CAPACITIVE SENSOR

Abstract

A dynamic capacitive sensor configuration is disclosed which imposes minimal force and resistance to motion on the moving electrode. Moving electrodes avoid adverse effects of large bias voltages such as pull-in instability, despite arbitrary levels of compliance. This configuration facilitates incorporation of highly compliant and thin electrode materials that present the least possible resistance to motion. This type of material is particularly useful for sensing sound. A large bias voltage can be applied without influencing its motion, e.g., 400 V. The electrical sensitivity to sound is high, e.g., approximately 0.5 volts/pascal, two orders of magnitude greater than typical acoustic sensors.

Claims

1. A capacitive sensor, comprising: at least two conductors, isolated from each other by at least one spatial gap, each respective conductor interacting with an electrostatic field occupying a region proximate to the at least two conductors and the at least one spatial gap, being electrically responsive to a perturbation of the electrostatic field; and a displaceable element configured to move along an axis of displacement having a directional component crossing the spatial gap selectively responsive to a sensed condition, and perturbing the electrostatic field corresponding to the movement, wherein over a range of the movement of the displaceable element, the electrostatic field does not substantially alter a responsivity of the displaceable element to the sensed condition or cause pull-in instability.

2. The capacitive sensor according to claim 1, wherein: the at least two conductors comprise a pair of fixed conductors, separated by a linear spatial gap, each of the pair of fixed conductors being maintained at a respective electric potential, to generate the electrostatic field in a space above the pair of fixed conductors having a major field vector component directed across the linear spatial gap dependent on a difference between the respective electric potentials of the pair of fixed conductors, and the displaceable element comprises a charged element configured with the axis of displacement having a vector component directed across the linear spatial gap, such that a force imposed on the displaceable element due to the electrostatic field is insensitive to a state of displacement of the displaceable element in response to the sensed condition.

3. The capacitive sensor according to claim 1, wherein the displaceable element comprises a cantilever supported diaphragm.

4-12. (canceled)

13. The capacitive sensor according to claim 1, wherein the displaceable element comprises at least one of a perforated diaphragm, a fiber mesh, a fiber mat, metallized electrospun fiber, carbon nanotubes, and graphene.

14-50. (canceled)

51. The capacitive sensor according to claim 1, wherein a potential at each of the conductive surfaces is maintained at ground predefined potential by a respective transimpedance amplifier while a change in charge is induced on the respective conductive surfaces by a movement of the displaceable element.

52-55. (canceled)

56. The capacitive sensor according to claim 1, wherein the capacitive sensor comprises a microphone, and the displaceable element has an amplitude of movement corresponding to an acoustic wave.

57-64. (canceled)

65. A method of sensing a vibration, comprising: providing at least two separated conductive surfaces, and a deflectable element, having an axis of deflection perpendicular to a force generated by the at least two separated conductive surfaces on the deflectable element; inducing a voltage potential on the deflectable element with respect to the at least two conductive surfaces; and sensing a change in induced charge on the at least two conductive surfaces resulting from deflection of the deflectable element along the axis of deflection, wherein the force generated by the at least two separated conductive surfaces on the deflectable element does not substantially alter a deflection of the deflectable element.

66-132. (canceled)

133. A sensor comprising: two conductors, each conductor of the two conductors having a planar surface; a displaceable electrode configured for movement relative to the two conductors in response to an external condition to be sensed; and a voltage source configured to supply a bias voltage, the bias voltage establishing an electrostatic field between the displaceable electrode and the planar surfaces of the two conductors; wherein: the displaceable electrode has a thickness in a direction along the movement and a length in a direction orthogonal to the movement; the length and the thickness are chosen such that the displaceable electrode is compliant; and the two conductors and the displaceable electrode are oriented such that the length of the displaceable electrode is orthogonal to the planar surfaces of the two electrodes.

134. The sensor of claim 133, wherein the displaceable electrode is positioned and oriented relative to the two conductors such that an electrostatic force arising from the electrostatic field acts as a restoring force to return the displaceable electrode to an equilibrium position.

135. The sensor of claim 133, further comprising a further conductor disposed between the two conductors.

136. The sensor of claim 135, wherein: the two conductors are biased via the voltage source at different voltages relative to the displaceable electrode; and the further conductor is biased via the voltage source at an intermediate potential.

137. The sensor of claim 135, wherein: the two conductors are biased via the voltage source such that the net electrostatic force is an attractive force on the displaceable electrode; and the further conductor is biased to establish a repulsive force on the displaceable electrode.

138. The sensor of claim 133, wherein the displaceable electrode is compliantly supported by a hinge.

139. The sensor of claim 133, wherein: the displaceable electrode comprises a planar diaphragm; and the planar diaphragm comprises a plurality of apertures such that the external condition to be sensed comprises viscous drag of air flowing through the plurality of apertures.

140. The sensor of claim 133, wherein the planar surfaces of the two conductors are coplanar.

141. The sensor of claim 133, wherein: the two conductors are separated by a gap; and the displaceable electrode has an equilibrium position disposed in the gap.

142. The sensor of claim 133, wherein the length of the displaceable electrode is orthogonal to the planar surfaces of the two conductors in an equilibrium position of the displaceable electrode.

143. The sensor of claim 133, wherein the bias voltage is applied to the two conductors.

144. The sensor of claim 133, further comprising an amplifier configured to produce a signal corresponding with the movement.

145. The sensor of claim 133, wherein the displaceable electrode is configured for movement relative to the two conductors selectively responding and corresponding to vibrations of an acoustic wave.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0132] FIGS. 1A-1D show conventional capacitive sensing schemes.

[0133] FIG. 2 shows a compliant electrostatic sensor.

[0134] FIG. 3 shows a photograph of the physical setup according to FIG. 2.

[0135] FIGS. 4A-4C show estimated first and second derivatives of the potential energy and charge sensitivity as a function of the tip displacement of the moving electrode for the electrode configuration of FIG. 2.

[0136] FIG. 5 shows a schematic representation for the characterization setup.

[0137] FIGS. 6A-6D shows measured results for the electrode configuration of FIG. 2.

[0138] FIGS. 7A-7B show measured results versus frequency for the electrode configuration of FIG. 2.

[0139] FIG. 8A shows a perforated plate diaphragm embodiment of the invention.

