# A Theory for Energy-Optimized Operation of Self-Adaptive Vibration Energy Harvesting Systems with Passive Frequency Adjustment

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## 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 ($k<\frac{1}{2}$, ${s}_{2}>1$): An adjustment bandwidth reduction improves nothing, but a widening is advantageous for $1<s<{s}_{2}$. This is only possible when the adjustment bandwidth in state A is not the potential maximum.
- Optimum harvester ($k=\frac{1}{2}$, ${s}_{2}={s}_{1}=1$): 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 ($\frac{1}{2}<k<1$, $0<{s}_{2}<1$): Narrowing is advantageous for ${s}_{2}<s<1$, widening never.
- Inefficient harvester ($k>1$): ${s}_{2}$ would be negative, which is not admissible for physical reasons. Narrowing is always advantageous ($0<s<1$) because the harvester is in a state in which it expends more energy on its adaptivity than it gains from it.

- $k<\frac{1}{2}$: $1<s<{s}_{2}$,
- $\frac{1}{2}<k<1$: ${s}_{2}<s<1$,
- $k\ge 1$: $s<\frac{1}{k}$.

## 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 ${f}_{\mathrm{a},8}$ should be avoided as ${W}_{\mathrm{T},8}{W}_{0,\hspace{0.17em}8}=0$. This would also save the adjustment energy ${W}_{\mathrm{T},1}$ from ${f}_{\mathrm{a},8}$ to ${f}_{\mathrm{a},1}$ because ${f}_{\mathrm{a},7}={f}_{\mathrm{a},1}$.
- Adjusting to ${f}_{\mathrm{a},4}$ should be avoided because the capacitor voltage decreases in hold phase 4, so ${W}_{\mathrm{T},4}{W}_{0,\hspace{0.17em}4}$. Notice that, in contrast to hold phase 8, the energy ${W}_{0,\hspace{0.17em}4}$ harvested in hold phase 4 would have exceeded the adjustment energy ${W}_{\mathrm{T},\hspace{0.17em}4}$ if the hold phase duration $\mathsf{\tau}$ had been longer.

#### 5.3. Influence of the Process Stationarity Time

- $\mathsf{\tau}<48\hspace{0.17em}\mathrm{s}$: the load cannot be supplied with ${P}_{\mathrm{L}}=2\hspace{0.17em}\mathrm{mW}$ continuously (Equation (15)).
- $48\hspace{0.17em}\mathrm{s}<\mathsf{\tau}<60\hspace{0.17em}\mathrm{s}$: narrowing the adjustment bandwidth always pays off, and ${s}_{\mathrm{opt}}=120\hspace{0.17em}\mathrm{s}/\mathsf{\tau}$.
- $60\hspace{0.17em}\mathrm{s}<\mathsf{\tau}<120\hspace{0.17em}\mathrm{s}$: a narrowing pays off for $1<s<60\hspace{0.17em}\mathrm{s}/(\mathsf{\tau}-60\hspace{0.17em}\mathrm{s})$, and ${s}_{\mathrm{opt}}=120\hspace{0.17em}\mathrm{s}/\mathsf{\tau}$.
- $\mathsf{\tau}\approx 120\hspace{0.17em}\mathrm{s}$: ABS does not improve the system.
- $\mathsf{\tau}>120\hspace{0.17em}\mathrm{s}$: widening the adjustment bandwidth is advantageous for $1>s<60\hspace{0.17em}\mathrm{s}/(\mathsf{\tau}-60\hspace{0.17em}\mathrm{s})$, and ${s}_{\mathrm{opt}}=120\hspace{0.17em}\mathrm{s}/\mathsf{\tau}$.

#### 5.4. Influence of a Frequency Dependence of the Adjustment Power

## 6. Summary and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Symbol | Meaning |

${f}_{\mathrm{a}}$; ${f}_{\mathrm{a},i}$ | ambient vibration frequency; i-th ambient vibration frequency |

${f}_{\mathrm{a},\mathrm{low}}$, ${f}_{\mathrm{a},\mathrm{high}}$ | lowest and highest ambient vibration frequency |

${f}_{\mathrm{r}}$; ${f}_{\mathrm{r},i}$ | resonance frequency of energy harvester; i-th resonance frequency |

$\Delta f$ | frequency spacing for possible adaptation step |

$\Delta {f}_{\mathrm{L}}$ | upper limit of $\Delta f$ for positive net output energy of harvester |

