High performance sealed-gap capacitive microphone with various gap geometries

Abstract

Some preferred embodiments include a microphone system for receiving sound waves, the microphone including a back plate, a radiation plate, first and second electrodes, first and second insulator layers, a power source and a microphone controller. The radiation plate is clamped to the back plate so that there is a hermetically sealed regular convex polygon-, ellipse-, or regular convex elliptic polygon-shaped gap between the radiation plate and the back plate. The first electrode is fixedly attached to a side of the back plate proximate to the gap. The second electrode is fixedly attached to a side of the radiation plate. The insulator layers are attached to the back plate and/or the radiation plate, on respective gap sides thereof, so that the insulator layers are between the electrodes. The microphone controller is configured to use the power source to drive the microphone at a selected operating point comprising normalized static mechanical force, bias voltage, and relative bias voltage level. Relevant dimensions of the gap, and a thickness of the radiation plate, are determined using the selected operating point so that a sensitivity of the microphone at the selected operating point is an optimum sensitivity for the selected operating point.

Claims

1. A microphone system for receiving sound waves, the microphone system comprising: a back plate; a radiation plate having a thickness t.sub.m, the radiation plate clamped to the back plate so that there is a sealed gap between the radiation plate and the back plate such that passage of gas into or out of the gap is prevented, the gap having a regular convex polygon shape and a gap height t.sub.g; the gap having a regular convex polygon shape with a number n4 sides, the gap having an apothem of length r.sub.n; a first electrode, either the first electrode being fixedly coupled to a side of the back plate proximate to the gap, or the first electrode comprising or contained within the back plate; a second electrode, either the second electrode being fixedly coupled to a side of the radiation plate, or the first electrode comprising or contained within the radiation plate; a first insulator layer of thickness t.sub.i1 and relative permittivity .sub.r_i1, and a second insulator layer of thickness t.sub.i2 and relative permittivity .sub.r_i2, the first and second insulator layers being disposed between the first and second electrodes, and the first and second insulator layers being disposed between the back plate and the radiation plate; a power source; and a microphone controller configured to use the power source to drive the microphone at an operating point, wherein F.sub.Peb is a net static force exerted on the radiation plate due to an ambient static pressure, F.sub.Peg is a uniformly distributed force required to displace a center of the radiation plate by an effective gap height t.sub.ge, and V.sub.C is a limit to bias voltage V.sub.DC for uncollapsed operation of the microphone system, the operating point comprising: a normalized static mechanical force F.sub.Peb/F.sub.Peg, a bias voltage of the first and second electrodes V.sub.DC, and a relative bias voltage level of the first and second electrodes V.sub.DC/V.sub.C; wherein $t_{\mathrm{ge}}=t_g+\frac{t_{i1}}{{\mathrm{.Math.}}_{r\_i1}}+\frac{t_{i2}}{{\mathrm{.Math.}}_{r\_i2}};$ wherein the apothem length r.sub.n, the gap height t.sub.g, and the radiation plate thickness t.sub.m are determined using the selected operating point so that an OCRV sensitivity of the microphone at the selected operating point is an optimum OCRV sensitivity for the selected operating point; and wherein the equivalent disc gap radius a.sub.eq is related to a minimum equivalent disc gap radius a.sub.eq_min corresponding to the optimum sensitivity at the operating point, and the radiation plate thickness t.sub.m is related to a minimum radiation plate thickness t.sub.m_min corresponding to the optimum sensitivity at the operating point, by a selected scaling constant K, such that a.sub.eq(K.sup.3)a.sub.eq_min, and t.sub.m=(K.sup.4)t.sub.m_min.

2. The microphone system of claim 1, wherein the gap comprises a hole machined into the substrate, and the back plate comprises a portion of the substrate forming a floor of the gap.

3. The microphone system of claim 1, wherein the apothem length r.sub.n is determined by determining a radius of an equivalent circle a.sub.eq, wherein $a_{\mathrm{eq}}=r_n\sqrt[4]{\frac{n}{}\tan \left(\frac{}{n}\right)}.$

4. The microphone system of claim 1, wherein the first electrode covers at least 80% of the area of the back plate on the side of the back plate proximate to the gap, and wherein the second electrode covers at least 80% of the area of the radiation plate on the side of the radiation plate proximate to the gap.

5. The microphone system of claim 1, wherein the sound waves are human-audible and the gap contains a vacuum.

6. The microphone system of claim 1, wherein both insulator layers are fixedly coupled to the radiation plate, or both insulator layers are fixedly coupled to the back plate, or the first insulator layer is fixedly coupled to the radiation plate and the second insulator layer is fixedly coupled to the back plate.

7. The microphone system of claim 1, wherein the equivalent disc gap radius a.sub.eq, the gap height t.sub.g, and the radiation plate thickness t.sub.m are determined using the operating point so that the microphone system will maintain uncollapsed, linear elastic operation.

8. The microphone system of claim 1, further comprising an electret configured to increase an effective bias voltage of the first and second electrodes.

9. The microphone system of claim 1, wherein the radiation plate comprises a selected solid material suitable for fabrication of a MEMS microphone; and wherein the particular selected solid material does not affect the optimum sensitivity, and does not affect a corresponding gap height or radiation plate thickness.

10. The microphone system of claim 1, wherein the operating point is a selected operating point, the selected operating point being selected by selecting up to three of the following: the equivalent disc gap radius a.sub.eq, the apothem r.sub.n, the radiation plate thickness t.sub.m, the effective gap height t.sub.ge, the optimum OCRV sensitivity, an SCRC sensitivity, the normalized static mechanical force F.sub.Peb/F.sub.Peg, the bias voltage V.sub.DC, or the relative bias voltage level V.sub.DC/V.sub.C.

11. The microphone system of claim 1, wherein multiple ones of the microphone systems are electrically connected in parallel.

12. The microphone system of claim 1, wherein the radiation plate comprises of one or multiple layers of a single material or multiple layers of a multitude of different materials, for which an equivalent single layer Young's modulus, Y.sub.eq and Poisson's ratio, .sub.eq can be calculated.

13. The microphone system of claim 1, wherein a.sub.n is a normalized radius of the regular convex polygon shaped gap, and a.sub.n is in the range:
a.sub.n14.2t.sub.ge_n2.84 for 0.2<t.sub.ge_n0.8
0.9t.sub.ge_n0.72<a.sub.n14.2t.sub.ge_n2.84 for 0.8<t.sub.ge_n6.8; wherein t.sub.m_n is a normalized thickness of the radiation plate, and t.sub.m_n is in the range:
t.sub.m_n36t.sub.ge_n7.2 for 0.2<t.sub.ge_n0.8
0.93t.sub.ge_n0.744<t.sub.m_n36t.sub.ge_n7.2 for 0.8<t.sub.ge_n6.8; wherein .sub.0 is a permittivity of free space, P.sub.0 is a static pressure difference between an ambient and the gap, and V.sub.DC_n is a normalized operating bias voltage such that: $V_{DC\_n}=\frac{3}{2}\sqrt{\frac{{\mathrm{.Math.}}_0}{P_0}}{V}_{DC};$ wherein t.sub.ge_n is a normalized effective gap height, and the equivalent disc radius a.sub.eq, the gap height t.sub.g, and the radiation plate thickness t.sub.m are: $t_{\mathrm{ge}}={V_{DC\_n}\left(\frac{V_{DC}}{V_C}\right)}^{-1}{t}_{\mathrm{ge}\_n}$ $t_{\mathrm{ge}\_n}\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)\frac{\sqrt{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}}}{\begin{array}{c}0.9961-1.0468\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}+\\ 0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2+0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6\end{array}}$ $a_{\mathrm{eq}}=\left(10\sqrt[4]{\frac{Y_0}{15\left(1-{}^2\right)P_0}}\right)V_{\mathrm{DC\_n}}{a}_n$ $t_m=5{V_{DC\_n}\left(\frac{V_{DC}}{V_C}\right)}^{-1}{t}_{m\_n}$ wherein Y.sub.0 is a Young's modulus of a material comprising the radiation plate and is a Poisson's ratio of the material comprising the radiation plate.

14. The microphone system of claim 13, wherein the normalized gap radius a.sub.n corresponds to a normalized minimum gap radius a.sub.n_min that is within the range for a.sub.n, the normalized radiation plate thickness t.sub.m_n corresponds to a normalized minimum radiation plate thickness t.sub.m_n_min that is within the range for t.sub.m_n, K is a selected scaling constant, X.sub.P is a static deflection of the center of the radiation plate, $g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)$ function of $\frac{X_P}{t_{\mathrm{ge}}},\mathrm{and}{g}^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)$ is a function which is the first derivative of $g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\text{:}$ $g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)=\frac{{\tanh }^{-1}\left(\sqrt{X_P/t_{\mathrm{ge}}}\right)}{\sqrt{X_P/t_{\mathrm{ge}}}}$ $g^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)=\frac{1}{2\frac{X_P}{t_{\mathrm{ge}}}}\left(\frac{1}{1-\frac{X_P}{t_{\mathrm{ge}}}}-g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\right)$ $\frac{a_{n\_min}}{t_{m\_n\_min}}=\sqrt[4]{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}/\frac{X_P}{t_{\mathrm{ge}}}}$ $\frac{V_{DC}^2}{V_C^2}{\left(0.9961-1.0468\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}+0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2+0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6\right)}^{2}2g^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)-3\left(\frac{X_P}{t_{\mathrm{ge}}}-\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)0\mathrm{for}\frac{X_P}{t_{\mathrm{ge}}}>\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}$ $a_n=\left(K^3\right)a_{n\_min}$ $t_{m\_n}=\left(K^4\right)t_{m\_n\_min}.$

15. A microphone system for receiving sound waves, the microphone system comprising: a back plate; a radiation plate having a thickness t.sub.m, the radiation plate clamped to the back plate so that there is a sealed gap between the radiation plate and the back plate such that passage of gas into or out of the gap is prevented, the gap having an elliptic shape with minor radius a.sub.1 and major radius a.sub.2 and a gap height t.sub.g; a first electrode, either the first electrode being fixedly coupled to a side of the back plate proximate to the gap, or the first electrode comprising or contained within the back plate; a second electrode, either the second electrode being fixedly coupled to a side of the radiation plate, or the first electrode comprising or contained within the radiation plate; a first insulator layer of thickness t.sub.i1 and relative permittivity .sub.r_i1, and a second insulator layer of thickness t.sub.i2 and relative permittivity .sub.r_i2, the first and second insulator layers being disposed between the first and second electrodes, and the first and second insulator layers being disposed between the back plate and the radiation plate; a power source; and a microphone controller configured to use the power source to drive the microphone at an operating point, wherein F.sub.Peb is a net static force exerted on the radiation plate due to an ambient static pressure, F.sub.Peg is a uniformly distributed force required to displace a center of the radiation plate by an effective gap height t.sub.ge, and V.sub.C is a limit to bias voltage V.sub.DC for uncollapsed operation of the microphone system, the operating point comprising: a normalized static mechanical force F.sub.Peb/F.sub.Peg, a bias voltage of the first and second electrodes V.sub.DC, and a relative bias voltage level of the first and second electrodes V.sub.DC/V.sub.C; wherein $t_{\mathrm{ge}}=t_g+\frac{t_{i1}}{{\mathrm{.Math.}}_{r\_i1}}+\frac{t_{i2}}{{\mathrm{.Math.}}_{r\_i2}};$ wherein the pair elliptic gap minor radius a.sub.1 and elliptic gap major radius a.sub.2, the gap height t.sub.g, and the radiation plate thickness t.sub.m are determined using the selected operating point so that an OCRV sensitivity of the microphone at the selected operating point is an optimum OCRV sensitivity for the selected operating point; and wherein a is a radius of a seed circle of the elliptic shaped gap, and .sub.e is an aspect ratio of the elliptic shaped gap, and the seed circle radius a is related to a minimum gap radius a.sub.min corresponding to the optimum sensitivity at the operating point, and the radiation plate thickness tin is related to a minimum radiation plate thickness t.sub.m_min corresponding to the optimum sensitivity at the operating point, by a selected scaling constant K, such that a=(K.sup.3)a.sub.min, and t.sub.m=(K.sup.4)t.sub.m_min, where $a_2=a\left(\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)$ $\mathrm{and}$ $a_1=a\left(\frac{1}{{}_e}\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right).$

16. The microphone system of claim 15, wherein the gap comprises a hole machined into the substrate, and the back plate comprises a portion of the substrate forming a floor of the gap.

17. The microphone system of claim 15, wherein the first electrode covers at least 80% of the area of the back plate on the side of the back plate proximate to the gap, and wherein the second electrode covers at least 80% of the area of the radiation plate on the side of the radiation plate proximate to the gap.

18. The microphone system of claim 15, wherein the sound waves are human-audible and the gap contains a vacuum.

19. The microphone system of claim 15, wherein both insulator layers are fixedly coupled to the radiation plate, or both insulator layers are fixedly coupled to the back plate, or the first insulator layer is fixedly coupled to the radiation plate and the second insulator layer is fixedly coupled to the back plate.

20. The microphone system of claim 15, wherein the pair elliptic gap minor radius a.sub.1 and elliptic gap major radius a.sub.2, the gap height t.sub.g, and the radiation plate thickness tin are determined using the operating point so that the microphone system will maintain uncollapsed, linear elastic operation.

