The paper presents a fast and reliable model identification technique for piezoelectric transducers based on an equivalent electromechanical circuit easily implementable on SPICE-like simulation tools. Model parameter extraction is simple and requires just standard and inexpensive laboratory equipment. Indeed, the equivalent circuit representation permits the evaluation of the response of a real energy harvesting system, where the electronic load is a synchronized switching converter which usually causes a significant feedback on the mechanical part of the system during energy extraction. Both simulation and measurements show that the damping effect is particularly important near resonance, where the adopted model is able to fit the experimental data and provides a more realistic description of the behavior of the system

Paganelli, Rudi Paolo

Romani, Aldo

Tartagni, Marco

English

ROMANI, ALDO

PAGANELLI, RUDI PAOLO

TARTAGNI, MARCO

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

Fast and Reliable Modeling of Piezoelectric Transducers for Energy Harvesting Applications

Open Access Repository

Procedia Engineering 25 (2011) 1345 – 13481877-7058 © 2011 Published by Elsevier Ltd.doi:10.1016/j.proeng.2011.12.332Available online at www.sciencedirect.com Proc. Eurosensors XXV, September 4-7, 2011, Athens, Greece Fast and Reliable Modeling of Piezoelectric Transducers for Energy Harvesting Applications Aldo Romani a*, Rudi Paolo Paganelli b, Marco Tartagni a a DEIS-ARCES, II School of Engineering, University of Bologna, Via Venezia 52, 47521 Cesena, Italy bCNR-IEIIT, c/o DEIS-University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy  Abstract This paper presents a fast and reliable model identification technique for piezoelectric transducers based on an equivalent electromechanical circuit easily implementable on SPICE-like simulation tools. Model parameter extraction is simple and requires just standard and inexpensive laboratory equipment.  The equivalent circuits of four cantilevered Q220-A4-303YB transducers from Piezo Systems with different tip masses were identified and several simulations were performed with different electric loading conditions; measurement results agree with model response predictions in all the experiments.  Indeed, the equivalent circuit representation permits the evaluation of the response of a real harvester system, where the electronic load is a synchronized switching converter which usually causes a significant feedback on the mechanical part of the system during energy extraction.  Both simulation and measurements show that the damping effect is particularly important near resonance, where the adopted model is able to fit the experimental data and provides a more realistic description of the behavior of the system.  © 2011 Published by Elsevier Ltd.  Keywords: piezoelectric devices; energy harvesting; equivalent electromechanical model; model identification;  1. Introduction The present interest on pervasive sensor networks and the steady development of electronic devices with low power consumption motivates the research on electronic systems capable of harvesting energy from the surrounding environment. Among many viable methods, the use of piezoelectric devices for converting vibrations into electric energy seems particularly effective. However, a correct estimation of the harvester performance is important to reliably evaluate in an early phase of system design the joint  * Corresponding author Tel : +39 0547 339128; fax: +39 0543 786926Open access under CC BY-NC-ND license.Open access under CC BY-NC-ND license.1346  Aldo Romani et al. / Procedia Engineering 25 (2011) 1345 – 1348performance of the piezoelectric transducer loaded by the devised power conversion circuit. This is particularly true when using synchronized switching converters [1-3]. Since such circuits are intrinsically non-linear and are triggered at specific time instants by smart control logic, the ability of simulating their performance with specific transducers is essential for system optimization. For this purpose, analytical or FEM models of piezoelectric devices produce quite accurate results but cannot be easily coupled to Spice simulations. On the other hand, many works on non-linear power conversion circuits are often based on excessively simplified assumptions, such as for example weak electromechanical coupling or out of resonance operation [1, 2]: these assumptions are particularly limiting in energy harvesting applications where mechanical structures are purposely excited near their resonant frequencies.                              (a)        (b) Fig. 1. (a) Equivalent electromechanical circuit of a piezoelectric transducer;  (b) open circuit reduced equivalent network 2. Equivalent circuit of a piezoelectric transducer Almost any piezoelectric transducer can be successfully modeled with a lumped elements circuit of the type illustrated in Fig. 1a and fully described in [4]. The circuital approach is based on the electro-mechanic analogy: currents are associated to mass velocities  and voltages are associated to forces; therefore, the model consists of two domains, the mechanical and the electrical one, interacting by means of a couple of controlled generators, as shown in Fig. 1a.  In the mechanical part (left side) the inductor LM accounts for the equivalent inertial mass, the capacitor CM is related to the equivalent mechanical compliance and the resistor RM models the structural losses associated to the specific mode of oscillation of the mechanical structure. This description works fairly well in the neighborhood of the resonant frequencies of the oscillating modes of interest.  The mechanical part is fed by the force generator F opposed to a feedback force generator controlled by the voltage developed on the device output capacitor CP (inverse piezoelectric effect). At the same time the mechanical velocity produces a forward current feeding both the output capacitor (direct piezoelectric effect) and any electric load connected to the transducer.  Hence, model identification involves the following independent six parameters: LM, CM, RM, CP, α and β.  In addition, it is sometimes handy to pull back the electrical impedance towards the mechanical domain, as illustrated in Fig. 1b.  2.1. Experimental setup and available measurements The modeled transducers are a set of Q220-A4-303YB piezoelectric beams from Piezo Systems with additional tip masses respectively of 6 g, 10 g, 12 g and 18 g (Fig. 1a). The transducers are mounted on an inexpensive custom shaker system based on a Ciare CW200Z woofer, providing the impressed vibration force FIN with arbitrary waveforms (Fig. 2b), while a Kionix KXP-84 accelerometer measures the actual input acceleration ÿ applied to the base of the cantilever device. A further mechanical quantity, the peak to peak displacement Δz of the tip mass during an AC steady-state sinusoidal regime, is measured by taking a shot with a 1 s exposure time with a standard digital camera and comparing the pixels swept by 1347Aldo Romani et al. / Procedia Engineering 25 (2011) 1345 – 1348the tip elongation with those in a known length, as shown in Fig. 2c. Finally, an Agilent 34401A digital multimeter and an oscilloscope gauge the open circuit output voltage and the short circuit current. However, since common multimeters are unable to resolve very low AC current values (such those produced in a shorted device), a load resistor RL = 98.2 Ω has been used to replace current acquisitions with voltage ones: the admittance of RL is at least two order of magnitude higher than that of CP at the frequencies of interest: RL emulates the short circuit and almost cancels the mechanical feedback αVP.             (a)               (b)        (c) Fig. 2. (a) A cantilever piezoelectric transducer;  (b) the experimental set-up;   (c) a digital picture for evaluating  tip displacement Table 1. Summary of the measurements required for model extraction  In this work, all the forces acting on the system are expressed in scaled units by assuming a unity equivalent mass meq. The measurements used for model identification can be classified according to the electrical loading condition (short or open) and to the type of mechanical excitation (pulse or sinusoidal). With sinusoidal excitations it is easy to sweep the frequency and detect the resonance peak. In a shorted configuration, the detected peak is at the true natural frequency fM of the pure mechanical mode, when the piezoelectric feedback is totally cancelled. In an open circuit configuration, the resonance frequency fO is shifted with respect to fM because: (1) the effect of the feedback force generator αVP and of the electrical capacitance CP results in