28 th ICPIG, July 15-20, 2007, Prague, Czech Republi c
Topic number: 05
Department of Electrical Technology Education, Faculty of Industrial Education and Technology
King Mongkut’s University of Technology Thonburi, Bangmod, Thungkru, Thailand
Tel:+66-2-470-8540, Fax:+66-2-4708541
E-mail: narong_kmutt@hotmail.com
In order to analyze current density effecting on plasma parameters, a cathode spot model is performed. The cathode spot model assumes that the collisionless sheath and the singly ionized collisional plasmas are directly connected .
It was found that current density increases with decreasing arc current. Dependent variables, cathode temperature, electron current fraction, plasma temperature, sheath voltage, plasma density and cathode electric field are increased with increasing current density. Alternatively, the cathode spot radius is inversely dependent to the current density. These dependent variables are related with to T-F emission and thermionic emission process . This cathode spot model of the present analysis which is used to investigate the cathode spot parameter in low-current vacuum arc may be valid for volatile materials.
1. Introduction
Vacuum arcs play a dominant role in a several range of industrial applications, i.e. for vacuum interrupters, thin film technologies. The cathode current fraction δ ( I )
flowing toward the anode are applied to obtain the eight dependent variables [3]-
[4]. The seven dependent variables and independent region is considered to be the most active region in the vacuum arcs. The cathode spots provide not only for current continuity but supply the medium for the discharge, viz the metal vapour [1]-[2].
During arc, the current density of cathode spot is very important parameter for arc cathode. The cathode current density affects directly to the T-F emission and the thermionic emission on cathode surface. Accordingly, extensive research has been performed on modeling of the vacuum arcs[3]-[4].
In order to investigate the behavior of the current density which affects to the plasma parameters, the variable are plotted with the current density.
Potential 0
Vp a
0 r
Cu e
Cu
+
Nn
N o e
Reverse Diffusion
X
Electrode Collisionl ess Collisiona l Diffused cathode spot model as shown in Fig. 1 is performed for an analysis of instability phenomena in low current vacuum arc. The cathode spot region is considered to be the collisionless space charge
Sheath Plasma
Fig. 1 Cathode Spot Model
Plasma sheath connected by the singly ionized collisional plasmas. In this paper, this model is used to analyze phenomena of the relation between the current density and other plasma parameters.
2.1 Nomenclature
1. Independent Variable
I Arc current (A)
2. Experimental Data
V eff
( ) Effective cathode heating voltage (V)
2. System of Equation [4]
The cathode spot model assumes that the collisionless sheath and the collisional plasma are directly connected. Eight equations are required in order to determine the eight dependent variables. For the lack of a simple exact formula by which to determine the sheath voltage Vp , some other means is required. The experimental data of effective heating voltage for cathode input V eff
( I ) and ion
( I ) Ion current fraction flowing toward the
anode
3. Dependent Variables
V p a
J
S
T
Sheath voltage (V)
Cathode spot radius (m)
Current density (A/m 2 )
Electron current fraction
Temperature of cathode spot surface (K)
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28 th ICPIG, July 15-20, 2007, Prague, Czech Republi c
F o
N o
Cathode electric field (V/m)
Plasma density (1/m 3 )
T e
H o
( T )
Electron temperature (K)
4. Physical Properties and Constant
Γ
P ev ev
Evaporation rate (kg/m 2 s)
Evaporation energy (W/m 2 s)
Heat of evaporation per atom (J/atom)
K
V i
Φ o
A
Φ ( F o
, T )
Thermal conductivity (W/mK)
Ionization voltage of Copper 7.73 (eV)
Work function of Copper 4.5 (eV)
Richardson’s constant 1.20×10 6 (A/m 2 K 2
Cooling effect of electron emission (eV)
)
M Mass of atom and ion of Copper (kg) m q
Electronic mass (kg)
Electronic
− q

Φ o
− qF o
4 πε o


SJ = AT
2 exp (4) kT
2.2.5 Equation of electric field
The equation of the electric field of the cathode surface is given by the Mackeown equation, including the effect of the space charge of the electrons returning from the collisional plasma to the sheath.
4
ε
0







M
2 q
1 − S
J − m
2 q
SJ



V p
−
2 kT e
ε o
N o



1 −
F
0
2 exp
 −
 qV kT e p
=







(5) k Boltzmann’s constant (J/K)
System of equations is divided in two region equations as sheath region equation and equation of the plasma region.
2.2 Equation of sheath region
2.2.1 Equation of current
(
1 − S
)
I
J
= π a
2
J (1)
2.2.6 Equation of energy balance
The equation (6) is the solution of heat of conduction at the boundary condition, which is as follows:
K
T
X
JV eff
r
a
(
(
2
V
K o
(
K
)
T
( 0 .
45 T
)
0
+ 348 )
)
=
−
8 a
JV eff
(6)
0
M
δ J
3 π
2 .
2.2
Equation of mass flow
Γ ev
( T ) − N


