Trans. Tianjin Univ. 2014, 20: 407-414 DOI 10.1007/s12209-014-2318-3
Complete Charging for Piezoelectric Energy Harvesting System * Fan Kangqi (樊康旗)，Xu Chunhui (徐春辉)，Wang Weidong (王卫东) (School of Mechano-Electronic Engineering, Xidian University, Xi’an 710071, China) © Tianjin University and Springer-Verlag Berlin Heidelberg 2014
Abstract：Under an in-phase assumption, the complete charging for an energy harvesting system is studied, which consists of a piezoelectric energy harvester (PEH), a bridge rectifier, a filter capacitor, a switch, a controller and a rechargeable battery. For the transient charging, the results indicate that the voltage across the filter capacitor increases as the charging proceeds, which is consistent with that reported in the literature. However, a new finding shows that the charging rate and energy harvesting efficiency decrease over time after their respective peak values are acquired. For the steady-state charging, the results reveal that the energy harvesting efficiency can be adjusted by altering the critical charging voltage that controls the transition of the system. The optimal energy harvesting efficiency is limited by the optimal efficiency of the transient charging. Finally, the relationship between the critical charging voltage and the equivalent resistance of the controller and rechargeable battery is established explicitly. Keywords：energy harvesting; mechanical vibration; piezoelectric energy harvester; charging rate; energy harvesting efficiency
Low-power devices and the related physical mechanisms have attracted interests over the past few years[1-3]. The related designs usually run on batteries that fail to meet the power demands in terms of energy density[4]. In addition, the batteries require periodical replacement or recharging, which can be a costly and cumbersome task. To overcome this deficiency, efforts have been devoted to harvesting energy from the ambient environment[5-8]. Such energy transducers are termed as microenergy harvesters or micropower generators[9,10]. There are several different energy sources available, such as solar energy, thermal gradient and mechanical vibration. Among them, mechanical vibration is ubiquitous, and hence it has drawn significant attention. Mechanical vibration can be converted into electrical energy as a power supply via electrostatic[11], electromagnetic[12], or piezoelectric transductions[8,9,13,14]. The electrical energy harvested by a vibrating piezoelectric element in a piezoelectric energy harvester (PEH) is in the form of alternating current (AC), while the low-power devices generally require a stabilized direct current (DC) supply. Therefore, an interface circuit is usually demanded to realize AC-DC conversion, and thus ensure the electrical compatibility
between a PEH and a low-power device. The standard interface circuit only involves a bridge rectifier[15]. Significant efforts have been devoted to optimizing the interface circuit. In particular, Lefeuvre et al[16,17] proposed a synchronous switch harvesting on inductor (SSHI), which increases the PEH output power significantly. To implement SSHI, some auxiliary devices such as displacement sensor are usually needed. However, the power consumption of these auxiliary devices can be even larger than the harvestable power[18]. Most lowpower devices adopt low duty cycle designs[19], for which a storage device is generally indispensable. The charging of storage devices is critical to maximizing the duty cycle of a low-power device[19,20]. The complete charging generally involves transient charging and steady-state charging. Shu and Lien[21,22] investigated the efficiency and output power of a PEH system operating in the steadystate charging, where the filter capacitor was employed to stabilize the voltage across the battery. Wu et al[19] studied the transient behavior of charging a capacitor by a PEH with a standard interface circuit, under the assumption that the PEH vibration amplitude kept constant. Wickenheiser et al[20] studied the effects of electrome-
Accepted date: 2014-01-24. *Supported by the National Natural Science Foundation of China (No. 51205302) and Fundamental Research Funds for the Central Universities (No. K5051304011). Fan Kangqi, born in 1979, male, Dr, associate Prof. Correspondence to Fan Kangqi, E-mail:
[email protected].
Transactions of Tianjin University
Vol.20 No.6 2014
chanical coupling on the energy storage through a piezoelectric energy harvesting system, which included a PEH, a standard interface circuit, and a capacitor. Fan et al[8] examined the dynamic characteristics of a piezoelectric energy harvesting system in the case that forward and backward electromechanical coupling coefficients differ from each other. This paper studies the complete charging of an energy harvesting system. Under the in-phase assumption, the analytical models concerning the charging rate, energy harvesting efficiency and electrical damping ratio are derived, respectively. The comparison among the results from the analytical models, numerical simulation and experimental observation indicates that the analytical models can predict the complete charging.