[0140] FIG. 8B shows a fiber mesh moving element embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0141] FIG. 2 shows a compliant electrostatic sensor according to the present technology.

[0142] The moving electrode according to a preferred embodiment examined herein is composed of a thin sheet of material that is extremely flexible in the direction normal to its plane surface. It is supported along one edge so that it can rotate or bend easily about that supporting line as illustrated in FIG. 2. The figure represents a two-dimensional cross section of the system which is unchanged throughout the dimension that is perpendicular to this section. The moving electrode consists of a thin flexible element of length L.sub.2 and thickness H.sub.2 shown deflected relative to the horizontal orientation by the angle α. The moving electrode is shown as a straight solid body that pivots about its attachment point. It could also consist of a flexible beam or string hung at one end having small enough bending stiffness that it is free to rotate in a manner similar to that shown in FIG. 2.

[0143] The fixed electrodes are oriented so that they create plane surfaces that are orthogonal to the plane of the moving electrode. Orienting the fixed and moving electrodes so that their surfaces are orthogonal helps to minimize the net electrostatic force on the moving electrode because the electric field will always be normal to the surface of the conductors. With proper arrangement of the positions and orientations of these electrodes, one can cause the electrostatic forces acting on the plane surfaces of the moving electrode to effectively cancel, leaving the comparatively small force that is applied normal to the free edge.

[0144] According to an exemplary embodiment, the moving electrode consists of a thin flexible element of length L.sub.2=6.2 mm and thickness H.sub.2=5 μm shown deflected relative to the horizontal orientation by the angle α. Two vertical, fixed electrodes of length L.sub.1=L.sub.3=2.5 mm and thickness H.sub.1=H.sub.3=200 μm are shown to the right of the moving electrode. The horizontal gap between the moving electrode and the two fixed electrodes is g=300 μm. They are separated by a gap in the vertical direction of g.sub.y=50 μm and are held at the same potential.

[0145] Sensing the position of the moving electrode is achieved by dividing the fixed electrode into two surfaces, denoted by electrodes 1 and 3, which are both held at the same voltage. The charge on these two fixed electrodes will vary with the moving electrode's position. For electrodes having practical dimensions, it is found that the electrostatic forces can be made to be negligible in comparison to those associated with the electrode's elastic properties.

[0146] The two vertical, fixed electrodes of length L.sub.1 and L.sub.3 and thicknesses H.sub.1 and H.sub.3 are shown to the right of the moving electrode in FIG. 2. As the moving electrode rotates, it changes the charge distribution and net capacitance on this set of electrodes. The voltage applied to the moving electrode will be set by a voltage source, V.sub.2. The two fixed electrodes will be set at a voltage of V=0. In this example, to simplify our calculations we will assume that the moving electrode maintains the shape of a straight line so that it moves as a rigid pendulum that is free to pivot about its attachment point Small deviations from this straight line shape due to bending will not significantly change the results. The length L.sub.2=6.2 mm and thickness H.sub.2=5 μm of the moving electrode ensure that the electrode is highly compliant Their values were chosen to correspond to the measured configuration described below.

[0147] A photograph of the fabricated device is shown in FIG. 3. The three electrodes are supported on insulating blocks that were attached to micromanipulators that allowed electrode 2 to be positioned close to the line separating electrodes 1 and 3. The distances between the electrodes and the overall dimensions were determined optically using a microscope.

[0148] To very roughly estimate the mechanical stiffness of a realizable moving electrode, the stiffness of a cantilever beam supported by a fixed boundary may be considered. Assume that this electrode is constructed of a polymer having a Young's modulus of elasticity of E=2×10.sup.9 N/m.sup.2. In order for the electrode to be conductive, it is coated with a very thin layer of aluminum, which is taken to be thin enough to not add appreciable stiffness. Considering the force to be applied uniformly along the length, the equivalent mechanical stiffness per unit width may be approximated by k≈8EI/L.sup.3/2 where I=H.sup.3/2/12. Since L.sub.2=6.2×10.sup.−3 m and H.sub.2=5×10.sup.−6 m, the mechanical stiffness per unit width is k≈0.7 N/m.sup.2. While this is a very approximate estimate, the results below indicate that the effective stiffness due to electrostatic forces is significantly less than this mechanical stiffness, and will thus have negligible influence on the motion.

[0149] FIG. 3 shows a photograph of the physical setup used to realize the concept shown in FIG. 2. Electrodes 1 and 3 are formed using thin strips of copper tape aligned vertically near the left, free end of electrode 2. Electrode 2 is clamped at its right end. The sound field was incident as shown from a direction normal to the plane of electrode 2.

[0150] While the electrode configuration of FIG. 2 does not lend itself to analysis by simple design equations as do parallel plate capacitive sensors, it is possible to estimate the charge distribution numerically for a given set of applied electrode voltages. Here a boundary element approach is utilized, which provides a numerical solution to the following integral equation:

[00010]$\begin{array}{cc}v\left(\stackrel{.fwdarw.}{r}\right)=\int \frac{\rho \left(\stackrel{.fwdarw.}{r}\right)}{\mathrm{ϵ4\pi }.Math..Math.R\left(\stackrel{.fwdarw.}{r},\stackrel{.fwdarw.}{r}.Math.\prime \right)}.Math.{\mathrm{ds}}^{\prime }.Math..Math.\mathrm{where}& \left(\mathrm{I0}\right)\\ R\left(\stackrel{.fwdarw.}{r},{\stackrel{.fwdarw.}{r}}^{\prime }\right)=\sqrt{\left(\stackrel{.fwdarw.}{r}-{\stackrel{.fwdarw.}{r}}^{\prime }\right).Math.\left(\stackrel{.fwdarw.}{r}-{\stackrel{.fwdarw.}{r}}^{\prime }\right)}& \left(\mathrm{II}\right)\end{array}$

[0151] is the distance between {right arrow over (r)} and {right arrow over (r)}′, which are any two points on the surface of the electrodes, v({right arrow over (r)}) is the given surface voltage specified at each position {right arrow over (r)}, and ρ({right arrow over (r)}) is the unknown surface charge density. ϵ=8.854 pF/m is the permittivity of the medium. When the domain is two dimensional, equation (10) becomes [16]

[00011]$\begin{array}{cc}v\left(\stackrel{.fwdarw.}{r}\right)=-\int \frac{\rho \left(\stackrel{.fwdarw.}{r}\right).Math.\log .Math..Math.R\left(\stackrel{.fwdarw.}{r},\stackrel{.fwdarw.}{r}.Math.\prime \right)}{\mathrm{ϵ2\pi }}.Math.{\mathrm{ds}}^{\prime }& \left(\mathrm{I2}\right)\end{array}$

[0152] Discretizing the surface into a finite number of areas enables one to solve for the charge distribution for any given electrode geometry. Knowing the charge density for a variety of positions of the moving electrode enables the calculation of the electrostatic potential energy as a function of the electrode position. These data may then be numerically differentiated to estimate the first and second derivatives which provide the electrostatic force and effective stiffness associated with the given motion.

