# Theoretical Model

The mass of the hard magnet pair, the stiffness of the supporting spring, and the magnetostatic coupling between the solenoids and the hard magnet pair determine the resonant vibration frequency and the output voltage of the energy harvester. The equivalent spring-mass system becomes a nonlinear oscillation system due to the magnetostatic coupling between the solenoids and the hard magnet pair. This nonlinear effect can be explained from the potential energy point of view, as shown in the previous section. The magnetostatic potential energy has two identical minimum values due to the coupling between the magnets and solenoids. These minimums occur when the magnets move a short distance up or down from the equilibrium position in the middle of the hard magnet pair. As a result, the
superposition of two different types of potential energies result in a nonlinear total potential, leading to a wider oscillation frequency range, as has been shown in Fig. 18.11.

As explained in the previous section, since the magnetic field magnitude varies along the solenoid axis, the open circuit voltage can be calculated by an integration through the solenoid.

V = 2d'(t) = 2dR{H[x, y(t) + 4uM[x, y(x, t)]}- A • dN

dt _ 2dR4*M[x, y(x, t)] • d4 • dN ’ (18.8)

dt

where A is the total cross-sectional area of the multilayer cores, dN is the number of loops in the infinitesimal unit length of the solenoid, and

dN = Nl • dx /dw

NL is the number of loop layers and dw is the copper wire diameter. Hence, the maximum output power, which happens when the load impedance equals the conjugate of the output impedance of the solenoid coil, is

where S is the number of layers in each core and A’ is the cross-section area of one layer. Equation (18.10) indicates that the output power increases as the resonance frequency increases, if all other parameters are kept constant. Moreover, at a particular frequency, the output power depends on the total magnetic flux change in the solenoid in one oscillation period, which is directly related to the permeability of the magnetic cores. The solenoid with a soft magnetic MuShield core, which has a high permeability, has a great potential for generating a high voltage output. Moreover, the multilayer structure of MuShield material is expected to generate a much larger flux change than a single layer. The application of a single layer was discussed in the earlier section.