Because of their high stiffness, compact size, and effectively

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1 1630 IEEE Tranaction on Ultraonic, Ferroelectric, and Frequency ontrol, vol. 60, no. 8, Augut 2013 harge Drive With Active D Stabilization for inearization of Piezoelectric Hyterei Andrew J. Fleming, Member, IEEE Abtract harge drive circuit can ignificantly reduce piezoelectric nonlinearity; however, they are rarely ued in practice becaue of their limited low-frequency performance, their dependence of voltage gain on the load capacitance, and their requirement for time-conuming tuning procedure. In thi report, a new charge drive circuit i propoed that ue a controlled current ource with voltage feedback to tabilize the low-frequency behavior. Thi approach eliminate many of the preent difficultie and allow extremely low tranition frequencie without a long tranient repone. Experimental reult demontrate that the propoed charge amplifier can effectively reduce piezoelectric hyterei and creep to le than 1.3% at can-rate of 10, 1, and 0.1 Hz. Manucript received Augut 23, 2012; accepted May 3, The author i with the entre for omplex Dynamic Sytem and ontrol, School of Electrical Engineering and omputer Science, Univerity of Newcatle, allaghan, NSW, Autralia ( Andrew.Fleming@ newcatle.edu.au). DOI I. Introduction Becaue of their high tiffne, compact ize, and effectively infinite reolution, piezoelectric actuator are employed in a wide range of indutrial, cientific, and commercial application. Example include canning probe microcope poitioning ytem [1], [2], fuel injection valve [3], and laer beam manipulation [4]. Although piezoelectric actuator have everal deirable characteritic, a major diadvantage i the hyterei exhibited at high electric field [5], [6]. To avoid poitioning error, many application require ome form of compenation to account for nonlinearity. The mot popular technique for compenation of hyterei i enor-baed feedback control uing integral or proportional-integral (PI) control [7], [8]. Such controller are imple but are diadvantaged by cot, complexity, limited bandwidth, and enor-induced noie [9]. In ome application, the requirement for a enor can be relaxed by elf-ening the poition from the actuator current [10], [11]. Alternatively, feedforward approache ue a model to invert nonlinearity [12], [13]. A urvey of feedforward and feedback compenation technique can be found in reference [14] and [15]. An alternative technique for reducing hyterei i to drive the actuator with charge or current rather than voltage [16], [17]. Simply by regulating the current or charge, the hyterei nonlinearity can be reduced from approximately 10% of the range to 1% [18] [20]. Thi technique wa originally reported by omtock in 1981 [17]. Following thi work, everal variation and improvement appeared; for example, reitive feedback to compenate for drift [18], grounded load [19], witched capacitor implementation [21], and dynamic compenation [22]. Although the circuit topology of a charge or current amplifier i much the ame a a imple voltage amplifier, the uncontrolled nature of the output voltage typically reult in the load capacitor being charged. To avoid thi problem, a reitive feedback network i commonly employed to tabilize the low-frequency behavior [18]. Unfortunately, thi introduce a number for problem: the voltage gain i inverely proportional to the load capacitance, the dc gain mut be tuned to match the ac gain, which can be an extremely low proce, the tranition frequency i fixed by the component value, and tranition frequencie below 1 Hz are not practical becaue of the extremely long tranient repone aociated with a long time contant. Becaue of thee practical difficultie, the potential benefit of driving a piezoelectric actuator with charge have been largely ignored in commercial application. To overcome thee difficultie, thi article propoe a new method for dc tabilization that utilize a controlled current ource. Thi method dramatically improve the eaeof-ue of a charge drive and ha the potential to directly replace a voltage amplifier in many application. In the following ection, preent charge drive circuit are dicued, followed by a decription of the active dc tabilization technique. The performance of the active dc tabilization technique i then examined experimentally by driving a tandard piezoelectric tack actuator. The key performance characteritic and advantage of active dc tabilization are ummarized in the concluion. II. Exiting harge Drive ircuit The chematic diagram of a floating-load charge drive circuit i hown in Fig. 1. The piezoelectric load, modeled a a capacitor and voltage ource v p, i haded in gray. The high-gain feedback loop work to equate the applied reference voltage in to the voltage acro a ening capacitor. Neglecting the reitance R and R, the charge q i q = v in. (1) /$ IEEE

