• M.J. Hagmann
  NewPath, Salt Lake City, Utah

We have shown that a group of sinusoidally-wound coaxial toroids can be used to determine the transverse distribution of a time-dependent current that passes through their common aperture. The current is expressed in a basis of chapeau (pulse) functions over an array of pixels, and matrix methods are used to determine the current in each pixel from measurements of the voltages that are induced on the toroids. Optimum configurations of pixels are used, for which the condition number of the matrix is bounded by the number of pixels. For example, with a resolution of 50 pixels, the fractional errors in determining the current at each pixel are approximately 50 times the fractional errors in the measurements of the induced voltages as well as imperfections in the fabrication of the toroids and their placement. Our algorithms were tested numerically by specifying the currents, calculating the voltages that would be induced on the toroids, adding Gaussian noise to these voltages, and then using the algorithms to calculate the currents from the simulated voltage measurements. These simulations confirm that the condition number of the matrix is bounded by the number of pixels.

• M.J. Hagmann
  NewPath Research L.L.C., Salt Lake City
• M.J. Hagmann
  NewPath, Salt Lake City, Utah

We have shown that a group of sinusoidally-wound non-ferrous coaxial toroids can be used to determine the transverse distribution of a time-dependent current that passes through their common aperture. A single current filament requires one uniformly-wound toroid, and two others having turn densities proportional to the sine and cosine of the azimuthal coordinate. Three simple algebraic equations give the magnitude and phase of the current and its position in terms of the voltages induced on the three toroids, and there is no ill-conditioning. Two current filaments require two additional toroids with turn densities proportional to the sine and cosine of two times the azimuthal coordinate, and the solution may be obtained by using steepest descent to minimize the residuals. Ill-conditioning makes it impractical to use more than two currents. We have tested our algorithms numerically by specifying the magnitudes and phases of the currents and their locations, calculating the voltages that would be induced on the toroids, adding Gaussian noise to these voltages, and then using the algorithms to calculate the currents and their locations from the simulated voltage measurements.

• M.J. Hagmann
  NewPath Research L.L.C., Salt Lake City
• M.J. Hagmann
  NewPath, Salt Lake City, Utah

We have shown that a group of sinusoidally-wound non-ferrous coaxial toroids can be used to determine the transverse distribution of a time-dependent current that passes through their common aperture. A single current filament requires one uniformly-wound toroid, and two others having turn densities proportional to the sine and cosine of the azimuthal coordinate. Three simple algebraic equations give the magnitude and phase of the current and its position in terms of the voltages induced on the three toroids, and there is no ill-conditioning. Two current filaments require two additional toroids with turn densities proportional to the sine and cosine of two times the azimuthal coordinate, and the solution may be obtained by using steepest descent to minimize the residuals. Ill-conditioning makes it impractical to use more than two currents. We have tested our algorithms numerically by specifying the magnitudes and phases of the currents and their locations, calculating the voltages that would be induced on the toroids, adding Gaussian noise to these voltages, and then using the algorithms to calculate the currents and their locations from the simulated voltage measurements.