$\overline{\Delta f}$; ${\overline{\Delta f}}_{\mathrm{A}}$, ${\overline{\Delta f}}_{\mathrm{B}}$ | average frequency spacing for periodic adjustment; same before (A) and after (B) adjustment-bandwidth scaling (ABS) |

$k$ | ratio between tuning and harvested power before ABS |

${k}_{\mathrm{eff}}$ | effective spring stiffness |

$m$ | effective mass |

${P}_{0}$; ${P}_{0,\mathrm{A}}$, ${P}_{0,\mathrm{B}}$ | average harvested power; same before (A) and after (B) ABS |

${P}_{0\mathrm{m}}$ | maximum of ${P}_{0}$ |

${P}_{\mathrm{L}}$ | load power |

${P}_{\mathrm{net}}$; ${P}_{\mathrm{net},\mathrm{A}}$, ${P}_{\mathrm{net},\mathrm{B}}$ | net available power; same before (A) and after (B) ABS |

$\Delta {P}_{\mathrm{net}}$ | gain in net available power of energy harvester |

${P}_{\mathrm{T}}$; ${P}_{\mathrm{T},\mathrm{A}}$, ${P}_{\mathrm{T},\mathrm{B}}$ | average tuning power; same before (A) and after (B) ABS |

${P}_{\mathrm{Tm}}$ | maximum tuning power |

$r$ | decimation ratio |

$s$ | scaling factor for adjustment bandwidth |

${s}_{1}$, ${s}_{2}$ | limit scaling factors with zero gain in net available power |

${s}_{\mathrm{opt}}$ | optimum scaling factor for maximum gain in net available power |

$\mathsf{\tau}$ | hold phase duration = average time span between adjustment steps = process stationarity time of ambient vibration |

${\mathsf{\tau}}_{0}$ | value of $\mathsf{\tau}$ at which net available power vanishes |

${\mathsf{\tau}}_{1}$ | value of $\mathsf{\tau}$ at which net available power equates load power |

