Patent Publication Number: US-8111059-B1

Title: Electric current locator

Description:
STATEMENT OF GOVERNMENTAL SUPPORT 
     The United States Government has rights in this invention pursuant to the employer-employee relationship of the Government to the inventors as U.S. Department of Energy employees and site-support contractors at the National Energy Technology Laboratory. 
    
    
     TECHNICAL FIELD 
     The disclosure herein provides an apparatus for location of a current vector in an electrical device, where the current vector has a known direction and a known relative magnitude to an input current supplied to the electrical device. The apparatus utilizes magnetic field sensors oriented to a sensing plane and provides current vector location based on the solution of Biot-Savart equations formulated for the specific electrical device. 
     BACKGROUND OF THE INVENTION 
     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. 
     SUMMARY OF INVENTION 
     The novel apparatus disclosed herein provides localization of one or more current vectors in an electrical device, where the one or more current vectors have a known direction and a known relationship to an input current supplied to the device. The electrical device may be any device which receives an input current and produces a magnetostatic field in response. 
     The novel apparatus utilizes one or more magnetic field sensors oriented on a sensing plane, where the sensing plane is oriented substantially perpendicularly to the one or more current vectors to be localized, and where the one or more magnetic field sensors sense the magnitude of the magnetic field in two directions on the sensing plane. The one or more magnetic field sensors and an input current measurement device is in data communication with a processor. The processor receives the data communication and localizes the one or more current vectors using Biot-Savart superposition equations applicable to the specific electrical device and the orientation of the novel apparatus disclosed. 
     The Biot-Savart superposition equations utilized by the processor are functions of the mathematical constants m x , m y , a, and b in addition to other measured parameters, as discussed supra. The mathematical constants m x , m y , a, and b are determined by a mathematical regression of calibration data, where the calibration data is comprised of the magnetic field in two directions at the location of the one or more magnetic field sensors, the location of the current vector producing that magnetic field, and the relative magnitudes of the input current and the current vector. The calibration data is taken over a finite number of points and fit to the Biot-Savart equation, such that the processor is able to describe the current vector location in a mathematically continuous expression during operation of the electrical device. 
     The calibration data determining the mathematical constants m x , m y , a, and b may be generated by finite element analysis of the electrical device and simulation of appropriate parameters, in order to determine the magnetic field expected at the one or more magnetic field sensors under a finite number of conditions. Alternatively, the calibration data may be determined by supplying an input current to the electrical device and physical manipulation of the electrical device such that a current vector having a known relationship to the input current exists at a specified point on the sensing plane. In the latter method, the magnetic field produced at the one or more magnetic field sensors is measured under a finite number of conditions. Using the calibration data thus determined, the mathematical constants m x , m y , a, and b result from mathematical regression of the calibration data to Biot-Savart superposition equations. The mathematical constants m x , m y , a, and b thus determined are defined for the specific electrical device, the location and orientation of the novel apparatus described herein, and the relationship between the input current and the current vector to be located. 
     In a particular embodiment, the electrical device is an arc furnace and the one or more current vectors to be localized is one or more electric arcs bridging the electrode gap of the arc furnace during operation. This embodiment provides a real-time location of the electric arc, so that the time-averaged distribution of the electric arc during arc furnace operation can be evaluated, and the arc furnace parameters can be adjusted to achieve the desired diffuse arc condition. 
     The disclosure herein thus provides an electric current locator where one or more current vectors may be localized in an electrical device directly from remotely sensed magnetic field parameters. The electric current locator provides localization without restricting of the current vector to well defined current paths within the electrical device. This allows localization in electrical devices that routinely experience poorly defined current patterns over the course of normal operation. This capability stems from treating sensed magnetic field parameters as arising from as Biot-Savart superposition equations, and the determination of appropriate calibration data such that the Biot-Savart superposition equations may be utilized as a mathematically continuous description of current vector location. 
     The novel apparatus and principles of operation are further discussed in the following description. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates an electrical device receiving an input current and generating a current vector and a superposition vector perpendicular to a sensing plane, as viewed in the x-z plane. 
         FIG. 2  illustrates an electrical device receiving an input current and generating a current vector and a superposition vector perpendicular to a sensing plane, as viewed in the x-y plane. 
         FIG. 3  illustrates an electrical device having poorly defined current paths receiving an input current, and generating a current vector at a plurality of locations. 
         FIG. 4  illustrates an electrical device having well-defined current paths and receiving an input current, and generating a current vector at a plurality of locations. 
         FIG. 5  illustrates an electric current locator operating on an electrical device. 
         FIG. 6  illustrates the components of an arc furnace. 
