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Modeling Piezoceramic Actuators for Smart Applications
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MODELING PIEZOCERAMIC ACTUATORS FOR SMART APPLICATIONS H. M. Saoulli, R. Ben Mrad 1 Department of Mechanical and Industrial Engineering University of Toronto, Toronto, Canada
Abstract: The development of a self-sensing piezoceramic actuator requires the development of a model, which completely characterizes the electrical and mechanical properties of piezoceramics. A model that characterizes the nonlinear mechanical and electrical coupling inherent to piezoelectric ceramic stack actuators is presented. The behaviour of the piezoceramic stack actuator is established through experimentation, where the charging and loss processes are modeled with the use of complex permittivity. The piezoceramic model presented characterizes the coupling of the mechanical and electrical properties of the piezoceramic actuator when subject to variable load conditions, under a range of voltage excitations at frequencies up to 5kHz. Copyright © 20021FAC Keywords: Modeling of piezoceramics, smart actuators, self-sensing actuators.
where T represents the stress applied, S the strain and sE denotes the compliance of the medium at constant field strength E. D denotes the dielectric displacement, ET is the permittivity of the medium at constant T and d is the piezoelectric charge constant that relates the mechanical and electrical domains. All variables in (1) and (2) are tensors. Certain ceramics can be made piezoelectric with the application of an electric field, such as Lead Zirconate Titanate (PZT).
1. INTRODUCTION Today, there is interest in the self-sensing ability of piezoelectric ceramics, which have the unique ability to simultaneously act as both a sensor and an actuator. Self-sensing involves extracting the sensing signal from that of the actuating signal (Sensor Technology, 1999; Jones et aI., 1994). The need for a self-sensing actuator originates from the need to control high precision positioners and structures, and the numerous applications range from vibration suppression in bridges and large scale structures to pressure regulation in micro pumps.
The direct piezoelectric effect, where a conversion from mechanical force or movement is made proportional to an electrical signal, can be found in acceleration and pressure sensors. The inverse piezo effect, where a conversion from an electrical signal is made proportional to a mechanical displacement or when the piezoelectric ceramic is utilized as an actuator, can be found in pneumatic and fuel-injection valves. Piezoelectric actuators offer high resolution down to sub-nanometre range, high stiffness, and fast response times, which makes them well suited for high precision and accuracy tracking and positioning systems (Physik Instrumente, 1998). Unfortunately, piezoceramics suffer from hysteretic behaviour inherent to non-linear dielectric materials, where a single input voltage corresponds to a range of output displacements (Hu and Ben Mrad, 2002).
A standards committee of the IEEE (1987) published a mathematical description of the piezoelectric phenomenon. IEEE (1987) adopted the linear theory of piezoelectricity, where the linear elasticity is coupled to charge by means of piezoelectric constants: S=sET+dE T
D=dT +E E
(1) (2)
I Corresponding Author: 5 King's College Road, University of Toronto, Toronto, Ontario M5S 3G8, Canada; Tel: 416-946-3597; Fax: 416-978-7753; e-mail: [email protected].
155
In a variety of PZT applications, including active vibration control of structures, pressure regulation in pumps and force control in micro-actuators, there is a need to develop a self-sensing ability in piezoelectric actuators, which would have the unique ability to simultaneously act as both a sensor and an actuator. Self-sensing in piezoelectric ceramics involves extracting the sensing signal from that of the actuating signal (Sensor Technology, 1999; lones et aI., 1994). The development of a self-sensing piezoceramic actuator requires the development of a model, which completely characterizes the electrical and mechanical properties of piezoceramics, and therefore allows the separation of the current generated in the piezoceramic due to the application of a force to it from the current in the piezoceramic arising from the application of a voltage to it. The development of such a model is pursued in this paper, where the authors present a method of tracking the nonlinear mechanical and electrical coupling inherent to piezoelectric ceramic stack actuators.
piezoelectric vibrator to couple mechanical motion, such as in the case of a self-sensing actuator, then these models are no longer sufficient and a mechanical terminal must be added to form an electromechanical equivalent model (Ballato, 2001).
2.2 Electromechanical Equivalent Models The application of the piezoelectric effect to selfsensing depends upon the motor-generator action of the crystal (Mason, 1950). Mason developed an electrical equivalent network circuit to represent the electrical and mechanical properties of the crystal by applying tradition'\l . electromechanical analogies, along with the piezJel~ctric equations and Newton's laws of motion (Mason, '1,948). ",.
One perceived problem with Mason's equivalent model is that it requires a negative capacitance at the electrical port (Sherrit et aI., 1999). Attempts by others in order to modify Mason's model have led to models that are too complex or "unphysical" (Ba11ato, 200 I; Sherrit et aI., 1999). In addition, the widely accepted linear piezoelectric constitutive equations, which formed the foundation of Mason's models, fail to explicitly describe both the hysteretic nonlinearities inherent to piezoelectric ceramics and the dynamical aspects of the electromechanical coupling. The author's believe that the modeling of a nonlinear piezoelectric system is better achieved through a phenomenological approach.
