1995 analysis of piezo actuators in translation constructions
1.
Analysis of pie20 actuators in transllation constructions
A. E. Holman, P. M. L. 0. Scholte, W. Chr. Heerens, and F. Tuinstra
Del’ University of Technology, Department of Applied Physics, Lorentzweg I, 2628 CT Del&
The Netherlands
(Received 26 October 1994; accepted for publication 13 February 1995)
A translation stage has been developed for generating displacements with nanometer accuracy and
a dynamic range of 10 pm. The stage uses piezo stacks as actuators and is equipped with capacitive
sensors which are able to measure displacements with subnanometer resolution. Because the
measurements are very accurate, the displacement properties of the piezo actuator used in the
translation stage can be recorded with high precision. This allows us to investigate the displacement
response of the piezo actuator when sinusoidal and triangular voltages are applied to it. These
measurements will be used to model the hysteresis behavior of the piezo actuator. It is observed that
the branches of the hysteresis curves can be described by a third-order polynome and that the
hysteresis curve has point symmetry properties. Also a model is presented for describing the general
behavior of a piezo actuator in a translation stage. 8 1995 American Institute of Physics.
i. INTRODUCTION single opamps are available to control the piezo.7 On the
other hand the low voltage types have a lower Curie tem-
Piezo actuators are becoming increasingly popular for
perature compared to the high voltage types. This limits the
generating small displacements in mechanical constructions.
temperature range in which these piezos can be used, since
By applying ,a voltage or current to a piezo it extends or
above the Curie temperature the stack becomes depolarized.
retracts due to the piezoelectric properties of the ceramic
At low temperatures the usefulness of a piezo is limited by
material. This extension is converted into a mechanical dis-
the piezosensitivity, since the piezosensitivity decreases with
placement by incorporating the piezo in a translation stage.
the temperature.’
In order to characterize the behavior of the piezo we have
Other factors that determine the type of piezos that can
developed a translation stage in which specially designed
be used in a translation stage, are the force that should be
capacitive sensors were incorporated.‘ With these sensors
,2
delivered, and the ambient conditions of the stage. The maxi-
we could detect displacements with subnanometer resolution
mum force depends for instance on the size and type of cas-
and therefore measure the behavior of the piezo stack with
ing of the piezo. If the piezo is to be used in an ultrahigh
high accuracy.
vacuum (UHV) setup, then the pidzo has to be coated with a
The translation stage is in general a mechanical con-
material with a low outgassing rate.
struction with elastic properties that influence the static and
The organization of the paper is as follows. In the next
dynamic behavior of the piezo. The accuracy with which
section a short description of the setup is given and experi-
displacements can be attained is limited by the design of the
mental results on a specific piezo are presented. Subse-
stage, the properties of the piezo, and the electronics used.
quently in Sec. III a model is introduced that describes the
For small displacements in the nanometer and subnanometer
general behavior of the piezo in a mechanical construction.
range the accuracy is limited by thermal drift, external vibra-
In Sec. IV an empirical model is given for the hysteresis
tions, and electronic noise from the high voltage amplifiers.
behavior of the piezo. This model is tested on experimental
Hysteresis, nonlinearity, and creep in the ceramic piezoelec-
data.
tric material are the dominant error sources at large displace-
ments on the order of 100 pm.1y3-6
il. EXPERIMENTAL SETUP
A difficulty in the design of translation stages with pi-
ezoelectric actuators is the experimental characterization of The translation stage used was developed for an IJHV
the piezobehavior. Especially in the range of small displace- scanning tunneling microscope (STM)2 and was capable of
ments, very few accurate experimental data are available. generating translations in two perpendicular directions (X
There are several types of piezos that can be used in a and Y) with high accuracy. For the actuators we used 20-
translation stage. The choice depends on the specifications of mm-long piezostacks.g These were specified for displace-
the translation stage, such as the required extension range, ments of maximal 20 pm for a total voltage range of 1000 V.