1
Model of energy harvesting system
1.1
Governing equations The energy harvesting system includes a PEH, a bridge rectifier, a filter capacitor, a switch, a controller and a rechargeable battery, as shown in Fig. 1. During the transient charging, the AC electrical energy harvested by the vibrating piezoelectric cantilever beam (i.e., PEH) is first converted into DC electrical energy by the bridge rectifier, and then stored in the filter capacitor. When the voltage across the filter capacitor (charging voltage) reaches the critical value determined by the controller and rechargeable battery, the switch is closed, and the system shifts into the steady-state charging. The harvested electrical energy is then accumulated in the battery, and the filter capacitor acts as an energy reservoir to smoothen the DC voltage across the battery. In addition, a controller is usually placed between the switch and battery to control the switch and regulate the output voltage.
fixed at the free end of the piezoelectric cantilever beam is caused to vibrate, an applied mechanical stress T1 along axis-1 is produced in the piezoelectric layer, which induces an electric displacement D3 along axis-3. Meanwhile, the applied electrical field E3 along axis-3 affects the mechanical strain S1 along axis-1. Usually, a storage circuit subsystem is connected to the PEH to collect the harvested energy, as shown in Fig. 1. The electromechanical behavior of this system can be described as follows[8,23]: Mz z Kz 1VP F (t ) (1) 2 z CPVP I
where M, η and K are the effective mass, mechanical damping coefficient and stiffness of PEH, respectively; z is the displacement of beam tip; VP is the voltage across the piezoelectric layers (piezoelectric voltage); F(t) is the external excitation originating from the mechanical vibration of the base; CP is the capacitance of the piezoelectric layers (intrinsic capacitor); I is the current flowing into the interface circuit; Θ1 and Θ2 are the effective piezoelectric coefficients, representing force factor and current factor, respectively. According to some previous studies, Θ1 was chosen to be equal to Θ2[19-21]; while for others, Θ1 was different from Θ2[24-26]. For example, a distributed parameter model[27] revealed that Θ1 is identical to Θ2 only if the single-mode mechanical vibration is considered; otherwise, Θ1 differs from Θ2. Therefore, the governing equations presented here can be considered as a generalized model in which Θ1 is equal to Θ2.
Fig. 2 Fig. 1
Schematic diagram of energy harvesting system
A PEH is usually composed of two piezoelectric layers and a metal shim, i.e., a triple-layer piezoelectric cantilever beam, as shown in Fig. 2. This triple-layer piezoelectric cantilever beam is designed to operate in the transverse mode (3-1 mode), which is the general case for the bending deformation. When the proof mass af—408—
(2)
1.2
Schematic diagram of PEH
In-phase assumption In-phase assumption supposes that the applied excitation F(t) is in phase with the vibration velocity ż(t) of the piezoelectric cantilever beam. Precisely, it is described as follows[21,28]: (3) F (t ) F sin t z (t ) z sin t (4)
Fan Kangqi et al: Complete Charging for Piezoelectric Energy Harvesting System
where z is the vibration amplitude of piezoelectric cantiThe average power harvested and stored in the filter lever beam; F and ω are the magnitude and angular fre- capacitor during the ith half cycle is[8,20] quency of the applied excitation, respectively. 2 ( 2 zi CCVC,i )( 2 zi CPVC,i ) Pi (8) Using the in-phase assumption, the coupling term Θ2 π(CC 2CP ) ż in Eq. (2) can be modeled as a current source IP, i.e., According to the energy balance between the me I P (t ) 2 z (t ) 2 z sin t I P sin t (5) chanical and electrical components, the beam tip deflection can be determined as[8] 1.3 Transient charging During the transient charging process, the switch is open, and the active components include a PEH, a bridge rectifier and a filter capacitor, as shown in Fig. 3. The charging voltage increases continuously, whereas the piezoelectric voltage generated by PEH vibrates periodically. In the ith half cycle defined by iπ ≤ ωt ≤ (i＋1)π, the piezoelectric voltage VP(t) is smaller in magnitude than the charging voltage VC(t) from iπ to χi, and thus the rectifying bridge is blocked, as shown in Fig. 4. At phase χi, the magnitude of VP(t) reaches that of VC(t), and then the rectifying bridge begins to conduct and the magnitudes of VP(t) and VC(t) increase simultaneously due to the continuous bending of the beam. At the end of the ith half cycle, the rectifying bridge is blocked again since the beam deflection and the generated piezoelectric voltage begin to decrease. Thus, this circuit topology results in the following conditions: , iπ ≤ t ≤ i ; 0 I (t ) (6) CCVC sgn z , i ≤ t ≤ (i 1)π
