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A 3D transient model of a cathode-spot crater and droplet formation in a vacuum arc is developed. The model includes mass, momentum, heat transfer (energy), current continuity and potential equations. Using the energy flux density, current density and pressure as external parameters, the shape of the cathode-spot crater and the temperature, velocity, potential and current density distributions at each time step are determined via numerical simulation. Under symmetrical conditions, the 3D simulation results are highly consistent with those from the previous 2D model. The new cathode spot tends to appear in the direction in which the protrusion is largest because the liquid-metal velocity is lower and the liquid-metal ridge radius is larger in this direction. The effect of the external transverse magnetic field is considered by using a symmetric space function to represent the pressure and energy flux density of the plasma cloud. Simulation results show that even small changes in the plasma cloud distribution have a significant impact on the cathode-spot crater process and droplet formation. Since the pressure is asymmetric, the crater becomes asymmetric and the new cathode spot tends to appear opposite to the direction of the Ampere force. Based on this phenomenon, a possible explanation for the retrograde motion phenomenon of cathode spots is proposed.

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