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Decay Modes of Anode Surface Temperature After Current Zero in Vacuum Arcs—Part II: Theoretical Study of Dielectric Recovery Strength Zhenxing Wang, Yunbo Tian, Hui Ma, Yingsan Geng, Member, IEEE, and Zhiyuan Liu, Member, IEEE Abstract— Anode surface temperature after current zero has a great impact on the interruption capacity of a vacuum interrupter. The objective of this paper is to theoretically investigate the relation between breakdown voltages and anode surface temperature after current zero. A heat conduction model was adopted to describe the temperature development of the anode, taking account of phase transition and evaporation. The breakdown voltages in certain metal vapor densities were obtained by the Particle-in-Cell/Monte Carlo collision (PIC-MCC) method. Finally, the Paschen curve for copper vapor was obtained using the PIC–MCC method and verified by the theoretical model. Moreover, the minimum breakdown voltage, 30 V, was obtained at a density of 1.3 × 1022 /m3 with a gap of 10 mm, which corresponded to a surface temperature of 1983 K. In order to ensure a successful interruption, anode surface temperature should not be higher than 1983 K at current zero, and the melting time should be kept as short as possible. Index Terms— Anode surface temperature, dielectric recovery strength, metal vapor density, Paschen curve, vacuum interrupter.
I. I NTRODUCTION
A
NODE surface temperature after interrupting a high-current vacuum arc has a significant impact on the decay of metal vapor density after current zero, which dominates the interruption capacity of a vacuum interrupter [1]–[3]. A breakdown may occur if the surface is still hot in a postarc dielectric recovery process. In order to achieve a successful interruption, the surface temperature has to be reduced to a certain level so that the contact gap rapidly recovers to a vacuum condition. For low-current interruptions, anode surface temperature is quite low because the vacuum arc is emitted from cathode spots on the cathode contact, and the anode acts as a passive absorber. For high-current interruptions, however, a stationary anode spot appears on the anode surface, which inputs energy into the surface, locally resulting in overheating. Thus, excess metal vapor is
Manuscript received November 28, 2014; revised June 21, 2015; accepted August 6, 2015. Date of publication August 24, 2015; date of current version October 7, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 51177122 and Grant 51407135 and in part by Shaanxi Province Innovation Team Funding under Grant 2012 KCT-07. The authors are with the State Key Laboratory of Electrical Insulation and Power Equipment, Department of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China (e-mail:
[email protected];
[email protected];
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPS.2015.2467158
continuously generated from the anode spot even after current zero. The higher the anode surface temperature is, the more the metal vapor that will be evaporated into the contact gap of the vacuum circuit breaker (VCB). Therefore, it is necessary to keep the anode surface temperature low and avoid heating the surface locally. In [4], the decay processes of the anode surface temperature after current zero exhibited two modes. 1) For a low current, the anode surface temperature decayed exponentially (mode I). 2) For a high current, the surface temperature did not drop smoothly and presented two stages in the decay process (mode II). The results also show that the anode began to activate in mode II. These results agree with the results obtained in [5] and [6], in which was introduced an indicator called melting time to describe the decay of the anode surface temperature after interrupting a high current, and the interruption capacity of a VCB was associated with the duration of melting time directly. However, we still do not explicitly know the effect of surface temperature on interruptions. There are many calculations on thermal transfer on an anode [7]–[9] and also many calculations on breakdown in a low pressure [10], [11], but the relation between surface temperatures and breakdowns in the postarc period, which is critical to understanding the physical processes of an interruption in a vacuum interrupter, is still unclear. The objective of this paper is to theoretically investigate the relation between breakdowns and anode surface temperature after current zero. Thus, a heat conduction model was used to calculate the anode surface temperature after interrupting a vacuum arc. This was followed by the use of the Particle-in-Cell/Monte Carlo collision (PIC-MCC) method to simulate the evolution of postarc breakdowns in metal vapor via temperature calculations. The physical meaning of the two decay modes observed by the previous experiments [4]–[6] was interpreted by the temperature calculations. The breakdown voltage curve was obtained by the PIC calculations. In addition, the effects of a molten surface on a breakdown after current zero were discussed. II. S URFACE T EMPERATURE S IMULATION M ODEL A. Physical Model A thermal model was established to describe a heat conduction process of an anode surface after interrupting a
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WANG et al.: DECAY MODES OF ANODE SURFACE TEMPERATURE AFTER CURRENT ZERO IN VACUUM ARCS
Fig. 1.
