**1. Introduction**

Vibration energy harvesters (VEHs) are devices that transduce vibration kinetic energy to electric power [1]. They have attracted much attention lately due to their potential as alternative power sources to batteries in vibration environments [2]. The vibration energy harvesters can be implemented with different operation principles, such as piezoelectric [3], electromagnetic [4], electrostatic [5], and triboelectric [6]. Among them, electromagnetic vibration energy harvesters (EVEHs) have the advantage of easy design and fabrication, relatively high output voltage and power, as well as good reliability [7]. The transduction of the EVEH is often realized by vibration-induced movement between a permanent magnet and a set of solenoid coils.

Traditional EVEHs can be implemented by fixing a macroscopic magnet or magnet array to the vicinity of a coil, which is a simple and robust approach [8]. However, the macroscopic EVEH is too bulky to power most of the smart electronic devices, giving rise to the strong demand to miniaturize the EVEHs. The typical methods to miniaturize the EVEHs are based on printed circuit board (PCB) [9–11] and microelectromechanical system (MEMS) technologies [12–15], where the coils are often fabricated with deposited planar coils. The main bottleneck for the planar coils is the limited number of turns, as the wires are confined within a single plane [16]. To cope with this limitation, the microfabricated vibrating proof mass is sometimes used with wound coils [17,18] However, such an approach has two issues: Firstly, the assembly process of the microcomponents is difficult; secondly, when the size of the coil is big, many of the turns in the coils cannot effectively cut the magnetic field generated with the micro magnet. In order to deal with these two issues, the authors have proposed a method based on stacked flexible coils, where high-density coils can be formed within the thickness range of 1–2 mm [19]. The low-cost repeating layers of flexible coils also simplifies the assembly and fabrication processes. It

has been shown that with flexible coils, the EVEH device was capable of generating an output voltage of 1.56 V, with a volume of 7.5 cm<sup>3</sup> .

Toward the goal of miniaturization, the fabrication of the vibrating proof mass has been shifted from PCB to MEMS technologies in a previous study [20]. In these designs, the permanent magnet is suspended by four folded springs, which is a classic type of layout for reducing the resonant frequency of MEMS structures (in addition to cantilever structures). However, it is discovered that these design strategies limit the miniaturization of the device. Therefore, a structure with a torsionally oscillating magnet is proposed in 2021, which has been proven to be as effective as the folded spring design in terms of output power, but with a relatively small footprint [21]. In this work, we will extend our conference contributions in [21] and systematically present the design, implementation, and results of a torsional EVEH device.

## **2. System Design**

#### *2.1. Structure of the EVEH*

The operation of the proposed EVEH device is based on the torsional movement of a disc magnet suspended by microfabricated torsional silicon springs, over a stack of high-density flexible coils, as shown in Figure 1a. Figure 1b shows the exploded view of the proposed EVEH device. A disc magnet is glued to the bottom surface of a silicon plate suspended by straight torsional springs. The magnetization direction of the disc magnet is along the z axis, as depicted by the coordinate in Figure 1a. Two types of torsional springs are presented in this work, i.e., the EVEH denoted as T1 consists of a straight torsional spring connecting to the magnet plate via a wide and short silicon beam (spring–plate connector), and device T2 consists of a folded torsional spring consists of three segments of straight springs, as shown in Figure 1c. For the fabrication of the silicon plate and torsional spring, a 200 µm thick monocrystalline silicon wafer is glued to a carrier wafer and etched through using inductively coupled plasma (ICP) reactive ion etching (RIE) process. A flexible coil stack is mounted below the disc magnet, and the coil–magnet distance is precisely controlled by the spacers between the coil and magnet (0.4 mm in this study). Each flexible coil layer consists of two oppositely wound spiral copper wires electroplated on the two sides of a 180 µm thick polyimide film, as shown in Figure 1d. The stack of flexible coils is aligned by the bolt holes and clamped together between two rigid FR4 layers, in order to be connected in series and form a coil with a large number of turns. Four M1 bolts are used to fix all the components. When the EVEH experiences external vibrations, the magnet oscillates torsionally (around the y axis in Figure 1) and generates electrical power in the coil. The torsional movement is mainly asserted on the long torsional beam, and the spring–plate connector will not deform due to its high stiffness. The main innovation of this technology is the torsional movement of the magnet and the formation of a high-density coil. The torsional movement of the magnet is capable of generating a large magnetic field gradient in the coils, with a reduced footprint. The overall footprint of the EVEH device is the same as the device reported in [20], but the functional area of the silicon layer (plate and spring area) is significantly reduced by using the torsional spring design. In this design, a much larger margin area is included in the device layer, which greatly facilitated the assembly process of the device. The stacked coils are formed by simple and low-cost repeating units of flexible coils. With proper technology development, the stacked coils may have the potential to be integrated into the fabrication of MEMS energy harvesters, enabling batch fabrication of the devices. The main design parameters of the EVEH device are listed in Table 1.

