Magnetic circuit modeling of chaotic vibration energy harvester

Abstract

The chaotic vibration energy harvester (CVEH) converts vibration of constructions to electric energy in wide frequency. It needs large computational cost for evaluation of the magnetic force between the moving magnets and magnetic material embedded in the stator coils of CVEH when using finite element method (FEM). This paper presents magnetic circuit modeling of CVEH to reduce computational burden. It is shown that the frequency responses of CVEH computed by the present method is in good agreement with those obtained by FEM. This result is validated through the experiment.

Keywords

Vibration energy harvester magnetic circuit chaotic oscillation genetic algorithm

1. Introduction

The vibration energy harvester (VEH) which converts vibration of constructions into electric energy. It has attained great attentions for the energy sources of low-powered wireless sensor devices composed of sensors, interface and communication ICs [1,2]. A conventional VEH works only at around the resonance frequencies, whereas the actual vibration can have wide frequency spectra. To overcome this difficulty, we have proposed the chaotic VEH (CVEH) which works in wider frequency ranges [3,4]. The potential energy of CVEH relevant to the magnetic force between the permanent magnets and the silicon steel plates embedded in the coil has double wells along the direction of displacement. The nonlinearity in the magnetic force with respect to the displacement enables wide-frequency operations. Moreover, when the design parameters are carefully tuned, it is found in numerical results and experiments that chaotic vibrations emerge so that the frequency band for operation is further broadened.

In design of CVEH, the magnetic force has been evaluated using three-dimensional finite element method (FEM) [5]. However, FEM which involves mesh generation for consideration of movement of the oscillator needs large computational cost. In this paper, we propose magnetic circuit modeling of CVEH to reduce computational burden. The circuit parameters, some of which depend on the displacement, are identified so that the magnetic potential is as close to that computed by FEM as possible. Once the circuit parameters are identified, we can easily compute the magnetic potential of CVEH for different design parameters using the magnetic circuit without time-consuming FEM. It will be shown that the chaotic motion of CVEH can be reproduced when using the present magnetic circuit.

2. Magnetic circuit modeling of CVEH

Figure 1 shows CVEH which consists of NdFeB magnets and coils [4]. The magnets are fixed to a cantilever which is oscillated by ambient vibrations. As a result, the magnetic flux across the coils temporally changes so that electromotive force is induced in the coils. The pieces of silicon steel are inserted into the bobbin hole of the coil so as to increase the magnetic flux interlinkage. The nonlinear magnetic force with respect to the displacement between the magnets and silicon steel in the bobbins gives rise to nonlinear oscillation of the VEH. This nonlinear oscillation makes it possible for CVEH to realize off-resonant operations. The double coils are introduced so that the magnetic flux forms a closed loop at the two extremes of the oscillation.

Fig. 1.

CVEH model.

2.1. Equation of motion of the CVEH

The equation of motion for CVEH is expressed as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle m\ddot{z}(t)+{\gamma}{\dot{z}}(t)+\frac{dE(z)}{dz}=-m\ddot{y} (t)\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle E(z)=E_{\text{mag}}(z)+\frac{1}{2}kz(t)^{2}\end{eqnarray}$ (2) where m, z, y, 𝛾, E and k are the oscillator mass, relative displacement between the coil axis and magnets, input acceleration, damping coefficient, potential energy and spring constant, respectively [6]. Moreover, E _mag represents the magnetic energy which is computed from either magnetic circuit model or FEM.

Fig. 2.

Cross sectional view of CVEH.

Fig. 3.

Magnetic circuit model of CVEH.

