PATENT DOCUMENT

Abstract:
A method of forming a miniature, surface micromachined, differential microphone, comprising depositing a sacrificial layer on a surface of a silicon wafer; depositing a diaphragm material on a surface of the sacrificial layer; etching the diaphragm material layer to isolate a diaphragm; and removing a portion of the sacrificial layer beneath the defined diaphragm. A diaphragm formed in the diaphragm material layer is supported by a hinge and otherwise isolated from a remaining portion of the diaphragm material layer by a slit adjacent a perimeter of the diaphragm. An enclosed back volume beneath the diaphragm has a depth defined by a thickness of the sacrificial layer, and communicates with an external region via the slit. A transducer may be provided for producing an electrical signal responsive to a displacement of the diaphragm.

Full Description:
FUNDED RESEARCH 
     This work is supported in part by the following grant from the National Institute of Health: R01DC005762-03. The Government may have certain rights in this invention. 
    
    
     RELATED APPLICATIONS 
     The present application is related to U.S. Pat. No. 6,788,796 for DIFFERENTIAL MICROPHONE, issued Sep. 7, 2004; and copending U.S. patent application Ser. No. 10/689,189 for ROBUST DIAPHRAGM FOR AN ACOUSTIC DEVICE, filed Oct. 20, 2003, and Ser. No. 11/198,370 for COMB SENSE MICROPHONE, filed Aug. 5, 2005, all of which are incorporated herein by reference. 
     FIELD OF THE INVENTION 
     The present invention pertains to differential microphones and, more particularly, to a micromachined, differential microphone absent a backside air pressure relief orifice, fabricatable using surface micromachining techniques. 
     BACKGROUND OF THE INVENTION 
     In typical micromachined microphones of the prior art, it is generally necessary to maintain a significant volume of air behind the microphone diaphragm in order to prevent the back volume air from impeding the motion of the diaphragm. The air behind the diaphragm acts as a linear spring whose stiffness is inversely proportional to the nominal volume of the air. In order to make this air volume as great as possible, and hence reduce the effective stiffness, a through-hole is normally cut from the backside of the silicon chip. The requirement of this backside hole adds significant complexity and expense to such prior art micromachined microphones. This present invention enables creation of a microphone that does not require a backside hole. Consequently, the inventive microphone may be fabricated using only surface micromachining techniques. 
     SUMMARY OF THE INVENTION 
     In accordance with the present invention, there is provided a differential microphone having a perimeter slit formed around the microphone diaphragm. Because the motion of the diaphragm in response to sound does not result in a net compression of the air in the space behind the diaphragm, the use of a very small backing cavity is possible, thereby obviating the need for creating a backside hole. The backside holes of prior art microphones typically require that a secondary machining operation be performed on the silicon chip during fabrication. This secondary operation adds complexity and cost to, and results in lower yields of the microphones so fabricated. Consequently, the microphone of the present invention requires surface machining from only a single side of the silicon chip. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A complete understanding of the present invention may be obtained by reference to the accompanying drawings, when considered in conjunction with the subsequent, detailed description, in which: 
         FIG. 1  is a top view of a micromachined microphone diaphragm in accordance with the invention; 
         FIG. 2  is a side, sectional, schematic view of a differential microphone of the invention; 
         FIGS. 3 and 4  are, respectively, schematic representations of the differential microphone of  FIG. 2  as a series of diaphragms without and with an indication of the motion thereof; 
         FIG. 5  is a diagram showing the orientation of an incident sound wave on the diaphragm of  FIG. 1 ; 
         FIGS. 6   a - 6   d  are schematic representations of the stages of fabrication of the inventive, surface micromachined microphone of the invention; 
         FIG. 7  is a side, sectional, schematic view of a differential microphone formed by removing a portion of a sacrificial layer of  FIG. 6   d ; and 
         FIG. 8  is a side, sectional, schematic view of an alternate embodiment of the microphone of  FIG. 2 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The present invention relates to a micromachined differential microphone formed by surface micromachining a single surface of a silicon chip. 
     The motion of a typical microphone diaphragm results in a fluctuation in the net volume of air in the region behind the diaphragm (i.e., the back volume). The present invention provides a microphone diaphragm designed to rock due to acoustic pressure, and hence does not significantly compress the back volume air. 
     An analytical model for the acoustic response of the microphone diaphragm including the effects of a slit around the perimeter and the air in the back volume behind the diaphragm has been developed. If the diaphragm is designed to rock about a central pivot, then the back volume and the slit has a negligible effect on the sound-induced response thereof. 
     Referring first to  FIGS. 1 and 2 , there are shown, respectively, a top view of a micromachined microphone diaphragm, including a slit around the perimeter of the diaphragm, and a side, sectional, schematic view of a differential microphone in accordance with the invention, generally at reference number  100 . A rigid diaphragm  102  is supported by hinges  104  that form a pivot point  106  around which diaphragm  102  may “rock” (i.e., reciprocally rotate). A back volume of air  108  is formed in a cavity  110  formed in the chip substrate  112 . A slit  114  is formed between the perimeter  103  of diaphragm  102  and the chip substrate  112 . 
     Diaphragm  102  rotates about the pivot point  106  due to a net moment that results from the difference in the acoustic pressure that is incident on the top surface portions  116 ,  118  that are separated by the central pivot point  106 . 
     In order to more readily examine the effects of the back volume  108  and the slit  114  around the diaphragm  102 , several assumptions are made. It is assumed that the pivot point  106  is centrally located and that diaphragm  102  is designed such that the rocking, or out-of-phase motion of diaphragm  102  is the result of the pressure difference on the two portions  116 ,  118  of the exterior surface thereof. Because diaphragm  102  is normally designed to respond to the difference in pressure on its two portions  116 ,  118 , microphone  100  is referred to it as a differential microphone. However, in addition to motion induced by pressure differences, it is also possible that diaphragm  102  will be deflected due to the average pressure on its exterior surface. Such pressure causes diaphragm  102  motion in which both portions  116 ,  118  of the diaphragm  102  separated by the pivot point  106  respond in-phase. 
     The air  108   a  in the slit  114  around the diaphragm  102  on each portion  116 ,  118  is assumed to have a mass ma. Consequently, diaphragm  102  responds like an oscillator. Hence, the two portions  116 ,  118  of the differential microphone  100 , along with the two masses of air  108 ,  108   a  can be represented by a system of diaphragms  120 ,  122 ,  124 ,  126  as shown in  FIG. 3 . Each of the diaphragms is identified as air  108  (reference number  120 ), microphone portion  116  (reference number  122 ), microphone portion  118  (reference number  124 ), and air  108   a  (reference number  126 ). The response of each diaphragm is governed by the following equation:
 
