Processing math: 100%
Advanced Search+
Yongpeng MO, Zongqian SHI, Shenli JIA. Study of post-arc residual plasma dissipation process of vacuum circuit breakers based on a 2D particle-in-cell model[J]. Plasma Science and Technology, 2022, 24(4): 045401. DOI: 10.1088/2058-6272/ac5235
Citation: Yongpeng MO, Zongqian SHI, Shenli JIA. Study of post-arc residual plasma dissipation process of vacuum circuit breakers based on a 2D particle-in-cell model[J]. Plasma Science and Technology, 2022, 24(4): 045401. DOI: 10.1088/2058-6272/ac5235

Study of post-arc residual plasma dissipation process of vacuum circuit breakers based on a 2D particle-in-cell model

More Information
  • Author Bio:

    Zongqian SHI, E-mail: zqshi@mail.xjtu.edu.cn

  • Received Date: July 16, 2021
  • Revised Date: February 04, 2022
  • Accepted Date: February 04, 2022
  • Available Online: December 15, 2023
  • Published Date: April 03, 2022
  • In order to get an insight into residual plasma radial motion during the post-arc stage, a two-dimensional (2D) cylindrical particle-in-cell (PIC) model is developed. Firstly, influences of a virtual boundary condition on the residual plasma motion are studied. For purpose of validating this 2D cylindrical particle-in-cell model, a comparison between one-dimensional particle-in-cell model is also presented in this paper. Then a study about the influences of the rising rate of transient recovery voltage on the residual plasma radial motion is presented on the basis of the 2D PIC model.

  • The post-arc dielectric recovery process has decisive influences on a successful current interruption in a vacuum circuit breaker. Therefore, the post-arc dielectric recovery process causes more and more concern. The post-arc dielectric recovery process begins at the moment when the current flowing a vacuum circuit breaker decreases to zero and it will last till the moment when the interelectrode region recoveries its dielectric strength.

    During the arcing stage, arc plasma, metal vapor and droplets are generated by the erosion of vacuum arc. When the interruption current decreases to zero, the residual plasma, metal vapor and droplets still exist between the contacts [1]. After current zero, a negative rapid increasing transient recovery voltage generated by the external circuit will be applied to the contacts. Then the ions and electrons are separated from each other. The electrons are driven to the post-arc anode quickly while the heavier ions stay behind. Then an ion sheath, the so-called post-arc sheath, forms and then it will develop toward the post-arc anode. During the above residual plasma dissipation process, the post-arc current forms when the electrons and ions are absorbed. The residual plasma dissipation process, the beginning of post-arc dielectric recovery process, is very important for a successful current interruption process in a vacuum circuit breaker. Therefore, it is of great significance to study this process.

    The continuous transition model (CTM) which is developed in the 1970s, has been being widely used in post-arc sheath simulations of vacuum circuit breakers [2]. However, CTM is not very self-consistent in dealing with the bounded plasma because the derivation of CTM is based on a semi-infinite plasma. In recent years, more and more refined models are developed to study the residual plasma dissipation process. Sarrailh et al [35] studied the effects of transient recovery voltage and residual plasma density on the post-arc sheath expansion with a one-dimensional hybrid Maxwell-Boltzmann (hybrid-MB) model. In addition, the effects of contact evaporation and secondary emissions were also studied by a one-dimensional hybrid-MB model combining with the Monte-Carlo collision (MCC) method.

    The one-dimensional hybrid-MB model takes less computing time but not so self-consistent as the full particle-in-cell (PIC) model because both ions and electrons in a full PIC model are treated as macro particles. Some researchers then adopted the PIC models to study the microphysical mechanism of post-arc dielectric recovery process [69]. It should be noted that the studies with 1D PIC models only can focus on the residual plasma axial motion.

    To gain further understanding of the residual plasma dissipation process, Sarrailh et al [10] developed a two-dimensional hybrid-MB model to study the influence of vacuum interrupter shield polarity on the post-arc residual plasma dissipation process. Takahashi et al [11] investigated the influence of the residual plasma on the post-arc breakdown process in a 168 kV/25 kA vacuum interrupter with a 2D PIC-MCC model. Wang [12] studied the influence of metal vapor on the post-arc breakdown process after current zero by a 2D PIC-MCC model. However, a 2D PIC-MCC model based on the full size of commercial vacuum interrupter is time-consuming. In order to reduce the computing time, an open wall boundary condition (zero axial electric field and specular reflection of ions) was adopted in the 2D hybrid-MB model [10]. Due to this, the simulation domain of the 2D hybrid-MB model can be reduced to a much smaller size than the commercial vacuum interrupters. It also should be noted that some charged particles would fly to the virtual boundary according to the simulations in [10]. Otherwise, these charged particles might also affect the potential distribution near the virtual boundary. Obviously, a proper boundary condition for PIC simulation will benefit the further study of residual plasma dissipation process. However, there is still few studies about the effects of boundary condition on the residual plasma dissipation process. What is more, the residual plasma radial motion and its effects on the residual plasma dissipation process still remain unclear.