21. The microphone system of claim 15, further comprising an electret configured to increase an effective bias voltage of the first and second electrodes.

22. The microphone system of claim 15, wherein the radiation plate comprises a selected solid material suitable for fabrication of a MEMS microphone; and wherein the particular selected solid material does not affect the optimum sensitivity, and does not affect a corresponding gap height or radiation plate thickness.

23. The microphone system of claim 15, wherein the operating point is a selected operating point, the selected operating point being selected by selecting up to three of the following: the pair elliptic gap minor radius a.sub.1 and elliptic gap major radius a.sub.2, the radiation plate thickness t.sub.m, the effective gap height t.sub.ge, the optimum OCRV sensitivity, an SCRC sensitivity, the normalized static mechanical force F.sub.Peb/F.sub.Peg, the bias voltage V.sub.DC, or the relative bias voltage level V.sub.DC/V.sub.C.

24. The microphone system of claim 15, wherein multiple ones of the microphone system are electrically connected in parallel.

25. The microphone system of claim 15, wherein the radiation plate comprises of one or multiple layers of a single material or multiple layers of a multitude of different materials, for which an equivalent single layer Young's modulus, Y.sub.eq and Poisson's ratio, .sub.eq can be calculated.

26. The microphone system of claim 15, wherein a.sub.1n is a normalized radius of the ellipse minor radius a.sub.1, a.sub.2n is a normalized radius of the ellipse major radius a.sub.2, and .sub.e is the aspect ratio of said ellipse, and a.sub.n is a normalized radius of the seed circle, and a.sub.n is in the range:
a.sub.n14.2t.sub.ge_n2.84 for 0.2<t.sub.ge_n0.8
0.9t.sub.ge_n0.72<a.sub.n14.2t.sub.ge_n2.84 for 0.8<t.sub.ge_n6.8; wherein t.sub.m_n is a normalized thickness of the radiation plate, and t.sub.m_n is in the range:
t.sub.m_n36t.sub.ge_n7.2 for 0.2<t.sub.ge_n0.8
0.93t.sub.ge_n0.744<t.sub.m_n36t.sub.ge_n7.2 for 0.8<t.sub.ge_n6.8; and wherein $a_{2n}=\left(\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)a_n,\mathrm{and}$ $a_{1n}=\left(\frac{1}{{}_e}\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)a_n;$ and wherein .sub.0 is a permittivity of free space, P.sub.0 is a static pressure difference between an ambient and the gap, and V.sub.DC_n is a normalized operating bias voltage such that: $V_{D\mathrm{C\_}n}=\frac{3}{2}\sqrt{\frac{{\mathrm{.Math.}}_0}{P_0}}{V}_{DC};$ wherein t.sub.ge_n is a normalized effective gap height, and the ellipse minor radius a.sub.1 and the ellipse major radius a.sub.2, the gap height t.sub.g, and the radiation plate thickness t.sub.m are: $t_{\mathrm{ge}}={V_{DC\_n}\left(\frac{V_{DC}}{V_C}\right)}^{-1}{t}_{\mathrm{ge}\_n}$ $t_{\mathrm{ge}\_n}\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)\frac{\sqrt{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}}}{\begin{array}{c}0.9961-1.0468\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}+\\ 0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2+0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6\end{array}}$ $a_1=\left(10\sqrt[4]{\frac{Y_0}{15\left(1-{}^2\right)P_0}}\right)V_{DC\_n}{a}_{1n}$ $a_2=\left(10\sqrt[4]{\frac{Y_0}{15\left(1-{}^2\right)P_0}}\right)V_{DC\_n}{a}_{2n}$ $t_m=5{V_{DC\_n}\left(\frac{V_{DC}}{V_C}\right)}^{-1}{t}_{m\_n}$ wherein Y.sub.0 is a Young's modulus of a material comprising the radiation plate and is a Poisson's ratio of the material comprising the radiation plate.

27. The microphone system of claim 26, wherein the normalized gap radius an corresponds to a normalized minimum gap radius a.sub.n_min that is within the range for a.sub.n, the normalized radiation plate thickness t.sub.m_n corresponds to a normalized minimum radiation plate thickness t.sub.m_n_min that is within the range for t.sub.m_n, the normalized ellipse major radius a.sub.2n corresponds to a normalized ellipse minimum major radius a.sub.2n_min, K is a selected scaling constant, X.sub.P is a static deflection of the center of the radiation plate, $g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)$ function of $\frac{X_P}{t_{\mathrm{ge}}},\mathrm{and}{g}^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)$ is a function which is the first derivative of $g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\text{:}$ $g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)=\frac{{\tanh }^{-1}\left(\sqrt{X_P\text{/}{t}_{\mathrm{ge}}}\right)}{\sqrt{X_P\text{/}{t}_{\mathrm{ge}}}}$ $g^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)=\frac{1}{2\frac{X_P}{t_{\mathrm{ge}}}}\left(\frac{1}{1-\frac{X_P}{t_{\mathrm{ge}}}}-g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\right)$ $\frac{a_{\mathrm{n\_min}}}{t_{\mathrm{m\_n}\mathrm{\_min}}}=\sqrt[4]{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}/\frac{X_P}{t_{\mathrm{ge}}}}$ $\frac{V_{\mathrm{DC}}^2}{V_C^2}(0.9961-1.0468\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}+0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2+{0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6)}^{2}2g^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)-3\left(\frac{X_P}{t_{\mathrm{ge}}}-\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)0\mathrm{for}\frac{X_P}{t_{\mathrm{ge}}}>\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}{a}_n=\left(K^3\right)a_{\mathrm{n\_min}}{a}_{2\mathrm{n\_min}}=\left(\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)a_{\mathrm{n\_min}}{t}_{\mathrm{m\_n}}=\left(K^4\right)t_{\mathrm{m\_n}\mathrm{\_min}}.$

28. A microphone system for receiving sound waves, the microphone system comprising: a back plate; a radiation plate having a thickness t.sub.m, the radiation plate clamped to the back plate so that there is a sealed gap between the radiation plate and the back plate such that passage of gas into or out of the gap is prevented; the gap having an elliptic polygon shape with a number n4 sides and a gap height t.sub.g, the elliptic polygon shaped gap having a minor apothem of length r.sub.n1 and a major apothem of length r.sub.n2, an equivalent ellipse of the elliptic polygon shaped gap having a minor radius a.sub.e1 and a major radius a.sub.e2 such that: $a_{\mathrm{eq}1}=r_{n1}\sqrt[4]{\frac{n}{}\tan \left(\frac{}{n}\right)},$ and $a_{\mathrm{eq}2}=r_{n2}\sqrt[4]{\frac{n}{}\tan \left(\frac{}{n}\right)};$ a first electrode, either the first electrode being fixedly coupled to a side of the back plate proximate to the gap, or the first electrode comprising or contained within the back plate; a second electrode, either the second electrode being fixedly coupled to a side of the radiation plate, or the first electrode comprising or contained within the radiation plate; a first insulator layer of thickness t.sub.i1 and relative permittivity .sub.r_i1, and a second insulator layer of thickness t.sub.i2 and relative permittivity .sub.r_i2, the first and second insulator layers being disposed between the first and second electrodes, and the first and second insulator layers being disposed between the back plate and the radiation plate; a power source; and a microphone controller configured to use the power source to drive the microphone at an operating point, wherein F.sub.Peb is a net static force exerted on the radiation plate due to an ambient static pressure, F.sub.Peg is a uniformly distributed force required to displace a center of the radiation plate by an effective gap height t.sub.ge, and V.sub.C is a limit to bias voltage V.sub.DC for uncollapsed operation of the microphone system, the operating point comprising: a normalized static mechanical force F.sub.Peb/F.sub.Peg, a bias voltage of the first and second electrodes V.sub.DC, and a relative bias voltage level of the first and second electrodes V.sub.DC/V.sub.C; wherein $t_{\mathrm{ge}}=t_g+\frac{t_{i1}}{{\mathrm{.Math.}}_{r\_i1}}+\frac{t_{i2}}{{\mathrm{.Math.}}_{r\_i2}};$ wherein the minor apothem length r.sub.n1 and the major apothem length r.sub.n2, the gap height t.sub.g, and the radiation plate thickness t.sub.m are determined using the selected operating point so that an OCRV sensitivity of the microphone at the selected operating point is an optimum OCRV sensitivity for the selected operating point wherein a is a radius of a seed circle of the equivalent ellipse, and .sub.e is the aspect ratio of said ellipse, and the seed circle radius a is related to a minimum gap radius a.sub.min corresponding to the optimum sensitivity at the operating point, and the radiation plate thickness t.sub.m is related to a minimum radiation plate thickness t.sub.m_min corresponding to the optimum sensitivity at the operating point, by a selected scaling constant K, such that a=(K.sup.3)a.sub.min, and t.sub.m=(K.sup.4)t.sub.m_min where $\begin{array}{cc}{a}_{\mathrm{eq}2}=a\left(\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)& \\ \mathrm{and}& \\ a_{\mathrm{eq}1}=a\left(\frac{1}{{}_e}\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right).& \end{array}$

29. The microphone system of claim 28, wherein the gap comprises a hole machined into the substrate, and the back plate comprises a portion of the substrate forming a floor of the gap.

30. The microphone system of claim 28, wherein the first electrode covers at least 80% of the area of the back plate on the side of the back plate proximate to the gap, and wherein the second electrode covers at least 80% of the area of the radiation plate on the side of the radiation plate proximate to the gap.

31. The microphone system of claim 28, wherein the sound waves are human-audible and the gap contains a vacuum.

32. The microphone system of claim 28, wherein both insulator layers are fixedly coupled to the radiation plate, or both insulator layers are fixedly coupled to the back plate, or the first insulator layer is fixedly coupled to the radiation plate and the second insulator layer is fixedly coupled to the back plate.

33. The microphone system of claim 28, wherein the pair equivalent ellipse minor radius a.sub.eq1 and equivalent ellipse major radius a.sub.eq2, the gap height t.sub.g, and the radiation plate thickness t.sub.m are determined using the operating point so that the microphone system will maintain uncollapsed, linear elastic operation.

34. The microphone system of claim 28, further comprising an electret configured to increase an effective bias voltage of the first and second electrodes.

35. The microphone system of claim 28, wherein the radiation plate comprises a selected solid material suitable for fabrication of a MEMS microphone; and wherein the particular selected solid material does not affect the optimum sensitivity, and does not affect a corresponding gap height or radiation plate thickness.

36. The microphone system of claim 28, wherein the operating point is a selected operating point, the selected operating point being selected by selecting up to three of the following: the pair equivalent ellipse minor radius a.sub.eq1 and equivalent ellipse major radius a.sub.eq2, the radiation plate thickness t.sub.m, the effective gap height t.sub.ge, the optimum OCRV sensitivity, an SCRC sensitivity, the normalized static mechanical force F.sub.Peb/F.sub.Peg, the bias voltage V.sub.DC, or the relative bias voltage level V.sub.DC/V.sub.C.

37. The microphone system of claim 28, wherein multiple ones of the microphone system are electrically connected in parallel.

38. The microphone system of claim 28, wherein the radiation plate comprises of one or multiple layers of a single material or multiple layers of a multitude of different materials, for which an equivalent single layer Young's modulus, Y.sub.eq and Poisson's ratio, .sub.eq can be calculated.

39. The microphone system of claim 28, wherein a.sub.1n is a normalized radius of the equivalent ellipse minor radius, a.sub.2n is a normalized radius of the equivalent ellipse major radius, and .sub.e is the aspect ratio of said ellipse, and a.sub.n is a normalized radius of the seed circle, and a.sub.n is in the range:
a.sub.n14.2t.sub.ge_n2.84 for 0.2<t.sub.ge_n0.8
0.9t.sub.ge_n0.72<a.sub.n14.2t.sub.ge_n2.84 for 0.8<t.sub.ge_n6.8; wherein t.sub.m_n is a normalized thickness of the radiation plate, and t.sub.m_n is in the range:
t.sub.m_n36t.sub.ge_n7.2 for 0.2<t.sub.ge_n0.8
0.93t.sub.ge_n0.744t.sub.m_n36t.sub.ge_n7.2 for 0.8<t.sub.ge_n6.8; and wherein $\begin{array}{cc}{a}_{2n}=\left(\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)a_n,& \\ \mathrm{and}& \\ a_{1n}=\left(\frac{1}{{}_e}\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)a_n& \end{array}$ and wherein .sub.0 is a permittivity of free space, P.sub.0 is a static pressure difference between an ambient and the gap, and V.sub.DC_n is a normalized operating bias voltage such that: $V_{\mathrm{DC}\_n}=\frac{3}{2}\sqrt{\frac{{\mathrm{.Math.}}_0}{P_0}{V}_{\mathrm{DC}}};$ wherein t.sub.ge_n is a normalized effective gap height, and the pair equivalent ellipse minor radius a.sub.e1 and equivalent ellipse major radius a.sub.e2, the gap height t.sub.g, and the radiation plate thickness t.sub.m are: $t_{\mathrm{ge}}={V_{\mathrm{DC}\_n}\left(\frac{V_{\mathrm{DC}}}{V_C}\right)}^{-1}{t}_{\mathrm{ge}\_n}$ $t_{\mathrm{ge}\_n}\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)\frac{\sqrt{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}}}{\begin{array}{c}0.9961-1.0468\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}+0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2+\\ 0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6\end{array}}$ $a_{\mathrm{eq}1}=\left(10\sqrt[4]{\frac{Y_0}{15\left(1-{}^2\right)P_0}}\right)V_{\mathrm{DC}\_n}{a}_{1n}$ $a_{\mathrm{eq}2}=\left(10\sqrt[4]{\frac{Y_0}{15\left(1-{}^2\right)P_0}}\right)V_{\mathrm{DC}\_n}{a}_{2n}$ $t_m=5{V_{\mathrm{DC}\_n}\left(\frac{V_{\mathrm{DC}}}{V_C}\right)}^{-1}{t}_{m\_n}$ wherein Y.sub.0 is a Young's modulus of a material comprising the radiation plate and is a Poisson's ratio of the material comprising the radiation plate.

