The magnetic inverse problem for two stacked layers of sources

TL;DR

The paper tackles reconstructing electronic current densities in a two-layer stack from magnetic-field measurements taken on planes above and below the stack. It develops a Fourier-domain framework using the Biot-Savart law and a continuation matrix to enable exact, non-invasive recovery of layer currents, demonstrated via numerical simulation. The approach highlights potential applications in battery cells and other layered devices, and sets the stage for generalizing to more layers with regularization or invasive measurement schemes. Overall, it provides a principled method for multi-layer current imaging with non-destructive diagnostic potential.

Abstract

We present calculations that reconstruct electronic current densities in two stacked layers at known depths, using magnetic field data. Solving this inverse problem requires knowledge of the magnetic field in two planes -- one above both current layers, one below -- corresponding to non-invasive measurements of the field. We corroborate the accuracy of current density reconstruction from the resulting system of equations using a numerical simulation. This method is anticipated to be applicable to non-destructive current imaging for quality assurance in a range of applications featuring two-layer geometries, including printed circuit boards, capacitors, fuel cells, and battery cells; we focus particularly here on battery cells, due to their rapidly increasing relevance for automotive applications. This method also offers a framework for generalising the model to more than two layers in future work.

Figures (2)

• Figure 1: A schematic of the problem, featuring two current layers, $\mathrm{S}_{1,2}$, respectively of thickness $\delta_{1,2}$, and two measurement planes, $\mathrm{M}_{1,2}$, all parallel with the $x$-$y$-plane and stacked in the $z$-direction. The separations between the current layers and measurement planes are measured with respect to the middle (in $z$) of each current layer. In this figure, $\delta_{2}>\delta_{1}$, but this is only for the sake of example and to emphasise that they need not be equal. It is implicit that $\delta_{1,2}<\Delta_{\mathrm{S}}$, but no other constraints are placed on $\delta_{1,2}$ within the model. All layers and planes are considered to be infinitely extended in $x$ and $y$, prior to performing spatial filtering on the problem. However, non-zero current densities will only be localised to small portions of $\mathrm{S}_{1,2}$.
• Figure 2: Results from a numerical simulation, in MATLAB, of the reconstruction of current densities in two layers, from magnetic flux densities in two different measurement planes -- one above the current stack and one below. (a) and (b) show the original $y$-components of the current densities, $J_{y, \mathrm{S_{1,2}}}$, in layers $\mathrm{S}_{1,2}$. (c) and (d) show the resultant $x$-components of the magnetic flux densities, $B_{x, \mathrm{M_{1,2}}}$, in planes $\mathrm{M}_{1,2}$; the contributions from the two layers partially overlap. (e) and (f) show the reconstructed $J_{y, \mathrm{S_{1,2}}}$, calculated by taking fast Fourier transforms (FFTs) of $B_{x, \mathrm{M_{1,2}}}$, then using \ref{['eq:j_y']}, before taking inverse FFTs of the result. Blackman-Harris and Hann functions were used to mitigate windowing effects caused by the FFTs used to implement this method, although some small aberrations remain, which increase the range of values for the reconstructed current densities. The currents (all in the $y$-direction) through $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ were $-0.8$ A and 0.7 A, respectively; the respective widths of the layers (in the $x$-direction) were 80 mm and 120 mm. The other parameters were: $\Delta_{1} = 14$ mm, $\Delta_{2} = 13$ mm, $\Delta_{\mathrm{S}} = 11$ mm, $\delta_{1} = 1$ mm, and $\delta_{2} = 1.2$ mm.