    In order to get a further understanding of the effects of boundary condition and the residual plasma radial motion on the residual plasma dissipation process, a 2D PIC model with cylindrical coordinate system is developed in this work due to the axisymmetric characteristic of vacuum interrupter based on an extension of our previous work [13]. Then an evaluation of this 2D PIC model is presented. Besides, because the transient recovery voltage is important to the residual plasma dissipation process, the influences of rising rate of transient recovery voltage on the residual plasma radial motion are also simulated and discussed.

    In this work, a 2D PIC model with the cylindrical coordinate r-z system, which does not take the magnetic field into consideration, is developed by the software VSim. Figure 1 presents a sketch map of cells with indices j, k. The ∆r and ∆z in directions r and z are uniform, respectively. With Gauss' law, the charges for the volume shown by dash lines centered on (rj, zk) can be obtained

    Qj,k=2πrj+1/2zEr,j+1/2,k-2πrj-1/2zEr,j-1/2,k+π(r2j+1/2-r2j-1/2)(Ez,j,k+1/2-Ez,j,k-1/2) (1)
    Figure  1.  The sketch map of cells with indices j, k.

    Where Qj,k is the charge, Er,j,k is the electric filed in r direction, and Ez,j,k is the electric filed in r direction.

    At the origin (j=0),

    Q0,k=2πr1/2zEr,1/2,k+πr21/2(Ez,0,k+1/2-Ez,0,k-1/2) (2)

    With ρj,k=Qj,k/Vj,k and Vj,k=π(r2)jz taking into consideration, the charge density is

    ρj,k=2(r2)j(rj+1/2Er,j+1/2,k-rj-1/2Er,j-1/2,k)+Ez,j,k+1/2-Ez,j,k-1/2z (3)

    The Er, j, k and Ez, j, k can be written as

    Er,j+1/2,k=-ϕj+1,k-ϕj,krj+1/2 (4)
    Ez,j,k+1/2=-ϕj,k+1-ϕj,kz (5)

    Then a five points finite-difference form of Poisson's equation can be obtained

    -ρj,k=2rj+1/2(r2)jrj+1/2(ϕj+1,k-ϕj,k)-2rj-1/2(r2)jrj-1/2×(ϕj,k-ϕj-1,k)+1z2(ϕj,k+1-2ϕj,k+ϕj,k-1)(j>0) (6)

    After that, the instantaneous position of electrons and ions can be solved by Newton's law.

    d2rdt2=qEr (7)
    d2zdt2=qEz (8)

    Where (r, z) are the charged particle position, qe and me are the charge and mass respectively for electrons, and qi and mi are those for positive ions, and (Er, Ez) are the electric field strengths at (r, z). More theory and numerical approach of PIC simulation can be founded in [14].

    Figure 2 presents the structure of a vacuum interrupter and the schematic diagram of the 2D PIC model. The computing time increases rapidly with the increase of computing area. In an ideal simulation, the 2D PIC model should be based on a full-size vacuum interrupter as shown in the upper picture of figure 2. However, this will cost a lot of computing time, which is not conducive to the further study of residual plasma dissipation process.

    Figure  2.  Structure of the vacuum interrupter (upper) and sketch diagram of the 2D PIC model (bottom).

    The residual plasma dissipation process is the major concern during the post-arc dielectric recovery process. Decreasing the computing area with a virtual boundary instead of a physical boundary is important for further study. Therefore, the 2D PIC model (bottom) only focuses on the shaded part of a vacuum interrupter (upper) as shown in figure 2. The radius of two contacts Lcr is 0.01 m. The shield is assumed to be 0.02 m away from the contact axis (R direction). The contact plate thickness Lpt is 0.0025 m. The contact gap between the post-arc cathode and post-arc anode is 0.005 m and the conductive rod radius Lrr is 0.006 m. Lc is the length between the right boundary and the left boundary (Z direction).