40. The microphone system of claim 39, wherein the normalized gap radius a.sub.n corresponds to a normalized minimum gap radius a.sub.n_min that is within the range for a.sub.n, the normalized equivalent ellipse major radius a.sub.2n corresponds to a normalized equivalent ellipse minimum major radius a.sub.2m_min, the normalized radiation plate thickness t.sub.m_n corresponds to a normalized minimum radiation plate thickness t.sub.m_n_min that is within the range for t.sub.m_n, K is a selected scaling constant, X.sub.P is a static deflection of the center of the radiation plate, $g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)$ function of $\frac{X_P}{t_{\mathrm{ge}}},\mathrm{and}{g}^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)$ is a function which is the first derivative of $\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\text{:}$ $g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)=\frac{{\tanh }^{-1}\left(\sqrt{X_P\text{/}{t}_{\mathrm{ge}}}\right)}{\sqrt{X_P\text{/}{t}_{\mathrm{ge}}}}$ $g^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)=\frac{1}{2\frac{X_P}{t_{\mathrm{ge}}}}\left(\frac{1}{1-\frac{X_P}{t_{\mathrm{ge}}}}-g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\right)$ $\frac{a_{\mathrm{n\_min}}}{t_{\mathrm{m\_n}\mathrm{\_min}}}=\sqrt[4]{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}/\frac{X_P}{t_{\mathrm{ge}}}}$ $\frac{V_{\mathrm{DC}}^2}{V_C^2}{\left(0.9961-1.0468\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}+0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2+0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6\right)}^{2}2g^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)-3\left(\frac{X_P}{t_{\mathrm{ge}}}-\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)0\mathrm{for}\frac{X_P}{t_{\mathrm{ge}}}>\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}{a}_n=\left(K^3\right)a_{\mathrm{n\_min}}{a}_{2\mathrm{n\_min}}=\left(\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)a_{\mathrm{n\_min}}{t}_{\mathrm{m\_n}}=\left(K^4\right)t_{\mathrm{m\_n}\mathrm{\_min}}.$

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The disclosed inventive subject matter will be described with reference to the accompanying drawings, which show important sample embodiments and which are incorporated in the specification hereof by reference, wherein:

(2) FIG. 1 schematically shows a cross-section of a prior art example of a pressure compensated MEMS microphone.

(3) FIG. 2 schematically shows a cross-section of a prior art example of a pressure compensated electret microphone.

(4) FIG. 3 schematically shows an example of a cross section view of a Micromachined Capacitive Microphone (MCM) with an undeflected radiation plate.

(5) FIG. 4 schematically shows an example of a cross section view of a MCM with a depressed radiation plate.

(6) FIG. 5 shows a graph of the relationship between the ratio of the bias voltage to the collapse voltage in a vacuum V.sub.DC/V.sub.r and the normalized static displacement of the center of the radiation plate 310 X.sub.P/t.sub.ge at the electromechanical equilibrium (the equilibrium point).

(7) FIG. 6 shows a graph of the relationship between normalized effective gap height t.sub.ge_n and normalized static mechanical force F.sub.Peb/F.sub.Peg for an MCM.

(8) FIG. 7 shows a lin-log semi-log graph of the relationship between the relevant maximum normalized radiation plate radius-to-thickness ratio

(9) ${\left(\frac{}{t_m}\right)}_{N\_\mathrm{ma}x}$
that enables an MCM to meet the elastic linearity constraint, and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C.

(10) FIG. 8A shows a log-lin semi-log graph of the relationship between normalized minimum relevant gap radius .sub.n_min that enables an MCM to meet the elastic linearity constraint, and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C.

(11) FIG. 8B shows a log-lin semi-log graph of the relationship between normalized minimum radiation plate thickness t.sub.m_n_min that enables an MCM to meet the elastic linearity constraint, and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C.

(12) FIG. 9 shows a semi-log graph of the relationship between normalized Open Circuit Receive Voltage Sensitivity (OCRV) and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C, where parasitic capacitance C.sub.P divided by clamped capacitance C.sub.0 equals zero (C.sub.P/C.sub.0=0).

(13) FIG. 10 shows a log-lin semi-log graph of the relationship between normalized input capacitance C.sub.in_n and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C, where the relevant normalized radius-to-thickness ratio equals the relevant maximum normalized radius-to-thickness ratio

(14) ${\left(\frac{}{t_m}\right)}_N,$
which enables linearly elastic operation.

(15) FIG. 11 shows a graph of the relationship between normalized Short Circuit Receive Current Sensitivity (SCRC) and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C.

(16) FIG. 12 shows a graph of the relationship between normalized Short Circuit Receive Current Sensitivity (SCRC) per square meter and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C.

(17) FIG. 13 shows a graph of the relationship between relevant normalized minimum gap radius .sub.n_min and normalized effective gap height t.sub.ge_n for various values of the relative bias voltage level V.sub.DC/V.sub.C.

(18) FIG. 14 shows a graph of the relationship between normalized minimum radiation plate thickness t.sub.m_n_min and normalized effective gap height t.sub.ge_n for various values of the relative bias voltage level V.sub.DC/V.sub.C.

(19) FIG. 15 shows a graph of the relationship between relevant normalized gap radius .sub.n and normalized effective gap height t.sub.ge_n for various values of the relative bias voltage level V.sub.DC/V.sub.C and various values of the scaling constant K.

(20) FIG. 16 shows a graph of the relationship between normalized radiation plate thickness t.sub.m_n and normalized effective gap height t.sub.ge_n for various values of the relative bias voltage level V.sub.DC/V.sub.C and various values of the scaling constant K.

(21) FIG. 17 shows an example view comparing multiple gap shapes.

(22) FIG. 18 shows an example view comparing multiple gap shapes.

(23) FIG. 19 shows a graph of the relationships between the minor axis a.sub.1 of an ellipse-shaped gap and the radius a of a seed circle gap, between the major axis a.sub.2 of the ellipse-shaped gap and the radius a of the seed circle gap, and between the area of the ellipse-shaped gap and the area of the seed circle gap.

(24) FIG. 20 shows an example view comparing multiple gap shapes.

DETAILED DESCRIPTION OF SAMPLE EMBODIMENTS

(25) The numerous innovative teachings of the present application will be described with particular reference to presently preferred embodiments by way of example, and not of limitation. The present application describes inventive scope, and none of the statements below should be taken as limiting the claims generally.

(26) The present application discloses new approaches to capacitive MEMS microphones with a sealed gap, and to design of such microphones.

(27) Some exemplary parameters will be given to illustrate the relations between these and other parameters. However it will be understood by a person of ordinary skill in the art that these values are merely illustrative, and will be modified by scaling of further device generations, and will be further modified to adapt to different materials or architectures if used.

(28) A capacitive MEMS microphone with a sealed gap is disclosed herein which is preferably an airborne microphone configured for off-resonance operation (described below with respect to FIG. 3). Such microphones are referred to herein as Micromachined Capacitive Microphones (MCM).

(29) The inventors have made the surprising discovery that MCMs can be constructed with gap and vibrating membrane dimensions that result in robust uncollapsed, linearly elastic operation with high sensitivity and little or no self-noisein some embodiments, an SNR of approximately 94 dBA can be achieved across the audible spectrum! Further, because MCMs are sealed, they are waterproof, in some embodiments down to tens of meters in depth.

(30) The inventors have also made the surprising discovery that certain MCM operating parameters and MCM gap and vibrating membrane dimensions are deterministically related, such that MCM dimensions which will result in high sensitivity (or optimal sensitivity for selected operating parameters) can be determined from selected operating parameters. In other words, microphone design can be performed backwards for MCMs, starting from selected performance requirements, which can be used to determine corresponding physical microphone dimensions which will result in those performance characteristics! Moreover, if an MCM microphone is made from solid materials suitable for MEMS device fabrication, the determined dimensions will generally be unaffected by the particular materials used!

(31) MCMs are related to CMUTs, but preferably operate in an audible range. MCMs can be used in, for example, airborne consumer and professional products, such as computers, ear phones, hearing aids, mobile phones, wireless equipment and wideband precision acoustic measurement and recording systems. Preferred MCM embodiments comprise a relatively simple structure, which can be fabricated at low cost using standard MEMS processes.

(32) In an MCM, dimensions of the microphone that optimize microphone sensitivity, SNR and other performance characteristics can be determined by selecting values for three operating parameters (an operating point): normalized static mechanical force F.sub.Peb/F.sub.Peg, bias voltage of electrodes V.sub.DC, and relative bias voltage V.sub.DC/V.sub.C. (When not specified, sensitivity herein refers to the Open Circuit Receive Voltage (OCRV) sensitivity). The operating point, including the collapse voltage V.sub.C, is further described below, along with the relationships between the operating point, MCM dimensions, MCM sensitivity and other MCM parameters. Further, the operating point can be used to determine normalized values for microphone dimensions, which are independent of properties of materials used in fabricating the microphone. De-normalized microphone dimensions (physical dimensions for fabrication) can then be determined from normalized dimensions using elastic properties (Young's modulus and Poisson's ratio) of a vibrating element (radiation plate), a static differential pressure between the gap and the ambient atmosphere (referred to herein as the ambient), and the permittivities of insulator layers connected to gap-facing sides of the radiation plate. These relationships are described below.

(33) A model relating various dimensions and properties of CMUTs is developed in H. Kymen, A. Atalar, E. Aydo{hacek over (g)}du, C. Kocaba, H. K. O{hacek over (g)}uz, S. Olum, A. zgrlk, A. nlgedik, An improved lumped element nonlinear circuit model for a circular CMUT cell, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 59, no. 8, pp. 1791-1799, August 2012, which is incorporated herein by reference (and referred to herein as the Circuit Model reference). This model is further developed in H. Kymen, A. Atalar and H. K. O{hacek over (g)}uz, Designing Circular CMUT Cells Using CMUT Biasing Chart, 2012 IEEE International Ultrasonics Symposium Proceedings pp. 975-978, Dresden, October, 2012 (the CMUT Design reference). As MCM structure is based on principles of CMUT operation, the model developed in the Circuit Model and CMUT Design references is relevant to MCM design. However, the relationships described herein enabling determination of MCM measurements and OCRV sensitivity from an operating point were not stated in the Circuit Model and CMUT Design references.

(34) A single capacitive microphone, such as an MCM, is also called a cell. A microphone system can comprise multiple cells.

(35) An MCM with a circular sealed gap, and processes for determining dimensions of such an MCM to produce an optimum OCRV when the MCM is operated at a particular operating point, are disclosed in U.S. patent application Ser. No. 15/939,077, which is incorporated herein by reference (and referred to herein as the Circular Gap reference).

(36) FIG. 3 schematically shows an example of a cross section of a Micromachined Capacitive Microphone 300 (MCM), comprising a capacitive electroacoustic microphone with a sealed gap 302. As shown in FIG. 3, an MCM 300 preferably comprises a circular gap 302 (or other shape of gap 302, as further described below) fabricated (e.g., machined or etched) into a surface of a substrate 304, with the substrate 304 at the bottom of the gap 302 forming a back plate 306. The back plate 306 is made of a solid material suitable for use in manufacturing MEMS microphones, such as a metal, a conducting, semi-conducting or insulating ceramic, or a crystalline or polycrystalline material. A bottom electrode 308 is formed over the back plate 306, e.g., using a metallization technique.

(37) A vibrating element in a microphone that is used to measure acoustic energy is generally called a membrane or a radiation plate depending on the vibrating element's radius-to-thickness ratio. If the vibrating element's radius-to-thickness ratio is less than a threshold (which different authorities specify as, for example, 40, 80 or 100), then the vibrating element is a radiation plate; otherwise, it is a membrane. MCMs 300 will generally use a vibrating element with a radius-to-thickness ratio less than 40. (This is discussed below with respect to FIG. 7, using the scaling constant term first described with respect to Equation 29). Therefore, the vibrating element in MCMs described herein is referred to as a radiation plate.