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Physical model for anode surface temperature calculation.
vacuum arc, taking into account the phase transition process as shown in Fig. 1. There were four regions in the model including an arc column, an anode sheath, a molten region, and a solid region. The shape of the arc was centrosymmetric corresponding to the arcs with an axial magnetic field (AMF) control. The energy was provided by the arc column with the same shape as the arc which heats the contact surface, resulting in melting. Thus, the contact was divided into a solid part and a molten part. The boundary can move according to the input and output of the energy. The energy can dissipate by thermal conduction and evaporation, despite the effects of radiation and convection. In order to simplify the model, the sheath, which adjusts the energy into the surface, was also neglected. In addition, copper was selected as the anode material. B. Thermal Conduction A 3-D transient model of the anode surface temperature was adopted to describe a thermal conduction by the following equations: ∂ (ρ H ) = ∇ · (k∇T ) (1) ∂t where T is the temperature on contact, ρ is the density of the contact material copper, k is the thermal conductivity, and H is the enthalpy. The enthalpy of the material is calculated by the following equation: H = h + βL
(2)
where h is the sensible enthalpy, β is the liquid fraction, and L is the latent heat of the material. The liquid fraction β is defined as ⎧ ⎪ if T < Ts ⎨β = 0, (3) β = 1, if T > Tl ⎪ ⎩ β = (T − Ts )/(Tl − Ts ), if Ts < T < Tl where Ts is the solidus temperature of the contact material and Tl is the liquidus temperature. C. Heating Energy The distribution of heat flux varied with time can be empirically expressed by the following equation [7]: H = H p × e−5000r × sin(100πt) 2
(4)
Fig. 2. Variations of saturated pressure and evaporated energy as functions of surface temperature Ts .
where r is the radial position at the anode surface, t is the simulation time, and H p is the peak value of heat flux density. In this paper, the value of H p that is closely related to the arc current was obtained from magnetohydrodynamic simulation of the vacuum arc [12]. The values of 5 × 108 , 8 × 108 , 1.2 × 109 , and 1.5 × 109 W/m2 correspond to arcing currents of about 20–50 kA (peak) with an AMF of 5 mT/kA [13]. D. Cooling Process by Evaporation Evaporation can vaporize anode material as well as take a lot of energy away from the anode surface. The energy carried away by evaporation per unit area can be expressed as [14] Ps (Ts ) L evap = cV √ Ts m
(5)
where c = 5.76 × 109 is a constant, V = 3.19 eV is the vaporization enthalpy per atom, Ts is the surface temperature, Ps is the saturated vapor pressure, and m is the atomic mass of the anode material (copper). The saturated vapor pressure for copper material can be expressed by the Clausius–Clapeyron equation vap H 1 1 − Ps = Patm exp − (6) R Ts Tboil where Patm = 105 Pa represents the atmospheric pressure, R is the gas constant, vapor H = 300.4 kJ/mol is the latent heat of vaporization, and Tboil = 2868 K is the boiling temperature of copper. Fig. 2 shows the calculated curves for saturated pressure and evaporated energy as functions of surface temperature Ts . III. B REAKDOWN S IMULATION M ODEL A. PIC–MCC Model The PIC–MCC was adopted to simulate the evolution of breakdowns in metal vapor. The PIC method is a simulation method for studying plasma physics, based on the evolution of plasma distribution function in phase space [15]. In some cases such as discharge, a short-range collision has to be considered. Thus, the MCC method is introduced to PIC’s scheme. A detailed discussion of the PIC–MCC can be found in [16], and the simulations in this paper were implemented by the code VSim, also known as VORPAL [17].
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Fig. 3.
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO. 10, OCTOBER 2015
Schematic of the breakdown model.
A model with two dimensions in physical space and three dimensions in velocity space (2d3v) was developed, as shown in Fig. 3. The cathode was designed with a rounded end for avoiding a distorted electric field near the edge, and a negative direct voltage was connected to it. In addition, the anode was connected to the ground. Only the monovalent ions were taken into account because we focused on the initial stage of breakdowns in which Cu+ ions are generated easily. Thus, there were three types of particles in the model including Cu vapor, Cu+ , and electrons. The particles were absorbed by the boundaries when they reached the boundaries or contact surfaces of the model. Elastic scatter, ionization, and excitation were taken into account, and the cross-sectional data were obtained from evaluated electron data library [18] Cu + e− → Cu + e− Cu + e− → Cu+ + 2e−
(7) (8)
Cu + e− → Cu∗ + e− .