    Because the vacuum arc mainly burns in the interelectrode region, we assume that the residual plasma mainly exists in the interelectrode region at current zero. The interelectrode region represents the region where R < 0.01 m and -0.0025 m < Z < -0.0025 m. The blue particles represent ions and the red particles represent electrons. According to some experimental results in [15, 16], the electron temperature ranges from 0.8 eV to 4.6 eV. Therefore, the residual plasma temperature is assumed to be 2 eV. The copper ions are presumed monovalent. The qe, me, qi and mi are -1.602176487×10−19 C, 9.10938215×10−31 kg, 1.602176487×10−19 C, and 1.0667×10−25 kg, respectively. Some impurity gases precipitated from the contacts are not taken into consideration. According to experimental studies [1720], the residual plasma density ranges from 3×1017 m−3 to 4×1018 m−3. Düning and Lindmayer [21] studied residual plasma at current zero by placing a pair of contact near the main contact. They had obtained the residual plasma density 0.75×1017 m−3 for 1.5 kA. After referring to the computation time and above relevant research results, a residual plasma density 1×1017 m−3 is chosen in the simulations. The grid spacing ∆z and ∆r are 2×10−5 m, which are smaller than the Debye length λD. The time step ∆t is 1×10−11 s. The rising rate of transient recovery voltage is -2 kV μs−1 in section 3. The post-arc anode is assumed to be zero potential. Besides, the shield potential is also assumed to be zero potential (connecting to post-arc anode). All simulations are converged.

    Because the electrons move to post-arc anode quickly and the heavier ions are left behind, an ion sheath forms in front of post-arc cathode and develops to post-arc anode as shown in figure 2. For reducing computation time, the collisions between charged particles, secondary electrons and emissions from both contact and shield, are still not taken into consideration in this model. When the electrons or ions arrive at the contacts, the conductive rod or the shield, they are removed from the 2D PIC model.

    Two virtual boundaries, the left boundary and right boundary as shown in the bottom picture of figure 2, are needed for the 2D PIC model. In a vacuum interrupter, the potential distribution is closely related to the contact plate structure, the residual plasma and the metal shield. The metal shield is parallel to conductive rod of contacts except the parts near two ends. For a 12 kV/1.6 kA commercial vacuum interrupter, the distance between two ends is about 0.2 m. The length of commercial vacuum interrupters will be longer for a higher voltage level.

    When two boundaries are relatively far away from the contact plates, the residual plasma in the interelectrode region can hardly fly to the left boundary or right boundary. Under such a condition, the residual plasma motion will show small effects on the potential distribution near the left boundary and the right boundary. Besides, when two boundaries are relatively far away from contact plates and the ends of metal shield, the structure of contact plates and the irregular ends of metal shield would show relatively small influence on these two boundaries. Therefore, the potential near two boundaries between the conductive rod and metal shield would not change along the direction Z. Because few charged particles will fly to two boundaries, we ignore the reflections and secondary electrons on two boundaries. All charged particles which fly to two boundaries are assumed to be absorbed and removed from simulations.

    When Lc is 0.100 m, two boundaries are about 0.045 m away from the contact plates and at least 0.04 m away from two ends of the metal shield. The influence of contact plate structure and irregular ends of metal shield on the potential distribution near the left boundary and right boundary is assumed to be small under this condition (about 20 times of the contact gap). According to the above analysis, a boundary condition Ez = 0, which means that the electric field strength does not change in direction R, is adopted to the left boundary. Referring to the boundary conditions in [10], the right boundary potential is assumed to be zero, which is equal to post-arc anode potential. The other parameters are introduced in section 2.

    Figure 3 shows the distribution of equipotential lines with Lc=0.100 m. The irregular distribution of equipotential lines in the interelectrode region at T=0.0 μs is mainly caused by the random distribution of limited macro particles. The equipotential lines with different potentials are irregular near the contact plate at T=0.3 μs. From figure 3 we can see that the equipotential lines between Z=-0.01 m and -0.02 m are irregular and gradually parallel to the contact axis. Figure 3 also shows that the equipotential lines on the left of Z=-0.02 m are almost parallel to the contact axis. From figure 3 we can see that Lc can be further reduced. Then Lc=0.05 m has been simulated. Figure 4 shows the distribution of equipotential lines with Lc=0.05 m. From the figure 4 we can see that the equipotential lines on the left of Z=-0.017 m are also parallel to the contact axis after T=0.3 μs. The equipotential lines between Z=-0.01 m and Z=-0.017 m are gradually parallel to the contact axis. From figures 3 and 4 we can see that the equipotential lines V=180 V with Lc=0.050 m and 0.100 m are both placed at R=0.015 m near the left boundary at T=0.9 μs. This indicates that the variations of the equipotential lines with Lc=0.050 m and 0.100 m are similar.

    Figure  3.  The distribution of the equipotential lines with Lc=0.100 m.
    Figure  4.  The distribution of the equipotential lines Lc=0.050 m.

