# Optimum Operating Conditions of (PbxX1−x)(ZryTizY1−y−z) Piezoelectric Transducer for Vibrational Energy Harvesting Applications

Authors: Funda Demir; Mustafa Anutgan
Journal: Shock and Vibration
 (2016)

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## Abstract

The electrical energy production capability of bimorph (PbxX1-x)(ZryTizY1-y-z) fiber composite piezoelectric transducer has been investigated for energy harvesting applications. The material has been analyzed under different frequencies, bending amounts, and temperatures. The operating conditions for maximum electrical energy outcome have been determined. The natural frequencies of oscillations in the macro dimensions have been found to be inversely proportional to the length of the material. On the other hand, the voltage output with respect to the oscillation frequency exhibits an interesting behavior such that the characteristic curve shifts to higher frequencies as the bending radius is decreased. This behavior has been interpreted as a result of possible overtone transitions of the oscillations to a stiffer mode. The increasing temperature has been observed to have a negative effect on the piezoelectric energy harvesting property. When the determined optimum conditions were utilized, the amount of electrical energy stored in 6300 s by an energy harvester circuitry has been found to be 0.8 J.


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## Body

## 1. Introduction

As the environmental concerns and limited resources decrease the popularity of the traditional energy production methods, clean and renewable techniques using solar, wind, geothermal, wave, and vibrational energies gain more importance [1–5]. Among these, energy harvesting from vibrations is promising especially for μW applications as it renders relatively high energy densities.Various principles utilized to transform vibrational energy into electrical energy can be cited as electromagnetism, electrostatics, and piezoelectricity. When compared with the formers, piezoelectricity leads to about 3 times greater energy density output [6, 7]. Furthermore, the power obtained in the μW scales when a piezoelectric material is operated under ordinary conditions can be improved toward mW scales via proper optimization procedures. These procedures include [8–10](i)
the use of different materials such as lead zirconate titanate ((PbxX1-x)(ZryTizY1-y-z) or (PZT)), polyvinylidene fluoride (PVDF), and macrofiber composite (MFC);(ii)
loading the piezoelectric material in directions parallel (d31) or perpendicular (d33) to the plane of electrodes;(iii)
the choice of geometrical shape (ring, triangle, square, etc.) of the piezoelectric material;(iv)
the production of the material in different types of morphology (unimorph, bimorph, s-morph, etc.);(v)
the design of the energy harvester circuitry of AC-DC rectifier (half-wave, full-wave, and bridge), DC-DC converter (buck, boost, and buck-boost), and the storage device (rechargeable battery, super capacitor, etc.).On the other hand, brittleness of the piezoelectric materials may be regarded as their main disadvantage [11].Among piezoelectric materials, PZT piezoceramic transducer has a high reputation [12–15]. Its piezoelectric charge coefficient (d), piezoelectric voltage coefficient (g), dielectric constant (ϵ), electromechanical coupling coefficient (k), and charge sensitivity are high, whereas its mechanical loss is low [16, 17]. Recent works regarding the PZT transducer mostly focus on ultrasonography technology and acoustic microscopy owing to its suitable physical properties [18]. Consequently, PZT transducer, which is capable of working over 30 MHz, is used in different branches like dermatology, intravascular, intracochlear implants [19–21], displays [22], and other optoelectronics devices [23–28] integrated with micro electromechanical systems (MEMS), micro spectroscopy [28–30], optical coherence tomography [31–35], microfluidity applications [36], electromechanical impedance (EMI) spectroscopy for real-time monitoring [37], biosensors sensitive at nanoscales [38], radio frequency (RF) applications [39], and lens-on-silicon ultrasound focusing units [40]. Furthermore, PZT thin film high frequency (~100 MHz) transducers were also studied well for ultrasound applications [41–43], biomicroscopy [44], and dermatologic and ophthalmologic diagnostics [45].In order to be used for practical energy harvesting purposes, the power output of a PZT transducer must be further improved. Among several types mentioned above, bimorph structure comprising a metal electrode sandwiched between two piezoelectric layers takes much attention due to its capability of energy production with higher density when compared with other types [18, 40, 46–48]. Also, replacing the standard Al electrode used in the bimorph structure with Ag or Pt was shown to affect the energy harvesting performance [36].In this work, a bimorph fiber composite PZT piezoelectric transducer has been investigated with particular emphasis on its energy harvesting performance. In this respect, the operating conditions for maximizing the electrical signal output have been determined. The quenching of the piezoelectric property with increasing temperature has been demonstrated. Although this material promises high energy yields, it has been shown that the operating conditions to be utilized for different applications must be predetermined with caution since the material exhibits a nonlinear behavior under systematically changing excitations. This behavior is attempted to be interpreted as a result of a change in the resonance frequency at the molecular level for different bending amounts.