2 fleming: charge drive with active dc tabilization for linearization of piezoelectric hyterei 1631 That i, the gain i coulomb/volt, which implie an input-to-output voltage gain of / /. The foremot problem aociated with charge drive are due to tray current, the finite output impedance, and the dielectric leakage, modeled by R. Thee effect caue the output voltage to drift at low frequencie. However, by etting the ratio of reitance equal to the ratio of capacitance, low-frequency error can be avoided. To maintain a contant voltage gain, the required reitance ratio i R R =. (2) The parallel reitance effectively turn the charge drive into a voltage amplifier below the tranition frequency: f T 1 = 2πR Hz. (3) Fig. 1. Simplified chematic diagram of a charge drive [18]. Although the parallel reitance act to tabilize the voltage gain at low frequencie, the amplifier now operate a a voltage ource below the tranition frequency and a charge drive above it. A conequence i that ignificant reduction of nonlinearity only occur at frequencie above the tranition frequency. Therefore, the tranition frequency mut be ignificantly lower than the minimum frequency of operation. The time contant of the R network i 1/2π f T and the 99% ettling time i approximately 5/2π f T, o a low tranition frequency will reult in a long tranient repone after turn-on or a tranient event. For example, to operate effectively at 1 Hz, the tranition frequency mut be 0.1 Hz or le. Thi reult in a time contant of 1.6 and a 99% ettling time of 8, which may be impractical for ome application. Therefore, to avoid exceively long tranient repone, the minimum operational frequency of a tandard charge drive i approximately 1 Hz. A further inconvenience arie from the fixed nature of the charge gain; the voltage gain of the amplifier i inverely proportional to the load capacitance, which i inconvenient if the load capacitance varie. intead of tuning the dc gain to match the ac gain, which i extremely low, the ac gain i tuned to match the dc gain, which i fat and traightforward, the tranition frequency can be varied freely to uit the application, and the tranition frequency can be extremely low becaue long tranient repone are eliminated. In Fig. 2, a tandard charge ource i connected to the piezoelectric load hown in gray. The dc ervo loop, hown in blue, conit of a voltage divider 1/α, a ummer, and the current ource I c. The voltage acro the load can be computed by uperpoition. Firt, neglecting the reitance R, the load voltage due to the charge ource i III. harge Drive With Active D Stabilization The difficultie encountered with charge drive are primarily due to the method in which the dc gain i controlled. In preent deign, the dc gain i controlled by the reitor R and R. In the following, thee reitor are replaced by a controlled current ource that regulate the low-frequency voltage gain and eliminate drift. The major benefit of thi approach are: the low-frequency voltage gain i fixed, rather than a function of load capacitance, the reitor for etting the dc gain are eliminated, Fig. 2. A charge drive with dc ervo control. The dc tabilization loop i hown in blue.

3 1632 IEEE Tranaction on Ultraonic, Ferroelectric, and Frequency ontrol, vol. 60, no. 8, Augut 2013 q = in, (4) the charge gain can be adjuted to equal the voltage gain. That i, the charge gain hould be uch that where i the gain of the charge ource. The econd contributor to the load voltage i the current ource I c, which i equal to I = k α, (5) c i in where k i i the gain of the current ource in amp/volt. By neglecting R and applying Ohm law ( = I c Z, where Z = 1/ ) the voltage due to the current ource i i k i( in i/ α) =, (6) where k i i the gain of the current ource. Hence, the total load voltage i which i equal to Thu, and = k ( / α) = +, in i in (7) k ki = +. (8) α in i in k i k 1 + = +, (9) α 1 ( 1 + ) ki α in in i in k i + 1 ( 1 + ) k i α in. (10) = α. (13) If the charge gain i properly adjuted to = α, the tranfer function i () = () α. + + β β + β = α (14) in That i, the voltage gain of the amplifier i α regardle of frequency. I. Practical Implementation The circuit diagram in Fig. 2 contain a grounded-load charge ource and a high-voltage current ource, neither of which are traightforward to contruct in practice. Although there everal method in which the chematic in Fig. 2 could be implemented, one imple method with deirable characteritic i hown in Fig. 3. The circuit in Fig. 3 i identical in function to Fig. 2. However, the load i now floating and the current ource appear in erie rather than in parallel with the load. The advantage of thi approach i that the current ource i both grounded and expoed only to low voltage. Thi i ignificant becaue a high-voltage current ource i difficult to contruct with the requiite performance for thi application that i, with low noie, low drift, high impedance, and low offet current. To allow variation of the charge gain, a gain of k q 1 i incorporated into the charge feedback loop. Thi decreae the overall charge gain to Therefore, the tranfer function from the input to the load voltage i () () = ( + ) + β α β ( + β), in (11) where β = k i /α i the tranition frequency in radian/ econd. In hertz, the tranition frequency i f T k i =, (12) 2πα which would typically be le than 1 Hz. The tranfer function (11) conit of two part, one related to the charge ource, which i effectively high-pa filtered, and another related to the dc tabilization loop, which act a a complementary low-pa filter. Becaue of thee complementary filter, the amplifier act like a charge ource above the tranition frequency and a voltage amplifier below it (with a gain of α). If the dc gain i fixed, Fig. 3. Practical implementation of a charge drive with dc tabilization circuit (hown in blue).