    The simulation results in figures 3 and 4 indicate that Lc seems to be able to be further reduced because the irregularity of equipotential lines mainly occurs in the region between Z=-0.01 m and Z=-0.02 m. Then we carried out another simulation with Lc=0.025 m. Figure 5 presents the distribution of equipotential lines with Lc=0.025 m. From figures 3 and 4 we can see that not all equipotential lines near Z=-0.0125 m are parallel to the contact axis, especially when R > 0.01 m. Figure 5 shows that all equipotential lines near the left boundary are forced parallel to the contact axis under the effect of the left boundary condition. The equipotential line V=180 V with Lc=0.025 m is placed at about R=0.016 m near the left boundary at T=0.9 μs. This indicates that there is a small difference between the distribution of equipotential lines with Lc=0.025 m and Lc=0.050 m, 0.100 m. Therefore, the left boundary condition shows certain unexpected effects on the residual plasma motion under these three conditions. The left boundary condition might further distort the distribution of equipotential lines and charged particle motion if Lc is further reduced, which may cause more unexpected calculation errors.

    Figure  5.  The distribution of the equipotential lines with Lc=0.025 m.

    For the purpose of getting a further study about the influence of left boundary condition, figure 6 presents the residual plasma motion when Lc is 0.025 m. The red ones are electrons and the blue ones are ions. From figure 6 we can see that the macro particles near R=0 m look sparse. A brief explanation is presented below. The cells of Z and R directions are uniform in the 2D PIC model. The macro particles in a cell represent the total electrons or ions in a ring whose volume increases with radius in the cylindrical coordinate system. This means that the total number of macro particles in a cell increases with the radius. Therefore, the residual plasma between the contact gap of the 2D PIC model is uniform despite macro particles near R=0 look uneven. However, it should be noted that there are only a small number of macro particles placed near the R=0 region, which may cause noise when we estimate the plasma density, potential distribution or post-arc sheath thickness. Even so, a possible strategy to deal with this issue is increasing the number of macro particles at the small radii.

    Figure  6.  Residual plasma motion at different moments when Lc=0.025 m.

    Figure 6 also shows that both ions and electrons will fly out of the interelectrode region during the residual plasma dissipation process. The ions mainly move toward the post-arc cathode and the electrons move toward the post-arc anode under the effect of transient recovery voltage. As a result, most ions and electrons in the interelectrode region are absorbed by the contact surface. Figure 6 also shows that the electrons or ions outside of the interelectrode region can fly to the lateral surface of contacts, the conductive rod or the metal shield and then they are absorbed. From figure 6 we can also see that some electrons could fly to the right boundary after T=0.6 μs. Figure 6 also indicates that the ions cannot fly to the left boundary under the effect of transient recovery voltage. The ions can fly to the position Z=-0.011 m at T=0.8 μs and then be absorbed by the conductive rod. However, some electrons can fly to the right boundary after T=0.6 μs.

    What is more, the number of ions and electrons flying outside the interelectrode region is also counted for evaluating the influence of the boundary condition quantitatively. The βe means the proportion of the number of electrons outside the interelectrode region to the number of electrons in the whole simulation domain. The βi means the proportion of the number of ions outside interelectrode region to the number of ions in the whole simulation domain. Figure 7 presents the proportion βi with Lc=0.025 m, 0.050 m and 0.100 m at different moments. Figure 8 presents the proportion βe with Lc=0.025 m, 0.050 m and 0.100 m at different moments. It can be seen from figure 7 that the boundary condition shows little effect on the ion radial motion when Lc is reduced to 0.025 m. It can also be seen from figure 8 that the boundary condition still shows little influence on the electron radial motion before 0.6 μs. At T=0.9 μs, more electrons would fly outside the gap with Lc=0.025 m compared with Lc=0.050 m and 0.100 m. The βe with Lc=0.025 m, 0.05 m and 0.1 m are about 10.38%, 9.75% and 9.62% at 0.9 μs, respectively. After the calculations follow [βe(Lc=0.025)-βe(Lc = 0.1)]/βe(Lc=0.1), the βe with Lc=0.025 m is about 6.5% higher than the βe with Lc = 0.050 m at 0.9 μs. The βe with Lc=0.025 m is about 7.9 % higher than the βe with Lc=0.1 m at 0.9 μs. On the whole, we propose that the boundary condition shows relatively limited effects on the plasma radial motion when Lc is reduced to 0.025 m. It also should be noted that the boundary condition shows certain effects on the radial motion of electron at the later stage when Lc is reduced to 0.025 m.

    Figure  7.  The proportion βi with Lc=0.025 m, 0.050 m and 0.100 m at different moments.
    Figure  8.  The proportion βe with Lc=0.025 m, 0.050 m and 0.100 m at different moments.

