# Optimal Structural Design of a Magnetic Circuit for Vibration Harvesters Applicable in MEMS

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^{2}

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## Abstract

**:**

## 1. Introduction

**M**; magnetic flux Φ; magnetic lines of force; oriented area

**S**enclosed by the coil thread; electric coil; and, character of the generator’s core motion with respect to the coil. The related Figure 2c then introduces a design version that minimizes the impact of external electromagnetic fields (non-stationary) on the principal function of the generator.

^{−2}). In the discussed concept, the resonance might vary, according to the origin of the vibrations, from the tuned resonance frequency f

_{r}by up to tens of percent.

## 2. Designing the MG

_{out}. The required output power P

_{out}of the optimal design depends on the type of the output load Z. The optimal arrangement of the MG is based on the concepts in Figure 1, Figure 2c, and Figure 3, with the magnetization orientation indicated. In terms of the mechanical properties, the device was discussed in dedicated papers and patents, such as [10,25]. Figure 4 presents details of the transformation process and electricity generation; the actual engineering approach adopted in solving these procedures then embodies the necessary precondition for the subsequent identification of the optimal design. The mathematical model outlined in [2] is, in a basic form, incorporated in the corresponding Formula (5), below.

_{res}and it constitutes the basis of the optimal approach. Such an arrangement allows for us to reach the maximum possible harvest rate and transform the field into an electric voltage; Figure 4 shows the corresponding preconditions.

## 3. Modeling the MG

**E**(t) denotes the electric field intensity vector,

**B**(t) is the magnetic flux density vector,

**v**(t) represents the generator core position drift in time (the instantaneous velocity) vector, S stands for the cross section of the area with magnetic flux Φ, and l denotes the curve along the boundary of the S. Figure 4 illustrates the change of the magnetic flux of the field (t

_{i1}, …, t

_{i4}) and also the resulting induction of the voltage u. The behavior of the voltage u(t) can be evaluated by following the steps that are indicated in Figure 4; this behavior assumes the validity of Equation (1), magnetic flux Φ configuration, and electric coil shape with an active surface S

_{c}.

_{k}and W

_{p}, respectively, which are related to the movement of the generator’s core, can be defined as

**n**denotes the normal vector; γ is the specific conductivity of the wire; ℓ is the length of the shift caused by the specific strength;

**B**denotes the magnetic flux density vector;

**J**represents the current density vector; V

_{Jc}stands for the coil wire volume; and, V

_{J}is the volume of the electrically conductive components. The MG system then also includes the braking forces

**v**of the moving parts, m represents the mass, k stands for the stiffness coefficient, l

_{c}is the damping coefficient, and F

_{z}is the forces affecting the moving parts. The simplified model is described as

**B**

_{br}is the braking magnetic flux density,

**J**

_{v}denotes the current density of the electrically conductive components,

**J**

_{circ}represents the current density in the coil winding, and i stands for the instantaneous value of the current through the coil. The geometrical model that is applied in ANSYS (Version 12, ANSYS inc., Houston, USA) is presented in sources [23], ref. [2] as well as Figure 3a and Figure 5a. Figure 6, as below, shows the typical analysis of the ANSYS numerical model. The novel (optimal) generator design was tested on both a pneumatic and an electrodynamic shaker to verify the magnetic independence of the proposed solution. The magnetic circuit is designed such that its structure is enclosed within the body of the generator, ensuring reduced sensitivity to the external magnetic field and its changes. This parameter is of interest for application in the periodic structure of the outlined design usable in MEMS. The assumptions embodied in the variant from Figure 2c were experimentally verified.

## 4. Selecting the MG Core Design

## 5. Critical Parameters of the MG Design

- mechanical dynamics;
- electromagnetic field; and,
- electronic systems (power management blocks).

_{res}. Regarding this task, an aspect of major importance consists in the nonlinear stiffness coefficient k in the entire generator system (Figure 8). If the factor is adequately considered, then the device is capable of providing an operational efficiency of approximately 90%; in such conditions, the MG will operate at its maximum efficiency with minimal vibrations. The nonlinearity of the stiffness coefficient k depends on the choice of principal approach (Figure 1a,b and also Figure 6, Figure 7 and Figure 8).

_{1}, t

_{2}, ϕD

_{1}− ϕD

_{3}.

## 6. Microstructures

_{c}is the effective area of the coil, Figure 2c; S

_{p}denotes the area of the pole extension; t

_{1}represents the thickness of the pole extension of the MG core; N

_{seg}is the number of the electrical winding segments; N

_{pol}is the number of the pole extensions of the core and shell; and, f (dg/dt), g (dg/dt) denote the time variation of the gravitational acceleration of the moving part of the microgenerator. The correct setting of the MG0 structure, Figure 9, Figure 10 and Figure 11, can be verified through measuring or evaluating the behavior of the output voltage on the terminals of the MG segment. The obtained instantaneous values of the patterns of the voltage u(t) are then applicable in expressing, via the indirect method and based on the above formula (5), the observed physical quantities of the model.