(9)
Assuming that the background metal vapor has a uniform unchanged temperature of T = 2000 K, the density N can be obtained by substituting the ideal gas law into (6), which results in the equation N = Ps /kT. Moreover, it was assumed that a breakdown was initialized by seed electrons emitted from the cathode surface. The Richardson–Dushman equation was adopted to describe the emission of the electrons W 4πme 2 T exp − J=M (10) h3 T where h is the Planck constant, T is the surface temperature, M is a constant parameter, m is the mass of an electron, and e is the charge of an electron. And W can be expressed as W = W0 −
e |Ef | 4πε0
(11)
where W0 is the work function of the contact material, ε0 is the vacuum permittivity, E is the electric field strength, and f is the field enhancement factor. B. Secondary Electron Emission Secondary electron emission from contact surfaces is another important physical effect that must be taken into account. Vaughan [19], [20] gave an empirical formula to
Fig. 4.
Secondary electron emission yield for copper.
estimate that secondary electron emission yield and can be expressed as θ2 × f (w, k) (12) δ(E inc , θ ) = δmax 1 + ks 2π where E inc is the energy of an incident particle, δmax is the maximum value of the secondary electron emission yield, θ is the angle between the incident particle and contact surface, and ks is the surface coefficient. Moreover
⎧ ⎪ ⎨(we(1−w) )k , k = 0.56, w ≤ 1 (13) f (w, k) = 0.25, 1 < w ≤ 3.6 ⎪ ⎩ −0.35 , w > 3.6 1.125w and w = (E inc − E min )/(E max − E min ), where E min is the minimum energy that can induce secondary electron emission from contact surface and E max is the maximum energy corresponding to the maximum value of the secondary electron emission yield. According to the data for copper in [15], the secondary electron emission yield for the contact material copper is shown in Fig. 4. IV. R ESULTS A. Temperature Simulation Results A 3-D transient model of anode surface temperature was established to obtain the evolution and distribution of an anode surface temperature, taking into account conduction, phase transition, and evaporation. The anode was made of copper with a diameter of 60 mm; the input and the output of the energy on the surface are described in (4) and (5), respectively. From the results of temperature simulations, the background metal vapor density can be estimated quantitatively, and the decay characteristics, which are correlated with the dielectric recovery process of a vacuum interrupter, can be obtained. Figs. 5 and 6 show the temperature distribution and evolution on the cross section and surface, respectively. Fig. 7 shows the variation of temperature along the radial axis from the center to the edge. The heat flux into the anode was a 50-Hz sine wave with a peak value of 1.2 × 109 W/m−2 . Apparently, the energy from the arc caused melting on the anode, and the melting lasted well after 10 ms when the arc was extinguished. The maximum temperature, depending on the distribution of
WANG et al.: DECAY MODES OF ANODE SURFACE TEMPERATURE AFTER CURRENT ZERO IN VACUUM ARCS
Fig. 5. Distribution and evolution of temperature on the cross section of the anode during the arcing period.
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Fig. 8. Evolution of the surface temperature at the center of the anode during arcing periods.
Fig. 9. Decay of surface temperature at the center of the anode after current zero.
Fig. 6. Distribution and evolution of temperature on the anode surface during the arcing period.
Fig. 7.
Variations of surface temperature along the radial axis.
energy input, appeared at the center of the anode surface and declined with the radial direction. The energy was conducted to the inside of the anode, causing the depth of melting area to increase. The whole melting area extended to half of the surface. It remained a large area after current zero with a temperature well above the melting temperature of the anode material. Fig. 8 shows the evolution of the surface temperature at the center of the anode. The heat fluxes into the anode were
50-Hz sine waves with peak values of 5 × 108 , 8 × 108 , 1.2 × 109 , and 1.5 × 109 W/m−2 , respectively. A higher energy input indicated a higher temperature curve, both the peak value and the value at current zero. The anode surface temperatures changed in accordance with sine waves, which were similar to the wave of an arcing current but with different frequencies. Furthermore, the temperature usually achieved the peak value later after 5 ms when the arcing current was at its peak value. This is because the thermal capacity of the material impeded the change in the temperature rapidly. For the heat flux density of 5 × 108 W/m−2 , the peak value during the arcing was about 1250 K, which was lower than the melting point of the anode material copper, 1356 K. For the heat flux of 8 × 108 , 1.2 × 109 , and 1.5 × 109 W/m−2 , however, the peak values were higher than the melting point, so melting occurred. Consequently, the rise in the surface temperature was limited because the melting process, which transferred the material from solid to liquid, required extra energy from the arcing current to be input into the surface. Furthermore, the temperature at current zero was still higher than the melting temperature for the heat flux density 1.2 × 109 and 1.5 × 109 W/m−2 , which led to a liquid state of the anode surface. Fig. 9 shows the decay of the surface temperature at the center of the anode after current zero. Although the initial temperatures at current zero were different, the anode temperature exponentially decayed after current zero for the heat flux density of 5 × 108 and 8 × 108 W/m−2 . The anode
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Fig. 10. Multiplication of electrons and ions varied with time during a breakdown. Fig. 11.
