Abstract:
A single housing with a non-ferromagnetic piezo-driven flexure has primary and secondary coil forms of different diameters, one coaxially inside the other, integrated in the flexure. The cylinders defining the planes of the primary and secondaries do not spatially overlap. The secondary coil forms may be wound in opposite directions and wired to provide a transformer device. Movement of the primary relative to the secondaries in the direction of the central axis of the coils can be differentially detected with high precision.

Description:
CROSS-REFERENCE OF RELATED APPLICATION 
       [0001]    This application is a continuation application of U.S. Ser. No. 12/587,947 filed Oct. 14, 2009, now U.S. Pat. No. 8,502,925 issued Aug. 6, 2013, which claims the benefit of U.S. Provisional Ser. No. 61/195,983 filed Oct. 14, 2008, the disclosure of which is herewith incorporated by reference in their entirety. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    The present invention relates generally to (i) linear variable differential transformers (LVDTs), devices that convert very small mechanical displacements, as small as those in the sub-nanometer level, into differential voltages (and vice versa), and (ii) integrating LVDTs into the structure of a scanning probe device such as the atomic force microscope (AFM) so that certain movements of the device may be conveniently sensed and corrected if desired. 
         [0003]      FIG. 1  shows an LVDT according to U.S. Pat. No. 7,038,443, Linear Variable Differential Transformers for High Precision Position Measurements, by some of the same inventors as here. This LVDT reflects the basic idea of these devices in the prior art that the mutual inductances between a moving primary and two secondaries change as a function of the position of a moving part. In commercial LVDTs available the LVDT of U.S. Pat. No. 7,038,443, the moving part was a ferromagnetic core and the positions of the primary and the secondaries was fixed. However, because of the use of non-ferromagnetic materials in its construction, the fact that the primary moves rather than being stationary and the advanced signal conditioning electronics controlling its operation, the LVDT of U.S. Pat. No. 7,038,443 provides sensitivity unavailable in previous LVDTs. The  FIG. 1  LVDT comprises a movable non-ferromagnetic coil form  114  around which a primary coil  115  is wound and a stationary non-ferromagnetic coil form or forms  110  around which two secondary coils  103  and  104  are wound. The coil forms can be made of plastic or paramagnetic material. The primary coil form  114  is mechanically connected to the object of interest (not shown) by a shaft  108 . The shaft  108  can transmit displacements of the object of interest on the order of microns or smaller. Alternatively the primary coil form could be stationary and the secondary coil forms could be movable with the object of interest mechanically connected to the secondary coil forms. The functionality of such a LVDT would be equivalent to that shown in  FIG. 1 . 
         [0004]    Excitation electronics  111  produce the current driving the primary coil  115 . As the position of the object of interest attached to shaft  108  changes, and therefore the position of the primary coil  115  with respect to the secondary coils  103  and  104  changes, the flux coupled to the two secondaries,  103  and  104 , also changes. These voltages are amplified with a differential amplifier  106  and converted to a voltage proportional to the core displacement by the signal conditioning electronics  112 . For small displacements, the signal is linear. The use of plastic or paramagnetic material in the construction of the  FIG. 1  LVDT lowers the sensitivity gain that would be provided by high permeability magnetic material, but eliminates Barkhausen noise. The elimination of Barkhausen noise permits the output of the excitation electronics  111  to be raised without causing a corresponding increase in output noise, thus increasing the sensitivity of the LVDT. 