    The residual plasma motion in figure 6 presents an intuitive impression of the influence of left boundary condition. As shown in figure 6, an ion sheath forms in front of the post-arc cathode. Therefore, the development of post-arc sheath can also offer a quantitative analysis of the influence of the boundary condition. The electrons will reverse to post-arc anode quickly under the effects of transient recovery voltage and there are few electrons in the post-arc sheath. The estimation of post-arc sheath thickness is obtained from the electron positions which are next to post-arc sheath. Figure 9 presents the development of post-arc sheath thickness with Lc=0.025 m, 0.050 m and 0.100 m at the positions of R=0.005 m and R=0.008 m. From figure 9 we can see that the development of sheath thicknesses with three different Lc is almost the same at R=0.005 m or R=0.008 m, which indicates that the left boundary condition still shows relatively small effects on the residual plasma dissipation even when Lc is reduced to 0.025 m.

    Figure  9.  The development of sheath thicknesses with different Lc at the positions of R=0.005 m and 0.008 m and a comparison with 1D PIC model.

    Although the influence of plasma radial motion could not be taken into consideration in a one-dimensional PIC model. However, it is useful for evaluating the effects of residual plasma radial motion. It should be noted that the same 1D PIC model had been evaluated in our previous work [7, 22]. Evaluation between 1D results and 2D results is also presented in figure 9. The contact gap, residual plasma density, residual plasma temperature and rising rate of transient recovery voltage of the 1D PIC model are the same as that chosen by the above 2D PIC model. Figure 9 shows that the development of post-arc sheath thickness obtained from the 2D PIC model is similar to that obtained from the 1D PIC model before T=0.6 μs, which means that the 2D PIC model developed in this study coincides well with the evaluated 1D PIC model.

    What is more, the electron density has also been calculated for further analysis. Figure 10 presents a comparison of electrons density at 0.6 μs (R=0.005 m) with Lc=0.025 m, 0.050 m and 0.100 m. It should be noted that noise exists in figure 10 due to limited macro particles adopted in the simulations. Then the average density between 0 m < Z < -0.001 m is calculated. The averaged densities with Lc=0.025 m, 0.050 m and 0.100 m are about 7.82×1016 m−3, 7.27×1016 m−3 and 8.32×1016 m−3, respectively. Therefore, the variation of electron densities with Lc=0.025 m, 0.050 m and 0.100 m can be assumed to be similar.

    Figure  10.  A comparison of electron density at 0.6 μs.

    According to the simulation results from figures 6 to 10, we conclude that the boundary condition (Ez=0 with particles absorption) is reasonable for this 2D PIC simulation even when Lc is reduced to 0.025 m. In the above simulations, the total number of particles in our simulation is about 78115. Besides, the simulation time for 0.025 m, 0.050 m and 0.100 m are 1 μs in our simulations. The total computation time in our simulations for 0.025 m, 0.050 m and 0.100 m are 97.3 h, 230.5 h and 416.25 h with 4 cores@3.6 GHz, respectively. Obviously, the boundary condition will benefit the further simulation of residual plasma dissipation process.

    From figure 9 we can see that the post-arc sheath at R=0.008 m grows faster than that at R=0.005 m after about T=0.7 μs. Besides, the post-arc sheath obtained from the 2D PIC model also develops faster than that obtained from the 1D PIC model after T=0.6 μs.

    From figure 6 we can see that only a small amount of residual plasma flies outside of the interelectrode region before T=0.3 μs. According to figures 8 and 9, the proportion βi and βe are about 6% and 5%, respectively. The amount of residual plasma flying outside of the interelectrode region continues to increase and more residual plasma leaves the interelectrode region at T=0.6 μs. The proportion βi and βe increase to 14% and 8%, respectively. Because the residual plasma moves outside the interelectrode gradually, its influence on the development of post-arc sheath increases at the later stage. After that the residual plasma density in the interelectrode region of the 2D PIC model is lower than that of the 1D model due to the residual plasma radial motion, the post-arc sheath in the 2D PIC model develops faster than that in the 1D PIC model in the later stage.

    As shown in figure 6, the residual plasma in the interelectrode region is assumed to be uniform and there is no residual plasma outside of the interelectrode region at current zero. Therefore, the residual plasma near R=0.010 m can move outside of the interelectrode region due to their thermal motion. This causes the decrease of residual plasma density near R=0.010 m. When the residual plasma density near R=0.010 m decreases, the plasma at R < 0.010 m continues to move to the region near R=0.010 m. As a result, the residual plasma density in the interelectrode region decreases along the R direction after a few microseconds. Therefore, the post-arc sheath near the edge of contact develops faster than that near the contact axis. This might be the reason why the post-arc sheath at R=0.008 m develops faster than that at R=0.005 m after a few microseconds.

