Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases
The new criterion of arc stability and instability is introduced, which enables one to find arc duration dependently on given circuit parameters and properties of contact material. The mathematical model of phase transformations inside electrodes during arcing is elaborated which describes dynamics...
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| Published in: | Электрические контакты и электроды |
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| Date: | 2010 |
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Інститут проблем матеріалознавства ім. І.М. Францевича НАН України
2010
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| Cite this: | Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases / S.N. Kharin, Yu.R. Shpady, A.T. Kulakhmetova // Электрические контакты и электроды. — К.: ИПМ НАН України, 2010. — С. 123-131. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859470919624294400 |
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| author | Kharin, S.N. Shpady, Yu.R. Kulakhmetova, A.T. |
| author_facet | Kharin, S.N. Shpady, Yu.R. Kulakhmetova, A.T. |
| citation_txt | Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases / S.N. Kharin, Yu.R. Shpady, A.T. Kulakhmetova // Электрические контакты и электроды. — К.: ИПМ НАН України, 2010. — С. 123-131. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
| container_title | Электрические контакты и электроды |
| description | The new criterion of arc stability and instability is introduced, which enables one to find arc duration dependently on given circuit parameters and properties of contact material. The mathematical model of phase transformations inside electrodes during arcing is elaborated which describes dynamics of arc erosion in metallic and gaseous arc phases. Increasing of arc duration and erosion with inductance occurs on account of enlarging of gaseous arc phase, while variation of metallic phase is relatively small.
Получен новый критерий стабилизации дуги, который позволяет найти зависимость продолжительности металлической фазы дуги от заданных параметров цепи и свойств контактного материала. Предложена математическая модель фазовых переходов внутри электродов в процессе горения дуги, которая описывает динамику дуговой эрозии в металлической и газовой фазах дуги. Установлено, что причвозрастания продолжительности дуги и эрозии контактов, происходящего с ростом индуктивности цепиявляется расширение газообразной фазы дуги относительно ее металлической фазы.
Отримано новий критерій стабілізації дуги, який дозволяє знайти залежність тривалості металічної фази дуги від заданих параметрів ланцюга і властивостей контактного матеріалу. Запропоновано математичну модель фазових переходів в середині електродів в процесі горіння дуги, яка описує динаміку дугової ерозії в металічній і газовій фазах дуги. Встановлено, що причиною зростання тривалості дуги і ерозії контактів, яке має місце із зростанням індуктивності ланцюга, є розширення газоподібної фази дуги відносно її металічної фази.
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| first_indexed | 2025-11-24T09:36:00Z |
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| fulltext |
UDC 517.958:[536.2+539.219.3]
Mathematical model of arc temperature
and conductivity at metallic and gaseous arc phases
S. N. Kharin, Yu. R. Shpady, A. T. Kulakhmetova
Institute of Mathematics of the National Academy of Sciences
of Kazakhstan
The new criterion of arc stability and instability is introduced, which enables one to find arc duration dependently
on given circuit parameters and properties of contact material. The mathematical model of phase transformations
inside electrodes during arcing is elaborated which describes dynamics of arc erosion in metallic and gaseous arc
phases. Increasing of arc duration and erosion with inductance occurs on account of enlarging of gaseous arc
phase, while variation of metallic phase is relatively small.
Keywords: mathematical model, arc erosion, arc phases.
Introduction
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.
Mathematical model of arc temperature and conductivity
at metallic arc phase
Equation for the temperature
The arc temperature θ in opening contacts just after ignition is less than the threshold value
required for gas ionization, , however it is sufficient to ionize metallic vapours in the contact gap,
which takes place at the temperature :
( , )r t
igθ
imθ
θim < θ < θig.
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
2θ 1 θ(λ ) σ rC r E
t r r r
W∂ ∂ ∂
= + −
∂ ∂ ∂
, (1)
© S. N. Kharin, Yu. R. Shpady, A. T. Kulakhmetova, 2010
123
where C and λ are thermal capacity and density, λ and σ are heat and electrical conductivities of the arc
plasma, is electrical field and is power loss due to arc radiation and heat conduction from arc
column to electrodes. The initial temperature distribution along radius
E rW
124
r
θ( ,0) ( )r f= (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) = θoJo(μ1r / rA) by parabola
2
0 2( ) θ (1 )
A
rf r
r
= −
or by the Bessel function
f(r) = θoJo(μ1r / rA) , (3)
where is the first root of the Bessel function and is the temperature maximum at the centre
of arc disc.