(38) A radiation plate 310 of total thickness t.sub.m (thickness of membrane) is clamped to the back plate 306 at the aperture of the gap 302 (the upper side of the gap 302, that is, the side distant from the back plate 306), preferably at the rim of the gap's 302 aperture, such that the gap 302 is sealed (Total thickness refers to t.sub.m being the sum of the thickness of the radiation plate 310, plus any electrodes or insulator layers, further described below, which are attached to it). To implement this clamping and seal, the substrate 304 and the radiation plate 310 are mechanically coupled, e.g., by bonding, wafer bonding or sacrificial layer processing. The gap 302 is preferably completely (hermetically) sealed, so that no air (or other gas, dust or other material) can pass between the gap 302 and the ambient. The radiation plate 310 can be made of a solid material generally suitable for MEMS manufacture, such as a metal, a conducting, semi-conducting or insulating ceramic, or a crystalline or polycrystalline material.

(39) The radiation plate 310 can comprise multiple layers of different materials, such as a metal layer (or layers) for an electrode, a layer for compliance (C.sub.RM), and an insulator layer. The elastic properties of one layer will generally be more significant than the elastic properties of the other layers, since the other layers will generally be comparatively thin. The combined effects of multilayer structures on elastic behavior of a vibrating element in a microphone are described by: M. Funding la Cour, T. L. Christiansen, J. A. Jensen, Fellow, IEEE, and E. V. Thomsen, Electrostatic and Small-Signal Analysis of CMUTs With Circular and Square Anisotropic Plates, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 62, no. 8, pp. 1563-1579, 2015 (the Anisotropic Plates reference). This reference provides an approach to treating a multilayered vibrating element as an equivalent single layer vibrating element, and determining a Young's modulus and Poisson's ratio for the equivalent single layer vibrating element.

(40) The radiation plate 310 can have an elliptic shape, corresponding to an ellipse-shaped gap 302. In this case the elasticity of the radiation plate 310 is modified by a term that is a function of the aspect ratio of the radiation plate 310. The aspect ratio .sub.e, of the radiation plate 310 is defined as the ratio of the major radius of the radiation plate 310 to the minor radius of the radiation plate 310. This modification is described by: A. W. Leissa, Vibration of Plates, Scientific and Technical Information Division, National Aeronautics and Space Administration, 1969, p. 39.

(41) Airborne MCMs 300 (MCMs operated in air) are preferably operated off-resonance. This is because an MCM 300 operated on-resonance would have a high sensitivity peak, but the bandwidth would be relatively narrow (in some embodiments, too narrow for typical consumer electronics implementations such as cellular phone microphones).

(42) The gap 302 has the same planar geometry as the radiation plate 310. Accordingly, the gap 302 has the same shape as the radiation plate 310 (in a plane parallel to the radiation plate 310), and the shape of the gap 302 can be described by the same values used to describe the shape of the radiation plate 310 (such as major and minor radii, or radius, or apothem, depending on the shape). The gap 302 is also described by a gap height t.sub.g. The gap height t.sub.g is the distance between the uppermost material at the bottom of the gap 302 and the lowermost material at the top of the gap 302 when the radiation plate 310 is undeflected. The radiation plate 310 is undeflected when the normalized static mechanical force F.sub.Peb/F.sub.Peg equals zero (generally, when there is no static pressure difference between the gap 302 and the ambient), and the bias voltage V.sub.DC is zero or the relative bias level V.sub.DC/V.sub.C equals zero. F.sub.Peb/F.sub.Peg, V.sub.DC/V.sub.C and the collapse voltage V.sub.C are further described below). A smaller gap 302 radius or a larger radiation plate 310 thickness t.sub.m will increase the stiffness of the radiation plate 310.

(43) FIG. 17 shows an example view comparing multiple gap 302 shapes. A circular (circle-shaped) gap 1700 is shown as a basis for comparison. The circular gap 1700 has a radius a. An octagonal gap 1710 has an apothem (also called inradius) r.sub.8. Herein, the apothem of a regular polygon refers to the length of a line segment from the polygon's center to the midpoint of one of its sides. The r refers to the apothem, and the 8 refers to the number of sides of the corresponding polygon. A square gap 1720 has an apothem r.sub.4. The octagonal gap 1710 has an equivalent radius a.sub.eq, which is the same as radius a. An octagon 1730 with apothem a.sub.eq is shown. The square gap 1720 also has equivalent radius a.sub.eq. A square 1740 with apothem a.sub.eq is shown. Regular convex polygonal (polygon-shaped) gaps 302 have an equivalent radius (or equivalent apothem) a.sub.eq. The equivalent radius a.sub.eq of the regular convex polygon can be used, similarly to the radius a of a circular gap 1700 as described in the Circular Gap reference, in determining MCM 300 dimensions which will enable operation of the MCM 300 at a corresponding operating point with an optimum OCRV. An equivalent radius a.sub.eq refers to the radius of a circle which would be equivalent to the polygon for purposes of determination of MCM 300 dimensions to optimize OCRV at a corresponding operating point. Accordingly (as further described below), a regular convex polygon-shaped gap 302 has an equivalent radius a.sub.eq which can be determined from the apothem of the gap 302, an area of the gap 302 (of the polygon), and the gap height t.sub.g.

(44) FIG. 18 shows an example view comparing multiple gap 302 shapes. An elliptical gap 1800 has a minor radius a.sub.1 1810 and a major radius a.sub.2 1820. A circular gap 1830 has a radius a. A major radius 1820 and a minor radius 1810 (or a major radius to minor radius aspect ratio) of an elliptical gap 1800 can be determined to enable operation of the MCM 300 at a corresponding operating point with an optimum OCRV, as further described below.

(45) FIG. 19 shows a graph of the relationships between the minor axis a.sub.1 of an ellipse-shaped gap 302 and the radius a of a seed circle gap 302, between the major axis a.sub.2 of the ellipse-shaped gap 302 and the radius a of the seed circle gap 302, and between the area of the ellipse-shaped gap 302 and the area of the seed circle gap 302. A seed circle is similar to an equivalent circle, and equivalent circle may sometimes be used instead of seed circle herein (but not vice versa). A seed circle of an MCM 300 with an ellipse-shaped gap 302 is the circle of radius a which enables an MCM 300 which has the same radiation plate 310 thickness t.sub.m, has the same effective gap height t.sub.ge, and is operated at the same operating point as the MCM 300 with an ellipse-shaped gap 302 to have the same OCRV sensitivity (determination of OCRV is further described below). A minimum seed circle with radius a.sub.min (the smallest seed circle with the same OCRV which enables operation in a linear elastic regime, as further described below) can also be determined. Seed circle scaling (corresponding to scaling of the ellipse-shaped gap 302) can be performed as described in Equations 29-32, below.

(46) FIG. 20 shows an example view comparing multiple gap 302 shapes. An elliptical (ellipse-shaped) gap 2000 is shown as a basis for comparison. The elliptical gap 2000 has a minor radius a.sub.eq1 2030 (or a.sub.1) and a major radius a.sub.eq2 2040 (or a.sub.2). A regular elliptic convex octagonal gap 2010 (an example of a regular elliptic convex polygonal or polygon-shaped gap 302; a rectangle 2020 is another example of a regular elliptic convex polygon-shaped gap 302) has a minor apothem r.sub.81 and a major apothem r.sub.82. Regular elliptic convex polygon-shaped gaps 302 have an equivalent minor radius a.sub.eq1 and an equivalent major radius a.sub.eq2. Equivalent radii a.sub.eq1 and a.sub.eq2 refer to the radii of an ellipse which would be equivalent to the polygon for purposes of determination of MCM 300 dimensions to optimize OCRV at a corresponding operating point. An equivalent major radius is the major radius of an equivalent ellipse, and an equivalent minor radius is the minor radius of an equivalent ellipse. Accordingly, an equivalent ellipse is an ellipse with a major radius to minor radius ratio which is the same as the regular elliptic convex polygon's major apothem to minor apothem ratio, and with major and minor radii which will result in the MCM 300 producing the same OCRV if the ellipse-shaped gap 302 is substituted for the regular elliptic convex polygon-shaped gap 302.

(47) The relevant gap 302 radius or relevant radiation plate 310 radius refers to the radius a if the gap 302 is circle-shaped, or the equivalent radius a.sub.eq if the gap 302 is regular convex polygon-shaped. Note that the gap 302 and the radiation plate 310 have the same radius (or major and minor radii, or equivalent radius).

(48) The relevant minor gap 302 radius .sub.1 or relevant minor radiation plate 310 radius .sub.1 refers to the minor radius a.sub.1 if the gap 302 is ellipse-shaped, or the equivalent minor radius a.sub.eq1 if the gap 302 is regular elliptic convex polygon-shaped.

(49) The relevant major gap 302 radius .sub.2 or relevant major radiation plate radius .sub.2 refers to the major radius a.sub.2 if the gap 302 is ellipse-shaped, or the equivalent major radius a.sub.eq2 if the gap 302 is regular elliptic convex polygon-shaped.

(50) A top electrode 312 is fixedly connected to the radiation plate 310, or can be the radiation plate 310 itself if the radiation plate 310 is made of a conductive material. The top electrode 312 can be formed on either surface of the radiation plate 310, or can be formed within the radiation plate 310 if the radiation plate 310 is made of a dielectric material. The top electrode 312 is preferably formed using a metallization technique (if the radiation plate 310 is not itself the top electrode 312). Preferably, the bottom electrode 308 fully covers the back plate 306 (the bottom of the gap 302; that is, the back plate 306 is fully electroded), and the top electrode 312 fully covers the portion of the radiation plate 310 that faces and touches the gap 302 (the radiation plate 310 is fully electroded). The voltage across the electrodes 308, 312 is a bias voltage V.sub.DC. Generally, at lower bias voltages V.sub.DC, better microphone performance is achieved if the back plate 306 and radiation plate 310 are fully electroded. Electrodes 308, 312 can also be smaller than the gap 302, down to 80% of the size of the gap 302, as further explained below. Electrodes 308, 312 which are smaller than the gap 302 are preferably concentric with the gap 302.

(51) There is preferably a first dielectric insulator layer 314 of thickness t.sub.i1 fixedly attached to and covering the gap 302 side of the bottom electrode 308, and a second dielectric insulator layer 316 of thickness t.sub.i2 fixedly attached to and covering the gap 302 side of the combination of the radiation plate 310 and the top electrode 312. In alternative embodiments, both of the dielectric insulator layers 314, 316 can be located on the gap 302 side of either the bottom electrode 308, or the combination of the radiation plate 310 and the top electrode 312. The insulating layers 314, 316 can be made of an insulating material suitable for use in a MEMS microphone (generally, any such material), such as an insulating ceramic, polymer, crystalline or polycrystalline material. One or both of the insulator layers 314, 316 can be electrets.

(52) Electrets and certain CMUT performance measurements are addressed in H. Kymen, A. Atalar, Itir Kymen, A. S. Tadelen, A. nlgedik, Unbiased Charged Circular CMUT Microphone: Lumped Element Modeling and Performance, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, Vol. 65, no. 1, pp. 60-71, Nov. 14, 2018, which is incorporated herein by reference (and referred to herein as the Electret and Performance reference). The Electret and Performance reference and the Anisotropic Plates reference show that noise (losses) in a CMUT (a capacitive MEMS microphone with a sealed gap) are very smallin some embodiments, approximately 0 dBA.

(53) An MCM 300 is a capacitive microphone. Capacitive microphone operation uses the fact that if a voltage (electric potential) is applied across two parallel conducting plates (the bottom and top electrodes 308, 312) separated by a gap 302, the parallel conducting plates 308, 312 will attract each other electrostatically via the electromechanical attraction force. The radiation plate 310 is clamped (fixedly connected) to the substrate 304 at the rim of the gap 302, and the top electrode 312 is attached to (fixedly connected to or comprised of) the radiation plate 310. Because the radiation plate 310 is clamped to the substrate 304 at the rim of the gap 302, the spring reaction (elastic restoring force) due to the elasticity of the radiation plate 310 resists the electromechanical force exerted by the top electrode 312. That is, the attraction between the electrodes 308, 312 pulls the radiation plate 310 down into the gap 302, and the elasticity of the radiation plate 310 pulls the radiation plate 310 back towards a resting position. The voltage across the electrodes 308, 312 is the bias voltage V.sub.DC. For a given bias voltage V.sub.DC, the electromechanical force and elastic restoring force are balanced when the center of the radiation plate 310 is displaced by an equilibrium displacement distance (also called the equilibrium point).

(54) As stated, the voltage across the electrodes 308, 312 is a bias voltage V.sub.DC. If the bias voltage V.sub.DC is increased beyond a limit for uncollapsed microphone operation called the collapse voltage V.sub.C, the elastic restoring force is unable to prevent the electromechanical force from causing the center of the radiation plate 310 to collapse into (make physical contact with) the bottom of the gap 302. In example embodiments as shown in FIG. 3, this would comprise the first insulator layer 314 touching the second insulator layer 316. Generally, microphone SNR is significantly decreased in collapsed operation. The ratio between the bias voltage V.sub.DC and the collapse voltage V.sub.C is called the relative bias level V.sub.DC/V.sub.C.

(55) Preferably, the sealed gap 302 contains a very low pressure environment (a vacuum, for example, less than 10 mbar). If the gap 302 contains a vacuum, there is a static pressure difference P.sub.0 between the ambient environment (on the other side of the radiation plate 310 from the gap 302) and the gap 302 which results in a net static force F.sub.Peb pushing the radiation plate 310 into the gap 302.