[0153] FIGS. 4A-4C show the estimated first and second derivatives of the potential energy as a function of the tip displacement of the moving electrode for the electrode configuration of FIG. 2. The effective electrostatic force is proportional to the first derivative of the potential energy shown in FIG. 4A and the electrostatic stiffness is proportional to the negative of the second derivative shown in FIG. 4B. The estimated force is always attractive and stiffening since it always acts to return the electrode to the equilibrium position at x=0. It is assumed that the bias voltage applied to the moving electrode is V=400 volts and the two fixed electrodes are at zero volts. The maximum magnitude of the charge sensitivity shown in FIG. 4C is approximately 4×10.sup.−8 coulombs/meter.

[0154] Because the domain is taken to be two dimensional, the results are for a unit length in the direction normal to the plane of FIG. 2. FIG. 2 shows that the force is always attractive, restoring the electrode to its single equilibrium position at x=0. The second derivative of the electrostatic energy provides the effective electrostatic stiffness at the equilibrium position. This electrostatic stiffness can be compared to an estimate of the mechanical stiffness of a cantilever beam as discussed above. The electrostatic stiffness is found to be approximately 0.4 N/m.sup.2. The mechanical stiffness is estimated above to be k≈0.7 N/m.sup.2. Note that this electrostatic stiffness is estimated with the bias applied to the moving electrode having a large value of 400 volts. This bias voltage is expected to have little noticeable effect on the electrode motion. This voltage is beyond what would be practical in a miniature microphone design. Precision microphones, however, (such as the Bruel and Kjaer 4138 used below) commonly employ a 200-volt bias. The use of this rather extreme voltage provides evidence that this electrode design is not adversely impacted by bias voltages likely to be used in practice.

[0155] The total charge Q.sub.i, for i=1, 2, 3, on each of the three electrodes can be computed knowing the charge density ρ on all surfaces, S.sub.i,

Q.sub.i=∫.sub.S.sub.iρ({right arrow over (r)})ds′  (13)

[0156] The output of the sensor will be taken to be the difference in charge between electrodes 1 and 3.

[0157] FIG. 4C shows the predicted charge sensitivity of the device which is computed knowing the difference Q.sub.1-Q.sub.3 for a range of displacements of the moving electrode 2.

[0158] The derivative of this charge difference with respect to the displacement of the tip of electrode 2 is then computed, giving the sensitivity in coulombs/meter as shown in FIG. 4C.

[0159] The overall sensor sensitivity can be expressed as a combination of the charge sensitivity, denoted by S.sub.Q in coulombs/meter, the electrical sensitivity, S.sub.e in volts/coulomb, and the mechanical sensitivity S.sub.m in meters/pascal. The over-all sensitivity will then be:

S=S.sub.Q×S.sub.e×S.sub.m volt/pascal  (14)

[0160] In the experimental results presented below, a transimpedance, or charge amplifier is used to obtain an electronic output. This is accomplished using a general purpose operational amplifier where the gain is set primarily through the effective feedback capacitance C.sub.f. The electrical sensitivity may then be approximated by:

[00012]$\begin{array}{cc}{S}_e\approx \frac{1}{C_f}.Math..Math.\mathrm{volt}.Math.\text{/}.Math.\mathrm{coulomb}& \left(\mathrm{I5}\right)\end{array}$

[0161] The mechanical sensitivity, S.sub.m will, of course, depend on the mechanical properties of the moving electrode 2. As a rough approximation, in a hypothetical ‘ideal’ sensor, the average motion of electrode 2 is sought to be very nearly the same as that of the air in a sound field, such as what has been demonstrated in [3]. Taking the sound field to consist of a plane wave traveling in one direction, the acoustic particle velocity is given by U=P/(ρ.sub.0c), where ρ.sub.0 is the nominal air density and c is the speed of propagation of an acoustic wave. The quantity ρ.sub.0c is the characteristic acoustic impedance of the medium [17]. Because the electrode is assumed to rotate about its fixed end, the free end, where the sensing occurs, will move with approximately twice the average displacement, which will occur at the center. For a harmonic wave at the frequency ω, the mechanical sensitivity of the displacement of the free end of the electrode in this idealized case can then be approximated by:

[00013]$\begin{array}{cc}{S}_m\approx \frac{2}{{\mathrm{\omega \rho }}_0.Math.c}.Math..Math.\mathrm{meter}.Math.\text{/}.Math.\mathrm{pascal}& \left(\mathrm{I6}\right)\end{array}$

[0162] To obtain experimental results for the electrode system of FIG. 2, a 5 μm thick polyethylene terephthalate film, metallized with a thin layer of aluminum was used to create electrode 2 (Goodfellow.com part No. ES301855). The fixed electrodes 1 and 3 were constructed using two strips of 2.5 mm wide copper tape. The assembly was supported on micromanipulators to enable adjustment of the nominal position of the moving electrode relative to the fixed electrodes. The moving electrode was driven acoustically by a loudspeaker placed roughly 1 meter away. Measurements were performed in an anechoic chamber. FIG. 5 shows a schematic representation for the characterization setup.

[0163] An electronic output was obtained through the use of transimpedance circuits connected to electrodes 1 and 3. These circuits were not optimized for performance and used a general purpose TL074 quad operational amplifier using 1 GΩ feedback resistors. Note that the impedance of such high-value resistors is often significantly influenced by parasitic capacitance, typically on the order of C.sub.f=1 pF, in parallel with the resistor, which can dominate the impedance over a wide range of frequencies. The circuit was realized using through-hole components on a prototype circuit board, which can also influence the parasitic capacitance.