the control voltage, the temperature range, etc. The most The piezostacks used were specified for a voltage range be-
widely used are piezoplates and piezostacks. A stack consists tween -750 and +250 V. Our high voltage amplifiers could
of several piezoplates packed together and creating a pillar of deliver a voltage between -350 and f350 V. This voltage
several millimeters length up to several centimeters. The range, combined with the translation stage, gives maximal
maximal extension range of such a stack runs from less than displacements of 10 pm in X and Y direction. The stage
a micron to more than 100 ,um. depending on the number itself will be discussed elsewhere2 but it basically consisted
and type of piezoplates that are used. Piezostacks are avail- of a system of leafsprings made out of one piece of stainless
able in high voltage (0 to -1000 V) and low voltage (0 to steel. Additional spiral springs gave extra preload forces to
-150 Vi types. The latter offer some advantages, since the system to improve the specifications of the stage. The
3208 Rev. Sci. Instrum. 66 (5), May 1995 0 1995 American institute of Physics
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2.
system was equipped with accurate capacitive sensorsl’ *O,ll
in two directions to measure positions with subnanometer
., ,; ,, ,.
resolution. The advantage of capacitive sensors is, that in
principle absolute measurements are possible if the capaci-
tive readout electronics are properly calibrated. All ~the mea-
surements were performed with a IIIUltiChaMei data acquisi-
tion station with an A/D converter of 12 bit.
There are some practical speed limitations in our mea-
suring system and in the translation stage. To prevent artifi-
cial hysteresis caused by the sequential scanning of the A/D
channels the conversion time between the readout of subse-
quent A/D channels is minimized (40 ps) given by the capa-
bilities of our acquisition system. For an error of 1 Lsb the 0 -50 -100 -150 -200 -250 -300 -350
frequency of the exciting voltage should not be more than a
few Hz otherwise this artificial hysteresis would show up in Piezovoltage[ V ]
the measurements. To get rid of this artifact one could use a
data acquisition station with a sample and hold at every FIG. 1. Experimental hysteresis curves with varying loop width. It is clearly
channel. visible that the average loop slope decreases if the loop width decreases.
Another practical speed limitation is the maximum cur-
rent i,, which our high voltage amplifier could deliver. Be- system,1’ typical hysteresis curves are obtained which are
2
cause the piezo can basically be considered as a capacitance shown in Figs. 1 and 2. Figure 1 shows that if the amplitude
C, the voltage on the piezo VPis given as follows: of the triangular wave is decreasing, the average slope of the
t2 hysteresis curve is also decreasing. This means that the sen-
sitivity of the piezo d, is a function of the displacement
(1)
J‘i dt.
t1 range. Another example of the hysteresis effect is given in
The maximum frequency which can be controlled with a Fig. 2 where some curves have been recorded with a constant
current limited amplifier can be derived from this formula amplitude of the exciting voltage. Changing the dc bias volt-
and is for a triangular wave, age for each curves first in an upward direction and then in a
downward direction, a set of hysteresis curves is obtained
that resemble a pseudo-hysteresis curve made out of single
fma=& loops.
- P
From these measurements it is seen that due to the hys-
and for a sinusoidal wave, teresis effect precise positioning with a piezo actuator is
complicated, since the actual position depends on the dis-
placement history of the piezo. Another effect which can be
observed in measurements of the dynamic situation is that
With a maximal current of 40 mA, a piezo capacitance the hysteresis curves are becoming tilted if the frequency of
of typically 80 nF, and a voltage range of 1000 V this would the excitation is increased as can be seen in Fig. 3. There are
result in a maximal frequency of 250 Hz for a triangular two possible reasons which we could think of to explain this
wave and 80 Hz for a sinusoidal wave. This is much higher behavior.
than the frequency limitations caused by the data acquisition
station and therefore causes no problem in our situation.
A third speed limitation is the resonance frequency f. of
the translation stage. For proper operation the working fre-
quency of the system should be lower than this frequency. In
our stage f. was approximately 2 kHz. In the experiments
we used frequencies less than 20 Hz, which is far enough
below fo. For the recordings of most hysteresis curves a
frequency of 0.1 Hz has been used. If higher frequencies are
needed one should realize that a triangular wave has a lot of
harmonics (e.g., 3f, 5f, 7f, etc.). For accurate reproduction
of this triangular shape the higher harmonics should also be
less than the resonance frequency of the stage and not forget
the bandwidth of the high voltage amplifier. Another speed
limitation is given in the discussion of the piezoseparation
section later on. Piezo voltage [ V ]
By applying a low frequency triangular voltage to the
piezo actuator and recording the resulting displacement of FIG. 2. Keeping the loopwidth constant but changing the DC voltage of the
the translation stage with our accurate capacitive sensor piezo results in a pseudo hysteresis curve made out of single loops.
