97 Figure 59: DFT of the squared acceleration amplitude of the new excitation and the average PHR-tuned energy generation behavior of the conventional vibration energy harvester and the coupled harvester with increased trial masses (total mass 54% larger than conventional). 98 Figure 60: DFT of the squared acceleration amplitude of the new excitation and the average PHR-tuned energy generation behavior of the conventional vibration energy harvester and the coupled harvester with increased test masses (total mass 118% larger than conventional).
Overview
Background and Motivation
Considering that solar and wind energy generators must be exposed to the relevant elements from which they obtain energy, this limits the available options for installation on the bridge. Conversely, since the bridge is mechanically excited during normal use, an electromechanical harvester that derives energy from bridge vibrations does not have this drawback.
Overview of Existing Research for Addressing Single Operational Frequency of Vibration Harvesters
Most of the approaches that exist in the literature can usually be grouped into one of two categories: (1) extending the generator power gain bandwidth using an array of multiple collectors with different natural frequencies; or (2) adjusting the natural frequency of the generator to match the dominant excitation frequency [19]. Other approaches to solving the single operating frequency problem of vibration generators are listed in references [19] and [18], which provide a thorough review of the literature.
Organization of Dissertational Work
It is precisely this gap, which is formed by the need for a better power metric, and the currently unrecognized need for stability evaluation in electrical load setting, that is addressed by the research described in this work.
MODELING APPROACH AND DEVELOPMENT OF PHR AND STABILITY ASSESSMENT
Modeling Approach
In this representation, is an excitation-dependent exogenous voltage source, is the harvester's source impedance (this source impedance includes both the mechanical and electrical components), and is the impedance of the harvester's electrical load. This simplifies the problem by creating a perception of an ideal voltage source that is not affected by the choice of the combine's electrical load, which in turn simplifies the analysis of how the electrical load affects power generation.
Stability Considerations
This fact can be used in conjunction with the Nyquist stability criterion to impose certain conditions on the electrical load selection to ensure the stability of the harvester. In this case, the Nyquist stability criterion specifies that the graph of the Nyquist contour must have as many counter-clockwise encirclements of the critical point ( 1 ∙ 0 as there are non-minimal phase zeros).
A Multifrequency Power Generation Metric – Power Harvesting Ratio
- Derivation
Equation (11) defines the mathematical relationship between the acceleration of the excitation squared and the average power generated by the harvester at a specific frequency. The normalization with respect to excitation facilitates the calculation of the amount of power harvested from each frequency component;.
EXPERIMENTAL VALIDATION
Validating PHR for Single Frequency Component Excitation
In the testing described in this section, the electrical load was limited to the passive components and selected based on the MPTT performed at the natural frequency of the harvester. The only variable that is not defined is , which represents the amplitude of the excitation shift.
Validating PHR for Excitation Comprised of Multiple Frequency Components
The energy harvesting ratio takes into account the power generated from all frequency components that make up the excitation, thereby attempting to maximize the accuracy of the estimate of the harvester's total energy generation. PHR can estimate average energy generation within 5% of experimentally measured values, while another third is within about 10.
Validation Using Real-world Bridge Vibration Data
Again, the magnitude of the errors between the PHR predicted and experimentally measured average energy generation values are quite small, never exceeding 5. Given the amount and breadth of experimental validation, the PHR average energy generation method was considered as an accurate analytical tool.
PHR AND MAXIMIZING POWER GENERATION
86 it was shown that a conventional harvester was capable of generating a significantly greater amount of average power than the harvester array or the coupled harvester for the excitation shown in Figure 16 in the previous section. The PHR curves of the three architectures, DFT of the excitation acceleration amplitude squared, and the resulting overall average power generation values for each architecture (see legend) are shown in Figure 19. By not dividing the test mass and using the mechanical natural frequency setting the device to the largest frequency component observed in the excitation DFT, maximum average power generation will be guaranteed.
CONCLUSION AND TECHNICAL CONTRIBUTIONS OF PRESENTED WORK
Summary
This tool works by analyzing a closed-loop system constructed using Laplace domain representations of the relevant harvester dynamics and the proposed electrical load. While it also uses the Laplace transform, this technique is based on incorporating the harvester, its excitation, and its electrical load into a Thévenin equivalent circuit.
Technical Contributions of Dissertational Work
- APPENDIX A: Analytical Tools for Investigating Stability and Power Generation of
- APPENDIX B: Assessing Stability and Predicting Power Generation of Electromagnetic Vibration
- APPENDIX C: Analysis and Optimization of Electromagnetic Vibration Energy Harvesters Using
Analysis of the effect that frequency shifts and amplitude reductions have on the total power generation of different harvesting architectures. PHR-based recommendations for the tuning approach when the excitation has a DFT whose components differ in frequency and/or amplitude.
Abstract
Introduction
Due to these disadvantages of electrostatic and piezoelectric power harvesting, as well as the relatively greater robustness and controllability [20], electromagnetic vibration energy harvesting is the best candidate for the application under consideration and is the focus of the research presented in this article. The proposed techniques still attempt to match the natural frequency of the harvester to the dominant frequency of the excitation source, but the tuning is achieved solely by an adjustable electrical load.
Modeling
A representation of the harvester in a single energetic domain can be easily achieved via a bond graph model of the system, shown in Figure 21. In this final representation of the harvester, the mechanical input excitation is translated into an exogenous voltage source.
Harvester Stability
- Case I: Z L (s) has no nonminimum-phase zeros
- Case II: Z L (s) has nonminimum-phase zeros
- Stability Case Study: Maximum Power Transfer Theorem
Briefly, the maximum power transfer theorem states that maximum power will be transferred to the load impedance when the load impedance is equal to the complex conjugate of the source impedance. Assuming all system parameters are positive, this particular load impedance has three non-minimum phase zeros; however, the phase of the system satisfies (28), which means that applying the maximum power transfer theorem at all frequencies will always result in unstable harvester dynamics.