4 fleming: charge drive with active dc tabilization for linearization of piezoelectric hyterei 1633 charge gain = /. (15) k q The gain k q alo reduce the equivalent voltage gain to /k q. Becaue k q only reduce the charge gain, hould be deigned to provide ufficient gain with the larget expected load capacitance. That i, α. (16) With a maller load capacitance, k q can be ued to match the dc and ac voltage gain. That i, the equivalent gain i: k q. Experimental Reult = α. (17) The efficacy of the propoed technique i demontrated by comparing the repone of a piezoelectric tack when driven with a tandard voltage amplifier and the propoed charge drive. The actuator i a Noliac SMAP02-10mm multilayer piezoelectric tack actuator (Noliac A/S, Kvitgaard, Denmark) with a full-cale voltage of 60, a range of 10.6 μm, and a capacitance of 5.6 μf. The cro-ection i 5 5 mm and the length i 10 mm. A hown in Fig. 4, the actuator i mounted horizontally with a polihed aluminum cube bonded to the top to provide a uitable enor target. The capacitive enor i a Microene 6810 active probe (Microene, owell, MA) with a enitivity of 2.5 μm/, a range of 50 μm, and a tand-off ditance of 50 μm. Becaue the load capacitance i 5.6 μf, a charge gain of = 120 μ/ wa elected. Thi provide a maximum voltage gain of 21.4, which i ufficient to achieve the deired voltage gain of α = 20. The ac gain of the amplifier wa calibrated by applying a (peak) ine-wave with an offet of The charge gain k q wa adjuted until Fig. 4. A horizontally mounted piezoelectric actuator facing a capacitive diplacement enor. the peak amplitude of the load voltage wa equal to the dc value. A major benefit of the propoed charge drive i that extremely low tranition frequencie are poible. To achieve a tranition frequency of 7.8 mhz, the required current gain i k i = 5.5 μa/. If reitive feedback wa ued, the reitance value required to obtain a tranition frequency of 7.8 mhz would be R 1 = = 170 kω, and R = 340. MΩ. 2πf T Therefore, with reitive feedback, the time-contant of the R network i 20.4, o the ettling time after turn-on or a tranient event i more than 100, which i not practical. With active dc tabilization, the long ettling time can be reduced to le than a econd by briefly increaing the current gain from k i = 5.5 μa/ to 1 ma/. To evaluate the linearity of the charge driven piezoelectric actuator, a 50- triangular canning pattern wa applied at 10, 1, and 0.1 Hz. The reulting actuator diplacement when driven by voltage and charge are plotted in Fig. 5. In thi plot, the input ignal wa normalized to allow a traightforward linearity comparion between the input and diplacement. With a 10 Hz input frequency, the maximum deviation from linear in Fig. 5 i ± 1.7% and the rm error i 0.97%. The actuator linearity can alo be oberved in Fig. 6, where the diplacement i plotted againt the input voltage and charge. The maximum nonrepeatability in the charge-driven cae i 102 nm at 10 Hz (1.2%) veru 840 nm (10.5%) with a voltage amplifier. Although the actuator i not perfectly linearized, the remaining nonlinearity i primarily tatic. That i, the reidual nonlinearity could be inverted by a polynomial, pline, or look-up table. When the can peed i reduced to 1 Hz, there i no ignificant change in performance. At 1 Hz, the maximum deviation from linear wa ±1.9% and the rm error wa 1.03%. The maximum nonrepeatability in the charge-driven cae wa 105 nm (1.3%) veru 820 nm (10.2%) for the voltage-driven cae. A dicued in Section II, the lowet practical frequency of operation for a charge amplifier with reitive feedback i approximately 1 Hz. However, becaue of the low tranition frequency of the propoed deign, operation at 0.1 Hz and below i feaible without lo of performance. At 0.1 Hz, the maximum deviation from linear wa ±1.8% and the RMS error wa 1.01%. Furthermore, the maximum nonrepeatability of the charge driven actuator wa 83 nm (1.0%) veru 788 nm (9.9%) for a voltage amplifier. Becaue of the low frequency of the 0.1-Hz can, a ignificant amount of creep i alo exhibited by the voltagedriven actuator. The full-cale diplacement at 10 Hz i 7.79 μm, wherea at 0.1 Hz, the diplacement increae to 8.40 μm, which i an increae of 7.3%. When the actuator i driven by charge, the full-cale diplacement only in-

5 1634 IEEE Tranaction on Ultraonic, Ferroelectric, and Frequency ontrol, vol. 60, no. 8, Augut 2013 Fig. 5. The diplacement reulting from a 50- triangular can pattern at 10, 1, and 0.1 Hz.

