*2.3. Simulation*

Based on the above discussion on the electromechanical modeling of the proposed energy harvester, we performed time domain simulations using an appropriate simulation tool (MATLAB) to predict the output voltage generated by both PE and EM transducers simultaneously while operated by low-frequency excitation (i.e., human-limb motion). The parameters used in the simulation were calculated from the geometry and material parameters of the device components and will be discussed in the following sections. In the simulation, it was considered that the system was excited periodically in the horizontal direction at 5 Hz frequency and 2 g (g = 9.8 ms<sup>−</sup>2) peak acceleration. It was also assumed that the sphere started moving from the left end of the rectangular channel and moved back and forth periodically in response to the applied excitation.

Figure 3 shows the simulated open circuit voltage waveforms generated by the PE and EM transducers. In each voltage waveform, two consecutive maximum peaks occurred due to the transverse impact when the sphere slid over the proof-mass during its forward and backward motion in one cycle and the process continued as long as the excitation existed. The positive half cycle of the acceleration waveform indicates the forward motion and the negative half cycle indicates the backward motion of the sphere. Since both transducers generated voltage simultaneously, the frequency of both open circuit voltage waveforms was the same, which was the resonant frequency of the vibrating piezoelectric beam. As seen in the figure, the amplitudes of the instantaneous voltage waveforms (in both cases) decayed exponentially with time due to mechanical damping and became almost zero until the next impact occurred.

**Figure 3.** Simulated open circuit voltage waveforms of the piezoelectric (**a**) and electromagnetic (**b**) transducers at 5 Hz excitation frequency and 2 g peak acceleration.

#### **3. Prototype and Test Setup**

## *3.1. Prototype Fabrication*

A macro-scale prototype of the proposed hybrid energy harvester was fabricated and tested. The PE transducer of the prototype comprised a piezoelectric (PZT) parallel bimorph (SMBA4510T05M, STEMiNC, Davenport, FL, USA), a neodymium (N52) cylinder magnet, and a cubic iron mass of 6 mm length with a 1 mm high parabolic top, both glued to the middle of either sides of the bimorph beam. A suitable assembly of a cylinder magne<sup>t</sup> and a 1000-turn coil (0.1 mm diameter laminated copper wire) attached to a printed circuit board (PCB) constituted the EM transducer. A 316 stainless steel ball was enclosed in a rectangular shaped aluminum channel (inner area 10.5 × 10.5 mm2) with a square (7 × 7 mm2) opening at the middle of its bottom wall, which was assembled on top of the PE transducer. The channel opening was occupied by the parabolic-top of the cubic mass with a 0.4 mm overlap with the ball. Figure 4 shows a photograph of the fabricated prototype device along with the schematics of the electrical connection of the piezoelectric bimorph and the magnet-coil assembly. The geometric parameters and material properties of the components are tabulated in Table 1.

**Figure 4.** Photographs of the prototype components (**a**), fabricated prototype (**b**), schematics of the parallel bimorph connection (**c**), and the magnet-coil assembly (**d**).


**Table 1.** Geometric parameters and materials properties of the harvester components.

#### *3.2. Human-Limb Motion Test Setup*

Our fabricated energy harvester was tested by human-limb motion to observe its power generation capability under a real-world situation. In order to achieve a robust test setup, it required convenient (small and portable) measuring equipment to record the characteristics (frequency and amplitude) of the excitation generated by human-limb motion. An EVAL-ADXL326Z (Analog Devices Inc., Norwood, MA, USA) tri-axial MEMS accelerometer kit (mounted on the harvester prototype) in conjunction with a XR5-SE (Pace Scientific Inc., Mooresville, NC, USA) data logger was used to record the excitation profile of human-limb motion for further analysis. The outputs of both PE and EM transducers were connected to a digital storage oscilloscope (TDS 5052B, Tektronix Inc., Beaverton, OR, USA) to observe and record the output responses. Furthermore, benchtop tests using an electrodynamic shaker were conducted to observe the damping behavior of both transducers and to determine the optimal overlap (±d) between the magne<sup>t</sup> and coil, as described in our previous work [24].

#### **4. Experimental Results and Discussion**

#### *4.1. Optimal Overlap and Damping Measurements*

In order to generate maximum possible voltage and power from the prototype, it was important to determine the optimum overlaps between the magne<sup>t</sup> and coil as well as between the freely-movable sphere and parabolic-top of the proof-mass. The optimum magnet-coil overlap was determined by a benchtop test setup [24] using an electrodynamic shaker whereas the overlap between the sphere and the mass-top was determined by the human-limb vibration test setup (due to the limitation of the shaker to generate low-frequency, large-amplitude excitation). As seen from Figure 5a, the optimum magnet-coil overlap was −1 mm. The lateral gap between the magne<sup>t</sup> and coil was also 1 mm. Since the absolute values were not primarily of interest in determining the optimum magnet-coil overlap, normalized values were used. Figure 5b shows the change in the open circuit voltages generated by both the PE and EM transducers with the change in the overlap between the sphere and the mass-top. As seen from the figure, the sphere could not make significant contact with the parabolic top when the overlap was 0.2 mm as the clearance between the ball and inner surface of the channel was 0.2 mm. On the other hand, the sphere could not slide over the mass-top and was captured in the middle when the overlap was 0.5 mm because the speed/force of the sphere was not sufficient to pass through. The 0.4 mm overlap between the sphere and mass-top was considered as the optimum value since the open circuit voltages were the maximum for both the PE and EM transducers. The error bars in Figure 5b indicate the range of voltages generated for multiple attempts as the characteristics of the excitation (frequency and amplitude) applied by human-limb were not always the same.

**Figure 5.** Normalized open circuit voltages for different magnet-coil overlaps (**a**) and open circuit voltages of the transducers for various overlaps between the freely movable sphere and the parabolic top of the proof-mass (**b**).

The damping behavior of both PE and EM transducers were determined by the impulse response test using an electrodynamic shaker [24]. A high amplitude impulse (30.3 ms<sup>−</sup><sup>2</sup> with 50 ms pulse period and 500 μs pulse width) was applied to the harvester. Then, the mechanical damping ratio (ζ*m*) and total damping ratio (ζ*T*) of both transducers were estimated from the open circuit and loaded impulse response signals, respectively. The logarithmic decrement method was used to calculate the damping ratio as

$$\zeta = \frac{1}{2\pi} \ln \left( \frac{a\_1}{a\_2} \right) \tag{11}$$

where *a*1 and *a*2 are the amplitudes of two consecutive peaks in the impulse response plot of the transducer. Subtraction of the mechanical damping ratio (ζ*m*) from the total damping ratio (ζ*T*) gives the electrical damping ratio (ζ*e*). By conducting this experiment, the mechanical damping ratio was found to be 0.011. On the other hand, the electrical damping ratio for the piezoelectric transducer and electromagnetic transducer were 0.017 and 0.016, respectively. It should be noted that the electrical damping values were determined by the impulse response across the corresponding optimum load resistances of the transducers, which were determined by measuring the voltage across various load resistors and calculating the power delivered to them. The power is experimentally equal to *V*<sup>2</sup> *<sup>p</sup>*−*<sup>p</sup>*/4*Rl*, where *Vp*−*<sup>p</sup>* is the peak–peak value of the measured voltage across each load resistance *Rl*.