## 2. Materials and Experiments

The fiber composite PZT energy harvester used in this work is schematically depicted in Figure1. The piezoelectric fibers were unidirectionally aligned in order to improve both device durability and charge collection efficiency. The electrical signal is observed when the device is flexed, bended, or vibrated. The electrical charges are taken to the battery by the metal electrode spread throughout the midst of the structure. Besides its contribution to the improvement of the charge collection efficiency, this metal electrode helps further prolong the lifetime of the device before it is mechanically broken.Figure 1
The schematical representation of the bimorph PZT piezoelectric energy harvester used in this work.Firstly, the analysis of the voltage signal output of our device was performed before connecting it to rechargeable battery. Figure2 shows the experimental setup used in this study. The setup was prepared to measure and determine the optimum conditions to obtain maximum electrical energy from the PZT transducer.Figure 2
Experimental setup depicting the method of data acquisition from the flexible PZT piezoelectric transducer.In order to gather the experimental data from the device, firstly one of its ends was fixed to a stationary support as shown in Figure2 since the PZT energy harvester at hand is of flexible type. Then, the device was flexed by a single push and released to let it freely oscillate. The electrical signal was collected by an oscilloscope (Figure 3) for different oscillation frequencies and radii of curvature. Then, it was passed through a data acquisition unit where AC signal produced by the PZT energy harvester was rectified by bridge diodes, filtered by a capacitor, and stored as DC signal. The produced analog voltage value was read by Arduino Mega 2560 microcontroller card and transferred to the data acquiring and analysis model developed by MATLAB/Simulink. Real-time measurement results were recorded by Arduino add-in of MATLAB/Simulink environment.Figure 3
The time dependence of the electrical signal produced by the PZT piezoelectric energy harvester after its release.


## 3. Results and Discussion

### 3.1. Basic Characteristics

It is seen from the voltage signal in Figure3 that the first two voltage peaks were more or less spiky. Therefore, we considered the third peak to be Vmax to render systematic comparisons between different measurements. After the third peak, the decay characteristics of the signal were almost perfectly sinusoidal exhibiting a single period.The decay kinetics of the oscillations are given in Figure4. A rapid decay was observed just after the release of the device during the first oscillations with relatively high amplitudes. At about 0.25 s, this anharmonic behavior is overcome by slower decay characteristics, with the latter being consistent with the simple model of damped harmonic motion. The voltage signal drops down to 10% of Vmax at ~0.7 s. The damping time of the signal could be increased by connecting a load resistor parallel to the outlet of the rectifier circuit [49]. However, since some portion of the produced power is lost on the load resistor, this method is not preferable despite its contribution to prolong the damping time.Figure 4
Decay curve of the voltage signal produced by the PZT energy harvester.


### 3.2. Length-Frequency Relation

In order to observe the relationship between the length and the frequency of the material, first the length of the material was kept constant and it was observed that the frequency did not change when the material was bent at different ratios. This phenomenon is called the natural frequency of the cantilever [50, 51]. As the material and its mass used in the experiments were not changed, the force applied to change the bending radius did not affect the oscillation frequency which is mainly determined by the inverse proportionality to the length of the PZT cantilever as shown in Figure 5.Figure 5
Change of the oscillation frequency with respect to the length of the PZT cantilever.