    The transient recovery voltage, which has an important effect on the electric field distribution, is an important factor of the residual plasma dissipation process in a vacuum interrupter. The rising rate of transient recovery voltage can be higher than 5 kV μs−1 in a vacuum circuit breaker [23]. In other studies, the rising rates of transient recovery voltage ranged from 0.1 kV μs−1 to 5 kV μs−1 [5]. Therefore, four different rising rates of transient recovery voltage 0.5 kV μs−1, 1.0 kV μs−1, 2.0 kV μs−1 and 3.0 kV μs−1 are adopted in the simulations. According to the above discussions in sections 3 and 4, Lc =0.025 m is adopted in the simulations. The other parameters are the same as that introduced in section 2. It should be noted that the low number of macro particles near R=0 region as shown in the above section may cause some inaccuracies when we simulated residual plasma radial behavior.

    Figures 11 and 12 present the number of ions and electrons outside the interelectrode region with different rising rates of transient recovery voltage before 0.9 μs, respectively. The residual plasma is expelled from the interelectrode region quickly with a high rising rate of transient recovery voltage [5]. It should be noted that almost all electrons are expelled from the interelectrode region and only a small number of ions are left at 0.9 μs with 3.0 kV μs−1 according to the simulation results as shown in figure 13. Figure 13 also indicated that there are no electrons in the whole simulation domain at 0.9 μs with 3.0 kV μs−1. From figures 11 and 12 we can see that the number of ions and electrons outside interelectrode region decreases when the rising rate of transient recovery voltage increases. From figures 11 and 12 we can also see that the number of electrons and ions outside the interelectrode region firstly increases and then decreases after several microseconds.

    Figure  11.  The number of ions outside interelectrode region with different rising rates of transient recovery voltage at different moments.
    Figure  12.  The number of electrons outside interelectrode region with different rising rates of transient recovery voltage at different moments.
    Figure  13.  The residual plasma motion with 3.0 kV μs−1 at 0.9 μs. The left picture represents ions (blue dots) and the right picture represents electrons (red dots).

    To get a further understanding of the residual plasma radial motion, figures 14 and 15 present the proportion βi and βe with different rising rates of transient recovery voltage, respectively. It should be noted that no electrons exist in the simulation domain at 0.9 μs with 3.0 kV μs−1 according to figure 13. Therefore, the βe at 0.9 μs with 3.0 kV μs−1 is not presented in figure 15. From figure 14 we can see that the proportions βi with different rising rates of transient recovery voltage are similar before 0.6 μs. The βi with 3.0 kV μs−1 is about 45.18% higher than βi with 0.5 kV μs−1 at 0.9 μs [βi (3.0 kV μs−1)-βi (0.5 kV μs−1)]/βi (0.5 kV μs−1). However, figure 15 shows that the proportion βe is lower with a high rising rate of transient recovery voltage. The βe with 0.5 kV μs−1 is about 138.41% higher than βe with 3.0 kV μs−1 at 0.6 μs.

    Figure  14.  The proportion βi with different rising rates of transient recovery voltage.
    Figure  15.  The proportion βe with different rising rates of transient recovery voltage.

    During the post-arc sheath expansion process, the residual plasma in the interelectrode region will fly outside due to their thermal motion. Firstly, the number of ions or electrons flying outside the gap increases over time. After several microseconds, the residual plasma between the gap decreases due to the absorption under the effect of transient recovery voltage and less charged particles will fly outside the gap. Meanwhile, the ions and electrons outside the gap will also be absorbed. Therefore, the number of ions and electrons outside of the interelectrode region decreases after several microseconds.

    Because the electrons and ions will fly to the contact surface more quickly under a high rising rate of transient recovery voltage [5], the number of ions and electrons flying outside of the interelectrode region decreases. Meanwhile, more electrons and ions which fly outside the gap are also absorbed by post-arc cathode and post-arc anode easily. Therefore, the number of ions and electrons outside the gap decreases when the rising rate of transient recovery voltage increases.

    An evaluation of virtual boundary condition on 2D PIC simulation is presented and the influence of transient recovery voltage on plasma radial motion is investigated in this work. Following conclusions can be drawn from this study:

    The virtual boundary shows small effects on both the potential distribution and the residual plasma motion even when the simulation domain decreases to 0.025 m in this study. The virtual boundary condition (Ez=0 with particles absorption) seems to be reasonable for these 2D PIC simulations. Besides, the virtual boundary condition also makes it possible for the simulation domain to decrease to a certain extent, which will be helpful for future studies.

    Part of residual plasma would fly outside of the interelectrode region due to the plasma radial motion after a certain time during the residual plasma dissipation process. The residual plasma density in the interelectrode region decreases, which has an obvious effect on the development of post-arc sheath expansion process in this region.

    The transient recovery voltage is important for post-arc residual plasma dissipation process, which also shows noticeable effect on the residual plasma radial motion. When the rising rate of transient recovery voltage increases, less charged particles will fly outside of interelectrode region and more charged particles will be absorbed by the contact surface.

    This work was supported in part by National Natural Science Foundation of China (Nos. 51807148 and U1866202), and in part by China Postdoctoral Science Foundation (No. 2019M653628).

