# Analysis of Cantilever Triple-Layer Piezoelectric Harvester (CTLPH): Non-Resonance Applications

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## Abstract

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## 1. Introduction

_{31}= 0.38 for PZT-5H) directed Tadmor et al to reform Weinburge and Wang equations by highlighting the effect of large electromechanical coupling coefficients in the equations [22]. Ismail et al corrected the model by considering the geometry of the beam [23]. A model for a multi-layer energy piezoelectric harvester using carbon and glass fibers was developed by Lu et al [24]. However, the analysis of this harvester is based excitation, which is suitable for high-frequency applications.

## 2. Principle of Operation

## 3. Theoretical Analysis

#### 3.1. Assumptions and Constitutive Equations

_{b}). This beam is deformed by a uniformly distributed body force of p(x) and an applied tip force of $F$. The tip force is caused by a repulsive force between the slider magnet and the tip magnet of the CTLPH. The coordinate system is in a way that x or 1 axis is in the length direction, y or 2 axis is in the width direction, and z or 3 axis is in the thickness direction of the CTLPH. The transverse vibration causes the deflection of w(x,t) in the y-direction. The beam has an arbitrary cross-section, A(x), and is made of three layers of piezoelectric–substrate–piezoelectric. Considering I (x) as the moment of inertia about the z-axis and E(x) as the effective elastic modulus of the triple-layer beam, the bending stiffness of the beam is EI(x). The CTLPH is loaded by a mechanical force that causes a distribution of shear force Q(x,t) and moment M(x,t) along the CTLPH. Figure 2b shows a side view of an infinitesimal element of the CTLPH positioned at x. Based on the selected coordinate system, the tip force and lateral vibration are in the y direction. The central layer is a substrate made of an elastic material with a thickness of t

_{s}. The substrate is coated with piezoelectric layers with thickness t

_{p}. As with [19,20], we assume that the layers are bonded strongly so that there is no slip between them. Moreover, each part of the CTLPH is in static equilibrium. The radius of curvature (R

_{o}(x)) induced by the applied tip force is much larger than the CTLPH’s thickness, and the cross-section of the CTLPH is constant in a rectangular shape. Figure 2c,d also indicates the different arrangements of the CTLPH in terms of the polarization direction of piezoelectric layers and their electrical connections. The CTLPH is called parallel if the piezoelectric layers are electrically connected, as shown in Figure 2c, and the piezoelectric layers have the same polarization direction. However, if electrical connections are similar to Figure 2d, and the piezoelectric layers have the anti-polarization direction, the CTLPH is called a series. The electrodes of piezoelectric layers are perpendicular to the y direction. Both length (L) and width (W

_{b}) of the CTLPH are much larger than the total thickness (t

_{s}+ 2t

_{p}). If a tip force is applied to the beam, the electromechanical equations for the piezoelectric layers are [18]:

**s**’ represents the substrate elastic elements, and ${e}_{s},$ ${\sigma}_{s}\mathrm{and}{E}_{s}$ are the strain, stress, and elastic modulus of the substrate layer.

#### 3.2. Bending Stiffness

_{NA}= 0. If the curvature radius of the neutral plane at position x is Ro(x), the strain of the

**ith**piezoelectric layer can be calculated from Equation (4) [25]:

#### 3.3. Output Voltage and Power

_{in}. With the use of Equation (30) and considering $\left(L{w}_{b}\right)$ as the surface of the electrode, the voltage across the storage capacitor can be calculated as [27]

_{out}) of the voltage regulator module, LTC3588. In Figure 4, the discharging voltage level of the C

_{out}, $\Delta {V}_{out},$ is the voltage difference between the fully charged condition (i.e., point A) and the partially discharged condition (i.e., point B). Based on the energy conservation principle, the stored energy in C

_{out}is equal to the dissipated energy on the external load, ${E}_{out}$, which can be calculated by Equation (36):

## 4. Methodology

_{in}on V

_{in}, the input section of LTC3588 should be simulated by using a diode bridge. Then, a germanium diode bridge (1N34A) was employed to rectify the CLTPH output and store it in various C

_{in}

_{,}(Figure 3a). LTC 3588 was also used to regulate the output voltage of the CLTPH (Figure 3b). To investigate the performance of the harvester, a reciprocated mechanism, depicted in Figure 3c, was employed. In this mechanism, a slider with a tip magnet is coupled with a Scotch yoke to generate a reciprocated motion with adjustable frequency. The variable low frequency (1–3.2 Hz) of the reciprocating motion is achievable by changing the energizing voltage of the DC motor. The specifications of the employed CTLPH are presented in Table 1. As shown in Figure 3c, a very sensitive laser displacement sensor, model HK-052, measures the vibration amplitude at the tip of the beam and the tip displacement (δ) without mechanical contact [28]. To collect the data, the National Instrument data acquisition card BNC-2110 was employed. To compare the results, the triple-layer harvester is composed of a brass strip (t

_{s}= 0.11 mm, E

_{s}= 110 GPa) that is sandwiched between two PZT-5H layers (t

_{s}= 0.225 mm, E

_{p}= 60 GPa). The material constants of PZH-5H are given by d

_{31}= −270 × 10

^{−12}C/N, k

_{31}= 0.38, and ${\epsilon}_{33}^{\sigma}/{\epsilon}_{0}=3500$ [29].