         FIG. 7  illustrates an electric current locator operating on an arc furnace. 
         FIG. 8  illustrates an electric current locator having a plurality of magnetic field sensors. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The following description is provided to enable any person skilled in the art to use the invention and sets forth the best mode contemplated by the inventor for carrying out the invention. Various modifications, however, will remain readily apparent to those skilled in the art, since the principles of the present invention are defined herein specifically to provide an apparatus for localizing one or more current vectors in an electrical device, where the one or more current vectors have a known direction and a known magnitude relative to an input current supplied to the electrical device. 
     The apparatus utilizes one or more magnetic field sensors oriented on a sensing plane, where the sensing plane is oriented substantially perpendicularly to the one or more current vectors to be located, and where the at least one magnetic field sensor senses the magnitude of the magnetic field in two directions on the sensing plane. The apparatus describes the location of the current vector by determining the point of intersection between an intersect vector coincident with the current vector and the sensing plane. The apparatus senses the magnetic field in two directions on the sensing plane and utilizes a processor in order to solve mathematically continuous Biot-Savart superposition equations based on calibration data determined for a specific electrical device. In one embodiment, the electrical device is an arc furnace, and the apparatus serves to provide a location of the electric arc exiting the electrode during arc furnace operation. The calibration data is generated using a finite number of current location points, and the Biot-Savart superposition equations are fit to the calibration data in order to provide a mathematically continuous description of current vector location based on the finite number of points. 
     These and other aspects and advantages of the apparatus will become better understood with reference to the following description and claims. 
     Principles of the Method 
       FIG. 1  illustrates an electrical device  100 . Electrical device  100  is a three-dimensional object oriented with respect to the x-y-z axis shown, where the y-axis is illustrated as proceeding out of the page. Electrical device  100  receives an input current  101 , and in response to input current  101 , electrical device  100  experiences current flow  102  across sensing plane  103 . Sensing plane  103  is a plane having dimension in the x-y axes only. 
     Electrical device  100  may be any device which receives an input current and generates a current flow  102  across sensing plane  103 . Additionally, although current flow  102  is illustrated in two-dimensions at  FIG. 1 , current flow  102  may be a three-dimensional current vector field having y-axis components. In application, it would typically be expected that current flow  102  is a three-dimensional current vector field. Additionally, although current flow  102  is represented as an essentially continuous gradient across sensing plane  103  for illustrative purposes, current flow  102  may also be a series of geometrically well-defined currents transiting sensing plane  103  while confined to rigorous conducting paths. For example, current flow  102  could be a plurality of current vectors flowing in accordance with an electrical schematic, so that each current vector comprising current flow  102  intersects sensing plane  103  at a well-defined and known point in the x-y-z coordinate system. 
     As is well known, current flow  102  will cause electrical device  100  to emanate a magnetic field. If the magnetic field is modeled by a magnetostatic approximation, where electrostatic charges are ignored and the magnetic field is presumed essentially constant with respect to time, then the magnetic flux density B of the magnetic field may be described at a point in space using the Biot-Savart law. Generally speaking, magnetostatics may serve as an adequate approximation even if current flow  102  is not static, provided that current flow  102  does not alternate rapidly. Electrical processes occurring in the 60 Hz range have been treated using the Biot-Savart law without significant error. See e.g., Griffiths et al,  Introduction to Electrodynamics , Prentice Hall, Englewood Cliffs, N.J. (1999). As a result, and assuming current flow  102  produces a magnetostatic field, the Biot-Savart law may be utilized to describe the magnetic flux density B arising from current flow  102  at point  104 , illustrated at  FIG. 1 , where point  104  is a location lying within sensing plane  103 . Further, the magnetic flux density at point  104  may be expressed vectorially as B=B x +B y +B z , where component subscripts indicate vector components with respect to the x-y-z axis shown. Using the nomenclature of  FIG. 1  therefore, the magnetic flux density B at point  104  is comprised of components B x  and B y  extending from point  104  and lying within sensing plane  103 , and component B z  extending from point  104  and having a perpendicular orientation to sensing plane  103 . 
     In the absence of extraneous magnetic fields in the surrounding environment, the components B x  and B y  of magnetic flux density B at point  104  results from superposition of all magnetic fields resulting from the current vectors making up current flow  102 . As discussed supra, the nature of electrical device  100  dictates the nature of current flow  102 . However, in terms of producing the magnetic field components B x  and B y  at point  104 , it is possible to represent the current flow  102  as current vector  105  and superposition vector  106 , where current vector  105  and superposition vector  106  are oriented perpendicular to sensing plane  103 , and where current vector  105  and superposition vector  106  have respective magnitudes such that B x  and B y  at point  104  result. Further, if the intersection of superposition vector  106  and sensing plane  103  is maintained at a constant position with respect to point  104 , then the Biot-Savart equations describing B x  and B y  at point  104  may be expressed as:
 