a capacitance CEQ instead of CM ;  (2) the peak of a low-pass response is a function of the resonator damping time constant τ . Equations (1) clarify this aspect, whereas (2) expose the method:    (1) Acquired parameter In which experiment? How is it determined? τ  damping time constant Pulse mechanical excitation; open circuit load The shaker applies a mechanical pulse;   exponentially damped oscillations are analyzed with an oscilloscope  fM mech. resonance frequency Swept sinusoidal excitation; pseudo-short circuit configuration VP(f) and  ÿ(f) are measured. The shaker sweeps frequency until a maximum of VP(f)/ÿ(f) is found fO open circuit peak frequency Swept sinusoidal excitation; open circuit configuration  VP(f) and  ÿ(f) are measured. The  shaker sweeps frequency until a maximum of VP(f)/ÿ(f) is found VP(fO) output voltage peak amplitude Steady-state sinusoidal excitation at fO;  open circuit configuration The digital multimeter reads the AC voltage while the shaker applies the sinusoidal acceleration Δz(fO) peak-to-peak tip displacement Steady-state sinusoidal excitation at fO;  open circuit configuration A digital camera takes a 1s shot of the beam tip with a background ruler. Lengths are measured on pixel basis. ÿ(fO) impressed base acceleration Steady-state sinusoidal excitation at fO;  open circuit configuration The accelerometer RMS is readout while the shaker applies the sinusoidal acceleration CP transducer electrical capacitance Electrical excitation;  clamped device condition  LCR-meter measurement at 10kHz: sinusoidal excitation well above cut-off of any important vibration mode  . 11    ; 2121    ; 1212PMEQEQMOMMM CCCCLfCLfαβτππ+=−==760 px 54 px 1348  Aldo Romani et al. / Procedia Engineering 25 (2011) 1345 – 1348  (2)  2.2. Experimental validation In a first set of experiments (Fig. 3a) the identified models accurately predict the frequency response of the piezoelectric transducers with different masses and loading conditions.  In a second set of experiments the synchronized charge extractor described in [3] was connected to the transducers. In case of sinusoidal input vibrations, during operation the circuit applies a periodic series of current pulses for tapping energy. At resonance this action results in a periodic force opposed to input vibrations damping the actual output voltage VP* down to half its nominal value (Fig. 3b) of twice the unloaded voltage VP as explained in [3].                        (a)                 (b)  Fig. 3. (a) Comparison of measured and simulated frequency responses; (b) Voltage is measured in open circuit configuration (VP) and when the power converter is activated (VP*). The plot shows the measured and simulated ratios VP*/VP  at different frequencies. 3. Conclusion The presented modeling technique allows a circuit designer to quickly and reliably evaluate the loading effect of the non-linear power converter on the piezoelectric transducer and to readily estimate overall system performance. The loading effect cannot be neglected for small sized cantilever beams with tip inertial masses when forcing base vibrations have frequency components close to resonance.  Acknowledgements This research was funded by EC FP7 under grant agreement NANOFUNCTION no. 257375 and by ENIAC Joint Undertaking under grant agreement END no. 120214. The authors thank Fondazione Cassa dei Risparmi di Forlì for financial support.  References [1]  Lefeuvre E, Badel A, Richard C, Petit L, Guyomar D, A comparison between several vibration-powered piezoelectric generators for standalone systems. Sensors and Actuators A: Physical; 126 405–16 (2006)  [2]  Lallart M, Garbuio L, Petit L, Richard C, Guyomar D, Double synchronized switch harvesting (DSSH): a new energy harvesting scheme for efficient energy extraction. IEEE transaction on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 10, october 2008 2119-30  [3]  Romani A, Paganelli RP, Tartagni M, A scalable micro-power converter for multi-source piezoelectric energy harvesting applications. Procedia Engineering 5, 782-85 (2010)  [4]  Paganelli RP, Romani A, Golfarelli A, Magi M, Sangiorgi E, Tartagni M, Modeling and characterization of piezoelectric transducers by means of scattering parameters. Part I: Theory. Sensors and Actuators A: Physical; 160 9–18 (2010)  .11;)2(1;2;)()(;)()(22−===⋅Δ=Δ⋅=MEQPMMMMMOOOINMOOPPCCCLfCRLffzfyRfzfVCβαπτπβ

