3D MHD vacuum arc model in VI design

Abstract

Contact geometry evaluation using a full 3D Magneto-Hydrodynamic vacuum arc model is presented. The 3D model uses a two-step approach. The time dependent magnetic field is calculated by Flux3D® using the features of the arc control in axial magnetic field (AMF) type vacuum interrupters (VIs). The arc is modeled by a single fluid description of the arc plasma in Fluent®. Current flow is addressed by Ohms law. The model is applied to 3 contact geometries used in commercial AMF VI’s. The general results of the model like ion density, flow fields and energy flow are represented. It is demonstrated that the external transverse magnetic field created by neighboring phases in a circuit breaker causes off-axis deflection of the heat flow to anode. The deflection in the anti-Amperian direction is attributed to the boundary condition of the current flow at the cathode side.

Keywords

3-D magnetohydrodynamic (MHD) model magnetic field (MF)numerical simulation vacuum arcs

1. Introduction

A vacuum circuit breaker, Fig. 1a, is an electromechanical protection device, which function is to interrupt an over current or a short-circuit current in electrical distribution. The vacuum interrupter, Fig. 1b, is an essential component of the circuit breaker. Vacuum interrupters need an arc control to interrupt high short circuit currents [1]. Arc control is based on additional applied magnetic fields which “cool” the contact surface. Transverse Magnetic Field (TMF) arc control pushes the arc forward across the contact surface, hence avoiding local overheating of the contact surface; Axial Magnetic Field (AMF) arc control diffuses the arc which spreads the heat across the contact surface. Here we focus on the latter arc control system.

For the AMF to be effective the magnetic field strength needs to exceed 4 mT/kA and maximum interruption current is then proportional to the surface [2]. Strict application of these rules allow the development of high current vacuum circuit breakers, like generator breakers [3]. The minimum condition for the magnetic field is unsatisfactory from an industrial point of view as it limits the reduction of size. Therefore a general trend is to develop more sophisticated tools to understand the processes that underpin the minimum magnetic field strength. Here we focus on the modeling of the arc behavior.

In [4,5] 2D vacuum arc models are proposed to describe the behavior of the arc. These models provide insight in the arcing processes but do not consider the complex magnetic field structure of commercial vacuum interrupters. The influence of current flow in circuit breakers on the 2D vacuum arc is discussed in [6]. Here we focus on a complete 3D description of the circuit breaker. The 3D magnetic field includes the fields generated by all current conductors inside and outside the arcing chamber and the 3D reaction of the arc itself [7].

Fig. 1.

(a) Medium Voltage Vacuum Circuit breaker with 3 poles; each pole houses a vacuum interrupter. Current flow in neighboring poles influence the arcing in one pole thru the magnetic field coupling. (b) Vacuum Interrupter is a hermetically sealed device containing 2 contacts one movable. The arcing contacts contain an arc control to assist the current interruption process. The housing consists of two ceramic cylinders.

2. Model

The 3D MHD vacuum arc model is based on the conservation equations of mass, momentum and energy and the generalized Ohm’s law as represented below by Eqs (1) to (4) and outlined in detail [7,8]. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{Conservation of mass}\quad {\nabla}\cdot (n_{i}\vec{u})=0\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{Conservation of momentum}\quad m_{i}n_{i}(\vec{u}\cdot {\nabla})\vec{u}={\nabla}(n_{i}kT_{i}+n_{e}kT_{e})+\vec{J}\times \vec{B}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{Conservation of energy}\quad {\nabla}\cdot \left(\left({\textstyle \frac{1}{2}}m_{i}n_{i}u^{2}+{\textstyle \frac{5}{2}}n_{i}kT_{i}\right)\vec{u}\right)=Z_{i}en_{i}\vec{E}\cdot \vec{u}+n_{e}m_{e}{\nu}_{ei}\vec{u}\cdot (\vec{v}-\vec{u})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +∼\frac{3kn_{e}m_{e}{\nu}_{ei}}{m_{i}}(T_{e}-T_{i})\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \text{Generalized Ohms Law}\quad -\vec{E}=-\frac{\vec{J}}{{\sigma}}+\vec{u}\times \vec{B}-\frac{1}{n_{e}e}\vec{J}\times \vec{B}+\frac{1}{n_{e}e}{\nabla}(n_{e}kT_{e}).\end{eqnarray}$ (4) The 3D MHD vacuum arc model uses a two-step approach as is visualized in Figs 2 and 3.