### 3.3. Effect of Temperature onVmax

Since the operating temperature of an energy harvester would vary from time to time, the performance of the PZT material at hand was measured at different temperatures. Figure6 gives the Vmax value of the material for temperatures ranging from 306 K to 353 K with constant frequency (20 Hz) and bending radius (12 cm). As the operating temperature increases, the voltage output decreases. This behavior is expected due to two main reasons. First, phonons with greater momenta are formed at higher temperatures which results in larger scattering of the free electrons in arbitrary directions reducing their probability of reaching the target electrode. Second, the dielectric function of the PZT material is relaxed at higher temperatures [52], and after a critical temperature, called the Curie temperature (which is 623 K for this material), the material becomes completely depolarized and loses its piezoelectric properties. For this reason, the operating temperature is advised to be lower than half of the Curie temperature [53–55].Figure 6
A sample curve showing the monotonously decreasing behavior ofVmax with increasing temperature. The data were collected at constant frequency of 20 Hz and bending radius of 12 cm.


### 3.4. Frequency Response Characteristics under Stress

The frequency response characteristics of a piezoelectric material are commonly determined by the measurement of the power and/or voltage output signal as a function of the frequency of some external mechanical excitation source. Such measurement methods are usually performed when the piezoelectric material is at rest or under equilibrium. However, in energy harvesting applications, the output signal from the piezoelectric material is collected when it is stressed or out of equilibrium. In this respect, the validity of the standard frequency response curve of a piezoelectric material must be reconsidered if that material is to be used for energy harvesting purposes.In Figure7, the mean peak-to-peak voltage output (Vpp) per unit length of the PZT energy harvester was recorded as a function of its oscillation frequency at different radii of curvature (r from 9.5 cm to 17.5 cm). Here, inverse of the radius of curvature should be regarded as a degree of strain, distortion, or nonequilibrium.Figure 7
Voltage output per length of the PZT cantilever plotted against oscillation frequency for different radii of curvature. Right shift of the frequency response curve at higher bending amounts is obvious. Solid lines are drawn for eye guide.Forr=17.5 cm, the PZT energy harvester was slightly flexed in reference to its fixing point (see Figure 2 for the experimental setup). It is seen that the voltage output produced by the PZT material first decreases as the oscillation frequency increases from 20 Hz to 23 Hz. As the frequency is further increased toward 30 Hz, the voltage output starts to increase and finally passes over a maximum at about 33 Hz (note here that this frequency is in good agreement with the common resonance frequency of that material measured with direct external mechanical excitation while the material is kept at rest). The voltage signal slowly decreases for higher frequencies. This “spoon-like” behavior is common for all measurement sets. The behavior around the minimum point in this curve seems to be related to the absorption of the vibrational energy and its subsequent dissipation as phonons throughout the structure without the production of any electrical signal. On the other hand, the behavior around the maximum is directly related to the conversion of the vibrational energy into the electrical energy by the matching of the oscillation frequency with the molecular resonance frequency (i.e., the vibrational frequency becomes some integer multiple of the molecular resonance frequency which leads to a constructive effect and boosts the polarization of the PZT molecules). Interestingly, this characteristic curve shifts to higher frequencies at lower radii of curvature (i.e., stronger bending). In other words, together with an increase in the maximum voltage output, the oscillation frequency at which this voltage output is reached also increases. It should be noted here that the characteristic “spoon-like” curve upshifts as a whole in the frequency axis at different radii of curvature; that is, the frequency corresponding to the minimum voltage also shifts in the same manner. However, assuming that the underlying reason for the upshift of the curves remains the same, a physical interpretation is proposed focusing on the frequency shift of only the “maximum” voltage output which is the relevant quantity for an energy harvester.


### 3.5. Interpretation of the Shift of the Resonance Frequency

To start with, the Morse Potential is taken as the interatomic interaction energy,U, between the constituent atoms of the PZT structure:(1)U∝1-exp⁡-βr-rc2.Here, β is a parameter related to the width of the potential well and rc is the equilibrium distance between two subsequent atoms. Although Hooke’s Law gives exact analytical solution to Schrödinger’s Equation in the simple model of quantum harmonic oscillator, Morse’s Law is more general as it takes both the anharmonicity and the overtone transitions into account. Indeed, Hooke’s Law is the second-order Taylor expansion of (1) and therefore can be assumed to be a simplified version of Morse’s Law.Within the frame of the physical principles, the upshift of the resonance frequency with increasing initial bending must be related to an increase in theβ factor in (1). Two of the possibilities for β to take different values at different bending amounts are discussed in the following two subsections.