_{s}represents the length of the MG element, and l

_{h}is the space between the elements of the periodically structured field of microgenerators. During the propagation of vibrations, the structure behaves such that the electric voltage is almost in phase at the output of the windings.

_{s}, then determines the achievable resonant frequency, harvesting efficiency, and adjustment of the harvester’s lower sensitivity limit. Importantly, each concrete application of the principle requires designing a suitable MEMS structure by using the above-shown models.

## 7. Comparing MG Concepts, Designs, and Structures

**B**into the air gap that is to contain the generator winding, Figure 6; such an analysis and presetting have to facilitate the maximum magnetic flux change in time and space, as formulated within Faraday’s law of induction (1) and to enable geometrical configuration of the winding shown in Figure 4. In the given context, it is advantageous to employ the double action system to facilitate a magnetic flux change, as indicated in Figure 5a and Figure 7.

_{out}.

^{3}]; this quantity enables us to express the effectivity of individual generator concepts and structures as regards harvesting quality. The last four lines indicate a comparable quantity for fossil and nuclear fuels, batteries, and supercaps.

## 8. Conclusions

_{r}= 10–50 Hz (frequent in the automotive and aeronautics sectors), are integrable into miniaturized microgenerator structures working within the range of G = 0.05 g–0.08 g. This concept could advantageously employ in practice the higher level of vibrations available compared to the design based on principle I [3]. The generators that employ principle I operate at vibration levels higher than G ≥ 0.15 g. Generally, the winding configuration variants convenient for the frequency ranges of f

_{v}= 1–10 Hz, f

_{v}= 50–150 Hz are demonstrated in Figure 2b,c.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The principal configuration of the core of the milli- or micro generator (MG): (

**a**) a beam version, principle I; (

**b**) a beam version, principle II [2].

**Figure 2.**The basic arrangement of the investigated rotationally symmetric geometry device exploiting Faraday´s induction law: (

**a**) the classic solution; (

**b**) the novel arrangement; and, (

**c**) the option with a closed magnetic circuit to minimize (optimize) the impact of external magnetic fields [2].

**Figure 3.**The basic arrangement of the investigated device exploiting Faraday´s induction law: (

**a**) the MG core; and (

**b**) the magnetic flux density distribution along the z axis, line A-A.

**Figure 4.**The electric voltage induction in the applied coil, (

**a**–

**g**), according to Faraday´s law of induction [2].

**Figure 5.**A geometrical model of the tested MG [2]: (

**a**) the core of MG; and, (

**b**) the functional sample subjected to a shaker-based test of the double-action winding.

**Figure 6.**A geometrical rotationally SYMMETRIC model of the MG0 based on principle II [2]: (

**a**) the ANSYS geometrical model; (

**b**) the core; and, (

**c**) the optimal design, detailed distribution of the magnetic flux density module B [T].

**Figure 7.**The MG core with a ferromagnetic circuit according to principle II [2]: (

**a**) a structurally simple variant, acceleration g within the interval of 0.01 g–0.3 g; (

**b**) a scheme of the configuration.

**Figure 8.**The applied stiffness characteristics [2], coefficient k; the behavior is nonlinear in both of the MG magnetic circuit principles.

**Figure 9.**A component diagram of the MG0 design with the linearized coefficient of stiffness k (based on principle II).

**Figure 10.**A component diagram of the MG0 design with the linearized coefficient of stiffness k (based on principle II); (

**a**) configuration A, and (

**b**) its output voltage U

_{out}.

**Figure 11.**A component diagram of the MG0 design with the linearized coefficient of stiffness k (based on principle II); (

**a**) configuration B, and (

**b**) its output voltage.

**Figure 12.**A component diagram of the MG0 design with the linearized coefficient of stiffness k (based on principle II); (

**a**) configuration C, and (

**b**) its output voltage.

**Figure 15.**The vibration microgenerator designed by Beeby et al. [12].

**Figure 16.**The electromagnetic design by Wang et al. [15].

**Figure 17.**The electromagnetic vibration energy harvester using cylindrical geometry, developed by Yang et al. [16].

**Figure 20.**The tested microgenerator [2] based on Principle I: (

**a**) MG I—the dimensions of 90 × 40 × 30 mm, U

_{out max}= 300 V; (

**b**) MG II—the dimensions of 50 × 27 × 25 mm, U

_{out max}= 20 V; and (

**c**) the instantaneous behavior of the output electrical voltage in MG I and MG II (the effect of the stiffness coefficient k—Figure 8).