**Figure 1.** Schematic and exploded illustration of the assembled EVEH. (**a**) schematic illustration of the EVEH device; (**b**) exploded view of the EVEH device; (**c**) two types of torsional spring; (**d**) a layer of flexible coils. **Figure 1.** Schematic and exploded illustration of the assembled EVEH. (**a**) schematic illustration of the EVEH device; (**b**) exploded view of the EVEH device; (**c**) two types of torsional spring; (**d**) a layer of flexible coils.

Magnet Diameter **(**mm) 4.00 4.04

Coil Numbers of turns 40 40

( ) ( ) () ( ) <sup>2</sup>

*m c kz t m Y t*

+ +=

\* Functional area indicates the total area occupied by the silicon plate and springs.

Thickness **(**mm) 3.00 3.04

Magnet–coil distance **(**mm) 0.40 0.45

2

ω

sin

 ω

(1)


**Table 1.** Main design parameters of the EVEH device.

\* Functional area indicates the total area occupied by the silicon plate and springs.

using a second-order spring mass damper system as follows:

2 d d

d d *zt zt*

*t t*

*2.2. Modeling of the EVEH* 

## *2.2. Modeling of the EVEH*

According to Newton's second law, the EVEH dynamic behavior can be described using a second-order spring mass damper system as follows:

$$m\frac{\mathbf{d}^2 z(t)}{\mathbf{d}t^2} + c\frac{\mathbf{d}z(t)}{\mathbf{d}t} + kz(t) = m\omega^2 Y \sin(\omega t) \tag{1}$$

$$2\pi f = \sqrt{\frac{k}{m}}\tag{2}$$

where *m* is the proof mass (magnet) of the EVEH device, *c* is the total damping coefficient including mechanical and electromagnetic damping, *k* is the effective stiffness of spring, *z*(*t*) is the relative displacement between the magnet and base of the z axis, *y*(*t*) = *Y*sin(*ωt*) is the external vibration applied to the base, and *f* is the resonant frequency of spring. According to Equation (2), the resonant frequency of the device can be reduced by decreasing the effective stiffness of the spring or increasing the mass of the magnet. For a torsional spring, the effective torsional stiffness can be described as follows [22]:

$$T = \frac{KG\theta}{L} \tag{3}$$

where *K* is the spring torsion constant dependent on the form and dimensions of the cross section, *T* is the twisting moment, *L* is the length of the spring on y direction in Figure 1, *G* is the shear modulus of the spring, and *θ* is the torsional angle of spring (twisting around the y direction). According to [22], the value of *K* is given by

$$K = ab^3 \left[ \frac{16}{3} - 3.36 \frac{b}{a} \left( 1 - \frac{b^4}{12a^4} \right) \right] \text{ for} \quad a \ge b \tag{4}$$

where *a* and *b* are half of the width of the spring and half of the thickness of the spring, respectively. When the magnet moves under external vibration, according to Faraday's law, the induced voltage in a coil within a changing magnetic field is given by

$$V = -N\frac{d\phi}{dt} = -NA\frac{d\mathcal{B}}{dz}\frac{d\mathcal{z}}{d\theta}\frac{d\theta}{dt} \tag{5}$$

where *N* is the number of turns of the solenoid coil, *Φ* is the magnetic flux, *t* is time, *A* is the area of the coil, and *B* is the magnetic flux density. Additionally, when the EVEH device is twisting around the y direction (in Figure 1) at a small angle, the value of *z* equals *θ*. The magnetic flux density of a disc magnet along the central axis (z axis) can be described as follows [23]:

$$B\_z = \frac{B\_r}{2} \left\{ \frac{d+t}{\sqrt{(d+t)^2 + R^2}} - \frac{d}{\sqrt{d^2 + R^2}} \right\} \tag{6}$$

where *B<sup>z</sup>* is the magnetic flux density of the z axis, *B<sup>r</sup>* is the remnant magnetic flux density, *R* and *t* are the radius and thickness of the disc magnet, and *d* is the distance from the surface of the magnet. For a torsional spring, Equation (6) can be written as

