Abstract:
A method to maintain a movable electrode of a variable capacitor in fixed relationship with a stationary electrode by electrostatic force feedback when one capacitor electrode is grounded. The invention exploits the high quiescent capacitance and capacitance-load sensitivity of a variable-area capacitance transducer in combination with the advantage of controlling the flexible electrode with a low voltage compared to transducers with parallel-plate capacitors. The invention can be used to accurately control the position of a surface, stylus, inertial mass, valve, electrical contact, electrical component; or an optical component such as a mirror, lens, grating, filter, or holographic element.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]    This application is a continuation-in-part of divisional application Ser. No. 09/482,119, Jan. 13, 2000, of application Ser. No. 09/037,733 of Mar. 10, 1998, now U.S. Pat. No. 6,151,967. The invention of the present application references art disclosed in continuation-in-part application Ser. No. 09/834,691, filed Apr. 13, 2001, Ser. No. 09/816,551, filed Mar. 24, 20001 and Ser. No. 09/794,198, filed Feb. 27, 2001, of divisional application Ser. No. 09/482,119. Each disclosure of the foregoing applications are incorporated herein by reference. All of the applications are assigned to the same assignee as the present application. 
     
    
     GOVERNMENT RIGHTS  
       [0002] This invention was made with Government support under contract N00024-97-C-4157 from the Naval Sea Systems Command. The Government has certain rights to this invention 
     
    
     
       FIELD OF THE INVENTION  
         [0003]    The present invention relates to force-balanced capacitive transducers including sensors and actuators. More specifically, it relates to a method to electrostatically force-balanced transducers that control the position of a surface, stylus, inertial mass, valve, electrical contact, electrical component; or an optical component such as a mirror, lens, grating, filter, or holographic element.  
         BACKGROUND OF THE INVENTION  
         [0004]    Electrostatically, force-balanced, capacitive transducers maintain a compliant member, such as a beam, diaphragm, or bridge at a fixed position or predetermined gereratrix. One example is an accelerometer with capacitor electrodes that sense the displacement of an inertial mass suspended on a compliant member in relation to stationary support structure. A simple capacitive accelerometer measures a change in spacing between a capacitor electrode affixed to a surface of an inertial mass in close proximity to a cooperating capacitor electrode affixed to adjacent stationary structure. The two electrodes form a displacement sensing, parallel-plate capacitor with a capacitance value inversely proportional to the plate spacing. A force acting on the mass causes it to be displaced in proportion to the stiffness of the compliant member. When an unconstrained axis of the mass is orientated with the gravity gradient, it will deflect to an equilibrium or quiescent position where restoring forces due to the bending moment of the compliant member balance the force of acceleration acting on the mass. Motion of the support structure modulates the capacitor plate spacing due to the inertia of the mass. The corresponding change in capacitance is detected and transduced to provide a voltage proportional to the square root of a change in force acting on the mass.  
           [0005]    The disadvantages of a simple capacitive accelerometer are: 1) a severely restricted dynamic measurement range due to a small capacitor gap, 2) a highly non-linear capacitance response with mass displacement, and 3) the requirement to dampen the response of the accelerometer at frequencies near the resonance of its spring-mass system. These disadvantages are avoided by operating a capacitive accelerometer in an electrostatic force-balanced feedback loop that maintains the inertial mass in a substantially fixed position under loading. An incremental displacement of the mass is capacitively detected, transduced, and amplified to provide a control voltage V across the capacitor plates to create an electrostatic force that restores the mass to its initial position. The electrostatic force F E  applied to a electrode of a parallel-plate capacitor varies as V 2 /d, where d is the effective plate spacing. Because the force F E  is independent of the polarity of the control voltage V, the compliant member of a simple capacitive transducer is required to be electrostatically biased or mechanically preloaded to allow the accelerometer to sense a bi-directional force. Accelerometers, actuators, and other capacitive transducers with one, force balancing capacitor are defined here as single-side, force-balanced transducers.  
           [0006]    In a similar manner, a simple electrostatic actuator applies an electrostatic force to an electrode affixed to a compliant member and/or attached payload to control the position of the member or payload. The payload can include a surface, stylus, inertial mass, valve, electrical contact, electrical component; or an optical component such as a mirror, lens, grating, filter, or holographic element. This method is of controlling the position of a compliant actuator member is equivalent to applying a voltage bias to the electrodes of a capacitive accelerometer to move its inertial mass to a new equilibrium position. For most actuator applications, a static force such as gravity is generally small compared to the applied electrostatic force and the reaction force of the compliant member and payload. A general disadvantage of a single-side, capacitive transducers (e.g., sensors and actuators) is that the effective spring constant and elastic restoring force of the preloaded compliant member is influenced by ageing and physical effects such as temperature.  
