On the dynamics of a capacitive electret-based micro-cantilever for energy harvesting
Introduction
Energy harvesting is to scavenge or harvest energy from a variety of ambient energy sources (e.g. solar, wind and vibration) and convert it into electrical energy for self-powered wireless electronic applications ranging from structural health monitoring to medical implants [[1], [2], [3]]. Vibration energy harvesting devices can convert ambient vibrations to electrical energy, and they are developed based on three basic energy conversion mechanisms including, electromagnetic [4,5], piezoelectric [[6], [7], [8]] and electrostatic (capacitive) [[9], [10], [11], [12]] transductions. Since the technology for manufacturing electrostatic transducers such as capacitive-based sensors and actuators (e.g., accelerometers, gyroscopes, comb drives) is well established, it is beneficial to utilize the same standard MEMS (Micro-electro-mechanical systems) technology for manufacturing capacitive energy harvesting devices. This allows the production of devices in large numbers at low cost [13]. The basis of electrostatic energy harvesting devices is a charged variable capacitor whose capacitance changes as a result of ambient vibrations. Any change in the capacitance leads to a charge rearrangement on the capacitor electrodes and consequently, a charge flow through the electrical circuit. Based on how the capacitance changes, capacitive energy harvesting devices are classified into three different types, including in-plane overlap in which the overlap area between electrodes varies, in-plane gap closing in which the gap between electrode fingers changes, and out-of-plane gap closing in which the gap between two electrodes varies [2]. These devices can operate in either switched or continuous scheme. In switched systems, some controlled switches are employed to change the condition of the capacitor discontinuously through voltage-constrained or charge-constrained cycles, however, continuous systems do not require any controlled switches to operate and any change in the biased capacitor causes a charge flow through the resistance. Since the utilization of switches requires some extra circuitry to control them, precious energy is consumed by that control circuitry, which makes continuous systems are more advantageous than switched ones [2]. Electrostatic generators require an initial voltage to charge the capacitor and initiate the energy conversion. This bias voltage can be provided by electret materials in continuous systems. Electrets are dielectric materials with a quasi-permanent charge which can provide an electric field for tens of years [14]. Various materials have been used as electrets in electrostatic energy harvesters such as Teflon [15] [11], and CYTOP [16]. Among these materials, CYTOP is reported to have a high charge density, which is also a MEMS-compatible perfluoropolymer [16].
Many researchers have studied the electret-based energy harvesters. Boisseau et al. [9] developed an out-of-plane energy harvester in which a cantilever beam with a tip mass is used as the movable electrode of the variable capacitor and an electret layer is employed to provide the bias voltage. The resonant system is modeled as a spring-mass to simulate the mechanical behavior of the device. Tsutsumino et al. [16] developed an in-plane overlap energy harvester, in which CYTOP is used as a high performance electret material to supply the bias voltage. The resonant system consists of a proof mass with parylene high-aspect-ratio springs which allows large amplitude oscillation and low response frequency. Tao et al. [14] proposed an out-of-plane energy harvesting device with dual-charged electret, in which the resonant system consists of a movable circular mass with a series of spiral spring and two LDPE (low-density polyethylene) electret plates with positive and negative charge are used as the voltage source. Wang et al. [17] presented an out-of-plane gap electrostatic energy harvester with a proof mass suspended by beam type springs. The device consists of a four wafer stack, and CYTOP is employed both as the electret material and adhesive interface between wafers. Peano et al. [18] introduced and optimized a capacitive electret-based energy harvesting device with in-plane overlap scheme using a nonlinear dynamical model. Sterken et al. [19] proposed, optimized and fabricated an electret micro generator with in-plane overlap scheme. The device consists of two bonded wafers as the electrodes of the variable capacitor. The system is analyzed using a lumped element model based on the equivalency between mechanical and electrical elements. Tvedt et al. [20] introduced an in-plane overlap energy harvesting device with an electret material to polarize the electrodes. The resonant system is modeled as a lumped equivalent circuit based on both linear and nonlinear models. Halvorsen et al. [21] designed and fabricated an in-plane overlap electret-based energy harvesting device. The resonant system consists of a proof mass supported by springs above an oxidized silicon substrate with patterned electret stripes. Kloub