# Design of a Weighted-Rotor Energy Harvester Based on Dynamic Analysis and Optimization of Circular Halbach Array Magnetic Disk

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overall Design of the WREH

#### 2.1. Weighted Rotor for Energy Harvester

_{1}, y

_{1}) and (x

_{0}, 0), respectively. The outer radius of the wheel is R

_{1}; the distance between the pivot of the weighted rotor and the wheel center is R

_{2}. The angular displacement, velocity, and acceleration of the wheel rotation are represented by $\text{\alpha}$, $\dot{\text{\alpha}}$, and $\ddot{\text{\alpha}}$, respectively. According to Wang et al. [17], the dynamic equation of a weighted rotor in a rotating wheel is written as:

_{T}

^{*}is the generalized damping constant (including electromagnetic and mechanical damping), g is the gravitational acceleration, and a

_{0}is the linear acceleration of the wheel center. The swing angle, angular velocity, and angular acceleration of the weighted rotor are represented by $\text{\theta}$, $\dot{\text{\theta}}$, and $\ddot{\text{\theta}}$, respectively. The characteristic length (L

^{*}) is expressed as:

_{a}and m

_{b}), and the distance (d

_{a}and d

_{b}) between the pivot point of the weighted rotor and the center of mass of off-center parts, such as the weighted block and the adjustable screw, which are expressed as:

_{0}= 0), Equation (1) becomes:

**Figure 2.**Schematic diagram of WREH. (

**a**) 3D schematic view of the WREH for TPMS. (

**b**) Weighted rotor composed of magnets, a weighted block, and an adjustable screw.

^{−6}kg·m

^{2}, g = 9.81 m/s

^{2}, and considering a 15-inch wheel to decide R

_{1}= 0.300 m, R

_{2}= 0.203 m. The characteristic length and generalized damping constant are 0.200 m and 1.15 N·s·kg

^{−1}·m

^{−1}, respectively, to simulate the time response of the WREH. Figure 3 shows the swing angle θ when the car speed is 50 km/h under the initial conditions $\text{\theta}\left(0\right)=0$ and $\dot{\text{\theta}}$(0) = 0.

_{max}is the maximum swing angle defined according to the widest swing angle in one period. Equation (4) describes the dynamic behavior of the weighted rotor at a constant car speed. Linearized sinθ by θ in Equation (5), the natural frequency ${w}_{n}$ of the weighted rotor can be written as:

**Figure 3.**Transient response of the swing angle at the car speed 50 km/h and initial conditions $\text{\theta}\left(0\right)=0$ and $\dot{\text{\theta}}$(0) = 0.

^{*}= R

_{2}causes the system to oscillate in resonance at any wheel speed [22]. Figure 4 shows the variation of the maximum angular speed of a WREH in a steady state at various characteristic lengths and wheel speeds with a total generalized damping constant of 1.10 N·s·kg

^{−1}·m

^{−1}. When the condition L

^{*}= R

_{2}= 0.203 m is met, the angular speed varies continuously at wheel speeds from 200 to 1000 rpm. The maximum angular speed for L

^{*}> R

_{2}is lower than that for L

^{*}= R

_{2}at any wheel speed, meaning that the power generation is low. When L

^{*}<< R

_{2}, such as L

^{*}= 0.198 m, the maximum angular speed exhibits a sudden discontinuous jump at a critical wheel rotation speed. When L

^{*}is close but slightly lower than R

_{2}, such as L

^{*}= 0.200 m, and the critical wheel speed is reached, the maximum angular speed decreases with the increment of wheel speed. The decreasing slope of the maximum angular speed for L

^{*}<< R

_{2}is higher than that for L

^{*}< R

_{2}once the critical wheel speed is achieved. Therefore, the characteristic length L

^{*}of a WREH should be designed to be approximately equal to R

_{2}. In this case, a WREH with a characteristic length from 0.202 to 0.204 m is suitable at wheel speeds from 200 to 1000 rpm for energy harvesting.

**Figure 4.**Variation of the steady-state maximum angular speed of WREH with various L

^{*}and wheel speed with C

_{T}

^{*}= 1.10 N·s·kg

^{−1}·m

^{−1}.

^{−1}·m

^{−1}, and the initial conditions are $\text{\theta}\left(0\right)=0$ and $\dot{\text{\theta}}$(0) = 0. The wheel rotation speed is represented by a yellow line shown in Figure 5. For the wheel radius of 0.300 m, the wheel rotation speed of 707 rpm is equal to the car speed of 80.2 km/h. As shown in Figure 5a, when the wheel angular acceleration speed is 73.7 rad/s

^{2}, the maximum swing angles of the three weighted rotors with different L

^{*}are close to those at the same constant wheel speed. Consider another example depicted in Figure 5b, when the high wheel angular acceleration speed of 148 rad/s

^{2}to the same final speed 707 rpm, the weighted rotor with L

^{*}= 0.191 m demonstrates a jump phenomenon, but this does not occur for the other two L

^{*}(0.198 m and 0.203 m). The weighted rotors with L

^{*}close to R

_{2}avoid the jump phenomenon even after a wheel angular acceleration is executed. The acceleration speed shown in Figure 5b is equal to that of a car accelerating from 22.6 to 80.2 km/h in 5 s, which represents the acceleration performance of a normal car on a highway. When a well-weighted rotor satisfies the requirement of L

^{*}= R

_{2}, the variation in maximum swing angle and maximum angular velocity is continuous, and no jump phenomenon occurs.