 m   i   {umlaut over (X)}   i   +k   i   X   i   =F   i   (1)
 
where: F i  is the net force acting on each diaphragm  120 ,  122 ,  124 ,  126  and X 4 , X 1 , X 2 , and X 3 , represent the motion of each respective diaphragm  120 ,  122 ,  124 ,  126 . As may be seen in  FIG. 4 , X 1  and X 2  represent the average motion of each portion  116 ,  118  of the diaphragm and X 3  and X 4  represent the motion of the air  108   a  in the slit  114 .
 
     A differential microphone without the slit  114  (i.e., a differential microphone of the prior art) can be represented by a two degree of freedom system with rotational response θ and translational response x:
 
 m{umlaut over (x)}+kx=F   (2a)
 
 I{umlaut over (θ)}+k   t   θ=M   (2b)
 
where: F is the net applied force, and M is the resulting moment about the pivot point. k and k t  represent the effective transverse mechanical stiffness and the torsional stiffness respectively, of the diaphragm and pivot  102 , and  106 .
 
     If d is the distance between the centers of each portion  116 ,  118  of the diaphragm  102 , then X 1  and X 2  may be expressed in terms of the generalized co-ordinates x and θ: 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       1 
                     
                     = 
                     
                       
                         x 
                         + 
                         
                           
                             d 
                             2 
                           
                           ⁢ 
                           θ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           and 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             X 
                             2 
                           
                         
                       
                       = 
                       
                         
                           
                             x 
                             - 
                             
                               
                                 d 
                                 2 
                               
                               ⁢ 
                               θ 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⇒ 
                           
                               
                           
                           ⁢ 
                           x 
                         
                         = 
                         
                           
                             
                               
                                 X 
                                 1 
                               
                               + 
                               
                                 X 
                                 2 
                               
                             
                             2 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           and 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     θ 
                     = 
                     
                       
                         
                           X 
                           1 
                         
                         - 
                         
                           X 
                           2 
                         
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     These relations may also be written in matrix form: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             X 
                             1 
                           