         [0005]      FIG. 2  shows a more detailed depiction of the digital excitation and signal conditioning electronics for the  FIG. 1  LVDT, taken from US Patent App. Pub. No. US20040056653, Linear Variable Differential Transformer with Digital Electronics, by some of the same inventors as here. The  FIG. 2  digital excitation and signal conditioning electronics are based on a digitally generated square wave, which when filtered produces a sine wave drive signal with more precisely defined amplitude and frequency, and lower noise, than a sine wave drive signal generated by an analog sine wave generator. This digitally generated square wave originates in a microprocessor  280 . The microprocessor could be a digital signal processor, a microcontroller or other similar microprocessors known to those skilled in the art. The square wave in turn is filtered by a low pass filter  224  that effectively removes all the harmonics of the square wave above the fundamental, resulting in a pure sine wave. The filter is optimized to be stable with respect to variations in temperature. The sine wave in turn is amplified by a current buffer  225  that directly drives the LVDT primary  215 . A sine wave generated by this excitation circuit has nearly perfect frequency and amplitude stability and has a high signal to noise ratio. 
         [0006]    In the embodiment of the excitation and signal conditioning electronics depicted in  FIG. 2 , one lead from each of the secondaries  103  and  104  is grounded and the other is connected to a high precision, low noise differential amplifier  106  which subtracts the input of one secondary from the input of the other and amplifies the difference mode signal. The differential amplifier is designed to produce low noise when coupled to a low impedance input source (such as a coil). The signal from the differential amplifier  106  is input to a buffer amplifier  231  and an inverting buffer amplifier  232 . The output of the buffer amplifier  231  is fed into a normally closed input of an analog switch  233  while the output of the inverting buffer amplifier  232  is fed into a normally open input of the same switch. This arrangement could be reversed with no loss of functionality as long as the two inputs of the switch are set so that one input is open when the other input is closed. The action of the analog switch  233  is controlled by a square wave originating in the microprocessor  280  which can be phase shifted relative to the square wave also originating in the microprocessor  280  which (when filtered and amplified) drives the LVDT primary  215 . Alternatively to a phase shift relative to the primary drive square wave originating in the microprocessor  280 , it is possible to shift the phase relative to the signal going into the primary drive current buffer  225 . All that matters is that the phase of the primary drive relative to the phase of the reference square wave is adjustable. Preferably, the opening of one input which occurs with the closing of the other input of switch  233  is 90 degrees out of phase with the output signal from amplifier  106 . The output of the analog switch  233  is fed into a stable, low noise, low pass filter  234 . The output of this filter provides a signal proportional to the position of the moving primary coil  215 . 
         [0007]    Scanning probe devices such as the atomic force microscope (AFM) can be used to obtain an image or other information indicative of the features of a wide range of materials with molecular and even atomic level resolution. As the demand for resolution has increased, requiring the measurement of decreasingly smaller forces and movements free of noise artifacts, the old generations of these devices are made obsolete. The preferable approach is a new device that addresses the central issue of measuring small forces and movements with minimal noise. 
         [0008]    For the sake of convenience, the current description focuses on systems and techniques that may be realized in a particular embodiment of scanning probe devices, the atomic force microscope (AFM). Scanning probe devices include such instruments as AFMs, scanning tunneling microscopes (STMs), 3D molecular force probe instruments, high-resolution profilometers (including mechanical stylus profilometers), surface modification instruments, NanoIndenters, chemical or biological sensing probes, instruments for electrical measurements and micro-actuated devices. The systems and techniques described herein may be realized in such other scanning probe devices, as well as devices other than scanning probe devices which require precision, low noise displacement measurements. 
         [0009]    An AFM is a device which obtains topographical information (and/or other sample characteristics) while scanning (e.g., rastering) a sharp tip on the end of a probe relative to the surface of the sample. The information and characteristics are obtained by detecting changes in the deflection or oscillation of the probe (e.g., by detecting small changes in amplitude, deflection, phase, frequency, etc.) and using feedback to return the system to a reference state. By scanning the tip relative to the sample, a “map” of the sample topography or other characteristics may be obtained. 
         [0010]    Changes in the deflection or oscillation of the probe are typically detected by an optical lever arrangement whereby a light beam is directed onto the side of the probe opposite the tip. The beam reflected from the probe illuminates a position sensitive detector (PSD). As the deflection or oscillation of the probe changes, the position of the reflected spot on the PSD also changes, causing a change in the output from the PSD. Changes in the deflection or oscillation of the probe are typically made to trigger a change in the vertical position of the base of the probe relative to the sample (referred to herein as a change in the Z position, where Z is generally orthogonal to the XY plane defined by the sample), in order to maintain the deflection or oscillation at a constant pre-set value. It is this feedback that is typically used to generate an AFM image. 