Rev. Sci. Instrum., Vol. 66, No. 5, May 1995 Pie20 actuators 3209
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3.
FIG. 4. Schematic translation stage used for the modeling of the quasistatic
Piezo voltage [ V ] and dynamic properties of the system.
FIG. 3. By applying a sinusoidal voltage to a piezo one can observe that the A. Quasistatic analysis
hysteresis curves are becoming more tilted at higher operating frequencies.
A schematic construction of a translation stage is shown
in Fig. 4 and will be used for the discussion. It consists of a
Piezos suffer from creep. If a step voltage is applied to piezo resting on a solid wall or frame, a mass M which must
the piezo then it will in first instance react almost immedi- be moved, and a spring fixed to a solid wall which delivers a
ately to the step followed by a delayed reaction which moves preload force F,, to the mass and the piezo. The whole sys-
the extension of the piezo to the desired end value. This tem has an angle /3 with respect to the gravitation vector and
effect can be compared to a step voltage which is filtered by is restricted only to translations along the x direction by
a low pass filter. If the frequency of the excitation voltage is means of its construction.
low enough for the piezo to settle down then there is no In the analysis of this construction some simplifications
problem. However if the frequency of the piezo is increased are being used. The walls and the mass are assumed to have
then it will go to a new position while the previous location infinite stiffness and the elastic behavior of the piezo and
has not yet been reached because of the delayed expansion. elastic elements can be described with Hookes law. Espe-
This will therefore result in a lower response amplitude com- cially if one is interested in nanometer displacements these
pared to the lower frequency and will show up as a tilted simplifications are not always allowed. The piezo has a stiff-
hysteresis curve. ness lcp and the spring a stiffness k, . The spring stiffness k,
The second possible explanation is more of an electronic represents the behavior of the elastic construction being
nature. If the high voltage amplifier which controls the piezo translated. It could represent for instance a simple spiral
has an output impedance R, than combined with the capaci- spring, flexural hinges, or a leafspring construction. The
tance C of the piezo it will form a low pass filter with time damping of the system is expressed through a damper with
constant RC. This attenuates the amp&de of the signal damping coefficient c. The total preload force Fe on the
more at higher frequencies resulting in a more tilted hyster- piezo is given by the sum of F,, of the spring and the gravi-
esis curve at these frequencies. We have not investigated tation component in the direction of the system translation
these effects in detail. axis
Fo=P,,,+Mg cos(p), (3)
III. MODELING THE PIEZO SYSTEM
where g is the gravitation constant.
The behavior of a piezo actuator in a translation stage is The extension xi of a piezo without a preload force is
always a complex interaction between the electrical, me- given as follows:
chanical, and thermal properties. Specifying for instance the
piezo properties requires extensive use of thermodynamics.i2 xl=dxVp+ aLoAT, (4)
Often a simpler model is sufficient. Due to its simplicity this where Lo is the length of the piezo at temperature To if no
model will not always be accurate but it serves well as a force and voltage is applied to it, d, is the piezo constant
rough estimate of the behavior of the system and can there- describing the extension sensitivity in the translation direc-
fore be used to foresee problems and design a system in such tion when applying a voltage VP to the piezo, LYthe thermal
a way to counteract or prevent these problems. Nowadays expansion coefficient of the piezo, and AT is equal to
piezos systems are specifically used for generating displace- T, - TO, the temperature difference referenced to T,-, .
ments with nanometer accuracy. Exact modeling of the piezo If the piezo is preloaded and tries to extend with an
is not very useful in this region. The influences of manufac- amount x1 then for the actual displacement x of the mass the
turing tolerances, contact surfaces, temperature dependen- following equation can be derived:
cies, electronic and electromagnetic noise, vibrations, degra-
dation of piezo properties, etc., are most of the time
unpredictable.
3210 Rev. Sci. Instrum., Vol. 66, No. 5, May 1996 Pie20 actuators
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4.
VP the piezo voltage. For a piezo operating at 100 Hz with a
piezo voltage of 1000 V and a capacitance of 80 nF this
would result in a heat source of 400 mW.