AbstractThis paper presents a fast and reliable model identification technique for piezoelectric transducers based on an equivalent electromechanical circuit easily implementable on SPICE-like simulation tools. Model parameter extraction is simple and requires just standard and inexpensive laboratory equipment. The equivalent circuits of four cantilevered Q220-A4-303YB transducers from Piezo Systems with different tip masses were identified and several simulations were performed with different electric loading conditions; measurement results agree with model response predictions in all the experiments. Indeed, the equivalent circuit representation permits the evaluation of the response of a real harvester system, where the electronic load is a synchronized switching converter which usually causes a significant feedback on the mechanical part of the system during energy extraction. Both simulation and measurements show that the damping effect is particularly important near resonance, where the adopted model is able to fit the experimental data and provides a more realistic description of the behavior of the system

Elsevier - Publisher Connector 

Fast and Reliable Modeling of Piezoelectric Transducers for Energy Harvesting Applications 

Procedia Engineering 25 (2011) 1345 – 1348
1877-7058 © 2011 Published by Elsevier Ltd.
doi:10.1016/j.proeng.2011.12.332
Available online at www.sciencedirect.com
 
Proc. Eurosensors XXV, September 4-7, 2011, Athens, Greece 
Fast and Reliable Modeling of Piezoelectric Transducers 
for Energy Harvesting Applications 
Aldo Romani a*, Rudi Paolo Paganelli b, Marco Tartagni a 
a DEIS-ARCES, II School of Engineering, University of Bologna, Via Venezia 52, 47521 Cesena, Italy 
bCNR-IEIIT, c/o DEIS-University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy 
 
Abstract 
This paper presents a fast and reliable model identification technique for piezoelectric transducers based on an 
equivalent electromechanical circuit easily implementable on SPICE-like simulation tools. Model parameter 
extraction is simple and requires just standard and inexpensive laboratory equipment.  The equivalent circuits of four 
cantilevered Q220-A4-303YB transducers from Piezo Systems with different tip masses were identified and several 
simulations were performed with different electric loading conditions; measurement results agree with model 
response predictions in all the experiments.  Indeed, the equivalent circuit representation permits the evaluation of the 
response of a real harvester system, where the electronic load is a synchronized switching converter which usually 
causes a significant feedback on the mechanical part of the system during energy extraction.  Both simulation and 
measurements show that the damping effect is particularly important near resonance, where the adopted model is able 
to fit the experimental data and provides a more realistic description of the behavior of the system. 
 
© 2011 Published by Elsevier Ltd. 
 
Keywords: piezoelectric devices; energy harvesting; equivalent electromechanical model; model identification;  
1. Introduction 
The present interest on pervasive sensor networks and the steady development of electronic devices 
with low power consumption motivates the research on electronic systems capable of harvesting energy 
from the surrounding environment. Among many viable methods, the use of piezoelectric devices for 
converting vibrations into electric energy seems particularly effective. However, a correct estimation of 
the harvester performance is important to reliably evaluate in an early phase of system design the joint 
 
* Corresponding author Tel : +39 0547 339128; fax: +39 0543 786926
Open access under CC BY-NC-ND license.
Open access under CC BY-NC-ND license.
1346  Aldo Romani et al. / Procedia Engineering 25 (2011) 1345 – 1348
performance of the piezoelectric transducer loaded by the devised power conversion circuit. This is 
particularly true when using synchronized switching converters [1-3]. Since such circuits are intrinsically 
non-linear and are triggered at specific time instants by smart control logic, the ability of simulating their 
performance with specific transducers is essential for system optimization. For this purpose, analytical or 
FEM models of piezoelectric devices produce quite accurate results but cannot be easily coupled to Spice 
simulations. On the other hand, many works on non-linear power conversion circuits are often based on 
excessively simplified assumptions, such as for example weak electromechanical coupling or out of 
resonance operation [1, 2]: these assumptions are particularly limiting in energy harvesting applications 
where mechanical structures are purposely excited near their resonant frequencies. 
 