         [0011]    AFMs can be operated in a number of different sample characterization modes, including contact modes where the tip of the probe is in constant contact with the sample surface, and AC modes where the tip makes no contact or only intermittent contact with the surface. 
         [0012]    Actuators are commonly used in AFMs, for example to raster the probe over the sample surface or to change the position of the base of the probe relative to the sample surface. The purpose of actuators is to provide relative movement between different parts of the AFM; for example, between the probe and the sample. For different purposes and different results, it may be useful to actuate the sample or the probe or some combination of both. Sensors are also commonly used in AFMs. They are used to detect movement, position, or other attributes of various components of the AFM, including movement created by actuators. 
         [0013]    For the purposes of this specification, unless otherwise indicated (i) the term “actuator” refers to a broad array of devices that convert input signals into physical motion, including piezo activated flexures; piezo tubes; piezo stacks, blocks, bimorphs and unimorphs; linear motors; electrostrictive actuators; electrostatic motors; capacitive motors; voice coil actuators; and magnetostrictive actuators, and (ii) the term “sensor” or “position sensor” refers to a device that converts a physical quantity such as displacement, velocity or acceleration into one or more signals such as an electrical signal, and vice versa, including optical deflection detectors (including those referred to above as a PSD), capacitive sensors, inductive sensors (including eddy current sensors), differential transformers (such as described in U.S. Pat. No. 7,038,443 and co-pending applications US Patent App. Pub. Nos. US20020175677, Linear Variable Differential Transformers for High Precision Position Measurements, and US20040056653, Linear Variable Differential Transformer with Digital Electronics, which are hereby incorporated by reference in their entirety), variable reluctance sensors, optical interferometry, strain gages, piezo sensors and magnetostrictive and electrostrictive sensors. 
     
    
     
       SUMMARY OF THE INVENTION 
       Brief Description of the Drawings 
         [0014]      FIG. 1 : Prior art showing a LVDT with a low-permeability core and a moving primary. 
           [0015]      FIG. 2 : Prior art showing excitation and signal conditioning electronics based on a synchronous analog switch. 
           [0016]      FIG. 3 : Preferred embodiment of integrated piezo flexure and LVDT. 
           [0017]      FIG. 4 : Preferred embodiment of digital excitation and signal conditioning electronics for integrated piezo flexure and LVDT. 
           [0018]      FIG. 4A : Preferred embodiment of field programmable gate array for digital excitation and signal conditioning electronics of  FIG. 4 . 
           [0019]      FIG. 4B : Alternative embodiment of primary drive of digital excitation and signal conditioning electronics of  FIG. 4 . 
           [0020]      FIG. 4C : Alternative embodiment of wiring for secondary of digital excitation and signal conditioning electronics of  FIG. 4 . 
       
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0021]    A piezo activated flexure may be used to move the sample in the X and Y directions relative to the tip of the probe of an AFM, that is to scan the sample in the XY plane, using a XY positioning stage like that shown in FIG. 20 of U.S. Pat. No. 7,038,443, by some of the same inventors as here. Similarly, as shown in FIG. 19 of that patent, such a flexure may be used to change the vertical position of the base of the probe relative to the sample (that is along the Z axis) using a Z positioning stage in order to maintain the deflection or oscillation of the probe tip at a constant pre-set value. In both situations a LVDT like that disclosed in U.S. Pat. No. 7,038,443 may be used to sense movement in the X, Y and Z directions and to make any desired corrections. For this purpose, what is of interest, the X position and the Y position of the XY positioning stage, and the Z position of the Z positioning stage, is each mechanically connected to a primary coil form  114  of a separate LVDT via the shaft  108  of the LVDT in question and each primary coil form therefore moves in tandem with any movement of the X position, Y position or Z position, as the case may be. The secondary coil forms  110  of each such LVDT in turn are fastened to the frame of the XY positioning stage in the case of the LVDTs pertaining to the X position or the Y position and to the frame of the AFM in the case of the LVDT pertaining to the Z position. The secondary coil forms  110  of each such LVDT therefore remain stationary relative to its primary coil form  114 . 