[0164] FIG. 5 shows a schematic representation for the characterization setup. The electrode motion was detected using a laser vibrometer. The sound field created by a loudspeaker was measured using a Bruel and Kjaer 4138 reference microphone. The electronic output was measured using charge/transimpedance amplifiers. All signals were recorded using a National Instruments PXI-1033 Data Acquisition System.

[0165] The velocity of the moving electrode was also measured using a Polytec laser vibrometer consisting of a Polytec OFV-534 compact sensor head and a Polytec OFV-5000 Vibrometer Controller. The sound pressure near the moving electrode was measured using a Bruel and Kjaer 4138 precision microphone having a 1/8-inch diameter pressure sensing diaphragm. A bias voltage of V.sub.2=400 volts was applied to electrode 2 using a M5-1000 DC-DC converter from American Power Designs.

[0166] FIGS. 6A-6D show the measured results for the electrode configuration of FIG. 2. The bias voltage applied to the moving electrode is V=400 volts and the two fixed electrodes are at zero volts. The moving electrode was driven by a sound field consisting of a 250 Hz tone produced by a loudspeaker having an amplitude of approximately 1 pascal. FIG. 6A shows the measured sound pressure (pascals) at the location of the moving electrode as measured by a Bruel and Kjaer 4138 reference microphone. FIG. 6B shows the measured velocity (mm/s) halfway between the clamped and free ends of the moving electrode as obtained using a laser vibrometer. The velocity is nearly proportional to, and in phase with the pressure as occurs when the sound field propagates as a plane wave. This design may be employed using the beam response above its first resonant frequency. Because the beam is then highly-compliant i.e., mass-dominated, as opposed to stiffness-dominated, the beam velocity is expected to be in phase with the acoustic velocity.

[0167] FIGS. 6C and 6D show the output voltages produced by the detection circuits, employing simple transimpedance amplifiers, that respond to the charge on electrodes 1 and 3. These signals are seen to be roughly out of phase with each other as would be expected given that, when the moving electrode moves toward one of the fixed electrodes, it moves away from the other. One could then subtract the two outputs to obtain an improved detection with increased sensitivity. The output voltages have an amplitude of approximately 250 mV so that the difference output would have a sensitivity of approximately 0.5 volts/pascal. The DC bias voltage applied to electrode 2 is 400 volts for the data shown in FIGS. 6A-6D.

[0168] The displacement corresponding to the velocity shown in FIGS. 6A-6D is approximately 2.5 microns for a sound pressure of approximately 1 pascal. Note that this displacement is measured at a point halfway between the clamped and free ends of the electrode. We can then estimate that the displacement of the free end will be approximately 5 microns for a 1 pascal sound field. The effective mechanical sensitivity may then be taken to be S.sub.m≈5×10.sup.−6 meter/pascal. This measured result can be compared to the rough estimate provided in equation (16) where ρ.sub.c≈415 pascal-second/meter and ω=2π250. Equation (16) then gives S.sub.m≈3×10.sup.−6 meter/pascal, in reasonable agreement with the measured result.

[0169] The estimated charge sensitivity shown in FIG. 4C is about S.sub.Q≈40 nanoCoulombs/meter. The electrical sensitivity given in equation (15) depends on the effective capacitance, C.sub.f, which as mentioned above, is estimated to be C.sub.f≈1 pF. The terms in equation (14) are evaluated:

S=S.sub.Q×S.sub.e×S.sub.m≈(40×10.sup.−9)×10.sup.12×3×10.sup.−6≈0.12 volt/pascal  (17)

[0170] The measured electrical output can be taken to be the difference in the signals shown in FIGS. 6C and 6D would give a measured signal having a peak voltage of about 0.5 volts for a 1 pascal sound field, which is greater than but within reasonable proximity to the rough approximation of 0.12 volts.

[0171] The data in FIGS. 6A-6D show that the sensor is capable of producing a sizable electronic output due to acoustic excitation. This is due to the use of a generous bias voltage applied to the moving electrode of 400 volts. The use of such a large bias voltage on a highly compliant electrode could normally be expected to have a marked influence on its motion.

[0172] FIGS. 7A and 7B show measured results versus frequency for the electrode configuration of FIG. 2. These results show that the bias voltage has negligible effect on the motion of the electrode while the electrical sensitivity is roughly proportional to the bias voltage over a wide range of frequencies per FIG. 7A. The measured electrode displacement amplitude as a function of frequency is independent of bias voltage for bias voltages of zero, 200 volts, and 400 volts. Also shown is the predicted air displacement amplitude for a 1 pascal plane sound wave. This shows that the electrode moves at least as much as the air in a plane wave, per FIG. 7B. The electrical sensitivity is taken to be the difference in output voltages obtained from electrodes 1 and 3 relative to the amplitude of the sound pressure at the moving electrode. This shows that the sensitivity roughly doubles for a doubling of the bias voltage, as expected.

[0173] In spite of the use of a rather large bias voltage, the results shown in FIGS. 7A-7B indicate that the electric field does not result in stiffening (or softening) of the motion of the moving electrode. The figure shows the measured electrode displacement amplitude as a function of frequency (FIG. 7A) along with the measured electrical sensitivity, defined as the difference in the output voltages acquired from electrodes 1 and 3 (shown in FIG. 2) relative to the incident sound pressure (FIG. 7B). Results are shown for bias voltages of zero, 200 volts, and 400 volts. While the response as a function of frequency is not ideal (i.e. not flat) owing to the effects of sound reflections from the fixture and from resonances of the moving electrode, it is clear that the measured electrode velocity is essentially unaffected by significant changes in the bias voltage. This indicates that the electrostatic force is negligible relative to other mechanical forces acting on the electrode, as expected from the data of FIGS. 4A-4C.

[0174] FIGS. 7A-7B also show the predicted amplitude of the fluctuating air displacement in a plane wave sound field. This indicates that the measured electrode displacement due to sound is generally higher than predicted for a plane sound wave. This also suggests that the electrode is highly compliant and its motion is unencumbered by either electrostatic or mechanical forces or stiffness.