#### 3.5.1. What Happens in the Vicinity of the Elastic/Plastic Deformation Regions?

Note thatβ is directly proportional to the modulus of elasticity, which is defined as the slope of the stress-to-strain curve in the elastic deformation region. The slope of that curve decreases at larger strain values and finally becomes zero where the material reaches its ultimate strength. In other words, a material starts to lose its ability to restore its original shape with increasing bending amounts and, after a critical point, the type of deformation changes from elastic to plastic. From microscopic point of view, some of the atomic bonds start to form defect states leading to a lower restoring force. Such behavior results in a decrease in β, and therefore it cannot be attributed to the phenomena observed in this work. Here, the previous works [56, 57] on piezoelectric MEMS reporting a decrease in the resonance frequency at higher acceleration values should not be confused since the concept of resonance frequency of those works was used for the oscillation frequency determined by the dimensions of the MEMS cantilever whose length-to-volume ratio is much larger. Probably, their piezoelectric cantilever experiences a more or less prominent nonlinear stress-to-strain curve. On the contrary, the resonance frequency in this work corresponds to the oscillations in the molecular level.


#### 3.5.2. Overtone Transitions to a Stiffer Mode

Although the experiments on the material of this work were performed with initial elastic deformation, the degree of anharmonicity increases with increasing bending according to (1) (Figure 8). For the correct interpretation of the increase in the resonance frequency with increasing bending, the dynamics at the very moment of the release are crucial since Vpp is determined from the second sine-wave of the oscillations. It is seen in Figure 4 that the PZT transducer firstly experiences a fast anharmonic decay; then a transition to a relatively slow harmonic decay occurs. The decay of the output voltage is a direct result of the decrease in the oscillation amplitude r-rc in (1). On the other hand, it should be kept in mind that the overall energy loss is mainly governed by possible phonon emission mechanisms where the temperature has a key role (Figure 6). Considering the constant temperature (i.e., room temperature) for all sets of measurements in Figure 7, no drastic change is expected in the main energy loss mechanism. In this regard, the decrease in the oscillation amplitude during the fast anharmonic decay just after the release cannot be totally attributed to the potential energy loss in the material. Instead, during the strong attempt to rearrange its oscillation behavior (i.e., from anharmonic to harmonic), the PZT material can make an overtone transition from a lower β state toward a higher β state. One of the underlying reasons for this strong attempt may be the instability imposed by the inequality between the repulsive force applied on the compressed half and the attractive force applied on the expanded half of the flexed material during an anharmonic oscillation. Regarding the conservation of energy arguments, such a transition is possible as shown in Figure 8. When the material is excited by bending, Morse’s curve in Figure 8 might be followed. The PZT material having such a potential curve can be said to be in a relatively lower β state as designated by the corresponding Hooke’s curve with β1. Assuming an instantaneous rearrangement just after the release, the PZT material can transit to Hooke’s curve with β2, where β2>β1. This assumption seems to be reasonable due to its good agreement with the “quick” transition from the anharmonic to the harmonic state. Obviously, at higher bending amounts, the material experiences a larger restoring force and consequently a faster anharmonic-to-harmonic transition leading to a greater difference β2-β1 and to a greater upshift of the resonance frequency.Figure 8
Potential energy curves plotted with Morse’s and Hooke’s Laws. A possible overtone transition from a lower to higherβ mode is shown by the arrow.