6 fleming: charge drive with active dc tabilization for linearization of piezoelectric hyterei 1635 Fig. 6. The diplacement veru input charge and voltage for 10, 1, and 0.1 Hz can rate.

7 1636 IEEE Tranaction on Ultraonic, Ferroelectric, and Frequency ontrol, vol. 60, no. 8, Augut 2013 TABE I. Practical omparion of harge Drive (With a Tranition Frequency of 0.01 Hz). haracteritic harge drive with reitive feedback harge drive with active dc tabilization oltage gain Baed on load capacitance Fixed D gain Mut be tuned to ac gain Fixed Tranition frequency Inflexible ariable A gain Inflexible ariable Recovery from tranient Slow (e.g., 80 ) Fat (e.g., 1 ) Speed of tuning gain ery low (minute) Fat (econd) Offet/bia voltage arie with gain Fixed creae 1.1% from 7.92 μm to 8.01 μm when the frequency i changed from 10 Hz to 0.1 Hz. The maximum repeatability error of 1.3% exhibited by the charge driven actuator may eliminate the need for cloed-loop control in application that require accurate periodic motion, uch a canning probe microcopy [20], [23]. I. oncluion Although charge drive can ignificantly reduce piezoelectric nonlinearity, they are rarely ued becaue of their limited low-frequency performance, their dependence of voltage gain on the load capacitance, and their requirement for time-conuming tuning procedure. In contrat to preent deign that ue reitive feedback, the propoed charge amplifier ue a controlled current ource with voltage feedback to tabilize the lowfrequency behavior. Thi approach eliminate many of the preent difficultie and allow extremely low tranition frequencie without a long tranient repone. A ummary of the improvement i given in Table I. Experimental reult demontrate that the propoed charge amplifier can effectively reduce piezoelectric hyterei and creep at 10, 1, and 0.1 Hz in a 60- piezoelectric tack actuator. The repeatability error, which i the maximum difference between the forward and backward can path, i ummarized in Table II. At all peed, the charge drive reduce error to le than 1.3% of the can range. Thi i approximately one-ninth the error experienced when uing a voltage amplifier. In many application, uch a canning probe microcopy, a can error of 1.3% may reduce or eliminate the neceity for cloed-loop control. Hence, the ue of a charge amplifier could ignificantly reduce the ize, complexity, and cot of piezoelectric poitioning ytem. TABE II. omparion of Repeatability Error in oltage- Driven and harge-driven Piezoelectric Actuator. Scan peed (Hz) oltage amplifier Repeatability (%) harge drive Reference [1] S. M. Salapaka and M.. Salapaka, Scanning probe microcopy, IEEE ontr. Syt. Mag., vol. 28, no. 2, pp , Apr [2] J. A. Butterworth,. Y. Pao, and D. Y. Abramovitch, A comparion of control architecture for atomic force microcope, Aian J. ontrol, vol. 11, no. 2, pp , Mar [3] D. Mehlfeldt, H. Weckenmann, and G. Stöhr, Modeling of piezoelectrically actuated fuel injector, Mechatronic, vol. 18, no. 5 6, pp , [4] S. Z. S. Haen, M. Heur, E. H. Huntington, I. R. Peteren, and M. R. Jame, Frequency locking of an optical cavity uing linear quadratic Gauian integral control, J. Phy. At. Mol. Opt. Phy., vol. 42, no. 17, p [5] S. H. hang,. K. Teng, and H.. hien, An ultra-preciion xyθ piezo-micropoitioner. I. Deign and analyi, IEEE Tran. Ultraon. Ferroelectr. Freq. ontrol, vol. 46, no. 4, pp , Jul [6] H. Jiang, H. Ji, J. Qiu, and Y. hen, A modified Prandtl Ihlinkii model for modeling aymmetric hyterei of piezoelectric actuator, IEEE Tran. Ultraon. Ferroelectr. Freq. ontrol, vol. 57, no. 5, pp , [7] P. Ge and M. Jouaneh, Tracking control of a piezoceramic actuator, IEEE