[0175] The observation that this thin electrode can move with a displacement that is similar to that of the air in a sound field is in line with what is predicted for the sound-induced motion of a thin, flexible wall [18]. While numerous additional effects influence the motion of the electrode examined here and it does not closely resemble the problem of predicting sound transmission through walls, it is clear that a thin, lightweight membrane can move with the air in a sound field. If we consider the incident sound to be a harmonic wave at the frequency ω, propagating normal to the plane of the membrane, one can calculate the ratio of the complex amplitude of the sound wave transmitted through the membrane, p.sub.t to that of the incident pressure, p.sub.1 [18],

[00014]$\begin{array}{cc}\frac{p_t}{p_1}=\frac{1}{1+\frac{\hat{i}.Math.{\rho }_w.Math.h.Math..Math.\omega }{2.Math.{\rho }_0.Math.c}}& \left(\mathrm{I8}\right)\end{array}$

[0176] where ρ.sub.w is the mass density of the membrane material, h is its thickness, ρ.sub.0c≈415 pascal-s/m is the product of the nominal air density ρ.sub.0 and the sound speed c. In a plane sound wave the ratio of the pressure to the acoustic particle velocity is equal to ρ.sub.0c This leads to:

[00015]$\begin{array}{cc}{u}_1=\frac{p_1}{{\rho }_0.Math.c},u_w=\frac{p_t}{{\rho }_0.Math.c}& \left(\mathrm{I9}\right)\end{array}$

[0177] where u.sub.1 is the complex amplitude of the acoustic particle velocity of the incident plane wave. Equations (18) and (19) give the ratio of the membrane velocity relative to the velocity of the acoustic medium if the membrane weren't present,

[00016]$\begin{array}{cc}\frac{U_w}{U_1}=\frac{1}{1+\frac{\hat{i}.Math.{\rho }_w.Math.h.Math..Math.\omega }{2.Math.{\rho }_0.Math.c}}& \left(20\right)\end{array}$

[0178] Because both velocities U.sub.w and U.sub.1 are related to the corresponding displacements by the same factor {circumflex over (.Math.)}ω, the ratio in equation (20) will also equal the ratio of the displacements. This ratio depends only on the factor, (ρ.sub.whω/2ρ.sub.0c). For the metalized polymer electrode used here, the density is estimated to be ρ.sub.w≈1380 kg/m.sup.3 and the thickness is h≈5 μm. Over the range of frequencies shown in FIGS. 7A-7B, this factor varies from approximately 0.003 at 50 Hz to unity at 20 kHz. Over this range of frequencies, it is thus plausible that this thin electrode can move with a displacement that is similar to that of the air in a sound field.

[0179] The fact that the electrode is highly compliant is, of course, a major reason that its motion is easily detected by this capacitive sensing scheme. The use of a highly compliant electrode can be effective as long as the sensing configuration does not itself introduce significant electrostatic forces that would affect the motion.

[0180] The measured electrical sensitivity is shown in FIG. 7B. Again, the frequency response is not ideal due to mechanical resonances but the sensitivity is in the range between 0.1 and 1 volt/pascal over the lower frequency range shown. An optimized electrode design and a more refined readout circuit would doubtless provide improved results over these measurements.

[0181] Comparing FIGS. 7A and 7B shows that an increase in the bias voltage increases the sensitivity at nearly all frequencies in proportion to the bias voltage change while having no noticeable effect on the measured motion. The electrode configuration examined here thus achieves a decoupling of the sensing approach from the mechanical design of the electrode; one does not need to design the electrode so that it will withstand the forces applied by the electric field. The designer is free to construct as compliant a moving electrode as desired to achieve a given sensitivity without concerns that the electrostatic forces will cause instability or will impede the motion.

[0182] In the foregoing, it has been assumed that the moving electrode consists of a flat planar member. However, in some cases, it may be beneficial that its free edge be curved. Further, the plane of the moving member could also be oriented so that it is not parallel to the gap between the fixed electrodes. In this case, motion of the electrode will result in its overlap area with one of the fixed electrodes to increase while the overlap area with the other fixed electrode decreases. This would cause it to function much like the embodiment shown in FIG. 1D, in which the charge on the fixed electrodes depends on overlap area rather than distance as in FIG. 1C. In this case, however, the overlap area is formed by only the free edge of the moving electrode rather than its planar surfaces. The motion occurs in a direction that is generally orthogonal to the fixed electrodes rather than parallel as shown in FIG. 1D. As in the other situation previously described, large motion causes a reduced force back to the equilibrium position, causing the system to be globally stable. One could achieve this effect also by using a flat, planar moving electrode and making the gap between the fixed electrodes not parallel to it or not straight in the direction normal to the plane of FIG. 2.

[0183] Reviewing FIG. 3, the free end of the prototype embodiment of electrode 2 is not perfectly straight. This may contribute to the lack of electrostatic stiffness seen in the data.

[0184] There are numerous sensing applications where it is very desirable that the moving element is driven with diminutive forces and must therefore be as lightweight and compliant as possible so that it provides the least possible resistance and subsequently responds with the largest possible displacement. In cases where the moving element is an electrode in a capacitive sensor, care must be taken to ensure that the forces associated with the electric field do not adversely affect the motion and subsequent sensor performance. The present electrode geometries minimize the electrostatic forces that actin the direction of motion.

[0185] In the electrode design examined here, the electrostatic potential energy is considered as a function of the electrode motion. If the potential energy is roughly constant as the electrode moves, the force will tend to be small since, for this conservative force, the force is equal to the derivative of the potential energy. By splitting the fixed electrode into two elements, one can retain the insensitivity of the potential energy to the electrode displacement while enabling one to sense the differences in charge on the two fixed electrodes. The result is an ability to sense the motion without imposing significant electrostatic forces that affect the motion.

