surface [45], I-effect [46]
or even by redoubling of Richardson constant [47]. However they could not
explain observed experimental data. High values of current density for such
materials like Cu, Fe, Ni similarly as for thermo-chemical cathodes Zr, Hf etc can be
150
explained by the phenomenon of electrochemical emission of electrons [31],
[33]. This phenomenon was observed first in experiments with thermo-chemical
zirconium cathode (TCC) used for plasma generators [47]. It is caused by the
formation of a solid or liquid electrolyte on the contact surface after its melting.
The anode oxidation of the material occurs on the outer side of the surface with
electrochemical ejection of electrons, while the cathode reduction to the initial
material takes place on the inner side of the surface. If the rates of
electrochemical oxidation and reduction are equal, then the constancy of
electron emission is provided. Herewith the energy required for electron
emission is equivalent to the energy of chemical reaction spent for the
generation of an electrolytic film. Thus, heat energy of chemical reactions spent
for generation of oxides is compensated by electrical energy for their
destruction.
Fig 21 depicts clearly outlined zones formed in TCC. External surface
(zone 1) consists of zirconium dioxide. The next (in depth ) zone 2 contains
smaller amount of oxygen, and this amount decreases even more in zone 3 due
to appearance of metallic zirconium globules in it. Zone 4 is identified as
zirconium nitride. Oxides of zones 2 and 3 are amorphous, that testifies to a
liquid aggregate state during operations, while the nitrides in zone 4 have
crystalline structure. Below oxide layer (zone 5) there is metallic zirconium
having no phase transformation during arcing because oxygen does not penetrate
into this zone.
Fig 22. Cross-section of zirconium cathode.
1 – crystalline zirconium dioxide ZrO2
2 – amorphous molten layer of ZrO2  Zr3O
3 – eutectic of ZrO2  Zr3O and globules
of metallic zirconium
4 – zirconium nitride ZrN
5 – zirconium Zr
151
A general model describing interdependent phenomena in electric arc
column, anode and cathode ionization zones sheathes and bulks during opening
of electrical contacts was presented in the papers [31], [33]. However chemical
reactions and diffusive phenomena inside cathode bulk were neglected in this
model. To take them into consideration the corresponding subsystem of
equations for cathode bulk has to be modified and presented in the form
1
div (  )  0
(3.3.26)

(CvT )
1
(3.3.27)
 div (T )  FT (C , T , X )  ( ) 2
t

C
(3.3.28)
 div ( DC )  FC (C , T , X ) ,
t
where  , T , C , and X  X 1 ,..., X n are electrical potential, temperature inside
cathode, concentration of diffusing gas, and the quantity of the substance of
each i -phase X i (r , z, t ) (mol / m 3 ) respectively,
n
FT (C , T , X )   riT
i 1
n
FC (C , T , X )   riC
i 1
X i
t
X i
t
and
are heat and concentration sources density of chemical
and riC are specific heat of reaction and stechiometric
reactions, riT
coefficients,  , Cv ,  , and D are electrical resistivity, thermal capacity, thermal
conductivity, and diffusion coefficient. In particular, for above considered
Zr  O system n  5 (See Fig. 22). Dynamics of X i quantities is describing by
the system of differential equations
dX i
 K i ( X i  X eq ,i )i
(3.3.29)
dt
the balance quantities of the i -th phase determined from a