### 3.6. Storage of the Electrical Energy

Besides extracting the maximum possible output voltage from the PZT material at hand, storing that voltage with minimum losses is also important. One of the efficient ways is that the sinusoidal electrical signal can be passed through a rectifying circuit and filtered before its storage in a battery (Figure2). Then, the stored energy would be ready to use at any time for any purposes. Figure 9 shows the time dependence of the stored voltage in a 0.1 F capacitor at an oscillation frequency of 20 Hz with bending radius of curvature of 9.5 cm. These parameters may be regarded as the optimum operating conditions for this PZT piezoelectric energy harvester. Another set of parameters for maximum voltage output would be formed by using 65 Hz as the oscillation frequency instead of 20 Hz (Figure 7). However, this would mean at least three times more frequent oscillations of a stiffer material at a large bending and, therefore, would not be a good choice in order to avoid faster fatigue and lower lifetime of the material. With the determined optimum operating conditions, the amount of energy stored in 6300 s at room temperature is 0.8 J. The maximum voltage output observed at the instant of the release under optimum conditions was slightly more than 200 V and the current at this high voltage was 53 μA which results in a power of about 10 mW.Figure 9
Charge storage characteristics of the PZT energy harvester measured at the optimum operating conditions.


## 3.1. Basic Characteristics

It is seen from the voltage signal in Figure3 that the first two voltage peaks were more or less spiky. Therefore, we considered the third peak to be Vmax to render systematic comparisons between different measurements. After the third peak, the decay characteristics of the signal were almost perfectly sinusoidal exhibiting a single period.The decay kinetics of the oscillations are given in Figure4. A rapid decay was observed just after the release of the device during the first oscillations with relatively high amplitudes. At about 0.25 s, this anharmonic behavior is overcome by slower decay characteristics, with the latter being consistent with the simple model of damped harmonic motion. The voltage signal drops down to 10% of Vmax at ~0.7 s. The damping time of the signal could be increased by connecting a load resistor parallel to the outlet of the rectifier circuit [49]. However, since some portion of the produced power is lost on the load resistor, this method is not preferable despite its contribution to prolong the damping time.Figure 4
Decay curve of the voltage signal produced by the PZT energy harvester.


## 3.2. Length-Frequency Relation

In order to observe the relationship between the length and the frequency of the material, first the length of the material was kept constant and it was observed that the frequency did not change when the material was bent at different ratios. This phenomenon is called the natural frequency of the cantilever [50, 51]. As the material and its mass used in the experiments were not changed, the force applied to change the bending radius did not affect the oscillation frequency which is mainly determined by the inverse proportionality to the length of the PZT cantilever as shown in Figure 5.Figure 5
Change of the oscillation frequency with respect to the length of the PZT cantilever.


## 3.3. Effect of Temperature onVmax

Since the operating temperature of an energy harvester would vary from time to time, the performance of the PZT material at hand was measured at different temperatures. Figure6 gives the Vmax value of the material for temperatures ranging from 306 K to 353 K with constant frequency (20 Hz) and bending radius (12 cm). As the operating temperature increases, the voltage output decreases. This behavior is expected due to two main reasons. First, phonons with greater momenta are formed at higher temperatures which results in larger scattering of the free electrons in arbitrary directions reducing their probability of reaching the target electrode. Second, the dielectric function of the PZT material is relaxed at higher temperatures [52], and after a critical temperature, called the Curie temperature (which is 623 K for this material), the material becomes completely depolarized and loses its piezoelectric properties. For this reason, the operating temperature is advised to be lower than half of the Curie temperature [53–55].Figure 6
A sample curve showing the monotonously decreasing behavior ofVmax with increasing temperature. The data were collected at constant frequency of 20 Hz and bending radius of 12 cm.