 B   x   =m   x   l   n [(sin Θ)/ d−a]   (1)
 
 B   y   =m   y   l   n [(cos Θ)/ d−b]   (2)
 
     where m x , m y , a, and b are mathematical constants, and where l n  is the magnitude of current vector  105 , d is the magnitude of a position vector from point  104  to the intersection of current vector  105  with sensing plane  103 , and Θ is the angle between the position vector and some reference axis which lies in sensing plane  103 , and which passes through point  104  and the intersection point between superposition vector  106  and sensing plane  103 . The value of the mathematical constants m x , m y , a, and b are dependent on the location of the superposition vector  106 , the electromagnetic nature of electrical device  100 , and the relationship between input current  101  and current vector  105 . 
     The terms of equations (1) and (2) are further illustrated at  FIG. 2 .  FIG. 2  shows a view of electrical device  200 . Electrical device  200  is directly analogous to electrical device  100  shown at  FIG. 1 , rotated in accordance with the x-y-z axis shown at  FIG. 2 .  FIG. 2  also illustrates point  204  lying within a sensing plane (not shown) having dimension in the x-y axes only, analogous to sensing plane  103 . A current vector  205  and a superposition vector  206  are illustrated having a direction parallel to the z-axis and coming out of the page. Position vector  207  extends from point  204  to the intersection of current vector  205  with the sensing plane, and reference axis  208  lies in the sensing plane, and passes through point  204 . With respect to equations (1) and (2), d is the magnitude of position vector  207 , and Θ is the angle  209  between position vector  207  and reference axis  208 . 
     By using equations (1) and (2) when values of B x , B y , l n , m x , m y , a, and b are known, it is possible to determine values for Θ and d, and determine the location where current vector  205  intersects the sensing plane with respect to point  204 . Inverting equations (1) and (2) will provide two solutions about point  204  for the intersection of current vector  205  and the sensing plane. The correct solution for the situation illustrated at  FIG. 2  can be determined by selecting the solution which places current vector  205  within the physical bounds of electrical device  200 . 
     When the values of m x , m y , a, and b are known, the principle may be extended for locating multiple current vectors, where the multiple current vectors each have a direction parallel to or coincident with current vector  205  and the multiple current vectors have a total summation current equal to l n . This is further illustrated at  FIG. 8 .  FIG. 8  shows a view of electrical device  800 . Electrical device  800  is directly analogous to electrical device  100  shown at  FIG. 1 , rotated in accordance with the x-y-z axis shown at  FIG. 8 .  FIG. 8  illustrates points  841  and  842  lying within a sensing plane (not shown) having dimension in the x-y axes only, analogous to sensing plane  103 . Current vectors  805  and  835 , and a superposition vector  806 , are illustrated having a direction parallel to the z-axis and coming out of the page. Position vector  843  and position vector  844  extend from point  841  to the intersection of current vectors  805  and  835  with the sensing plane. Reference axis  845  lies in the sensing plane and passes through point  841 . Subtending angle  846  and subtending angle  847  lie between reference axis  845  and position vector  843  and position vector  844  respectively. 
     Biot-Savart equations describing B x  and B y  at, for example, point  841  can be expressed as: 
     
       
         
           
             
               
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     where m x , m y , a, and b are mathematical constants specific to point  841 . The summation is conducted over the quantity n of current vectors, and j, is the fraction of the total current l n  carried by the i th  current vector. Θ i  is the angle between the reference axis  845  and a position vector from point  841  to the i th  current vector, for example, subtending angle  846  when the i th  current vector is current vector  805 . The equations may be further simplified by assuming that current vectors  805  and  835  represent equivalent fractions of the total current l n , giving equations (3) and (4): 
     
       
         
           
             
               
                 
                   
                     