2. MODELS OF PIEZOELECTRIC CERAMICS
2.1 Electrical Equivalent Models Butterworth showed that an electrical equivalent circuit could represent a mechanically vibrating system with a single degree of freedom, driven by exposure to the electric field of a capacitor (Ballato, 200 I). Van Dyke also presented a circuit that consisted of constant value elements, which turned out to be identical to Butterworth's (Ballato, 2001). The Butterworth-Van Dyke (BVD) circuit is depicted in Fig. I .
2.3 Modern Phenomenological Models Two models based on physical principles have been developed recently (Adriaens et aI., 2000; Goldfarb and Celanovic, 1997). These models consist of an electrical and mechanical domain, as well as a connection between the two. Both proposed models describe the hysteretic non linearity of the piezoelectric ceramic.
-ilr
In this paper the authors present a model of piezoelectric actuators based on (Adriaens et aI., 2000) and (Goldfarb and Celanovic, 1997), where the behaviour of the PZT stack actuator is established through experimentation. The authors introduce a model where the charging and loss processes are represented with the use of complex permittivity. The PZT model presented characterizes the coupling of the mechanical and electrical properties of the piezoceramic actuator when subject to variable load conditions, under a range of voltage excitations at frequencies up to 5kHz.
Fig. I: Butterworth-Van Dyke (BVD) circuit. The representation of a piezoelectric vibrator by the BVD circuit is useful only if the circuit parameters are constant and independent of frequency, which only occurs for a narrow range of frequencies near the resonance frequency and only if the mode in question is sufficiently isolated from other modes (IEEE, 1987). The close proximity of several modes of vibration may be represented by adding additional branches in parallel to the R-L-C branch. These models represent equivalent electric circuits of piezoelectric vibrators and they only depict the "electric behaviour". If it is desired to use the
156
3. EXPERIMENTAL ASSESSMENT OF PIEZOELECTRIC CERAMICS' ELECTRICAL PROPERTIES
(6) A BM532 PZT stack actuator was used (Sensor Technology, 1999). Two physical loading conditions, namely a free case and a blocked case, as illustrated in Fig. 2, were investigated. Under the two conditions, the admittance was examined under variable peak voltage conditions, namely 20V, 50V, and lOOV, at frequencies up to 5kHz.
3.1 Motivation/or Experimentation
The models presented in (Adriaens et aI., 2000) and (Goldfarb and Celanovic, 1997) make the assumption of a constant capacitance to represent the electrical domain. The capacitance of nonlinear dielectrics, such as piezoceramics, is nonlinear with respect to the operating voltage and is mostly related to changes occurring within the material during operation (Sensor Technology, 1999; Jones et aI., 1994). Making reference to the ideal parallel-plate capacitor given by (3), the variance in the capacitance is due to a variance in £, the dielectric constant (Sensor Technology, 1999; Jones et aI., 1994; Sadiku, 1995): M C=-
IIII .LOCKED
Fig. 2: Two physical loading conditions.
(3)
d
3.2 Experimental Results
where C is the capacitance of the parallel plate capacitor, where the plates, with a surface area of A, are separated by a distance of d.
The frequency variation of the complex permittivity of the PZT stack was determined, when subject to variable loading conditions, under a range of voltage excitations at frequencies up to 5kHz. The experimental results agree with the authors' prediction that the models presented by (Adriaens et aI., 2000) and (Goldfarb and Celanovic, 1997) are a good starting point for a phenomenological model based on physics, unfortunately, both models make the wrong assumption of a constant capacitance to represent the electrical domain with no resistive component. The results (Fig. 3 and Fig. 4) show the admittance has a strong dependence upon force, peak voltage, and frequency, with the electrical domain modeled as a resistor in parallel with a capacitor. In addition, the results show a strong need for a resistive component to be included in the electrical domain modeling of PZTs.
In a parallel plate capacitor, when material is inserted between the plates the capacitance is increased (Hench and West, 1990). The dielectric constant k is defined as the ratio of capacitance of a capacitor with a dielectric between the plates to that of merely a vacuum (Hench and West, 1990):
C MId k=-=--=£I£o Co £oAld
(4)
where £ is the permittivity of the dielectric material and t:o is the permittivity of free space. In this paper, complex permittivity is pursued in order to improve upon previous attempts of modeling piezoelectric actuators. The complex permittivity is expressed as: k
"
=£ /£0 =
,. , k -lk
The results presented in Fig. 3 show that the magnitude of the capacitance depends on applied voltage and load, where a decrease in applied voltage and an increase in applied force have a similar effect of decreasing the magnitude of the capacitance. The results demonstrate that the capacitance is constant with respect to frequency over the frequency range studied.
(5)
where k' and k" represent the real and imaginary components of the complex permittivity, respectively.
The results shown in Fig. 4 indicate that the resistance depends on frequency and applied voltage, where an increase in both the frequency and peak input voltage have a parallel effect of decreasing the magnitude of the resistance. The results indicate that there is an insignificant dependence of applied load on the resistance.