Power Harvesting Ratio: A Power Generation Metric
Equation (35) defines the mathematical relationship between the squared excitation acceleration and the average power generated at a given frequency. The reason for using excitation acceleration as opposed to displacement is that the ability of the harvester to generate power at different frequencies is properly portrayed.
Experimental Setup and Results
- Description of the Experimental Setup
- Experimental Testing and Results
- Single-frequency Excitation
- Multifrequency Excitation
A larger voice coil (BEI LAH28-52-000A linear voice coil) was used to produce crop excitation. These experiments involved varying the electrical load of the harvester, and the excitation frequency and amplitude of the components from which the harvester generated power.
Conclusions
Acknowledgements
This paper presents the use of the Power Harvesting Ratio (PHR) approach to evaluate the ability of an electromagnetic vibration energy harvester to harvest power. The remainder of this manuscript is organized as follows: Section 3 provides a review of the two analytical tools introduced in [43].
Review of Stability Assessment and the Power Harvesting Ratio
- Modeling
- Stability Analysis
- Power Harvesting Ratio
The harvester's stability can be determined by plotting the Nyquist contour of the open-loop transfer function shown in (46) and using the Nyquist Stability Criterion, shown in (47). Equation (49) defines the normalized average power generated by the harvester at a frequency; the normalization is done with respect to the squared amplitude of the input acceleration's component at that frequency.
Excitation Measurement and Characterization
Finally, the modal certainty criterion (MAC) [36] was used to assess the linear independence of the identified modes. The bridge response to single vehicle excitation was obtained for use in energy harvester analysis.
Experimental Setup
- Description of Harvester
- Description of the Electrical Loads
The function of the voice coil is to simulate bridge vibrations, which excites the second stage – the harvester – which generates electric current from this mechanical excitation. The excitation used in obtaining the solution was the bridge acceleration time series data set A, shown in Figure 36 – the same excitation used to optimize the 23.6 Ω passive load.
Methodology
- Stability Assessment of Active Loads
- Electrical Load Prescribed by the Maximum Power Transfer Theorem
- Trial Active Electrical Load
- Average Power Generation Estimation Using PHR
In fact, the resulting unstable behavior, when used as the electrical load, applies to any harvester of the form described by the lumped parameter model shown in Figure 30. The closed-loop transfer function of the harvester (ie, function of the system shown in Figure 32).
Experimental Results
Overall, the experimental results show that the PHR method accurately predicts the average power delivered by the harvester to the load, as no experimental run showed an error exceeding 6. The experimental load voltage matches the theoretical load voltage whenever it can/when not is clipped by servo amplifier voltage limits set to prevent damage to the experimental setup.
Conclusions
The theoretical load voltage is defined as the voltage that would theoretically develop across the stripper leads in response to the excitation if the electrical load were to behave as described in (51). The technique produced estimates that never showed more than 6% deviation from actual (i.e. experimentally measured) delivered power values; the testing included both passive and active loads. Using real bridge vibration data to validate: 1) the harvester's stability assessment tool, and 2) PHR, serves to show that both of these analytical tools are appropriate and practical.
Acknowledgements
This suggests that formal electrical load optimization, which can be guided by the analytical tools discussed in this manuscript, can be used to improve current generation of vibrational energy harvesters subjected to multifrequency excitation. Analysis and Optimization of Electromagnetic Vibration Energy Harvesters Using the Power Harvesting Ratio Performance Metric.
Validation of the Power Harvesting Ratio on Typical Bridge Vibration Data
Specifically, several simple nut-bolt-washer clamps were attached to the harvester's compliant mechanisms as shown in Figure 50. Each data set was run three times and the resulting average force delivered to the load during each experimental run was calculated using
PHR-based Analsys of Choice of Harvester Architecture on Power Output
- Introduction of Considered Architectures, Architecture Embodiments, and Assumptions
- Basic Embodiment
- PHR-tuned Embodiment
- MPTT Embodiment
- Simulated Power Generation Results of the Architecture/Embodiment Combinations
- Effect of Variable Excitation on Analysis Results
- Effect of Frequency Shifts
- Effect of Amplitude Changes
The PHR curves of the PHR-tuned and MPTT embodiments of the conventional harvester and coupled harvester with the larger test masses are shown in Figure 59 and Figure 60. All the previous simulation results showed that the performance of the conventional harvester is in many ways better than that of the coupled harvester and the harvester array for the chosen generation.
Overarching Conclusions from PHR-based Analysis of Power Generation from Vibrations
- Excitation with a Static DFT
- Excitation with a Varying DFT
Therefore, relying on the average power output on the square of the amplitudes of the excitation frequency components becomes somewhat of a counterargument to the previous assertion that the test mass should not be distributed. The PHR method for power generation estimation gives accurate results as long as the (linear) dynamics of the harvest, (linear) dynamics of the electric charge and the DFT of the excitation are known.
Simulink Diagrams
MATLAB Code
- Code for Running Experimental Setup Simulink and Analyzing Results
- Code for Comparing Power Outputs of the Three Architectures and Their Three Embodiments . 118
Powarray=max(max(max(max(theor_avg_pow)))) % Total pow for konv tuning af harv-array [RL1_index,RL2_index,k1_index,k2_index] = ind2sub(size(theor_avg_pow),find(theor_avg_pow == . Powarray)) ;. Powarraymax=max(max(max(max(theor_avg_pow)))); % Tot-effekt for PHR-tunet harv-array [RL1_index,RL2_index,k1_index,k2_index] = ind2sub(size(theor_avg_pow),find(theor_avg_pow == . Powarraymax));.