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     Written only with respect to point  841  and describing current vectors  805  and  835 , and formulated with knowledge of B x , B y , m x , m y , a, and b at point  841 , equations (3) and (4) are insufficient to determine the values of Θ i  and d i  and thereby determine the locations of current vectors  805  and  835 , since such a formulation would provide 2 equations with 4 unknowns. However, knowledge of B X , B y , m x , m y , a, and b at a second point, for example point  842 , allows formulation of equations (3) and (4) with respect to point  842  and reference axis  848 . Because the location of point  841  and point  842  are known with respect to each other, and because reference axes  845  and  848  are geometrically defined, the Θ i  and d i  terms describing a given current vector in both formulations can be related through a common geometric reference. As a result, for the situation illustrated at  FIG. 8 , when B X , B y , M x , m y , a, and b are known at both point  841  and point  842 , equations (3) and (4) can be formulated at both point  841  and point  842  to produce 4 equations and 4 unknowns, and the locations of current vectors  805  and  835  may be determined with respect to the common geometric reference. 
     The methodology described above may be applied to determine the location of any number of current vectors of equal current magnitude passing through the sensing plane, provided that an at least equal number of locations points are present and that B X , B y , m x , m y , a, and b at each of the locations points is known. 
     The novel apparatus of this disclosure utilizes the principle that B x  and B y , arising from current flow  102  and as represented in equations (1) and (2), may be considered as arising from one or more current vectors, such as current vector  105  or current vectors  805  and  835 , and a superposition vector, such as superposition vector  106  or superposition vector  806 , where the one or more current vectors and the superposition vector have an orientation substantially perpendicular to the sensing plane. The novel apparatus utilizes determined values for the mathematical constants m x , m y , a, and b determined at each sensing location and inverts equations (3) and (4) to localize the one or more current vectors within an electrical device. As discussed supra, the values of m x , m y , a, and b are dependent on the location of the superposition vector  106 , the electromagnetic nature of electrical device  100 , and the relationship between input current  101  and the total current l n . In the novel approach disclosed herein, an apparatus senses the components B x  and B y  at one or more sensing locations in a sensing plane and localizes one or more current vectors using determined values of the mathematical constants m x , m y , a, and b at the one or more sensing locations, where the number of sensing locations is at least equal to the number of current vectors localized. 
     The mathematical constants m x , m y , a, and b may be determined through mathematical regression. For example, referencing  FIG. 1 , the regression can be conducted by determining the values of B x  and B y  at point  104  which arise for a plurality of current vector  105  locations described by Θ and d, where the magnitude of current vector  105  relative to input current  101  is maintained substantially constant at each location in the plurality of locations. Determining these values for the plurality of locations will yield a plurality of data points, each having a respective value for B x , B y , Θ, and d. Selecting a value for l n  and fitting these data points to equations (1) and (2) then allows a determination of the m x , m y , a, and b values at point  104  for the given magnitude ratio between the input current and current vector  105 . Having determined the m x , m y , a, and b values at point  104 , equations (1) and (2) may then be utilized as continuous expressions relating B x  and B y  at point  104  to the location of current vector  105 , as described by Θ i  and d i , provided the magnitude of current vector  105  relative to input current  101  substantially matches the magnitude ratio utilized to generate the calibration points. Further, the same methodology may be conducted to determine the m x , m y , a, and b values at multiple sensing points, such as points  841  and  842 . When multiple sensing points are utilized, the m x , m y , a, and b values determined for a single current vector may be utilized to determine the location of a quantity of current vectors by using the same m x , m y , a, and b values in equations (3) and (4), provided that the quantity of sensing points at least equals the quantity of current vectors, and further provided that the total current magnitude of the quantity of current vectors relative to electrical device input current substantially matches the magnitude ratio utilized to generate the calibration points using the single current vector. Under those conditions, the m x , m y , a, and b values determined at multiple sensing points for a single current vector may be utilized to determine the location of a quantity of current vectors, as described by Θ i  and d i  in equations (3) and (4). 
     In the novel apparatus described herein, a processor receives inputs representing B x  and B y  as expressed in equations (3) and (4) from one or more magnetic field sensors, and localizes one or more current vectors by solving equations (3) and (4) for the location parameters Θ i  and d i , where the total current magnitude of the one or more current vectors within the electrical device is known based on the magnitude of the input current received by the electrical device, and where the quantity of magnetic field sensors at least equals the number of current vector locations. An exemplary electrical device for the apparatus is an arc furnace, where a given amount of input power would be expected to produce a quantity of current arcs across the electrode gap, and where the total current magnitude of the current arcs can be related to the given amount of input power. 
     As stated, in order to conduct the mathematical regression and determine the mathematical constants m x , m y , a, and b for a given electrical device and a given magnitude ratio between an input current and the current vector to be located, a set of B x , B y , Θ, and d data points must be generated. One exemplary method of generating the B x , B y , Θ, and d data points utilizes finite element analysis of the electrical device. As an example, consider electrical device  300  shown at  FIG. 3 . Electrical device  300  receives input current  301  and generates current passing perpendicularly through sensing plane  303 . Point  304  is a known location in sensing plane  303 . In the situation illustrated at  FIG. 3 , the B x , B y , Θ, and d data for determination of the mathematical constants m x , m y , a, and b for a certain ratio of input current  301  to a current vector to be sensed may be generated by dividing electrical device  300  into a finite element network which further includes point  304 . The properties of the finite element network may be manipulated such that when input current  301  is applied, a current vector of known magnitude passes perpendicularly through sensing plane  303  at a sensing point, where the sensing point is located on sensing plane  303  and within an electrical device boundary, where the electrical device boundary is the intersection of electrical device  300  