(56) At equilibrium, when sound (a time varying pressure signal) is incident on the radiation plate 310 (accordingly, received by the MCM 300), the radiation plate 310 vibrates and the displacement of the radiation plate 310 changes (e.g., oscillates) around the equilibrium point. This movement causes variation of the microphone capacitance (the capacitance between the top and bottom electrodes 308, 312). Variation in the microphone capacitance, combined with the charge stored on the capacitance due to the bias voltage V.sub.DC, causes a voltage across the output terminals of the microphone to vary in proportion to the incident sound pressure signal. This output voltage can be amplified, measured, stored, and used to reproduce (play back) the sound originally received by the microphone (the MCM 300).

(57) An operating point is defined herein as a triplet of selected values comprising the applied bias voltage V.sub.DC, the relative bias level V.sub.DC/V.sub.C, and the normalized static mechanical force F.sub.Peb/F.sub.Peg (further described below with respect to FIG. 4 and Equation 8). As disclosed below, an operating point uniquely determines dimensions of an MCM 300 that will result in optimal sensitivity of the MCM 300 at that operating point. For example, an operating point can be used to determine an MCM's 300 relevant gap 302 radius or relevant major and minor gap 302 radii .sub.2 and .sub.1, radiation plate 310 thickness t.sub.m, and effective gap 302 height t.sub.ge (as described below with respect to, for example, Equations 11-21). Alternatively, the operating point can be used to determine an MCM's relevant minimum gap 302 radius .sub.min or relevant minimum major and minor gap 302 radii .sub.2_min and .sub.1_min, minimum radiation plate 310 thickness t.sub.m_min, and effective gap 302 height t.sub.ge, along with a range for relevant radiation plate 310 radius-to-thickness ratio /t.sub.m (described below) enabling the MCM 300 to maintain elastic linear operation (operation within the elastic linearity constraint, as described below with respect to FIGS. 7, 8A and 8B). Dimensions as determined yield a resulting (and optimal) open circuit receive voltage (OCRV) sensitivity at a corresponding operating point. Dimensions can then be adjusted to enable robust elastic linear operation without compromising the OCRV sensitivity. These results take advantage of the very low noise floor (in some embodiments, approximately 0 dBA) and high SNR (in some embodiments, approximately 94 dBA) in airborne sealed gap 302 MCMs 300 as disclosed herein.

(58) The relevant radius-to-thickness ratio /t.sub.m or relevant radiation plate 310 radius-to-thickness ratio /t.sub.m refers to the radius-to-thickness ratio a/t.sub.m if the gap 302 is circle-shaped; the equivalent-radius-to-thickness ratio a.sub.eq/t.sub.m if the gap 302 is regular convex polygon-shaped; the major-radius-to-thickness ratio a.sub.2/t.sub.m if the gap 302 is ellipse-shaped; or the equivalent-major-radius-to-thickness ratio a.sub.eq2/t.sub.m, if the gap 302 is regular elliptic convex polygon-shaped.

(59) The operating point can be selected: for example, to minimize bias voltage V.sub.DC, and/or to correspond to a selected OCRV sensitivity, relevant gap radius (or other physical dimension), or other desired performance characteristic. Selectable operating point values, and optimality of results with respect to the selected operating point, are not limited by materials to be used in fabrication of the radiation plate 310 or insulator layers 314, 316. Such components in an MCM 300 can be made out of materials suitable for manufacture of similar components in MEMS devices (in preferred embodiments, any such materials). Normalized dimensions of the MCM 300, which are not dependent on material properties, can be determined directly from the operating point. De-normalized dimensions used before MCM 300 fabrication can then be determined using properties of materials selected for use in MCM 300 components. As a result, dimensions, sensitivity and other properties of the MCM, including relevant gap radius and radiation plate 310 thickness t.sub.m, effective gap 302 height t.sub.ge, and Open Circuit Receive Voltage Sensitivity (OCRV), as well as other microphone performance parameters, are independent of the particular material(s) used to fabricate the radiation plate 310 and the insulator layers 314, 316.

(60) Also described herein are conditions enabling the relevant gap 302 radius , the radiation plate 310 thickness t.sub.m, and the ratio between the relevant gap 302 radius and the radiation plate 310 thickness /t.sub.m to be rescaled, within ranges and with relationships determined by the operating point, while maintaining the optimal OCRV sensitivity for that operating point.

(61) FIG. 4 shows an example visual representation 400 of an analytical model for a Micromachined Capacitive Microphone 300 (MCM), using a cross-section of the MCM 300. As shown in FIG. 4, the effective gap 302 height t.sub.ge, which is an electrical dimension of the gap 302 used in modeling the MCM 300, depends on the gap height t.sub.g, the relative permittivity of the first insulator layer 114 .sub.r_i1, and the relative permittivity of the second insulator layer 116 .sub.r_i2. These relative permittivities are the ratios between the respective permittivities of the insulator layers 314, 316 and the permittivity of free space (Permittivity is the resistance of a medium to forming an electric field in that medium. The gap 302 preferably contains a vacuum, which has a relative permittivity of 1). The effective gap height t.sub.ge is determined as shown in Equation 1. Note that if the entire gap 302 height t.sub.g and insulator height (t.sub.i1 plus t.sub.i2) comprised vacuum, the effective gap height t.sub.ge would equal the gap height t.sub.g.

(62) $\begin{array}{cc}{t}_{\mathrm{ge}}=t_g+\frac{t_{i1}}{{\mathrm{.Math.}}_{r\_i1}}+\frac{t_{i2}}{{\mathrm{.Math.}}_{r\_i2}}& \mathrm{Equation}1\end{array}$

(63) Insulator layer 314, 316 thicknesses and materials (corresponding to permittivities) can be selected after the effective gap 302 height t.sub.ge is determined. That is, appropriate materials for insulator layer 314, 316 fabrication can be selected to keep insulator layer 314, 316 thickness (t.sub.i1, t.sub.i2) small relative to the gap 302 height t.sub.g. Once effective gap 302 height t.sub.ge is determined, then gap 302 height t.sub.g can be determined such that gap 302 height t.sub.g is greater than the static displacement of the center of the radiation plate 310 X.sub.P, plus a margin for production tolerances and insulator layer 314, 316 thicknesses using selected insulator materials. The static displacement of the center of the radiation plate 310 X.sub.P is the deflection distance of the center of the radiation plate 310 from the effective gap height t.sub.ge at the equilibrium point. Higher relative permittivities of insulator layers 314, 316 generally correspond to thinner insulator layers 314, 316. The effective gap 302 height t.sub.ge is determined from the operating point as shown below in Equations 11 through 15.

(64) Microphones are more sensitive when the bias voltage V.sub.DC is larger. The effective gap 302 height t.sub.ge determines the level of bias voltage V.sub.DC that can be used, because higher bias voltages increase the deflection of the radiation plate 310, and sufficiently high bias voltages V.sub.DC will cause the radiation plate 310 to collapse. Voltage available on a device also limits bias voltage V.sub.DC. For example, some mobile phones are limited to about 14 volts available to mobile phone components. Electrets can provide, for example, 150 volts to 200 volts bias voltage. The Electret and Performance reference is relevant to implementation of electrets in a capacitive MEMS microphone with a sealed gap.

(65) In an MCM 100, the bias voltage V.sub.DC, the static displacement of the center of the radiation plate 310 X.sub.P, and the net static force on the radiation plate 310 due to the ambient static pressure F.sub.Peb are related, in static electromechanical equilibrium (at the equilibrium point), as shown in Equation 9 (below).

(66) The relationship shown in Equation 9 is dependent on various properties of the MCM 300 (which are explained below), including the shape function of a deflected clamped circular plate g(X.sub.P/t.sub.ge) (also referred to as g(u)), which is proportional to the capacitance of the MCM 300; the transduction force (proportional to g(u)), the first derivative of g(u)); the collapse voltage in vacuum V.sub.r (a reference voltage); the normalized static mechanical force F.sub.Peb/F.sub.Peg; the Young's modulus Y.sub.0 (stiffness) and Poisson's ratio (signed ratio of transverse strain to axial strain) of the radiation plate 310; the differential pressure P.sub.0 between the ambient static pressure and the pressure in the gap 302; the clamped capacitance C.sub.0, and the compliance of the radiation plate 310 C.sub.Rm (the inverse of the stiffness of the radiation plate 310).

(67) The transduction force is the force generated on the radiation plate 310 when a bias voltage V.sub.DC is applied. Equation 3 expresses the transduction force in terms of the effect the bias voltage V.sub.DC has on the shape of the radiation plate 310 (rather than in terms of the bias voltage V.sub.DC). The variable u corresponds to the ratio of the static displacement to the effective gap height X.sub.P/t.sub.ge.

(68) $\begin{array}{cc}g\left(u\right)=\frac{{\tanh }^{-1}\left(\sqrt{u}\right)}{\sqrt{u}}& \mathrm{Equation}2\\ g^{}\left(u\right)=\frac{1}{2u}\left(\frac{1}{1-u}-g\left(u\right)\right)& \mathrm{Equation}3\\ g^{}\left(u\right)=\frac{1}{2u}\left(\frac{1}{{\left(1-u\right)}^2}-3g^{}\left(u\right)\right)& \mathrm{Equation}4\end{array}$

(69) An MCM 300 which has a regular convex polygon-shaped gap 302 having n sides can be approximated by an MCM 300 having a circle-shaped gap 302 with equivalent radius a.sub.eq given in Equation 5:

(70) $\begin{array}{cc}{a}_{\mathrm{eq}}=r_n\sqrt[4]{\frac{n}{}\tan \left(\frac{}{n}\right)}& \mathrm{Equation}5\end{array}$
wherein r.sub.n is the apothem (or inradius) of the regular convex polygon. The equivalent radius a.sub.eq is the radius of a circle of area equal to the geometric mean of the area of the polygon's in circle (inscribed circle) and the area of the polygon. Examples with n=4 and n=8 are depicted in FIG. 17, together with the equivalent circle of radius a.sub.eq.

(71) An MCM 300 comprising a regular elliptic convex polygon shaped gap 302 having n sides is approximated to an MCM 300 comprising an elliptic gap 302 with minor radius a.sub.eq1 and major radius a.sub.eq2 given in Equation 6A and 6B.

(72) $\begin{array}{cc}{a}_{\mathrm{eq}1}=r_{n1}\sqrt[4]{\frac{n}{}\tan \left(\frac{}{n}\right)}& \mathrm{Equation}6A\\ \mathrm{and}& \\ a_{\mathrm{eq}2}=r_{n2}\sqrt[4]{\frac{n}{}\tan \left(\frac{}{n}\right)}& \mathrm{Equation}6B\end{array}$
wherein r.sub.n1 is the minor apothem (inradius) of the regular convex polygon and r.sub.n2 is the major apothem (inradius) of the regular convex polygon. Examples with n=4 and n=8 are depicted in FIG. 20 together with the equivalent ellipse of minor radius a.sub.eq1 and major radius a.sub.eq2.

(73) Equation 9 shows the collapse voltage in vacuum V.sub.r for a fully electroded MCM 300. V.sub.r depends on dimensions of the MCM 300 and properties of the radiation plate 310. This model is also valid for MCMs 300 using electrodes 308, 312 which are between 80% and 100% of the size of the gap 302 area, if g(u) and its derivatives (that is, the terms used to determine the transduction force and the shape function of the radiation plate 310) are modified as shown in the Circuit Model reference.

(74) $\begin{array}{cc}{V}_r=8\frac{t_{\mathrm{ge}}^{3/2}{t}_m^{3/2}}{{}_2^2}\sqrt{\frac{1}{8}\left(3\frac{{}_2^4}{{}_1^4}+2\frac{{}_2^2}{{}_1^2}+3\right)}\sqrt{\frac{Y_0}{27{\mathrm{.Math.}}_0\left(1-{}^2\right)}}& \mathrm{Equation}7\end{array}$
wherein .sub.1 (minor radius) and .sub.2 (major radius) are equal to gap radius a if the gap 302 has a circular shape; wherein .sub.1 and .sub.2 are equal to equivalent gap radius a.sub.eq if the gap 302 has a regular convex polygon shape;
wherein .sub.1 is equal to minor gap radius a.sub.1 and .sub.2 is equal to major gap radius a.sub.2 if the gap 302 has an elliptic shape;
and wherein .sub.1 is equal to equivalent minor gap radius a.sub.eq1 and .sub.2 is equal to equivalent major gap radius a.sub.eq2 if the gap 302 has a regular elliptic convex polygon shape.

(75) As previously stated, P.sub.0 is the differential pressure between the ambient static pressure and the pressure in the gap 302. For example, if the gap 302 contains a vacuum and the ambient static pressure equals Standard Atmospheric Pressure (SAP), then P.sub.0 equals SAP.

(76) As previously stated, F.sub.Peb is the net static force on the radiation plate 310 due to the ambient static pressure, that is, the force on the radiation plate 310 due to the differential static pressure between the ambient static pressure and the pressure in the gap 302 P.sub.0. F.sub.Peg is the uniformly distributed force required to displace the center of the radiation plate 310 by the effective gap height t.sub.ge (that is, to cause the radiation plate 310 to collapse). Because t.sub.get.sub.g in uncollapsed operation (depending on whether there is an insulator layer 314, 316 between the electrodes 308, 312, see Equation 1), the normalized static mechanical force F.sub.Peb/F.sub.Peg1. The normalized static mechanical force F.sub.Peb/F.sub.Peg is given in Equation 8.