           [0007]    Precision capacitive transducers use at least a pair of differential parallel-plate capacitors to sense and force-balance an inertial mass or to sense and position of a compliant member of an actuator. Planar electrodes are affixed to opposing surfaces of the mass and/or a compliant member which are located in close proximity to cooperating planar electrodes affixed to adjacent stationary structure. The electrodes form one or more pairs of differential displacement sensing and force generating capacitors. When the mass of an accelerometer or payload of an actuator moves, the capacitance of one displacement sensing capacitor increases while that of a second capacitor decreases by substantially an equal amount. One advantage of a differential capacitive transducer is that the compliant member is not required to be preloaded for bi-directional force or position control. Another advantage is that an incremental capacitance change can be detected with a bridge circuit to minimize errors associated with unmatched electronic components and variations of supply and reference voltages.  
           [0008]    Capacitance transducers with variable-gap, parallel-plate electrodes comprise a well-know and crowded art. The general disadvantage of prior-art capacitive transducers arise from the limitations imposed by parallel-plate capacitors: low-quiescent capacitance, low capacitance-load sensitivity, non-linear response, and the requirement to form and accurately maintain a precision structure with narrow electrode spacing. Variable-area capacitance transducers with a flexible electrode responsive to a physical effect are less known and less appreciated for their ability to provide an order of magnitude and greater increase in quiescence capacitance, capacitive-load sensitivity and linear dynamic range. Variable-area capacitors of U.S. Pat. No. 6,151,967 and those simply illustrated in FIGS. 1 and 2 are constructed by sandwiching a thin dielectric layer between a flexible electrode and a cooperating rigid electrode with a curved surface. The dielectric layer maintains a region of fixed capacitance spacing between mutually opposed areas of the rigid electrode and flexible electrode. The region of fixed capacitive spacing increases in extent as the flexible electrode deflects in response to a physical effect. The flexible electrode can be of electrically conducting material, or it can be comprise an electrically conducting layer affixed to at least one surface of a compliant member of dielectric material.  
           [0009]    A variable-area capacitor with a flexible electrode comprising a compliant cantilever beam was described by Carter et al., “Measurement of Displacement and Strain by Capacity Methods”,  Proc. J. Mech. Engr . (152) 1945, pp. 215-221. The Carter transducer is generally of the design shown in FIG. 1. The use of this transducer as a displacement sensor was described by Frank,  Electrical Measurement Analysis , McGraw-Hill, NY, 1959. Variable-area capacitors of U.S. Pat. No. 6,151,967, can be fabricated with flexible electrodes that include circular and rectilinear diaphragms. Transducers of this general design are in FIGS. 2 and 3.  
           [0010]    The electrostatic deflection of a cantilever beam is a well known art that was recently reviewed by Legtenberg, et al., “Electrostatic Curved Electrode Actuators,”  J. Micro Electro Mech Syst . vol. 6, no. 3, 1997. The electrostatic deflection of circular and rectilinear diaphragms for a loudspeaker is an invention of Kyle, U.S. Pat. No. 1,644,387, Oct. 4, 1927. The Kyle invention is further discussed by Ford, et al., “The Kyle Condenser Loud Speaker,”  P. Inst. Radio Engr ., vol. 17, no. 7, 1929. Since this early work, a rich art has developed for electrostatically controlled actuators. None of the above cited prior-art references teach or suggest using a variable-area capacitor to develop an electrostatic force to position an actuator and the same capacitor to sense the position of the actuator for closed-loop electrostatic position control.  
           [0011]    Accordingly, an object of the present invention is to provide a method to control the displacement of a flexible electrode of variable-area capacitive sensors and actuators by electrostatic force feedback.  