where X eq ,i
respective state diagram.
Similar phenomena were observed on the copper cathode of plasma
generator with arc root on the oxidized surface, because the electron work
function from this place is less then from the pure copper.
The copper oxides have strong polar covalent coupling and properties of a
semi-conductor. They are able to furnish the ion electrical conductivity in thin
films for providing of the emission current. The oxidation on the heated copper
takes place accordingly to the scheme
2Cu  O2  2CuO , 4Cu  O2  2Cu 2 O
Since isobaric-isothermal potential g for Cu 2 O is equal to  146.2  103 J / mol ,
while for CuO it is equal to  118.7  103 J / mol , so generation of CuO is more
preferable from thermodynamic point of view. From the other side
electrochemical reduction of copper oxides to metal can occur on the interface
between cathode surface and plasma accordingly to the scheme:
152
2CuO  4e  2Cu 0  O2  4e ,
2Cu 2 O  4e  4Cu 0  O2  4e
It means that relay-race transmission of electrons from the cathode to the
plasma occurs at the electrochemical reduction of copper oxides to metal. Thus
side by side with thermionic component of electron current density j em there
exists electrochemical component jec , and the total current density jet is defined
by the formula
jet  jem  jec
(3.3.30)
Consequently the mathematical model presented in the paper [8] has to be
corrected by replacing the expressions for energy of emitted electrons
1
3
2
meVem  kTc  eU c ,
2
2
for electron current density
jem  nem eVem ,
as well as Richardson-Dushman formula for thermionic emission
jem  ATc exp( 
2
e eff
kTc
)
(3.3.31)
with corresponding new expressions taking into account also electrochemical
mechanism of electron emission:
1
3
2
meVem  kTc  eU c  e ec ,
2
2
jet  nem eVem  nec eVec ,
(3.3.32)
(3.3.33)
and with formula (3.43) for total current density.
The electrochemical component of current density jec can be evaluated
using Faraday’s law and Arrhenius law, which in simplest case is defined as
K  K 0 exp( 
Ea
).
RTc
Then
jec  jec exp( 
0
Ea
),
RTc
(3.3.34)
where
jec 
0
ZFCK 0
rc 2
(3.3.35)
It is possible to simplify the expressions (3.43)-(3.48) if we suggest that
activation energy E a is equivalent to the energy required for destruction of
oxides:
153
Ea  e p
where  p  
g
is the potential required for the electrochemical mechanism of
ZF
electron emission.. Using the value g one can estimate the corresponding value
of this potential At the standard conditions  p  0.53 V for CuO and  p  0.76 V
for Cu 2 O . Taking into account the theory of mirror image for the electron
emission from the cathode surface to plasma these values must be doubled. Thus
finally  p  1.06 V for CuO and  p  1.52 V for Cu 2 O .
The analysis of the solution shows that this potential  p can be considered
as correction of effective work function in Richardson-Dushman formula (3.44)
to take into account electrochemical mechanism of electron emission:
jet  ATc exp[ 
2
e( eff   p )
RTc
]
(3.3.36)
In this case the interval of corrected potential of electron emission
 et   eff   p for Cu  contacts is 3.0  3.5 V , and that is enough to maintain arc
and explain experimental observed electron current density.
3.4 Arc to glow transition
For some contact materials at certain circuit parameters (low current and
high inductance) the arc instability may lead to the transformation of arc to glow
discharge rather than to the arc extinction. Fig. 14 depicts such transformation
for all stages of arc root immobility and arc running.
t t=43ms
tG
tA
i(t)
10ms
u(t)
1A
UG
5 ms, d0,6mm
13 ms, d1,6mm
IG
29 ms, d3,5mm
250V
30 ms, d3,6mm
32 ms, d3,8mm
Fig. 14 The unstable arc to glow transition when nickel was used.
This phenomenon was described in detail in [13]. It was found that such
transition appears in low current inductive circuits and accompanied by a step of
spasmodic voltage increase and current decrease with duration 107 106 s. (Fig.
154
15).
UA, V
IA, A
0.25
500
400
UA
0.2
0.15
300
IA
200
0.1
0.05
100
0
1
2
3
4
0
t,  s
Fig. 15 Transition voltage and current. AgCdO contacts,
I  0.22 A Dashed- calculation, solid- experiment
cr
At certain conditions arc stage duration becomes much smaller than glow
stage duration. The problem is to find criteria and optimal choice of
interdependent parameters (material properties, current, voltage, resistance,
inductance, pressure, opening velocity etc.) providing arc instability and
controlled arc-to-glow transformation.
Such information is very important, because new resources for diminution
of failure and for enhancement of time life and reliability of electrical contacts
may be found due to reduction of arc duration at the expense of enlarging of
glow duration, which burns practically without erosion. The conditions of arc
instability from electrical point of view in terms of circuit parameters are
discussed in [13].
3.5 Dependence of the arc erosion on the current frequency
The rate of the arc erosion in opening electrical contacts at ac current has
nonmonotonic dependence on the frequency f . It increases in the range
50Hz  f  500Hz and then decreases for f  500Hz . This phenomenon can be
explained by the redistribution of the components of the arc heat flux between
anode and cathode during the transition from the metallic arc phase to the
gaseous arc phase [34], [48]. Shifting of the arc temperature maximum in time
entails the change of the time duration t when the arc temperature is greater
than the temperature of gas ionization (  50000 K ) at which evaporation is more
intensive (Table 2)
155
Table 2. Dependence of t on the current frequency
f , Hz
t , ms
50
49
250
67
500
78
750
62
1000
36
The table 3 gives the information about relationship between current
frequency and evaporated domain.
Table 3. Dependence of erosion on the current frequency
Frequency
Hz
Mass, mg
50
0.54
250
0.73
500
0.75
750
0.48
1000
0.23
The comparison of these values with the experimental data [34] confirms
the conclusion about the non-monotonic dependence of arc erosion on the
current frequency.
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164
APPENDIX 1. MODELLING AND CALCULATION
OF THE CONTACT RESISTANCE COMPONENTS
A1.1 Constriction resistance. The contact resistance components are defined in terms of electrical
conductivity components
 c , m , f
by the expression (1.1.5). To calculate these conductivity
components, we use to consider both contact surfaces as planes covered with a multitude of spherical
asperities with the same radius rm and the variances of height distributions  1 and  2 for each of the
contacting surface ( Fig. A-1.1).
z
r0
rm  z  d
x
rm
d
rm  z
0
Fig A-1.1 Simplified schematic model of a micro-asperity
The contact between such rough surfaces can be replaced by the contact of plane surface with model rough
surface, if the variance of height distribution  of the model surface is chosen as
   12   2 2
165
(A.1.1)
n of contact spots per unit area of the nominal
Assuming Gaussian asperity height distribution, the number
contact surface can be expressed as

d
 2
,
 
z
n
N
 2
z
z
2
 exp( 2
2
)dz 
d
N
(erfc   erfc   ) ,
2

( A.1.2 )
 2
where N is the total number of asperities defined by the function of the surface finish, d is the distance

( gap ) between the reference planes, z is the maximal asperity height of the model plane . The special
function in the right side is determined by the expression

2
exp( z
 
erfc  
2
)dz
If the spots are uniformly distributed over the contact surface, and the distance 2r between two
individual spots is much greater than the radius of a spot r0 , then the radius r of the constriction for each
individual spot is given as
r
1
2 n
 [2 N (erfc   erfc   ]1/ 2
(A.1.3)
and the constriction resistance can be expressed as [1]:
Rc ( z ) 
  ( 1  2 ) / 2 , 1
where
Thus
and
2
N
c 
 2
From Fig. A-1.1 one can derive
arctan

r
arctan
 r0
r0
,
(A.1.4)
are electrical resistivity of the contacting members.
z
1
z
exp( 2 )dz
2
c ( z)
R
d
1
r0  ( z  d )(2rm  z  d ) , and since r  r0 , z  d  2rm , then
r 
and

r0 2
r0  2rm ( z  d )
(A.1.5)
Therefore, we obtain for the constriction conductivity the following expression
c  4 N
2rm
3 
( 2)1/ 2 (     ) exp[(  )2 ]  o(    )
(A.1.6)
A1.2 Micro-asperity resistance. Bearing in mind that the asperity radius at the point x (Fig. A-1.1) is
h  (rm  x)1/ 2 (rm  x)1/ 2 , the resistance Rm ( z ) of an asperity of the height z is given as

Rm ( z ) 

rm  z  d

rm  z
z (2rm  z  d )
dx


ln
(rm  x)(rm  x) 2 rm ( z  d )(2rm  z )
Hence, the conductivity of asperities
m
is given as
166
(A.1.7)
m 
z
N
 

d
1
z2
exp( 2 )dz
Rm ( z )
2
This integral can be approximated if
(A.1.8)
( z   d )  d . Then the expression (A.1.8) becomes
 0 2
(2  0   )(     )
exp(  2 ) li (
)
 m  4 N
 (2  0   )
0 
2
where
0 
rm
 2
and
dt
ln t
0
li ( x)  
is the integral logarithmic function .
The contact resistance of an oxide film for a single spot
Rf 
ft
 r0
2

