REVIEW OF SCIENTIFIC INSTRUMENTS
VOLUME 71, NUMBER 4
APRIL 2000
Creep characteristics of piezoelectric actuators Hewon Junga) and Dae-Gab Gweon Department of Mechanical Engineering ME3265, Korea Advanced Institute of Science and Technology, 373-1 Kusung-Dong, Yusong-Gu, Taejon 305-701, Korea
共Received 15 October 1998; accepted for publication 21 December 1999兲 A major limitation of piezoelectric translator 共PZT兲 actuators is their lack of accuracy originated from the hysteresis and creep. Nevertheless the creep phenomenon is an important factor in many applications of PZT actuators, but it has been investigated less frequently in comparison with the displacement hysteresis. In this article, we present a basic creep model with some parameters that have hysteresis properties which make it possible to predict an open loop response of PZT actuators based on these properties. © 2000 American Institute of Physics. 关S0034-6748共00兲02804-5兴
I. INTRODUCTION
vious dynamic one. Generally, it has been known that the creep response has a logarithmic shape over time. This can be represented by the following equation:6
Piezoelectric translator 共PZT兲 actuators, with their high stiffness, fast frequency response, and high resolution, are increasingly being used in micropositioning applications.1 However, they are fundamentally nonlinear in their displacement to an applied electric field because PZT materials are ferroelectric.2 Nonlinearities in PZT can be classified in two categories. First is the hysteresis relation between the applied electric fields and the displacements. Second is the creep phenomenon, which is the drift of the displacement of PZT for a constant applied electric field. There have been many efforts to analyze and compensate for the hysteresis effects as the use of PZT actuators increased.3–5 However, creep phenomenon has been investigated less frequently compared with hysteresis, while its phenomenon is a very important factor in many application of the PZT actuator. For instance, in a surface measurement system which can be a laser interferometry or a scanning probe type, the measuring sample should not only be positioned precisely, but also keep its position without any movement while the measuring operation is going on. In many other cases as well as this case, analyzing slow drift of PZT displacement after applying a constant electric field is very important for improving the precision positioning. The purpose of this article is to investigate the PZT actuator’s creep characteristics. We suggest that the PZT actuator’s creep has hysteretic properties and therefore it can be possible to predict an open loop response of a PZT actuator based on these properties.
冋
L 共 t 兲 ⫽Lo 1⫹ ␥ log10
冉 冊册
t , 0.1
共1兲
where L(t) is a PZT actuator’s displacement for any fixed input voltage, Lo is a nominal constant displacement value which is the displacement of 0.1 s after applying the input voltage, and ␥ is a creep factor that determines the rate of the logarithm. At t⫽0.1 s in Eq. 共1兲, the response displacement reaches the nominal displacement Lo, causing this equation to exclude a PZT dynamic response. We already know that the nominal displacement Lo has a hysteresis property about the applied voltage loops and it also has a different hysteresis shape, depending on which kinds of PZT actuators and mechanical loads are used. In the case of ␥, it is known only that ␥ lies between 0.01 and 0.02, and it changes according to an applied voltage. However, ␥ is not defined exactly for the specific input voltage. Figure 2 shows that the rates of creep ␥ are different from each other according to the input voltage. To represent the difference of the creep rates, dis-
II. CREEP MODELING
Figure 1 illustrates a typical PZT response for a step input voltage. When applying any specified input voltage, PZT shows its step response with a dynamic transient behavior within a few milliseconds followed by the creep response, which is a much slower drift response than the preFIG. 1. A PZT actuator’s creep pattern over time. Upon applying the input voltage, PZT shows its dynamic transient behavior within a few milliseconds followed by a creep phenomenon which is a slower response than the previous dynamic one.
a兲
Author to whom correspondence should be addressed; electronic mail:
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© 2000 American Institute of Physics
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placements at 0.1 s after applying the input voltage for each response curve is shifted to 0. Moreover, even if the final applied voltages are the same, the value of the parameter ␥ is still different from the others according to the history of the past applied voltages. This is shown in Fig. 3. It is surprisingly similar to the phenomenon known as hysteresis. To determine whether this phenomenon has any relation to hysteresis, we performed some experiments. In this article we extract Lo and ␥ values from the experiments that have different sets of input voltages, and investigate these two parameters’ properties. III. EXPERIMENTAL SETUP
FIG. 2. The rates of PZT creep are different from each other according to an applied input voltages. Symbol 共↑兲 means that an applied voltage is in the ascending loop and symbol 共↓兲 means that an applied voltage is in the descending loop.