## 3.4. Frequency Response Characteristics under Stress

The frequency response characteristics of a piezoelectric material are commonly determined by the measurement of the power and/or voltage output signal as a function of the frequency of some external mechanical excitation source. Such measurement methods are usually performed when the piezoelectric material is at rest or under equilibrium. However, in energy harvesting applications, the output signal from the piezoelectric material is collected when it is stressed or out of equilibrium. In this respect, the validity of the standard frequency response curve of a piezoelectric material must be reconsidered if that material is to be used for energy harvesting purposes.In Figure7, the mean peak-to-peak voltage output (Vpp) per unit length of the PZT energy harvester was recorded as a function of its oscillation frequency at different radii of curvature (r from 9.5 cm to 17.5 cm). Here, inverse of the radius of curvature should be regarded as a degree of strain, distortion, or nonequilibrium.Figure 7
Voltage output per length of the PZT cantilever plotted against oscillation frequency for different radii of curvature. Right shift of the frequency response curve at higher bending amounts is obvious. Solid lines are drawn for eye guide.Forr=17.5 cm, the PZT energy harvester was slightly flexed in reference to its fixing point (see Figure 2 for the experimental setup). It is seen that the voltage output produced by the PZT material first decreases as the oscillation frequency increases from 20 Hz to 23 Hz. As the frequency is further increased toward 30 Hz, the voltage output starts to increase and finally passes over a maximum at about 33 Hz (note here that this frequency is in good agreement with the common resonance frequency of that material measured with direct external mechanical excitation while the material is kept at rest). The voltage signal slowly decreases for higher frequencies. This “spoon-like” behavior is common for all measurement sets. The behavior around the minimum point in this curve seems to be related to the absorption of the vibrational energy and its subsequent dissipation as phonons throughout the structure without the production of any electrical signal. On the other hand, the behavior around the maximum is directly related to the conversion of the vibrational energy into the electrical energy by the matching of the oscillation frequency with the molecular resonance frequency (i.e., the vibrational frequency becomes some integer multiple of the molecular resonance frequency which leads to a constructive effect and boosts the polarization of the PZT molecules). Interestingly, this characteristic curve shifts to higher frequencies at lower radii of curvature (i.e., stronger bending). In other words, together with an increase in the maximum voltage output, the oscillation frequency at which this voltage output is reached also increases. It should be noted here that the characteristic “spoon-like” curve upshifts as a whole in the frequency axis at different radii of curvature; that is, the frequency corresponding to the minimum voltage also shifts in the same manner. However, assuming that the underlying reason for the upshift of the curves remains the same, a physical interpretation is proposed focusing on the frequency shift of only the “maximum” voltage output which is the relevant quantity for an energy harvester.


## 3.5. Interpretation of the Shift of the Resonance Frequency

To start with, the Morse Potential is taken as the interatomic interaction energy,U, between the constituent atoms of the PZT structure:(1)U∝1-exp⁡-βr-rc2.Here, β is a parameter related to the width of the potential well and rc is the equilibrium distance between two subsequent atoms. Although Hooke’s Law gives exact analytical solution to Schrödinger’s Equation in the simple model of quantum harmonic oscillator, Morse’s Law is more general as it takes both the anharmonicity and the overtone transitions into account. Indeed, Hooke’s Law is the second-order Taylor expansion of (1) and therefore can be assumed to be a simplified version of Morse’s Law.Within the frame of the physical principles, the upshift of the resonance frequency with increasing initial bending must be related to an increase in theβ factor in (1). Two of the possibilities for β to take different values at different bending amounts are discussed in the following two subsections.


### 3.5.1. What Happens in the Vicinity of the Elastic/Plastic Deformation Regions?

Note thatβ is directly proportional to the modulus of elasticity, which is defined as the slope of the stress-to-strain curve in the elastic deformation region. The slope of that curve decreases at larger strain values and finally becomes zero where the material reaches its ultimate strength. In other words, a material starts to lose its ability to restore its original shape with increasing bending amounts and, after a critical point, the type of deformation changes from elastic to plastic. From microscopic point of view, some of the atomic bonds start to form defect states leading to a lower restoring force. Such behavior results in a decrease in β, and therefore it cannot be attributed to the phenomena observed in this work. Here, the previous works [56, 57] on piezoelectric MEMS reporting a decrease in the resonance frequency at higher acceleration values should not be confused since the concept of resonance frequency of those works was used for the oscillation frequency determined by the dimensions of the MEMS cantilever whose length-to-volume ratio is much larger. Probably, their piezoelectric cantilever experiences a more or less prominent nonlinear stress-to-strain curve. On the contrary, the resonance frequency in this work corresponds to the oscillations in the molecular level.