R( z ) is structure-
Cu2O this resistance can be expressed as:
dependent. For conductive oxide films, such as
f
(A.1.9)
x
A1.3 Film resistance.
where
,
ft
2 rm ( z  d )
 o(
1
),
zd
(A.1.10)
t is its thickness. Hence, the oxide film conductivity is
is the resistivity of oxide film and
z
Nr  2
N 2 rm
z2
f 
(
z

d
)
exp(

)dz  m
(ierfc   ierfc  * ) , (A.1.11)
2

2
ft
 2  f t d
where
ierfc   i erfc  
1
2


 ( z   ) exp( z
2
)dz 
e 
2

  erfc  ,
and in general
i erfc  

2
1
( z   ) exp( z 2 )dz


(


1)


for any real  .
In the case of thin (adhesive or passivating) films whereby the conduction occurs via tunneling, the film
conductivity can be expressed as
f 
N rm 2
f
( ierfc   ierfc   ) ,
(A.1.12)
where  f is the tunnel resistivity.
For practical use of the above described model it is essential to know the values for  and d . The
variance of the height distribution of asperities  can be determined from the surface profile-metric traces,
while d can be derived from the expression relating deformation z  d of the i  th asperity with the
initial height
z by load Pci [10]:
E rm
4
,
Pci  ( z  d )3/ 2
3
1  2
(A.1.13)
where E is the elastic modulus,  is the Poisson ratio.
It has to be noted that this formula combined with the formula (A.1.5) gives well-known Hertz formula
for a plane-spherical contact:
r0  1.2
3
Pc
rm (1   2 )
E
167
Hence, the total load
Pc acting on a unit area of the contacting surface can be derived by integration of
(A.1.13) with Gaussian distribution density:

3/ 2
z
2 rm 3/ 2
4 NE rm
z2
4 NE
3/ 2
Pc 
(
z

d
)
exp(

)
dz

(i erfc  -i3/2erfc  * )
2
2
2 
3   (1   ) d
2
3
(1   )
(A.1.14)
The graph of the function
f (  )  i erfc  is given on the Fig. A-1.
3/ 2
f ( )
0.4
0.3
0.2
0.1

0
0.4
0.6
0.8
Fig. A-1 The graph of the function
Hence if the applied contact force
1.0
1.2
f (  )  i3/ 2erfc 
Pc is given, then the values for  ,  and d can be derived from this
tra
168
Scientific publication
Stanislav Nikolaevich Kharin
APPENDIX 2. SOLUTION IN SPHERICAL MODEL
A.2.1 Radial temperature and potential distribution. The current density in the spherical model is
j
I
,
2 r 2
(A.2.1)
thus Joule heat power per unit volume is
1

grad 2   j 2 
I 2
4 2 r 4
The equation (1.2.22) for a stationary temperature distribution (
(A.2.2)

 0 ) in spherical co-ordinates taking into
t
account expression (A.2.2) can be written in the form
1 d 2 d
I 2
(
r
)

0
r 2 dr
dr
4 2 r 4
(A.2.3)
The boundary conditions for the equation (A.2.3) are very simple. The surface of ideal sphere r  b is a natural
adiabatic boundary between two contact pieces due temperature symmetry, therefore
d (b)
(A.2.4)
0
dr
while outside the constriction contact zone ( r   ) the super-temperature (i.e. the real temperature minus
temperature of the surroundings) remains the same like before heating:
 ()  0
(A.2.5)
If we assume the linear temperature dependence of resistivity
  0 (1   )
(A.2.6)
then the solution of the equation (A.2.3) with the boundary conditions (A.2.4) and (A.2.5) is
 (r ) 
1

cos

2
[
cos
(1  b / r )

 1]
(A.2.7)
2
where

0
 b 
I
(A.2.8)
2
169
Using this formula the gradient of potential can be represented as
I
d 0 (1   ) I

 0 2
2
dr
2 r
2 r

cos
2
cos
(1  b / r )

2
and after integration we obtain the expression for the potential in the form

0 I sin 2 (1  b / r )
 (r )  2

 b
cos
(A.2.9)
2
A.2.2 Simplified models.
In the range of relatively small temperature electrical resistivity can be averaged
and considered as constant:  ( )    const . Then corresponding expressions for the temperature  0 (r )
and the potential
:
 0 (r )
can be derived from (A.2.7) and (A.2.9) using limit as
I 2 1
b
(1  )
2
4 b r
2r
I
b
 0 (r ) 
(1  )
2 b
r
 0 (r ) 
The value of

and replacing
0
by
(A.2.10)
(A.2.11)
can be obtained by comparison of the real constriction resistance with modelled constriction
resistance for the constant averaged
The resistance
 0
.
R(r ) of a spherical shell between two hemispheres with radii b and r respectively is
R(r ) 
 (r )
I

0 sin 2 (1  b / r )
 2

 b
cos
(A.2.12)
2
thus the resistance of one contact member
R ( ) 
0
 2b

tan

2
(A.2.13)
and the total constriction resistance
Rc  2 R() 
Similar values for the averaged

2 0
 2b
tan


2
(A.2.14)
can be calculated from (A.2.11) as
R0 () 
 0 ( )
I


2 b
170
(A.2.15)
2 0 () 

I
b
R0c  2 R0 () 
(A.2.16)
Equating (A.2.14) and (A.2.16) we get the expression for the replaced averaged resistivity
1  2  0
If

(hence
tan
 in the form

tan

2
(A.2.17)

2  1  1 ( / 2) 2 and one can derive from (A.2.8), (A.2.10)
 / 2
3
 as well) is small, then
the well known formula [1]
2
3
 2  0 [1   0 (b)]
(A.2.18)
However much better result, especially for high current, can be achieved by replacing
2
3
3  0 [1   (b)]
 0 (b) for  (b) :
(A.2.19)
It should be noted that temperature calculation using expressions (A.2.7) and (A.2.10) gives very similar results,
especially in the range of not too high temperature.
A.2.3 Kohlraush’s law If  (r ) and  (r ) are the temperature and electrical potential corresponding to a
hemisphere of the radius r , then equating power of heat flow passing through the surface of hemisphere and
electrical power generating this heat flow we obtain:
  2 r 2
d
 I   (r )
dr
This balance formula can be checked also by direct substitution of the expressions (A.2.7) - (A.2.9). Multiplying
both sides by the factor
 ( )
2 r 2
and taking into account that