         [0022]    The present invention uses piezo activated flexures to move the sample in the X, Y and Z directions relative to the tip of the probe of an AFM, that is to scan the sample in the XY plane, and to move the vertical position of the sample relative to the base of the probe, that is to move the sample or base of the probe in the Z plane, in order to maintain the deflection or oscillation of the probe tip at a constant pre-set value. These piezo activated flexures are part of a scanner module of an AFM. An AFM for which these piezo activated flexures may be used is described in a co-pending application, U.S. patent application Ser. No. 60/______, Modular Atomic Force Microscope, by some of the same inventors 
         [0023]      FIG. 3  depicts one of these piezo activated flexures, the one that is used to change the vertical position of the sample relative to the base of the probe. Piezo activated flexures similar to that depicted in  FIG. 3  may be used to change to scan the sample in the XY plane. The piezo  301  of  FIG. 3  is of a stack design known to those versed in the art with top and bottom indentations to accommodate a top ball bearing  302  and a bottom ball bearing  303 . The flexure  304  is of a tube design with interior threads to provide for a threaded disk insert (or contiguous disk inserts) at the top  305 , together with a threaded sample support plate  307 , and another threaded disk insert at the bottom  306 . The disk inserts have top and bottom indentations corresponding to those of the piezo  301  to accommodate the top and bottom ball bearings  302  and  303 . The interior threads of the flexure  304  and the threaded disk inserts  305  and  306  permit the piezo  301  via the ball bearings  302  and  303  to be locked into place within the flexure  304 . 
         [0024]    The bottom disk insert  306  of the flexure  304  taken together with the design of the flexure itself serve as a cap and permits very little motion along the Z axis in the direction of the bottom of the flexure. The top disk insert  305  again taken together with the design of the flexure itself permits free movement of the flexure  304  along the Z axis in the direction of the top of the flexure in accordance with vertical expansion and contraction of the piezo  301 . The cut-outs or recesses  308  in the flexure  304  constrain this movement to the Z plane, and permit very little motion in the X and Y planes. 
         [0025]    When the piezo  301  is locked into place within the flexure  304  the top disk insert  305  is tightened somewhat more than is necessary to lock the piezo in place as a means of preloading the flexure  304 . The cut-outs or recesses  308  in the flexure  304  transform this additional tightening into movement of the flexure, together with the sample support plate  307  and thereby the sample (not shown), along the Z axis in the direction of the top of the flexure. When the piezo  301  is contracted (using the appropriate electrical charge) the cut-outs or recesses  308  of the flexure  304  transform this contraction into movement of the portion of the flexure  304  above the cut-outs or recesses  308 , together with the sample support plate  307  and thereby the sample, along the Z axis in the direction of the bottom of the flexure. When the piezo  301  is expanded (using the appropriate electrical charge) the cut-outs or recesses  308  of the flexure  304  transform this expansion into movement of the portion of the flexure  304  above the cut-outs or recesses  308 , together with the sample support plate  307  and the sample, along the Z axis in the direction of the top of the flexure. As noted this motion is accompanied by very little motion in the X and Y planes. 
         [0026]    As already noted, LVDTs like those disclosed in U.S. Pat. No. 7,038,443 may be used in an AFM to sense and correct movement in the X, Y or Z directions when the sample is scanned in the XY plane or when the vertical position of the sample relative to the base of the probe is moved in the Z plane. As shown in that patent, this is achieved by mechanically connecting the primary and secondaries of LVDTs to the parts of the AFM relevant for the purpose. 