  • [1]
    Schade E 2005 IEEE Trans. Plasma Sci. 33 1564 doi: 10.1109/TPS.2005.856530
    [2]
    Andrews J G and Varey R H 1971 Phys. Fluids 14 339 doi: 10.1063/1.1693433
    [3]
    Sarrailh P et al 2010 Plasma Sources Sci. Technol. 19 065020 doi: 10.1088/0963-0252/19/6/065020
    [4]
    Sarrailh P et al 2009 J. Appl. Phys. 106 053305 doi: 10.1063/1.3204969
    [5]
    Sarrailh P et al 2008 J. Phys. D: Appl. Phys. 41 015203 doi: 10.1088/0022-3727/41/1/015203
    [6]
    Mo Y P et al 2016 Phys. Plasmas 23 053506 doi: 10.1063/1.4948422
    [7]
    Mo Y P et al 2015 Phys. Plasmas 22 023511 doi: 10.1063/1.4913677
    [8]
    Wang Z X, Geng Y S and Liu Z Y 2011 Collision effects on sheath development after interrupting a vacuum arc Proc. of the 1st Int. Conf. on Electric Power Equipment—Switching Technology (Xi'an(IEEE) 2011) 27
    [9]
    Wang Z X et al 2016 J. Appl. Phys. 120 083301 doi: 10.1063/1.4961420
    [10]
    Sarrailh P et al 2008 IEEE Trans. Plasma Sci. 36 1046 doi: 10.1109/TPS.2008.924502
    [11]
    Takahashi S et al 2007 IEEE Trans. Plasma Sci. 35 912 doi: 10.1109/TPS.2007.901970
    [12]
    Wang Z X et al 2013 Simulation of breakdown in Cu-Cr metal vapor after vacuum arc extinctions Proc. of the 2013 2nd Int. Conf. on Electric Power Equipment—Switching Technology (Matsue(IEEE) 2013) 1
    [13]
    Mo Y et al 2018 Study of post-arc sheath expansion process with a two-dimensional particle-in-cell model Proc. of the 2018 28th Int. Symp. on Discharges and Electrical Insulation in Vacuum (Greifswald(IEEE) 2018) 239
    [14]
    Birdsall C K and Langdon A B 1985 Plasma Physics Via Computer Simulation (New York: McGraw-Hill)
    [15]
    Klajn A 2005 IEEE Trans. Plasma Sci. 33 1611 doi: 10.1109/TPS.2005.856485
    [16]
    Schneider A V et al 2011 IEEE Trans. Plasma Sci. 39 1349 doi: 10.1109/TPS.2011.2134849
    [17]
    Farrall G A 1968 Proc. IEEE 56 2137 doi: 10.1109/PROC.1968.6826
    [18]
    Dullni E, Schade E and Gellert B 1987 IEEE Trans. Plasma Sci. 15 538 doi: 10.1109/TPS.1987.4316750
    [19]
    Lins G 1989 IEEE Trans. Plasma Sci. 17 672 doi: 10.1109/27.41179
    [20]
    Lins G 1991 IEEE Trans. Plasma Sci. 19 718 doi: 10.1109/27.108404
    [21]
    Duning G and Lindmayer M 1999 IEEE Trans. Plasma Sci. 27 923 doi: 10.1109/27.782261
    [22]
    Jia S L et al 2017 Phys. Plasmas 24 103511 doi: 10.1063/1.5004180
    [23]
    Smeets R P P and Van der Linden W A 2003 IEEE Trans. Plasma Sci. 31 852 doi: 10.1109/TPS.2003.818438
  • Related Articles