## 5. Results and Discussion

#### 5.1. Tip Force

_{in}) on generated voltage by the harvester. The values of all of the parameters can be found in the catalog of the piezoelectric material manufacturer or can be measured. Figure 5 shows the tip deflection of the CTLPH when the slider reciprocates at various low frequencies. The maximum deflection for all frequencies is about 2.7 mm. In the next step, considering the values in Table 1, bending stiffness, EI, can be computed as 0.0088 N·m

^{2}using Equation (16). By replacing the bending stiffness value (EI = 0.0088 N·m

^{2}) and maximum tip deflection ${\delta}_{max}=2.7\mathrm{mm}$, in (33), the maximum applied tip force can be specified, $F=3EI\times \delta /{L}^{3}=3\times 0.0088\times 0.0027/{0.035}^{3}=1.66\mathrm{N}$.

#### 5.2. Storage or Input Voltage

_{in}), the voltage across the storage capacitors (V

_{in}) was measured. Figure 6 shows V

_{in}versus time for a variety of C

_{in}values; the higher C

_{in}, the lower the storage voltage. Figure 7 shows V

_{in}versus C

_{in}at a given time of 2 s. This figure confirms that the stored voltage (V

_{in}) has a reverse relationship with C

_{in}, as Equation (35) suggests.

#### 5.3. Output Voltage and Power with LTC3588

_{in}), output capacitors (C

_{out}), and resistive loads were investigated. The rectified voltage (after the germanium diode bridge) is stored in the storage capacitor (C

_{in}). In each reciprocated motion, generated by the Scotch Yoke, the bending deflection of the CTLPH increases the accumulated charges on the C

_{in}, which is proportional to V

_{in}.

#### 5.3.1. Effect of Frequency

_{out}terminals when the CTLPH is bent by variable frequency. Since the output voltage of LTC3588 is set at 3.3 V, the output voltage cannot exceed this value. This figure highlights that higher frequency causes faster charging and reaches the set value of 3.3 V. The generated charge in low frequency (e.g., f = 1.62 and 1 Hz) does not generate a voltage in the monitoring period of 120 s.

#### 5.3.2. Effect of Storage and Output Capacitors

_{in}, and output capacitor, C

_{out,}on the output voltage (V

_{out}) is manifested in Figure 9 when the slider moves at 2.88 Hz in an open circuit condition. For the same charge generated by the CTLPH, a lower C

_{in}, causes a larger V

_{in}. By increasing the C

_{in}, more charges need to be stored to reach the threshold voltage value. In other words, a higher charging period causes later charge transportation to the output capacitor. Comparison of Figure 9a–c for any C

_{in}(e.g., for C

_{in}= 22 $\mathsf{\mu}$F) show lower C

_{out}, which leads to higher V

_{out}.

#### 5.3.3. Effect of Resistive Load

_{dis}. Figure 10 emphasises again the role of C

_{out}to provide output voltage with a lower ripple or smaller $\Delta {V}_{out}$.

_{dis}, where larger resistance increases T

_{dis}. Changing load value affects both the $\Delta {V}_{out}$ and T

_{dis}. By substituting the measured values of $\Delta {V}_{out}$ and T

_{dis}in Equation (37), the generated power of each resistive load,${R}_{load}$, can be calculated (Figure 12). When the load is small, the voltage drop across the load is low (e.g., in a short circuit condition the output voltage is zero). By increasing the external resistive load, the output voltage (voltage drop across the external load) varies. Furthermore, higher excitation frequency causes a faster storage or voltage raise. Figure 12 shows that the power will reach its maximum value in a specific resistance and mitigates in the higher values. As an example, $\Delta {V}_{out}$ was measured at 1.9 V for R = 20 kΩ at the frequency of 3.2 Hz. By substituting the measured value of $\Delta {V}_{out}$ in (36), the discharge energy from the output capacitor (Cout = 47 µF) is 84.83 $\mathsf{\mu}\mathrm{J}$. Considering T

_{dis}= 4.9 s, it causes a power of 17.31 µW. Similarly, the maximum power for 2.44 and 2.88 Hz at 20 kΩ would be 2.23 and 5.48 µW, respectively. In other words, the maximum output power of the CTLPH combined with LTC3588 module is realised when it is connected to a 20 kΩ external load. In other words, the proposed CTLPH in this research with 17.31 µW output power is a good candidate to empower microelectronic sensors or biomedical implants (e.g., 30 µW pacemakers). Compared with other harvesters presented in Table 2, the highest power density of this work (i.e., 77.2 µW/cm

^{3}) can be related to many factors, such as piezoelectric coefficient (d

_{31}) or amplitude of applied tip force (as shown in Equation (35)). Although higher applied force can generate higher voltage, it leads to more microstructural cracks and lower longevity of the harvester. Therefore, a trade-off between lifespan and power density should be considered for real-life applications.