         [0027]    The present invention uses LVDTs to sense and correct movement in the X, Y or Z directions in an AFM, but in a very different way than shown in U.S. Pat. No. 7,038,443. Instead of mechanically connecting the primary and secondary coil forms of LVDTs to the parts of the AFM relevant for the purpose, here the primary and secondary coil forms are integral to the parts themselves. As shown in  FIG. 3 , a channel  309  which serves as the primary coil form for the LVDT is formed into the top of the flexure  304  just above the cut-outs or recesses  308  in the flexure  304 . As described above this portion of the flexure, and therefore the channel  309 , moves as the piezo  301  is contracted or expanded. Similarly a pair of channels  310  which serve as the secondary coil forms for the LVDT are formed into a stationary sleeve  311  which is fastened to the flexure  304  below the cut-outs or recesses  308 . As described above the portion of the flexure  304  to which the sleeve  311  is attached, and therefore the sleeve, does not move as the piezo  301  is contracted or expanded. 
         [0028]    Within the limits imposed by the requirement for preloading the flexure  304 , loosening or tightening the top disk insert  305  can be used to center the channel  309  which serves as the primary coil form for the LVDT relative to the channels  310  which serve as the secondary coil forms. 
         [0029]    The flexure  304  provides conduits whereby electrical connections may be established with the primary coil, the secondary coils and the piezo  301 .  FIG. 3  shows the exterior portion of one of these conduits  312 . 
         [0030]    As noted in U.S. Pat. No. 7,038,443 non-ferromagnetic coil forms are an important contributor to making a sensitive LVDT. For this purpose, the coil forms could be made of plastic or paramagnetic material. In the present invention the flexure  304 , in which the channel  309  which serves as the primary coil form is integrated, is preferably fabricated from a high-yield-stress non-ferromagnetic aluminum such as 7075 aluminum. Alternatively they could be fabricated from a ceramic material. The stationary sleeve  311 , in which the pair of channels  310  which serve as the secondary coil forms are integrated, is preferably fabricated from a plastic material such as PEEK. Again, they could also be fabricated from a ceramic material. 
         [0031]      FIG. 4  shows a preferred embodiment of digital excitation and signal conditioning electronics for the LVDT of the present invention in which the primary and secondary coil forms of the LVDT are integral to the relevant moving parts of a piezo activated flexure like that depicted in  FIG. 3  and the coil forms are formed from a non-ferromagnetic material. These electronics may be used with LVDTs of other designs, for example the LVDT of  FIG. 1  where they would replace the digital excitation and signal conditioning electronics depicted in  FIG. 2 . 
         [0032]    The embodiment of the digital excitation and signal conditioning electronics of  FIG. 4  are based on a digitally generated sine wave drive signal with much more precisely defined amplitude and frequency, and lower noise, than a sine wave generated by an analog sine wave generator. This sine wave drive signal originates in a direct digital synthesizer  401 , implemented within the field programmable gate array  412 , some of the components of which are shown separately in  FIG. 4A . The sine wave is then routed though a digital gain stage  402 , also implemented within the FPGA  412 , which permits the user to control the amplitude of the wave. At this point the sine wave has the following form: 
         [0000]        A  sin ω t  
 
         [0033]    The sine wave is then converted to analog form by a digital to analog converter  403  and amplified by a buffer  404  that directly drives the LVDT primary  405  of the present invention typically at a +10V to −10V voltage range, but other voltages may be used. The voltages driving the primary  405  may be doubled through another embodiment depicted in  FIG. 4B . In that embodiment, the output of the digital to analog converter  403  is split and sent both to a gain stage  420  and a negative gain stage  421 . The output of each stage is in turn connected to one of the leads of the LVDT primary  405 , and the primary is differentially driven at twice the original voltage range. 