                         
                       
                       
                         
                           
                             X 
                             2 
                           
                         
                       
                       
                         
                           
                             X 
                             3 
                           
                         
                       
                       
                         
                           
                             X 
                             4 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 d 
                                 / 
                                 2 
                               
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 
                                   - 
                                   d 
                                 
                                 / 
                                 2 
                               
                             
                             
                               1 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               1 
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               θ 
                             
                           
                           
                             
                               x 
                             
                           
                           
                             
                               
                                 X 
                                 3 
                               
                             
                           
                           
                             
                               
                                 X 
                                 4 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         [ 
                         T 
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               θ 
                             
                           
                           
                             
                               x 
                             
                           
                           
                             
                               
                                 X 
                                 3 
                               
                             
                           
                           
                             
                               
                                 X 
                                 4 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     If the dimensions of the air cavity  110  ( FIG. 2 ) behind the diaphragm  102  are much smaller than the wavelength of sound, it may be assumed that the air pressure in the back volume  108  is spatially uniform within the air cavity. The air  108  in this back volume (i.e., cavity  110 ) then acts as a linear spring. It is necessary to relate the pressure in the back volume air  108  to the displacement of the diaphragm  102  to estimate the stiffness of this spring. If the mass of the air in back volume  108  is assumed to be constant, then the motion of the diaphragm  102  results in a change in the density of the air  108  in cavity  110 . The relation between the acoustic, or fluctuating density, ρ a  and the acoustic pressure, p, is the equation of state:
 
p=c 2 ρ a   (5)
 
where: c is the speed of sound.
 
     The total density of air is the mass divided by the volume, ρ=M/V. If the volume fluctuates by an amount ΔV due to the motion of diaphragm  102 , then the density becomes ρ=M/(V+ΔV)=M/V(1+ΔV/V. For small changes in the volume, this can be expanded in a Taylor&#39;s series            ρ≈(M/V(1−ΔV/V). The acoustic fluctuating density is then ρ a =−ρ 0 ΔV/V, where the nominal density is ρ 0 =M/V. The fluctuating pressure in the volume V due to the fluctuation ΔV, resulting from an outward motion, x, of the diaphragm  102  is then given by:
 
 P   d =−ρ 0   c   2   ΔV/V=−ρ   0   c   2   Ax/V   (6)
 
where: A is half the area of the diaphragm.

     This pressure in the back volume  108  exerts a force on the diaphragm  102  given by:
 
 F   d   =P   d   A−ρ   0   c   2   A   2   x/V=−K   d   x   (7)
 
where: K d =ρ 0 c 2 A 2 /V is the equivalent spring constant of the air  108  with units of N/m.
 
     The force due to the back volume of air  108  adds to the restoring force from the mechanical stiffness of the diaphragm  102 . Including the air in the back volume  108 , Equation (2) becomes:
 
 m{umlaut over (x)}+kx+k   d   x=−PA   (8)
 
     The negative sign on the right hand side of Equation (8) is attributed to the convention that a positive pressure on the diaphragm&#39;s exterior causes a force in the negative direction. From Equation (8), the mechanical sensitivity at frequencies well below the resonant frequency is given by S m =A/(k+K d ) m/Pa. 
     The air  108   a  in the slit or vent  114  is forced to move due to the fluctuating pressures both within the space  110  behind the diaphragm  102  and in the external sound field, not shown. Again, it may be assumed that the dimensions of the volume of moving air in the slit  114  to be much smaller than the wavelength of sound and hence it may be approximately represented as a lumped mass ma. An outward displacement, x a , of the air  108   a  in the slit  114  causes a change in the volume of air in the back volume  108 . A corresponding pressure similar to Equation (6) is given by:
 
 P   aa =−ρ 0   c   2   A   a   x   a   /V   (9)
 
where: A a  is the area of the slit  114  on which the pressure acts.
 