Fig. 3
Equivalent structure of transient charging
Fig. 4
ith half cycle of transient charging
2 2 2 2 B 2 B 16 ACCCP CC CP 1 2VC, i zi 2 A 2
(9)
where A and B are given respectively by A 4(CC CP )1 2 π CC (CC 2CP ) B πFCC (CC 2CP ) 41VC, i (CP 2 CC 2 ) During the transient charging, the harvested energy varies with time. For each half cycle, the energy harvesting efficiency can be defined as the ratio of the electrical energy WC, i stored in the filter capacitor to the input mechanical energy WF, i, i.e., ef , i
WC, i WF, i
1 CC (VC,2 i 1 VC,2 i ) 2 (i 1) π / ( ) ( )d F t z t t iπ /
CC 2 (VC,2 i 1 VC,2 i ) 2 π 2 zi 1 (CP CC )(VC,2 i 1 VC,2 i )
(10)
It can be seen from Eq. (10) that the energy harvesting efficiency varies as the charging proceeds. The optimal energy harvesting efficiency can be determined by Eq. (10). Even if the PEH structure has been fixed, the energy harvesting efficiency could be regulated by varying the critical charging voltage VC that controls the transition of the system, which will be discussed further in Sect. 2. Williams and Yates[29] argued that the conversion of mechanical energy to electrical energy was similar to a linear damper in the conventional mass-spring system. Furthermore, Shu and Lien[22] proposed that the total damping ratio of a piezoelectric energy harvesting system can be decomposed, and the energy harvesting efficiency can be re-defined as e ef (11) m e
where m / 2 KM is the mechanical damping ratio; In each half cycle, the charging voltage can be de- e is the electrical damping ratio caused by the extraction scribed by[20] of mechanical energy from the vibrating system by the CC CP 2 2 piezoelectric transduction. Thus, from Eqs. (10) and VC, i 1 VC, i zi (7) CC CP CC CP (11), the electrical damping ratio e can be expressed by —409—
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e, i CC 2 (VC,2 i 1 VC,2 i ) 2 KM (1CC
1CP CC 2 )(VC,2 i 1 VC,2 i ) π 2 zi2
1
(12)
Similarly, the electrical damping ratio e, i also varies as the charging proceeds. 1.4 Steady-state charging When the charging voltage increases to the critical value determined by the controller and battery, the switch is closed, and the system enters into the steady-state charging, as shown in Fig. 5, where the filter capacitor is used to smoothen the voltage across the battery. Note that the controller and the battery are usually replaced by an equivalent resistor R to facilitate the design analysis[21,30], and the critical charging voltage VC maintains essentially constant, provided that CC is large enough and the time constant RCC is much longer than the vibration period of the PEH [28]. Moreover, the net current flowing into the filter capacitor is zero in this process. Thus, integrating Eq. (2) over a half cycle leads to the relationship between critical charging voltage VC and vibration ampli tude z , i.e., 2 R 2 VC z (13) π 2 RCP
Fig. 5
Equivalent structure of steady-state charging
For the steady-state charging, the energy increase in the intrinsic capacitor and filter capacitor during each half cycle is zero as the critical charging voltage VC maintains essentially constant. As a result, the harvested energy is transferred to R, and then the energy harvesting efficiency ef is ef
WR ,i WF,i
VC2 T ( i 1)π / R 2 ( ) ( )d F t z t t iπ /
4 2VC F (π 2 RCP )
(17)
Using Eqs. (11) and (17), the electrical damping ratio e in the steady-state charging can be determined by 2 2VC e (18) KM [ F (π 2 RCP ) 4 2VC ]