In order to estimate the electrical properties of the piezoceramic material, the PZf stack is modeled with the input to the system as the voltage across the PZf stack and the output of the system as the current through the PZT stack. The PZT stack is represented as a capacitance C in parallel with a resistor R. The transfer function representing the PZT then becomes the admittance, namely:
157
S.ooE-07
~
7.ooE-07
tt;;;~::=~~~;::;;;;~~
6.ooE-07 5.ooE-07
E U
The authors agree with (Goldfarb and Celanovic, 1997) in that the hysteresis inherent to piezoelectric ceramics lies solely in the electrical domain between the applied actuator voltage and resulting charge. Fig. 5 shows the experimental results of the voltage versus charge data. Fig. 6 illustrates the linear relationship between charge and displacement. These results are used to validate the form of the model to be presented in a subsequent section.
.... ............................................................................. ........... .
4.ooE-07 3.ooE-07 2.ooE-07 l.ooE-07
+------__,_-----,--------1
O.ooE+oo
10
lOO
1000
10000
Frequeacy (HI)
------------------
1-+-20Vpp __ 50Vpp _Ioovppl 100
(a)
so
8.ooE-07 ,........ .... ................ ..................... .. . 7.00E-07 6.ooE-07 5.00E-07
E
4.00E-07
U
3.00E-07
20
2.00E-07 O~~~-,_-,_-__,_-_r-_r-_r-~
1.00E-07
O.E+oo I.E-OS 2.E-OS 3.E-OS 4.E-OS S.E-OS 6.E-OS 7.E-OS S.E-OS
O.OOE+OO +-----~----~------; 10 100 1000 10000
Cha .... (C)
Fig. 5: Hysteretic voltage versus charge behaviour.
Frequency (HI)
I·· •. 20Vpp .. • .
50Vpp .. d· · 100vprl
S.ooE-OS T ...... ·· .......................... · .. ·.... ·..·....· .. ..............................................................................,
(b)
7.ooE-OS
Fig_ 3: Measured capacitance versus frequency, where Vpp is the peak -to-peak voltage_ (a) Free; (b) Blocked. I~~··
...... ······.. ······...... ··············· ·· ············· .....................................................................
6.ooE-OS ~
S.ooE-OS
l-
4.00E·OS
e
,
3.ooE-OS 2.ooE·OS l.ooE·OS
100000
O.ooE+OO .Jo"----~--~--~--~---i O.ooE+OO l.ooE-06 2.ooE-06 3.ooE-06 4.ooE-06 5.ooE-06
IQooO
Displacement (m)
1000
Fig. 6: Charge versus displacement.
lOO 10+-----_r-----~----~
10
100
1000
10000
3.3 Explanation of Experimental Results
Frequency [Hz) 1-+-20Vpp --SOVpp --Ioovppl
As revealed by (4), dielectric capacitors can store more charge than their ideal parallel-plate vacuum capacitor counterparts. The physical quantity corresponding to the stored charge per unit area is the electric displacement D, where it is related to the electric field E by (7) (Sadiku, 1995):
(a) I~T
.. ·..······.. ·.. · ····················· .. ···· ..···········._
................................................................... .
100000
I '"
10000
I()()()
D=EoE+ P
(7)
lOO
where P represents polarization. 10+------,------~----~
IQ
100
IQoo
10000
The definition of dielectric displacement for free space is a special case of that in equation (7) for P = 0 (Sadiku, 1995). Therefore the total electric displacement of a condenser is the sum of the electric field, if the material was not present, plus the phenomena known as polarization (Hench and West, 1990; Uchino, 1997). There are four primary mechanisms of polarization, namely electronic or optical, atomic, dipole or orientation, and interfacial
Froqu.acy [Hz)
I... .20Vpp ..... SOVpp
"0"
loovppl
(b)
Fig. 4: Measured resistance versus frequency, where Vpp is the peak-to-peak voltage_ (a) Free; (b) Blocked.
158
concluded that the shape of the experimental results follows the theoretical prediction of Debye, dictated by dipole polarization.
or space charge polarization (Uchino, 1997). The total polarization may be thought of as the contribution of individual polarizabilities, each arising from a particular mechanism of polarization (Hench and West, 1990). The frequency dependence of the polarization is depicted in Fig. 7, where it can be seen in the frequency range of interest, dipolar polarization is the main contributor to the polarizability of the dielectric material, such as piezoelectric ceramics.
In addition, the shift in the curves with a change in applied peak voltage, observed in the experimental results, is due to polarization. For some dielectrics it is expected that P would vary directly as the applied electric field E, given by (9) (Uchino, 1997):
(9) where X, is known as the electric susceptibility of the material (Sadiku, 1995). Hence as the voltage is increased the dipolar polarization P increases, which corresponds to a shift in Fig. 7 and Fig. 8 or an increase in k' and k' , equivalent to increasing C and decreasing R, as observed.
a j nterfac i al i
...... ........ .......... .. ... .... .... ... .
~-------
ex a a tom ic
(l
, ele c tr o ni c 4. ELECTROMECHANICAL MODEL
10
An electromechanical model of PZT stack actuators is presented that explicitly accounts for the electrical and mechanical domains of the piezoelectric ceramic. The two domains are separated and connected through an electromechanical transducer. The proposed model describes the hysteretic nonlinearity of the piezoelectric ceramic with a hysteresis operator, based on a generalized Maxwell resistive capacitor model (Goldfarb and Celanovic, 1997). The dynamical mechanical behaviour is also explicitly accounted for by using a mechanical operator M that considers the piezoelectric stack actuator as a distributed parameter system, where the mass of the PZT is distributed evenly over the element (Adriaens et aI., 2000).