The experimental setup is shown in Fig. 4. We used a stack-type PZT actuator 共AE0505D08, TOKIN兲 which has nominal displacements of 6 m for 100 V. The guide mechanism was designed and made in our laboratory with a flexible hinge mechanism that gives a spring load to the PZT and guides the actuator linearly by eight leaf flextures.7 The displacement sensor is a noncontact capacitive sensor 共3890 system, ADE兲 with a 2.5 nm resolution. IV. ESTIMATION OF Lo , ␥
To investigate whether the creep phenomenon has any relation to the hysteresis, we applied a series of voltage steps shown in Fig. 5 共it is similar to a ladder兲. Each step has a 5 V height and 20 s duration. In each 20 s duration, the PZT actuator may expand its length approximately with proportion to the applied voltage and this response shows the same pattern shown in Fig. 1. Each step response data is measured by the displacement sensor and stored with an interval of 0.2 s. Figure 6 shows the step displacement response of PZT. In each step, we can see a slow creep response after a short transient response. In this slow creep region of each step, Lo
FIG. 3. All response curves are under the 30 V input voltage. But each curve has a different input history. 0-40-20-30 means that 0, 40, 20, and 30 V input voltage was applied sequentially. This means that even if the final applied voltages had the same values, the values of parameters are different from each other according to the history of past applied voltages.
FIG. 4. Experimental setup. It consists of PZT, a flexible guide mechanism, and a gap sensor.
FIG. 5. An applied input voltage series. Each step has a 5 V height and 20 s duration: 共a兲 ladder-like input voltages, 共b兲 input from 90 s to 130 s.
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FIG. 6. PZT output displacements. A slow creep response is shown in PZT output displacement after a short transient response: 共a兲 ladder output displacements, 共b兲 output from 90 to 130 s.
and ␥ parameters can be estimated from Eq. 共1兲 with the least square method. Figure 7 shows how well this estimated curve fits the response curve. It has been taken when the input voltage is 25 V. In this figure, Lo is estimated at 1.84 m and ␥ is estimated at 0.0137, and their uncertainties are 0.000 74 m for Loand 0.000 29 for ␥. This was calculated as follows. Equation 共1兲 may be represented by a linear simple regression model y i ⫽a⫹bx i ,
共2兲
where y i is L(t i ), a is Lo, b is Lo ␥ , and x i is log10(t i /0.1). Lo and Lo ␥ are two parameters of a linear simple regression model. Therefore uncertainties u 共generally standard deviation 兲 are derived as follows:8
FIG. 7. Fitted curve vs time response curve. The peak shown in front is the PZT dynamic response, which is excluded in the fitting procedure because it is not creep effect.
FIG. 8. Uncertainty error of Lo and ␥ when ascending input voltages o represents uncertainty and when descending input voltages * represents uncertainty: 共a兲 uncertainty error of Lo; 共b兲 uncertainty error of ␥.
u 2 共 Lo 兲 ⫽ 2 共 Lo 兲 ⫽
s 2 兺 x 2i ⌬
u 2 共 Lo ␥ 兲 ⫽ 2 共 Lo ␥ 兲 ⫽n
,
共3兲
s2 , ⌬
共4兲
where ⌬⫽n s 2⫽
兺
x 2i ⫺
冉兺 冊
兺 关 y i ⫺y 共 x i 兲兴 2
n⫺2
2
xi ,
共5兲
,
共6兲
FIG. 9. 2v test 共reduced 2 test 2v ⫽s 2 / 2 兲. s 2 is the variance of the fit, 2 is the parent variance of the displacement data. When ascending voltages are applied the 2v value is represented by o, and when descending voltages are applied the 2v value is represented by *.
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FIG. 11. Hysteresis property of PZT creep. Not only Lo but also ␥ shows hysteresis property when PZT has some input history: 共a兲 Lo hysteresis property, 共b兲 ␥ hysteresis property.
0–7 m and uncertainty of Lo has a range of 7.0 ⫻10⫺4 – 13⫻10⫺4 m, which is almost 0.001% of the calculated Lo. ␥ has a range of ⫺0.2–0.02, and uncertainty of ␥ gas a range of 1.0⫻10⫺4 – 7.0⫻10⫺4 , which is almost 0.1% of the calculated ␥ value. Figure 9 shows the reduced 2 test 共 2v test 2v ⫽s 2 / 2 兲. s 2 is the variance of the fit, 2 is the parent variance of the displacement data which were derived from the displacement data acquired after 30 min from the application of the input voltage. Acquisition after 30 min guarantees the removal of the creep effect in the response. Therefore, the displacement data acquired after 30 min can be thought to bear the pure parent variance. In this figure 2v is between 0.8 and 2.8, which guarantees that this fitting method is reasonable. V. RESULTS FIG. 10. Loop characteristics of the PZT creep. Lo and ␥ show a hysteresis loop which keeps its properties even when many loops have passed: 共a兲 Lo loop for ladder-like input voltage; 共b兲 ␥ loop for ladder-like input voltage; 共c兲 Lo loop after many loops have been passed; 共d兲 ␥ loop after many loops have been passed.
