ISSN:
1089-7550
Source:
AIP Digital Archive
Topics:
Physics
Notes:
Exact analytical solutions for the surface potential due to a current source are available only for special volume conductors. Here we derive approximate analytical solutions exploiting the fact that for the upper convexity of the head corrections to the spherical approximation are small compared to its radius. First we approximate the real surface, defined in terms of an angular dependent "radius," by a finite sum of spherical harmonics, regarding everything but the zeroth component (the sphere) as a small perturbation. Inserting this formulation into the standard surface integral equation allows us to analytically construct and invert the integral operator as a Taylor expansion with respect to the perturbation. Remarkably, for finite order of perturbation the integral operator and its inverse, expressed as a matrix in the basis of spherical harmonics, is sparse. Furthermore, without loss of generality the solution due to a current dipole can be expressed as a set of sums over a single index. Explicit formulas and examples will be presented for one shell, for an approximation of the surface up to second order of spherical harmonics, and up to first order of the perturbation. By comparing the results to the exact solution for a prolate spheroid we estimate the performance for realistic deformations. © 1999 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.371128