${W}_{0}$ | harvested energy |

${W}_{\mathrm{T}}$ | tuning energy |

${\tilde{W}}_{\mathrm{T}}$ | tuning energy per unit frequency |

## Appendix A

## References

- Werner-Allen, G.; Ruiz, M.; Marcillo, O.; Johnson, J.; Lees, J.; Welsh, M. Deploying a wireless sensor network on an active volcano. IEEE Internet Comput.
**2006**, 10, 18–25. [Google Scholar] [CrossRef] - Yang, J.; Zhou, J.; Lv, Z.; Wei, W.; Song, H. A Real-time monitoring system of industry carbon monoxide based on wireless sensor networks. Sensors
**2015**, 15, 29535–29546. [Google Scholar] [CrossRef] [PubMed] - Magno, M.; Polonelli, T.; Benini, L.; Popovici, E. A low cost, highly scalable wireless sensor network solution to achieve smart LED light control for green buildings. IEEE J.
**2015**, 15, 2963–2973. [Google Scholar] [CrossRef] - Rawat, P.; Deep Singh, K.; Chaouchi, H.; Bonin, J.M. Wireless sensor networks: A survey on recent developments and potential synergies. J. Supercomput.
**2014**, 68, 1–48. [Google Scholar] [CrossRef] - Matiko, J.; Brabham, N.J.; Beeby, S.P.; Tudor, M.J. Review of the application of energy harvesting in buildings. Meas. Sci. Technol.
**2014**, 25, 012002. [Google Scholar] [CrossRef] - Mitcheson, P.; Yeatman, E.M.; Rao, G.K.; Holmes, A.S.; Green, T.C. Energy harvesting from human and machine motion for wireless electronic devices. Proc. IEEE
**2008**, 96, 1457–1486. [Google Scholar] [CrossRef] - Cepnik, C.; Lausecker, R.; Wallrabe, U. Review on electrodynamic energy harvesters—A classification approach. Micromachines
**2013**, 4, 168–196. [Google Scholar] [CrossRef] - Guyomar, D.; Lallart, M. Recent progress in piezoelectric conversion and energy harvesting using nonlinear electronic interfaces and issues in small scale implementation. Micromachines
**2011**, 2, 274–294. [Google Scholar] [CrossRef] - Xu, Z.; Xi Wang, X.; Shan, X.; Xie, T. Parametric analysis and experimental verification of a hybrid vibration energy harvester combining piezoelectric and electromagnetic mechanisms. Micromachines
**2017**, 8, 189. [Google Scholar] [CrossRef] - Zhu, D.; Tudor, M.J.; Beeby, S.P. Strategies for increasing the operating frequency range of vibration energy harvesters: A review. Meas. Sci. Technol.
**2010**, 21, 189. [Google Scholar] [CrossRef] - Tang, L.; Yang, Y.; Soh, C.K. Toward broadband vibration-based energy harvesting. J. Intelligent Mater. Syst. Struct.
**2010**, 21, 1867–1897. [Google Scholar] [CrossRef] - Pellegrini, S.; Tolou, N.; Schenk, M. Bistable vibration energy harvesters: A review. J. Intelligent Mater. Syst. Struct.
**2012**, 24, 1303–1312. [Google Scholar] [CrossRef] - Gieras, J.; Oh, J.H.; Hauzmezan, M.; Sane, H.S. Electromechanical Energy Harvesting System. WO2007070022(A2), WO2007070022(A3). 21 June 2007. [Google Scholar]
- Wu, X.; Lin, J.; Kato, S.; Zhang, K.; Ren, T.; Lui, L. A frequency adjustable vibration energy harvester. In Proceedings of the PowerMEMS, Sendai, Japan, 9–12 November 2008; pp. 245–248. [Google Scholar]
- Leland, E.; Wright, P. Resonance tuning of piezoelectric vibration energy scavenging generators using compressive axial preload. Smart Mater. Struct.
**2006**, 15, 1413–1420. [Google Scholar] [CrossRef] - Hu, Y.; Hue, H.; Hu, H. A piezoelectric power harvester with adjustable frequency through axial preloads. Smart Mater. Struct.
**2007**, 16, 1961–1966. [Google Scholar] [CrossRef] - Eichhorn, C.; Goldschmidtboeing, F.; Woias, P. A frequency tunable piezoelectric energy converter based on a cantilever beam. In Proceedings of the PowerMEMS, Sendai, Japan, 9–12 November 2008; pp. 309–312. [Google Scholar]
- Peters, C.; Maurath, D.; Schock, W.; Mezge, F.; Manoli, Y. A closed-loop wide-range tunable mechanical resonator for energy harvesting systems. J. Micromech. Microeng.
**2009**, 19, 094004. [Google Scholar] [CrossRef] - Eichhorn, C.; Tchagsim, R.; Wilhelm, N.; Woias, P. A smart and self-sufficient frequency tunable vibration energy harvester. J. Micromech. Microeng.
**2011**, 21, 104003. [Google Scholar] [CrossRef] - Challa, V.; Prasad, M.G.; Shi, Y.; Fisher, F.T. A vibration energy harvesting device with bidirectional resonance frequency tunability. Smart Mater. Struct.
**2008**, 17, 015035. [Google Scholar] [CrossRef] - Zhu, D.; Roberts, S.; Tudor, M.J.; Beeby, S.P. Closed-loop frequency tuning of a vibration-based micro-generator. In Proceedings of the PowerMEMS, Sendai, Japan, 9–12 November 2008; pp. 229–232. [Google Scholar]
- Zhu, D.; Roberts, S.; Tudor, M.J.; Beeby, S.P. Design and experimental characterization of a tunable vibration-based electromagnetic micro-generator. Sens. Actuat. A
**2010**, 158, 284–293. [Google Scholar] [CrossRef] - Ayala-Garcia, I.; Mitcheson, P.D.; Yeatman, E.M.; Zhu, D.; Beeby, S.P. Magnetic tuning of a kinetic energy harvester using variable reluctance. Sens. Actuat. A
**2013**, 189, 266–275. [Google Scholar] [CrossRef] - Challa, V.; Prasad, M.G.; Fisher, F.T. Towards an autonomous self-tuning vibration energy harvesting device for wireless sensor network applications. Smart Mater. Struct.
**2010**, 20, 025004. [Google Scholar] [CrossRef] - Hoffmann, D.; Willmann, A.; Hehn, T.; Folkme, B.; Manoli, Y. A self-adaptive energy harvesting system. Smart Mater. Struct.
**2016**, 25, 035013. [Google Scholar] [CrossRef] - Sun, W.; Jung, J.; Seok, J. Frequency-tunable electromagnetic energy harvester using magneto-rheological elastomer. J. Intellignt Mater. Syst. Struct.
**2015**, 27, 959–979. [Google Scholar] [CrossRef] - Zhou, Y.; Apo, D.J.; Shashank, P. Dual-phase self-biased magnetoelectric energy harvester. Appl. Phys. Lett.
**2013**, 103, 192909. [Google Scholar] [CrossRef] [Green Version] - Roundy, S.; Zhang, Y. Toward self-tuning adaptive vibration based micro-generators. Proc. SPIE
**2004**, 5649, 373–384. [Google Scholar] - Roundy, S. On the effectiveness of vibration-based energy harvesting. J. Intelligent Mater. Syst. Struct.
**2005**, 16, 809–823. [Google Scholar] [CrossRef] - Beeby, S.; Torah, R.N.; Tudor, M.J.; Glynne-Jones, P.; O’Donnell, T.; Saha, C.R.; Roy, S. A micro electromagnetic generator for vibration energy harvesting. J. Micromech. Microeng.
**2007**, 17, 1257–1265. [Google Scholar] [CrossRef] [Green Version] - Cepnik, C.; Wallrabe, U. Approaches for a fair comparison and benchmarking of electromagnetic vibration energy harvesters. Micromachines
**2013**, 4, 286–305. [Google Scholar] [CrossRef] - Hoffmann, D.; Willmann, A.; Folkmer, B.; Manoli, Y. Tunable vibration energy harvester for condition monitoring of maritime gearboxes. J. Phys. Conf. Ser.
**2014**, 557, 012099. [Google Scholar] [CrossRef] - Bronstein, I.N.; Semendyayev, K.A.; Musiol, G.; Mühlig, H. Handbook of Mathematics, 6th ed.; Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Rao, S.S. Mechanical Vibrations, 5th ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2011. [Google Scholar]