Integration from b to
I  ( ) d
we get

2 r 2
dr
d
d

dr
dr
r gives well-known Kohlraush’s law
 (b )

 d 
 (r ) 2
2
 (r )
Since super-temperature
 (r )  T (r )  T0 ,
(A.2.20)
T (r ) and T0 are contact and ambient temperature
where
respectively, we get
T (b )

 dT 
 (r ) 2
T (r )
Using Wiedemann – Franz law
conclude that
  LT ,
where
2
L  2.4 108V 2 K 2 is the Lorenz constant, one can
L[T (b)2  T (r )2 ]   (r )2
171
(A.2.21)
In particular for the whole constriction zone of one contact member ( r
 , T ()  T0 ,  ()  U k / 2 ) the
final formula, so-called the temperature – potential relationship, takes the form [1]
L[T (b)2  T0 ]  U c / 4
2
2
(A.2.22)
It will be shown later that this formula keeps to be valid not for the spherical model but in general case as well.
APPENDIX 3. Reducing the system of dual integral equation (21) to a Fredholm equation.
System (21) is :

2

0 ( x  W )  A( x)  J 0 ( xr )dx  2 , 0  r  f
(A.1)

 A( x)  J
0
( xr )  xdx  0 ,
f r 
0
After substitution
A( x)  x  C ( x)

this system is represented in the form
 [1  g ( x)]  C ( x)  J
0
( xr )dx 
0


0r  f ,
(A.2)

 C ( x)  J
0
( xr )dx  0 ,
f r 
0
where
g ( x)  
2 1

W x
Let be
f
C ( x)  x   (t )  cos( xt )dt
(A.3)
0
Then the second equation of the system (A.2) is satisfied for every differentiable function
 (t ) . Indeed ,


f

0
0
 C ( x)  J 0 ( xr )dx   x  J 0 ( xr )dx   (t )  cos( xt )dt   J 0 ( xr ) [ ( f )  sin( fx) 
0
0
f

f

o
0
0
0
   '(t )  sin( xt )dt ]dx   ( f )   J 0 ( xr )  sin( fx)dx    '(t )dt  J 0 ( xt )  sin( xt )dx  0
if r  f , in accordance with the formula

J
0
( xr )  sin( xt )dx  0 ,
if 0  t  r ,
0

J
1
, if 0  r  t
t  r2
The first equation of the system (A.2) may be written in the form
0
( xr )  sin( xt )dx 
2
0
172

 C ( x)  J 0 ( xr )dx 
0



  C ( x)  g ( x)  J o ( xr )dx ,
0
0r  f
This equation together with the second equation enables to write the Bessel integral

 C ( x)  J
0
( xr )dx  F (r )
0
where
F (r ) 



  C ( x)  g ( x)  J 0 ( xr )dx , if 0  r  f
0
if
F (r )  0 ,
Converting this Bessel integral we obtain
(A.4)
f r 
f
C ( x)  x  F (  )  J 0 ( x ) d
0
Using the formula

2 cos( xt )
J 0 ( x )  
 0 2 t2
we get
f
f
2x
F (  )
C ( x) 
cos( xt )dt 
d

 0
2 t2
t
and after comparison this expression for C (x) with (A.3) one can write
 (t ) 
2
F (  )
f

2 t2
t
d
or taking into account (A.4)
f
f
2
d
2
 (t ) 

K (t , t1 )   (t1 )dt1
 t  2  t 2  0
(A.5)
where
f
K (t , t1 )  
t
2

W
f

t
d
2 t2
d
2 t2

 x  g ( x)  J

J
0
( x )  cos( xt1 )dx 
0
0
( x )  cos( xt1 )dx
0

Since  J 0 ( x )  cos( xt1 )dx   2  t1 , if   t1
2
0
173

J
0
( x )  cos( xt1 )dx  0
,
if   t ,
0
therefore
K (t , t1 )  
2
W
d
f

  t   2  t1 2
2
t
2

ln
W
f 2 t2 
K (t , t1 )  
2
W
2

ln
W
f 2  t1
t 2  t1
d
  t   2  t1 2
2
t1
f 2  t1 
2
, if t  t1
2
2
f


2

2
, if
f 2 t2
t  t1
t1  t 2
2
hence
f 2  t1 
f 2 t2
2
2
K (t , t1 )  
ln
W
t1  t 2
2
and equation (A.5) finally is written in the form :
2
 (t ) 
4
W

t
 ln
f 2 t2 
f 2 t2 
0
t 2  t1
f 2  t1
2
2
  (t1 )dt1
,
(22)
APPENDIX 4. GENERAL MHD ARC MODEL
A. Equations for the cathode region.
1. Cathode surface ( z  0)
Energy balance equation
qi  qie  qem  q  qr  qev ( A.1 )
consisting of :
- Ion bombardment component
qi  ji ( uc  i   eff 
where
ji is the ion current density ,
uc is the cathode fall ,
174
5 kTc
) , (A.2)
2 e
 i is the ionization voltage ,
 eff is the effective work function ,
Tc is the cathode spot temperature ,
( Tc  c (r , z , t ) at z  0 , 0  r  rc (t ) )
k is the Boltzmann constant ,
e is the electron charge .
- Back-diffusion electrons component
qie  jie ( eff 
where
5 kTe
) , ( A.3 )
2 e
jie is the back-diffusion electron
current density ,
Te is the electron temperature .
- Electron emission component
qem  jem ( eff 
where
jem is the electron emission current density .
-Heat conduction energy flux component
q    c
where
5 kTc
) , ( A.4 )
2 e
c
 c
z
at
z  0 , ( A.5 )
is the thermal conductivity of the cathode material .
-Arc column radiation component
where
T
4
4
qr  T ( A  Tc ) , ( A.6 )
is the Boltzmann's constant .
-Evaporation energy flux component qev could be evaluated using simple de Boer equation [1,p.430] or the
equation proposed by M.H.Kogan and H.K.Makashev [21] . We shall use the Frenkel kinetic theory of
evaporation . Accordingly to this theory the rate of evaporation Vev in front of the cathode depends on the
cathode temperature
Tc as
Vev  Vo exp ( 
) , ( A.7 )
where
V0  0.82
c
Tc
 0
( r / ma )
1/ 3
,
c 
c
k
,
o 
is the corrected frequency of the
normal lattice oscillations ,
ma is the atom mass of the cathode
material ,
c is the atom bond energy .
Thus the evaporation energy flux component is
qev  Lc  cVem , ( A.8 )
where
Lc is the latent heat of vaporization ,
c
of the cathode material ,
is the cathode material density .
175
2.
Sheath (0  z  d1 )
 Equation for the
electrical potential U :
e
U 
0
( ni  nem  nie ) ,
(A.9 )
where
U 
2U 1 U  2U