and sensing plane  303 . For example, the finite element network may be manipulated such that current vector  305  passes through sensing point  312  when input current  301  is supplied. The finite element analysis can then provide the resulting B x  and B y  at point  304  when input current  301  is supplied and current vector  305  passes through sensing point  312 , as illustrated at  FIG. 3 . Similarly, the finite element network may be manipulated such that when the same input current  301  is supplied, current vector  305  passes through sensing point  313 , and B x  and B y  at point  304  when current vector  305  passes through sensing point  313  can be determined. Similar finite element network manipulation could be conducted so that current vector  305  passes through sensing point  314  in response to input current  301 . If such an analysis is conducted and B x , B y , Θ, and d data is available for the three sensing points  312 ,  313 , and  314 , then the B x , B y , Θ, and d data may be mapped to equations (1) and (2) in order to determine suitable values for the mathematical constants m x , m y , a, and b at point  304 . Suitable values for the mathematical constants m x , m y , a, and b at additional points, for example point  349 , may be determined similarly. 
     Within this methodology, as discussed supra, the magnetic field arising at a certain point in response to input current supplied to an electrical device is equated to a magnetic field arising from a current vector and a superposition vector. As a result, the superposition vector represents the response of an electrical device when the current vector occurs at a certain location on a sensing plane. For example, referring to  FIG. 1 , when input current  101  is supplied to electrical device  100  and current vector  105  passes through sensing plane  103  at the location shown, superposition vector  106  represents the response of the electrical device  100  to the current vector  105  at that location. It is understood, therefore, that manipulation of a finite element network in order to generate data for determination of the mathematical constants m x , m y , a, and b is most effective when the manipulation reflects realistic operating points for the electrical device. For example, if the electrical device  100  is an arc furnace, current vector  105  is a singular arc bridging the arc furnace electrode gap, and the sensing plane  103  passes through the electrode gap, it would be most effective to manipulate the properties of the air gap in order to force the singular arc across the gap at a desired location, rather than the physical properties of the electrode itself, or some other component that provides a relatively strong contribution to the magnetic field in the sensing plane. Manipulation of the physical properties of the electrode itself rather than the air gap properties would have a much larger impact on the resulting B x  and B y  values simulated, and would have a corresponding impact on the accuracy of any resulting mathematical constants m x , m y , a, and b determined. 
     It is further understood that the validity of the mathematical constants m 1 , m 2 , a and b determined is dependent on the relative magnitude of input current  101  and current vector  105 . Those skilled in the art understand that if the mathematical constants are determined for a given scaling factor, where the scaling factor is equal to the magnitude of current vector  105  divided by the magnitude of input current  101 , then the mathematical constants m x , m y , a and b describe current vector  105  location most accurately when electrical device  100  operates with a substantially equivalent scaling factor. 
     Alternatively, another method of generating data points for determination of the mathematical constants m x , m y , a, and b is through physical manipulation and measurement of the electrical device. For example, the electrical device may have well defined current paths through the sensing plane as defined by a circuit schematic, such as electrical device  400  illustrated at  FIG. 4 . Electrical device  400  receives input current  401  and is designed to operate in accordance with schematic  410 . Schematic  410  is comprised of electrical components, such as electrical component  411 , and provides well defined paths where the schematic  410  currents intersect sensing plane  403 , shown as sensing point  412 , sensing point  413 , and sensing point  414 . Magnetic field sensor  415  is positioned at point  404  and oriented to sense B x  and B y  in sensing plane  403 . In the situation illustrated at  FIG. 4 , the B x , B y , Θ, and d data for determination of the mathematical constants m x , m y , a, and b for a certain ratio of input current  401  to the current vector to be sensed could be generated by physical manipulation of electrical component  411  or other electrical components in schematic  410 , such that when input current  401  is applied, a current vector of known magnitude results through one of the sensing points  412 ,  413 , or  414 , and magnetic field sensor  415  provides the resulting B x  and B y . Conducting the evaluation over the plurality of sensing points  412 ,  413 , and  414  provides B x , B y , Θ, and d data which may be mapped to equations (1) and (2) in order to determine suitable values for the mathematical constants m x , m y , a, and b. Note also that this methodology is not restricted to electrical devices having well defined and pre-determined current paths through the sensing plane, as illustrated at  FIG. 4 . The same methodology could be employed in, for example, an arc furnace, where the current vector represents the arc bridging the electrode gap and intersecting the sensing plane at the electrode gap. In such an application, the location of the arc might not be predictable, however physical manipulation of the system, for example, shorting across the electrode gap at a plurality of locations, could provide sufficient B x , B y , Θ, and d data such that the mathematical constants m x , m y , a, and b can be determined. Suitable values for the mathematical constants m x , m y , a, and b at additional points, for example point  449 , may be determined similarly. 
     Additionally, it is understood that the validity of the mathematical constants m 1 , m 2 , a and b determined is dependent on a given scaling factor for input current  401  and current vector  405 , where the scaling factor is equal to the magnitude of current vector  405  divided by the magnitude of input current  401 , and that the mathematical constants m 1 , m 2 , a and b describe current vector  405  location most accurately when electrical device  400  operates with a substantially equivalent scaling factor. 
     The novel apparatus of this disclosure senses the magnetic field components B x  and B y  at one or more magnetic field sensors and utilizes equations (3) and (4) to describe the location of a quantity of current vectors in an electrical device receiving an input current. The novel apparatus utilizes the mathematical constants m x , m y , a, and b as determined for the electrical device and a specific ratio of the input current to the current vector sensed. The novel apparatus utilizes a mathematically continuous description of B x  and B y , such that at all points on the sensing plane within the electrical device can be described based on calibration data from a finite plurality of locations. 
     Description of the Apparatus 