B. Dynamic analysis
The dynamic behavior of the system for sinusoidal wave
excitation is given by the following differential equation if
we use the schematic construction of Fig. 4 as a representa-
tion of our system, where x is the displacement of the mass
referenced to its equilibrium position
. ,.,/, ..,,... . . . ..t. ,,..I,. M~“+~~‘
+(k~+k,)x=k~d~V~ cos(ot) (8)
~.-..I.........i...,l....i.,..;....l....;....I....;....i....,i
with w= 25-f, f the frequency of the exciting wave, and c
Piezo voltage the internal damping coefficient of the piezo or from a delib-
erately connected damping system. For instance in some
translation stages the moving part of the translation stage is
FIG. 5. Simulation of the change of behavior of a piezo in a translation embedded in silicon rubber which functions as the damping
stage. The displacement response of a “free” piezo changes if it is inserted
into a elastic construction. In general this will result in a change in offset
system. The term to the right-hand side of the equal sign is
and a decrease of piezo sensitivity. the forced vibration generated by a sinusoidal wave voltage
VP = V. cos( wt) applied to the piezo. For a triangular wave
this term would consist of the series expansion of the wave.
Because the piezo is preloaded with a force F,, the piezo Solving this differential equation gives the following so-
will become shorter according to Hookes law with an lution which describes the resulting displacement of the pi-
amount FO/(kp+kks). In the following equation these effects ezo:
are combined, giving the total length L of the piezo:
&.+W,Vo
x=c e’
lf+C &-2f+
1 2
L=Lo- - Fo
k,+k, + -
kP
k,+k, (Wp+~WT). M2(w;-02)2+(cco)2
Xcos(wt-cp) (94
This equation describes the general behavior of a piezo with the resonance frequency o, given as
(stack) in a practical construction for quasistatic translations
and ignoring hysteresis.
Simulating this behavior results in Fig. 5. In this simu- (9b)
lation we used the experimental hysteresis data of Fig. 1 and
and the phase shift as
applied Eq. (6) to this data using parameter values that illus-
trate clearly the behavior of the model. Compared to a “free
piezo,” preloading the piezo in a construction results in gen-
eral in a change in zero offset and a decrease of sensitivity
(tilting of the hysteresis curves). Another important implica- and with ri, r2 the roots of the characteristic equation of Eq.
tion observed here is that, due to preloading of the piezo the &V,
hysteresis curve becomes flatter and therefore more linear.
Also the effective temperature dependency is diminished as (94
can be seen from Eq. (6).
The thermal term in Eq. (6) should not be underesti- C, and Ca are given by the initial conditions of the system.
mated. The thermal expansion coefficient a has in general a It is favorable to design the system in such a way that the
value of around 10m5. If a piezo stack of 2 cm length is exponential functions can be considered as a transient solu-
heated by only 1 “C the extension due to thermal expansion tion which quickly dies away after a short time. Then the
is 200 nm, which is enormous if one is interested in displace- resulting steady state displacement response caused by the
ments on the order of nanometers. At higher operating fre- applied voltage is given by Eq. (9a) where the two exponen-
quencies a relatively large amount of heat is generated in the tial terms can be omitted.
piezo itself resulting in an increase of its temperature. The In this analysis we have assumed that the mass M is
thermal power generated in the piezo can be estimated as much larger than the mass mp of the piezo and the mass of
follows:‘ 3 the spring. If this is not true then the mass M must be re-
placed by an effective mass which takes into account that the
Pm tan( S)fC$, (7) piezo consists of distributed masses and actuators where each
actuator disk must lift not only the mass M but also the mass
where 6is the power loss of the piezo and is about 0.05, f is of the piezo above it and is given by M,,=M + m,/2. In the
the operating frequency, C the capacitance of the piezo, and specifications of the piezo actuator the resonant frequency is
Rev. Sci. Instrum., Vol. 66, No. 5, May 1995 Piezo actuators 3211
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5.
given. One should realize that this frequency only applies for L, and if it is made of a material with an elasticity modulus
an unloaded piezo stack. Every additional mass will lower its E then the following equation can be used to estimate its
resonance frequency. stiffness k, :
EAm
6. Piezo-mass separation k,,=y-, (12)
m
It is important to note that if the mass M is not ftxed where A,,, is assumed to be the contact area between the
rigidly to the piezo then it is possible that due to the large piezo and the mass. For typical values of E =2.5x lOlo,
acceleration of the end face of the piezo, the piezo and mass L,=5 cm and a piezo stack with a diameter of 1 cm the
may become separated from each other. After a while they stiffness of the mass is around 15X lo7 N m-t. Compared to
will come into contact again. How serious this effect is de- the stiffness of our piezo of 5X lo7 N m-l, the mass is only
pends on the retraction speed of the piezo, the shape and 3’times stiffer than the piezo itself. Looking at Eq. (12) the
material type of the contact surface between the piezo and stiffness can be increased by placing the contact surface of
the mass, and how the mass is connected to the piezo (glued, the piezo and the preload spring as close to each other as
magnetic, kinematically connected, etc.). The effect could be possible thereby reducing the length L, .