Fig. 7: Frequency dependence of polarizabilities. Debye extended the concept of the frequency dependence of dipolar polarization and developed equations, which describe the frequency-dependent relationships of the charging and loss constants and the loss tangent of the dielectric material (Hench and West, 1990). The graphical relation of the Debye equations is depicted in Fig. 8.
In addition, in the model presented, the capacitance is a function of applied voltage and load, and it is assumed to be constant with respect to frequency. A resistive component in the electrical domain is used in the modeling of the PZf. This resistance is a function of frequency and applied voltage. The electromechanical model is presented in Fig. 9.
Frequency
Fig. 8: k' and k" versus frequency as described by the Debye equations.
The piezoelectric transformer ratio
Referring back to equations (4) and (5), an alternative derivation of the PZT admittance can be derived that takes into account k' and k' . The expression is:
R(u pt a' 0)
effect is represented by a whereas C(u pta ' F) and
T,m'
characterize the electrical domain.
q
is
the current flow through the PZT actuator, and iR is the current flow through the resistor, while q p is the transducer charge. Furthermore,
(8)
up
voltage due to the piezo effect, and
symbolizes the
uh
is the voltage
across the Maxwell resistive capacitor, whereas u pta
It can be easily verified by relating the terms in (6) and (8) that C is proportional to k ' and R is inversely proportional to k'. By noting this relation and comparing Fig. 8 with Fig. 3 and Fig. 4, it can be
is the total voltage across the PZf. Fp is the force due to the inverse piezo effect, while Ft is the externally applied force. The elongation of the PZf
159
(10)
is denoted by y . As in (Adriaens et aI., 2000), the dynamic mechanical relation between force and elongation is related by M, while as in (Goldfarb and Celanovic, 1997) the hysteric behaviour is compensated with a hysteresis operator H. The complete set of electromechanical equations that describes the actuator is as follows:
= H(q)
(11 )
q = C(u pea ' F)u p + q p + fi Rdt
(12)
Uh
. 1 qp = - - y Tem Fp =T,mup
y =M(Fp - F,)
(13) (14) (15)
Fp F. : :
uh
Hysteresis Operator
..... ........ ... .. ... .... .... .
..... .... .. ... .... .. .. .. ...... qp
upe'
(upe., F)
M
L+y
R(up
Up
Electromechanical Coupling
Fig. 9: Electromechanical model. Hu, H. and R. Ben Mrad, (2002), On the classical Preisach model of hysteresis in piezoceramic actuators. Mechatronics, in press. IEEE (1987). An American National Standard: IEEE Standard on Piezoelectricity. Jones, L., E. Garcia, and H. Waites (1994) . SelfSensing Control as Applied to a PZT Stack Actuator Used as a Micropositioner. Smart Mater. Struct., vol. 3, pp. 147-156. Mason, W. P. (1948). Electromechanical Transducers and Wave Filters, 2nd ed.. D. Van Nostrand Company, Inc., Toronto. Mason, W. P. (1950). Piezoelectric Crystal and Their Application to Ultrasonics. D. Van Nostrand Company, Ltd., Toronto. Physik Instrumente (1998). NanoPositioning. Germany. Elements of Sadiku, M. N. O. (1995). Electromagnetics, 2nd ed., Oxford University Press, New York. Sensor Technology (1999). Piezoceramics Product Catalogue and Application Notes (BM Hi-Tech Division). Canada. Sherrit, S., S. P. Leary, B. P. Dolgin, and Y. BarCohen (1999). Comparison of the Mason and KLM Equivalent Circuits for Piezoelectric Resonators in the Thickness Mode. IEEE Ultrasonics Symposium, pp. 921-926. Uchino, K. (1997). Piezoelectric Actuators and Ultrasonic Motors. Kluwer Academic Publishers, Boston.
CONCLUSIONS An electromechanical model for piezoelectric ceramics is presented. The model uses a resistive component in the electrical domain modeling, and accounts for the effects of load, peak voltage, and frequency in the model. The model characterizes hysteresis, electromechanical coupling as well as the effects of the dynamic forces in the mechanical domain.
REFERENCES Adriaens, H. J. M. T. A., W. L. de Koning, and R. Banning (2000) . Modeling Piezoelectric Actuators. IEEElASME Trans. on Mechatronics, vol. 5, no. 4, pp. 331-341. Ballato, A. (2001). Modeling Piezoelectric and Piezomagnetic Devices and Structures via Equivalent Networks. IEEE Trans. on Ultrasonics and Freq. Cont., vol. 48, no. 5, pp. 1189-1240. Goldfarb, M. and N. Celanovic (1997). Modeling Piezoelectric Stack Actuators for Control of Micromanipulation. IEEE Control Systems, vol. 17, no. 3, pp. 69-79. Hench, L. L. and 1. K. West (1990) . Principles of Electronic Ceramics. John Wiley & Sons, Inc. , Canada.