Spatial evolution of ions during a breakdown.
B. Breakdown Simulation Results
Fig. 12.
Spatial evolution of electrons during a breakdown.
A model with two dimensions in physical space and three dimensions in velocity space (2d3v) was developed. There were three types of particles in the model including Cu vapor, Cu+ , and electrons. The model accounted for elastic scatter, ionization, and excitation. From the results, the detailed process of a breakdown can be obtained, and the relation between the background metal vapor densities and breakdown voltages can also be obtained. Consequently, the relation between surface temperatures and breakdown voltages can be derived based on the assumption that the background metal vapor density depends on the surface temperature at an equilibrium state. Fig. 10 shows the multiplication of electrons and ions varied with time during a breakdown process. The background neutral vapor was copper with a density of 1022/m3 , an applied voltage of 100 V, and a gap distance of 10 mm. The curves for electrons and ions developed in the breakdown have the same features. From the starting point to 1.5 μs, the primary electrons emitted by the fields ionized the neutral background vapor, and the number of the particles increased with a relatively slow rate. After that, the number accelerated at a high rate and exponentially increased due to electron avalanche. The curve for ions had a smooth feature compared with that for electrons since copper ion mass is much greater than electron mass. Thus, the ions remained in the calculation
region, but the electrons were absorbed by the boundaries of the region. For the same reason, the number of the ions was larger than that of the electrons. Figs. 11 and 12 show the spatial evolution of ions and electrons, respectively, during a breakdown. The background neutral vapor was copper with a density of 1022/m3 , an applied voltage of 100 V, and a gap distance of 10 mm. The evolutions for ions and electrons presented different characteristics in developing a breakdown. The neutral metal vapor near the cathode was ionized by the electrons emitted from its surface in the initial stage of the breakdown. Then the gap was gradually filled with ions, and the density also increased. Since the electrons moved faster than the ions, they moved to the anode by the electric force and concentrated in the region. Then they expanded from the anode to the cathode. The evolutions combined with the multiplication shown in Fig. 10 can be adopted to judge the time of the breakdown. If electrons were uniformly distributed across the gap while multiplying, this meant that a discharge channel was formed. Fig. 13 shows the evolution of electric potential distribution across the gap during a breakdown. The background neutral vapor was copper with a density of 1022 /m3 , an applied voltage of 100 V, and a gap distance of 10 mm. In the initial stage,
temperature cannot quickly change from the temperature at current zero to the ambient temperature before arcing because of the thermal capacity of the anode material. For the heat flux density of 1.2 × 109 and 1.5 × 109 W/m−2 , the surface presented two stages in the decay periods. In the first stage, the temperature decayed from the initial value at current zero to the melting point of the contact material and held for a while. In the second stage, the temperature exponentially decayed to the ambient temperature. The discontinuous curves were caused by the material recovering from a liquid to a solid state and releasing excess energy out of the surface. The released energy also impeded the decay of the anode temperature, so the temperature stayed at the melting point for several milliseconds.
WANG et al.: DECAY MODES OF ANODE SURFACE TEMPERATURE AFTER CURRENT ZERO IN VACUUM ARCS
Fig. 13. Evolution of electric potential distribution across the gap during a breakdown.
Fig. 15.
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Typical curves of particle number without a breakdown.
Fig. 16. Comparison between breakdown time points for 1- and 10-mm gaps varied with applied voltages.
Fig. 14.
Energy and current absorbed by the anode and cathode surfaces.
the electric potential was uniformly distributed across the gap since the number of the particles was too low to affect the distribution. However, the electric field strength adjacent to the cathode began to increase after 1500 ns, corresponding to the electron distribution that expanded from the anode to the cathode shown in Fig. 12. Eventually, a sheath formed adjacent to the cathode, which resulted in a strong electric field that accelerated ions to bomb the cathode. The ions with high energy led to an intensive secondary electron emission and acted as a major electron source instead of a primary electron emission by the fields. Fig. 14 shows the energy and current absorbed by the anode and cathode surfaces, respectively. The background neutral vapor was copper with a density of 1022/m3 , an applied voltage of 100 V, and a gap distance of 10 mm. The energy absorbed by the surfaces for both particles gradually increased along with the current amplitude. The amount of electron current absorbed by the anode was greater than that of ions absorbed by the cathode, but it was still in the same order of magnitude. However, the ion energy absorbed by the cathode was slightly greater than that of electron because the sheath formed adjacent to the cathode giving more energy to ions from the electric field.