           [0012]    The transducer of Cadwell, U.S. Pat. 4,584,885, and other early capacitive transducers used a first pair of capacitors to sense the displacement of a mass and a second pair of capacitors to apply an electrostatic force to the required side of the inertial mass to maintain it in a substantially fixed position. The disadvantage of this arrangement is that the surface area of the inertial mass is divided to accommodate two, smaller area capacitor electrodes. The accelerometer of Sherman et al., U.S. Pat. No. 5,540,095, and other force-balance capacitive transducers, are constructed with electronics that allow a single pair of capacitors to be used for both displacement sensing and force-balancing. The accelerometer of Suzuki et al., U.S. Pat. No. 5,095,752, and Yazdi, et al., U.S. Pat. No. 6,035,714, utilize only three electrodes to perform the displacement sensing and force balancing function. For these accelerometers, two differential capacitors are formed by affixing two planar electrodes on rigid support structure adjacent to opposite sides of an inertial mass. If inertial mass is of electrically conducting material, opposing surfaces of the mass serve as a common movable electrode for both capacitors. Alternately, electrodes affixed to opposing surfaces of the mass can be connected electrically in parallel to provide a common electrode. A disadvantage of prior art, differential capacitive transducers and associated capacitance detection electronics is that a common electrode cannot be electrically grounded. Grounding the common electrode minimizes the susceptibility of a transducer to electromagnetic interference and signal loss due to the attenuation (e.g., signal division) of stray capacitance. The total capacitance of 0.1 pF disclosed for the accelerometer of Sherman is representative of the values of other micromachined capacitive transducers. This value is small compared to the input capacitance of active electronics devices used for signal amplification.  
           [0013]    Accordingly, still another object of the present invention is to provide a method that allows an electrode of a capacitive transducer controlled by electrostatic feedback to be grounded.  
           [0014]    Variable-area capacitance sensors and actuators can be fabricated by many of the methods used to construct parallel-plate capacitive transducers with planar capacitor electrodes. U.S. Pat. No. 5,095,752, Suzaki et al., is one example of an accelerometer with two stationary planar electrodes and a common movable electrode. The movable electrode includes a thin section connected to an inertial mass that is bulk micromachined from single-crystal silicon. The invention of Sherman is a second example of a capacitive accelerometer with stationary planar electrodes and a common movable electrode. This accelerometer is surface micromachined by the well known steps of sacrificial layer etching. The profile and sidewalls of the movable electrode of polycrystalline silicon is formed over a sacrificial layer, typically of silicon dioxide (SiO 2 ) or oxynitride (SiO X N X ) formed on a silicon substrate. The movable electrode is then released by etching the underlying silicon dioxide with hydrofluoric acid. U.S. Pat. No. 6,199,871 B1, Galvin et al., is a third example of a capacitive accelerometer with planar electrodes formed by surface micromachining of silicon using the method of multi-step, deep reactive ion etching (DRIE). The details of these and other micromachining methods are described Elwenspoek, et al.,  Silicon Micromachining , Cambridge University Press, Cambridge, UK, 1998.  
           [0015]    Accordingly, another object of the present invention is to provide a method to control by electrostatic force feedback the displacement of a flexible electrode of a variable-area capacitor fabricated at least in part by the processing steps of bulk and surface micromachining.  
           [0016]    The above cited capacitance sensors and actuators require a means to detect an incremental change of capacitance between at least two electrodes or a pair of differential capacitor electrodes. These inventions are representative of the many different capacitive transducers that can exploit the advantages of the capacitive detection circuits of U.S. pat. application Ser. Nos. 09/834,691 and 09/794,198. Specific advantages include: high DC stability, low impedance inputs, wide dynamic range, a linear output, the ability to ground a common capacitor electrode, and an option to provide a high-resolution digital signal output. The advantages of both circuit inventions are realized by variable capacitors with variable-area electrodes, variable-gap electrodes, and a combination of variable-area and variable gap electrodes. A further advantage of these preferred circuit inventions is that a bridge excitation signal applies a small periodic vibratory force to a flexible electrode to overcome and average any micro-frictional and stiction forces. The frequency of the vibratory force is higher than the frequencies of forces being detected or motion being controlled.  
           [0017]    Accordingly, a further object of the present invention is to provide a method to control the displacement of a moveable electrode of a capacitive sensor or actuator using a capacitance detection circuit that provides current feedback to actively null a bridge network used to provide a voltage to electrostatic force balance the movable electrode.  
         SUMMARY OF THE INVENTION  
         [0018]    The present invention is a method of providing electrostatic force feedback to maintain the displacement of a movable electrode of a variable capacitor in fixed relationship with a cooperating stationary electrode.  
           [0019]    A preferred embodiment is a method to control a flexible electrode of a capacitive transducer by electrostatic force feedback comprising the steps of providing a capacitor with a flexible electrode and at least one cooperating stationary electrode; providing a capacitance detection circuit to detect a value of capacitance between the flexible and at least one stationary electrode; measuring and amplifying a change in capacitance of the capacitor to generate a feedback control voltage; and applying the feedback control voltage across the flexible and at least one stationary electrode to maintain the flexible electrode at a fixed position or predetermined generatrix. The compliant electrode can be a beam, diaphragm, or flexible bridge that supports a payload.  