r 2 r r z 2
for a two-dimensional model , and
d 2U
U 
dz 2
for a one-dimensional model
 Boundary conditions :
U  0 at z  0 , U  uc at z  d1 ( A.10 )
 Equations for the number densities of emitted electrons
nem , ions ni , and back-diffusion inverse electrons
nie .
At the point
z
inside the sheath with the corresponding potential U above mentioned equations are :
ji 2
3
[ ( euc  eU  kTc )]1/ 2 ( A.11 )
2
e mi
j
2
3
( A.12 )
nem  em [ ( eU  kTc )]1/ 2
2
e me
j 2 3
nie  ie [ ( kTie  euc  eU )]1/ 2 (A.13)
e me 2
ni 
These expressions are obtained from the
equations :
nem 
jem
j
j
, ni  i , nie  ie ( A.14 )
eVem
eVi
eVie
where the velocities Vem , Vi , and Vie for
emitted electrons, ions , and back-diffusion inverse electrons can be evaluated using energy balance equations at
the position z , U :
1
3
2
meVem  kTc  eU
( A.15 )
2
2
1 2 2 3
mi V i  kTc  e( uc  U ) ( A.16 )
2
2
1
3
2
meVie  kTie  e( uc  U ) ( A.17 )
2
2
Integration of the one-dimensional differential equation (A.9) gives the expression for the electric field in the
sheath
176
 dU 
E 

 dz 
2
z  d1

2
z 0
( u c  u1  u1 ) 
4e
0
[(
j ie
e
ji
e
mi
j
 em
2e
e
me
)
2e
me
( u 2  u 2  u c )]
2e
( A.18 )
where
3 kTe
3 kTie me jie 2
, u2 

(
) ,
2 e
2 e
2e enie 0
u1 
(A.19 )
nie is the number density of the back-diffusion inverse electrons on the boundary z  d1 given below .
0
3). Boundary between the sheath and the ionization zone
( z  d1 ) .
Equations for the components of the current density :
j  ji  jem  jie
( A.19 )
with components
- Ion current density
ji 
1 0
1 0 8kTc
eni vi  eni
, (A.20 )
4
4
mi
0
where ni is the ion number density ,
mi is the ion mass
-
Back-diffusion electrons current density
-
jie 
1
0 8 kTe
enie
, ( A.21 )
4
me
0
where nie is the back-diffusion electron number
density ,
me is the electron mass .
- Electron emission current density is determined by the Richardson-Dushman equation
jem  Ac Tc exp ( 
2
e eff
kTc
) ( A.22 )
with the Schottky correction
 eff   0 
and
Ac 
Where
 0 is the work function of the cathode
Ec is the electric field strength at the
eE c
, ( A.23 )
4  0
4 k 2 eme
h3
cathode spot ,
177
0
is the dielectric constant ,
me is the electron mass ,
h is Plank's constant ;
It has to be noted that theoretical value of A is
Ac  120  104 A  m2  K 2 ,
however , the practical value of A is different for different materials . Particularly , for Cu
Ac  60 A  m2  K 2 , and for Mo Ac  55 A  m2  K 2 .
Equations for the particles density
at z  d1 .
0
- Electron emission number density nem at
z  0 can be obtained from the equation
( A.11) by substitution U  uc :
nem 
0
jem 2
3
[ ( euc  kTc )]1/ 2 (A.24)
2
e me
- Back-diffusion electron number density
nie  ( ni  nem ) exp ( 
0
0
euc
) (A.25)
kTe
4).Ionization zone (d1  z  d2 )
 Motion equations for electrons ( index
"e"),ions ( index "i" ) and neutral atoms
( index "g" )
 me
dVe
1 dpe
  eE 

dz
ne dz
 M eg eg (Ve  Vg )  M ei ei (Ve  Vi )
( A.26 )
mV
i i
dVi
1 dpi
 eE 

dz
ni dz
 Mie ie (Vi  Ve )  Mig ig (Vi  Vg )
( A.27 )
 mgVg
dVg
dz

1 dpg

ng dz
 M ge ge (Vg  Ve )  M gi  gi (Vg  Vi )
( A.28 )
Here
and
Meg  M ge , Mie  Mie ,
M ge  Meg
eg   ge , ie  ei
 ge  eg
are reduced masses and frequencies of collisions respectively
;
178
pe , pi , pg and Ve , Vi , Vg
are partial pressure and velocities of electrons , ions , and
neutral atoms respectively .
 Energy equations for particles
- For electrons :
d
dT
3
d
(  e e )  ( e i  kTe ) ( neVe ) 
2
dz
dz
dz
3
3
dV
 ne k   es es ( Te  Ts )  ne kTe e 
2
2
dz
s i , g
 ne  es
s i , g
me ms
( e   s ) 2
2
2
me  ms
- For ions ( s  i ) and for atoms (

( A.29 )
s  g) :
d
dT
3
(  s s )   ei ns es k ( Te  Ts )
dz
dz
2
( A.30 )
where
 es 
2me ms
2
2
me  ms
is the portion of the s - particle's energy
transferred during collision with electrons ;
 e ,  s are thermal conductivity .
 Ionization and recombination equations :
d
3
( eneVe )  ne ng  ne ( A.31 )
dz
d
3
( eniVi )  ne ng  ne
dz
where

and

( A.32 )
are the ionization and recombination rates
 Condition of quasi-neutral plasma
equilibrium
ne  ni  ng (A.33 )
 Dalton's Low for the pressure
p:
p  ne kTe  ni Ti  ng Tg (A.34 )
 Equation for the current density :
j  eneVe  enV
i i  const ( A.35 )
 Modified Saha equation
ne ni 2 Zi ( 2 me k  A ) 3/ 2
  i

exp (  i
)
3
ng
Z0
h
kTe
179
( A.36 )
where the lowering of the ionization energy is given by
i 
e2
4  0
[ e2 ne (1 
Te 1/ 2
)] , ( A.37 )
Tc
where
Zi is the statistical ion state function ,
Z0 is the statistical atom function .
4. The boundary between the
ionization zone and the arc column
( z  d2 )
 Energy flux W into plasma
:
5 kTc
5 kTc
)  ji (  i 
)
2 e
2 e
,
5 kTe
 jie ( uc 
)
2 e
Wa  jem ( uc 
( A.38 )
and if the plasma is almost fully ionized,
then [14]
Wa 
kTe
m 2
[ 0  5  e ( i  5)] j  K  A
2e
mi k  A
( A.39 )
where
 0 is the thermal diffusion factor ,
K is the thermal conductivity of the
plasma gas
5. Cathode bulk
(   z  0) .
 Equation for the temperature
c ( r , z , t )
 c 1 