The features of the arc control, the magnetic field coil used in axial magnetic field (AMF) type vacuum interrupters (VIs) is created on a design platform ProEngineer^®. This geometry is imported into Flux3D^® or ANSYS Maxwell^® [2]. The time dependent magnetic field is calculated by a harmonic solver for 50 Hz for the arcing volume for each time instant and for each contact gap. An additional External Transverse Magnetic Field (ETMF) due to the time dependent magnetic field created by the neighboring phases in a 3-phase vacuum circuit breaker, is superimposed. These magnetic field values are supposed not to change during the subsequent calculation of the arc model.

The arc is modeled by a single fluid description of the arc plasma in Fluent^® using the conservation equations of mass, momentum and energy. Current flow is addressed by Ohms law. A complete description of the model can be found in [7,8]. The magnetic field produced by the arc control is coupled as boundary condition and the magnetic field inside the plasma is regenerated using the magnetic diffusion equation. The equations governing the arc (1) to (4) are given. The arcing volume is confined by the upper and lower contact and strictly limited to radial extent of these contacts; the arcing volume presents hence a cylinder with the radial dimension of the arcing contacts, here 60 mm in diameter and between 1 and 10 mm in axial direction depending on the momentary contact stroke represented by a green hashed line in Fig. 3. The meshing of the arcing volume is performed by Gambit^® with hexahedral elements. The boundary conditions for the arc are in short: - at the cathode, a homogeneous influx of mass and electronic current; - at the anode, free flow of mass and free flow of current; - across the radial arc boundary, no flow of mass or current.

The formulation results in a weak coupling between magnetic field and arc.

Fig. 2.

Software structure.

Fig. 3.

Physical model of the high current vacuum arc under the influence of the combined magnetic fields created by the contact structure and the circuit breaker.

3. Contact geometries

Three AMF type contact structures used in commercial VIs are compared, see Fig. 4 from left to right. The AMF is generated by a D-coil with 3 parallel branches, by a C-coil with 6 inclined branches and by an A-coil with 14 inclined branches. Contact diameter is set at 60 mm for all. The here analyzed contact structures have about the same height. Each VI has two identical contacts.

Fig. 4.

Three axial magnetic field type contact structures used in commercial vacuum interrupter.

The D-coil [9] is essentially machined from a flat disc of copper. The disc is topped by a contact disc of arc resistant material like the copper-chromium alloy CuCr25. The thickness is adapted to the current flowing thru the branches. The C-coil [10] is machined in the cylindrical edge of a cup shaped copper part. A contact disc tops the open part of the coil. The A-coil [11] is an inverted C-coil with the copper bottom upside. The contact disc tops the closed side of the coil system.

Magnetic field strength generated in the coils is decreasing in the order D, C to A. Magnetic field homogeneity is increasing in the order D, C to A. The AC magnetic field creates eddy current in the contact surface which cause a phase-lag of the magnetic field with respect to the main current and has a negative impact on the interruption performance. The D-coil and C-coil are equipped with slots to reduce the eddy currents and hence the phase-lag of the magnetic field. The A-coil contact structure with a copper support disc underneath the contact disc yields a low over-all contact resistance, which favors the application for high nominal currents. These and other differences are resumed in Table 1. All calculations are based on a contact separation of 10 mm and an arc current of 25 kA.