1μ 2, 405= 0θ
The temperature on the interface r = rA between ambient air and arc plasma should be equal to
threshold of metal ionization
θ(rA, t) = θmi. (4)
It should be noted that thermal and electrical plasma conductivities, and σ , depend essentially on the
temperature and this dependence can not be averaged. In contrast the arc radiation can be neglected
for metallic arc phase, which temperature is relatively low : < ≈ 5000 oC (fig. 1).
λ
rW
imθ igθ
Equation for electrical conductivity σ
To solve the heat equation (1) we use the Kirchhoff’s substitution
mi
θ
θ
(θ) (θ) θS = ∫ λ d . (5)
Then the equation (1) transforms to
21 ( ) σ
λ r
C S Sr E
t r r r
W∂ ∂ ∂
= + −
∂ ∂ ∂
. (6)
Solving the equation (5) with respect to θ , get
miθ θ ( )g S= + . (7)
Since the function σ = σ (θ) is given (fig. 1), can write this function in term of using (6), i. e.
. Linearization of this function gives the expression (fig. 2)
σ
σ σ ( )S=
σ bS= ; g
gi
σ
tan φb
S
= = , (8)
where is given electrical conductivity at the transition from metallic arc phase to gaseous arc phase,
when θ = θgi, and
gσ
mi
θgi
θ
λ(θ) θgiS = ∫ d . (9)
5 6 7 8 9 10
0
1
2
3
4
5
6
1
2
3
3, 10T K
, , rC Wλ
5 6 7 8 9 10
0
1
2
3
4
5
6
1
2
3
3, 10T K
, , rC Wλ
S
( )Sσ
ϕ
0
gσ
giS S
( )Sσ
ϕ
0
gσ
giS
Fig. 1. Temperature dependence of λ, σ and
Wr: 1 — , Wm-1 K-1; 2 — C, ·102 Jm-3 K-1;
3 — Wr, ·1011 Wm-3 [6].
λ
Fig. 2. Linear approximation of . giσ( )S
Substituting (8) in (7) and using notation
2; τ
λ
x Cr
E bE b
= = , (10)
can write the equation with respect to σ
2
2
σ σ 1 στ
t x x x
∂ ∂ ∂ σ= + +
∂ ∂ ∂
. (11)
It should be noted that can be considered as constant because the thermal diffusivity is
approximately constant (fig. 1). The domain for this equation is
τ 2 / λa C=
00 x x< < , where 0 Ax r E b= .
The boundary conditions (2)—(4) transform to the type
σ( ,0) ( )x F x= (12)
with
mi
( / )
θ
( ) (θ) θ
f x E b
F x b d= ∫ λ , . (13) 0σ( , ) 0x t =
The solution of the problem (11)—(13) can be found in the form of Fourier-Bessel series
2
0 0
1
σ( , ) exp[ ( 1) / τ] ( )n n
n
x t C k t J k
∞
=
= − −∑ x , (14)
where
0
02 2
0 1 0
2 ( ) ( )
(μ )
x
n n
n
C F x J k x xdx
x J
= ∫ 0μ /n nk x, = ,
and μ are roots of the Bessel function: n 0 (μ ) 0, 1,2,3,...nJ n= = .
For approximation (3)
0 0 1 0σ( ,0) σ (μ / )x J x x=
and the solution (14) takes the simple form
2
1
0 02
0
μσ( , ) σ exp[ ( 1) / τ] (μ / )1 0x t t J
x
= − − x x ,
125
Taking into account (10), we get finally the expression for arc electrical conductivity in the form
2
2 2 2
0 1 0 12σ( , ) σ exp[ (μ ) ] (μ / )A A
A
a tr t E br J r r
r
= − − . (15)
The arc temperature can be found now from the expressions (5) and (8).
Let us introduce the criterion of arcing
2 2
1ξ AE br 2μ= − . (16)
We should distinguish three cases (fig. 3):
1) ξ . Rise of arc conductivity, power and temperature due to Joule heating. 0>
2) ξ . Maximum value of arc conductivity and power. 0=
3) ξ . Arc conductivity, power and temperature decrease, thus the arc should extinguish. 0<
Interaction between arc and contact surface
At the first stage of contact opening and then changes the sign. To find the critical point ξξ 0> 0=
we need to know the dynamics of arc radius rA, which expands during arcing. Then using formula
2π σA
IE
r
= (17)
and the expression (16) we can find the critical time crt t= at which ξ 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:
: 0 ( ), 0 σ ( , )b b bD r r t z r t≤ ≤ ≤ ≤ .
2. The zone of melted material:
Dm : σb (r, t) ≤ z ≤ σm (r, t), if 0 ≤ r ≤ rb (t), 0 ≤ z ≤ σm (r, t), if rb (t) ≤ r ≤ rm (t).