( cr c ) 
t r r
r
, ( A.40 )
2

 c
Jc (r , z , t )
(
)
z c z
 c (c )
c
where
c is the cathode thermal capacity,
Jc (r , z, t ) is the current density in the
cc  c
cathode bulk ,
.
 c (c ) is the cathode electrical
conductivity depending on the
 Equation for the electrical potential
c
c
1 
c

c
[ r c (c )
]  [  c (c )
]0
r r
r
z
z
( A.41 )
 Relationship between current density and potential
180
Jc (r , z )   c (c ) ( c / r ) 2  ( c / z ) 2
( A.42 )
 Initial condition for the temperature
c (r , z, tb )  gc (r , z)
(A.43 )
is determined as the solution of the same equation ( A.40 ) at the time of the bridge rupture
 Boundary conditions for the temperature
a) At z  0
 c
 q , if r  rc (t ) , ( A.44-1 )
z
 c
 0 , if r  rc (t ) ( A.44-2 )
z
where rc is the cathode spot radius .
 c
z  
b) At
where
c  T0 ,
T0 is the ambient temperature .
 Boundary conditions for the electrical potential
a). At z  0
c  0 , if
 a
0
z
b). At
, if
r  rc (t )
( A.45-1 )
r  rc (t )
( A.45-2 )
z  
a  U c
, ( A.46 )
U
where
c is the voltage in the cathode bulk .
B. Equations for the arc
 Heat equation
cA  A
 A
 A 1 


(  Ar A ) 
t
r r
r
1
j 
[ q  p ]
l (t )
,
1 2
( A.47 )
where
c A is the arc plasma thermal capacity ,
A
A
A
is the arc plasma density ,
is the arc plasma thermal
conductivity ,
is the arc plasma electric
conductivity ,
181
tb .
q is the heat flux into cathode ,
p is the heat flux into anode .
The dependencies of cA ,  A ,  A ,  A on the temperature  A are given usually by graphs .
Heat fluxes q  and p can be evaluated from
the formulas ( A.1 ) and ( A.54 )
 Boundary conditions
 A
0
r
A  T
where
at
r0
at
r  rA ( A.49 )
( A.48 )
T  determined from the equation
 A (T  )  0
,
( A.50 )
1
rA ( t )  [ rc ( t )  ra ( t )] ,
2
I
.
rc ( t ) 
j ( t )
( A.51 )
( A.52 )
 Ohm's low
I
where
2
l (t )
rA ( t )

 ( A )rdr
, ( A.53 )
0
l ( t ) is the arc length .
 Equations of quasi-neutral plasma condition , Dalton's Low , current density , Saha equation are quite the
same as for the ionization zone given by the expressions (A.33)- ( A.37 ) .
C. Equation for the anode region .
1) . Anode surface ( z  l (t )) .
 Energy balance equation
pe  pr  pev  p
( A.54 )
consisting of
- Electron energy flux component
pe  je ( ua   0 
where ua is the anode fall ;
- Heat conduction energy flux component
p    a
 a
z
at
5 kTe
) , ( A.55 )
2 e
z  l (t ) , 0  r  ra (t )
( A.56 )
a
where
is the thermal conductivity of the anode material ;
- Arc column radiation energy flux component
182
4
4
pr  T ( A  Ta ) ,
( A.57 )
where
Ta  a (r , z, t ) at z  l (t ) , 0  r  ra (t )
- Evaporation energy flux component
pev  La  aVev , Vev  V0 exp ( 
a
Ta
)
(A.58 )
2) Anode bulk
(l (t )  z  )
 Equation for the temperature
a ( r , z , t )
 a 1 


( ar a ) 
t r r
r
2
J (r , z, t )


( a a )  a
z
z
 a (a )
ca  a
( A.59 )
where ca is the anode thermal capacity,
Ja (r , z, t ) is the current density in the
 a (a )
anode bulk ,
is the anode electrical
conductivity depending on  a ,
 Equation for the electrical potential  a
1 
a

a
[ r a (a )
]  [  a (a )
]0
r r
r
z
z
( A.60 )
 Relationship between current density and potential
Ja (r , z)  a (a ) ( a / r ) 2  ( a / z) 2
( A.61 )
 Initial condition for the temperature
a (r , z, tb )  ga (r , z)
( A.62 )
is determined similarly like for the cathode ,
 Boundary conditions for the temperature
z  l (t )

  a a  p , if r  ra ( A.63 - 1 )
z
 a
0 ,
if r  ra , ( A.63 - 2 )
z
where ra is the anode spot radius .
a). At
b). At
z
a  T0
( A.64 )
183
 Boundary conditions for the electrical
potential
z  l (t )
a). At
a  0 , if
 a
 0 , if
z
r  ra (t )
( A 65 - 1 )
r  ra (t )
( A.65 - 2 )
z
b). At
a  U a ,
( A 66 )
where U a is the voltage in the anode bulk .
3). Simplified model
 Equations for the heat fluxes
r2
] , ( A.67 )
rs ( t ) 2
r2
0
pm ( r , t )  pm ( t ) exp[ 
] , ( A.68 )
rm ( t ) 2
where index s means i , ie, r , em, ev ,
qs ( r , t )  qs ( t ) exp[ 
0
and index
m
means e, r , ev .
 Equation for the current densities
me
4 jie
8kTc
jem [
 ( 4 ji
mi
8kTc

2
3
eu
( euc  kTc )]1/ 2 ) exp (  c )
me
2
kTc
( A.69 )
 Equation for the total current
rc ( t )
2p
т
[ ji (r , t ) + jem (r , t ) - jie (r , t )]rdr = I
0
 Simplified equation for the temperature
2
 a
1  a 2
2  
 aa ( 2a 

)
t
r
r r z 2
where
aa 
2
( A. 71 )
a
ca  a
 Equation for the electric potential
184
( A.70 )
 2  a 1  a  2  a