         [0034]    The signal conditioning electronics of the digital excitation and signal conditioning electronics for the LVDT of the present invention are depicted in  FIG. 4 . As shown there, one of the secondaries  408  and  409  may be wound in the opposite direction from the winding of the other and the adjoining leads from the oppositely wound secondaries wired together. The other lead from one of the secondaries, here secondary  409 , is grounded and the other lead from the second secondary, here secondary  408 , is connected to an analog gain stage  410 . 
         [0035]      FIG. 4C  shows another equivalent arrangement of the secondaries  408  and  409  and the analog gain stage  410  of  FIG. 4  where the secondary windings are wound in the same direction, but are wired to produce the same effect as with secondaries wound in opposite directions as in  FIG. 4 . Either arrangement offers a significant improvement in the signal-to-noise ratio of the signal conditioning electronics for the LVDT of the present invention relative to such electronics of other LVDTs, for example the electronics depicted in  FIG. 2 . One reason for this improvement is the self-cancelling feature of the arrangement. In the signal conditioning electronics of the  FIG. 2  LVDT, the signal coupled to one secondary by the primary  215  is subtracted in the differential amplifier  106  from the signal coupled to the other secondary in order to determine a voltage proportional to the displacement of the primary coil form  114  and therefore the displacement of the object of interest which is mechanically connected to the coil form. Indeed the differential amplifier  106  is making this calculation even when the primary is centered exactly between the two secondaries and there is no displacement to measure. With the signal conditioning electronics for the LVDT of the present invention however the signal coupled to the secondaries  408  and  409  are wound or wired such that either the currents induced by the coupled signal in the case of the arrangement shown in  FIG. 4  or the voltages so induced in the case of the arrangement shown in  FIG. 4C  oppose each other in the secondaries themselves, thereby obviating the need for a differential amplifier. This self-opposing phenomenon for example results in a zero signal from the secondaries  408  and  409  or  425  and  426 , as the case may be, when the primary  405  is centered exactly between the two secondaries. 
         [0036]    The signal-to-noise ratio of the signal conditioning electronics for LVDTs using a differential amplifier  106  like that of the  FIG. 2  LVDT is inherently lower than is desirable because of the voltage rails that are part of such amplifiers. These rails limit the voltages in the secondaries that can be accommodated by the amplifier to low levels and thus limit the possible signal-to-noise ratio. Furthermore as the voltage in the secondaries rises, so does Johnson noise, and therefore the signal-to-noise ratio of the signal conditioning electronics declines. 
         [0037]    The self-opposing phenomenon of the signal conditioning electronics for the LVDT of the present invention makes it possible to use much higher voltages and thus boost the signal-to-noise ratio. One method of doing this is to increase the voltage (or current) driving the primary  405  and therefore the voltages (or currents) induced in the secondaries  408  and  409  or  425  and  426 , as the case may be. As noted above, the embodiment depicted in  FIG. 4B  shows a method for doubling the primary voltage (or current) that is used. Another method substantially increases the voltages (or currents) induced in the secondaries  408  and  409  or  425  and  426 , as the case may be, by the voltage (or current) of the primary  405  by increasing the turns ratio of the secondaries relative to the primary. As is well known, the voltages (or currents) induced in the secondaries  408  and  409  or  425  and  426 , as the case may be, can be increased many times over by this method. However, whether the voltages (or currents) induced in the secondaries  408  and  409  or  425  and  426 , as the case may be, are increased by increasing the voltages (or currents) driving the primary  405  or by increasing the turns ratio of the secondaries relative to the primary, or both, the self-opposing phenomenon of the signal conditioning electronics for the LVDT of the present invention passes along to the gain stage  410  only the difference between the voltage (or current) induced in one secondary and that induced in the other. Thus, a large increase in the signal-to-noise-ratio may be obtained because the magnitude of the drive or the turns ratio of secondary to primary are no longer limited by the electronic input range of the differential amplifier imposed by its voltage rails. 