     Again, the pressure due to motion of air  108   a  in the slit  114  applies a restoring force on the mass thereof given by:
 
 F   aa   =P   aa   A   a =−ρ 0   c   2   A   a   2   X   a   /V=K   aa   x   a   (10)
 
     Since the pressure in the back volume  108  is nearly independent of position within the back volume, a change in the pressure due to motion of the air  108   a  in the slit  114  exerts a force on the diaphragm  102  given by:
 
 F   ad   =P   aa   A=−ρ   0   c   2   A   a   Ax   a   /V=K   ad   x   a   (11)
 
     Similarly, the motion of the diaphragm causes a force on the mass of air  108  given by:
 
 F   da   =P   d   A   a =−ρ 0   c   2   AA   a   x/V=−K   da   x   (12)
 
     From Equations (6), (10), (11) and (12), it may be seen that the forces add to the restoring forces due to mechanical stiffness in the system of Equation (1). Hence the volume change due to motion of each co-ordinate is given by ΔV i =A i X i  and F i =PA i . Now, the total pressure due to the motion of all co-ordinates is given by: 
     
       
         
           
             
               
                 
                   
                     P 
                     tot 
                   
                   = 
                   
                     
                       
                         - 
                         
                           
                             
                               ρ 
                               0 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           V 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               A 
                               1 
                             
                             ⁢ 
                             
                               X 
                               1 
                             
                           
                           + 
                           
                             
                               A 
                               2 
                             
                             ⁢ 
                             
                               X 
                               2 
                             
                           
                           + 
                           
                             
                               A 
                               3 
                             
                             ⁢ 
                             
                               X 
                               3 
                             
                           
                           + 
                           
                             
                               A 
                               4 
                             
                             ⁢ 
                             
                               X 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             
                               ρ 
                               0 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                           
                           V 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           i 
                         
                         ⁢ 
                         
                           
                             A 
                             i 
                           
                           ⁢ 
                           
                             X 
                             i 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The force due to this pressure on the jth coordinate in this model (indicating the motions of  120 ,  122 ,  124 , and  126  in  FIG. 3 ) is then given by: 
                     F   j     =         P   tot     ⁢     A   j       =         (       -         ρ   0     ⁢     c   2       V       ⁢       ∑   i     ⁢       A   i     ⁢     X   i           )     ⁢     A   j       =     -       ∑   i     ⁢       K   ij     ⁢     X   i                       (   14   )               
where:
 
     
       
         
           
             
               K 
               ij 
             
             = 
             
               
                 - 
                 
                   
                     
                       ρ 
                       0 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                   V 
                 
               
               ⁢ 
               
                 A 
                 i 
               
               ⁢ 
               
                 
                   A 
                   j 
                 
                 . 
               
             
           
         
       
     
     Equation (14) may be written as: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             F 
                             1 
                           
                         
                       
                       
                         
                           
                             F 
                             2 
                           
                         
                       
                       
                         
                           
                             F 
                             3 
                           
                         
                       
                       
                         
                           
                             F 
                             4 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       - 
                       
                         [ 
                         
                           
                             
                               
                                 K 
                                 11 
                               
                             
                             
                               
                                 K 
                                 12 
                               
                             
                             
                               
                                 K 
                                 13 
                               
                             
                             
                               
                                 K 
                                 14 
                               
                             
                           
                           
                             
                               
                                 K 
                                 21 
                               
                             
                             
                               
                                 K 
                                 22 
                               
                             
                             
                               
                                 K 
                                 23 
                               
                             
                             
                               
                                 K 
                                 24 
                               
                             
                           
                           
                             
                               
                                 K 
                                 31 
                               
                             
                             
                               
                                 K 
                                 32 
                               
                             
                             
                               
                                 K 
                                 33 
                               
                             
                             
                               
                                 K 
                                 34 
                               
                             
                           
                           
                             
                               
                                 K 
                                 41 
                               
                             
                             
                               
                                 K 
                                 42 
                               
                             
                             
                               
                                 K 
                                 43 
                               
                             
                             
                               
                                 K 
                                 44 
                               
                             
                           
                         
                         ] 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               X 
                               1 
                             
                           
                         
                         
                           
                             
                               X 
                               2 
                             
                           
                         
                         
                           
                             
                               X 
                               3 
                             
                           
                         
                         
                           
                             
                               X 
                               4 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Combining Equations (4) and (15), in terms of the coordinates θ and x of the differential microphone, the force is represented as: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             F 
                             1 
                           
                         
                       
                       
                         
                           
                             F 
                             2 
                           
                         
                       
                       
                         
                           
                             F 
                             3 
                           
                         
                       
                       
                         
                           
                             F 
                             4 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       - 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   K 
                                   11 
                                 
                               
                               
                                 
                                   K 
                                   12 
                                 
                               
                               
                                 
                                   K 
                                   13 
                                 
                               
                               
                                 
                                   K 
                                   14 
                                 
                               
                             