**Figure 21.**The devices based on Principle II [2]: (

**a**) MG III—the dimensions of 50 × 25 × 25 mm, U

_{out max}= 10 V; (

**b**) MG IV—the dimesnions of 50 × 35 × 25 mm, U

_{out max}= 20 V; and, (

**c**) the instantaneous behavior of the output electrical voltage in MG III and MG IV (the effect of the stiffness coefficient k—Figure 8).

Reference | Permanent Magnet Type | Generator Body Size x, y, z [m] | Resonant Frequency f_{r} [Hz] | Amplitude Mech. Part A [m] | Output Power P_{out} [W] | Output Voltage U_{out} [V] | Load Resistance R [Ω] | Acceleration G, g = 9.81 [m/s] | Effective Power Density [W/m^{3}] |
---|---|---|---|---|---|---|---|---|---|

Beeby et al. [12], 2007 | − | 375 mm^{3} | 52 | − | 2 × 10^{−6} | 0.428 RMS | 4000 | 0.06 g | ≈6 |

Zhu et al. [13], 2010 | FeNdB | 2000 mm^{3} | 67.6–98 | 0.6 × 10^{−3} | 61.6–156.6 × 10^{−6} | − | − | 0.06 g | ≈30–80 |

Kulkarni et al. [11], 2008 | FeNdB | 3375 mm^{3} | 60–9840 | 1.5 × 10^{−3} | 0.6 × 10^{−6} | 0.025 | 52,700 | 0.398–4 g | ≈0.2 |

Wang et al. [15], 2007 | FeNdB | 256 mm^{3} | 121.25 | 0.738 × 10^{−3} | - | 0.06 | - | 1.5 g | - |

Lee et al. [17], 2012 | FeNdB | 1.4 × 10^{−4} m^{3} | 16 | − | 1.52 × 10^{−3} | 4.8 | 5460 | 0.2 g | ≈10 |

Yang et al., [16], 2014. | − | 50,000 mm^{3} | 22–25 | 13.4 × 10^{−3} | 0.7–2.0 | 110 | 0.6 g | ≈270 | |

Elvin et al., [14], 2011 | − | 15,000 mm^{3} | 112 | − | 4 × 10^{−6} | 0.007 | 986 | - | ≈0.26 |

MG I [2], 2006 | FeNdB | 90, 40, 30 mm | 20–35 | 50 × 10^{−6}–400 × 10^{−6} | 70 × 10^{−3} | 4–60 (300) p-p | 7500 | 0.15–0.4 g | ≈650 |

MG II [2], 2006 | FeNdB | 50, 27, 25 mm | 17–25 | 50 × 10^{−6}–400 × 10^{−6} | 19.5 × 10^{−3} | 6−15 | 5000 | 0.1–0.7 g | ≈60 |

MG III | FeNdB | 50, 25, 25 mm | 21–31.5 | 50 × 10^{−6}–400 × 10^{-6} | 5.0 × 10^{−3} | 1.0–2.5 | 600 | 0.05–0.4 g | ≈15 |

MG IV | FeNdB | 50, 35, 25 mm | 21–31.5 | 50 × 10^{−6}–400 × 10^{−6} | 8.0 × 10^{−3} | 1.0–2.5 | 1200 | 0.05–0.4 g | ≈18 |

*Lith. battery [19], 2018 | ≈40 × 10^{6} | ||||||||

*supercap [20], 2010 | ≈3–5 | ||||||||

*fuel | ≈4 × 10^{9} | ||||||||

*U_{235} | ≈9 × 10^{16} |

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**MDPI and ACS Style**

Szabó, Z.; Fiala, P.; Zukal, J.; Dědková, J.; Dohnal, P.
Optimal Structural Design of a Magnetic Circuit for Vibration Harvesters Applicable in MEMS. *Symmetry* **2020**, *12*, 110.
https://doi.org/10.3390/sym12010110

**AMA Style**

Szabó Z, Fiala P, Zukal J, Dědková J, Dohnal P.
Optimal Structural Design of a Magnetic Circuit for Vibration Harvesters Applicable in MEMS. *Symmetry*. 2020; 12(1):110.
https://doi.org/10.3390/sym12010110

**Chicago/Turabian Style**

Szabó, Zoltán, Pavel Fiala, Jiří Zukal, Jamila Dědková, and Přemysl Dohnal.
2020. "Optimal Structural Design of a Magnetic Circuit for Vibration Harvesters Applicable in MEMS" *Symmetry* 12, no. 1: 110.
https://doi.org/10.3390/sym12010110