It is understood that if all currents in a system are known, then the magnetic field can be determined from the currents by the Biot-Savart equation.  The Biot-Savart equation allows the determination at a fixed point by integration over the path of the currents to find the total magnetic field at that point.  The application of this law implicitly relies on the superposition principle for magnetic fields, i.e. the fact that the magnetic field is a vector sum of the field created by each infinitesimal section of the wire individually.  However, magnetic inverse problems, where the determination is approached in reverse order so that a current distribution may be determined based on the magnetic field produced, are significantly more problematic.  The prediction of physical parameters based on the sensed magnetic fields of an electrical device tends to result in highly nonlinear formulations, and it becomes difficult to construct effective inversion algorithms.  In general, these formulations do not have a unique solution, and regularization techniques are needed. 
In spite of these difficulties, the remote sensing of electrical device current is an area of highly active interest due to the clear advantage afforded by non-obtrusive measurement.  Typically the applications are performed in such a way that the mathematical issues arising in inverse problems are avoided.  For example, magnetic current imaging as used heavily in the semiconductor industry relies on the magnetic fields produced in current carrying paths, and conducts inspection by comparing the magnetic field images produced by devices under inspection with the images produced by fault-free devices, as well as comparison with the circuit schematic.  This technique allows identification of currents occurring outside of design pathways and a rough estimation of magnitudes depending on the resolution of the magnetic field sensor involved, however the nature of the process precludes an exact solution for the location and magnitude of any offending current vector.  Additionally, the technique is based on a finite number of well defined current paths in the fault-free devices, and is not suited to a situation where an electrical device might experience significant deviation in the current pattern from one finished item to the next.  For example, the crucible section in an arc furnace, where currents would be expected during the normal course of operation, but where the location and magnitude of the current vectors comprising any formulation of the overall current pattern would also be expected to continually change based on both the operating condition and inherent inhomogeneity existing from one furnace to the next.  Such a system precludes approaches that rely on comparison with a pre-determined and definable normal operating condition. 
There is also a history of utilizing sensed magnetic fields in order to troubleshoot machinery in operation, and identify possible causes of observed abnormalities.  These methods are typically utilized for diagnosis of electromagnetic motors and generators, where common faults can be inserted and the resulting impact on the electromagnetic signature sensed at a specific location can be evaluated.  In application, the operating device is periodically monitored and the electromagnetic signature sensed is compared with an existing library of magnetic signatures to indicate possible sources of an abnormality.  This is a widely used technique which continues to be refined.  See, e.g., Bui et al, “Non Invasive Faults Monitoring of Electrical Machines by Solving Steady State Magnetic Inverse Problem,” Knowledge-Based Intelligent Information and Engineering Systems: KES 2007-WIRN 2007, Springer (2007), among others.  However, the methodology is similar to magnetic current imaging in that diagnosis is limited to defined fault locations, and is thus restricted to either diagnosis of common faults or identification of a general region where a fault may be located.  The method does not provide a manner in which the sensed magnetic field may be interpreted in a mathematically continuous way, so that a remotely sensed parameter may serve to locate one or more current vectors with precision within an electrical device. 
There are also known devices such as clamp-on ammeters which indicate a magnitude of current passing through a conductor by sensing the magnetic field generated by the conductor, and determining the current magnitude by assuming the current is centered in the conductor and flowing in a straight line over the region where the magnetic field is sensed.  The validity of the devices is limited to the conditions assumed, and location of the current vector producing the magnetic field is strictly limited to the center of the conductor.  Such devices do not allow interpreting a sensed magnetic field in a mathematically continuous way, so that one or more current vectors having undefined location may be located with precision within an electrical device. 
Thus, the methods fall short in applications where it becomes necessary or desired to utilize a remotely sensed magnetic field to locate one or more current vectors in an electrical device where the current vectors may routinely operate outside defined paths, or where the nature of the electrical device itself precludes analysis of well defined current paths within the device.  It would be advantageous to provide a system where the magnetic field of an electrical device could be monitored and the location of one or more current vectors within the electrical device could be precisely located.  It would be particularly useful in applications where the location of the current vectors is expected to routinely alter over the course of normal operation of the electrical device.  For example, where the electrical device is an arc furnace, and where the one or more current vectors represent the arc across the electrode arc gap.  It would further be advantageous to provide a system where exact definition of the currents occurring outside the current vector in question is not required, so that magnetic inversion issues do not arise and precise location of the current vector can be determined without necessary determination of the remaining current field. 
Accordingly, it is an object of this disclosure to provide a system whereby an exact solution for the location of a quantity of current vectors in an electrical device can be determined directly from remotely sensed magnetic field parameters. 
Further, it is an object of this disclosure to provide a system whereby the exact solution for the location of a quantity of current vectors does not rely on restriction of the current vector to well defined current paths within the electrical device. 
Further, it is an object of this disclosure to provide a system whereby the exact solution for the location of a quantity of current vectors may be determined in an electrical device that experiences poorly defined current patterns over the course of normal operation. 
Further, it is an object of this disclosure to provide a system whereby a sensed magnetic field may be interpreted in a mathematically continuous way, so that a remotely sensed parameter may serve to locate a quantity of current vectors with precision within an electrical device. 
These and other objects, aspects, and advantages of the present disclosure will become better understood with reference to the accompanying description and claims.