Experimental validation of a new magnetic method applied to diagnose a low voltage breaker arc

In this paper, we present an original method, which allows reconstructing the current density in a low voltage breaker arc. This method is based on the deconvolution of magnetic induction measurements and a function depending on the experimental device geometry. For this study, we propose to use an iterative deconvolution method using minimisation by gradient.


Introduction
Manufacturers of circuit breakers need to know the arc's size, current density, and whether it is diffuse or concentrated, because the mode of the arc (diffuse or concentrated) influences the erosion of the contacts [1][2].The size of arc can be characterized by optical or magnetic methods.The optical methods give an image of the arc size [3] but their use requires the introduction of transparent walls, which can influence the physicalchemical properties of the arc.Bouvier [4] proposed a magnetic method to determine the current density in plasma.Although this method is not disruptive, it presents the inconvenience to suppose that the arc moves with a constant speed; that is not always realistic.In this paper, we propose a non-intrusive magnetic method based on deconvolution of magnetic signals and a function depending on the experimental device geometry.At a given moment, it reconstructs the current density, going into a low voltage breaker arc, allowing us to determine the arc size, to characterize its mode and to study its dynamic.A validation of the method is experimentally obtained using the reconstruction of the current density flowing in a stationary conductor.

Experimental Device
The experimental device used in this study is already presented in [5][6].It is constituted of (figure 1):  breaking device,  power source,  64 Hall Effect sensors placed on both sides of the moving electric arc.These are used to measure magnetic induction along the Oz direction,  four multiplexing cards which allow conditioning and switching of measurements,  Acquisition system to store and treat measurements.

Principle of the method
The method consists of reconstructing the current density distribution along the Oz direction in low voltage breaker arc.For this purpose, we use a set of vertical measurements of the magnetic induction, which occur on both sides of the electric arc.In order to simplify the mathematical form, the column of the arc is modeled as a rectangular current sheet, composed of a mix of threadlike currents segments; the other elements of the breaking device are compared to rectilinear threadlike segment.This assumption is justified by the fact that the magnetic sensors are placed at a quite distance from the breaking device, and it was experimentally confirmed in earlier work [3,7].Initially we suppose that the electric arc is a rectilinear current segment placed in the xOz plane at the mark Z (figure 2).Crossed by a current I, this circuit creates a magnetic induction at a point S(0,ys,z), located at a distance ys in the xOz plane.See figure 3. The component bzt of this induction in the Oz direction results from passage of the current I in a circuit consisting of the two current feeds, noted "circuit a", and the electric arc of variable geometry noted "circuit b".The vertical electrodes do not have a magnetic contribution along the Oz direction.
Now, let us consider that the arc is composed of N parallel and rectilinear segments.Each segment placed at mark Z j is crossed by a current i(Z j ).In this case b zt is equal to .
So the induction created by the electric arc is a convolution product of the function g(z) by the current i(z).The function g(z) represents the magnetic induction of circuit 2 when the current segment is at Z = 0 and crossed by a current of 1A.Using the usual deconvolution methods, the current density i(z) may be found.Magnetic induction is measured in the Oz direction using a step of 1.5 mm.In Equation ( 4) the variable z can be replaced by the variable k, with k=z/1.5 and K=Z/1.5.The following relation is obtained: where (k) b zb is the magnetic induction created by the arc, (k) b zt is the total magnetic induction created by the breaker device and measured by the sensor which is placed at z = kx1.5 mm, and I is the total current.The relation (5) represents the total magnetic induction created by N current segments at one point of the Oz axis.For N points the following matrix is obtained: where zb B is the magnetic induction vector, and I is the current density vector, which is to be determined.The calculation of the current density i(k) consists of solving the system (6).The matrix G is a Toeplitz's matrix, and all of its elements are located on parallels at the diagonal and all are identical.Moreover G is a symmetrical and square matrix.The conditioning factor of this matrix, defined by: x ) Cond( is much higher than 1.Thus the matrix G is ill-conditioned.Due to the noisy data, a direct calculation of the matrix inverse 1 G  will lead to problems linked to the instability of the solutions.To solve this system we need to use optimization methods, such as iterative method based on the minimization by gradient [8][9].The principle of this method starts with an estimate 0 I of the vector I to determine the successive vectors 1 I , 2 I etc.The starting vector 0 I presents a uniform distribution of the current divided into N elements and covers the interelectrode space.Each element represents a current segment crossed by a current equal to I N , where I is the total current.The solution of the system requires that the criterion [8]: be minimal.