0
r 2 r r
z 2
( A.72 )
 Analytical solutions
a (r , z, t ) 
arctan[
I
1

2 ra ( t ) ca  a  a
1
ra ( t ) 2
s  s2  4ra ( t )( z  l 2 ( t )) ]
2
( A.73 )
where
2
s  r 2  ra (t )  [ z  l (t )]2 ( A.74 )
a ( r , z , t )  a1 ( r , z , t )  a 2 ( r , z , t ) 
a 3 ( r , z , t )
( A.75 )
where
t
ra ( t )
0
0
 a1 (r , z, t )   dt1
 [G( z  l (t ), r, r , t  t ) 
1
1
1
( A.76 )
G ( z  l (t1 ), r , r1 , t  t1 )] p (r1 , t1 )r1dr1
is the component of the anode temperature due to arc heat flux ,
t
 a 2 (r , z, t )   dt1
0



dz1   a J a (r , z ) 
l ( t1 )
1
2
0
[G ( z  z1 , r , r1 , t  t1 ) 
G ( z  z1  2l (t1 ), r , r1 , t  t1 )]r1dr1
( A.77 )
is the Joule sources component of the anode temperature ,
 a 3 (r , z, t ) 


l (t )

dz1  [G ( z  z1 , r , r1 , t ) 
0
G ( z  z1  2l (t ), r , r1 , t )]g a (r1 , t1 )r1dr1
( A.78 )
is the component of the anode temperature caused by the initial condition ( A.62 )
G ( z , r , r1 , t ) 
1
4aa  a t 3

( A.79 )
z 2  r 2  r1
rr
 exp ( 
) I0 ( 1 )
2
2 aa t
4 aa t
and I 0 is the modified Bessel function .
2
The expression for the anode current density
Ja (r , z ) is given by the equation ( A.61 ) and the contact gap ( arc
length ) l ( t ) is
l (t )  lb  V0t ,
( A80 )
where V0 is the contact opening velocity .
185
 The temperature at the centre of the anode and the cathode spots ( r  0 ) :
t
Ta (0, t ) =
pm (t )rm (t )
dt
2
rm (t ) + 4a (t - t ) t - t
те
m
0
( A81 )
2
q ( )rs ( )
d
Tc (0, t )    2 s
2
t 
0 s rs ( )  4a (t   )
2
t
where


and
m
s
electrode , and sign
( A.82 )
are considered as algebraic sums with sign " " for the flux components entering the
" " for the flux components leaving the electrode .
 Equation for the average arc temperature
 c A Al (t ) rA 2 (t )[
Iu A  2
rA ( t )

dTA
2

(TA  T  )] 
dt
rA (t )
( A.83 )
[ q ( r , t )  p ( r , t )]rdr
0
where
TA (t ) 
2
2
rA (t )
rA ( t )

 A (r , t )rdr
( A.84 )
0
rA is the effective arc radius ,
u A is the arc voltage ,
T  is given by the formulas ( A.49 ) -( A.50 ).
 Equations for approximate effective radii of heat flux components
rs (t )  rc (t ) ,
s  ev ( A.85 )
rm (t )  ra (t ) ,
m  ev ( A.86 )
Tc (rev , t )  Tb for cathode rev ( t ) ( A.87 )
Ta (rev , t )  Tb for anode rev ( t ) , ( A.88 )
where Tb is the boiling temperature of the
contact material
 Equation for the critical time
t cr
Ta (0, tcr )  Tc (0, tcr )
Appendix 5 .
( A.89
The temperature field in electrical contacts
Each member of a contact pair can be considered as a body occupying the cylindrical halfspace region D(0  r  , 0  z  ) .
186
The region occupied by arc at blow-off repulsion can be considered as a cylindrical disk
DA : 0  r  rb (t ), 0  z  x(t ) interacting with contact surface, which radius rb (t ) is much
greater than contact gap x(t ) (Fig. A.1). This interaction results into phase transformations of
contact material and formation of three zones:
4) The zone of evaporated material Db : 0  r  rb (t ), 0  z   b (r , t ) ,
5) The zone of melted material
Dm :  b (r, t )  z   m (r, t ), if 0  r  rb (t ), and 0  z   m (r, t ) if rb (t )  r  rm (t )
6) The solid zone Ds :  m (r, t )  z  , if 0  r  rm (t ), and 0  z   if rm (t )  r  
r
x(t)
Ds
rm (t )
z   m (r , t )
Dm
Db
rb (t )
2rA0
0
zb (t )
zm (t )
z   b (r , t )
z
DA
Movable contact
Fixed contact
r
Fig. A.1 The arc and contacts geometry: arc region DA ,
evaporated zone Db , melted zone Dm and solid zone Ds
The dynamics of contact heating can be described by the heat equations for temperature Ti
(indexes i  0,1 and 2 correspond to the zones Db , Dm and Ds relatively):
Ci
Ti
1
 div(i gradTi )  grad 2i
t
i
(A.1)
where Ci , i , and i are thermal capacitance, thermal conductivity and electrical resistivity
respectively. The electrical potential i (r , z, t , rC ) can be expressed as [5], [12]
187
i (r , z, t , rC ) 
I (t ) i
 (r , z, rC )
arctan
2 rC
rC
(A.2)
where I (t ) is the electrical current, and rC is the current conducting contact radius. The
function  (r , z, rC ) can be found from the equation
r2
z2

1
2
2
rC   2 
i.e.
 (r , z, rC ) 
1
2
s  s 2  4rC z 2 ,
2
s  r 2  z 2  rC
2
(A.3)
The boundary conditions on the interface z   b (r , t ) can be expressed as
T0
z  b ( r ,t )
 T1 z 
b ( r ,t )
 Tb
1
(A.4)
T1
z
 PC (r , t )  Pb (r , t )
(A.5)
z  b ( r ,t )
Here Tb is the boiling temperature of contact material, PC (r , t ) is the total heat flux (arc power
per unit contact area) entering the surface z   b (r , t ) from heated vapour. It consists of
many components, such as heating due to ion bombardment, electron emission cooling,
radiation etc. which are different for anode and for cathode. They are described and defined in
[12]. To a first approximation one can suggest that total arc power should be divided into
I (t )U (t )
equal parts for anode and for cathode [1]. In this case PC (r , t ) 
2 . The flux component
2 rC
Pb (r , t ) is the portion of total flux consumed for evaporation of contact material, which can be
found by the expression
Pb (r , t )  Lb
 b (r , t )
t
(A.6)
where Lb and is specific heat for evaporation,  is density of contact material.
The boundary conditions on the interface z   m (r , t ) are similar if we replace boiling index
b by melting index m :
T1 z 
m ( r ,t )
 T2
z  m ( r ,t )
 Tm
(A.7)
1
T1
z
 2
z  m ( r ,t )
T2
z
 Pm (r , t )
(A.8)
z  m ( r ,t )
with heat flux consumed for melting
Pm (r , t )  Lm
 m (r , t )
t
Each stage of contact separation should be considered separately.
188
(A.9)
1. Initial stage of contact separation. At this stage zones Db and Dm vanish, no arcing
occurs, thus the right side of (A.5) is equal to zero. The solution of the equation (A.1) for
averaged values of C and  can be presented for this stage in the form