Tran. ontr. Syt. Technol., vol. 4, no. 3, pp , May [8] A. J. Fleming, S. S. Aphale, and S. O. R. Moheimani, A new method for robut damping and tracking control of canning probe microcope poitioning tage, IEEE Tran. NanoTechnol., vol. 9, no. 4, pp , Sep [9] A. J. Fleming and K. K. eang, Integrated train and force feedback for high performance control of piezoelectric actuator, Sen. Actuator A, vol. 161, no. 1 2, pp , Jun [10] A. Badel, J. Qiu, and T. Nakano, Self-ening force control of a piezoelectric actuator, IEEE Tran. Ultraon. Ferroelectr. Freq. ontrol, vol. 55, no. 12, pp , Dec [11] A. J. Fleming and S. O. R. Moheimani, ontrol oriented ynthei of high performance piezoelectric hunt impedance for tructural vibration control, IEEE Tran. ontr. Syt. Technol., vol. 13, no. 1, pp , Jan [12] A. A. Eielen, J. T. Gravdahl, and K. Y. Petteren, Adaptive feedforward hyterei compenation for piezoelectric actuator, Rev. Sci. Intrum., vol. 83, no. 8, art. no , [13] Y. Wu and Q. Zou, Iterative control approach to compenate for both the hyterei and the dynamic effect of piezo actuator, IEEE Tran. ontr. Syt. Technol., vol. 15, no. 5, pp , Sep [14] K. K. eang, Q. Zou, and S. Devaia, Feedforward control of piezoactuator in atomic force microcope ytem, IEEE ontrol Syt. Mag., vol. 29, no. 1, pp , Feb [15] S. Devaia, E. Eleftheriou, and S. O. R. Moheimani, A urvey of control iue in nanopoitioning, IEEE Tran. ontr. Syt. Technol., vol. 15, no. 5, pp , Sep [16].. Newcomb and I. Flinn, Improving the linearity of piezoelectric ceramic actuator, IEE Electron. ett., vol. 18, no. 11, pp , May [17] R. omtock, harge control of piezoelectric actuator to reduce hyterei effect, U.S. Patent , Apr. 21, [18] K. A. Yi and R. J. eillette, A charge controller for linear operation of a piezoelectric tack actuator, IEEE Tran. ontr. Syt. Technol., vol. 13, no. 4, pp , Jul [19] A. J. Fleming and S. O. R. Moheimani, Senorle vibration uppreion and can compenation for piezoelectric tube nanopoition-

8 fleming: charge drive with active dc tabilization for linearization of piezoelectric hyterei 1637 er, IEEE Tran. ontr. Syt. Technol., vol. 14, no. 1, pp , Jan [20] A. J. Fleming and K. K. eang, harge drive for canning probe microcope poitioning tage, Ultramicrocopy, vol. 108, no. 12, pp , Nov [21]. Huang, Y. T. Ma, Z. H. Feng, and F. R. Kong, Switched capacitor charge pump reduce hyterei of piezoelectric actuator over a large frequency range, Rev. Sci. Intrum., vol. 81, no. 9, art. no , [22] G. M. layton, S. Tien, S. Devaia, A. J. Fleming, and S. O. R. Moheimani, Invere-feedforward of charge controlled piezopoitioner, Mechatronic, vol. 18, no. 5 6, pp , Jun [23] A. J. Fleming, Quantitative SPM topographie by charge linearization of the vertical actuator, Rev. Sci. Intrum., vol. 81, no. 10, art. no , Oct Andrew J. Fleming graduated from The Univerity of Newcatle, Autralia (allaghan campu) with a Bachelor of Electrical Engineering degree in 2000 and Ph.D. degree in Dr. Fleming i a Senior ecturer and Autralian Reearch ouncil fellow at the School of Electrical Engineering and omputer Science, The Univerity of Newcatle, Autralia. Hi reearch include nano-poitioning, high-peed canning probe microcopy, nanofabrication, and micro-cantilever enor. Dr. Fleming reearch award include the IEEE Tranaction on ontrol Sytem Technology Outtanding Paper Award, The Autralian ontrol onference Bet Student Paper Award, The Univerity of Newcatle Reearcher of the Year Award, and the Faculty of Engineering and Built Environment Award for Reearch Excellence. He i the co-author of two book, everal patent application, and more than 100 journal and conference paper.