Fig. 15 shows the typical multiplication processes of ions and electrons without a breakdown. The background neutral vapor was copper with a density of 1022 /m3 , an applied voltage of 40 V, and a gap distance of 10 mm. The numbers of electrons and ions linearly increased in the initial stage, but soon reached equilibrium between the generation and dissipation of the particles. This equilibrium could be achieved after several microseconds or more time, which should be noted to judge a breakdown. Even if there were no breakdowns, the number of particles would exponentially increase in the initial stage. Thus, it is necessary to take enough time to estimate the state of equilibrium and breakdowns. Fig. 16 shows the comparison between breakdown time points for 1- and 10-mm gaps with applied voltages. The background neutral vapor was copper with a density of 1022/m3 , and the gap distances were 1 and 10 mm, respectively. The applied voltages were 50, 100, 200, 300, and 400 V. It took more time to break down the gap with a larger distance compared with a smaller one. The differences between the two cases were significant when the applied voltages were low. However, they were reduced to about 1 μs with an increase in the applied voltages and remained at that level. The larger gap distance required more time for particles to be absorbed by the surfaces leading to secondary emissions, which is an important mechanism to induce a discharge. However, the increase in applied voltages reduced
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Fig. 17. densities.
IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 43, NO. 10, OCTOBER 2015
Breakdown voltages varied with the background metal vapor
the difference caused by the variation of gap distances for the voltages. It also accelerated the particles that were then rapidly absorbed at the surface. Fig. 17 shows the breakdown voltages varied with the background metal vapor densities. The gap was fixed at a distance of 10 mm. To make computation more efficient, the breakdown voltages were obtained by attempting to impose the applied voltages at an interval of 5 V. In the low-density regions, the breakdown voltages declined with the increase in the metal vapor densities. After the density of 1.3 × 1022 /m3 , the breakdown voltage remained until 2 × 1022 /m3 and, after that, began to increase with a slow rate. The breakdown voltage for the density 1.3 × 1022 /m3 was 30 V, which was the minimum voltage of breakdown voltage in copper vapor. The corresponding surface temperature was 1983 K derived from (6) and the ideal gas law, assuming a constant metal vapor temperature of 2000 K. This was also the critical temperature for breakdowns. V. D ISCUSSION The dielectric recovery strength after current zero plays a decisive role in the current interruption process of a vacuum interrupter [13]. Usually, there are many factors that can affect the process such as the dissipation of metal vapor, ions, and droplets [21]. These particles, which are mainly produced during the arcing time, remain in the switching contact gap and decay with time under the effect of a transient recovery voltage that is applied upon the gap after current zero. Moreover, it is considered that the decay of metal vapor dominates the recovery process, and the dielectric recovery behaviors can be estimated according to it since different metal vapor densities correspond to different breakdown voltages. Furthermore, the boundary of breakdowns is adopted to describe the dielectric recovery rate that indicates the interruption capacity of a vacuum interrupter [22]. In addition, the surface temperature on the anode is also a significant factor that cannot be neglected, especially after interrupting a high current [23]. The temperature on the cathode rapidly recovers to a low level after current zero because of homogeneous heating in the arcing period. On the contrary, the anode is still high due to inhomogeneous heating evaporating a great deal of particles. The former anode in
Fig. 18.
Dissipate of metal vapor after current zero as a function of time.