important if the connection between these two is made with
for instance a steel ball, as is often done for constructions to
comply with kinematic principles and to prevent torsion mo- IV. EMPIRICAL MODELING OF PIE20 HYSTERESIS
ments (which can break the piezo). If the contact surface is CURVES
more diffuse, the effect becomes less important. With a vi-
In the previous section the general behavior of a practi-
brating piezo the continuously banging of the mass on the
cal piezo construction has been analyzed. In the next para-
piezo caused by the separation could damage it especially
graph we try to model the shape of the hysteresis curves
with larger excursions and relatively high frequencies. It is
themselves.
even possible that the piezo vibrates itself out of the con-
The problem with hysteresis is that the exact displace-
struction.
ment of the piezo depends on its history. Unless one continu-
Ignoring hysteresis and assuming the spring is tightly
ously records its displacement it will almost be impossible to
fixed to the mass and the solid wall, the condition for keep-
predict exactly its next location if the movements are irregu-
ing the piezo and the mass connected is given as follows:
lar. For that reason we will limit our model to harmonic
MY>-Mg cos(/?)-ks-Fso. ilo) movements like sinusoidal or triangular shaped displace-
Combining this equation with the steady state solution of ments. This will cover a large part of the applications of
scanning stages. Such scanning movements can be divided
J%-(9a) gives the following condition: into a dc component and a harmonic component. The dc
C&W&k,)<Mg COS(~)+F,~ (llaj location is then a scanning offset or a scan at another loca-
with C, defined as, tion. Our discussion will be limited to hysteresis curves with
a fixed dc offset.
k&Jo First we will analyze the shape of the hysteresis curves
C,= ill’
4 of Fig. 1.
~M2(w~-~2)2+(cw)2’
Observing the general shape of the curves the simplest
If the system complies with Eq. (11) then the piezo and form of description which comes into mind is a third-order
mass will not become separated. For a given system this polynome. Fitting this polynome to the measured upper and
results in a theoretical maximum frequency f,, . If the ex- lower branch and displaying the difference between the fit
citation frequency is higher than f,, , the piezo and the mass and the actual data gives Fig. 6 and demonstrates that there is
becomes separated. Using some practical values, c = 4 X 1 03, indeed a very good fit. Ignoring the end points, the overall
kp=5x107, k,=1x107, M=0.3, v,= 1000, p=o, deviation is less than 5 nm. Relative to the displacement
fso = 10, d,= 20 X 1 O”, gives us a maximum operating fre- range of 5.4 pm of the analyzed loop, the deviation error is
quency of 885 Hz. In our separation model we have assumed less than 0.1%. Looking at Fig. 6 one can observe that there
only one surface which could become separated. In real situ- is still some structure visible in the deviation plot. This sug-
ations there are often more surfaces present. This will usually gests that a better fitting model is still possible. The noise
result in a lower f,, than the one calculated from Eq. (11). band visible is the combined noise of mechanical vibrations,
A lower f,,,, is also observed during our experiments. electronic noise, and quantization errors.
In the discussion we have assumed that the solid walls For the fitting process we have not used all the data of
and the mass M have infinite stiffness. In reality this is not the recorded loop. In the fits the end points are omitted. The
the case. Even if the solid wall has infinite thickness the reason for doing so lies in the observation that the exact
contact surface between, e.g., the piezo and the wall will still shape of the end points are not very well delved and are
react elastically and the elastic displacement can easily be of often not directly related to the piezo itself. The shape of the
nanometer magnitude. The exact elastic response of the wall end points depends for instance on the mechanical properties
is difficult to calculate due to the uncertainties of the contact of the translation stage (resonance frequency, limited band-
surface. For the mass M it is easier to give a more specific width, etc.) and the speed of the electronics. For this reason
analysis. If for instance the thickness of the mass is equal to the accuracy of the model at the end points is less good. In
3212 Rev. Sci. Instrum., Vol. 66, No. 5, May 1995 Pie20 actuators
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6.
