A Theory for Energy-Optimized Operation of Self-Adaptive Vibration Energy Harvesting Systems with Passive Frequency Adjustment
Abstract
:1. Introduction
2. Adjustment Modes
3. Derivation of Design Rules
3.1. Single Adjustment Steps
3.2. Periodic Adjustment
4. Optimization of Net Available Power
4.1. Omission of Adjustment Steps
4.2. Scaling of the Adjustment Bandwidth
4.2.1. Upper Limit for the Frequency Spacing
4.2.2. Rules for a Periodic-Adjustment System
- Efficient harvester, but too narrowband (, ): An adjustment bandwidth reduction improves nothing, but a widening is advantageous for . This is only possible when the adjustment bandwidth in state A is not the potential maximum.
- Optimum harvester (, ): Limiting case, no change in the adjustment bandwidth can improve the system efficiency, because it is already at the potential maximum.
- Efficient harvester, but too wideband (, ): Narrowing is advantageous for , widening never.
- Inefficient harvester (): would be negative, which is not admissible for physical reasons. Narrowing is always advantageous () because the harvester is in a state in which it expends more energy on its adaptivity than it gains from it.
- : ,
- : ,
- : .
5. Validation by Application to an Implemented System
5.1. System Description and Analysis
5.2. Results for a Fixed-Process Stationarity Time
- Adjusting to should be avoided as . This would also save the adjustment energy from to because .
- Adjusting to should be avoided because the capacitor voltage decreases in hold phase 4, so . Notice that, in contrast to hold phase 8, the energy harvested in hold phase 4 would have exceeded the adjustment energy if the hold phase duration had been longer.
5.3. Influence of the Process Stationarity Time
- : the load cannot be supplied with continuously (Equation (15)).
- : narrowing the adjustment bandwidth always pays off, and .
- : a narrowing pays off for , and .
- : ABS does not improve the system.
- : widening the adjustment bandwidth is advantageous for , and .
5.4. Influence of a Frequency Dependence of the Adjustment Power
6. Summary and Outlook
Author Contributions
Funding
Conflicts of Interest
Abbreviations
Symbol | Meaning |
; | ambient vibration frequency; i-th ambient vibration frequency |
, | lowest and highest ambient vibration frequency |
; | resonance frequency of energy harvester; i-th resonance frequency |
frequency spacing for possible adaptation step | |
upper limit of for positive net output energy of harvester | |
; , | average frequency spacing for periodic adjustment; same before (A) and after (B) adjustment-bandwidth scaling (ABS) |
ratio between tuning and harvested power before ABS | |
effective spring stiffness | |
effective mass | |
; , | average harvested power; same before (A) and after (B) ABS |
maximum of | |
load power | |
; , | net available power; same before (A) and after (B) ABS |
gain in net available power of energy harvester | |
; , | average tuning power; same before (A) and after (B) ABS |
maximum tuning power | |
decimation ratio | |
scaling factor for adjustment bandwidth | |
, | limit scaling factors with zero gain in net available power |
optimum scaling factor for maximum gain in net available power | |
hold phase duration = average time span between adjustment steps = process stationarity time of ambient vibration | |
value of at which net available power vanishes | |
value of at which net available power equates load power | |
harvested energy | |
tuning energy | |
tuning energy per unit frequency |
Appendix A
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Mode of Operation | P0/mW | PT/mW | Pnet/mW | τ0/s | τ1/s | Opt. Solution If |
---|---|---|---|---|---|---|
Non-adjusting system | 0.85 | 0.85 | τ < 26 s | |||
Adjusting system | ||||||
Strictly periodic (Reference [25]) | 10.1 | 8.7 | 1.3 | 61 | 76 | |
This work | ||||||
Skip adjustment to fa,8 | 10.1 | 6.2 | 3.8 | 44 | 55 | 112 s < τ |
Skip adjustment to fa,4 and fa,8 | 9.0 | 4.6 | 4.4 | 36 | 46 | 79 s < τ < 112 s |
Narrowed range, s = 0.58 | 6.9 | 2.3 | 4.7 | 23 | 33 | 26 s < τ < 79 s |
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Mösch, M.; Fischerauer, G. A Theory for Energy-Optimized Operation of Self-Adaptive Vibration Energy Harvesting Systems with Passive Frequency Adjustment. Micromachines 2019, 10, 44. https://doi.org/10.3390/mi10010044
Mösch M, Fischerauer G. A Theory for Energy-Optimized Operation of Self-Adaptive Vibration Energy Harvesting Systems with Passive Frequency Adjustment. Micromachines. 2019; 10(1):44. https://doi.org/10.3390/mi10010044
Chicago/Turabian StyleMösch, Mario, and Gerhard Fischerauer. 2019. "A Theory for Energy-Optimized Operation of Self-Adaptive Vibration Energy Harvesting Systems with Passive Frequency Adjustment" Micromachines 10, no. 1: 44. https://doi.org/10.3390/mi10010044