Revisiting the Bloch equation through averaging

A novel approach to finding an approximate analytic solution to the Bloch equation is developed in this paper. The method is based on time scaling and averaging of the Bloch equation after transformation to a rotating frame of reference. In order to accomplish the scaling, a novel time scaled magnetisation vector is introduced. The resultant time scaled system is subsequently approximated through averaging, a technique that to the best of our knowledge, has not previously been applied in the nuclear magnetic resonance context. Our proposed method of approximating the solution to the Bloch equation is valid for continuous wave excitation as well as the traditional pulse excitation with an arbitrary envelope, making this a widely applicable technique unlike previously proposed methods. Comparison of the approximate analytic solution and simulation results clearly indicates that the error is negligible when the field inhomogeneities are small compared to the excitation field amplitude. Extremum seeking techniques may be applied to determine the optimal excitation, given the form of the approximate solution. This result is applicable to a range of research areas including nuclear magnetic resonance, magnetic resonance imaging and optical resonance problems.