     The principles as discussed may be utilized in an apparatus such as the electric current locator  516  illustrated at  FIG. 5 . Electric current locator  516  is comprised of input current measurement device  517 , one or more magnetic field sensors shown as magnetic field sensor  515  and magnetic field sensor  550 , and processor  518 . Electric current locator  516  is capable of determining a location where one or more current vectors, illustrated as current vector  505  and current vector  535 , intersect sensing plane  503  in electric device  500  in response to input current  501 . The quantity of current vectors localized by electric current locator  516  is limited by the quantity of magnetic field sensors, here illustrated as two, however electric current locator  516  may have any quantity of magnetic field sensors and may thus locate any quantity of current vectors. 
     The direction of current vectors  505  and  535  are known, and the electric current locator  516  species the point where current vectors  505  and  535  intersects sensing plane  503  by solving equations (3) and (4) for the quantities Θ i  and d i , where d i  is a position vector originating from either field sensing location  504  or field sensing location  542 , and Θ i  is an angle between the position vector d i  and a sensing reference axis passing through either field sensing location  504  or field sensing location  542 . 
     Sensing plane  503  is not a physical construct, but rather represents a two-dimensional plane having a substantially perpendicular orientation with respect to the current vectors  505  and  535  to be sensed. As a result, the direction of current vectors  505  and  535  determine the orientation of sensing plane  503  and directly impacts other geometric parameters specified for electric current locator  516 , as discussed infra. At  FIG. 5 , sensing plane  503  is a plane having dimension on the x-y axes only, and current vectors  505  and  535  have direction parallel to the z axis. Electric current locator  516  provides results assuming that current vector  505  and  535  are perpendicular to sensing plane  503 , so deviations from the perpendicular impact the accuracy of electric current locator  516 . The allowable deviation from the perpendicular for a given accuracy is dependent on the application utilizing electric current locator  516 , as is understood in the art. Further, electric current locator  516  provides results assuming that current vector  505  and  535  geometrically intersect sensing plane  503 . Those skilled in the art recognize that electric current locator  516  may be utilized to provide an indication of current vector  505  and  535  locations when current vectors  505  and  535  do not geometrically intersect sensing plane  503 , but that the accuracy may be compromised. For example, as applied to an electric arc furnace, electric current locator  516  could be positioned with sensing plane  503  slightly above or below the electrode gap and still provide an indication of electric arc location within the gap, however optimal accuracy would be expected when sensing plane  503  passed through the electrode gap. 
     Similarly, field sensing locations  504  and  542  are not physical constructs, but rather points on sensing plane  503  where magnetic field sensors  515  and  550  are substantially located. Electric current locator  516  provides results assuming that a magnetic field sensed by magnetic field sensor  515  is occurring at field sensing locations  504  and  550 , so the proximity of magnetic field sensor  515  and  550  to field sensing locations  504  and  550  respectively impacts the accuracy of electric current locator  516 . The necessary degree of proximity for a given accuracy is dependent on the application utilizing electric current locator  516 , as is understood in the art. Similarly, the sensing reference axis for a given field sensing location is not a physical construct, but rather is a line lying in the sensing plane and passing through the given field sensing location. The orientation of the sensing reference axis outside of these constraints is arbitrary for a given field sensing location, however once the orientation is determined, the sensing reference axis for the given field sensing location remains fixed during operation of electric current locator  516 . In a preferred embodiment, the sensing reference axis is oriented such that expected values for Θ i  within equations (3) and (4) are less than approximately 60 degrees. 
     Current measurement device  517  determines at least the magnitude of input current  501  supplied to electric device  500 . Additionally, current measurement device  517  has current signal output  519  for transmission of a signal representative of the magnitude of input current  501 . 
     Magnetic field sensor  515  is a field sensor capable of measuring a magnetic field in a first direction and in a second direction, and is substantially located at field sensing location  504 . Further, magnetic field sensor  515  is oriented with respect to sensing plane  503  such that the first direction and the second direction lay within the sensing plane. In a preferred embodiment, the first direction and the second direction are orthogonal. Additionally, magnetic field sensor  515  has first output  520  for transmission of a signal representative of the magnitude of the magnetic field sensed in the first direction, and magnetic field sensor  515  has second output  521  for transmission of a signal representative of the magnitude of the magnetic field sensed in the second direction. Similarly Magnetic field sensor  550  is a field sensor capable of measuring a magnetic field in a first direction and in a second direction, and is substantially located at field sensing location  542 . It is not necessary that the first direction of magnetic field sensor  550  correspond to the first direction of magnetic field sensor  515 , or that the second direction of magnetic field sensor  550  correspond to the second direction of magnetic field sensor  515 . Further, magnetic field sensor  550  is oriented with respect to sensing plane  503  such that the first direction and the second direction lay within the sensing plane. In a preferred embodiment, the first direction and the second direction are orthogonal. Additionally, magnetic field sensor  550  has first output (not shown) for transmission of a signal representative of the magnitude of the magnetic field sensed in the first direction, and magnetic field sensor  550  has second output (not shown) for transmission of a signal representative of the magnitude of the magnetic field sensed in the second direction 
     Processor  518  is a computing device in data communication with current measurement device  517 , magnetic field sensor  515 , and magnetic field sensor  550 . Processor  518  receives current signal output  519  from current measurement device  517 , and receives first outputs and second outputs from magnetic field sensors  515  and  550 . Processor  518  formulates, for both magnetic field sensors  515  and  550 , Biot-Savart superposition equations having the form, 
                 B   1     =       m   1     ⁢     l   ⁡     [       (       ∑     i   =   1     n     ⁢           ⁢     sin   ⁢           ⁢       Θ   Mi     /     (     nd   Mi     )           )     -   a     ]           ,     
     ⁢       B   2     =       m   2     ⁢     l   ⁡     [       (       ∑     i   =   1     n     ⁢           ⁢     cos   ⁢           ⁢       Θ   Mi     /     (     nd   Mi     )           )     -   b     ]           ,         
where
 