the arcing period becomes the postarc cathode after current zero whose surface dominates the following breakdowns. Therefore, the condition on the postarc cathode, the former anode, is also a factor that has an impact on breakdowns. One reason is that a higher temperature on the surface leads to a higher probability to emit electrons from it; the other reason is that a protrusion, which alters the local electric field a lot, is much easier to be formed on a liquid surface under the effect of a transient voltage [24]. Next, we will discuss the effect of temperature in detail from the aspects of dissipation, breakdown, and the state of surface. A. Dissipation of Metal Vapor The dissipation of metal vapor plays a critical role in understanding the dielectric recovery after current zero. Although the pressure, compared with the pressure in gas discharges, is not very high after current zero, the vapor density has a significant impact on breakdowns when the gap is applied to a transient voltage after current zero. The recovery processes are considered complete when the density of metal vapor drops below a level of the electron mean free path in the vapor, which is of the order of the gap length [22]. The Rich–Farrall model [25] is widely used to describe the dissipation of metal vapor after current zero. The model not only considers the variation of metal vapor density with a vacuum arc current but also calculates the decay of the metal vapor with time. This model was adopted to predict the dissipation of metal vapor. The expression for the metal vapor density at the center of the gap volume at time t after current extinction can be described as 1 L 1 L − 12 (14) n 0, , t = n 0 1 − e α erf 2 2 R α where α = t/R(2k B T /m)1/2 , R is the radius of the contact, L is the gap length, m is the mass of a metal atom, kb is √ 2 x the Boltzmann constant, and erf(x) = 2/ π 0 e−τ /2 dτ represents an error function. Fig. 18 shows the dissipation of metal vapor after current zero as a function of time using the Rich–Farrall model. The metal vapor density decayed smoothly and dropped by an order of magnitude compared with the initial density over 50 μs. The results indicate that metal vapor decayed
WANG et al.: DECAY MODES OF ANODE SURFACE TEMPERATURE AFTER CURRENT ZERO IN VACUUM ARCS
with a time constant of the order of microseconds. However, Takahashi et al. [26] measured the metal vapor density after interrupting a vacuum arc, and their experimental results show that the dissipation of metal vapor is slower than that predicted by the Rich–Farrall model. The time constant of the dissipation was of the order of hundreds of microseconds. If this model is suitable for estimating the dielectric recovery, there may be other factors impeding the dissipation of the metal vapor. On the other hand, surface temperature is an important factor that postpones the dissipation of metal vapor. The results shown in Figs. 5, 6, and 9 show that the surface temperature decayed with a time constant of the order of milliseconds. That means the surface temperature will stay nearly constant in a short period of several microseconds. A hot surface will impede the dissipation of metal vapor because it will absorb the vapor as well as cause evaporation. In a quantitative analysis, the relation between surface temperature and metal vapor can be described by (6). Although the dissipation of metal vapor is not in an equilibrium state, thermodynamics will force the metal vapor to approach to an equilibrium state. Furthermore, Sarrailh et al. [27] gave the estimation on the transfer time, from the beginning to an equilibrium state, of the order of several microseconds by the Direct Simulation of Monte Carlo method. That means the transfer time is much shorter than the decay time of temperature, and the pressure of the gap will quickly achieve an equilibrium state depending on the decay of the surface temperature. Therefore, it is reasonable to take account of the effect of surface temperature to estimate a dielectric recovery process, especially after interrupting a high current when the surface temperature is also high. B. Breakdown in Metal Vapor The other important issue is the relation between metal vapor densities and breakdown voltages. When the vapor density drops to a quite low value after the temperature cooling down, the breakdown is supposed to be a vacuum breakdown instead of a gas breakdown. The vacuum breakdown needs to model the process of metal vapor generations, but an initial vapor density can be assumed in a gas breakdown calculation. Moreover, the gas discharge theory is still valid to estimate the breakdown after high-current interruptions because the vapor density is still large. In addition, the PIC–MCC, adopted to simulate the evolution of particles, is a good choice for estimating the breakdown in low-density vapor. Unfortunately, the data for metal vapor are not available from experiments, so it is difficult to confirm the simulation results directly. For gas discharges, the Paschen law is commonly used to describe the relation between vapor densities and breakdown voltages, and also many experimental results are obtained to verify the law. However, it is difficult for metal vapor, especially for materials with high vaporization temperatures, to be vaporized and accurately controlled. Thus, our simulation results cannot be directly verified by experiments, but can be compared with other theoretical models. Burm [28] gave a set of equations that can be adopted to describe the Paschen curves in metal vapors and also verified the theory
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Fig. 19. Breakdown voltages as a function of pressure × gap distance. The solid line was obtained from Burm’s equations and the dot from PIC–MCC simulations.
