Estimation of charge distribution on a bulky solid dielectric using regularization technique

We have been studying a multi‐point charge measurement method using an electrostatic probe. In this technique, charge densities x must be estimated from the probe outputs b by an inverse calculation based on an equation A x = b . The matrix A is obtained by applying a numerical field calculation technique. When the matrix A is in ill‐condition, the solution often makes no sense, including extremely large errors. Therefore, we apply the regularized least squares method (RLS) with a penalty term to perform the inverse calculation stably for the ill‐conditioned matrix. The penalty term imposes some constraints on the solutions. In this paper, first, we have analyzed the accuracy of the charge distribution estimated by the inverse calculation. Although the perturbation bound of the solution errors has already been proposed for the least squares method, it has not yet been given for the RLS. We have derived the equations that express the perturbation bound of the solution errors in applying the RLS to evaluate the estimation accuracy. Second, we have applied the above equations to an experimental result for a cylindrical dielectric solid, and estimated the charge distribution represented by 10,140 unknowns. We have utilized an iteration technique and the symmetric configuration of the measured arrangement so as to reduce the amount of operations and memory capacity required for the inverse calculation. © 2008 Wiley Periodicals, Inc. Electr Eng Jpn, 163(2): 1–13, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20412