Analytical Force and Flux for a 1-D Electromagnetic Vibration Energy Harvester

Abstract

A Vibration Energy Harvester (VEH) converts the kinetic energy of a moving source into electrical energy. Here we consider a one-dimensional electromagnetic vibration energy harvester (1D-EMVEH) which consists of three coaxial cylindrical permanent magnets enclosed in a tube, such that the middle magnet is levitating. The resulting movement of the middle magnet can then induce an electromotive force (emf) in one or more surrounding coils. Using an analytical model, we derive expressions for the 1D-EMVEHs characteristic frequency and output power by using Fourier space approach. First, the magnetostatic energy of the system as a function of the position of the levitating magnet is calculated. Its spatial gradient gives the force acting on a magnet, which drives its dynamics. Next, more accurate magnetic flux and emf expressions are obtained. The results are compared with experimental measurements, revealing an excellent agreement.

The full phase space dynamics of a magnetically levitated electromagnetic vibration harvester

“…We have implicitly assumed that the magnetization is uniform within each magnet and independent of the magnetic field experienced by the magnets. However, we emphasize that this procedure for computing the force expression has been experimentally validated 16 , and that these two assumptions have been verified to be realistic enough for the system under consideration.…”

“…In this section we consider the interaction between the floating magnet and the two fixed magnets. The force between two coaxial cylindrical uniformly-magnetized magnets has previously been obtained through a Fourier approach 16 – 19 . This approach consists of employing the Fourier transform to evaluate over the reciprocal space the integral of the dipole-dipole interaction energy density between the magnets using a specific shape function.…”

“…This approach consists of employing the Fourier transform to evaluate over the reciprocal space the integral of the dipole-dipole interaction energy density between the magnets using a specific shape function. The force between one of the fixed magnets and the floating magnet is directed along the z -axis and is given by 16 Here and are the remanent magnetisations of the floating and fixed magnet respectively, is the relative distance between the floating and fixed magnet and is the dipolar coupling integral: Here is the Bessel function of first kind, q is the reciprocal space variable, and are the radii of the magnets, and and are the height-to-radius ratios of the magnets, . This force expression is similar to other expressions presented in Ref.…”

“…By using again the principle of Fourier transform, an experimentally verified analytical expression for the flux through a single turn of a coil (i.e. a ring) can be obtained 16 . It should be noted that two different expressions appear depending on whether the magnet is inside or outside the ring, where in the former case, the magnetisation, M , contributes to the flux, , in addition to the demagnetising field H .…”

“…Our hypothesis is that by understanding this influence, the harvested power and the frequency harvesting range can be optimized, which is needed for EMVEHs to be truly useful for a broad range of applications. To investigate this, we here present an analysis of an EMVEH system using a numerical model based on experimentally validated expressions for the magnet-magnet restoring force and the induced electromotive force 16 . We explore the full dynamics of the system to investigate how the initial conditions of the floating magnet influence the harvested power and compare this to the standard approach of approximating the magnet-magnet force with a polynomial expression.…”