$$B\_z = \frac{B\_r}{2} \left\{ \frac{r\sin\theta + d\_0 + t}{\sqrt{\left(r\sin\theta + d\_0 + t\right)^2 + R^2}} - \frac{r\sin\theta + d\_0}{\sqrt{\left(r\sin\theta + d\_0\right)^2 + R^2}} \right\} \cos\theta \tag{7}$$

where *r* is the distance between the center of the magnet and torsional spring, and *d*<sup>0</sup> is the initial distance between magnet and coil. Figure 2 shows the analytical calculated magnetic flux density along the z-axis as a function of *θ*, and the magnetic field gradient (d*B*/d*θ*) is linear at a small angle.

**Figure 2.** Curve of magnetic flux density along the z axis as a function of *θ.* **Figure 2.** Curve of magnetic flux density along the z axis as a function of *θ.*

Equation (4) indicates that under a certain external vibration, the following parameters can be increased in order to enhance the induced voltage: the number of coil turns (*N*), the area of the coil (*A*), the magnetic field gradient (d*B*/d*θ*) around the coil, or the torsional spring design (d*θ*/d*t*). However, when designing a miniaturized EVEH device, the coil area (*A*) and magnetic flux gradient (d*B*/d*θ*) are always limited by the device design (size and materials) and fabrication technologies. Therefore, a more effective way to increase induced voltage is by increasing the number of turns in the coil and the vibration amplitude of the EVEH device's movable part. The vibration amplitude can be improved by different spring designs. In comparison, the number of coil turns (based on the planar Equation (4) indicates that under a certain external vibration, the following parameters can be increased in order to enhance the induced voltage: the number of coil turns (*N*), the area of the coil (*A*), the magnetic field gradient (d*B*/d*θ*) around the coil, or the torsional spring design (d*θ*/d*t*). However, when designing a miniaturized EVEH device, the coil area (*A*) and magnetic flux gradient (d*B*/d*θ*) are always limited by the device design (size and materials) and fabrication technologies. Therefore, a more effective way to increase induced voltage is by increasing the number of turns in the coil and the vibration amplitude of the EVEH device's movable part. The vibration amplitude can be improved by different spring designs. In comparison, the number of coil turns (based on the planar coil technology) is difficult to improve in most of the reported MEMS EVEH devices.

coil technology) is difficult to improve in most of the reported MEMS EVEH devices. In order to study the dynamic characteristics of the EVEH device with a torsional spring, a FEM modal analysis of the proposed EVEH was performed with COMSOL Multiphysics software, as shown in Figure 3. In the FEM model, the simplification of the anchoring frame structure was carried out by applying fixed constraints to the end of the springs. This simplification is valid as the torsional spring is an in-plane spring. The material parameters of the silicon used in the FEM model were Young's Modulus of 170 GPa and Poisson's ratio of 0.28. The torsional vibration mode (the main resonance mode during operation, as the device is subjected to a uniform z-direction acceleration, and other resonant modes have a very low possibility be excited with the z-axis vibration) of the EVEHs is shown in Figure 3: T1 device twisting mode around the x-axis at 128.49 Hz, and T2 device twisting mode around the x-axis at 107.97 Hz. However, the FEM simulation of the torsional mode is not accurate, for the mode is influenced by the residual stress of the spring, and the complex stress distribution is difficult to include in the FEM model. The In order to study the dynamic characteristics of the EVEH device with a torsional spring, a FEM modal analysis of the proposed EVEH was performed with COMSOL Multiphysics software, as shown in Figure 3. In the FEM model, the simplification of the anchoring frame structure was carried out by applying fixed constraints to the end of the springs. This simplification is valid as the torsional spring is an in-plane spring. The material parameters of the silicon used in the FEM model were Young's Modulus of 170 GPa and Poisson's ratio of 0.28. The torsional vibration mode (the main resonance mode during operation, as the device is subjected to a uniform z-direction acceleration, and other resonant modes have a very low possibility be excited with the z-axis vibration) of the EVEHs is shown in Figure 3: T1 device twisting mode around the x-axis at 128.49 Hz, and T2 device twisting mode around the x-axis at 107.97 Hz. However, the FEM simulation of the torsional mode is not accurate, for the mode is influenced by the residual stress of the spring, and the complex stress distribution is difficult to include in the FEM model. The factor also ignored in the FEM model is the slight geometrical asymmetry caused by the device assembly process (e.g., the eccentric position of the magnet caused by the gluing process).

factor also ignored in the FEM model is the slight geometrical asymmetry caused by the device assembly process (e.g., the eccentric position of the magnet caused by the gluing

process).