160
[: 0
Copyright (:) IFAC Mechatronic Systems, California, USA, 2002
[>
Publications www.elsevier.comllocatelifac
MODELING PIEZOCERAMIC ACTUATORS FOR SMART APPLICATIONS H. M. Saoulli, R. Ben Mrad 1 Department of Mechanical and Industrial Engineering University of Toronto, Toronto, Canada
Abstract: The development of a self-sensing piezoceramic actuator requires the development of a model, which completely characterizes the electrical and mechanical properties of piezoceramics. A model that characterizes the nonlinear mechanical and electrical coupling inherent to piezoelectric ceramic stack actuators is presented. The behaviour of the piezoceramic stack actuator is established through experimentation, where the charging and loss processes are modeled with the use of complex permittivity. The piezoceramic model presented characterizes the coupling of the mechanical and electrical properties of the piezoceramic actuator when subject to variable load conditions, under a range of voltage excitations at frequencies up to 5kHz. Copyright © 20021FAC Keywords: Modeling of piezoceramics, smart actuators, self-sensing actuators.
where T represents the stress applied, S the strain and sE denotes the compliance of the medium at constant field strength E. D denotes the dielectric displacement, ET is the permittivity of the medium at constant T and d is the piezoelectric charge constant that relates the mechanical and electrical domains. All variables in (1) and (2) are tensors. Certain ceramics can be made piezoelectric with the application of an electric field, such as Lead Zirconate Titanate (PZT).
1. INTRODUCTION Today, there is interest in the self-sensing ability of piezoelectric ceramics, which have the unique ability to simultaneously act as both a sensor and an actuator. Self-sensing involves extracting the sensing signal from that of the actuating signal (Sensor Technology, 1999; Jones et aI., 1994). The need for a self-sensing actuator originates from the need to control high precision positioners and structures, and the numerous applications range from vibration suppression in bridges and large scale structures to pressure regulation in micro pumps.
The direct piezoelectric effect, where a conversion from mechanical force or movement is made proportional to an electrical signal, can be found in acceleration and pressure sensors. The inverse piezo effect, where a conversion from an electrical signal is made proportional to a mechanical displacement or when the piezoelectric ceramic is utilized as an actuator, can be found in pneumatic and fuel-injection valves. Piezoelectric actuators offer high resolution down to sub-nanometre range, high stiffness, and fast response times, which makes them well suited for high precision and accuracy tracking and positioning systems (Physik Instrumente, 1998). Unfortunately, piezoceramics suffer from hysteretic behaviour inherent to non-linear dielectric materials, where a single input voltage corresponds to a range of output displacements (Hu and Ben Mrad, 2002).
A standards committee of the IEEE (1987) published a mathematical description of the piezoelectric phenomenon. IEEE (1987) adopted the linear theory of piezoelectricity, where the linear elasticity is coupled to charge by means of piezoelectric constants: S=sET+dE T
D=dT +E E
(1) (2)
I Corresponding Author: 5 King's College Road, University of Toronto, Toronto, Ontario M5S 3G8, Canada; Tel: 416-946-3597; Fax: 416-978-7753; e-mail: [email protected].
155
In a variety of PZT applications, including active vibration control of structures, pressure regulation in pumps and force control in micro-actuators, there is a need to develop a self-sensing ability in piezoelectric actuators, which would have the unique ability to simultaneously act as both a sensor and an actuator. Self-sensing in piezoelectric ceramics involves extracting the sensing signal from that of the actuating signal (Sensor Technology, 1999; lones et aI., 1994). The development of a self-sensing piezoceramic actuator requires the development of a model, which completely characterizes the electrical and mechanical properties of piezoceramics, and therefore allows the separation of the current generated in the piezoceramic due to the application of a force to it from the current in the piezoceramic arising from the application of a voltage to it. The development of such a model is pursued in this paper, where the authors present a method of tracking the nonlinear mechanical and electrical coupling inherent to piezoelectric ceramic stack actuators.
piezoelectric vibrator to couple mechanical motion, such as in the case of a self-sensing actuator, then these models are no longer sufficient and a mechanical terminal must be added to form an electromechanical equivalent model (Ballato, 2001).
2.2 Electromechanical Equivalent Models The application of the piezoelectric effect to selfsensing depends upon the motor-generator action of the crystal (Mason, 1950). Mason developed an electrical equivalent network circuit to represent the electrical and mechanical properties of the crystal by applying tradition'\l . electromechanical analogies, along with the piezJel~ctric equations and Newton's laws of motion (Mason, '1,948). ",.
One perceived problem with Mason's equivalent model is that it requires a negative capacitance at the electrical port (Sherrit et aI., 1999). Attempts by others in order to modify Mason's model have led to models that are too complex or "unphysical" (Ba11ato, 200 I; Sherrit et aI., 1999). In addition, the widely accepted linear piezoelectric constitutive equations, which formed the foundation of Mason's models, fail to explicitly describe both the hysteretic nonlinearities inherent to piezoelectric ceramics and the dynamical aspects of the electromechanical coupling. The author's believe that the modeling of a nonlinear piezoelectric system is better achieved through a phenomenological approach.
