V. CONCLUSION AND TECHNICAL CONTRIBUTIONS OF PRESENTED WORK
3. Modeling
Figure 20 shows a typical lumped-parameter schematic model of an electromagnetic vibration energy harvester (“harvester”), consisting of a proof mass connected to a source of excitation (e.g. a bridge) through a linear spring (stiffness: ) and damper (damping: ). A linear motor with electromechanical transduction (back EMF) constant is placed between the input excitation and the proof mass, in parallel with the mass and spring. Relative velocity between the proof mass and the input excitation generates back EMF (voltage) in the motor coil (electrical resistance: , electrical inductance: ) which is coupled to an electrical load (represented as a generalized electrical impedance ) attached across the leads of the motor.
The overall dynamics of the harvester can be described by a system of two equations representing the coupled mechanical and electrical dynamics:
(19)
, (20)
where is the displacement of the input excitation, is the displacement of the proof mass, ≝ is the relative displacement between the input excitation and the proof mass, is the coil current, and is the voltage across the electrical load. Since power is ultimately harvested in the electrical domain, however, an
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analysis of the effect of the electrical load on power generation requires that the harvester’s dynamics be cast entirely into the electrical domain [30].
Figure 20: Lumped parameter representation of a vibration energy harvester: i) mechanical components; ii) electrical components
A representation of the harvester in a single energetic domain is easily realized through a bond graph model of the system, shown in Figure 21. The bond graph representation explicitly shows the flow of power through the harvester, making it particularly well suited to the analysis of harvester power generation. The input excitation is modeled as an ideal mechanical flow (velocity) source, which assumes that the dynamics of the harvester do not appreciably influence the input excitation. This assumption is valid as long as the mass of the harvester is substantially less than that of the source of the input excitation, as would be the case for a harvester attached to a bridge. The linear motor is modeled as an ideal electromechanical gyrator, which is consistent with the lumped-parameter model of a motor. In the bond graph, all mechanical dynamics of the harvester appear to the left of the gyrator, and all electrical dynamics appear to the right of the gyrator.
Proceeding from Figure 21, a purely electrical-domain representation of the harvester is created by reflecting the mechanical dynamics across the gyrator, into the electrical domain, as shown in Figure 22.
Figure 21: Bond graph representation of vibration energy harvester
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Figure 22: Bond graph representation of vibration energy harvester, cast entirely into the electrical domain
From the perspective of the electrical domain, the mechanical flow source from Figure 21 appears as an electrical effort (voltage) source in Figure 22. Furthermore, the behavior of the mechanical energy storage elements is inverted in the electrical domain view: a mechanical inertia (mass) appears as an electrical capacitor; a mechanical capacitor (spring) appears as an electrical inertia (inductor). The energetic behavior of dissipative elements, however, is unchanged.
Based on the single-domain representation in Figure 22, Figure 23 shows an electrical circuit with equivalent dynamics to the harvester, which allows the harvester to be studied using standard circuit analysis techniques. As previously suggested by [39], this equivalent circuit approach enables analysis of harvester power generation by means of the Thévenin equivalent circuit. Since the system is linear, it is convenient for analysis to proceed in the frequency domain, where the dynamics of electrical components can be described in terms of complex impedances. Whereas previous research has employed the Fourier transform [[40]] in modeling and lamented the complexity of the resulting power expressions, this paper employs the Laplace transform in order to facilitate a novel analysis of system stability and power generation, as discussed below. Furthermore, though not explicitly discussed in this paper, use of the Laplace transform allows for analysis of the transient response, which is not possible using the Fourier transform.
Figure 23: Electric circuit representation of vibration energy harvester
Using standard Laplace notation, where denotes the Laplace variable, the impedance of an inductor is given by ≝ , and the impedance of a capacitor is given by ≝ 1⁄ . The impedance of a
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resistor is purely real: ≝ . The circuit in Figure 23 can therefore be equivalently represented as the Thévenin circuit in Figure 24, which matches the findings of [39].
Figure 24: Thévenin equivalent circuit representation of the vibration energy harvester
In this final representation of the harvester, the mechanical input excitation is translated into an exogenous voltage source,
, (21)
where is the Laplace transform of . The presence of a second-order mechanical filter in
makes explicit the effect of the mass-spring-damper to amplify the input excitation over a narrow frequency band. Other combined mechanical and electrical dynamics of the harvester are represented in Figure 24 by a single source impedance:
. (22)
A key benefit of this representation of the harvester is the separation of all internal dynamics of the harvester from the dynamics of the external electrical load. Assuming that the load impedance comprises only linear elements such that
, (23)
where and are the Laplace transforms of and , respectively, the overall harvester dynamics can be written as a transfer function:
1 . (24)
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A transfer function between the exogenous voltage and the load voltage follows directly from (23) and (24):
. (25)
As an alternative to the preceding bond graph analysis, harvester dynamics in the form of (21), (22), (24), and (25) can be derived by combining the Laplace transforms of (19) and (20), eliminating , and substituting (23).