     n=the quantity of current vectors to be localized, illustrated as two at  FIG. 5 , 
     B 1  is the first output from magnetic field sensors  515  or magnetic field sensor  550 , 
     B 2  is the second output from the magnetic field sensor providing B 1 , 
     l is the combined total current of the two current vectors  505  and  535 , as determined based on the current signal output  519 . 
     d Mi  is an unknown variable representing the length of a sensing point vector originating at the field sensing location of the magnetic field sensor providing B 1  and terminating where the i th  current vector intersects the sensing plane, where the i th  current vector is either current vector  505  or current vector  535 , 
     Θ Mi  is an unknown variable representing an angle between d Mi  and the sensing reference axis for the magnetic field sensor providing B 1 , and 
     m 1 , m 2 , a and b are mathematical constants specific to electrical device  500  and specific to the magnetic field sensor providing B 1 , as determined through a procedure generating calibration data. 
     Processor  518  is further capable of expressing all Θ Mi  and d Mi  referenced to a common point based on the relative locations of field sensing locations  542  and  504 , of all individual 
     sensing locations, such that processor  518  receives data from magnetic field sensors  515  and  540  and produces a set of formulated equations having a quantity of equations at least equal to a quantity of unknowns. For the situation illustrated at  FIG. 5 , processor  518  utilizes the Biot-Savart superposition equations to produce four equations with four unknowns. Processor  518  then simultaneously solves the formulated equations to describe the location of each current vector in the quantity of current vectors based on the resulting values of Θ Mi  and d Mi . In this manner, as illustrated at  FIG. 5 , processor  518  provides location data for current vectors  505  and  535 . When additional magnetic field sensors are employed such that an increased quantity of current vectors may be localized, processor  518  is in similar data communication with all the magnetic field sensors, and processor  518  operates similarly to provide location data for each current vector in the increased quantity of current vectors. 
     As discussed, the mathematical constants m 1 , m 2 , a and b specific to each magnetic field sensor may be determined by physical measurement of the magnetic field at field sensing locations  504  and  542  when input current  501  is supplied and a current vector having a magnitude equal to the combined current of the quantity of current vectors—represented by current vectors  505  and  535  at FIG.  5 —is present in electrical device  500  at known locations described by Θ M  and d M . Alternatively, the mathematical constants m 1 , m 2 , a and b specific to each magnetic field sensor may be determined using finite element analysis of electrical device  500 , by simulating input current  501  into the finite element network, manipulating the finite element network such a current vector having a magnitude equal to the combined current of the quantity of current vectors passes through sensing plane  503  at known locations, and determining the magnitude of the magnetic field at field sensing locations  504  and  542 . The determined values of m 1 , m 2 , a and b specific to each magnetic field sensor are then utilized by processor  518  to provide location data for a quantity of current vectors. For the arrangement illustrated at  FIG. 5 , processor  518  could provide locations data for up to two current vectors, such as current vectors  505  and  535 , provided that the relative magnitude of the combined total current of current vectors  505  and  535  as compared to the input current  501  producing those current vectors is substantially equivalent to the relative magnitude between the single current vector utilized in the calibration and the input current which produced that single current vector. 
     Detailed Description of an Embodiment 
     The apparatus disclosed has been applied to the location of an electric arc in a Vacuum Arc Furnace (VAR). A schematic of VAR  600  is shown at  FIG. 6 . VAR  600  has ram  622  which translates in a direction parallel to the z axis of the x-y-z axis shown. Ram  622  is connected to stinger  623 . Stinger  623  is typically an expendable connection threadably engaged to electrode  624 . Electrode  624  is comprised of the material to be melted in the VAR, and is consumed during the process. Electrode  624  extends into copper crucible  626 , which is maintained in place by water-cooled jacket  625 . Vacuum enclosure  627  provides a mechanical feed through to ram  622  and acts in conjunction with crucible  626  to provide a vacuum environment within the enclosure formed by crucible  626  and vacuum enclosure  627 . 
     In operation, power supply  628  is connected to ram  622  and copper crucible  626 . Current driven by power supply  628  passes through ram  622 , stinger  623 , and electrode  624 , and produces a quantity of electric arcs  605  bridging electrode gap  634 , between electrode  624  and molten pool  632 . Molten pool  632  and solid ingot  633  are comprised of material from electrode  624  melted by the action of the quantity of electric arcs  605 . During the melting, the level of molten pool  632  and solid ingot  633  increases, and ram  622  is adjusted to maintain electrode gap  634 . 
     The location of the quantity of electric arcs  605  in electrode gap  634  during the melting process has significant importance. Because the quantity of electric arcs  605  is the primary source of heat in VAR  600 , an even distribution of arc positions versus time can minimize any propensity toward defect formation in solid ingot  633 . The ideal arc condition is that of diffuse arcs, achieved when the quantity of electric arcs  605  distributes evenly across electrode  624  surface on a time-averaged basis. However, assessing a diffuse arc condition is difficult during VAR  600  operation, because the quantity of electric arcs  605  is not visible. As a result, a real-time non-invasive measurement system that can provide a location of the quantity of electric arcs  605  during operation would be of great value. The novel apparatus described herein may be utilized in this capacity, such that the quantity of electric arcs  605  may be located in real time, and the operating parameters of VAR  600  may be adjusted in response to maintain the diffuse arc condition. 
       FIG. 7  illustrates the novel apparatus described herein operating on VAR  700 . VAR  700  is operating such that electric arc  705  bridges the electrode gap between the electrode  724  and molten pool  732  in response to power from power supply  728 . Sensing plane  703  has dimension on the x and y axes of the x-y-z axis shown, and extends between electrode  724  and molten pool  732  as illustrated. Field sensing location  704  is located on sensing plane  703 . 
     In the particular embodiment shown, power supply  728  provides DC power up to 30,000 Amps and 100 Volts. In a typical operation of VAR  700 , power supply  728  supplies 1200-4000 Amps at 20-30 Volts. 
     Electric current locator  716  includes current measurement device  717 . Current measurement device  717  determines at least the magnitude of current supplied from power supply  728 , and current measurement device  717  has current signal output  719  for transmission of a signal representative of the magnitude of current supplied from power supply  728 . 
     Electric current locator  716  further includes magnetic field sensor  715 . Magnetic field sensor  715  is a multi-axis Hall Effect sensor located at field sensing location  704  and oriented to detect magnetic fields in the x direction and y direction in sensing plane  703 . Magnetic field sensor  715  has first output  720  for transmission of a signal representative of the magnitude of the magnetic field sensed in the x direction, and magnetic field sensor  715  has second output  721  for transmission of a signal representative of the magnitude of the magnetic field sensed in the y direction. It is preferable that field sensing location  704  be located at a point where the gradient of the magnetic flux in the x and y direction is minimal over the respective dimensions of magnetic field sensor  715 . Additionally, because magnetic field sensor  715  responds to magnetic fields arising from all sources, it is advantageous if field sensing location  704  has a close proximity to VAR  700 , so that the influence of magnetic fields arising from background sources may be minimized. Once placed at field sensing location  704 , magnetic field sensor  715  is electronically zeroed with respect to magnetic fields arising outside of VAR  700  operation, such as the earth&#39;s magnetic field or other background magnetic fields expected in the VAR  700  operating environment. 