**Figure 5.**Maximum swing angle under various L

^{*}. (

**a**) After angular acceleration of 73.7 rad/s

^{2}. (

**b**) After angular acceleration of 148 rad/s

^{2}.

#### 2.2. Output Voltage and Electromagnetic Damping

_{L}) on the external load is expressed as:

_{L}and R

_{C}are the resistance of the external load and coil wires, respectively. The output (cross) voltage on the external load is measured using an oscilloscope and the angular velocity (dθ/dt) is calculated according to the rotor turns per second. In this case, R

_{L}= 550 Ω and R

_{C}= 85 Ω, based on Equations (6) and (7), the average value of dφ/dθ was calculated as 0.059 V·s in different angular velocity experiments. The value of R

_{L}is decided to make the damping constant approach to 1.10 N·s·kg

^{−1}·m

^{−1}. The torque T

_{e}caused by the electromagnetic damping to the weighted rotor can be expressed as:

_{e}

^{*}can be expressed as:

#### 2.3. Optimization of Halbach Array Magnetic Circuit

_{m}as the parameter to be optimized is defined by the ration between the axial-direction sector angle a to the radial-direction sector angle b. Because the relative motion between the circular Halbach array disk and coil sets is on the x–z plane, the y component of the magnetic flux density above the circular Halbach array disk should be analyzed. The coil rotation region is located between the inner and outer radius of the magnetic disk and is above the upper surface of the magnetic disk from 2 to 5 mm, as shown in Figure 7a. The magnetic field strength was calculated using COMSOL 4.2a software (COMSOL, Burlington, MA, USA) with a finite element module and the residual magnetism of each magnet set at 1.4 T. The relative permeability of the Neodymium-Iron-Boron (NdFeB) magnets and air are set at 1.05 and 1.00, respectively. The Y component of the magnetic flux density in the different altitudes from the upper surface of the magnetic disk to the altitude plane is shown in Figure 7b. Considering the uniform distribution of coil turns in each layer, the Y component of magnetic flux density for any r

_{m}in EMF calculation is determined according to the average value from the altitude planes of 2–5 mm (the thickness of the coil set), as shown in Figure 7c.

_{m}is applied as the objective function. The magnetic flux $\text{\phi}$ from each coil rotation in the x–z plane is expressed as v

_{t}·D·[B

_{y}·(R

_{c}+ D/2) − B

_{y}·(R

_{c}− D/2)], where v

_{t}, D, B

_{y}, and R

_{c}are the velocity of the coil, diameter of the coil, Y component of magnetic flux density, and position of the coil center, respectively. In the same layer, the diameter of each coil should be considered individually. Figure 8 shows the EMFs for different sector-angle ratios based on the same magnetic disk volume with an angular speed of 1 rps and the parameters listed in Table 1. The sector-angle ratios are plotted on logarithmic coordinates along the horizontal axis, and the sector angles are also indicated in Figure 8. A higher r

_{m}causes the circular Halbach array magnetic disk to approach the multipolar magnetic disk magnetized in tangential directions. Conversely, a lower r

_{m}causes the circular Halbach array magnetic disk as the multipolar magnetic disk magnetized in axial directions. The optimal r

_{m}for the largest EMF is located between the highest and the lowest r

_{m}. Moreover, r

_{m}= 1 is not the optimal design. As shown in Figure 8, r

_{m}= 19/26 provides the largest EMF as the optimal value of the sector-angle ratio for the circular Halbach array magnetic disk applied to the energy harvester for dimensions of the magnetic disk in this paper.

**Figure 7.**Simulations of the Y component of magnetic flux density. (

**a**) The definition of the analyzed region. (

**b**) The Y component of magnetic flux density for sector angle ratio (25/20) in different planes. (

**c**) The average Y component of magnetic flux density for sector angle ratio (25/20).

**Figure 8.**Peak value of electromotive force for magnetic disks with different sector angle ratios r

_{m}.

