A simple and precise formula for magnetic forces in nonlinear piezoelectric energy harvesting

The traditional linear piezoelectric energy harvesters (PEHs) suffer from a narrow operating bandwidth. To achieve broadband energy harvesting and to improve the energy harvested efficiency, researchers have shown an increased interest in nonlinear PEHs. One of the most straightforward methods for achieving this purpose is to introduce a nonlinear magnetic interaction. Many works have been conducted on the performance of nonlinear PEHs and related theories. However, the nonlinear magnetic force is only an approximation in the previous theoretical analysis. It is still different from the results obtained through accurate numerical modeling or testing. To reduce this error, two high-precision magnetic force formulas for cylindrical and cuboidal magnets are proposed in this work. These formulas are based on the previous magnetic formula by introducing two correction parameters. The correction process is very simple, but the relative error of the magnetic force can be less than 1%. Based on the magnetic field theory, we also give explicit expressions for the correction parameters.