           [0020]    This invention exploits the advantages of variable-area capacitive sensors and actuators: high quiescent capacitance and capacitance-load sensitivity and the ability to displace a flexible electrode at low voltages compared to the voltage required to move a planar electrode of a parallel-plate capacitor. One aspect of the method of the present invention includes providing a capacitive transducer and a capacitance detection circuit that allows one of the two electrodes of a single-side variable capacitor to be grounded. For the preferred capacitance detection circuit of the present invention, one electrode of a variable capacitor is connected to a ground potential and a second terminal is connected to a first-side node of an actively nulled capacitive bridge. A reference capacitor is connected between a second-side node of the bridge and said ground potential. Alternately, capacitive changes can be detected with less accuracy using an actively nulled, half-bridge network connected to a reference voltage. For transducers with a pair of differential variable capacitors, one electrode of each capacitor is connected to opposing sides of an actively nulled bridge. Operational descriptions and optional designs are disclosed for full bridge and half bridge networks in U.S. patent application Ser. Nos. 09/482,119, 09/794,198, and 09/816,551.  
       
    
    
     DESCRIPTION OF THE DRAWINGS  
       [0021]    In the drawings,  
         [0022]    [0022]FIG. 1, is a simplified sectional illustration of a variable-area capacitor with a flexible electrode comprising a compliant cantilever beam;  
         [0023]    [0023]FIG. 2, is a sectional view of a variable-area capacitor with flexible electrode comprising a diaphragm;  
         [0024]    [0024]FIG. 3, is a sectional view of a differential variable-area capacitor with a flexible electrode that suspends a payload;  
         [0025]    [0025]FIG. 4, is a plan view of the electrode structure a variable-area capacitive transducer formed by surface micromachining;  
         [0026]    [0026]FIG. 5, is a plot of capacitance vs. pressure for a variable-area capacitor with a silicon diaphragm;  
         [0027]    [0027]FIG. 6, is a plot of capacitance vs. pressure for a variable-area capacitor with a metallized polycarbonate diaphragm;  
         [0028]    [0028]FIG. 7, is a plot of capacitance vs. voltage applied across a variable-area capacitor with a metallized polycarbonate diaphragm;  
         [0029]    [0029]FIG. 8 is a schematic block diagram of a preferred method of the present invention to electrostatically force-balance a single-side variable-area capacitor.  
         [0030]    [0030]FIG. 9 is a schematic block diagram of a preferred method of the present invention to electrostatically force-balance a differential variable-area capacitor. 
     
    
     DETAILED DESCRIPTION  
       [0031]    Further objects and advantages of the present invention will become apparent from the following description. Well known techniques for processing semiconductor materials and fabricating micromachined devices are referred to without elaboration so not to obscure the description of the invention with unnecessary detail. All the drawings are schematic in nature and the features shown are not drawn to relative scale; like reference numbers designate similar parts or elements with similar functions.  
         [0032]    [0032]FIG. 1, is a simplified sectional illustration a variable-area capacitor generally shown by reference numeral  10  with a flexible electrode  12  comprising a compliant cantilever beam that may include a payload. Flexible electrode  12  is affixed to an edge region of a cooperating rigid electrode  14  with a curved surface region  16  with a dielectric layer  18 . Dielectric layer  18  electrically insulates and maintains a fixed capacitance spacing d between regions of mutually opposed areas of flexible electrode  12  and rigid electrode  14  as regions of said fixed capacitance spacing increases as flexible electrode  12  deflects in response to a difference between an applied stress S A  and a counteracting electrostatic stress S E . For purposes of illustration, the applied and electrostatic stresses are shown concentrated as forces F A  and F E  respectively. The curvature of surface region  16  of rigid electrode  14  facing flexible electrode  12  is selected to govern the rate of change in capacitance of capacitor  10  with deflection of electrode  12 . A variable-area capacitor with substantially equivalent performance can be constructed by adhering dielectric layer  18  to flexible electrode  12 .  
         [0033]    [0033]FIG. 2, is a sectional view of a variable-area capacitor generally shown by reference numeral  20 . For the construction illustrated, flexible electrode  22  comprises a metal layer  24  deposited on a diaphragm  26  of dielectric material. The rigid electrode of capacitor  20  comprises a second metal layer  28  deposited over a rotationally symmetric surface contoured region  30  formed over and in a planar surface  32  of a support member  34 . A cavity  36  connects a central region of contoured region  30  to an opposing surface  38  of support member  34 . Metal layer  28  is additionally formed on a wall  40  of cavity  36  and on surface  38  to provide a location  42  for bonding an electrical lead.  