*Micromachines* **2021**, *12*, x FOR PEER REVIEW 6 of 13

**Figure 3.** FEM modal analysis of the EVEH: (**a**) T1 device (128.49 Hz); (**b**) T2 device (107.97 Hz). **Figure 3.** FEM modal analysis of the EVEH: (**a**) T1 device (128.49 Hz); (**b**) T2 device (107.97 Hz). model of the simplified magnet–coil structure. In the figure, M and C represent the mag-

Figure 4 shows the distribution of the magnetic flux over the coil at different magnet– coil distances: (a) 0.15 mm and (b) 0.75 mm. It was obtained from a 2D axisymmetric FEM model of the simplified magnet–coil structure. In the figure, M and C represent the magnet and coil, respectively, and the arrowed lines indicate the magnetic flux lines of the magnet with z-axis magnetization. When the distance between magnet and coil is increased from 0.15 mm to 0.75 mm, the magnetic flux density along the z axis is decreased from 0.34 T to 0.29 T. This change of the magnetic flux in the coil represents the origin of the induction process of the EVEH under vibration excitation. Figure 4 shows the distribution of the magnetic flux over the coil at different magnet– coil distances: (a) 0.15 mm and (b) 0.75 mm. It was obtained from a 2D axisymmetric FEM model of the simplified magnet–coil structure. In the figure, M and C represent the magnet and coil, respectively, and the arrowed lines indicate the magnetic flux lines of the magnet with z-axis magnetization. When the distance between magnet and coil is increased from 0.15 mm to 0.75 mm, the magnetic flux density along the z axis is decreased from 0.34 T to 0.29 T. This change of the magnetic flux in the coil represents the origin of the induction process of the EVEH under vibration excitation. net and coil, respectively, and the arrowed lines indicate the magnetic flux lines of the magnet with z-axis magnetization. When the distance between magnet and coil is increased from 0.15 mm to 0.75 mm, the magnetic flux density along the z axis is decreased from 0.34 T to 0.29 T. This change of the magnetic flux in the coil represents the origin of the induction process of the EVEH under vibration excitation.

Figure 5 shows the assembly process of the proposed EVEH. As shown in Figure 5a, the flexible coil layers were stacked onto a rigid FR4 frame, which contained electrical **Figure 4.** The magnetic flux of the magnet–coil structure based on a 2D axisymmetric FEM model **Figure 4.** The magnetic flux of the magnet–coil structure based on a 2D axisymmetric FEM model at different magnet–coil distances: (**a**) 0.15 mm; (**b**) 0.75 mm.

#### feedthroughs on the bottom. Subsequently, four bolts were used to fix the stacked coils at different magnet–coil distances: (**a**) 0.15 mm; (**b**) 0.75 mm. **3. Experimental**

#### and bottom package, as shown in Figure 5b. Afterward, the disc magnet was glued to a *3.1. Assembly Process*

*3.1. Assembly Process* 

circular silicon plate connected to the torsional spring, as shown in Figure 5c. The intermediate package and the silicon layer device were then installed with four bolts, as shown in Figure 5d. Finally, the top cover of the package was installed by using four bolts, as shown in Figure 5e. The package of the EVEH device in this work was an anodized aluminum case for both mechanical protection and electrical insulation. **3. Experimental**  *3.1. Assembly Process*  Figure 5 shows the assembly process of the proposed EVEH. As shown in Figure 5a, Figure 5 shows the assembly process of the proposed EVEH. As shown in Figure 5a, the flexible coil layers were stacked onto a rigid FR4 frame, which contained electrical feedthroughs on the bottom. Subsequently, four bolts were used to fix the stacked coils and bottom package, as shown in Figure 5b. Afterward, the disc magnet was glued to a circular silicon plate connected to the torsional spring, as shown in Figure 5c. The intermediate

minum case for both mechanical protection and electrical insulation.

the flexible coil layers were stacked onto a rigid FR4 frame, which contained electrical

and bottom package, as shown in Figure 5b. Afterward, the disc magnet was glued to a circular silicon plate connected to the torsional spring, as shown in Figure 5c. The intermediate package and the silicon layer device were then installed with four bolts, as shown in Figure 5d. Finally, the top cover of the package was installed by using four bolts, as shown in Figure 5e. The package of the EVEH device in this work was an anodized alu-

package and the silicon layer device were then installed with four bolts, as shown in Figure 5d. Finally, the top cover of the package was installed by using four bolts, as shown in Figure 5e. The package of the EVEH device in this work was an anodized aluminum case for both mechanical protection and electrical insulation. *Micromachines* **2021**, *12*, x FOR PEER REVIEW 7 of 13

