Feedback-Based Control Over the Spatio-Temporal Distribution of Arcs During Vacuum Arc Remelting via Externally Applied Magnetic Fields

Abstract

Ampere Scientific’s VARmetric ^TM measurement system for Vacuum Arc Remelting (VAR) furnaces passively monitors the distribution of arcs over time during VAR in real time. The arc behavior is known to impact both product yield and quality and can pose potentially catastrophic operating conditions. Arc position sensing with VARmetric ^TM enables a new approach to control the heat input to the melt pool. Transverse external magnetic fields were applied to push the arcs via the Lorentz force while measuring the arc location to control the arc distribution over time. This has been tested on Ampere Scientific’s small-scale laboratory arc furnace with electromagnets used for control for up to 60 seconds while monitoring the arc location with VARmetric ^TM. The arc distributions were shown to be significantly different from the uncontrolled distributions with distinct thermal profiles at the melt pool. Alternatively, this type of control can be periodically applied to react to undesirable arc conditions.

Appendix A—VARmetric TM

VARmetric ^TM capitalizes on the Biot–Savart Law relating the magnitude and direction of a magnetic field to the electric current source generating it. As magnetic fields behave under the principal of superposition, the current source of interest can be obtained by isolating that source from other sources in the area:

$$ \vec{B} = \vec{B}_{A} + \vec{B}_{C} + \vec{B}_{E} + \vec{B}_{I} + \vec{B}_{F} $$

(A1)

For us, $\vec{B}_{A}$ is the magnetic flux density due to the current, I is flow within an arc at a location P, while $\vec{B}_{C} ,\; \vec{B}_{E} ,\; \vec{B}_{I} \;{\text{and}}\;\vec{B}_{F}$ are external sources not of interest and include the fields from the crucible, electrode, ingot, and other external sources. For the purpose of this analysis, we recognize that the other sources are static with the exception of small changes in current distribution as a function of changes in the arc position. The arc is moving much faster than the other sources and so we can ignore the magnetic field flux due to these sources. In the plane of the arc, $\vec{B}_{A}$ is then given in cartesian coordinates as $\vec{B}_{A} = B_{x} {\hat{\text{i}}} + B_{y} {\hat{\text{j}}}$ by treating the arc as a current line source:

$$ B_{x} = m_{x} I \left( {\frac{\sin \left( \theta \right)}{r} - a} \right) ;\;\;\;\;B_{y} = m_{y} I \left( {\frac{\cos \left( \theta \right)}{r} - b} \right) $$

(A2)

where B _x and B _y are the x and y components of the magnetic flux at P. We note here that in these equations, the coefficients m _x, m _y, a, and b are relaxed from the field of a current line source, where m _x = m _y and a = b; this has the effect of coupling the crucible’s and electrode’s current dependence on arc location back into the arc field. These equations can be solved analytically for r and θ, if the coefficients m _x, m _y, a, and b are known:

$$ r = \left[ {\left( {\frac{{B_{y} }}{{m_{y} I}} + b} \right)*\left( {\left[ {\frac{{\left( {\frac{{B_{x} }}{{m_{x} I}} + a} \right)}}{{\left( {\frac{{B_{y} }}{{m_{y} I}} + b} \right)}}} \right]^{2} + 1} \right)^{{\frac{1}{2}}} } \right]^{ - 1} $$

(A3)

$$ \tan \left( \theta \right) = \frac{{\left( {\frac{{B_{x} }}{{m_{x} I}} + a} \right)}}{{\left( {\frac{{B_{y} }}{{m_{y} I}} + b} \right)}} $$

(A4)

As previously reported for Arc Position Sensing,[12] the coefficients are evaluated based upon a computational sweep of potential geometrical changes in the system (e.g., differing arc locations across the arc gap) where a curve fit is utilized to provide a functional relationship for m _x, m _y, a, and b, based upon the arc location. In practice, some of the external field sources are naturally included in the computational model, so that these coefficients account for all fields produced by the furnace.