Using the in-phase assumption, the vibration ampliEqs. (17) and (18) indicate that the energy harvest tude z in Eq. (13) can be obtained from an energy baling efficiency ef and the electrical damping ratio e can ance between the mechanical and electrical components, be optimized by regulating the equivalent resistance R. i.e., However, varying R can result in the change of the preset 2 2 critical charging voltage VC. In fact, after the critical F R 2 R 2 F 81VC z (14) charging voltage is fixed, the energy harvesting effi2 R 2 ciency and the electrical damping ratio will keep constant Then, substituting Eq. (13) into (14) yields the followduring the steady-state charging, which will be discussed ing expression for vibration amplitude z , further in the next section. F ( 2 RCP ) 2 z (15) ( 2 RCP ) 2 8 R1 2 According to Eqs. (13) and (15), the critical charging voltage VC can be related to the equivalent resistance R by VC
2 FR 2 (π 2 RCP ) (π 2 RCP ) 2 8R1 2
(16)
It is clear from Eq. (16) that the critical charging voltage VC can be altered by regulating the equivalent resistance R. Since the energy harvesting efficiency of the transient charging varies with the charging voltage as shown in Eq. (10), it can also be changed by regulating the equivalent resistance.
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2
Results and discussion
We now reveal the charging behavior of this piezoelectric energy harvesting system during the complete charging. The considered PEH prototype was made from a two-layer sheet of PSI-5A4E ceramic with a brass center shim. The structural parameters are listed in Tab. 1. Initially, the intrinsic capacitor and the filter capacitor are completely discharged, and the charging process is activated at 1.4 s after an external sinusoidal excitation with amplitude 8.7 m·N and frequency 392 rad/s (open-circuit frequency) is applied.
Fan Kangqi et al: Complete Charging for Piezoelectric Energy Harvesting System Tab. 1
Structural parameters of energy harvesting system Parameter
Value
Structure size/mm3
28.575×3.175×0.190
2
/((N·s)·m ) -1
0.011,5
M/g
1.1
CP /nF
13.48
CC /μF
13.29
The variation of VC with time t is shown in Fig. 6. During the transient charging, VC increases as the charging proceeds. To validate the analytical models presented in this paper, Eq. (6) is substituted into Eqs. (1) and (2) to simulate the charging behavior shown in Fig. 1. Apparently, the numerical simulation agrees well with the results predicted by the analytical models. This process was also observed experimentally by Wickenheiser et al[20] using the same structural parameters as those in this paper. It can be seen that the charging voltage predicted by the analytical models is also in reasonable agreement with that obtained by the experiment. The deviations may be due to the diode loss and conduction loss, which are ignored in the current analysis. In practical applications, the switch may be closed at a certain critical charging voltage (e.g., 17 V), and the system will shift into the steady-state charging when the filter capacitor is charged at the critical voltage level, as shown in Fig. 6(b). As expected, the charging voltage remains constant, and the
generated current flows into the battery via the controller, as shown in Fig. 1. The variation of PEH vibration amplitude z with time t is shown in Fig. 7. Initially, the bridge rectifier is blocked, and the PEH vibrates in an open-circuit condition. Upon initiating the charging, a fast drop in the vi bration amplitude z can be observed. As the charging proceeds, the harvested power Pi and electrical damping ratio e decrease, as shown in Figs. 8 and 9, respectively. As a result, z increases as the charging proceeds. This trend was also revealed by Wickenheiser et al[20], who argued that the fast drop in the PEH vibration amplitude was caused by the change in the system’s natural frequency, when the energy harvesting system was altered from open-circuit condition to short-circuit condition. In this paper, it is further suggested that the electrical damping can also reduce the vibration amplitude significantly since e is large even after 6 s, as shown in Fig. 9. After the system shifts into the steady-state charging, the harvested power during each half cycle keeps constant, resulting in fixed electrical damping ratio and PEH vibration amplitude.