 
 
 
 
 
                       (a)        (b) 
Fig. 1. (a) Equivalent electromechanical circuit of a piezoelectric transducer;  (b) open circuit reduced equivalent network 
2. Equivalent circuit of a piezoelectric transducer 
Almost any piezoelectric transducer can be successfully modeled with a lumped elements circuit of the 
type illustrated in Fig. 1a and fully described in [4]. The circuital approach is based on the electro-
mechanic analogy: currents are associated to mass velocities Ī and voltages are associated to forces; 
therefore, the model consists of two domains, the mechanical and the electrical one, interacting by means 
of a couple of controlled generators, as shown in Fig. 1a.  In the mechanical part (left side) the inductor 
LM accounts for the equivalent inertial mass, the capacitor CM is related to the equivalent mechanical 
compliance and the resistor RM models the structural losses associated to the specific mode of oscillation 
of the mechanical structure. This description works fairly well in the neighborhood of the resonant 
frequencies of the oscillating modes of interest.  The mechanical part is fed by the force generator F 
opposed to a feedback force generator controlled by the voltage developed on the device output capacitor 
CP (inverse piezoelectric effect). At the same time the mechanical velocity produces a forward current 
feeding both the output capacitor (direct piezoelectric effect) and any electric load connected to the 
transducer.  Hence, model identification involves the following independent six parameters: LM, CM, RM, 
CP, α and β.  In addition, it is sometimes handy to pull back the electrical impedance towards the 
mechanical domain, as illustrated in Fig. 1b.  
2.1. Experimental setup and available measurements 
The modeled transducers are a set of Q220-A4-303YB piezoelectric beams from Piezo Systems with 
additional tip masses respectively of 6 g, 10 g, 12 g and 18 g (Fig. 1a). The transducers are mounted on an 
inexpensive custom shaker system based on a Ciare CW200Z woofer, providing the impressed vibration 
force FIN with arbitrary waveforms (Fig. 2b), while a Kionix KXP-84 accelerometer measures the actual 
input acceleration ÿ applied to the base of the cantilever device. A further mechanical quantity, the peak 
to peak displacement Δz of the tip mass during an AC steady-state sinusoidal regime, is measured by 
taking a shot with a 1 s exposure time with a standard digital camera and comparing the pixels swept by 
1347Aldo Romani et al. / Procedia Engineering 25 (2011) 1345 – 1348
the tip elongation with those in a known length, as shown in Fig. 2c. Finally, an Agilent 34401A digital 
multimeter and an oscilloscope gauge the open circuit output voltage and the short circuit current. 
However, since common multimeters are unable to resolve very low AC current values (such those 
produced in a shorted device), a load resistor RL = 98.2 Ω has been used to replace current acquisitions 
with voltage ones: the admittance of RL is at least two order of magnitude higher than that of CP at the 
frequencies of interest: RL emulates the short circuit and almost cancels the mechanical feedback αVP. 
 
 
 
 
 
 
      (a)               (b)        (c) 
Fig. 2. (a) A cantilever piezoelectric transducer;  (b) the experimental set-up;   (c) a digital picture for evaluating  tip displacement 
Table 1. Summary of the measurements required for model extraction 
 
In this work, all the forces acting on the system are expressed in scaled units by assuming a unity 
equivalent mass meq. The measurements used for model identification can be classified according to the 
electrical loading condition (short or open) and to the type of mechanical excitation (pulse or sinusoidal). 
With sinusoidal excitations it is easy to sweep the frequency and detect the resonance peak. In a shorted 
configuration, the detected peak is at the true natural frequency fM of the pure mechanical mode, when the 
piezoelectric feedback is totally cancelled. In an open circuit configuration, the resonance frequency fO is 
shifted with respect to fM because: (1) the effect of the feedback force generator αVP and of the electrical 
capacitance CP results in a capacitance CEQ instead of CM ;  (2) the peak of a low-pass response is a 
function of the resonator damping time constant τ
 
. Equations (1) clarify this aspect, whereas (2) expose 
the method:  
 