Table 1

Main characteristics of the coil structures and the contacts used in this study

	D-coil	C-coil	A-coil
Coil
Number of parallel arms	3	6	14
Angular extension of arms [degrees]	115°	100°	70°
Height of coil [mm]	15	15	15
Diameter of coil [mm]	60	60	60
Contact
Material	CuCr25	CuCr25	CuCr25
Number of Slots	3	6	–
Characteristics
Magnetic field strength	*****	****	***
Magnetic field homogeneity	***	****	*****
Phase-lag due to Eddy Currents	****	****	**
Overall Contact Resistance	**	**	****

(*, **, ***, ****, *****) increasing level of preference;

4. Results

Figure 5 shows the ion density in xoz-plane for the 3 geometries; for each geometry two ion density distributions are given: one without ETMF (above) and one with ETMF (below). The ion density decreases from the arc center to edge in both cases and the radial gradient of the ion density is much larger than the axial gradient. Under the influence of the external transverse magnetic field, the ion density distribution is deflected from the symmetrical axis. As the arc current flows from anode to cathode (in −z direction) and the external transverse magnetic field is in +y direction, the Amperian deflection direction is +x —right hand rule—. The deflection of the ion density is more significant at the anode. As a quasi-neutral plasma flow is assumed, the electron number density is proportional to the ion number density. The proportionality factor Z is assumed to be 1.8. The A-coil has the maximum ion density and highest arc pressure; ion temperature is about constant among the three contacts.

Fig. 5.

Ion density distribution at 25 kA. Upper diagram without ETMF and lower diagram with 40 mT of ETMF.

Figure 6 shows the velocity vector projection and its magnitude distribution in the xoy-plane at anode side. Figure 6a shows that in absence of an ETMF the flow is dominated by the axial magnetic field structure and its intensity; the maximum velocity is proportional to the AMF in increasing order A, C to D. The arc plasma’s rotation is almost axi-symmetric. Notice the inward directed vortex of the ion-rotation. The vortex confines the ions in the A-coil and the C-coil. At the periphery, there is an outflow of ions in A-coil and, to a lesser extent, in the D-coil.

Fig. 6.

(a) Velocity vector’s projection and its magnitude distribution on anode side without ETMF. (b) Velocity vector’s projection and its magnitude distribution on anode side with 40 mT of ETMF.

Figure 6b shows the influence of the ETMF of 40 mT. As the arc plasma deflects to right side (+x direction, Fig. 5), on left side (−x direction) the ion density decreases and the axial pressure gradient increases, which causes a significant increase in the ion velocity. The magnetic field intensity has great impact on the plasma kinetic behavior, which explains why the geometries are less to more affected in the sequence from D-coil, A-coil to C-coil. The foot point of the arc is not affected as its position is imposed by the boundary condition of the current flow.

Figure 7 shows the total heat flux to anode surface for the 3 arc controls to the anode with 40 mT of ETMF. Heat flux to anode is carried by electrons and ions. The energy deposited consists of kinetic and thermal energy, ionization and evaporation energy of condensing ions, and the work function of absorbed electrons. In absence of ETMF in Fig. 7a, most of the heat flux deposits near anode center as the current concentrates there and the distribution pattern of the heat flux density is like that of axial magnetic field; compare the heat distribution of the D-coil to the magnetic field profile in [2]. With the ETMF, Fig. 7b, the heat flux to anode surface concentrates to some part away from anode center in the anti-Amperian direction, and the maximum value of the heat flux increases. For higher external transverse magnetic fields, the arc may deflect to the contact edge. Excessive heating erodes the contacts and is source of interruption failure. For the D-coil the maximum heat-flux is lower than for C-coil and A-coil.

Fig. 7.

(a) The energy flow to the anode without ETMF. (b) The energy flow to the anode with 40 mT of ETMF.