     Electric current locator  716  further includes processor  718 . Processor  718  is a computing device which receives first output  720  and second output  721  from magnetic field sensor  715 , and current signal output  719  from current measurement device  717 . Processor  718  solves Biot-Savart superposition equations having the form of equations (3) and (4) discussed infra in order to determine Θ i  and d i  and describe the location of electric arc  705  with respect to field sensing location  704 . In the particular embodiment shown, processor  718  is a data acquisition system and a computer, where the data acquisition system is a NATIONAL INSTRUMENTS data acquisition system capable of digitizing  32  analog inputs and having a maximum 100 MHz transfer rate from the inputs to the computer. 
     As discussed infra, the use of the electric current locator requires that the relative magnitudes of an input current and a current vector to be localized be known. With respect to electric current locator  716 , the input current arises from power supply  728  and is indicated by current measurement device  717 . The current vector to be localized is electric arc  705 . With regard to VAR  700 , the relative magnitude of the input current indicated by current measurement device  717  and the current through electric arc  705  is assumed as a constant value. 
     In a particular embodiment, the mathematical constants m 1 , m 2 , a and b of equations (3) and (4) were determined by finite element analysis of VAR  700  using a finite element software package, such as COMSOL MULTIPHYSICS. In the particular embodiment, the geometry of VAR  700  and electric current locator  716  was drawn with sensing plane  703  passing through the electrode gap. The entire model was meshed to create discrete finite elements, with a finer mesh region in sensing plane  703 . An even finer mesh was utilized for magnetic field sensor  715 . 
     The finite element analysis was conducted for a plurality of electric arc  705  locations in sensing plane  703  in order to determine expected values for the magnetic field sensed by magnetic field sensor  715  in the x and y directions. Each location in the plurality of electric arc  705  locations was described by the terms Θ and d with respect to field sensing location  704 . An input current from power supply  728  was simulated, the magnitude of the current passing through electric arc  705  was determined based on the input current, and the finite element network was manipulated to force electric arc  705  through each location in the plurality of locations. In a preferred environment, the finite element network was manipulated by modeling extremely low conductivity in the electrode gap surrounding a desired location of electric arc  705 , so that substantially all current in electrode  724  was forced through that desired location of electric arc  705  at the electrode gap. Seven discrete locations for electric arc  705  were simulated, and simulated magnetic field measurements in the x and y directions at field sensing location  704  were determined. Regression was conducted on the data to determine the resulting mathematical constants m 1 , m 2 , a and b for equations (3) and (4). 
     In another embodiment, the mathematical constants m 1 , m 2 , a and b of equations (3) and (4) were determined by physical manipulation of VAR  700  and measurement of the resulting magnetic fields at field sensing location  704 . In this particular embodiment, a physical short was placed within the electrode gap at a plurality of locations, operating current was supplied to VAR  700 , and the resulting magnetic field in the x direction and y direction was measured by magnetic field sensor  715 . Regression was then conducted on the recorded data to determine the resulting mathematical constants m 1 , m 2 , a and b for equations (3) and (4). 
     Having determined the mathematical constants m 1 , m 2 , a and b, electric current locator  716  can be utilized to provide, a real-time non-invasive measurement for location of electric arc  705  during VAR  700  operation. In this manner, the distribution of electric arc  705  on a time-averaged basis may be determined, and the operating parameters of VAR  700  may be adjusted in response to maintain a diffuse arc condition. 
     As discussed supra, it is understood that the validity of the mathematical constants m 1 , m 2 , a and b determined is dependent on the relative magnitude of the input current and the current vector to be sensed by electric current locator  716 . Those skilled in the art understand that if the mathematical constants are determined for a given scaling factor, where the scaling factor is the magnitude of the current vector divided by the magnitude of the input current, then electric current locator  716  will operate most accurately when the current sensed by current measurement device  717  and the current through electric arc  705  substantially express the same given scaling factor, and that accuracy can be expected to degrade as deviations away from the given scaling factor increase. Further, it is understood that the same concept applies for all electrical devices analyzed by the electrical current locator described herein. 
     Further, as discussed supra, it is understood that additional magnetic field sensors may be utilized around VAR  700 , and that the mathematical constants m 1 , m 2 , a and b may be determined for each of the magnetic field sensors as discussed herein, so that electric current locator  716  may provide location data for a quantity of current vectors in VAR  700 . For example, rather than modeling electric arc  705  as a single arc, it may be preferable to model the arcing across the electrode gap as being comprised of four separate arcs, and determine the location of those four separate arcs using an electric current locator  716  having at least four magnetic sensors. The electric current locator  716  may be utilized to localize any quantity of arcs across the electrode gap of VAR  700 , provided that the quantity of magnetic field sensors is at least equal to the quantity of arcs localized. 
     Thus, the disclosure herein provides an electric current locator 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, the disclosure herein provides an electric current locator whereby the exact solution for the location of the quantity of current vectors does not rely on restriction of the current vector to well defined current paths within the electrical device. 
     Further, the disclosure herein provides an electric current locator whereby the exact solution for the location of the quantity of current vectors may be determined in an electrical device that experiences poorly defined current patterns over the course of normal operation. 
     Further, the disclosure herein provides an electric current locator 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. 
     Having described the basic concept of the invention, it will be apparent to those skilled in the art that the foregoing detailed disclosure is intended to be presented by way of example only, and is not limiting. Various alterations, improvements, and modifications are intended to be suggested and are within the scope and spirit of the present invention. Additionally, the recited order of elements or sequences, or the use of numbers, letters, or other designations therefore, is not intended to limit the claimed processes to any order except as may be specified in the claims. Accordingly, the invention is limited only by the following claims and equivalents thereto. 
     All publications and patent documents cited in this application are incorporated by reference in their entirety for purposes to the same extent as if each individual publication or patent document were so individually denoted.