2.2. Equivalent magnetic circuit of CVEH

Figure 3 shows the proposed equivalent magnetic circuit of CVEH shown in Fig. 2 in which F, R _m and R _a represent the magnetic force, magnetic resistances of the silicon steel and air, respectively. The magnetic resistances of the air gap R _ai, 3 ≤ i ≤ 6, which depend on the oscillator displacement z, are expressed as $\begin{eqnarray}\displaystyle R & = & \displaystyle \frac{l_{i}}{{\mu}S}\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle l_{i} & = & \displaystyle \left\{\begin{array}{@{}lc@{}}\sqrt{l_{\text{gap}}^{2}+(l_{\text{m}i}-z)^{2}}, & \quad (i=3,5),\\[7.5pt] \sqrt{l_{\text{gap}}^{2}+(l_{\text{m}i}+z)^{2}}, & \quad (i=4,6)\end{array}\right.\end{eqnarray}$ (4) where 𝜇 and S denote the permeability of the core and the cross-sectional area of magnetic flux path. By determining the magnetic flux 𝛷 in the magnetic circuit, E _mag is computed from $\begin{eqnarray}\displaystyle E_{\text{mag}}(z)=\mathop{\sum }_{i=1}^{4}\frac{1}{2}R_{\text{m}i}{\Phi}_{\text{m}i}^{2}+\mathop{\sum }_{i=1}^{6}\frac{1}{2}R_{\text{a}i}{\Phi}_{\text{a}i}^{2} & & \displaystyle\end{eqnarray}$ (5) which is inserted into (1) for dynamic analysis of CVEH. The parameters in magnetic circuit, F, R _mi and R _ai are determined so that E _mag computed from the magnetic circuit is as close to that obtained by FEM as possible. For this purpose, we solve the optimization problem defined by $\begin{eqnarray}\displaystyle \min F=\mathop{\sum }_{i}\left|\frac{E_{i}^{\text{FEM}}-E_{i}^{\text{MC}}}{E_{i}^{\text{MC}}}\right| & & \displaystyle\end{eqnarray}$ (6) using genetic algorithm (GA) where E _i ^FEM and E _i ^MC are the potential energy E computed form FEM and the equivalent magnetic circuit at the i-th sampling point. The proposed structure of the equivalent circuit model in Fig. 3 will be verified by comparing the potential energy computed from it with that obtained from FEM in Section 3.

Fig. 4.

Magnetic paths in air gap.

Fig. 5.

Magnetic circuit model of air gap.

2.3. Improvement of accuracy in magnetic circuit model

The magnetic circuit should give accurate results for different air gaps because they are tuned so that CVEH provides the maximum output for the given spectra of ambient vibration. It is found, however, that the magnetic potentials computed from the magnetic circuit with the variable magnetic resistances of the air gap R _ai given by (3) and (4) result in significant errors when the gap length changes. To improve the accuracy, the expression of R _ai is modified as in the following way.

When determining the circuit parameters, the leakage magnetic flux dependent on the mutual displacement has to be taken into account. To do so, we consider the three different magnetic paths $l_{i}^{k}$ , k = 1, 2, 3 in the air gap as shown in Figs 4 and 5, where $l_{i}^{1}$ is the mean distance between the magnet and the silicon steel in the coil axis.

When the air gap between the coil axis and magnets increases, the magnetic flux in the air gap extends, which leads to increase in the cross-sectional area of the magnetic flux. Thus, the spatial distribution of the magnetic flux in the air gap is considered. From Eq. (3), it is clear that the value of the magnetic resistance is inversely proportional to the cross-sectional area S. This effect is considered in the evaluation of R _ai.

From (i) and (ii), R _ai, 3 ≤ i ≤ 6 are now expressed by $\begin{eqnarray}R_{\text{a}i}=\frac{R_{\text{a}i}^{1}R_{\text{a}i}^{2}R_{\text{a}i}^{3}}{R_{\text{a}i}^{1}R_{\text{a}i}^{2}+R_{\text{a}i}^{2}R_{\text{a}i}^{3}+R_{\text{a}i}^{3}R_{\text{a}i}^{1}}\times \left\{1+c\left(\frac{l_{\text{gap}}-l_{\text{gap}}^{\text{min}}}{l_{\text{gap}}^{\text{min}}}\right)\right\}^{-1}\end{eqnarray}$ (7) where c is a constant relevant to the dependence of the magnetic resistance on l _gap. The minimum value of the flux path lengths of the air gap $l_{\text{gap}}^{\text{min}}$ is set to 1 mm. In the identification of the circuit parameters, l _gap is assumed to be 1 mm and 1.6 mm The optimization problem is modified to $\begin{eqnarray}\displaystyle \min F=\mathop{\sum }_{\stackrel{i}{l_{\text{gap}}=1∼\text{mm}}}\left|\frac{E_{i}^{\text{FEM}}-E_{i}^{\text{MC}}}{E_{i}^{\text{MC}}}\right|+\mathop{\sum }_{\stackrel{i}{l_{\text{gap}}=1.6∼\text{mm}}}\left|\frac{E_{i}^{\text{FEM}}-E_{i}^{\text{MC}}}{E_{i}^{\text{MC}}}\right|. & & \displaystyle\end{eqnarray}$ (8)

There are deviations in the resultant circuit parameters obtained by solving (8) when starting from different random seeds in GA. We choose the best solution among them. The convergence history to obtain the best solution is shown in Fig. 6.

Fig. 6.

Convergence history of GA.