et al. [22] optimized an in-plane overlap micro capacitive energy harvesting device by maximizing the capacitance variation. The device is composed of a seismic mass suspended by beams and modeled as a spring-mass-damper system. Miki et al. [23] developed and fabricated an in-plane electret energy harvester with a proof mass suspended by hybrid high-aspect ratio parylene springs. Genter et al. [24] fabricated an out-of-plane electret energy harvester in which Parylene-C is used as electret layer and spring material. The device is composed of two silicon wafers as base chip and resonator chip with a proof mass suspended by a flexural spring from the resonator chip. Tao et al. [25] designed and fabricated an out-of-plane energy harvesting device, in which the resonant structure is composed of a proof mass attached to a spiral spring and modeled as a mass-spring-damper. In this design, two opposite-charged electret layers are integrated at the top and bottom of the proof mass, which considerably increase the performance of the energy harvester in comparison to single electret layer configuration. Zhang et al. [26] reported design and fabrication of an out-of-plane electret-based energy harvester, in which they optimized the air damping between the movable electrode, a proof mass suspended from a fixed frame by four beams, and the fixed substrate in order to reach broad bandwidth and high power density at the same time. Yamamoto et al. [27] proposed and simulated the performance of an out-of-plane electret-based energy harvester, in which the proof mass is suspended from a nonlinear bi-stable spring, which causes the bandwidth of the device to increase compared to linear case. Gao et al. [28] analyzed and fabricated an out-of-plane electret-based energy harvesting, in which a doubly-clamped beam with a mass placed is used as the movable electrode. They modeled the beam as a mass-spring-damper system and optimized the length of the electret layer to reach the maximum output power.
To the authors' knowledge, most of the researchers on the electret-based energy harvesters have modeled the resonant system as a mass-spring model and mainly focused on the electrical behavior of the device. In this paper, an electret-based energy harvester with out-of-plane gap closing scheme is proposed, in which a micro-cantilever is used as the moving electrode of the variable capacitor. There are a few researchers which proposed a cantilever-based energy harvester and they have modeled the vibrating system as a spring-mass-damper [9]. In this study, the micro-cantilever is modeled as a continuous system based on Euler-Bernoulli beam theory and the characteristics of the device are investigated with a focus on the mechanical behavior of the energy harvester. An electret layer is attached to the stationary electrode (substrate) to supply the bias voltage. This variable capacitor is in series with a resistance which consumes the harvested energy. The simplicity of the presented device enables a low-cost manufacturing process with MEMS technology.
Section snippets
Modeling
The schematic representation of the proposed energy harvester is shown in Fig. 1. The device is composed of a variable capacitor which is formed by a fixed substrate and a micro cantilever of length L, width b and thickness h. Since capacitive energy harvesters require a voltage source for charging the electrodes to scavenge energy from the ambient vibrations, an electret layer is attached to the substrate to provide the bias voltage. This capacitor is in series with a resistance in an electric
Solution procedure
The Galerkin method is employed to discretize the integral-partial differential equation of the system. To this end, the transverse deflection of the micro-cantilever is considered in the form of the following expansion series:where N is the number of modes, qj(t) is the generalized coordinate of jth mode and φj(x) is jth linear mode shape of the micro-cantilever in free vibration. Multiplying Eq. (13) by [k – w(x,t)]2, inserting Eq. (16) into resulting expression,
Results and discussion
Since the silicon is the main material in MEMS technology and it possesses excellent thermo-mechanical properties [29], the micro-cantilever is assumed to be made of silicon. The mechanical, electrical and geometrical properties of the studied model are given in Table 1. Also, CYTOP® is used as the electret material which has a breakdown voltage of 90 kV/mm [31].
The variation of dimensionless deflection of the micro-cantilever tip with electret surface voltage is illustrated in Fig. 3 for
Conclusion
A capacitive energy harvester is developed in this paper. The system is made up of a micro cantilever and a fixed substrate that form a variable capacitor together. An electret layer is attached to the substrate which provides the required bias voltage of the device. The variable capacitor is in series with a resistance and the whole system mounted on a package that the ambient vibration is applied to it as a harmonic excitation. Unlike the previous studies, the resonant system is modeled
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