         [0034]    [0034]FIG. 3 is a sectional view of a differential variable-area capacitance transducer generally shown by reference numeral  50 . Transducer  50  has a flexible electrode  52  that comprises a diaphragm or membrane of low resistivity material such as doped single-crystal silicon. Electrode  52  can include a central integral hub  56 , or it can suspend an affixed inertial mass or other type of payload. Differential transducer  50  is constructed by sandwiching flexible electrode  52  between two dielectric layers  56  and  56 ′ formed on opposing surface contoured regions  58  and  58 ′ of rigid electrodes  60  and  60 ′ respectively. A recessed groove  62  exposes a surface region  64  of flexible electrode  52  for bonding an electrical terminal.  
         [0035]    [0035]FIG. 4A, is a plan view of the electrode structure of a variable-area capacitive transducer generally shown by reference numeral  70  micromachined in relief over a planar surface of a wafer of doped, low resistivity, single-crystal silicon. Electrode structure  70  comprises a stationary electrode member  72  and a movable electrode member  74  shown in FIG. 4 deflected a distance ΔX by an electrostatic force. Stationary electrode member  72  includes a frame  76  with four, curved sidewall regions  78 . Movable electrode member  74  is shown including four, flexible beam segments  80  connected between four anchor posts  82  that remain connected to the bulk silicon substrate and a central crossbeam  84  with an optionally wider section  86 . The width of section  86  is selected by design to established the desired total mass of movable electrode  74  and its corresponding natural frequency of oscillation. FIG. 4B is an exploded plan view  87  of a region of crossbeam  84  that shows an array of apertures  88  formed to facilitate an etching step used to release movable member  74  from the underlying silicon substrate. In FIG. 4A, electrode structure  70  is shown for a capacitor transducer with four, variable-area capacitor elements  89  comprising a portion of curved sidewalls regions  78  and adjacent sidewall regions of flexible beams  80 . It should be apparent by viewing the general design of electrode structure  70  of FIG. 4A that other electrode structures with at least one capacitor element  89  can be constructed. The total number of elements  89  can be selected in part by the desired size of the transducer and the total quiescent capacitance of a group of capacitor elements  89  electrically connected in parallel.  
         [0036]    The profile and sidewalls of electrode members  72  and  74  and apertures  88  are formed by anisotropic, deep reactive ion etching (DRIE) through a lithographically defined etch masking layer. A thin PECVD dielectric layer  90  of silicon dioxide, nitride, or oxynitride is conformally deposited over the top surface and sidewalls of the relieved features of electrode structure  70 , as well as on the floor regions and the bottom of trenches etched in the bulk silicon. A second substantially isotropic DRIE etch is performed to further etchback the silicon floor below the depth of protective dielectric layer  90  on the sidewalls of the relieved features. This etch undercuts the silicon beneath movable electrode member  74 , freeing it from the silicon substrate. The undercutting of wide sections is facilitated by apertures  88 . The dielectric layer  90  is then selectively removed from the surface and sidewalls of electrode structure  70  except at or near curved sidewalls regions  78  and adjacent walls of flexible beams  80 . Next, a metal layer such as aluminum, or gold over chrome, is deposited to form conducting surface regions on electrode members  72  and  74  and contact sites  92  and  94  to bonded terminals  96  and  98  respectively. The metallization is restricted from regions near capacitor elements  89  and is etched from floor regions  100  and the bottom of trenches to electrically isolate electrode members  72  and  74  Terminal  96  is connected to a capacitance detection circuit and a feedback control voltage and terminal  98  shown is grounded. Alternately, the terminals can be rearranged and terminal  98  connected to ground. The feedback control voltage can include a bias or voltage offset to provide a quiescent electrostatic force to move electrode member  72  a distance ΔX. The magnitude of this quiescent force determines the length l of the mutual area of surface contact between sidewall regions  78  and beams  80 . The majority of the quiescent capacitance of capacitor elements  89  is proportional to length I and the permittivity of dielectric layer  90 .  
         [0037]    More specific details of DRIE silicon micromachining and the associated process steps used to fabricate transducers with parallel-plate electrode structures are taught by Galvin, et al, and described by Elwenspoek, et al.,  Silicon Micromachining,  1998, both references are incorporated herein in their entirety. Capacitive transducers with variable-area electrodes having the general shape and profile of electrode structure  70  of FIG. 4 also can be surfaced micromachined in polycrystalline silicon by sacrificial layer etching. The accelerometer of Sherman is manufactured by this method. The steps of this micromachining method are reviewed by Elwenspoek, et al.  