 (1) 
Acquired parameter In which experiment? How is it determined? 
τ
 
 
damping time constant 
Pulse mechanical excitation; 
open circuit load 
The shaker applies a mechanical pulse;   exponentially 
damped oscillations are analyzed with an oscilloscope  
fM 
mech. resonance frequency 
Swept sinusoidal excitation; 
pseudo-short circuit configuration 
VP(f) and  ÿ(f) are measured. The shaker sweeps 
frequency until a maximum of VP(f)/ÿ(f) is found 
fO 
open circuit peak frequency 
Swept sinusoidal excitation; 
open circuit configuration  
VP(f) and  ÿ(f) are measured. The  shaker sweeps 
frequency until a maximum of VP(f)/ÿ(f) is found 
VP(fO) 
output voltage peak amplitude 
Steady-state sinusoidal excitation 
at fO;  open circuit configuration 
The digital multimeter reads the AC voltage while the 
shaker applies the sinusoidal acceleration 
Δz(fO) 
peak-to-peak tip displacement 
Steady-state sinusoidal excitation 
at fO;  open circuit configuration 
A digital camera takes a 1s shot of the beam tip with a 
background ruler. Lengths are measured on pixel basis. 
ÿ(fO) 
impressed base acceleration 
Steady-state sinusoidal excitation 
at fO;  open circuit configuration 
The accelerometer RMS is readout while the shaker 
applies the sinusoidal acceleration 
CP 
transducer electrical capacitance 
Electrical excitation;  
clamped device condition  
LCR-meter measurement at 10kHz: sinusoidal excitation 
well above cut-off of any important vibration mode  
. 
11
    ; 
21
2
1
    ; 
1
2
1
2
PMEQEQM
O
MM
M CCCCL
f
CL
f αβ
τππ
+=−==
760 px 
54 px 
1348  Aldo Romani et al. / Procedia Engineering 25 (2011) 1345 – 1348
 
 (2) 
 
2.2. Experimental validation 
In a first set of experiments (Fig. 3a) the identified models accurately predict the frequency response of 
the piezoelectric transducers with different masses and loading conditions.  In a second set of experiments 
the synchronized charge extractor described in [3] was connected to the transducers. In case of sinusoidal 
input vibrations, during operation the circuit applies a periodic series of current pulses for tapping energy. 
At resonance this action results in a periodic force opposed to input vibrations damping the actual output 
voltage VP* down to half its nominal value (Fig. 3b) of twice the unloaded voltage VP as explained in [3].  
 
 
 
 
 
 
 
               (a)                 (b)  
Fig. 3. (a) Comparison of measured and simulated frequency responses; (b) Voltage is measured in open circuit configuration (VP) 
and when the power converter is activated (VP*). The plot shows the measured and simulated ratios VP*/VP  at different frequencies. 
3. Conclusion 
The presented modeling technique allows a circuit designer to quickly and reliably evaluate the 
loading effect of the non-linear power converter on the piezoelectric transducer and to readily estimate 
overall system performance. The loading effect cannot be neglected for small sized cantilever beams with 
tip inertial masses when forcing base vibrations have frequency components close to resonance.  
Acknowledgements 
This research was funded by EC FP7 under grant agreement NANOFUNCTION no. 257375 and by ENIAC Joint Undertaking 
under grant agreement END no. 120214. The authors thank Fondazione Cassa dei Risparmi di Forlì for financial support.  
References 
[1]  Lefeuvre E, Badel A, Richard C, Petit L, Guyomar D, A comparison between several vibration-powered piezoelectric 
generators for standalone systems. Sensors and Actuators A: Physical; 126 405–16 (2006)  
[2]  Lallart M, Garbuio L, Petit L, Richard C, Guyomar D, Double synchronized switch harvesting (DSSH): a new energy 
harvesting scheme for efficient energy extraction. IEEE transaction on ultrasonics, ferroelectrics, and frequency control, 
vol. 55, no. 10, october 2008 2119-30  
[3]  Romani A, Paganelli RP, Tartagni M, A scalable micro-power converter for multi-source piezoelectric energy harvesting 
applications. Procedia Engineering 5, 782-85 (2010)  
[4]  Paganelli RP, Romani A, Golfarelli A, Magi M, Sangiorgi E, Tartagni M, Modeling and characterization of piezoelectric 
transducers by means of scattering parameters. Part I: Theory. Sensors and Actuators A: Physical; 160 9–18 (2010)  
.
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Fast and Reliable Modeling of Piezoelectric Transducers for Energy Harvesting Applications

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