-150 -200 -250 -300 -350
-1 -0.5 0 0.5 1
Pie20 voltage [ V ]
X
PIG. 6. Fitting a third-order polynome to the measured branches of the
hysteresis curve and subtracting the fit from the measured data results in a PIG. 8. This figure shows simulated contours which can be described with
deviation plot for the upper branch and for the lower branch. 5.4 nm is equal the point symmetry model.
to a full scale error of 0.1%. The noise band visible is a combined effect of
vibration, electronic, and quantization noise.
f(x)=a+bx-ax2+(1-bjx3, (14)
Fig. ‘ we have plotted as an example the difference between
7 g(xj=-a+bx+ax2+(1-b)x”. il.9
a hysteresis curve with sinusoidal and triangular excitation of
comparable amplitude. From physical considerations one must limit the first de-
The next step is to relate the shape of the upper branch rivative of f(x) in the point (1,l) between 0 and 1 and the
with the shape of the lower branch. If one compares the two root of the second derivative of f(x) must be equal or larger
branches of a hysteresis curve it suggests some form of point than 1. This results in a limiting range of values for a and b,
symmetry. Taking this as a model then the point symmetry OGaS0.75 and (1-a/3)Gb<(1.5-a). Some possible
operation will convert the contour of, e.g., the upper branch contours, which this model can describe, are shown in Fig. 8.
f(x) in the contour of the lower branch g(x). The model can The parameter a describes more or less the width of the
be developed as follows. Assume the model is defined in the hysteresis curve while the parameter b describes some form
region - 1 =G& 1 and - 1 GY=G 1 and the third-order poly- of skewness of the loop. For converting this model to the real
nome is defined as follows: world, offset and scaling factors can be applied to these
equations.
a+bx+cx”+dx3. (13) An important variable which can be derived from this
If the hysteresis curve has point symmetry around the model is the displacement shift between the upper and lower
origin then the upper and lower branch can be represented as branch.
follows: This shift S is given by the difference between f(x) and
g(x) as follows:
S=2a(l-x2). (16)
E. ‘
. a+: -1 This displacement shift is only dependent on one parameter
,.... and has a simple quadratic form,
z
u71-- -..-I r-r:--
To test the point symmetry model on our measurements
2
-- .,...- “‘.I......“y
-+- Triangle excitati
we took the normalized fitted representations of the upper
fj
23
8
and lower branch because we saw earlier that the deviation
from the measured contour is less than 0.1%. By applying
B
F
the point symmetry operation on the representation of the
3
%
!x
lower branch the two branches could be compared as is
shown in Fig. 9. The inset figure gives the difference AY
between the two curves. If there was perfect point symmetry
than the result would be a complete overlap of the two
curves. As can be seen there is some deviation between the
-240 -250 -260 -270 -280 -290 -300 curves. The maximum deviation is found around x= -0.12
Pie20 voltage [ V ]
and is equal to 0.46% of the full scale and referenced to the
average curve between the two contours. Although this error
FIG. 7. The measured end points of a hysteresis curve are different for
is larger than the polynome fit error of the two branches we
sinusoidal and triangular excitation voltages. The sharpness of the end
points of the hysteresis curve depends among others on the mechanical can conclude that the assumption of point symmetry holds
properties of the stage. quite good in practice.
Rev. Sci. Instrum., Vol. 66, No. 5, May 1995 Pie20 actuators 3213
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7.
FIG. 9. Converting the fitted lower branch using the point symmetry opera- PIG. 11. Normalizing the fitted representations of the hysteresis curves by
tion and plotting it in the same figure as the fitted upper branch gives a mapping them into the model space gives an indication how the shape of the
comparison of how well the assumed point symmetry between the branches curve changes as function of loop width. It is observed that the curves with
is valid. In the inset the difference AY between the two curves are plotted. smaller loop width are becoming hatter.