2. MODELS OF PIEZOELECTRIC CERAMICS
2.1 Electrical Equivalent Models Butterworth showed that an electrical equivalent circuit could represent a mechanically vibrating system with a single degree of freedom, driven by exposure to the electric field of a capacitor (Ballato, 200 I). Van Dyke also presented a circuit that consisted of constant value elements, which turned out to be identical to Butterworth's (Ballato, 2001). The Butterworth-Van Dyke (BVD) circuit is depicted in Fig. I .
2.3 Modern Phenomenological Models Two models based on physical principles have been developed recently (Adriaens et aI., 2000; Goldfarb and Celanovic, 1997). These models consist of an electrical and mechanical domain, as well as a connection between the two. Both proposed models describe the hysteretic non linearity of the piezoelectric ceramic.
-ilr
In this paper the authors present a model of piezoelectric actuators based on (Adriaens et aI., 2000) and (Goldfarb and Celanovic, 1997), where the behaviour of the PZT stack actuator is established through experimentation. The authors introduce a model where the charging and loss processes are represented with the use of complex permittivity. The PZT model presented characterizes the coupling of the mechanical and electrical properties of the piezoceramic actuator when subject to variable load conditions, under a range of voltage excitations at frequencies up to 5kHz.
Fig. I: Butterworth-Van Dyke (BVD) circuit. The representation of a piezoelectric vibrator by the BVD circuit is useful only if the circuit parameters are constant and independent of frequency, which only occurs for a narrow range of frequencies near the resonance frequency and only if the mode in question is sufficiently isolated from other modes (IEEE, 1987). The close proximity of several modes of vibration may be represented by adding additional branches in parallel to the R-L-C branch. These models represent equivalent electric circuits of piezoelectric vibrators and they only depict the "electric behaviour". If it is desired to use the
156
3. EXPERIMENTAL ASSESSMENT OF PIEZOELECTRIC CERAMICS' ELECTRICAL PROPERTIES
(6) A BM532 PZT stack actuator was used (Sensor Technology, 1999). Two physical loading conditions, namely a free case and a blocked case, as illustrated in Fig. 2, were investigated. Under the two conditions, the admittance was examined under variable peak voltage conditions, namely 20V, 50V, and lOOV, at frequencies up to 5kHz.
3.1 Motivation/or Experimentation
The models presented in (Adriaens et aI., 2000) and (Goldfarb and Celanovic, 1997) make the assumption of a constant capacitance to represent the electrical domain. The capacitance of nonlinear dielectrics, such as piezoceramics, is nonlinear with respect to the operating voltage and is mostly related to changes occurring within the material during operation (Sensor Technology, 1999; Jones et aI., 1994). Making reference to the ideal parallel-plate capacitor given by (3), the variance in the capacitance is due to a variance in £, the dielectric constant (Sensor Technology, 1999; Jones et aI., 1994; Sadiku, 1995): M C=-
IIII .LOCKED
Fig. 2: Two physical loading conditions.
(3)
d
3.2 Experimental Results
where C is the capacitance of the parallel plate capacitor, where the plates, with a surface area of A, are separated by a distance of d.
The frequency variation of the complex permittivity of the PZT stack was determined, when subject to variable loading conditions, under a range of voltage excitations at frequencies up to 5kHz. The experimental results agree with the authors' prediction that the models presented by (Adriaens et aI., 2000) and (Goldfarb and Celanovic, 1997) are a good starting point for a phenomenological model based on physics, unfortunately, both models make the wrong assumption of a constant capacitance to represent the electrical domain with no resistive component. The results (Fig. 3 and Fig. 4) show the admittance has a strong dependence upon force, peak voltage, and frequency, with the electrical domain modeled as a resistor in parallel with a capacitor. In addition, the results show a strong need for a resistive component to be included in the electrical domain modeling of PZTs.
In a parallel plate capacitor, when material is inserted between the plates the capacitance is increased (Hench and West, 1990). The dielectric constant k is defined as the ratio of capacitance of a capacitor with a dielectric between the plates to that of merely a vacuum (Hench and West, 1990):
C MId k=-=--=£I£o Co £oAld
(4)
where £ is the permittivity of the dielectric material and t:o is the permittivity of free space. In this paper, complex permittivity is pursued in order to improve upon previous attempts of modeling piezoelectric actuators. The complex permittivity is expressed as: k
"
=£ /£0 =
,. , k -lk
The results presented in Fig. 3 show that the magnitude of the capacitance depends on applied voltage and load, where a decrease in applied voltage and an increase in applied force have a similar effect of decreasing the magnitude of the capacitance. The results demonstrate that the capacitance is constant with respect to frequency over the frequency range studied.
(5)
where k' and k" represent the real and imaginary components of the complex permittivity, respectively.
The results shown in Fig. 4 indicate that the resistance depends on frequency and applied voltage, where an increase in both the frequency and peak input voltage have a parallel effect of decreasing the magnitude of the resistance. The results indicate that there is an insignificant dependence of applied load on the resistance.