**Figure 5.** Assembly process of: (**a**) the stacked flexible coils; (**b**) the bottom package; (**c**) the magnet and device layer; (**d**) the intermediate package; (**e**) the package top cover. **Figure 5.** Assembly process of: (**a**) the stacked flexible coils; (**b**) the bottom package; (**c**) the magnet and device layer; (**d**) the intermediate package; (**e**) the package top cover. **Figure 5.** Assembly process of: (**a**) the stacked flexible coils; (**b**) the bottom package; (**c**) the magnet and device layer; (**d**) the intermediate package; (**e**) the package top cover.

#### *3.2. Measurement Setup 3.2. Measurement Setup*

*3.2. Measurement Setup*  Figure 6 shows the measurement setup for characterizing the dynamic behavior of the fabricated EVEH. In the setup, the EVEH device was mounted on a vibration shaker, which was excited by a sinusoidal signal produced by the signal generator and amplified by a power amplifier. The real-time excitation acceleration was monitored by an accelerometer mounted underneath the EVEH by a customized installation fixture. Experimental data including the EVEH output and applied acceleration signals were collected simultaneously by a NI USB-6211 data acquisition board and LabVIEW software, with a sampling rate of 1000 samples per second. A fourth-order Butterworth low pass filter with Figure 6 shows the measurement setup for characterizing the dynamic behavior of the fabricated EVEH. In the setup, the EVEH device was mounted on a vibration shaker, which was excited by a sinusoidal signal produced by the signal generator and amplified by a power amplifier. The real-time excitation acceleration was monitored by an accelerometer mounted underneath the EVEH by a customized installation fixture. Experimental data including the EVEH output and applied acceleration signals were collected simultaneously by a NI USB-6211 data acquisition board and LabVIEW software, with a sampling rate of 1000 samples per second. A fourth-order Butterworth low pass filter with a cut-off frequency of 500 Hz was used to remove the high-frequency noise, implemented in the LabVIEW software. The acquired voltage in this work has a measurement error of ±0.05 mV, derived from the standard deviation of 594 periods of sinusoidal voltage output. Figure 6 shows the measurement setup for characterizing the dynamic behavior of the fabricated EVEH. In the setup, the EVEH device was mounted on a vibration shaker, which was excited by a sinusoidal signal produced by the signal generator and amplified by a power amplifier. The real-time excitation acceleration was monitored by an accelerometer mounted underneath the EVEH by a customized installation fixture. Experimental data including the EVEH output and applied acceleration signals were collected simultaneously by a NI USB-6211 data acquisition board and LabVIEW software, with a sampling rate of 1000 samples per second. A fourth-order Butterworth low pass filter with a cut-off frequency of 500 Hz was used to remove the high-frequency noise, implemented in the LabVIEW software. The acquired voltage in this work has a measurement error of ±0.05 mV, derived from the standard deviation of 594 periods of sinusoidal voltage output.

**Figure 6.** Schematic illustration of the measurement setup. **Figure 6.** Schematic illustration of the measurement setup.

**4. Results and Discussions** 

*4.1. Fabrication Results of the EVEH Device* 

**4. Results and Discussions** 

**Figure 6.** Schematic illustration of the measurement setup.

*4.1. Fabrication Results of the EVEH Device* 

The designed EVEH device (T1 device) was successfully fabricated and assembled,

excitation, the relative position of the magnet and the coil changed to generate a changing magnetic field, and an electrical voltage was induced in the coil. The stacked flexible coils

The designed EVEH device (T1 device) was successfully fabricated and assembled, as shown in Figure 7a. Additionally, for the T2 device, the difference is the spring type shown in Figure 1c. The NdFeB disc magnet was glued to the bottom of the circular silicon

as shown in Figure 7a. Additionally, for the T2 device, the difference is the spring type shown in Figure 1c. The NdFeB disc magnet was glued to the bottom of the circular silicon microplate as the proof-mass (not visible in the photo). The magnet proof-mass was suspended by a long torsional beam above a stack of flexible coils. Under external vibration excitation, the relative position of the magnet and the coil changed to generate a changing magnetic field, and an electrical voltage was induced in the coil. The stacked flexible coils