t

TJ (r , z, t , rC )   dt1  dz1  G(r , r1 , z  z1 , t  t1 )
0

0
1

[( / r1 ) 2 ( / z1 ) 2 ]r1dr1
(A.10)
where G(r , r1 , z, t ) is the Green’s function defining by the formula
G(r , r1 , z, t ) 
r 2  r1  z 2
rr
) I 0 ( 12 )
2
4a t
2a t
2
1
2a  t 3
exp(
(A.11)
and a 2   / C is the thermal diffusivity of contact material.
In a time range, for which dimensionless time (Fourier criterion) Fo 
a 2t
2 is sufficiently
rC
large, Fo  1 , the quasi-stationary model for Joule heating can be applied, and then the
formula (A.6) transforms into more simple expression [5], [10]
TJ (r , z, t , rC )  T2 ( , t , rC ) 
cos[2 (t , rC ) ]
 1}
 cos[2 (t , rC ) / 2]
1
{
(A.13)
where
  arctan
 (r , z, rC )
rC
(A.14)
2 (t , rC ) 
I (t )
2 rC
20 2
2
(A.15)
 20 is electrical resistivity before heating and  2 is its temperature coefficient for the solid
material.
It should be noted that the introduction of the new variable (A.14) transforms the half-infinite
region 0     into segment 0     / 2 .
The initial stage comes to the end at the time t  tm when the temperature at the contact
surface reaches the melting value Tm . This time can be found from the equation
T2 (0, tm , rC (tm ))  Tm or from oscillograms as the time corresponding to the melting voltage.
The estimation of Fourier criterion shows that quasi-stationary model can be applied as well
for following contact separation up to arc ignition.
2. Melting. The melting stage of contact separation is characterised by appearance of two
zones, melted Dm and solid Ds . The quasi-stationary solution of the equation (1) gives the
temperature distribution in the melted zone Dm in the form
T1 ( , t , rC ) 
cos[1 (t , rC ) ]
 1Tm  1}
1 cos[1 (t , rC ) m ]
1
{
(A.16)
189
This function satisfies the stationary heat equation and the conditions
T1

0
(A.17)
T1    Tm
m
z 0
(A.18)
Similarly the expression
T2 ( , t , rC ) 
(1   2Tm )sin[2 ( / 2   )]  sin[2 (   m )]  sin[ 2 ( / 2   m )]
 2 sin[2 ( / 2   m )]
(A.19)
satisfies the stationary heat equation for the solid zone Ds and the conditions
T2    Tm
(A.20)
m
T1   / 2  0
(A.21)
The melting isotherm  m (t ) can be found from the equation (A.8) by substitution of the
expressions (A.16) and (A.19) and replacing of  m (r , t ) by  m (t ) .
3. Evaporation. The temperature distribution in melted zone after the time t  tb , when the
temperature at the contact spot   0 becomes equal to boiling value Tb , becomes the form
T1 ( , t , rC ) 
[1  1 (Tb  Tm )]  sin[1 ( m   )]  sin[1 (   b )]
1
 Tm 
1 sin[1 ( m   b )]
1
(A.22)
while the temperature in the solid zone describes by the previous expression (A.19).
The boiling isotherm should be calculated using the equation (A.5).
4. Arcing. The Fourier criterion at arcing is not sufficiently large due to sharp increase of
current conducting radius, therefore the quasi- stationary model ceases to remain correct and
non-stationary solution of the equation (A.1) should be found. It can be represented in the
form
TC (r, z, t , rC )  TJ (r, z, t , rC )  TS (r, z, t , rC )
(A.23)
where TJ (r , z, t , rC ) and TS (r , z, t , rC ) are the temperature components due to volumetric Joule
heating and due to heating of contact surface by arc heat flux respectively. The first term in
the right side is defined by the expressions (A.19) and (A.22) for solid and liquid zones, while
the second term can be written as
t

0
0
TS (r , z, t , rC )   dt1  G(r , r1 , z, t1 ) PC (r1 , t1 )r1dr1
190
(A.24)
with the Green’s function G(r , r1 , z, t ) defined by the expression (A.11). Calculation shows
that the role of contact heating by flux is more significant. If the heat fluxes Pc (r , t ) Pb (r , t ) ,
Pm (r , t ) obeys the normal Gauss’s radial distribution
r2
r2
r2
Pc (r , t )  Pc (t ) exp(
),
Pb (r , t )  Pb (t ) exp(
) , Pm (r , t )  Pm (t ) exp(
),
rA (t )2
rA (t )2
rA (t )2
(A.25)
then the integral with respect to r in the formula (A.24) can be calculated and the expression
for the contact temperature becomes more simple form
t
[ Pc (t1 )  Pb (t1 )  Pm (t1 )]rA (t1 )2
a
z2
r2
TS (r , z, t ) 
exp[


]d
4a 2 (t  t1 ) rA (t1 )2  4a 2 (t  t1 )
  0 [rA (t1 )2  4a 2 (t  t1 )] t  t1
(A.26)
191
Scientific publication
Kharin Stanislav Nikolaevich
MATHEMATICAL MODELS OF PHENOMENA IN ELECTRICAL
CONTACTS
Monograph
Editor Z.V. Skok
________________________________________________________________
Signed in print 07. 11.2017
Scope 11,9 pub. sheet-copies.
Paper Size 60 × 84 1/16
Circulation of 300
Printed in CJSC RIC «PRice-courier»
630090, Novosibirsk, Akad. Lavrentiev str., 6, Ph. 330-72-02
Order No. 61
192
Научное издание
Харин Станислав Николаевич
МАТЕМАТИЧЕСКИЕ МОДЕЛИ ПРОЦЕССОВ
В ЭЛЕКТРИЧЕСКИХ КОНТАКТАХ
Монография
Редактор З.В. Скок
________________________________________________________________
Подписано в печать 07. 11.2017
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Формат бумаги 60 × 84 1/16
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