In order to estimate the electrical properties of the piezoceramic material, the PZf stack is modeled with the input to the system as the voltage across the PZf stack and the output of the system as the current through the PZT stack. The PZT stack is represented as a capacitance C in parallel with a resistor R. The transfer function representing the PZT then becomes the admittance, namely:
157
S.ooE-07
~
7.ooE-07
tt;;;~::=~~~;::;;;;~~
6.ooE-07 5.ooE-07
E U
The authors agree with (Goldfarb and Celanovic, 1997) in that the hysteresis inherent to piezoelectric ceramics lies solely in the electrical domain between the applied actuator voltage and resulting charge. Fig. 5 shows the experimental results of the voltage versus charge data. Fig. 6 illustrates the linear relationship between charge and displacement. These results are used to validate the form of the model to be presented in a subsequent section.
.... ............................................................................. ........... .
4.ooE-07 3.ooE-07 2.ooE-07 l.ooE-07
+------__,_-----,--------1
O.ooE+oo
10
lOO
1000
10000
Frequeacy (HI)
------------------
1-+-20Vpp __ 50Vpp _Ioovppl 100
(a)
so
8.ooE-07 ,........ .... ................ ..................... .. . 7.00E-07 6.ooE-07 5.00E-07
E
4.00E-07
U
3.00E-07
20
2.00E-07 O~~~-,_-,_-__,_-_r-_r-_r-~
1.00E-07
O.E+oo I.E-OS 2.E-OS 3.E-OS 4.E-OS S.E-OS 6.E-OS 7.E-OS S.E-OS
O.OOE+OO +-----~----~------; 10 100 1000 10000
Cha .... (C)
Fig. 5: Hysteretic voltage versus charge behaviour.
Frequency (HI)
I·· •. 20Vpp .. • .
50Vpp .. d· · 100vprl
S.ooE-OS T ...... ·· .......................... · .. ·.... ·..·....· .. ..............................................................................,
(b)
7.ooE-OS
Fig_ 3: Measured capacitance versus frequency, where Vpp is the peak -to-peak voltage_ (a) Free; (b) Blocked. I~~··
...... ······.. ······...... ··············· ·· ············· .....................................................................
6.ooE-OS ~
S.ooE-OS
l-
4.00E·OS
e
,
3.ooE-OS 2.ooE·OS l.ooE·OS
100000
O.ooE+OO .Jo"----~--~--~--~---i O.ooE+OO l.ooE-06 2.ooE-06 3.ooE-06 4.ooE-06 5.ooE-06
IQooO
Displacement (m)
1000
Fig. 6: Charge versus displacement.
lOO 10+-----_r-----~----~
10
100
1000
10000
3.3 Explanation of Experimental Results
Frequency [Hz) 1-+-20Vpp --SOVpp --Ioovppl
As revealed by (4), dielectric capacitors can store more charge than their ideal parallel-plate vacuum capacitor counterparts. The physical quantity corresponding to the stored charge per unit area is the electric displacement D, where it is related to the electric field E by (7) (Sadiku, 1995):
(a) I~T
.. ·..······.. ·.. · ····················· .. ···· ..···········._
................................................................... .
100000
I '"
10000
I()()()
D=EoE+ P
(7)
lOO
where P represents polarization. 10+------,------~----~
IQ
100
IQoo
10000
The definition of dielectric displacement for free space is a special case of that in equation (7) for P = 0 (Sadiku, 1995). Therefore the total electric displacement of a condenser is the sum of the electric field, if the material was not present, plus the phenomena known as polarization (Hench and West, 1990; Uchino, 1997). There are four primary mechanisms of polarization, namely electronic or optical, atomic, dipole or orientation, and interfacial
Froqu.acy [Hz)
I... .20Vpp ..... SOVpp
"0"
loovppl
(b)
Fig. 4: Measured resistance versus frequency, where Vpp is the peak-to-peak voltage_ (a) Free; (b) Blocked.
158
concluded that the shape of the experimental results follows the theoretical prediction of Debye, dictated by dipole polarization.
or space charge polarization (Uchino, 1997). The total polarization may be thought of as the contribution of individual polarizabilities, each arising from a particular mechanism of polarization (Hench and West, 1990). The frequency dependence of the polarization is depicted in Fig. 7, where it can be seen in the frequency range of interest, dipolar polarization is the main contributor to the polarizability of the dielectric material, such as piezoelectric ceramics.
In addition, the shift in the curves with a change in applied peak voltage, observed in the experimental results, is due to polarization. For some dielectrics it is expected that P would vary directly as the applied electric field E, given by (9) (Uchino, 1997):
(9) where X, is known as the electric susceptibility of the material (Sadiku, 1995). Hence as the voltage is increased the dipolar polarization P increases, which corresponds to a shift in Fig. 7 and Fig. 8 or an increase in k' and k' , equivalent to increasing C and decreasing R, as observed.
a j nterfac i al i
...... ........ .......... .. ... .... .... ... .
~-------
ex a a tom ic
(l
, ele c tr o ni c 4. ELECTROMECHANICAL MODEL
10
An electromechanical model of PZT stack actuators is presented that explicitly accounts for the electrical and mechanical domains of the piezoelectric ceramic. The two domains are separated and connected through an electromechanical transducer. The proposed model describes the hysteretic nonlinearity of the piezoelectric ceramic with a hysteresis operator, based on a generalized Maxwell resistive capacitor model (Goldfarb and Celanovic, 1997). The dynamical mechanical behaviour is also explicitly accounted for by using a mechanical operator M that considers the piezoelectric stack actuator as a distributed parameter system, where the mass of the PZT is distributed evenly over the element (Adriaens et aI., 2000).