         [0038]    Performance of Variable-Area Sensors and Actuators  
         [0039]    The sensitivity and dynamic range of variable-area capacitive transducers with, or without, force-balanced feedback is at least an order of magnitude higher than comparable sized transducers with parallel-plate capacitors. FIGS. 5, 6, and  7  are plots of the response of three different transducers having the general structure shown in FIG. 2. The surface contoured region of each transducer had a radius of 6.2 mm and a maximum depth of 72 microns. The overall shape of the contour is generally defined by the coordinates shown in the Table of FIG. 8A for curve  54  of FIG. 8B of U.S. Pat. No. 6,151,967. The measurement data plotted in FIGS. 5, 6, and  7  were acquired from mesoscale size transducers, but comparable response curves are realizable for micromachined transducers with electrodes having 50 micron and smaller dimensions and thinner dielectric layers.  
         [0040]    [0040]FIG. 5, is a capacitive-pressure response curve of a pressure transducer with a 0.483-mm thick silicon diaphragm anodically bonded to a support member of Corning 7740 glass. FIG. 6, is a response curve of a pressure transducer with a 2-micron thick polycarbonate diaphragm, with a vacuum deposited aluminum film, that was thermally bonded to a polycarbonate support member with a single-point diamond machined surface counter. The less than linear response of the polycarbonate pressure sensor is a result of the generatrix of the loaded diaphragm being primarily determined by tensile stress. A different surface contour is required to obtain a more linear response when the sensing electrode is a membrane. The full-scale change in capacitance shown in FIGS. 5 and 6 exceed those of conventional capacitive sensors by a factor of 20 to 50. Capacitance changes of several thousand percent were obtained when the diaphragms were fully deflected.  
         [0041]    [0041]FIG. 7, is a plot of capacitance as a function of voltage applied to the electrodes of a variable-area capacitance with a metallized polycarbonate diaphragm. The large capacitance change and electrode deflection of the polycarbonate transducer at low voltages demonstrates the ability to force-balance external loads over a wide dynamic range compared to prior-art transducers.  
       PREFERRED EMBODIMENTS  
       [0042]    [0042]FIG. 8, a simplified schematic block diagram of a preferred method of the present invention. While this invention is directed to a method to electrostatically force-balance variable-area capacitance transducers with a flexible electrode, it also provides an improved method to force-balance gap sensing capacitive transducers with substantially parallel-plate electrodes. FIG. 8 shows a variable-area capacitance transducer  100  with a quiescent capacitance value C T  connected in a circuit arrangement that includes a capacitance detection circuit  102  and a high-gain amplifying means  104 . Transducer  100  is shown schematically as having a flexible electrode  106  and a curved rigid electrode  108  connected to ground or another reference potential. The circuit arrangement of FIG. 8 includes two negative feedback loops. A first loop internal to capacitance detection circuit  102  feeds back current to actively null capacitance bridge network  116 . An output voltage V C  of circuit  102  is proportional to an incremental change between the quiescent capacitance C T  of transducer  100  and a reference capacitor C R . In the second feedback loop, voltage V C  is amplified at high gain by amplifying means  104  to provide a feedback voltage V 0  to electrostatically force-balance flexible electrode  106 , thereby maintaining it at a substantially fixed position of deflection. Flexible electrode  106  is connected to a common node  110  connected to a first side of a coupling capacitor C C . It is preferable that the capacitance of coupling capacitor C C  be at least 10 times larger than the quiescent capacitance C T  of transducer  100 . A second side of capacitor C C  is connected to an internal node  112  connected to a first-side node  114  of actively nulled capacitor bridge network  116 . A second-side node  118  of bridge network  116  is connected to reference capacitor C R  returned to a ground or to said reference potential. A positive voltage V +  is connected to a first terminal  120  of bridge network  116  and a second terminal  122  of bridge network  116  is connected to ground or to said reference potential. Alternately, the second-side node  118  of bridge network  116  can be directly connected (without reference capacitor C R ) to a reference voltage to form a half-bridge circuit arrangement with reduced measurement precision. Node  112  is also connected to internal node  124  connected to a first input of a differential integrating means  126 . The second-side node  118  of bridge network  116  is connected to a second input of opposing polarity of differential integrating means  126 . An output of integrating means  126  is connected to internal node  128  connected to an input of a voltage-controlled current sourcing means  130 . An output of current sourcing means  130  is connected back to first-side node  114  of bridge network  116  via connections between nodes  124  and  112  to provide a feedback current to actively null a difference between the running averages of periodic voltages at first- and second-side nodes  114  and  118  of bridge network  116  respectively.  