Fig. 7
(a) Transient charging
(b) Complete charging
Fig. 6
Charging voltage VC versus time t
PEH vibration amplitude z versus time t
For the transient charging, an undesired result revealed in this paper is that the charging rate slows down over time after the peak value is acquired, as shown in Fig. 8. This originates from the increasing time for the piezoelectric voltage VP to reach the present voltage VC of the filter capacitor during each half cycle, i.e., χi → (i＋ 1)π as i → ∞, as the transient charging proceeds. As a result, the net power stored in the filter capacitor during each half cycle decreases, leading to decreasing charging rate. This result can also be illustrated by the electrical damping ratio e versus charging time t, as shown in Fig. 9. A similar trend can be further demonstrated by the variation of energy harvesting efficiency ef with time t, as shown in Fig. 10. In summary, the maximum charging rate is corresponding to the maximum electrical damping ratio, as well as the optimal energy harvesting efficiency, —411—
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indicating that the energy harvesting system possesses an optimal operating point during the transient charging.
Fig. 8
harvesting efficiency of the steady-state charging is limited by the optimal efficiency of the transient charging, as shown in Fig. 10.
Harvested power Pi versus time t Fig. 11 Energy harvesting efficiency ef versus charging voltage VC
Fig. 9
Fig. 10
Electrical damping ratio e versus time t
Energy harvesting efficiency ef versus time t
The variation of energy harvesting efficiency ef with charging voltage VC during the transient charging shown in Fig. 11 can be beneficial for the optimization of energy harvesting system. It can be seen that the energy harvesting efficiency varies with the charging voltage, and the optimal energy harvesting efficiency of 0.475 can be attained when the charging voltage reaches 10.86 V. Additionally, during the steady-state charging, the system operates with a constant energy harvesting efficiency that is corresponding to the efficiency at the end of the transient charging. Therefore, the energy harvesting efficiency of the steady-state charging can be adjusted by varying the critical charging voltage. According to Eq. (16), varying the critical charging voltage can be realized by regulating the equivalent resistance. However, the optimal energy —412—
The electromechanical coupling characteristic can be demonstrated by the variation of the PEH vibration amplitude z with charging voltage VC, as shown in Fig. 12. Due to the added electrical damping and the shift in the system’s natural frequency, z decreases as VC increases from 0 to 10.86 V. As the charging proceeds, the energy removed from the mechanical vibration during each half cycle decreases, resulting in decreasing electri cal damping. Therefore, z increases gradually after the filter capacitor is charged to 10.86 V.
Fig. 12 PEH vibration amplitude z versus charging voltage VC
3
Conclusions
In this paper, the complete charging of an energy harvesting system is modeled and analyzed under the inphase assumption. For the transient charging, it is found that the system possesses an optimal operating point, at which the charging rate, energy harvesting efficiency and electrical damping ratio can be maximized simultaneously. For the steady-state charging, the energy harvesting efficiency can be optimized by varying the critical
Fan Kangqi et al: Complete Charging for Piezoelectric Energy Harvesting System
charging voltage. However, the optimal energy harvestSystems and Structures, 2012, 23(2): 135-139. ing efficiency of this process is limited by the optimal ［10］ Stewart M, Weaver P M, Cain M. Charge redistribution in efficiency of the transient charging. Finally, the relationpiezoelectric energy harvesters [J]. Applied Physics ship between the critical charging voltage and the equivaLetters, 2012, 100(7): 073901. lent resistance of the controller and rechargeable battery ［11］ Boisseau S, Despesse G, Sylvestre A. Optimization of an is established explicitly. electret-based energy harvester [J]. Smart Materials and The standard interface circuit used in this paper Structures, 2010, 19(7): 075015. shows limited potential in boosting the energy harvesting ［12］ Zorlu Ö, Topal E T, Külah H. A vibration-based efficiency, and nonlinear interface circuits with higher electromagnetic energy harvester using mechanical efficiency will be used in our future work. frequency up-conversion method [J]. IEEE Sensors Journal, 2011, 11(2): 481-488.
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(Editor: Wu Liyou)