Fig. 7: Frequency dependence of polarizabilities. Debye extended the concept of the frequency dependence of dipolar polarization and developed equations, which describe the frequency-dependent relationships of the charging and loss constants and the loss tangent of the dielectric material (Hench and West, 1990). The graphical relation of the Debye equations is depicted in Fig. 8.
In addition, in the model presented, the capacitance is a function of applied voltage and load, and it is assumed to be constant with respect to frequency. A resistive component in the electrical domain is used in the modeling of the PZf. This resistance is a function of frequency and applied voltage. The electromechanical model is presented in Fig. 9.
Frequency
Fig. 8: k' and k" versus frequency as described by the Debye equations.
The piezoelectric transformer ratio
Referring back to equations (4) and (5), an alternative derivation of the PZT admittance can be derived that takes into account k' and k' . The expression is:
R(u pt a' 0)
effect is represented by a whereas C(u pta ' F) and
T,m'
characterize the electrical domain.
q
is
the current flow through the PZT actuator, and iR is the current flow through the resistor, while q p is the transducer charge. Furthermore,
(8)
up
voltage due to the piezo effect, and
symbolizes the
uh
is the voltage
across the Maxwell resistive capacitor, whereas u pta
It can be easily verified by relating the terms in (6) and (8) that C is proportional to k ' and R is inversely proportional to k'. By noting this relation and comparing Fig. 8 with Fig. 3 and Fig. 4, it can be
is the total voltage across the PZf. Fp is the force due to the inverse piezo effect, while Ft is the externally applied force. The elongation of the PZf
159
(10)
is denoted by y . As in (Adriaens et aI., 2000), the dynamic mechanical relation between force and elongation is related by M, while as in (Goldfarb and Celanovic, 1997) the hysteric behaviour is compensated with a hysteresis operator H. The complete set of electromechanical equations that describes the actuator is as follows:
= H(q)
(11 )
q = C(u pea ' F)u p + q p + fi Rdt
(12)
Uh
. 1 qp = - - y Tem Fp =T,mup
y =M(Fp - F,)
(13) (14) (15)
Fp F. : :
uh
Hysteresis Operator
..... ........ ... .. ... .... .... .
..... .... .. ... .... .. .. .. ...... qp
upe'
(upe., F)
M
L+y
R(up
Up
Electromechanical Coupling
Fig. 9: Electromechanical model. Hu, H. and R. Ben Mrad, (2002), On the classical Preisach model of hysteresis in piezoceramic actuators. Mechatronics, in press. IEEE (1987). An American National Standard: IEEE Standard on Piezoelectricity. Jones, L., E. Garcia, and H. Waites (1994) . SelfSensing Control as Applied to a PZT Stack Actuator Used as a Micropositioner. Smart Mater. Struct., vol. 3, pp. 147-156. Mason, W. P. (1948). Electromechanical Transducers and Wave Filters, 2nd ed.. D. Van Nostrand Company, Inc., Toronto. Mason, W. P. (1950). Piezoelectric Crystal and Their Application to Ultrasonics. D. Van Nostrand Company, Ltd., Toronto. Physik Instrumente (1998). NanoPositioning. Germany. Elements of Sadiku, M. N. O. (1995). Electromagnetics, 2nd ed., Oxford University Press, New York. Sensor Technology (1999). Piezoceramics Product Catalogue and Application Notes (BM Hi-Tech Division). Canada. Sherrit, S., S. P. Leary, B. P. Dolgin, and Y. BarCohen (1999). Comparison of the Mason and KLM Equivalent Circuits for Piezoelectric Resonators in the Thickness Mode. IEEE Ultrasonics Symposium, pp. 921-926. Uchino, K. (1997). Piezoelectric Actuators and Ultrasonic Motors. Kluwer Academic Publishers, Boston.
CONCLUSIONS An electromechanical model for piezoelectric ceramics is presented. The model uses a resistive component in the electrical domain modeling, and accounts for the effects of load, peak voltage, and frequency in the model. The model characterizes hysteresis, electromechanical coupling as well as the effects of the dynamic forces in the mechanical domain.
REFERENCES Adriaens, H. J. M. T. A., W. L. de Koning, and R. Banning (2000) . Modeling Piezoelectric Actuators. IEEElASME Trans. on Mechatronics, vol. 5, no. 4, pp. 331-341. Ballato, A. (2001). Modeling Piezoelectric and Piezomagnetic Devices and Structures via Equivalent Networks. IEEE Trans. on Ultrasonics and Freq. Cont., vol. 48, no. 5, pp. 1189-1240. Goldfarb, M. and N. Celanovic (1997). Modeling Piezoelectric Stack Actuators for Control of Micromanipulation. IEEE Control Systems, vol. 17, no. 3, pp. 69-79. Hench, L. L. and 1. K. West (1990) . Principles of Electronic Ceramics. John Wiley & Sons, Inc. , Canada.
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