         [0038]    As shown in  FIG. 4 , the signal conditioning electronics of the present invention routes the output from the secondaries  408  and  409  or  425  and  426 , as the case may be, through an analog gain stage  410 , at which point the output is a 125 kHz sine wave with the following form: 
         [0000]        B  sin(ω t +Φ)
 
         [0039]    A typical operating frequency might be 125 kHz, although other frequencies could be used. This sine wave is then converted into digital form with an analog to digital converter  411  and sent to the FPGA  412 , the components of which are shown separately in  FIG. 4A . The digitized signal is then fed through a DC blocking filter  430 , implemented within the FPGA  412 , the output of which goes to a digital multiplier circuit  431 , also implemented within the FPGA  412 . The other input to the multiplier circuit  431  is the digital sine wave driving the LVDT primary  405  after its phase offset has been adjusted. As indicated above, this sine wave originates in the DDS  401 , implemented within the FPGA  412 . The phase offset of the digital sine wave driving the LVDT primary is adjusted by the phase offset adjustment circuit  432 , also implemented within the FPGA  412 . The output from the multiplier circuit  431  has the following form: 
         [0000]        A  sin(ω t )× B  sin(ω t +Φ)=( A×B )/2(sin(2 ωt +Φ)+sin Φ)
 
         [0040]    In order to increase the resolution provided by the signal conditioning electronics for the LVDT of the present invention, the ADC  411  used to covert the sine wave output from the secondaries  408  and  409  or  425  and  426 , as the case may be, is preferably at least an 18-bit converter sampling at least at a 2 MHz rate. Using such an ADC, this output, which for example is a sine wave at 125 kHz after having been passed through the analog gain stage  410  which intervenes between the secondaries  408  and  409  or  425  and  426 , as the case may be, and the ADC  411 , is sampled at a rate of 16 samples per cycle, several times the minimum rate required to capture a sine wave digitally. However the 18-bit resolution for each sample provided by the ADC  411  is insufficient to overcome quantization effects and measure displacement at the subnanometer dynamic ranges required for the LVDT of the present invention. The solution to this difficulty is found in the fact that the ADC  411  is sampling at a 2 MHz rate, a rate much faster than the rate required for correcting movement of the piezo flexure of the present invention. Accordingly, some samples are used to create additional resolution of the sine wave, a result that may be referred to as bit growth. The output of the ADC  411  sent to the FPGA  412 , therefore, is the sine wave output from the secondaries  408  and  409  or  425  and  426 , as the case may be, in high resolution digital form. 
         [0041]    The output from the multiplier circuit  431  is routed through a low pass filter  433  which filters out the sin(2ωt+Φ) term, leaving the dc term (A×B)/2 sin Φ. This dc term of the signal is proportional to the change in position of the piezo flexure of the present invention and may be used to correct that position to the position desired. 
         [0042]    Although only a few embodiments have been disclosed in detail above, other embodiments are possible and the inventors intend these to be encompassed within this specification. The specification describes specific examples to accomplish a more general goal that may be accomplished in another way. This disclosure is intended to be exemplary, and the claims are intended to cover any modification or alternative which might be predictable to a person having ordinary skill in the art. 
         [0043]    Also, the inventors intend that only those claims which use the words “means for” are intended to be interpreted under 35 USC 112, sixth paragraph. Moreover, no limitations from the specification are intended to be read into any claims, unless those limitations are expressly included in the claims. The computers described herein may be any kind of computer, either general purpose, or some specific purpose computer such as a workstation. The computer may be a Pentium class computer, running Windows XP or Linux, or may be a Macintosh computer. The computer may also be a handheld computer, such as a PDA, cellphone, or laptop. 
         [0044]    The programs may be written in C, or Java, Brew or any other programming language. The programs may be resident on a storage medium, e.g., magnetic or optical, e.g. the computer hard drive, a removable disk or media such as a memory stick or SD media, or other removable medium. The programs may also be run over a network, for example, with a server or other machine sending signals to the local machine, which allows the local machine to carry out the operations described herein.