[0186] In addition to designing the sensor to maintain a nearly constant potential energy for the range of motion of interest, because the electric field is orthogonal to the surface of a conductor, the fact that the moving electrode is thin and oriented orthogonally to the fixed electrodes causes the force between them to be small. By maintaining geometric symmetry about the nominal position of the moving electrode, the electrostatic forces applied normal to its surface will approximately cancel. This enables the design of moving electrodes having altogether negligible mechanical stiffness in their primary direction of motion. A negligible stiffness can be achieved by supporting the moving electrode by a hinge that has virtually no resistance to rotation or by making the moving electrode out of an extremely thin material that has negligible resistance to bending. If the material is thin enough, one could configure it to resemble a cantilevered beam, which is fully-fixed to the supporting structure and free at its other end. The mechanical restoring stiffness need only be sufficient to resist any other environmental forces that may act on it, such as gravity.

[0187] Because electrostatic forces don't affect the motion, the bias voltage applied to the moving electrode can be set to a high value which improves the overall electrical sensitivity. In the results provided here, a highly compliant moving electrode is used that readily moves in response to acoustic pressure. The electrode configuration enables the use of a relatively large bias voltage of 400 volts while having negligible effect on the electrode motion. This produces an output electrical sensitivity of approximately 0.5 volts/pascal.

[0188] Another desirable characteristic of the capacitive sensor is the assurance of stability for the entire range of possible motions and bias voltages. As shown in FIGS. 4A-4C, when the moving electrode undergoes large motions, the restoring force will always act to return it to the equilibrium position, ensuring global stability, despite having very small resistance to small excursions from the equilibrium position.

[0189] The motion of the moving electrode is essentially unaffected by changes in the bias voltage while the overall electrical output sensitivity to sound is increased as expected.

[0190] The sensor may be designed as a microphone which achieves an equivalent acoustic pressure noise floor of 20 dBA, with a frequency response will be flat±3 decibels over the frequency range of 20 Hz to 20 kHz.

[0191] While a cantilevered plate-shaped element has been described above as the transducing element for the acoustic waves in air to mechanical motion of a charge, it is also possible to employ one or more fibers, which have the advantage of a high aerodynamic drag to mass ratio. According to the present technology, since the electrostatic interaction of the sensing plates and the moving element does not substantially deflect the element nor materially alter its stiffness, the technology permits sensing of the approximate particle motion in the air surrounding the fiber by viscous drag, as compared to the pressure difference induced deflection of a plate as is more typically measured. Further, the sensor is not limited to a single fiber, and therefore a plurality of fibers may be provided, either as independently moving parallel elements each interacting with the sensing electrodes, or formed into a loose mat or mesh, so that all fibers move together. See, [4, 30, 41, 42, 3]. For example, the fibers may be spider silk coated with 80 nm gold, or electro spun poly methyl methacrylate.

[0192] The operation of a viscous drag moving element can be analyzed based on differences in pressure from a plane traveling acoustic wave acting on its two plane surfaces. One may construct an approximate, qualitative model by considering the moving element to be an elastic beam. Focusing attention on response at a single frequency, ω, the beam deflection at a point along its length x, at time t, w(x, t), may be calculated by solving the following standard partial differential equation,

[00017]$\begin{array}{cc}\mathrm{EI}.Math.\frac{{\partial }^4.Math.w}{\partial x^4}+\mathrm{pbh}.Math.\stackrel{\mathrm{.Math.}}{w}=P\left(e^{\hat{i}.Math.\omega .Math..Math.t.Math.\text{/}.Math.2}-e^{-\hat{i}.Math.\omega .Math..Math.t.Math.\text{/}.Math.2}\right).Math.{\mathrm{be}}^{\hat{i}.Math.\omega .Math..Math.t}+C\left({\mathrm{Ue}}^{\hat{i}.Math.\omega .Math..Math.t}-\stackrel{.}{w}\right)& \left(2.Math.I\right)\end{array}$

[0193] where E is Young's modulus of elasticity, I is the area moment of inertia, ρ is the density of the material, b is the width, his the thickness, P is the plane wave sound pressure amplitude, k=ω/c is the wave number with c being the wave propagation speed, d is the effective distance that sound would travel between the two plane surfaces of the beam, and C is a viscous damping coefficient. U is the complex amplitude of the acoustic particle velocity.

[0194] As the beam becomes sufficiently thin (i.e., as h and b become small), all of the terms in equation (21) become negligible in comparison to the viscous damping force, C (Ue.sup.{circumflex over (.Math.)}ωt−{dot over (w)}) [4] because C has a very much weaker dependence on h and b than all other terms. In addition, for an isolated fiber or beam, the effective separation distance dis approximately equal to b so the acoustic pressure difference term, the first term on the right hand side, also becomes small. Consequently, in this limiting case where the viscous term dominates, the relative motion between the fiber and the air becomes negligible leading to {dot over (w)}≈Ue.sup.{circumflex over (.Math.)}ωt [4]. Therefore, with suitable design of the sensing element so that viscous forces dominate, the sensing element will move with the acoustic medium.

[0195] Extremely thin, compliant materials are widely available for constructing these sensing electrodes, such as graphene [1, 2], and carbon nanotubes or nanotube yarn. Flow sensing has also been accomplished with electrospun polymer fibers [41]. These very thin structures have such low bending stiffness, however, that it is not possible to incorporate them into conventional capacitive microphone designs without having their motion be strongly influenced by the electrostatic forces which occur normal to their long axis. FIG. 8A shows a planar diaphragm having an array of apertures. This design senses drag of moving air through the diaphragm, but such a design has a significant stiffness, and therefore a presumption that all terms of equation (21) are fully dominated by the viscous drag term is not generally satisfied. In some cases, however, a perforated diaphragm represents an acceptable sensor. In one example, a diaphragm may be formed of multilayer graphene. This diaphragm may also be formed of polycrystalline silicon or silicon nitride in a microelectromechanical system (MEMS) design. The diaphragm may be intrinsically conductive or metallized, such as with a layer of gold. The diaphragm, or more generally the moving element, may be formed of an electret material.

[0196] A typical silicon microfabrication process to create the thin velocity-sensing film begins with a bare silicon wafer on which a one-micron oxide is grown through wet oxidation. This oxide film provides an etch stop for a through wafer etch used to create an open air space behind the film. A silicon nitride film having thickness approximately 0.5 micron is then deposited using a low pressure chemical vapor deposition (LPCVD) furnace. The silicon nitride is patterned through optical lithography to define the holes to achieve porosity and to define the electrode edges. Portions of the nitride film are made to be conductive by depositing and patterning a thin (approximately 80 nm) layer of phosphorous doped silicon using a LPCVD process. The film is then annealed to form polycrystalline silicon. A through-wafer backside reactive ion etch (RIE) is performed to expose the backside of the silicon electrode. The electrode is released by removing the thermal oxide, using buffered hydrofluoric acid. The fabrication of the sensing electrodes is performed by depositing conductive films around the perimeter of the moving electrode.