For the second test we used Eq. (16). First we normal- nome is not acceptable for this interpolating function. The
ized the fitted representations by mapping the two branches associated average loop slope is given in the inset figure of
into our model space by applying appropriate offset and lin- Fig. 10. Knowing this behavior for a given system, the end
ear scaling transformations. Subtracting the two representa- points of the hysteresis curves can be calculated for a desired
tions gives men the measured Smeas(x) function from which loop width by applying appropriate scaling factors derived
we extract the parameter a. Putting a into the Eq. (16) gives from this function.
the Smodel(x). Investigating the difference between Smeas(x) To investigate the shape of the individual curves as func-
and S,,&X) we find a maximal deviation from the model of tion of the loop width the nine hysteresis curves have been
0.1% which is comparable with the error found from Fig. 6. normalized and are plotted Fig. 11. From this figure we ob-
The fact that there is still some deviation present suggests serve that there is a systematic change of the shape ~of the
that the model could be improved. In most cases an error of contours as function of the loop width. Reducing the loop
0.1% is more than sufficient so we have not tried to refine the width flattens the hysteresis curve.
model any further. Comparing the factor a derived from the upper branch
From Fig. 1 we saw that the average loop slope de- a, with the one derived from the lower branch a- we see
creases if the loop width is decreasing. In Fig. 10 the calcu- that the a - is systematically smaller then a + . If we plot the
lated end points from the fits of nine hysteresis curves have difference a + - a- as function of the loop width we get a
been plotted. As can be seen the end points follow a well- curve as shown in Fig. 12 which suggests a strong relation of
defined path which can be represented by an interpolating the parameter a with the loop slope, see Fig. 10. To investi-
function. From our data we found that a third-order poly- gate the dependence of the parameter b as function of the
I. J
m
0.15 I+? z +.:. ‘
:
0.1 ”
-a-- +a, .i
-t -a.
0.05 4- +a+-a.
I. i
ot,‘ 150
,l”“‘
,.,,l”,,““‘l,i
I I , . I.
-200 -250 -300 -350
200 250 300 350
Piezo Voltage [ V ]
Loopwidth [V ]
PIG. 10. Plotting the calculated end points of several measured hysteresis FIG. 12. Measuring the parameter a, for the upper branch a+ , for the lower
curves givesa curve as shown here. The associated average loopslope is branch a- and their difference gives an indication how the shape of a
given in the inset figure. hysteresis curve changes as function of loop width.
3214 Rev. Sci. Instrum., Vol. 66, No. 5, May 1996 Pie20 actuators
Downloaded 11 May 2010 to 131.155.135.0. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/rsi/copyright.jsp
8.
loop width we have examined the shape of the function [f(x) ‘ Libioulle, A. Ronda, M. Taborelli, and J. M. Gilles, J. Vat. Sci. Tech-
L.
+&x)1/2 for different experimental curves. This function nol. B 9, 655 (1991).
“S. M. Hues, C. E Draper, K. P. Lee, and R. J. Colton, Rev. Sci. Instrum.
should only be dependent of the parameter b, see Eqs. (14)
65, 1561 (1994).
and (15). No significant loop width dependence for the dif- 7Burr-Brown Corporation, Opamps 3583, 3584; APEX MicmtechnoIogy
ferent hysteresis curves was found. For all the hysteresis Corporation. Opamps PA 815, PA8U.
curves this function looked practically the same. This means ‘ G. Vandervoort, R. K. Zasadzinski, G. G. Galicia, and G. W. Crabtree,
K
that in our piezo system the parameter b can be considered Rev. Sci. In&turn. 64, 896 (1993).
9PI Physik Instrumente GmbH & Co, Waldbronn, Germany Type P178.20
constant. with UHV option.
low. Chr. Heerens, J. Phys. E: Sci. Instrum. 19, 897 (1986).
‘ R Holman, W. Chr. Heerens, and E Tuinstra, Sensors and Actuators A
A. “W. Chr. Heerens, Journal A, Benelux Quarterly, Journal on Automatic
36, 37 (1993). Control 32, 52 (1991).
‘ E. Holman(unpublished).
A. “Takura Ikeda, Fundamentals of Piezoelectricity (Oxford University, New
‘ W. Basedow and T. D. Cocks, J. Phys. E: Sci. Instrum. 13,840 (1980).
R. York, 1990).
4L E. C. van de Leemput, P. H. H. Rongen, G. H. Timmerman, and H. van l3 PI Physik Instrumente GmbH & Co Waldbron, Germany Catalogue, Prod-
Kempen, Rev. Sci. Instrum. 62, 989 (1991). ucts for micropositioning.
Rev. Sci. Instrum., Vol. 66, No. 5, May 1995 Pie20 actuators 3215
Downloaded 11 May 2010 to 131.155.135.0. Redistribution subject to AIP license or copyright; see http://rsi.aip.org/rsi/copyright.jsp
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