3. The solid zone:
: σ ( , ) , if 0 ( ), 0 if ( ) s m m mD r t z r r t z r t r≤ ≤ ∞ ≤ ≤ ≤ ≤ ∞ ≤ ≤ ∞ .
The contact temperature can be presented as the sum ( , , )CT r z t
( , , ) ( , , ) ( , , )C J ST r z t T r z t T r z t= + , (18)
where and 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 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 .
(19)
( , , )JT r z t ( , , )ST r z t
( , , )JT r z t
1 1 1 1 1 1 1 1 1 1
0 0
( , , ) [ ( , ) ( , ) ( , )] ( , , , )
t
S c b mT r z t dt P r t P r t P r t G r r z t t r dr
∞
= − − −∫ ∫ 1
Here is the total heat flux (power per unit area) entering the contact surface during arcing, ( , )cP r t
( , )bP r t and ( , )mP r t are portions of this flux consumed for
126
Fig. 3. Evolution of arc conductivity.
evaporation and melting of contact material, which
can be found by the expressions
σ ( , )( , ) b
b b
r tP r t L
t
∂
=
∂
γ ; σ ( , )( , ) m
m m
r tP r t L
t
∂
=
∂
γ ; (20)
0ξ <
0ξ =
0ξ >
(0, )tσ
0σ
t
0ξ <
0ξ =
0ξ >
(0, )tσ
0σ
t
σ(0, t)
ξ > 0
σo
ξ = 0
ξ < 0
where and are specific heat for evaporation and melting, bL mL γ is density of contact material.
It reasonable to assume that the isothermal surfaces σ ( , )bz r t= and mσ ( , )z r t= are ellipsoids of
revolution that can be found from the equations
2 2
2 2 1;
( ) ( )b b
r z
r t z t
+ =
2 2
2 2 1
( ) ( )m m
r z
r t z t
+ = ;
in other words
2 2σ ( , ) ( ) 1 / ( ) ;b b br t z t r r t= − 2
mσ ( , ) ( ) 1 / ( )mr t z t r r t= − 2
m . (21)
The functions , , and should be found from the equations ( )br t ( )bz t ( )mr t ( )mz t
C ( ( ),0, )b bT r t t T= ; T zC (0, ( ), )b bt t T= ; T r ; C ( ( ), 0, )m mt t T=
mC (0, ( ), )mT z t t T= , (22)
where T is the melting temperature of the contact material. m
If the heat fluxes P r , obeys the normal Gauss’s radial distribution ( , )c t ( , )bP r t ( , )mP r t
2
2( , ) ( ) exp( )
( )c c
A
rP r t P t
r t
= − ;
2
2( , ) ( ) exp( )
( )b b
A
rP r t P t
r t
= − ;
2
2( , ) ( )exp( )
( )m m
A
rP r t P t
r t
= − , (23)
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
2 2 2
1 1 1 1
2 2 22 2
1 1 10 1 1 1
[ ( ) ( ) ( )] ( )( , , ) exp[ ] τ. (24)
4 ( ) ( ) 4 ( )λ π [ ( ) 4 ( )]
t
c b m A
S
AA
P t P t P t r ta z rT r z t d
a t t r t a t tr t a t t t t
− −
= − −
− + −+ − −∫
t
The heat flux 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 W directly from the measured values of arc voltage U , arc current
( )cP t
A ( )A ( )AI t and then
arc heat flux entering contact is
2
( ) ( ) ( )( )
2π ( ) 2π ( )
A A A
c
A A
I t U t P tP t
r t r t2
⋅
= = . (25)
127
r
z0
x(t)
Fixed contact Movable contact
( ) Ar t
sD
mD
bD
AD
( , )mz r tσ=
( , )bz r tσ=
( )mz t( )bz t
( )mr t
( )br t
r
z0
x(t)
Fixed contact Movable contact
( ) Ar t
sD
mD
bD
AD
( , )mz r tσ=
( , )bz r tσ=
( )mz t( )bz t
( )mr t
( )br t
Fig. 4. The arc and contacts geometry: arc
region AD , evaporated , melted
and solid
bD mD
sD zone.
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 and arc power . ( )Ar t ( )AP 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 0 14 V,U = current inductance
opening velocity [2]. One can see that critical time in this case is
0 20 A,I =
47,5 mH,L = 0,2 m / sV = cr 10 mst = ,
however the maximum of arc temperature occurs a little bit later, at 15 mst = due to thermal inertia.
Fig. 5. Dynamics of arc
power . ( )AP t
Fig. 6. Arc temperature:
1 — experimental data
[2]; 2 — calculation.