[0197] FIG. 8B shows a fiber mesh element which forms a loose plate which generally moves as a unit by viscous drag of moving air against the fibers. The mesh is designed to have high stiffness due to forces applied in the plane of the mesh while having high compliance when out of plane forces (such as those due to the acoustic flow) are applied. Because the mesh consists of a large number of loosely arranged individual fibers, dimensional precision and in plane stiffness at the edge nearest the sensing electrodes is difficult to assure if the fibers in proximity to the sensing electrodes are free. This, in turn, impairs repeatable sensitivity and resistance to in-plane electrostatic attractive forces. Therefore, a thin solid frame or binding may be provided attached to the mesh at the edge nearest the sensing electrodes. Suitable fibers include metallized electrospun PMMA and carbon nanotubes, or both in combination.

[0198] The fibers may be less and 1μ, and for example may be about 500 nm diameter.

[0199] The device according to the present technology may be used not only as a sensor, but also as an actuator. In this case, for example, we may apply a small time-varying differential voltage to electrodes which will effectively modulate the system's equilibrium position about a null position. A voltage applied to the moving element electrode may be set to a value that adjusts the electrostatic stiffness to nearly any value desired, leaving the motion to be limited only by the mechanical stiffness and mass of the moving electrode.

[0200] The use of an extremely compliant and lightweight moving electrode material, such as for example, graphene, would enable actuation with very small driving voltage. This configuration permits a wide range of adjustment of the equilibrium position as a function of small changes in the driving voltage. The response of the moving electrode to changes in voltage is linear, instead of quadratic, as might otherwise be expected for a parallel plate actuator. Further, in a 4-electrode embodiment which has three static electrodes instead of two as described above, may also be used. In this case, the additional electrode provides additional ability to adjust the effective electrostatic stiffness of the moving electrode. Note that the forces may be repulsive rather than attractive as discussed in various embodiments above.

[0201] The various embodiments described above can be combined to provide further embodiments. These and other changes can be made to the embodiments in light of the above-detailed description. In general, in the following claims, the terms used should not be construed to limit the claims to the specific embodiments disclosed in the specification and the claims, but should be construed to include all possible embodiments along with the full scope of equivalents to which such claims are entitled. Accordingly, the claims are not limited by the disclosure.