The arc diameter variation with time is explained by Figs 8 and 9. The 3D MHD model presented is a transient model with respect to the magnetic field interaction of the current carrying conductors and the solid inductors of the structure. Figure 8 shows the sinusoidal variation of the current through the arc for a half period of the power frequency (50 Hz); the maximum current is 25 kA_peak.

Fig. 8.

Variation of current and By:ETMF created by current flow in VI and CB and Bx:ETMF created by current in neighboring phases.

Fig. 9.

Arc Diameter D50 for D-coil; Sinusoidal current of 25 kA peak and linear contact gap opening at 1 m/s.

The ETMF generated in the arc due to the current flow in the VI and the CB, By, is given and amounts to 40 mT at current peak. The ETMF generated in the arc due to neighboring phases, Bx, is maximum 25 mT at current zero. The variation of the contact gap with contact separation at current zero and opening speed of 1 m/sec gives a contact distance of 5 mm maximum at contact peak and a contact separation of 10 mm after 10 millisec. The variation of the arc diameter D50 at the anode is shown in Fig. 9 for the D-coil. The arc diameter D ₅₀ [12] is defined from the section of the arc S ₅₀ around the center of the arc wherein 50% of the total current is flowing then: $\begin{eqnarray}D_{50}=\left(\frac{4}{{\pi}}S_{50}\right)^{0.5}.\end{eqnarray}$ The quantity D ₅₀ is a measure of the concentration of the arc current. For reference, the model supposes a homogeneous current density at the cathode; so, for a contact diameter of 60 mm, the arc diameter D ₅₀ at the cathode side is 42.4 mm. Figure 9 scales vertically from 15 mm to 42.4 mm. The latter corresponds to the diameter D ₅₀ of the homogeneous current density at the cathode. The thick points show the simulation points, for each point the corresponding momentary current value is given; the dotted curve presents a curve fitted through. Notice that the curve is asymmetric with respect to the current peak. Increasing contact distance decreases the arc diameter; compare for instance the equivalent currents at 1 ms and 9 ms and also at 3 ms and 7 ms. Experimental values for S ₅₀ based on optical measurements are given in [12]. At peak current the arc diameter is largest as found; but the relative value of S ₅₀ = 16–20% of S _total. So D ₅₀ ^experimental = 12–14 mm compared to 36 mm for the model calculation.

5. Discussion

Without ETMF the model calculations show no significant differences in the heat flux to the anode between the 3 contact geometries studied.

In the presence of an ETMF, the model predicts that the center of the arc with highest pressure shifts in the Amperian direction. Due to the boundary condition on the current inflow at the cathode side, which is fixed, the plasma column bends creating an increased pressure gradient on the anti-Amperian side of the arc center and a reduced pressure gradient on the Amperian side of the arc center, Fig. 5. This pressure drop has a large influence on the rotational plasma flow and surprisingly, because not confirmed in practice, focuses the heat flow on the anti-Amperian side of the arc center. With ETMF the peak heat flux increases from 550 MW/m² (see Fig. 7a) to resp. 620, 640 and 670 MW/m² for the D-coil, C-coil and A-coil (see Fig. 7b). The stronger magnetic field of the D-coil is more effective in maintaining a homogeneous heat flux to the anode than the C-coil and A-coil.

The arc diameter D50 predicted by the model is much larger than the measured one [12]. This discrepancy can be (partially) due to the fixed current density distribution at the cathode side.

6. Conclusion

The model gives realistic results for contact geometries producing a strong axial magnetic field. For weak AMF contacts the weak coupling between the magnetic field calculation and the plasma flow calculation leads to an increasing mismatch in the continuity of the current at the plasma contact interfaces. This induces a systematic error in the prevailing magnetic field in the plasma. To solve this a strong coupling is necessary between the magnetic field solver and Navier–Stokes solver. Furthermore, relaxation of the cathode boundary condition for the current flow is necessary to enable the arc to move to its preferred position on the contact according the magnetic field topology.

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