(77) $\begin{array}{cc}\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}=\frac{8}{\left(3\frac{{}_2^4}{{}_1^4}+2\frac{{}_2^2}{{}_1^2}+3\right)}3\frac{P_0\left(1-{}^2\right)}{16Y_0}\frac{{}_2^4}{t_m^4}\frac{t_m}{t_{\mathrm{ge}}}& \mathrm{Equation}8\end{array}$
wherein .sub.1 and .sub.2 are equal to gap radius a if the gap 302 has a circular shape;
wherein .sub.1 and .sub.2 are equal to equivalent gap radius a.sub.eq if the gap 302 has a regular convex polygon shape;
wherein .sub.1 is equal to minor gap radius a.sub.1 and .sub.2 is equal to major gap radius a.sub.2 if the gap 302 has an elliptic shape;
and wherein .sub.1 is equal to equivalent minor gap radius a.sub.eq1 and .sub.2 is equal to equivalent major gap radius a.sub.eq2 if the gap 302 has a regular elliptic convex polygon shape.

(78) A circle-shaped gap 302 (and radiation plate 310) is a special case of an ellipse-shaped gap 302. This special case occurs when .sub.1 and .sub.2 are equal. In this case, Equation 8 simplifies to Equation 8A and the normalized static mechanical force F.sub.Peb/F.sub.Peg for this special case is abbreviated as F.sub.b/F.sub.g as described in the Circular Gap reference:

(79) $\begin{array}{cc}\frac{F_b}{F_g}=3\frac{P_0\left(1-{}^2\right)}{16Y_0}\frac{{}_2^4}{t_m^4}\frac{t_m}{t_{\mathrm{ge}}}& \mathrm{Equation}8A\end{array}$

(80) In an MCM 300 in uncollapsed operation in which the gap 302 contains a vacuum, the normalized static mechanical force F.sub.Peb/F.sub.Peg can assume values between 0 (if the ambient static pressure is zero, so that differential static pressure P.sub.0=0; or if the radiation plate 310 is infinitely stiff, meaning

(81) 0$\frac{1}{C_{\mathrm{Rm}}}.fwdarw.)$
and the ratio between the gap 302 height and the effective gap 302 height t.sub.g/t.sub.ge. The limiting case F.sub.Peb/F.sub.Peg=1 means that the center of the radiation plate 310 is displaced by the effective gap 302 height t.sub.ge, which is not physically possible when there is an insulator layer 314, 316 between the electrodes 308, 312. (F.sub.Peb/F.sub.Peg is also zero in pressure compensated MEMS microphones.)

(82) The normalized static mechanical force F.sub.Peb/F.sub.Peg will generally be relatively low in an MCM 300 with a stiff radiation plate 310 (large

(83) $\frac{1}{C_{\mathrm{Rm}}},$
or with a compliant radiation plate 310 (large C.sub.Rm) and a large effective gap 302 height t.sub.ge. F.sub.Peb/F.sub.Peg will generally be relatively high if the ambient static pressure displaces the radiation plate 310 by a significant fraction of the effective gap 302 height t.sub.ge, which can occur, for example, in a MCM 300 with a compliant radiation plate 310, or with a stiff radiation plate 310 and a relatively small effective gap height t.sub.ge.

(84) FIG. 5 shows a graph 500 of the relationship between the ratio of the bias voltage to the collapse voltage in a vacuum V.sub.DC/V.sub.r and the normalized static displacement of the center of the radiation plate 310 X.sub.P/t.sub.ge at the electromechanical equilibrium (the equilibrium point). In FIG. 5, the solid curves correspond to operational domains in which the microphone will be in uncollapsed operation; the dotted line marks the transition between uncollapsed operation and collapsed operation; and the dotted curves correspond to operational domains in which the microphone will be in collapsed operation. The ratio of the bias voltage to the collapse voltage V.sub.DC/V.sub.r is given in Equation 9.

(85) $\begin{array}{cc}\frac{V_{\mathrm{DC}}}{V_r}=\sqrt{\frac{3\left(\frac{X_P}{t_{\mathrm{ge}}}-\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}{2g^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)}}\mathrm{for}\frac{X_P}{t_{\mathrm{ge}}}\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}& \mathrm{Equation}9\end{array}$

(86) Equation 9 shows that the static displacement of the center of the radiation plate 310 X.sub.P is equal to t.sub.ge(F.sub.Peb/F.sub.Peg) when the plate is electrically unbiased, so that V.sub.DC=0. This can also be viewed as the normalized static displacement of the center of the radiation plate 310 X.sub.P/t.sub.ge being equal to the normalized static mechanical force F.sub.Peb/F.sub.Peg when no bias voltage is applied, so that V.sub.DC=0.

(87) The collapse voltage V.sub.C depends on the normalized static mechanical force F.sub.Peb/F.sub.Peg, as well as the stiffness of the radiation plate 310 and the effective gap 302 height t.sub.ge. When the radiation plate 310 is displaced by ambient static pressure (accordingly, the MCM 300 is not in a vacuum), the collapse voltage V.sub.C is decreased from the collapse voltage in a vacuum V.sub.r. As shown in Equation 10, the collapse voltage V.sub.C, normalized to V.sub.r, depends only on F.sub.Peb/F.sub.Peg.

(88) $\begin{array}{cc}\frac{V_C}{V_r}0.9961-1.0468\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}+0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2+0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6& \mathrm{Equation}10\end{array}$

(89) As shown in Equations 11 through 21 below, the MCM 300 dimensions, that is, relevant gap radius , radiation plate 310 thickness t.sub.m and effective gap height t.sub.ge, can be expressed in terms of the operating point: normalized static mechanical force F.sub.Peb/F.sub.Peg, relative bias level V.sub.DC/V.sub.C, and bias voltage V.sub.DC.

(90) The effective gap 302 height t.sub.ge is determined as shown in Equation 11.

(91) $\begin{array}{cc}{t}_{\mathrm{ge}}=\frac{3}{2}\sqrt{\frac{{\mathrm{.Math.}}_0}{P_0}}{V}_r\sqrt{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}}=\frac{3}{2}\sqrt{\frac{{\mathrm{.Math.}}_0}{P_0}}{V_{\mathrm{DC}}\left(\frac{V_{\mathrm{DC}}}{V_C}\right)}^{-1}\left[{\left(\frac{V_C}{V_r}\right)}^{-1}\sqrt{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}}\right]& \mathrm{Equation}11\end{array}$

(92) Equation 11 can be rewritten to express the effective gap height t.sub.ge in terms of the normalized bias voltage V.sub.DC_n and the normalized effective gap height t.sub.ge_n, as shown in Equation 12. V.sub.DC_n is defined as shown in Equation 15.

(93) $\begin{array}{cc}{t}_{\mathrm{ge}}={V_{D\mathrm{C\_}n}\left(\frac{V_{DC}}{V_C}\right)}^{-1}{t}_{\mathrm{ge}\_n}& \mathrm{Equation}12\end{array}$

(94) The normalized effective gap height t.sub.ge_n is a function of normalized static mechanical force F.sub.Peb/F.sub.Peg, as shown in Equation 13.

(95) $\begin{array}{cc}{t}_{\mathrm{ge}\_n}\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)={\left(\frac{V_C}{V_r}\right)}^{-1}\sqrt{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}}& \mathrm{Equation}13\end{array}$

(96) FIG. 6 shows a graph 600 of the relationship between normalized effective gap height t.sub.ge_n and normalized static mechanical force F.sub.Peb/F.sub.Peg for a MCM 300, as described in Equation 14. Equation 13 can be rewritten using Equation 10 so that the normalized effective gap height t.sub.ge_n depends only on normalized static mechanical force F.sub.Peb/F.sub.Peg, as shown in Equation 14. Equations 12, 14 and 15 can be used to determine the effective gap height t.sub.ge using the operating point, independent of material properties. Equation 1 can be used to determine the gap height t.sub.g using the effective gap height t.sub.ge and permittivities of selected insulator layer 314, 316 materials.

(97) $\begin{array}{cc}{t}_{\mathrm{ge}\_n}\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)\frac{\sqrt{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}}}{\begin{array}{c}0.9961-1.0468\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)+0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2+\\ 0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6\end{array}}& \mathrm{Equation}14\end{array}$

(98) The normalized bias voltage V.sub.DC_n is related to the bias voltage V.sub.DC as shown in Equation 15. The normalized bias voltage V.sub.DC_n is approximately 1.410.sup.8 V.sub.DC (meters) for a sealed gap 302 containing vacuum when the ambient pressure is SAP.

(99) $\begin{array}{cc}{V}_{DC\_n}=\frac{3}{2}\sqrt{\frac{{\mathrm{.Math.}}_0}{P_0}}{V}_{DC}& \mathrm{Equation}15\end{array}$

(100) The radiation plate 310 thickness t.sub.m is related to the normalized static mechanical force F.sub.Peb/F.sub.Peg and the relative bias level V.sub.DC/V.sub.C using the relevant normalized radiation plate 310 radius-to-thickness ratio

(101) ${\left(\frac{}{t_m}\right)}_N$
and the normalized bias voltage V.sub.DC_n (see Equation 15), as shown in Equation 16.

(102) 0$\begin{array}{cc}{t}_m=5{V_{\mathrm{DC\_n}}\left(\frac{V_{\mathrm{DC}}}{V_C}\right)}^{-1}{\left(\frac{}{t_m}\right)}_N^{-4}{\left(\frac{V_C}{V_r}\right)}^{-1}{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^{3/2}& \mathrm{Equation}16\end{array}$
Relevant normalized values refer to the respective relevant values, normalized to remove dependence on material properties. For example, a relevant normalized gap 302 radius is the relevant gap 302 radius, after being normalized as described. Accordingly, the relevant normalized radius-to-thickness ratio

(103) ${\left(\frac{}{t_m}\right)}_N$
refers to the normalized radius-to-thickness ratio

(104) ${\left(\frac{a}{t_m}\right)}_N$
if the gap 302 is circle-shaped; the normalized equivalent-radius-to-thickness ratio

(105) ${\left(\frac{a_{\mathrm{eq}}}{t_m}\right)}_N$
if the gap 302 is regular convex polygon-shaped; the normalized major-radius-to-thickness ratio

(106) ${\left(\frac{a_2}{t_m}\right)}_N$
if the gap 302 is ellipse-shaped; and the normalized equivalent-major-radius-to-thickness ratio

(107) ${\left(\frac{a_{\mathrm{eq}2}}{t_m}\right)}_N$
if the gap 302 is regular elliptic convex polygon-shaped.

(108) The relevant normalized radiation plate 310 radius-to-thickness ratio

(109) ${\left(\frac{}{t_m}\right)}_N$
is related to the relevant radiation plate 310 radius-to-thickness ratio

(110) $\left(\frac{}{t_m}\right)$
as shown in Equation 17. The non-dimensional scaling constant

(111) $\sqrt[4]{\frac{16Y_0}{15\left(1-{}^2\right)P_0}}$
used in Equation 17 is dependent on the elastic properties of the radiation plate 310 (Young's modulus Y.sub.0 and Poisson's ratio ) and the static pressure difference P.sub.0 between the gap 302 and the ambient.

(112) $\begin{array}{cc}{\left(\frac{}{t_m}\right)}_N=\left(\frac{}{t_m}\right){\left(\sqrt[4]{\frac{16Y_0}{15\left(1-{}^2\right)P_0}}\right)}^{-1}{\left(\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\right)}^{-1}& \mathrm{Equation}17\end{array}$
wherein .sub.e=1 and is the gap 302 radius a if the gap 302 is circle-shaped; wherein .sub.e=1 and is the equivalent gap 302 radius a.sub.eq if the gap 302 is regular convex polygon-shaped;
wherein .sub.e is an aspect ratio a.sub.2/a.sub.1, and is equal to the major gap 202 radius a.sub.2 and the minor gap 302 radius is a.sub.1=a.sub.2/.sub.e if the gap 302 is ellipse-shaped;
wherein .sub.e is an aspect ratio a.sub.eq2/a.sub.eq1, and is equal to the equivalent major gap 302 radius a.sub.eq2 and the equivalent minor gap 302 radius a.sub.eq1=a.sub.eq2/.sub.e if the gap 302 is regular convex elliptic polygon-shape;

(113) The radiation plate 310 thickness t.sub.m can also be written in terms of the normalized radiation plate 310 thickness t.sub.m_n as shown in Equation 18.