REFERENCES

[0202] Each of the following its expressly incorporated herein by reference in its entirety: [0203] [1] Y.-M. Chen, S.-M. He, C.-H. Huang, C.-C. Huang, W.-P. Shih, C.-L, Chu, J. Kong, J. Li, and C.-Y. Su, “Ultra-large suspended graphene as a highly elastic membrane for capacitive pressure sensors,” Nanoscale, vol. 8, no. 6, pp, 3555-3564, 2016. [0204] [2] C. Berger, R. Phillips, A. Centeno, A. Zurutuza, and A. Vijayaraghavan, “Capacitive pressure sensing with suspended graphene-polymer heterostructure membranes”, Nanoscale, vol. 9, no. 44, pp. 17439-17449, 2017. [0205] [3] J. Zhouand R. N. Miles, “Sensing fluctuating airflow with spider silk,” Proceedings of the National Academy of Sciences, p. 201710559, 2017. [0206] [4] R. Miles and J. Zhou, “Sound-induced motion of a nanoscale fiber,” Journal of Vibration and Acoustics, vol. 140, no. 1, p. 011009, 2018, [0207] [5] R. Miles and F. Degertekin, “Optical sensing in a directional mems microphone,” Nov. 2, 2010, U.S. Pat. No. 7,826,629. [0208] [6] B. Bicen, S, Jolly, K. Jeelani, C. T. Garcia, N. A. Hall, F. L. Degertekin, Q. Su, W. Cui, and R. N. Miles, “Integrated Optical Displacement Detection and Electrostatic Actuation for Directional Optical Microphones With Micromachined Biometric Diaphragms,” IEEE Sensors Journal, vol. 9, no. 12, pp. 1933-1941, December 2009. [0209] [7] R. N. Miles, Q. Su, W. Cui, M. Shetye, F. L. Degertekin, B. Bicen, C. Garcia, S. Jones, and N. Hall, “A low-noise differential microphone inspired by die ears of the parasitoid fly Ormia ochracea,” Journal of the Acoustical Society of America, vol. 125, no. 4, Part I, pp. 2013-2026, April 2009. [0210] [8] M. L. Kuntzman, N. N. Hewa-Kasakarage, A. Rocha, D. Kim, and N. A. Hall, “Micromachined in-plane pressure-gradient piezoelectric microphones,” IEEE Sensors Journal, vol. 15, no. 3, pp. 1347-1357, 2015. [0211] [9] B.-H. Kim and H.-S. Lee, “Acoustical-thermal noise in a capacitive mems microphone,” IEEE Sensors Journal, vol. 15, no. 12, pp. 6853-6860, 2015. [0212] [10] L. K. Baxter, “Capacitive sensors,” Ann Arbor, vol. 1001, p. 48109, 2000. [0213] [11] D. K. Miu, Mechantronics: electromechanics and contromechanics. Springer Science & Business Media, 2012. [0214] [12] R. Miles, “Comb sense microphone,” Jun. 9, 2009, U.S. Pat. No. 7,545,945. [0215] [13]R. N. Miles, W. Cui, Q. T. Su. and D. Homentcovschi, “A mems low-noise sound pressure gradient microphone with capacitive sensing,” Journal of Microelectromechanical Systems, vol. 24, no. 1, pp. 241-248, 2015. [0216] [14] S. Towfighian, S. He, and R. B. Mrad, “A low voltage electrostatic micro actuator for large out-of-plane displacement,” in ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society of Mechanical Engineers, 2014, pp. V004T09A015-V004T09A015. [0217] [15] S. He, R. B. Mrad, and J. Chang, “Development of a high-performance microelectrostatic repulsive-force rotation actuator,” Journal of Microelectromechanical Systems, vol. 19, no. 3, pp. 561-569, 2010. [0218] [16] D. Poljak and C. A. Brebbia, Boundary element methods for electrical engineers. WIT Press, 2005, vol. 4. [0219] [17] P. Morse and K. Ingard, Theoretical Acoustics. Princeton Univ Pr. 1986. [0220] [18]A. London, “Transmission of reverberant sound through single walls,” J. Research Nat. Bur. of Stand. vol. 42, no. 605, p. 2, 1949. [0221] [19] R. N. Miles, W. Cui, Q. T. Su, and D. Homentcovschi, “A mems low-noise sound pressure gradient microphone with capacitive sensing,” Journal of Microelectromechanical Systems, vol. 24, no. 1, pp. 241-248, 2015. [0222] [20] T. T. Bringley. Analysis of the immersed boundary method for Stokes flow. PhD thesis, New York University, 2008. [0223] [21] K. K. Charaziak and C. A. Shera, Compensating for ear-canal acoustics when measuring otoacoustic emissions. The Journal of the Acoustical Society of America, 141(1):515-531, 2017. [0224] [22] R. Cox. The motion of long slender bodies in a viscous fluid part I. general theory. Journal of Fluid mechanics, 44(4):791-810, 1970. [0225] [23] W. Cui, B. Bicen, N. Hall, S. Jones, F. Degertekin, and R. Miles. Optical sensing in a directional MEMS microphone inspired by the ears of the parasitoid fly, Ormia ochracea. In MEMS 2006: 19th IEEE International Conference on Micro Electro Mechanical Systems, Technical Digest, Proceedings: IEEE Micro Electro Mechanical Systems Workshop, pages 614-617, 2006., Istanbul, TURKEY, Jan. 22-26, 2006. [0226] [24] H. Droogendijk, J. Casas, T. Steinmann, and G. Krijnen. Performance assessment of bio-inspired systems: flow sensing mems hairs. Bioinspiration & biomimetics, 10(1):016001, 2014. [0227] [25] T. Götz. Interactions of fibers and flow: asymptotics, theory and numerics. PhD thesis, Technical University of Kaiserslautern, 2000. [0228] [26]K. Grosh and R. J. Littrell. Piezoelectric mems microphone, Dec. 26, 2017. U.S. Pat. No. 9,853,201. [0229] [27] D. Homentcovschi, R. Miles, P. Loeppert, and A. Zuckerwar. A microacoustic analysis including viscosity and thermal conductivity to model the effect of the protective cap on the acoustic response of a mems microphone. Microsystem technologies, 20(2):265-272, 2014. [0230] [27] W.-X. Huang, S. J. Shin, and H. J. Sung. Simulation of flexible filaments in a uniform flow by the immersed boundary method. Journal of Computational Physics, 226(2):2206-2228, 2007. [0231] [28] G. Kämper and H.-U. Kleindienst. Oscillation of cricket sensory hairs in a low-frequency sound field. Journal of Comparative Physiology A, 167(2): 193-200, 1990. [0232] [29] R. Miles. Comb sense capacitive microphone, Jun. 9, 2009. US Patent App. 2009/0262958. [0233] [30]R. N. Miles. “A Compliant Capacitive Sensor for Acoustics: Avoiding Electrostatic Forces at High Bias Voltages.” IEEE Sensors Journal 18, no. 14 (2018): 5691-5698. [0234] [31] C. Motchenbacher and J. Connelly. Low-Noise Electronic System Design. John Wiley & Sons, Inc., 1993. [0235] [32] L. Rosenhead. Laminar Boundary Layers: An Account of the Development, Structure, and Stability of Laminar Boundary Layers in Incompressible Fluids, Together with a Description of the Associated Experimental Techniques. Clarendon Press, 1963. [0236] [33] M. J. Shelley and T. Ueda. The stokesian hydrodynamics of flexing, stretching filaments. Physica D: Nonlinear Phenomena, 146(1):221-245, 2000. [0237] [34] R. Sisto, L. Cerini, F. Sanjust, and A. Moleti. Intensimetric detection of distortion product otoacoustic emissions with ear canal calibration. The Journal of the Acoustical Society of America, 142(1):EL13-EL17, 2017. [0238] [35] G. G. Stokes. On the effect of the internal friction of fluids on the motion of pendulums, volume 9. Pitt Press, 1851. [0239] [36] A.-K. Tornberg and K. Gustaysson. A numerical method for simulations of rigid fiber suspensions. Journal of Computational Physics, 215(1):172-196, 2006. [0240] [37] A.-K. Tornberg and M. J. Shelley. Simulating the dynamics and interactions of flexible fibers in stokes flows. Journal of Computational Physics, 196(1):8-40, 2004. [0241] [38] S. Towfighian and R. N. Miles. A new approach to capacitive sensing. Repulsive sensors, Sep. 1, 2016. NSF Grant I608692. [0242] [39] B. Varanda, R. Miles, and D. Warren. Characterization of the dominant structural vibration of hearing aid receivers. Journal of Vibration and Acoustics, 138(6):061009, 2016. [0243] [40] E. Wente. A condenser transmitter as a uniformly sensitive instrument for the absolute measurement of sound intensity. Physical Review, 10(1):39, 1917. [0244] [41] J. Zhou, B. Li, J. Liu, W. Jones Jr., and R. Miles. Highly-damped nanofiber mesh for ultrasensitive broadband acoustic flow detection, 2018. to appear Journal of Micromechanics and Microengineering. [0245] [42] J. Zhou and R. Miles. Directional sound detection by sensing acoustic flow, 2018. to appear IEEE Sensors. [0246] [43] J. M. Crowley, Fundamentals of applied electrostatics. John Wiley & Sons, 1986. [0247] [44] N. Jonassen, Electrostatics. Springer Science & Business Media, 2013. [0248] [45]T. F. Eibert and V. Hansen, “On the calculation of potential integrals for linear source distributions on triangular domains,” IEEE Transactions on Antennas and Propagation, vol. 43, no. 12, pp. 1499-1502, 1995. [0249] [46] J. Zhou, R. N. Miles, and S. Towfighian, “A novel capacitive sensing principle for microdevices,” in ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015, pp. V004T09A024-V004T09A024.