Parameter | Circular Halbach Array Disk |
---|---|

Dimensions of individual magnet | r_{out} = 13.0 mm |

r_{in} = 7.0 mm | |

t_{m} = 3.0 mm | |

Residual magnetism of magnet by axial magnetization | B_{r} = 1.4 T |

Coil wire diameter | d = 0.1 mm |

Coil turns in a layer | N_{r} = 20 |

Coil layers in a set | N_{t} = 30 |

Thickness of coil set | t_{c} = 3.0 mm |

Inside diameter of the coil set | D_{in} = 2.0 mm |

Outside diameter of the coil set | D_{out} = 6.0 mm |

Air gap between magnetic disk and coil sets | t_{p} = 1.0 mm |

Number of coil turn | 600 |

## 3. Experiments and Results

_{m}= 19/26, as shown in Figure 10b. The generalized damping includes electromagnetic and mechanical damping. The mechanical damping attributed to the ball bearing friction and air-dragging force disperses the energy and is unavoidable. The mechanical damping can be estimated by measuring the output voltage of a WREH in the open circuit in a steady state. For this prototype, the mechanical damping measured in the experiments was 0.20 N·s·kg

^{−1}·m

^{−1}. In the experiments, a 550 Ω external resistor was connected in series with the WREH. The voltage across the external resistor (V

_{L}) was measured as the experimental output voltage, using an oscilloscope. Figure 11a provides the instant output voltage results when the plate-rotation speed was accelerated from 200 to 500 rpm in 5 s, as indicated by the blue line. The maximum voltages of simulations and experiments are represented by red and green dots, respectively. When plate-rotation speeds were below 430 rpm, two additional small peaks appeared in the output voltage wave for one period, as shown in Figure 11b, which shows the voltage measurement from 10 to 11 s. This phenomenon is attributed to the swing angle of the weighted rotor being wider than the one period of the magnetic pole arrangement of 45°. An entire swing period of the weighted rotor is still the same as the plate-rotation period. Within the acceleration region from 20 to 25 s, both the simulation and experimental output voltage vary according to the plate-rotation speed and are accompanied by a transient response of 20–30 s. The differences between the simulation and experimental voltage at low plate-rotation speeds are smaller than those at high speeds.

**Figure 10.**Prototype of the WREH with mass moment of inertia 1.725 × 10

^{−6}kg·m

^{2}. (

**a**) The weighted rotor assembled with the coil sets. (

**b**) The circular Halbach array magnetic disk of sector angle ratio r

_{m}= 19/26.

**Figure 11.**Instant output voltage of the WREH from 200 to 500 rpm. (

**a**) Simulation and experimental results. (

**b**) Experimental output voltage of the WREH from 10 to 11 s.

_{0}to t

_{1}, the electromagnetic damping obtained from Equation (9) as well as the square of the angular velocity of the weighted rotor:

Plate rotation speed (rpm) | Average power generation simulation (μw) | Average voltage simulation (V) | Average power generation experiment (μw) | Average voltage experiment (V) |
---|---|---|---|---|

300 | 404 | 0.471 | 399 | 0.468 |

350 | 454 | 0.499 | 445 | 0.494 |

400 | 507 | 0.528 | 491 | 0.520 |

450 | 555 | 0.552 | 526 | 0.538 |

500 | 585 | 0.567 | 535 | 0.541 |

## 4. Conclusions

^{*}is equal to R

_{2}, then the natural frequency of a well-weighted rotor matches the rotation frequency of the car wheel. Consequently, resonance occurs at any wheel speed, and the well-weighted rotor demonstrates no jump phenomenon at either the constant or acceleration speeds of the rotation plate. To provide a suitable L

^{*}, the weighted condition can be fine-adjusted using the adjustable screw. The extremely high angular acceleration of the plate causes only a transient response within 2–5 s and does not affect the steady-state response of a well-weighted rotor. Therefore, oscillation at a wide angle and high angular velocity is feasible for generating high power. The mathematical model of output voltage and power generation employed in numerical simulation demonstrated favorable agreement with the experimental results. In the experiments, when the plate-rotation speed is 300–500 rpm—which is equal to normal car speeds of 33.9–59.5 km/h—the power generation of a WREH is between 399 and 535 μW. These results demonstrate that a WREH has the potential to be a power source for a TPMS.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, Y.-J.; Hao, Y.-T.; Lin, H.-Y.
Design of a Weighted-Rotor Energy Harvester Based on Dynamic Analysis and Optimization of Circular Halbach Array Magnetic Disk. *Micromachines* **2015**, *6*, 375-389.
https://doi.org/10.3390/mi6030375

**AMA Style**

Wang Y-J, Hao Y-T, Lin H-Y.
Design of a Weighted-Rotor Energy Harvester Based on Dynamic Analysis and Optimization of Circular Halbach Array Magnetic Disk. *Micromachines*. 2015; 6(3):375-389.
https://doi.org/10.3390/mi6030375

**Chicago/Turabian Style**

Wang, Yu-Jen, Yu-Ti Hao, and Hao-Yu Lin.
2015. "Design of a Weighted-Rotor Energy Harvester Based on Dynamic Analysis and Optimization of Circular Halbach Array Magnetic Disk" *Micromachines* 6, no. 3: 375-389.
https://doi.org/10.3390/mi6030375