(114) 0$\begin{array}{cc}{t}_m=5{V_{\mathrm{DC\_n}}\left(\frac{V_{\mathrm{DC}}}{V_C}\right)}^{-1}{t}_{m\_n}& \mathrm{Equation}18\end{array}$

(115) The normalized radiation plate 310 thickness t.sub.m_n is defined in Equation 19 in terms of the relevant normalized radius-to-thickness ratio

(116) ${\left(\frac{}{t_m}\right)}_N$
and the normalized static mechanical force F.sub.Peb/F.sub.Peg. The ratio of the collapse voltage to the collapse voltage in a vacuum V.sub.C/V.sub.r can be substituted for using Equation 10. Note that there is an inverse relationship between the size of the normalized radiation plate 310 thickness t.sub.m_n and the normalized ratio between the relevant gap radius and the radiation plate 310 thickness

(117) ${\left(\frac{}{t_m}\right)}_N.$

(118) $\begin{array}{cc}{t}_{m\_n}={\left(\frac{}{t_m}\right)}_N^{-4}{\left(\frac{V_C}{V_r}\right)}^{-1}{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^{3/2}& \mathrm{Equation}19\end{array}$

(119) The relevant gap radius is determined, as shown in Equation 21, using the relevant normalized gap radius .sub.n. The relevant normalized gap radius .sub.n is defined in Equation 20 in terms of the relevant normalized radiation plate 310 radius-to-thickness ratio

(120) ${\left(\frac{}{t_m}\right)}_N.$
The ratio between the collapse voltage and the collapse voltage in a vacuum V.sub.C/V.sub.r can be substituted for using Equation 10. Note that there is an inverse relationship between the relevant normalized gap radius .sub.n and the normalized ratio between the relevant gap 302 radius and the radiation plate 310 thickness

(121) ${\left(\frac{}{t_m}\right)}_N.$

(122) $\begin{array}{cc}{}_n={\left(\frac{}{t_m}\right)}_N^{-3}\left[{\left(\frac{V_C}{V_r}\right)}^{-1}{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^{3/2}\right]& \mathrm{Equation}20\end{array}$

(123) As shown in Equation 21, relevant gap radius is determined by the elastic constants of a selected radiation plate 310 material, and the operating point. The normalized bias voltage V.sub.DC_n is given in Equation 15.

(124) $\begin{array}{cc}\begin{array}{c}=\left(\sqrt[4]{\frac{16Y_0}{15\left(1-{}^2\right)P_0}}\right)\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}{\left(\frac{}{t_m}\right)}_N{t}_m\\ =\left(10\sqrt[4]{\frac{Y_0}{15\left(1-{}^2\right)P_0}}\right)\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}\\ {V_{\mathrm{DC}\_n}\left(\frac{V_{\mathrm{DC}}}{V_C}\right)}^{-1}{}_n\end{array}& \mathrm{Equation}21\end{array}$
wherein .sub.e=1 and is the gap 302 radius a if the gap 302 is circle-shaped;
wherein .sub.e=1 and is the equivalent gap 302 radius a.sub.eq if the gap 302 is regular convex polygon-shaped;
wherein .sub.e is an aspect ratio a.sub.2/a.sub.1, and is the major gap 302 radius a.sub.2 and the minor gap 302 radius is a.sub.1=a.sub.2/.sub.e if the gap 302 is ellipse-shaped;
wherein .sub.e is an aspect ratio a.sub.eq2/a.sub.eq1, and is the equivalent major gap 302 radius a.sub.eq2 and the equivalent minor gap 302 radius a.sub.eq1=a.sub.eq2/.sub.e if the gap 302 is regular convex elliptic polygon-shaped;

(125) The equations set forth herein, particularly (but not only) Equations 10, 13, 19, 20 and 22-24, show that the normalized bias voltage V.sub.DC_n, the normalized gap height t.sub.ge_n, the relevant normalized gap 302 radius , the normalized radiation plate 310 thickness t.sub.m_n, and the relevant normalized radius-to-thickness ratio

(126) ${\left(\frac{}{t_m}\right)}_N$
are independent of material properties.

(127) Boundary conditions are discussed below for the relevant radius-to-thickness ratio

(128) $\left(\frac{}{t_m}\right),$
relevant gap 302 radius , radiation plate 310 thickness t.sub.m, and corresponding normalized values, with respect to FIGS. 7, 8A and 8B and Equations 22 through 28. Scalability of the relevant radius-to-thickness ratio while maintaining optimum sensitivity at a selected operating point is discussed below with respect to Equations 29 through 32. Together, these Figures and Equations demonstrate a certain degree of flexibility in selection of particular MCM 300 dimensions for use at a selected operating point while maintaining optimum sensitivity (subject to the relationships described herein).

(129) FIG. 7 shows a lin-log semi-log graph 700 of the relationship between a maximum normalized relevant radiation plate 310 radius-to-thickness ratio

(130) 0${\left(\frac{}{t_m}\right)}_{N\_\max }$
that enables an MCM 300 to meet the elastic linearity constraint (explained below), and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C. The elastic linearity constraint can be explained using Hooke's law. Hooke's law defines the behavior of linearly elastic structures under stress. Hooke's law states that the displacement in a spring (or other linearly elastic structure) is proportional to a force which stretches or compresses it. If the force is doubled, the displacement of the spring will be doubled. However, once a real spring is sufficiently displaced (stretched), doubling the force will not double the displacement, deviating from Hooke's law. This is due to elastic non-linearity. Beyond an upper bound for applied force and for displacement, Hooke's law no longer holds and the relationship between applied force and spring displacement is no longer linear.

(131) Hooke's law applies to clamped membranes as long as linearly elastic operation holds. Studies on applied mechanics classify the linearly elastic range, that is, the displacement range in which Hooke's law is applicable to a clamped circular radiation plate, as corresponding to the center deflection of the radiation plate being less than 20% of the plate thickness, that is, X/t.sub.m<0.2. The sensitivity of an MCM 300 will decrease when elastic linearity fails, accordingly, when X/t.sub.m0.2. This limit for linearly elastic behavior of an MCM 300 is referred to herein as the elastic linearity constraint.

(132) When a clamped elliptic plate deflects under uniformly distributed force, the deflection profile, which has elliptic equal displacement contours, is similar to that of a clamped circular disc, which has circular equal displacement contours.

(133) The elastic linearity constraint can be used to determine a maximum value for the radiation plate 310 relevant radius-to-thickness ratio

(134) $\left(\frac{}{t_m}\right)$
at which an MCM 300 at a particular operating point will exhibit linearly elastic behavior. This maximum relevant radius-to-thickness ratio

(135) ${\left(\frac{}{t_m}\right)}_{\max }$
corresponds to minima for the relevant gap radius and the radiation plate 310 thickness t.sub.m. Accordingly, as shown in Equations 15 and 18-21 herein, there is an inverse relationship between (1) the size of the relevant gap radius and radiation plate 310 thickness t.sub.m (the radiation plate 310 dimensions), and (2) the relevant radiation plate 310 radius-to-thickness ratio

(136) $\left(\frac{}{t_m}\right).$

(137) Elastic linearity of CMUT cells is described in A. Unlugedik, A. S. Tasdelen, A. Atalar, and H. Koymen, Designing Transmitting CMUT Cells for Airborne Applications, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, Vol. 61, pp. 1899-1910, 2014, which is incorporated herein by reference.

(138) The maximum radiation plate 310 relevant radius-to-thickness ratio

(139) ${\left(\frac{}{t_m}\right)}_{\max }$
is found using a maximum normalized relevant radius-to-thickness ratio

(140) ${\left(\frac{}{t_m}\right)}_{N\_\max },$
which is related to the normalized static displacement of the center of the radiation plate 310 X.sub.P/t.sub.ge as shown in Equation 22. In Equation 22, the normalized static displacement of the center of the radiation plate 310 X.sub.P/t.sub.ge is expressed as

(141) $X_{\mathrm{PN}}\left(\frac{V_{\mathrm{DC}}}{V_C},\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right),$
that is, as a function of the relative bias V.sub.DC/V.sub.C and the normalized static mechanical force F.sub.Peb/F.sub.Peg.

(142) $\begin{array}{cc}{\left(\frac{}{t_m}\right)}_N<{\left(\frac{}{t_m}\right)}_{N\_\max }=\sqrt[4]{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}/X_{\mathrm{PN}}\left(\frac{V_{\mathrm{DC}}}{V_C},\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}& \mathrm{Equation}22\end{array}$
wherein is the gap 302 radius a if the gap 302 is circle-shaped;
wherein is the equivalent gap 302 radius a.sub.eq if the gap 302 is regular convex polygon-shaped;
wherein is the major gap 302 radius a.sub.2 if the gap 302 is ellipse-shaped;
and wherein is the equivalent major gap 302 radius a.sub.eq2 if the gap 302 is regular convex elliptic polygon-shaped.

(143) The normalized static displacement of the center of the radiation plate 310

(144) $X_{\mathrm{PN}}\left(\frac{V_{\mathrm{DC}}}{V_C},\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)$
is obtained in Equation 23 by solving Equation 9, and substituting for the collapse voltage in a vacuum V.sub.r using Equation 10.

(145) $\begin{array}{cc}\left(\frac{V_{\mathrm{DC}}^2}{V_C^2}\right)(0.9961-1.0468\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}+0.06972{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}-0.25\right)}^2{+0.01148{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^6)}^{2}2g^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)-3\left(\frac{X_P}{t_{\mathrm{ge}}}-\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)0\mathrm{for}\frac{X_P}{t_{\mathrm{ge}}}\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}& \mathrm{Equation}23\end{array}$

(146) As shown in FIG. 7,

(147) 0${\left(\frac{}{t_m}\right)}_{\mathrm{N\_max}}1;$
that is, the maximum value of the relevant maximum normalized radius-to-thickness ratio

(148) ${\left(\frac{}{t_m}\right)}_{\mathrm{N\_max}},$
which is unity (one), is reached at F.sub.Peb/F.sub.Peg=1 (see description of FIG. 4 and Equation 8 regarding normalized static mechanical force F.sub.Peb/F.sub.Peg). The relevant maximum normalized radius-to-thickness ratio

(149) ${\left(\frac{}{t_m}\right)}_{\mathrm{N\_max}}$
that enables an MCM 300 to meet the elastic linearity constraint decreases as normalized static mechanical force F.sub.Peb/F.sub.Peg decreases or as relative bias level V.sub.DC/V.sub.C decreases.

(150) The first part of the scaling constant term relating relevant radius to relevant normalized radius .sub.n (see Equations 17 and 21),

(151) $\left(\sqrt[4]{\frac{16Y_0}{15\left(1-{}^2\right)P_0}}\right),$
is 35.6 for a silicon radiation plate 310, if Young's modulus Y.sub.0 is 14910.sup.9 Pa, Poisson's ratio is 0.17, and the static pressure difference P.sub.0 between the gap 302 and the ambient equals SAP (101.325 kPa). The second part of the scaling constant term,

(152) $\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)},$
depends on the geometry of the radiation plate 310, and equals to 1 if the radiation plate 310 is circle-shaped or regular convex polygon-shaped. For a silicon radiation plate 310, the relevant radius-to-thickness ratio

(153) $\left(\frac{}{t_m}\right)$
will therefore, to maintain linearly elastic operation, be kept less than 35.6 at large normalized static mechanical force F.sub.Peb/F.sub.Peg if the static pressure differential P.sub.0 is equal to SAP. This upper limit for relevant radius-to-thickness ratio

(154) $\left(\frac{}{t_m}\right)$
for elastic linear operation decreases as the normalized static mechanical force F.sub.Peb/F.sub.Peg decreases. The minimum relevant radius-to-thickness ratio

(155) $\left(\frac{}{t_m}\right)$
is about 8 for F.sub.Peb/F.sub.Peg=0.001. Because there is an inverse relationship between the radiation plate dimensions ( and t.sub.m) and the relevant radius-to-thickness ratio

(156) $\left(\frac{}{t_m}\right),$
the elastic linearity constraint suggests that the lower the normalized static mechanical force F.sub.Peb/F.sub.Peg, the larger the relevant gap 302 radius should be.

(157) Accordingly, a maximum relevant normalized radius-to-thickness ratio

(158) ${\left(\frac{}{t_m}\right)}_{\mathrm{N\_max}}$
implies minimum values for the relevant gap 302 radius .sub.min and the radiation plate 310 thickness t.sub.m_min that enable an MCM 300 to operate in the linearly elastic regime at a selected operating point (F.sub.Peb/F.sub.Peg, V.sub.DC, V.sub.DC/V.sub.C). The minimum relevant gap 302 radius .sub.min corresponds to the narrowest gap 302 that enables linearly elastic operation at a selected operating point. The minimum radiation plate 310 thickness t.sub.m_min corresponds to the thinnest radiation plate 310 that enables linearly elastic operation at a selected operating point.

(159) Equation 24 shows the relationship between minimum relevant radius .sub.min and normalized minimum relevant radius .sub.n_min, which is found using the relationship between relevant gap 302 radius and normalized relevant gap 302 radius .sub.n as described by Equation 21.

(160) 0$\begin{array}{cc}{}_{\min }=\left(10\sqrt[4]{\frac{Y_0}{15\left(1-{}^2\right)P_0}}\right){V_{\mathrm{DC\_n}}\left(\frac{V_{\mathrm{DC}}}{V_C}\right)}^{-1}\sqrt[4]{\frac{1}{8}\left(3{}_e^4+2{}_e^2+3\right)}{}_{\mathrm{n\_min}}& \mathrm{Equation}24\end{array}$

(161) FIG. 8A shows a log-lin semi-log graph 800 of the relationship between minimum normalized relevant gap 302 radius .sub.n_min that enables an MCM 300 to meet the elastic linearity constraint, and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C. Equation 25 defines normalized minimum relevant gap 302 radius .sub.n_min in terms of relevant maximum normalized radiation plate 310 radius-to-thickness ratio

(162) ${\left(\frac{}{t_m}\right)}_{\mathrm{N\_max}},$
using the relationship between normalized relevant gap 302 radius .sub.n and relevant normalized radiation plate 310 radius-to-thickness ratio

(163) ${\left(\frac{}{t_m}\right)}_N$
as described by Equation 20. Note that the ratio between the clamp voltage and the collapse voltage in a vacuum V.sub.C/V.sub.r depends only on the normalized static mechanical force F.sub.Peb/F.sub.Peg, as shown in Equation 10.