by comparing the theoretical results with experiments for Hg and Zn [29]. According to his derivations, the breakdown voltages can be expressed as Vbreakdown =
B0 pd ln(A0 pd) − ln(ln[γ + 1/γ ])
(15)
where A0 and B0 are constants for breakdowns depending on the type of vapor, d is the gap distance, p is the vapor pressure in the gap, and γ = 0.1 [30] is the second Townsend coefficient. The constants A0 and B0 can be written as 3 e4 (16) 2 k T 16πε02 E 12 B h 5 πar2 E 12 (17) B0 = 4 k B Th e where e is the electron charge, ε0 is the dielectric constant of vacuum, E 12 = 7.72 eV [31] is the first step ionization energy, k B is the Boltzmann constant, Th is the vapor temperature, and ar is the atomic radius. For the minimum point of the curve, the minimum breakdown voltage and corresponding pressure multiplied by the gap distance ( p × d) can be derived as B0 γ +1 (18) Vmin = exp(1) ln A0 γ exp(1) γ +1 . (19) ( p × d)min = ln A0 γ A0 = 0.224
According to the set of equations and parameters described above, the breakdown curve for copper vapor is shown in Fig. 19. The minimum point of the curve was p × d = 2.45 Pa · m corresponding to 44 V. On the left side of the point, the breakdown voltages rapidly increased with the decrease in p × d. On the contrary, the increased rate of breakdown voltages is relatively slow on the right side of the point. The ideal gas law was used to transform the unit of data in Fig. 17, and the gas temperature was assumed to be 2000 K. Therefore, the minimum point obtained from the PIC–MCC results was p × d = 3.61 Pa · m corresponding to 30 V. The minimum point from the PIC calculations is lower than that from Burm’s equations. This is partly because only the first step ionization was considered in Burm’s equations, which reduced the possibility of ionizations in vapor by
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neglecting processes such as excitations and elastic scatters. Moreover, the PIC model used the secondary emission yield as shown in Fig. 4 in contrast with the fixed value 0.1 used in Burm’s equations, and the lower fixed yield also gave a lower possibility to emit secondary electrons. Furthermore, the vapor can be removed or absorbed by the boundary in the PIC model, which reduced the vapor density compared with a fixed value p × d in Burm’s equation, and then the minimum point from the PIC calculations may shift right giving a larger p × d value. Although the minimum points were different, the two curves shared an identical shape. The discrepancy between the two curves was acceptable after the uncertain, and simplifications in the calculation were taken into account. Schade and Dullni [3] gave addition support to these calculations. They proposed a critical value of metal vapor 3 × 1021 /m3 at a gap distance of 10 mm. This value can be transformed to 0.83 Pa · m by assuming the vapor temperature to be 2000 K. However, they did not completely sure about the value and pointed out that the result was of the order of magnitude [13]. Thus, the minimum value can vary within the range of 0.17–4.15 Pa · m. Therefore, the results from PIC–MCC are still within a reasonable range.
period will become a postarc cathode. The postarc cathode will easily emit electrons if the surface is in a liquid state. Sandolache and Rowe [32] observed a deformation process of a liquid surface under the influence of a voltage and found that a tip would appear on the surface. Therefore, there are good reasons to believe that a liquid surface will reduce the interruption ability of a VCB significantly. Thus, the melting time should be kept as short as possible for a successful interruption.
C. Effect of Molten Anode Surface As shown in Figs. 5–9, the surface temperature may be above the melting point of the material. For the heat flux density of 1.2 × 109 and 1.5 × 109 W/m−2 , the surface temperature dropped to the melting point after current zero and stayed for several milliseconds. As stated before, this is because a solidification process released excess energy stored in the melting surface and impeded the whole process. The phase transition from a liquid state to a solid state retarded the decay of surface temperature and kept the surface in a liquid state for several milliseconds. Watanabe et al. [5] and Niwa et al. [6] measured the melting time lasting from 2 to 14 ms, which is coincided with our previous results [4]. A hot surface also is a source of evaporation after current zero. As mentioned in the previous section, although metal vapor dissipates from a gap at current zero, the evaporation from the surface will impede the dissipation until achieving an equilibrium between the surface temperature and the pressure in the gap. For a low temperature, the effect of evaporation is not significant. However, it is impossible to neglect the effect after interrupting a high current. In our simulation, the critical temperature is 1983 K corresponding to the minimum point of the breakdown curve shown in Fig. 17. If the temperature is higher than 1983 K, the breakdown voltages will remain in a relatively low level, which indicates a higher probability to lead a breakdown. The temperature that corresponds to the critical value proposed in [3] was 2000 K, which is another support of the simulation. Moreover, a liquid surface caused by arcing can also increase the probability of breakdowns after current zero even if the temperature drops below the critical value. The liquid surface is easily deformed by an electric field caused by a transient recovery voltage, enhancing the local electric field. Hence, the polarity of the voltage applied upon a gap is reversed after current zero, and an anode in the arcing