    [1]Guanghui ZHU, Qing LI, Xuan SUN, Jianyuan XIAO, Jiangshan ZHENG, Hang LI. Particle simulations on propagation and resonance of lower hybrid wave launched by phased array antenna in linear devices[J]. Plasma Science and Technology, 2022, 24(7): 075102. DOI: 10.1088/2058-6272/ac5f80
    [2]A A ABID, Quanming LU (陆全明), Huayue CHEN (陈华岳), Yangguang KE (柯阳光), S ALI, Shui WANG (王水). Effects of electron trapping on nonlinear electron-acoustic waves excited by an electron beam via particle-in-cell simulations[J]. Plasma Science and Technology, 2019, 21(5): 55301-055301. DOI: 10.1088/2058-6272/ab033f
    [3]Hong LI (李鸿), Xingyu LIU (刘星宇), Zhiyong GAO (高志勇), Yongjie DING (丁永杰), Liqiu WEI (魏立秋), Daren YU (于达仁), Xiaogang WANG (王晓钢). Particle-in-cell simulation for effect of anode temperature on discharge characteristics of a Hall effect thruster[J]. Plasma Science and Technology, 2018, 20(12): 125504. DOI: 10.1088/2058-6272/aaddf2
    [4]Weili FAN (范伟丽), Zhengming SHENG (盛政明), Fucheng LIU (刘富成). Particle-in-cell/Monte Carlo simulation of filamentary barrier discharges[J]. Plasma Science and Technology, 2017, 19(11): 115401. DOI: 10.1088/2058-6272/aa808c
    [5]Yi CHEN (陈毅), Fei YANG (杨飞), Hao SUN (孙昊), Yi WU (吴翊), Chunping NIU (纽春萍), Mingzhe RONG (荣命哲). Influence of the axial magnetic field on sheath development after current zero in a vacuum circuit breaker[J]. Plasma Science and Technology, 2017, 19(6): 64003-064003. DOI: 10.1088/2058-6272/aa65c8
    [6]ZHANG Ya (张雅), LI Lian (李莲), JIANG Wei (姜巍), YI Lin (易林). Numerical Approach of Interactions of Proton Beams and Dense Plasmas with Quantum-Hydrodynamic/Particle-in-Cell Model[J]. Plasma Science and Technology, 2016, 18(7): 720-726. DOI: 10.1088/1009-0630/18/7/04
    [7]GUO Jun (郭俊), YANG Qinglei (杨清雷), ZHU Guoquan (朱国全), and LI Bo (李波). A Particle-in-Cell Simulation of Double Layers and Ion-Acoustic Waves[J]. Plasma Science and Technology, 2013, 15(11): 1088-1092. DOI: 10.1088/1009-0630/15/11/02
    [8]WU Mingyu (吴明雨), LU Quanming (陆全明), ZHU Jie (朱洁), WANG Peiran (王沛然), WANG Shui (王水). Electromagnetic Particle-in-Cell Simulations of Electron Holes Formed During the Electron Two-Stream Instability[J]. Plasma Science and Technology, 2013, 15(1): 17-24. DOI: 10.1088/1009-0630/15/1/04
    [9]Vahid Abbasi, Ahmad Gholami, Kaveh Niayesh. Three-dimensional Simulation of Plasma Deformation during Contact Opening in a Circuit Breaker, including Analyses of Kink and Sausage Instabilities[J]. Plasma Science and Technology, 2012, 14(11): 996-1001. DOI: 10.1088/1009-0630/14/11/07
    [10]WU Junhui, WANG Xiaohua, MA Zhiying, RONG Mingzhe, YAN Jing. Numerical Simulation of Gas Flow during Arcing Process for 252kV Puffer Circuit Breakers[J]. Plasma Science and Technology, 2011, 13(6): 730-734.
  • Cited by

    Periodical cited type(7)

    1. Wang, Z., Li, R., Cao, B. et al. 3D hybrid simulation of postarc sheath expansion with nonuniform residual plasmas. Journal of Vacuum Science and Technology A: Vacuum, Surfaces and Films, 2024, 42(5): 053008. DOI:10.1116/6.0003859
    2. Cheng, X., Chen, H., Ge, G. et al. Influence of the vapor shield potential control on the post-arc sheath development in vacuum interrupters. Vacuum, 2024. DOI:10.1016/j.vacuum.2024.112957
    3. Zhang, W., Cheng, X., Ge, G. et al. PIC Numerical Calculation of the Effect of Self-Voltage Sharing Configuration on the Post Arc Particle Transport Characteristics in High-Voltage Vacuum Interrupters. 2024. DOI:10.1109/ICEPE-ST61894.2024.10792559
    4. Wang, L., Chen, Z., Wang, D. et al. Two-Dimensional Particle-in-Cell/Monte Carlo Collisional Simulation of the Post-Arc Breakdown in Vacuum Circuit Breakers. IEEE Transactions on Plasma Science, 2024, 52(8): 3228-3236. DOI:10.1109/TPS.2024.3449271
    5. Chen, H., Cheng, X., Ge, G. et al. Influence of Main Shield Voltage Distribution Configuration on the Post-arc Sheath Development of Vacuum Interrupters. Lecture Notes in Electrical Engineering, 2024. DOI:10.1007/978-981-99-7413-9_6
    6. Tang, Z., Xu, Z. Research on Three-Dimensional Numerical Simulation Method of Low Voltage Cable Arc. Lecture Notes in Electrical Engineering, 2023. DOI:10.1007/978-981-99-3404-1_76
    7. Shi, Q., Yang, P., Ye, J. et al. Particle Simulation of Near-Cathode Sheath in Vacuum Arc. Lecture Notes in Electrical Engineering, 2023. DOI:10.1007/978-981-99-0357-3_85

    Other cited types(0)

Catalog

    Figures(15)

    Article views (102) PDF downloads (136) Cited by(7)

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return