3. Analysis of the CVEH

3.1. Evaluation of potential energy

We compute E _mag using the proposed magnetic circuit and FEM. In this analysis, we set damping coefficient k to 1000 N∕m. Figure 7 shows the profiles of the resultant potential energy E (z) for different values of l _gap, where conventional MC and proposed MC represent the results obtained based on (6) and (8), respectively. As we can see in Fig. 7, the potential profiles obtained by the proposed magnetic circuit are in good agreement with those obtained by FEM. It is remarkable that the proposed method is valid even when l _gap changes between the two extremes at which the circuit parameters are identified. It would be possible, therefore, to accurately simulate the dynamics of CVEH with small computational cost using the proposed magnetic circuit. To compute these potential profiles, FEM analysis requires a few hours, whereas the magnetic circuit requires a few seconds.

Fig. 7.

Potential energy profiles.

3.2. Frequency response

We employ the Newton Raphson method to solve (1) to compute the oscillator velocity ${\dot{z}}(t)$ . In this analysis, m, k, 𝛾, l _gap and y(t) are set to 17.5 g, 0.1, 1200 N∕m, 1.0 mm and Asin(ωt), respectively. The input acceleration A is set to 0.1 G and 1.5 G, where G is gravitational acceleration. The frequency characteristics of CVEH obtained by the proposed method and FEM are shown in Fig. 8. We can find from Fig. 8 that the frequency response obtained by the proposed method is in good agreement with that obtained by FEM. When the input acceleration is 0.1 G, the maximum value of velocity suddenly goes up at 18 Hz and gradually decreases as shown in Fig. 8(a). This suggests that the oscillator is trapped to one of the wells. When the input acceleration is increased to 1.5 G, the velocity has the random variations from 5 Hz to 13 Hz and 39 Hz to 42 Hz, and it increases linearly with frequency from 14 Hz to 38 Hz and 43 Hz to 66 Hz, as shown in Fig. 8(b). Moreover, it suddenly goes down at 38 Hz and 66 Hz.

Fig. 8.

Frequency characteristics.

Fig. 9.

Time variations of z computed by proposed magnetic circuit when acceleration is 1.5 G.

Fig. 10.

Trajectories on phase plane $(z,{\dot{z}})$ by proposed magnetic circuit when acceleration is 1.5 G.

Figures 9 and 10 show that the time variations of z and trajectories on phase plane $(z,{\dot{z}})$ , respectively, when the input acceleration is 1.5 G. It is suggested from these results that the chaotic, interwell and intrawell vibrations are induced at 10 Hz, 30 Hz, and 70 Hz, which are also observed in the measurement [3]. That is, the oscillator makes transition between two equilibrium positions of the potential energy when frequency is smaller than 66 Hz, whereas it traps to one of the wells of the potential energy when frequency is higher than 67 Hz. To characterize the vibration at 10 Hz, we evaluate the Lyapunov exponent using the Rosenstein method [7]. Its mean value is about 23, from which we conclude that this vibration is chaotic.

4. Comparison with experiment

To verify the numerical result, we perform experiment using the manufactured CVEH device shown in Fig. 11 [4], which has a cantilever, made of phosphor-bronze, electrical steel sheets, JIS 50H400, in the coil bobbin. The frame is produced using a 3D printer (Roland, ARM-10). The turn of the coil wire, 0.1 mm in diameter, is set to 1200. The parameters m, k and l _gap are set to 17 g, 500 N∕m and 1.4 mm, respectively. A resistive load, 100 kΩ, is connected to the coil, and the load voltage is measured by an oscilloscope. The input acceleration is fixed to 0.1 G, which is a typical acceleration observed in infrastructure constructions [8].

We compare the frequency characteristic of VEH observed in the experiment with that computed by the proposed magnetic circuit. Figures 12 and 13 show the computed potential energy and the frequency responses. We can see in Fig. 13 that the computed frequency response is the almost the same as the measured response. These vibration modes correspond to the intrawell oscillation. The computed profile near the resonance frequency is narrower than the measured one. One of the reasons for this difference would be due to the manufacturing error.

Fig. 11.

Manufactured VEH.

Fig. 12.

Potential energy.

Fig. 13.

Measured and computed frequency response.

5. Conclusions

In this paper, we have proposed a magnetic circuit model of CVEH which can be used for its dynamic analysis. We have improved the accuracy in the magnetic circuit model by considering the change in the air gap length and effective area of the magnetic flux. The proposed method has been shown to be accurate even when the air gap length is changed. The motion of CVEH computed based on the magnetic circuit has chaotic, interwell and intrawell modes being dependent on the input frequency and acceleration. The computed frequency response for 0.1 G has almost the same resonant frequency as the measured one.

The proposed magnetic circuit is effective to tune the response of CVEH to the spectrum of given ambient vibration. Moreover, it is also useful to optimize the design parameters of CVEH to maximize the performance.

Footnotes

Acknowledgements

This work was supported in part by KAKENHI 15H02976 and 18H0166408.

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