         [0043]    U.S. patent application Ser. No. 09/482,119, No. 09/794,198, and No. 09/816,551 teach the benefits and detailed operation of alternate embodiments of capacitor detection circuit  102 . Preferably, differential integrating means includes a low-pass filter at each input to an operational amplifier with only a small-value feedback capacitor to maintain stability of the first feedback loop near the high frequency limits of the amplifier. This provides a capacitance detection circuit  102  with very high DC stability. Bridge network  116  includes a pulse generator that provides an excitation voltage waveform to periodically charge reference capacitor C R  and capacitor C T  through capacitor C C . Accordingly, capacitors C R  and C T  are charged to a voltage V +  during short periods of time T 1  and then discharged toward ground (or said reference potential) during longer periods of time T 2 . An incremental change in capacitance between capacitor C T  of transducer  100  and reference capacitor C R  causes an error to develop between running averages of periodic voltage waveforms at nodes  114  and  118 . This differential error signal is low-pass filtered and amplified by differential integrating means  126  to provide a voltage V C  to control feedback current sourced or sinked from current sourcing means  130  to null bridge network  116  to high accuracy over a wide deflection range of flexible electrode  106 . Voltage-controlled current sourcing means  130  can be a resistor, or depending upon the polarity of the input connections to differential integrating means  126 , it can be an inverting voltage-controlled current source or a non-inverting voltage-controlled current conveyer. Voltage V C , that is linearly proportion to a change in capacitor C T , is amplified at high gain by amplifying means  104  to provide a feedback voltage V O  to electrostatically force balance transducer  100  via said second feedback loop.  
         [0044]    An input to amplifying means  104  is connected to node  128  and an output is connected to common node  132  connected to an output terminal  134 . A conducting lead  136  is connected between node  132  and a resistor R connected to node  110 . Resistor R, or an amplifying means with an equivalent output impedance, is required to minimize loading of the periodic voltage at node  114  by a low impedance amplifier output. Amplifying means  104  can be a high gain amplifier or an integrator with a small feedback capacitor to provide high gain at low frequencies including DC. The quiescent capacitance C T  and deflection position or generatrix of flexible electrode  106  can be adjusted by applying a bias voltage to current sourcing means  130 , amplifying means  104 , or bridge network  116  to offset voltage V C , as taught in patent application Ser. No. 09/794,198 and No. 09/816,551.  
         [0045]    The electrostatic force F E  between the plates of a capacitor varies as ΔV 2 , where V is an applied voltage. Accordingly, output voltage V of amplifying means  104  is proportional to the square root of the force required to force-balance transducer  100  when flexible electrode  106  deflects in response to a physical effect. The exact analytical determination of the relationship between F E  and V F  for large deflections of electrode  106  depends in part upon the curvature of rigid electrode  10 , the geometry and material properties of electrode  106 , and the properties of an intermediate fixed dielectric layer, not shown. Rakesh, et al., “Extension of the Boundary Element Method to Systems with Conductors and Piece-Wise Constant Dielectrics,”  J. Microelectromech Syst . vol. 5. September 1996, calculates the response of an electrostatically deflected actuator having the general configuration of the transducer of FIG. 1. Rakesh, et al., elected to call this device a “variable gap actuator” (VGA), a name commonly reserved for capacitive transducers with parallel-plate electrodes. The name, “variable-area capacitor” or “varying area condenser”, first used by Carter, et al, more appropriately describes the fundamental electromechanical behavior of this transducer. Wang, et al., “Computation of Static Shapes and Voltages for Micromachined Deformable Mirrors with Nonlinear Electrostatic Actuators,” also in  J. Microelectromech Syst . vol. 5, September 1996, provides a general analytical approach to calculating the response of actuators with parallel-plate electrodes. This analytical approach can be generally used to calculate the response of variable-area capacitors with much less severe nonlinearities because of substantially constant electrode spacing with electrode deflection. Neither article suggests measuring the capacitance between the electrodes of an electrostatically positioned actuator to provide a feedback voltage for closed-loop, electrostatic control of a movable or deformable electrode.  
         [0046]    The method of the present invention can be practiced using prior art capacitive detection circuits, amplifying means, and gap varying capacitors with substantially parallel-plate electrodes before deflection of a compliant electrode in response to a force or pressure.  