(164) $\begin{array}{cc}{}_{n\mathrm{\_min}}={\left(\frac{}{t_m}\right)}_{\mathrm{N\_max}}^{-3}\left[{\left(\frac{V_C}{V_r}\right)}^{-1}{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^{3/2}\right]& \mathrm{Equation}25\end{array}$

(165) Equation 26 shows the relationship between minimum radiation plate 310 thickness t.sub.m_min and normalized minimum radiation plate 310 thickness t.sub.m_n_min, which is found using the relationship between radiation plate 310 thickness t.sub.m and normalized radiation plate 310 thickness t.sub.m_n as described by Equation 18.

(166) $\begin{array}{cc}{t}_{\mathrm{m\_min}}=5{V_{\mathrm{DC\_n}}\left(\frac{V_{\mathrm{DC}}}{V_C}\right)}^{-1}{t}_{\mathrm{m\_n}\mathrm{\_min}}& \mathrm{Equation}26\end{array}$

(167) FIG. 8B shows a log-lin semi-log graph 802 of the relationship between normalized minimum radiation plate 310 thickness t.sub.m_n_min that enables an MCM 300 to meet the elastic linearity constraint, and normalized static mechanical force F.sub.Peb/F.sub.Peg, for example values of the relative bias voltage level V.sub.DC/V.sub.C. Equation 27 defines normalized minimum radiation plate 310 thickness t.sub.m_n_min in terms of relevant maximum normalized radius-to-thickness ratio

(168) ${\left(\frac{}{t_m}\right)}_{N\_\max },$
using the relationship between normalized minimum radiation plate 310 thickness t.sub.m_min and normalized relevant radius-to-thickness ratio

(169) ${\left(\frac{}{t_m}\right)}_N$
as described by Equation 19. Note that the ratio between the clamp voltage and the collapse voltage in a vacuum V.sub.C/V.sub.r depends only on the normalized static mechanical force F.sub.Peb/F.sub.Peg, as shown in Equation 10.

(170) $\begin{array}{cc}{t}_{m\_n\_\min }={}_{n\_\min }{\left(\frac{}{t_m}\right)}_{N\_\max }^{-1}={\left(\frac{}{t_m}\right)}_{N\_\max }^{-1}{\left(\frac{V_C}{V_r}\right)}^{-1}{\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}\right)}^{3/2}& \mathrm{Equation}27\end{array}$

(171) The scaling constant term in Equation 24 is determined for silicon in Equation 28, taking Young's modulus Y.sub.0 to be 14910.sup.9 Pa, Poisson's ratio to be 0.17, and the static pressure difference P.sub.0 between the gap 302 and the ambient to equal SAP (101.325 kPa).

(172) $\begin{array}{cc}10\sqrt[4]{\frac{Y_0}{15\left(1-{}^2\right)P_0}}=177.95& \mathrm{Equation}28\end{array}$

(173) This normalization parameter for the relevant minimum gap 302 radius .sub.min is non-dimensional and contains only the elastic constants of the radiation plate 310 material and the differential static pressure P.sub.0. The normalized minimum relevant gap 302 radius .sub.n_min and radiation plate 310 thickness t.sub.m_n_min are independent of material and ambient physical properties and the bias voltage V.sub.DC. The normalized minimum relevant gap 302 radius .sub.n_min and radiation plate 310 thickness t.sub.m_n_min are instead determined by normalized static mechanical force F.sub.Peb/F.sub.Peg and relative bias voltage V.sub.DC/V.sub.C, as shown in Equations 10, 22, 23, 25 and 27.

(174) Using Equations 29-32, a relevant normalized radiation plate 310 radius-to-thickness ratio

(175) ${\left(\frac{}{t_m}\right)}_N$
can be chosen (within the limitations described by the equations) that is less than the relevant maximum radius-to-thickness ratio

(176) 0${\left(\frac{}{t_m}\right)}_{N\_\max },$
that is, less than the value of the relevant normalized radius-to-thickness ratio

(177) ${\left(\frac{}{t_m}\right)}_N$
at the elastic linearity limit. The smaller the relevant normalized radius-to-thickness ratio

(178) ${\left(\frac{}{t_m}\right)}_N$
of an MCM 300 operated at a selected operating point (F.sub.Peb/F.sub.Peg, V.sub.DC, V.sub.DC/V.sub.C), the larger the relevant gap 302 radius , the thicker the radiation plate 310 (larger t.sub.m), and the more robust the linearly elastic operation (less prone to variations in operation removing the MCM 300 from the linearly elastic regime) of the MCM 300 operated at the selected operating point; without changing the OCRV sensitivity corresponding to that operating point. Further, increased relevant normalized radius-to-thickness ratio

(179) ${\left(\frac{}{t_m}\right)}_N$
(within the limitations described in the equations) results in increased input capacitance C.sub.in of the MCM 300, which is advantageous for pre-amplification electronics. Also, the larger the clamped capacitance C.sub.0, the smaller the relative effect of parasitic capacitance on MCM 300 performance. Accordingly, a choice of relevant normalized radius-to-thickness ratio

(180) ${\left(\frac{}{t_m}\right)}_N$
can be made while retaining the same optimal OCRV sensitivity at the selected operating point.

(181) A scalar K is defined in Equation 29, relating the relevant normalized radius-to-thickness ratio

(182) ${\left(\frac{}{t_m}\right)}_N$
to the relevant maximum normalized radius-to-thickness ratio

(183) ${\left(\frac{}{t_m}\right)}_{N\_\max }.$
K, as expressed in Equation 30, is defined to satisfy the elastic linearity constraint. Accordingly, K is larger than unity, that is, K>1.

(184) $\begin{array}{cc}{\left(\frac{}{t_m}\right)}_N=\frac{1}{K}{\left(\frac{}{t_m}\right)}_{N\_\max }& \mathrm{Equation}29\\ {\left(\frac{}{t_m}\right)}_N<{\left(\frac{}{t_m}\right)}_{N\_\max }& \mathrm{Equation}30\end{array}$

(185) The normalized relevant radius .sub.n can be expressed in terms of the minimum normalized relevant gap 302 radius .sub.n_min and the scalar K as shown in Equation 31. The normalized thickness t.sub.m_n of the radiation plate 310 can be expressed in terms of the minimum normalized thickness of the radiation plate 310 t.sub.m_n_min and the scalar K as shown in Equation 32. A larger K means a radiation plate 310 that is thicker relative to the relevant gap 302 radius . There is an upper limit for K, approximately K<5, above which the radiation plate 310 becomes too thick for the model to be valid. Further, in some embodiments comprising an MCM 300 fabricated from typical materials and intended for use in an air environment, K<2.5 is preferable. Microphones with K over 2.5 will have relevant gap 302 radius much larger than radiation plate 310 thickness t.sub.m. This can make the MCM 300 difficult and/or expensive to manufacture, and potentially fragile in operation.
.sub.n=(K.sup.3).sub.n_minEquation 31
t.sub.m_n=(K.sup.4)t.sub.m_n_minEquation 32

(186) For a particular selected operating point triplet (F.sub.Peb/F.sub.Peg, V.sub.DC, V.sub.DC/V.sub.C), changes in K (within boundaries as described) will not affect the MCM 300 sensitivity or the effective gap 302 height t.sub.ge.

(187) Open Circuit Receive Voltage (OCRV) sensitivity of an MCM 300 is obtained, in volts (V) per Pascal (Pa), as shown in Equation 33. (Particular units are used herein by way of example only; other units can be used.) The OCRV sensitivity is represented by S.sub.VO. S.sub.VO=V.sub.OC/p. V.sub.OC is the voltage across the electrical terminals (not shown) of the MCM 300 when the terminals are in open circuit, and p represents incident pressure, meaning that V.sub.OC/p describes the strength (V.sub.OC) of the voltage induced between the terminals of a microphone circuit by a pressure wave of magnitude p incident on the radiation plate 310. Equation 33 assumes that the MCM 300 is mounted on a rigid baffle and operated off-resonance, and ignores radiation impedance (losses from radiation impedance are discussed in the Background, above).

(188) $\begin{array}{cc}{S}_{\mathrm{VO}}=\frac{V_{\mathrm{OC}}}{p}=-\left[\frac{3}{8}\sqrt{\frac{2}{5}}\sqrt{\frac{1-{}^2}{{\mathrm{.Math.}}_0{Y}_0}}\left(\frac{{}^2}{t_m}\right)\sqrt{\frac{t_{\mathrm{ge}}}{t_m}}\right]h_{\mathrm{oce}}\left(\frac{V_{\mathrm{DC}}}{V_C},\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}},\frac{C_p}{C_0},e\right)V\text{/}\mathrm{Pa}& \mathrm{Equation}33\end{array}$

(189) Equation 33 can be rewritten so that OCRV sensitivity is expressed in terms of the operating point parameters (F.sub.Peb/F.sub.Peg, V.sub.DC, V.sub.DC/V.sub.C). This is done using expressions for effective gap 302 height t.sub.ge, radiation plate 310 thickness t.sub.m, and relevant gap 302 radius , in Equations 12, 18 and 21, respectively. Expressions for input capacitance C.sub.in and clamped capacitance C.sub.0 in terms of the operating point are provided in Equations 40 and 41, respectively.

(190) $\begin{array}{cc}{S}_{\mathrm{VO}}=\frac{V_{\mathrm{OC}}}{p}=-\left[\frac{9}{2}\sqrt{\frac{1}{5}}\left(\frac{V_{\mathrm{DC}}}{P_0}\right)\right]h_{\mathrm{oce}}\left(\frac{V_{\mathrm{DC}}}{V_C},\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}},\frac{C_p}{C_0},e\right)V\text{/}\mathrm{Pa}& \mathrm{Equation}34\end{array}$

(191) The dimensionless normalized OCRV sensitivity h.sub.oce is given as shown in Equation 35. The dimensionless normalized OCRV sensitivity h.sub.oce is a function of the parasitic capacitance C.sub.P, the aspect ratio .sub.e, and the operating point parameters voltage bias level V.sub.DC/V.sub.C and normalized static mechanical force F.sub.Peb/F.sub.Peg. The functions g(u), g(u), and g(u) are shown and described with respect to Equations 2-4 (above). Preferably, the parasitic capacitance C.sub.P is relatively small compared to the input capacitance C.sub.0, for example, small enough that the effects of the parasitic capacitance can be ignored and/or do not prevent meeting design performance specifications. The ratio of the collapse voltage to the collapse voltage in a vacuum V.sub.C/V.sub.r can be substituted for using Equation 10. The dimensionless normalized OCRV sensitivity h.sub.oce is evaluated at the static equilibrium shown in Equation 23, but does not explicitly depend on the dimensions of the MCM 300 (such as relevant gap 302 radius ) or material properties (such as Poisson's ratio). The normalized static displacement of the center of the radiation plate 310 X.sub.PN equals the ratio of the static displacement X.sub.P to the effective gap height t.sub.ge at the operating point, as shown in Equation 36. Also, as shown in Equation 36, the normalized static displacement X.sub.PN depends only on the relative bias level V.sub.DC/V.sub.C and the normalized static mechanical force F.sub.Peb/F.sub.Peg.

(192) 0$\begin{array}{cc}{h}_{\mathrm{oce}}\left(\frac{V_{\mathrm{DC}}}{V_C},\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}},\frac{C_p}{C_0},{}_e\right)=\frac{\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}}{g}^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)}{\begin{array}{c}{\left(\frac{V_{\mathrm{DC}}}{V_r}{g}^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\right)}^2+\left(e\frac{C_p}{C_0}+g\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\right)\\ \left(\frac{3}{4}-\frac{1}{2}{\left(\frac{V_{\mathrm{DC}}}{V_r}\right)}^2{g}^{}\left(\frac{X_P}{t_{\mathrm{ge}}}\right)\right)\end{array}}& \mathrm{Equation}35\end{array}$
wherein .sub.e=1 if the gap 302 is circle- or regular convex polygon-shaped;
wherein .sub.e is an aspect ratio a.sub.2/a.sub.1, if the gap 302 is ellipse-shaped;
wherein .sub.e is an aspect ratio a.sub.eq2/a.sub.eq1, if the gap 302 is regular convex elliptic polygon-shaped;

(193) $\begin{array}{cc}{X}_{\mathrm{PN}}=\frac{X_P}{t_{\mathrm{ge}}}\mathrm{at}\mathrm{the}\mathrm{operating}\mathrm{point}\left(\frac{F_{\mathrm{Peb}}}{F_{\mathrm{Peg}}},\frac{V_{\mathrm{DC}}}{V_C}\right)& \mathrm{Equation}36\end{array}$