128
3 4 5 6 7 8
0.1 1 10 100, t ms
3, 10A KT
1
2
0 5 10 15 20 25 30 35 400 50 100 150 200 250 300 350 400
, m st
( ) , WAP t
0
t, ms
ТА, ·10-3 К
t, ms
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 μm, anode temperature rises very sharp in comparison with cathode temperature.
Fig. 7. Dynamics of anode and cathode
temperature at the centre of arc root.
4
129
It can be explained by the fact that in a short arc, which length is comparable with the length of ionization
zone, electron temperature is much greater than ion temperature Ті, therefore kinetic energy of electrons
bombarding anode,
eT
3
2
e
e
kT j
e
, exceeds significantly kinetic energy of ions entering cathode, 3
2
i
i
kT j
e
. Moreover,
calculation shows that in this range of contact gap electron component of current density ej is much greater
than ion component ij , 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
corresponds to change the direction of material transfer for inverse and to beginning of
compensation arc stage, which continues up to
0,15 msact =
1 1,8 mst = 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
. 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
1 2 mst =
0
1
2
3
3, , 10 Ka cT T ⋅
aT
cT
0 0,1 0,2 0.3
, mst
0 0.4 0.5
130
g. 8. Anode and ca-
:
Fi
thode mass transfer.
From bottom to top
1, 2 — ecM (calculated
and experi ent [2]);
3. 4 — ea
m
M (experi-
ment [2] an calcu-
lated).
metalli
d
c phase are sligh y greater than experimental data. It can be explained by recycling phenomenon,
d m , ati m
Influence of inductance on arc duration
Similar calculations were e in the range from 1 to 400
mH
arc phase, while
vari
Fig. 9. Arc
sus
Conclusions
Dynamics of temperature and electrical con e described and analysed satisfactorily by
the
es one to find arc duration
dep
ng arcing is elaborated which
desc
account of enlarging of gaseous arc
pha
1. rowne T. E. Circuit Interruption, Theory and Techniques. — New York and Basel: Marcel Dekker, 1984. —
, μsAt
tl
i. e. re-deposition of evaporate aterial which is ignored in the mathem cal odel.
-30 -25 -20 -15 -10 -5 0 5
0,001 0,01 0,1 1 10 100
7, 10 gecM M −⋅
, mst
ea
carried out for different values of inductanc
. It was found that arc duration increases proportionally inductance and depends on current at
relatively small values of inductance (fig. 9). However for inductance greater than 10 mH this dependence
becomes negligible. This result correlates with experimental data observed in work [2].
Increasing of arc duration with inductance occurs on account of enlarging of gaseous
ation 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.
duration ver
inductance and
current 1 — Io =
= 0,6 A, 2 — Io =
= 1 A and Io =
= 20 A.
ductivity can b
model based on a non-linear problem for axisymmetric heat equation.
The new criterion of arc stability and instability is introduced, which enabl
endently on given circuit parameters and properties of contact material.
The mathematical model of phase transformations inside electrodes duri
ribes dynamics of arc erosion in metallic and gaseous arc phases.
Increasing of arc duration and erosion with inductance occurs on
se, while variation of metallic phase is relatively small.
B
580 р.
2. Kharin S. N., Nouri H. and Davies T. Influence of Inductance on the arc evolution in AgMeO electrical
contacts // Proc. of the 48th IEEE Holm conf. on Electrical Contacts. — Orlando, Florida, USA, 2002.
3. Slade P. Electrical Contacts. Principles and Applications / Marcel Dekker ed. Basel, Switzerland, 1999.
0,
1
10
100
1000
10000
100000
0,01 0,1 1 10 100, mHL
1
2
3
1
0,001
1
2
3
4
131
ntacts
al stage // Proc. of 43th IEEE Holm
Математическая модель температуры и электрической проводимости дуги в
С. Н. Харин, Ю. Р. Шпади, А. Т. Кулахметова
Получен новый критерий стабилизации дуги, который позволяет найти зависимость продолжительности
иной
,
Механічна модель температури та електричної провідності дуги
С. М. Харін, Ю. Р. Шпади, А. Т. Кулахметова
римано новий критерій ст ривалості металічної фази
тематична модель, дугова ерозія, дугові фази.
4. Kharin S. N., Nouri H. and Bizjak M. Effect of vapour force at the blow-open process in double-break co
// IEEE Transactions on Components and Packaging Technologies. — 2009. — 32, No. 1. — P. 180—190.
5. Kharin S. N., Nouri H., Amft D. Dynamics of electrical contact floating in vacuum // Proc. of the 48th IEEE
Holm conf. on Electrical Contacts. — Orlando, Florida, USA, October 21—23, 2002.