### 3.5.2. Overtone Transitions to a Stiffer Mode

Although the experiments on the material of this work were performed with initial elastic deformation, the degree of anharmonicity increases with increasing bending according to (1) (Figure 8). For the correct interpretation of the increase in the resonance frequency with increasing bending, the dynamics at the very moment of the release are crucial since Vpp is determined from the second sine-wave of the oscillations. It is seen in Figure 4 that the PZT transducer firstly experiences a fast anharmonic decay; then a transition to a relatively slow harmonic decay occurs. The decay of the output voltage is a direct result of the decrease in the oscillation amplitude r-rc in (1). On the other hand, it should be kept in mind that the overall energy loss is mainly governed by possible phonon emission mechanisms where the temperature has a key role (Figure 6). Considering the constant temperature (i.e., room temperature) for all sets of measurements in Figure 7, no drastic change is expected in the main energy loss mechanism. In this regard, the decrease in the oscillation amplitude during the fast anharmonic decay just after the release cannot be totally attributed to the potential energy loss in the material. Instead, during the strong attempt to rearrange its oscillation behavior (i.e., from anharmonic to harmonic), the PZT material can make an overtone transition from a lower β state toward a higher β state. One of the underlying reasons for this strong attempt may be the instability imposed by the inequality between the repulsive force applied on the compressed half and the attractive force applied on the expanded half of the flexed material during an anharmonic oscillation. Regarding the conservation of energy arguments, such a transition is possible as shown in Figure 8. When the material is excited by bending, Morse’s curve in Figure 8 might be followed. The PZT material having such a potential curve can be said to be in a relatively lower β state as designated by the corresponding Hooke’s curve with β1. Assuming an instantaneous rearrangement just after the release, the PZT material can transit to Hooke’s curve with β2, where β2>β1. This assumption seems to be reasonable due to its good agreement with the “quick” transition from the anharmonic to the harmonic state. Obviously, at higher bending amounts, the material experiences a larger restoring force and consequently a faster anharmonic-to-harmonic transition leading to a greater difference β2-β1 and to a greater upshift of the resonance frequency.Figure 8
Potential energy curves plotted with Morse’s and Hooke’s Laws. A possible overtone transition from a lower to higherβ mode is shown by the arrow.


## 3.5.1. What Happens in the Vicinity of the Elastic/Plastic Deformation Regions?

Note thatβ is directly proportional to the modulus of elasticity, which is defined as the slope of the stress-to-strain curve in the elastic deformation region. The slope of that curve decreases at larger strain values and finally becomes zero where the material reaches its ultimate strength. In other words, a material starts to lose its ability to restore its original shape with increasing bending amounts and, after a critical point, the type of deformation changes from elastic to plastic. From microscopic point of view, some of the atomic bonds start to form defect states leading to a lower restoring force. Such behavior results in a decrease in β, and therefore it cannot be attributed to the phenomena observed in this work. Here, the previous works [56, 57] on piezoelectric MEMS reporting a decrease in the resonance frequency at higher acceleration values should not be confused since the concept of resonance frequency of those works was used for the oscillation frequency determined by the dimensions of the MEMS cantilever whose length-to-volume ratio is much larger. Probably, their piezoelectric cantilever experiences a more or less prominent nonlinear stress-to-strain curve. On the contrary, the resonance frequency in this work corresponds to the oscillations in the molecular level.