Estimation of the arc size and position
We seek to compute (an approximation to) the true plate position from the reconstructed data.The problem can be expressed as follows: Given data values on an interval from   L 0, , find the step function that best approximates the data with the following constraints on the step function: 1) It is zero on  

Fig. 4. Estimation of the plate position and width
In this work, we defined the "best" approximation in a natural way, that is, as the best in a least squares sense.I.e., we minimize the sum of squares of the differences between the step function and the data.Actually, for any given values of a Z and b Z , we can find * i immediately.
Since we are seeking the least squares solution, * i must be the average value of the data over the interval.This is an elementary computation, and it is documented in [10], which shows that the least squares line (even a constant one like we have here) passes through the centroid of the data.So the least squares search is now reduced to a search over only two variables, a Z and b Z .

Experimental validation of the method
Before studying the electric arc, we have decided to test the method by reconstructing the current density in stationary conducting plates.These plates are welded in a known position between two vertical electrodes.
In the following we present three typical tests.This allows studying the validity of the method for concentrated arc mode.Test 3: using two conducting Plates: This test studies two current sheets, which are traversed by different value currents, and simulates the re-strike phenomena in a lowvoltage breaking device.We placed two plates in parallel, each one having a width of  = 8 mm.The first one is made of stainless steel and is welded between the heights Z a1 = 24 mm and Z b1 = 32 mm, and the other one is made of copper and is welded between the heights Z a2 = 54 mm and Z b2 = 62 mm.The maximum value of the total current I is 1300 A.

Reconstructed current densities
Using the 64 effect Hall sensor, we have measured magnetic induction created by the conductor plates for the three tests, using the iterative deconvolution method we obtain the reconstructed current densities.Figures 5,  6, and 7, respectively, display the reconstructed current density and the theoretical current density for each test.It should be noted that the theoretical current density is assumed to have a uniform current distribution in the plate.Its value is equal to the ratio between the total current I, at a given time, and the plate width .

Plate position and size
Using the estimation method presented in paragraph 2.4 and the reconstructed current densities, the size and the position of the plates, for each test are evaluated.Table 1 regroups these values.

Uncertainties analysis
In order to discuss the result and to study the method precision, we evaluated, for each test, the relative uncertainties of the position and the size on the plate and value of the current density.

Precision on the value of the current density:
The precision on the current density in the plate is calculated with the following relation:

Precision of the plate position and size:
The estimated values of the plate positions and sizes are then compared to the true values of the position and the size of the plates.The differences between the estimated value and the true ones present the relative uncertainties (Z a , Z b ) on the plates position, and () on the plate size.The results are presented in Table 3. Table 3. Precision on the plate size and position.

Discussion
The reconstructed data show the position, the size and the current density values of the conductor with reasonable accuracy.However (figure 5, 6 and 7) the data fluctuate in space with a magnitude up to 10%, resulting in identification non-zero current densities outside the region where electric current is actually flowing, this is probably result from measurement uncertainties.In practice there are two sources of measurement uncertainty can be inherent to the magnetic induction measurements and these are: the uncertainty in knowing the position where the magnetic induction is measured, and the uncertainty in the magnetic induction measurement itself.It was shown in [11] that these two errors affect strongly the uncertainty of the arc geometry and position reconstruction.Note that the method is valid under the assumption that the current density is constant through the plate along the direction of the electric current, and has a rectangular shape.Departures from these two ideal states may result in error in the result.The result can be improved by using others deconvolution method such as wavelet transform method.

Fig. 2 .
Fig. 2. A simplified electrical circuit for the breaking device.

Fig. 3 .
Fig.3.An equivalent diagram of the figure 2 circuit.Thus b zt (z) = b za (z) + b zb (z) where b za (z) is the induction created by "circuit a" along Oz direction and b zb (z) is the induction created by "circuit b" along the Oz direction.These two inductions are calculated from Biot-Savart's law, for the calculation details see [5-6], b zb depends only on the relative position z-Z of the arc with the respect to the measurement point S; and b za depends only on the position of the point S.
4).This means that finding the best approximating step function is reduced to the problem of finding the three values, the best-fitting step function.

Test 1 :
using a wide conducting Plate: A copper plate with a width of  = 15 mm is welded to the two electrodes between the heights Z a = 57 mm and Z b = 72 mm.The electrode spacing is equal to L = 40 mm (figure 8).A current from the power source of maximum value 940A flows through the plate.The wide plate simulates the diffuse mode of the arc.Test 2: using a narrow conducting plate: This second test was carried out with a narrow copper plate with a width of  = 7.5 mm, welded to the electrodes between the heights Z a = 60 mm and Z b = 67.5 mm.It was traversed by a current of maximum value I = 1015 A.

Fig. 7 .
Fig.7.Results for the two current sheets.

Table 1 .
) Plates size and position resulting.

Table 2 .
Precision on the current density value.