6. Engelsht W. S. Dynamics of Electrical Arc. — Ilim ed., Bishkek, 1988.
7. Kharin S. N. Mathematical model of the short arc phenomena at the initi
conf. on Electrical Contacts, 1997. — Philadelphia, USA. — P. 289—305.
металлической и газовой фазах
металлической фазы дуги от заданных параметров цепи и свойств контактного материала. Предложена
математическая модель фазовых переходов внутри электродов в процессе горения дуги, которая
описывает динамику дуговой эрозии в металлической и газовой фазах дуги. Установлено, что прич
возрастания продолжительности дуги и эрозии контактов, происходящего с ростом индуктивности цепи
является расширение газообразной фазы дуги относительно ее металлической фазы.
Ключевые слова: математическая модель, дуговая эрозия, дуговые фазы.
в металічній і газовій фазах
От абілізації дуги, який дозволяє знайти залежність т
дуги від заданих параметрів ланцюга і властивостей контактного матеріалу. Запропоновано математичну
модель фазових переходів в середині електродів в процесі горіння дуги, яка описує динаміку дугової ерозії в
металічній і газовій фазах дуги. Встановлено, що причиною зростання тривалості дуги і ерозії контактів,
яке має місце із зростанням індуктивності ланцюга, є розширення газоподібної фази дуги відносно її
металічної фази.
слова: маКлючові
Fig. 6. Arc temperature:
1 — experimental data [2]; 2 — calculation.
|
| id | nasplib_isofts_kiev_ua-123456789-28894 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | XXXX-0085 |
| language | English |
| last_indexed | 2025-11-24T09:36:00Z |
| publishDate | 2010 |
| publisher | Інститут проблем матеріалознавства ім. І.М. Францевича НАН України |
| record_format | dspace |
| spelling | Kharin, S.N. Shpady, Yu.R. Kulakhmetova, A.T. 2011-11-25T15:39:00Z 2011-11-25T15:39:00Z 2010 Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases / S.N. Kharin, Yu.R. Shpady, A.T. Kulakhmetova // Электрические контакты и электроды. — К.: ИПМ НАН України, 2010. — С. 123-131. — Бібліогр.: 7 назв. — англ. XXXX-0085 https://nasplib.isofts.kiev.ua/handle/123456789/28894 517.958:[536.2+539.219.3] The new criterion of arc stability and instability is introduced, which enables one to find arc duration dependently on given circuit parameters and properties of contact material. The mathematical model of phase transformations inside electrodes during arcing is elaborated which describes dynamics of arc erosion in metallic and gaseous arc phases. Increasing of arc duration and erosion with inductance occurs on account of enlarging of gaseous arc phase, while variation of metallic phase is relatively small. Получен новый критерий стабилизации дуги, который позволяет найти зависимость продолжительности металлической фазы дуги от заданных параметров цепи и свойств контактного материала. Предложена математическая модель фазовых переходов внутри электродов в процессе горения дуги, которая описывает динамику дуговой эрозии в металлической и газовой фазах дуги. Установлено, что причвозрастания продолжительности дуги и эрозии контактов, происходящего с ростом индуктивности цепиявляется расширение газообразной фазы дуги относительно ее металлической фазы. Отримано новий критерій стабілізації дуги, який дозволяє знайти залежність тривалості металічної фази дуги від заданих параметрів ланцюга і властивостей контактного матеріалу. Запропоновано математичну модель фазових переходів в середині електродів в процесі горіння дуги, яка описує динаміку дугової ерозії в металічній і газовій фазах дуги. Встановлено, що причиною зростання тривалості дуги і ерозії контактів, яке має місце із зростанням індуктивності ланцюга, є розширення газоподібної фази дуги відносно її металічної фази. en Інститут проблем матеріалознавства ім. І.М. Францевича НАН України Электрические контакты и электроды Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases Математическая модель температуры и электрической проводимости дуги в металлической и газовой фазах Механічна модель температури та електричної провідності дуги в металічній і газовій фазах Article published earlier |
| spellingShingle | Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases Kharin, S.N. Shpady, Yu.R. Kulakhmetova, A.T. |
| title | Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases |
| title_alt | Математическая модель температуры и электрической проводимости дуги в металлической и газовой фазах Механічна модель температури та електричної провідності дуги в металічній і газовій фазах |
| title_full | Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases |
| title_fullStr | Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases |
| title_full_unstemmed | Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases |
| title_short | Mathematical model of arc temperature and conductivity at metallic and gaseous arc phases |
| title_sort | mathematical model of arc temperature and conductivity at metallic and gaseous arc phases |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/28894 |
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