VI. C ONCLUSION The relationship between breakdown voltages and anode temperatures after current zero was investigated theoretically. A heat conduction model was established, and the PIC–MCC was used to obtain breakdown voltages in certain metal vapor densities corresponding to surface temperatures. 1) The Paschen curve for copper vapor was obtained by the PIC–MCC and verified by the theoretical model. 2) The minimum breakdown voltage, 30 V, was obtained at a density of 1.3 × 1022 /m3 with a gap of 10 mm. 3) The minimum breakdown voltage corresponded to a surface temperature of 1983 K. 4) To ensure a successful interruption, the anode surface temperature should not be higher than 1983 K at current zero, and the melting time should be kept as short as possible. ACKNOWLEDGMENT The authors would like to thank Prof. H. Wang of BrightTech Information Technology Company for the helpful discussions. R EFERENCES [1] E. Dullni, B. Gellert, and E. Schade, “Electrical and pyrometric measurements of the decay of the anode temperature after interruption of high-current vacuum arcs and comparison with computations,” IEEE Trans. Plasma Sci., vol. 17, no. 5, pp. 644–648, Oct. 1989. [2] H. Schellekens and M. B. Schulman, “Contact temperature and erosion in high-current diffuse vacuum arcs on axial magnetic field contacts,” IEEE Trans. Plasma Sci., vol. 29, no. 3, pp. 452–461, Jun. 2001. [3] E. Schade and E. Dullni, “Recovery of breakdown strength of a vacuum interrupter after extinction of high currents,” IEEE Trans. Dielectr. Electr. Insul., vol. 9, no. 2, pp. 207–215, Apr. 2002. [4] Z. X. Wang, H. Ma, G. Kong, Z. Liu, Y. Geng, and J. Wang, “Decay modes of anode surface temperature after current zero in vacuum arcs-part I: Experimental study,” IEEE Trans. Plasma Sci., vol. 42, no. 5, pp. 1464–1473, May 2014. [5] K. Watanabe et al., “The anode surface temperature of CuCr contacts at the limit of current interruption,” IEEE Trans. Plasma Sci., vol. 25, no. 4, pp. 637–641, Aug. 1997. [6] Y. Niwa et al., “The effect of contact material on temperature and melting of anode surface in the vacuum interrupter,” in Proc. 19th Int. Symp. Discharges Electr. Insul. Vacuum, Xi’an, China, Sep. 2000, pp. 524–527. [7] L. Wang, S. Jia, D. Yang, K. Liu, G. Su, and Z. Q. Shi, “Modelling and simulation of anode activity in high-current vacuum arc,” J. Phys. D, Appl. Phys., vol. 42, no. 14, p. 145203, Jul. 2009. [8] K. N. Ul’yanov, “A physical model of anode spot formation in a high-current vacuum arc,” High Temperature, vol. 41, no. 2, pp. 135–140, Mar. 2003. [9] A. Shashurin, I. I. Beilis, and R. L. Boxman, “Heat flux to an asymmetric anode in a hot refractory anode vacuum arc,” Plasma Sour. Sci. Technol., vol. 19, no. 1, p. 015002, Feb. 2010. [10] D. Loffhagen, D. Uhrlandt, and K. Hencken, “Monte Carlo simulation of the breakdown in copper vapour at low pressure,” in Proc. 24th Int. Symp. Discharges Elect. Insul. Vacuum, Braunschweig, German, Aug./Sep. 2010, pp. 7–10.
WANG et al.: DECAY MODES OF ANODE SURFACE TEMPERATURE AFTER CURRENT ZERO IN VACUUM ARCS
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Zhenxing Wang was born in 1983. He received the B.S. and Ph.D. degrees in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2006 and 2013, respectively. His current research interests include vacuum arc, dielectric recovery after current zero in vacuum interrupters, and operating mechanisms for vacuum circuit breakers.
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Yunbo Tian was born in Shaanxi, China, in 1989. He received the B.S. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2012, where he is currently pursuing the Ph.D. degree. He is involved in research on vacuum arcs and anode phenomena in vacuum interrupters.
Hui Ma was born in Shaanxi, China, in 1987. He received the B.S. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2010, where he is currently pursuing the Ph.D. degree. His current research interests include vacuum arcs and vacuum circuit breakers.
Yingsan Geng (M’98) was born in Henan, China, in 1963. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Xi’an Jiaotong University (XJTU), Xi’an, China, in 1984, 1987, and 1997, respectively. He is currently a Professor with the State Key Laboratory of Electrical Insulation and Power Equipment, Department of Electrical Engineering, XJTU. His current research interests include theory and application of low voltage circuit breaker and high voltage vacuum circuit breakers.
Zhiyuan Liu (M’01) was born in Shenyang, China, in 1971. He received the B.S. and M.S. degrees from the Shenyang University of Technology, Shenyang, in 1994 and 1997, respectively, and the Ph.D. degree from Xi’an Jiaotong University (XJTU), Xi’an, China, in 2001, all in electrical engineering. He was with the General Electric Company Research and Development Center, Shanghai, China, from 2001 to 2002. Since 2003, he has been with the State Key Laboratory of Electrical Insulation and Power Equipment, Department of Electrical Engineering, XJTU, where he is currently a Professor. He is involved in research and development of high voltage vacuum circuit breakers. He has authored over 100 technical papers. Dr. Liu is a member of Current Zero Club and the CIGRE Working Group WGA3.27–The Impact of the Application of Vacuum Switchgear at Transmission Voltages.
