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This paper presents a novel tunable multi-frequency hybrid energy harvester (HEH). It consists of a piezoelectric energy harvester (PEH) and an electromagnetic energy harvester (EMEH), which are coupled with magnetic interaction. An electromechanical coupling model was developed and numerically simulated. The effects of magnetic force, mass ratio, stiffness ratio, and mechanical damping ratios on the output power were investigated. A prototype was fabricated and characterized by experiments. The measured first peak power increases by 16.7% and 833.3% compared with that of the multi-frequency EMEH and the multi-frequency PEH, respectively. It is 2.36 times more than the combined output power of the linear PEH and linear EMEH at 22.6 Hz. The half-power bandwidth for the first peak power is also broadened. Numerical results agree well with the experimental data. It is indicated that magnetic interaction can tune the resonant frequencies. Both magnetic coupling configuration and hybrid conversion mechanism contribute to enhancing the output power and widening the operation bandwidth. The magnitude and direction of magnetic force have significant effects on the performance of the HEH. This proposed HEH is an effective approach to improve the generating performance of the micro-scale energy harvesting devices in low-frequency range.

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The 2-D ballooning transform, devised to study local high toroidal number (n) fluctuations in axisymmetric toroidal system (like tokamaks), yields a well-defined partial differential equation for the linear eigenmodes. In this paper, such a ballooning equation of the second kind is set up for ion temperature gradient driven modes pertinent to a 2-D non-dissipative fluid plasma; the resulting partial differential equation is numerically solved, to calculate the global eigenvalues, and the 2-D mode structure is presented graphically along with analytical companions. The radial localization of the mode results from translational symmetry breaking for growing modes and is a vivid manifestation of spontaneous symmetry breaking in tokamak physics. The eigenmode, poloidally ballooned at # ¼ 6p=2, is radially shifted from associated rational surface. The global eigenvalue is found to be very close to the value obtained in 1-D parameterized (k ¼ Çp=2) case. The 2-D eigenmode theory is applied to estimate the toroidal seed Reynolds stress [Y. Z. Zhang, Nucl. Fusion Plasma Phys. 30, 193 (2010)]. The solution obtained from the relatively simplified ballooning theory is compared to the solution of the basic equation in original coordinate system (evaluated via FFTs); the agreement is rather good.

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