## 6. Conclusions

_{in}) and output capacitor of the energy storage module, LTC3588, is investigated. An impractical measurement method without mechanical contact is employed to specify the applied tip force. The performance of the CTLPH in low frequencies (<3.3 Hz) for various resistive loads is also investigated. It was demonstrated that both the excitation frequency and external resistance load have an impact on the maximum generated power. The developed CTLPH generates the optimum power of 17.31 µW at the external resistance of 20 kΩ, which is suitable for micropower implanted devices operating in environments with a minimum vibration frequency of 3.2 Hz.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The CTLPH (

**a**) A schematic, (

**b**) bending deflection in a reciprocating motion, and (

**c**) a real mechanism.

**Figure 2.**(

**a**) A cantilever triple-layer piezoelectric harvester in transverse vibration. (

**b**) Free-body diagram of a small element. (

**c**) Parallel connection. (

**d**) Series connection.

**Figure 3.**Performance evaluation. (

**a**) Germanium diode bridge (1N34A) of the input section of LTC3588. (

**b**) Connection of energy regulator module, LTC3588. (

**c**) Driving and measurement setup.

**Figure 4.**Discharging voltage versus time when LTC3588 is connected to the external load R

_{load}= 20 k$\Omega $; Cin = 22 $\mathsf{\mu}$F; Cout = 47 $\mathsf{\mu}$F; and f = 2.88 Hz.

**Figure 7.**Effect of the storage capacitor (C

_{in}) on the storage voltage (V

_{in}) after 2 s; f = 3.3 Hz.

**Figure 8.**Frequency effect on the open circuit output voltage of LTC3588; C

_{in}= 22 $\mathsf{\mu}$F, C

_{out}= 47 $\mathsf{\mu}$F.

**Figure 9.**The output voltage of LTC3588 versus time for different values of C

_{in}and for open circuit; f = 2.88 Hz (

**a**) C

_{out}= 47 µF, (

**b**) C

_{out}=220 µF, and (

**c**) C

_{ou}

_{t}= 330 µF.

**Figure 10.**The output voltage of LTC3588 versus time for different values of C

_{out}

_{,}when LTC3588 is connected to the external load R

_{load}= 20 k$\Omega $; Cin = 22 $\mathsf{\mu}$F; and f = 2.88 Hz.

**Figure 11.**Sample of discharging voltage of LTC3588 when connected to the various external load; C

_{in}= 22 $\mathsf{\mu}$F; C

_{out}= 47 $\mathsf{\mu}$F; and f = 2.88 Hz.

**Figure 12.**Generated power versus external resistance load for different motion frequencies; C

_{in}= 22 $\mathsf{\mu}$F; and C

_{out}= 47 $\mathsf{\mu}$F.

Substrate Layer | Piezoelectric Layer | Tip Mass | Tiple-Layer Beam | |
---|---|---|---|---|

Material | Brass | PZT-5H | NdFeB-N35 | PZT-Brass-PZT |

Dimensions (mm) | 40 × 10 × 0.11 | 40 × 10 × 0.225 | $\varnothing $ 5 × 5 | 40 × 10 × 0.56 |

Elastic modulus (GPa) | 110 | 60 | 38 | ---- |

Density (kg/m^{3}) | 7800 | 6500 | 7500 | ---- |

Mass (kg) | 0.00034 | 0.00058 | 0.00075 | 0.0015 |

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**MDPI and ACS Style**

Ghodsi, M.; Mohammadzaheri, M.; Soltani, P.
Analysis of Cantilever Triple-Layer Piezoelectric Harvester (CTLPH): Non-Resonance Applications. *Energies* **2023**, *16*, 3129.
https://doi.org/10.3390/en16073129

**AMA Style**

Ghodsi M, Mohammadzaheri M, Soltani P.
Analysis of Cantilever Triple-Layer Piezoelectric Harvester (CTLPH): Non-Resonance Applications. *Energies*. 2023; 16(7):3129.
https://doi.org/10.3390/en16073129

**Chicago/Turabian Style**

Ghodsi, Mojtaba, Morteza Mohammadzaheri, and Payam Soltani.
2023. "Analysis of Cantilever Triple-Layer Piezoelectric Harvester (CTLPH): Non-Resonance Applications" *Energies* 16, no. 7: 3129.
https://doi.org/10.3390/en16073129