where n⫽number of data. ␥ can be represented by Lo, Lo ␥ as Eq. 共7兲:
␥⫽
Lo ␥ . Lo
共7兲
Then the uncertainty of ␥ can be calculated by means of the combined standard uncertainty.9 Figure 8 shows uncertainty errors of Lo and ␥ with respect to the input voltages shown in Fig. 5. Lo has a range of
Lo and ␥ with respect to the applied voltages in Fig. 5 are shown in Fig. 10. As shown in the figures, Lo and ␥ shows the hysteresis loop and it keeps its properties even when many loops have passed. Two meaningful features can be derived from these figures. First, as for Lo, it is similar to a conventional PZT actuator’s hysteresis loop which is already known. The conventional hysteresis loop is obtained by the way that specific displacement data points in the hysteresis loop are acquired at fixed time interval after applying a specific voltage to the PZT actuator. However, since the PZT actuator has creep characteristics in its expansion, its displacement data for a fixed applied voltage have different values according to the different sampling intervals. Therefore, the conventional hysteresis loops have different shapes according to the different sampling intervals. However, in Fig. 10共a兲, each Lo point in the hysteresis loop has been
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Rev. Sci. Instrum., Vol. 71, No. 4, April 2000
H. Jung and D.-G. Gweon
acquired from all the displacement data points in each step through the Creep model and the least square method stated earlier. Therefore, the Lo hysteresis loop is uniform regardless of the sampling interval. Second, the creep factor ␥ shows a loop and hysteresis characteristics like Figs. 10 and 11. Until now, the creep factor ␥ has been known to have some value between 0 and 0.02, but its exact value was not known. But as shown in Figs. 10 and 11 the creep factor ␥ under any voltage input can be defined definitely by the hysteresis loop. But Fig. 12 shows that creep factor ␥ has an asymmetric property, while Lo has symmetric property as an input voltage is ascending or descending. However, using this creep hysteresis characteristic, PZT creep response can be treated like the displacement hysteresis except for the asymmetric property. That is, it can be represented by the mathematical model.4,5,10 Therefore, if these two factors can be estimated exactly, it is possible to predict not only the nominal displacement of PZT for any arbitrary input voltage but also the drift of response after the PZT, reaching the nominal displacement by the mathematical model, which of course will be our future work.
ACKNOWLEDGMENTS
This work was supported by Grant No. KOSEF 9620100-001-2 from the Korea Science and Engineering Foundation. A. Slocum, Precision Machine Design 共Prentice–Hall, Englewood Cliffs, NJ, 1992兲, pp. 666–674. 2 P. Chen and S. Montgomery, Ferroelectrics 23, 199 共1980兲. 3 E. Crawely and E. Anderson, Proceedings of 30th Structures, Structural Dynamics and Materials Conference, Mobile, AL 共AIAA, New York, 1989兲, pp. 2000–2010. 4 S. Jung and S. Kim, Precis. Eng. 16, 49 共1994兲. 5 P. Ge and M. Jouaneh, IEEE Trans. Control Syst. Technol. 4, 209 共1996兲. 6 S. Vieira, IBM J. Res. Dev. 30, 553 共1986兲. 7 J. Y. Shim, H. Chung, and D. G. Gweon, Proceedings from ASPE 1998 Annual Meeting, 1998, Vol. 18, pp. 457–461. 8 P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences 共McGraw-Hill, New York, 1969兲. 9 Guide to the Expression of Uncertainty in Measurement 共Internation Organization for Standardization, Switzerland, 1993兲. 10 I. Mayergoyz, Mathematical Models of Hysteresis 共Springer, New York, 1991兲. 1
FIG. 12. Asymmetrical property of PZT creep. Creep factor ␥ has an asymmetric property, while Lo has a symmetric property as input voltage is ascending or descending: 共a兲, 共b兲 Lo and ␥ values when input voltage is ascending from 20, 40, 60, 80, and 100 V to 0 V; 共c兲, 共d兲 Lo and ␥ values when input voltage is descending from 0 V to 20, 40, 60, 80, and 100 V.
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