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1995 analysis of piezo actuators in translation constructions


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1995 analysis of piezo actuators in translation constructions

  1. 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 Downloaded 11 May 2010 to Redistribution subject to AIP license or copyright; see
  2. 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 Downloaded 11 May 2010 to Redistribution subject to AIP license or copyright; see
  3. 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 Downloaded 11 May 2010 to Redistribution subject to AIP license or copyright; see
  4. 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 Downloaded 11 May 2010 to Redistribution subject to AIP license or copyright; see
  5. 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 Downloaded 11 May 2010 to Redistribution subject to AIP license or copyright; see
  6. 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 Downloaded 11 May 2010 to Redistribution subject to AIP license or copyright; see
  7. 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 Redistribution subject to AIP license or copyright; see
  8. 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 Redistribution subject to AIP license or copyright; see