         [0047]    [0047]FIG. 9 is a schematic block diagram of a preferred method of the present invention to electrostatically force-balance a differential capacitive transducer  150 . Transducer  150  comprises two differential capacitor elements C 1  and C 2  with stationary electrodes  152  and  154  respectively located on opposing sides of a common flexible or movable cooperating electrode  156  connected to ground or another reference potential. As electrode  156  moves in response to an applied force or stress, the capacitance of one capacitor element (C 1  or C 2 ) will increase and the capacitance of the second capacitor element (C 2  or C 1 ) will decrease substantially by the same amount. Stationary electrode  152  is connected to node  158  that is connected to a first side of a first coupling capacitor C C1  and stationary electrode  154  is connected to node  160  that is connected to a first side of a second coupling capacitor C C2 . A second side of capacitors C C1  and C C2  is connected to terminals  114  and  118  of a capacitance detection circuit  102  respectively. Referring to FIG. 8, it can be seen that reference capacitor C R  connected to capacitive detection circuit  102  is replaced in the circuit arrangement of FIG. 9 by capacitor element C 2  electrically connected in series with second coupling capacitor C C2 . Coupling capacitor C C2  allows the value of capacitor element C 2  to be differentially sensed with respect to capacitor element C 1 . Flexible electrode  156  is maintained at a neutral position at which the net physical and electrostatic forces acting on electrode  156  are substantially zero. Since the electrostatic force F E  on the electrodes of a capacitor with a voltage V is proportional to V 2 , a force of attraction will be created by a voltage of either polarity. Accordingly, to maintain flexible electrode  156  at a fixed position of deflection or displacement, it is necessary to use a feedback steering means  162  to apply feedback voltage V to the appropriate electrode  152  or  154  to provide a downward acting electrostatic force F D  or an upward acting electrostatic force F U  as required to maintain flexible electrode  156  at a fixed position. A output of circuit  102  is connected to a common node  164  connected to an input of a high-gain amplifying means  104  with an output connected to common node  166  connected to output terminal  168 . Node  164  is also connected to a common node  170  connected to output terminal  172  and to an input terminal  174  of a voltage comparator  176  of feedback steering means  162 . Feedback steering means  162  is shown representatively as comprising comparator  176  that controls the position of two, CMOS single-pole double-throw switches (SPDT)  178  and  180  as a function of the polarity of voltage V C  applied to terminal  174  of comparator  176 . A first pole terminal  182  of switch  178  is connected to a resistor R connected node  166  and a second pole terminal  184  of switch  180  is connected to ground or said reference potential. Resistor R can be eliminated if amplifying means  104  has an equivalent driving point impedance. A first switch terminal  186  connects a first pair of an open and a closed switch contacts to node  158  and a second switch terminal  188  connects a second pair of a closed and an open pair of switch contacts to node  160 . When no external forces or stresses are applied to electrode  156 , it remains in a quiescent or neutral position of deflection or displacement, and voltage V C  is substantially zero, whereby the output voltage V of amplifying means  104  is also zero and substantially zero feedback voltage is applied to stationary electrodes  152  or  154 . Comparator  176  can incorporate positive feedback and different high and low voltage thresholds to provide hysteresis in the switching response, as in the case of a Schmitt trigger circuit. Because the product of the closed-loop gain of capacitive detection circuit  102  and the gain of amplifying means  104  is very high and also because an electrostatic force F E  is dependent upon V 2 , a small value of hysteresis, several times higher than the peak-to-peak noise of voltage V, will not significantly effect the resolution of the displacement, position, or flexure control of electrode  156 .  
         [0048]    The embodiment of the method of FIG. 9 can also be performed by other types of capacitance detection circuits, amplifying means, and feedback steering means for differential capacitive transducers with multiple variable-area or variable-gap electrodes. Transducer  150  of FIG. 9 is shown constructed with two capacitor elements with three electrodes. Many prior art differential parallel-plate capacitor transducers are constructed with independent differential capacitor elements, each with two electrodes, e.g., U.S. Pat. No. 5,540,095 of Sherman. Prior art transducers frequently used a separate pair of electrodes for force rebalancing, e.g., Cadwell, U.S. Pat. 4,584,885. Because the differential embodiment of the present invention detects capacitive changes with signals coupled through capacitors C C1  and C C2 , a separate set of force rebalancing electrodes is not required and a common electrode to each capacitor element can be grounded, thereby only three electrodes are required for transducer  150 .  
         [0049]    The performance of certain prior art capacitive transducers that use a separate pair of force rebalancing electrodes can be improved by connecting the sensing and force capacitors in parallel. The larger parallel capacitor can more accurately sense a capacitance change and provide larger electrostatic force. Since one electrode of differential capacitor elements combined in parallel can be grounded, only three connections are required for prior art transducers formerly requiring up to eight connections.  
         [0050]    The specific details of the embodiments described above are not intended to limit the scope of the appended claims and their legal equivalents.