## 3.5.2. Overtone Transitions to a Stiffer Mode

Although the experiments on the material of this work were performed with initial elastic deformation, the degree of anharmonicity increases with increasing bending according to (1) (Figure 8). For the correct interpretation of the increase in the resonance frequency with increasing bending, the dynamics at the very moment of the release are crucial since Vpp is determined from the second sine-wave of the oscillations. It is seen in Figure 4 that the PZT transducer firstly experiences a fast anharmonic decay; then a transition to a relatively slow harmonic decay occurs. The decay of the output voltage is a direct result of the decrease in the oscillation amplitude r-rc in (1). On the other hand, it should be kept in mind that the overall energy loss is mainly governed by possible phonon emission mechanisms where the temperature has a key role (Figure 6). Considering the constant temperature (i.e., room temperature) for all sets of measurements in Figure 7, no drastic change is expected in the main energy loss mechanism. In this regard, the decrease in the oscillation amplitude during the fast anharmonic decay just after the release cannot be totally attributed to the potential energy loss in the material. Instead, during the strong attempt to rearrange its oscillation behavior (i.e., from anharmonic to harmonic), the PZT material can make an overtone transition from a lower β state toward a higher β state. One of the underlying reasons for this strong attempt may be the instability imposed by the inequality between the repulsive force applied on the compressed half and the attractive force applied on the expanded half of the flexed material during an anharmonic oscillation. Regarding the conservation of energy arguments, such a transition is possible as shown in Figure 8. When the material is excited by bending, Morse’s curve in Figure 8 might be followed. The PZT material having such a potential curve can be said to be in a relatively lower β state as designated by the corresponding Hooke’s curve with β1. Assuming an instantaneous rearrangement just after the release, the PZT material can transit to Hooke’s curve with β2, where β2>β1. This assumption seems to be reasonable due to its good agreement with the “quick” transition from the anharmonic to the harmonic state. Obviously, at higher bending amounts, the material experiences a larger restoring force and consequently a faster anharmonic-to-harmonic transition leading to a greater difference β2-β1 and to a greater upshift of the resonance frequency.Figure 8
Potential energy curves plotted with Morse’s and Hooke’s Laws. A possible overtone transition from a lower to higherβ mode is shown by the arrow.


## 3.6. Storage of the Electrical Energy

Besides extracting the maximum possible output voltage from the PZT material at hand, storing that voltage with minimum losses is also important. One of the efficient ways is that the sinusoidal electrical signal can be passed through a rectifying circuit and filtered before its storage in a battery (Figure2). Then, the stored energy would be ready to use at any time for any purposes. Figure 9 shows the time dependence of the stored voltage in a 0.1 F capacitor at an oscillation frequency of 20 Hz with bending radius of curvature of 9.5 cm. These parameters may be regarded as the optimum operating conditions for this PZT piezoelectric energy harvester. Another set of parameters for maximum voltage output would be formed by using 65 Hz as the oscillation frequency instead of 20 Hz (Figure 7). However, this would mean at least three times more frequent oscillations of a stiffer material at a large bending and, therefore, would not be a good choice in order to avoid faster fatigue and lower lifetime of the material. With the determined optimum operating conditions, the amount of energy stored in 6300 s at room temperature is 0.8 J. The maximum voltage output observed at the instant of the release under optimum conditions was slightly more than 200 V and the current at this high voltage was 53 μA which results in a power of about 10 mW.Figure 9
Charge storage characteristics of the PZT energy harvester measured at the optimum operating conditions.


## 4. Conclusion

In this study, the energy production capability of fiber reinforced bimorph PZT piezoelectric transducer was examined systematically. The electrical signal was collected by a high yield energy harvester circuitry during the self-oscillations of the PZT material after an externally applied bending. The sinusoidal voltage signal created by the mechanical oscillations exhibited an initial fast anharmonic decay followed by a relatively slower harmonic decay. The frequency of these oscillations could be adjusted by changing the length of the piezoelectric cantilever. The ambient temperature, which is crucial especially for outdoor energy harvesting applications, has a strong effect on the produced electrical signal. It is advised that the temperature of the PZT material should not exceed half of the Curie temperature in order to keep the phonon scattering and the dielectric relaxation at low levels. The frequency response curve shows an upshift in the frequency axis with increasing bending amounts. Regarding the energy conservation principle, this behavior was attributed to an increase in theβ factor of the PZT material in the form of a transition to a stiffer oscillation mode. The optimum room temperature operating conditions of the PZT energy harvester at hand were determined to be 20 Hz oscillation frequency and 9.5 cm radius of curvature. At these optimum conditions, the maximum voltage (Vmax) and the corresponding electrical current obtained are around 200 V and 53 μA, respectively, producing an electrical power of about 10 mW. Although this power level seems to be low compared to the conventional renewable energy systems, it was obtained from a single PZT material and can be boosted by a proper configuration including multiple number of PZT materials. Also, such power levels in the order of milliwatts are sufficient for applications such as biosensing, MEMS thermal imaging, and wireless network systems.



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*Source: 9425273-2016-12-18.xml*