2 kT e
π M


1
2
= q
M (2)
The temperature dependence on thermal conductivity of copper is considered [5]. The heat loss due to thermal conduction into the cathode is as follows:
The first term of the left-hand side of equation (2) is atom flux due to evaporation from the cathode, and the second term is the return flux of ions from the plasma to the cathode. The right hand-side of the equation (2) is mass flow to the anode provided by the ion current.
2..2.3 Equation of ion current
The ion current density (1S ) J in the pace charge sheath is assumed to be equal to the ion saturation current density of collisional plasma. Thus, equation
(3) is concluded as
= qN o

 kT e
2 π M


1
2
2.2.4 Equation of electron emission current
The electron emission current from the cathode is determined primarily via thermionic mechanism, together with the Shottky effect.
JV eff
=
(
1 − S
)
J p
+ V i
− Φ o
+ H o
SJ Φ ( F
0
, T ) − P ev
( T )
(7)
The first term of the right-hand side of the equation (7) is the input due to the ion bombardment, the second term is power dissipated by the electron emission, and the third term is the power dissipated by vaporization.
2.3 Equation of the plasma region
2.3
.
1 Particle conservation
The equation of particle conservation is as same the equation (2).
(3)
2 .
3 .
2 Energy conservation of the collisional plasma.
kT e q
J + 2 δ − S + qV i
Γ ev
M
= 0 .
851 a η J
2
(8)
The first term of the left hand–side of the equation
(8) is the energy flow into the cathode and the
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28 th ICPIG, July 15-20, 2007, Prague, Czech Republi c anode, and the second term is the power required by ionization. The right-hand side is the input power to the plasma by joule heating, where η is the plasma resistance expressed by the Spitzer formula.
3. The experimental data
The experimental data for effective cathode heating voltage V eff
( )
obtained using the calorimetric method [3]. The ion current fraction
( I ) is set to
0.1 [6].
4. Numerical results and discussions
The simultaneous algebraic equation (1) – (8) are solved numerically using a bisection method. The eight dependent variables are obtained for the independent variable of arc currents ranging 19 – 70
A, as shown in Fig. 2-9.
As Fig.2., it is seen that the current density increases with decreasing arc current. As Fig. 3-8., it was found that cathode temperature, electron current fraction, plasma temperature, sheath voltage, plasma density and cathode electric field are increased with increasing current density. Alternatively, the cathode spot radius is inversely dependent to the current density. These dependent variables are related significantly with to the T-F emission and the thermionic emission process . Also, these results were similar to obtained data in [7], indicating that the cathode spot model of the present analysis which is used to investigate the cathode spot parameters in low-current vacuum arc may be valid for volatile materials.
80
70
60
50
40
30
20
10
0
0 2 4 6 8 10
Current density J(10 9 )[A/m 2 ]
Fig. 2. Arc current vs. Current density
12
3
2.5
2
1.5
1
0.5
0
0
0.3
0.25
0.2
0.15
5000
4800
4600
4400
4200
4000
3800
3600
3400
3200
3000
0 2 4 6 8 10
Current density J (10 9 )[A/m 2 ]
Fig. 3. Cathode temperature vs. Current density
12
0.4
0.35
0.1
0.05
0
0 2 4 6 8 10 12
Current density J (10 9 )[A/m 2 ]
Fig. 4. Electron current fraction vs. Current density
2 4 6 8
Current density J (10 9 )[A/m 2 ]
10 12
Fig. 5. Plasma temperature vs. Current density
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28 th ICPIG, July 15-20, 2007, Prague, Czech Republi c
12
10
8
6
4
2
0
0
10
9
8
7
6
3
2
1
0
5
4
0
30
25
20
15
10
5
2 4 6 8
Current density J (10 9 )[A/m 2 ]
10
Fig. 6. Sheath voltage vs. Current density
2 4 6 8
Current density J (10 9 ) [A/m 2 ]
10
Fig. 7. Plasma density vs. Current density
12
12
18
16
14
12
10
8
6
4
2
0
0 2 4 6 8 10 12
Current density J (10 9 )[A/m 2 ]
Fig. 9. Cathode spot radius vs. Current density
5 References
[1] A. Farrall, “Current zero phenomena” in
Vacuum Arcs: Theory and Application, J.M.
Lafferty, Ed. New York : Jhon Wiley and Sons, pp.
197-199, 1980
[2] G. Eker, “Theoretical aspects of the vacuum arc,” in Vacuum Arcs: Theory and Application, J.M.
Lafferty, Ed. New York : Jhon Wiley and Sons, pp.
265-268, 1980
[3] O.Morimiya, S.Suzuki and K. Watanabe, “An analysis of instability phenomena of a low current vacuum arc ,” (in Japanese) T.IEE Japan., vol. 119-
A, No. 2, pp. 190-196 , 1999
[4] N. Mungkung O.Morimiya, S.Suzuki and T.
Kamikaji , “An analysis of instability phenomena of a low current vacuum arc for copper cathodes,”
IEEE Trans. Plasma Sci.,vol.31, No. 5, pp. 963-967,
2003
[5] H.S.Carslow and J.C.Jaeger, Conduction of
Heat in solids. New York: Oxford, 1959, pp.214-217
[6] J.E. Daalder., “Cathode spot and vacuum arcs,”
Physica, vol. 10C4, pp. 91-106, 1981
[7] B. Juttner, V.F. Puchkarev, E. Hantzsche and I.
Beilis., “Cathode spots,” in Handbook of Vacuum
Arc Science and Technology, R. l. Boxman,P.J.
Martin, and D.M. Sanders, Eds. Park Ridge, Nj:
Noyes, 1995, ch7.
0
0 2 4 6 8
Current density J (10 9 )[A/m 2 ]
10 12
Fig. 8. Cathode electric field vs. Current density
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