**Figure 1.**The available power P

_{net}for periodic adjustment as a function of the time span τ between adjustment steps.

**Figure 2.**Omission of adjustment steps. The black bars mark a sequence of random ambient vibration frequencies, and the black and grey arrows respectively represent successive and omitted resonance frequency jumps of the energy harvester. (

**a**) The harvester tracks the ambient resonance closely, and every possible adjustment step is performed (r = 1). (

**b**) Every other step is omitted (r = 1/2).

**Figure 3.**The possible harvested energy W

_{0}and required adjustment energy W

_{T}as functions of the frequency spacing Δf between the harvester resonance frequencies before and after adjustment.

**Figure 5.**Two adjustment-range strategies. (

**a**) State A. (

**b**) State B, with a narrowed adjustment bandwidth relative to state A (s < 1).

**Figure 6.**The change ΔP

_{net}in net available power as a function of the adjustment-bandwidth scaling (ABS) factor s with the power ratio k (=adjustment power divided by power harvested prior to the ABS) as a parameter.

**Figure 7.**P

_{net,B}after ABS with a factor of s with the power ratio k (=adjustment power divided by power harvested prior to the ABS) as a parameter.

**Figure 8.**Regions in the s-k-domain for which the condition ΔP

_{net}>0 is satisfied (hatched from northwest to southeast) and for which the condition P

_{net,B}>0 is satisfied (hatched from southwest to northeast). The crosshatched region marks all parameter combinations associated with an energy harvester the energy output of which may be improved by ABS. The optimum performance is achieved for parameter combinations on the dashed line marked with s

_{opt}.

**Figure 10.**Voltage of the energy storage and coil output voltage of the physical system in Reference [25].

**Figure 11.**The capacitor energy for various operating modes of the self-adaptive harvester from Reference [25]. (

**a**) Periodic adjustment with step 8 omitted. (

**b**) Periodic adjustment with steps 4 and 8 omitted.

**Figure 13.**The available power for various operating modes of the self-adaptive harvester from Reference [25].

**Figure 14.**The adjustment energy per frequency interval, ${\tilde{W}}_{\mathrm{T}}(f)$, for the harvester used as validation case (computed from data in [25]). The dash-dotted rectangle marks the frequencies to which the adjustment with the ABS factor s = 0.58 was allowed.

**Table 1.**Powers (@ τ = 70 s) and minimum process stationarity times τ

_{0}, τ

_{1}for various operating modes of the self-adaptive harvester from Reference [25].

Mode of Operation | P_{0}/mW | P_{T}/mW | P_{net}/mW | τ_{0}/s | τ_{1}/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 f_{a,8} | 10.1 | 6.2 | 3.8 | 44 | 55 | 112 s < τ |

Skip adjustment to f_{a,4} and f_{a,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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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Mösch, 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