                             
                               
                                 
                                   K 
                                   21 
                                 
                               
                               
                                 
                                   K 
                                   22 
                                 
                               
                               
                                 
                                   K 
                                   23 
                                 
                               
                               
                                 
                                   K 
                                   24 
                                 
                               
                             
                             
                               
                                 
                                   K 
                                   31 
                                 
                               
                               
                                 
                                   K 
                                   32 
                                 
                               
                               
                                 
                                   K 
                                   33 
                                 
                               
                               
                                 
                                   K 
                                   34 
                                 
                               
                             
                             
                               
                                 
                                   K 
                                   41 
                                 
                               
                               
                                 
                                   K 
                                   42 
                                 
                               
                               
                                 
                                   K 
                                   43 
                                 
                               
                               
                                 
                                   K 
                                   44 
                                 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           T 
                           ] 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             θ 
                           
                         
                         
                           
                             x 
                           
                         
                         
                           
                             
                               X 
                               3 
                             
                           
                         
                         
                           
                             
                               X 
                               4 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Equation (16) may be rewritten in terms of the average force acting on the differential microphone  100  and the net moment acting on the pivot point  106 . This is given by: 
     
       
         
           
             
               F 
               = 
               
                 
                   
                     
                       
                         F 
                         1 
                       
                       + 
                       
                         F 
                         2 
                       
                     
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   M 
                 
                 = 
                 
                   
                     
                       
                         ( 
                         
                           
                             F 
                             1 
                           
                           - 
                           
                             F 
                             2 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         d 
                         2 
                       
                     
                     ⇒ 
                     
                       F 
                       1 
                     
                   
                   = 
                   
                     F 
                     + 
                     
                       
                         M 
                         d 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                     
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               F 
               2 
             
             = 
             
               F 
               - 
               
                 M 
                 d 
               
             
           
         
       
     
     What follows therefrom is: 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             M 
                           
                         
                         
                           
                             F 
                           
                         
                         
                           
                             
                               F 
                               3 
                             
                           
                         
                         
                           
                             
                               F 
                               4 
                             
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     d 
                                     / 
                                     2 
                                   
                                 
                                 
                                   
                                     
                                       - 
                                       d 
                                     
                                     / 
                                     2 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     1 
                                     / 
                                     2 
                                   
                                 
                                 
                                   
                                     1 
                                     / 
                                     2 
                                   
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   1 
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     F 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     F 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     F 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     F 
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
 
                         
                         ⇒ 
                         
                           ( 
                           
                             
                               
                                 M 
                               
                             
                             
                               
                                 F 
                               
                             
                             
                               
                                 
                                   F 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   F 
                                   4 
                                 
                               
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             
                               - 
                               
                                 
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             d 
                                             / 
                                             2 
                                           
                                         
                                         
                                           
                                             
                                               - 
                                               d 
                                             
                                             / 
                                             2 
                                           
                                         
                                         
                                           0 
                                         
                                         
                                           0 
                                         
                                       
                                       
                                         
                                           
                                             1 
                                             / 
                                             2 
                                           
                                         
                                         
                                           
                                             1 
                                             / 
                                             2 
                                           
                                         
                                         
                                           0 
                                         
                                         
                                           0 
                                         
                                       
                                       
                                         
                                           0 
                                         
                                         
                                           0 
                                         
                                         
                                           1 
                                         
                                         
                                           0 
                                         
                                       
                                       
                                         
                                           0 
                                         
                                         
                                           0 
                                         
                                         
                                           0 
                                         
                                         
                                           1 
                                         
                                       
                                     
                                     ] 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             K 
                                             11 
                                           
                                         
                                         
                                           
                                             K 
                                             12 
                                           
                                         
                                         
                                           
                                             K 
                                             13 
                                           
                                         
                                         
                                           
                                             K 
                                             14 
                                           
                                         
                                       
                                       
                                         
                                           
                                             K 
                                             21 
                                           
                                         
                                         
                                           
                                             K 
                                             22 
                                           
                                         
                                         
                                           
                                             K 
                                             23 
                                           
                                         
                                         
                                           
                                             K 
                                             24 
                                           
                                         
                                       
                                       
                                         
                                           
                                             K 
                                             31 
                                           
                                         
                                         
                                           
                                             K 
                                             32 
                                           
                                         
                                         
                                           
                                             K 
                                             33 
                                           
                                         
                                         
                                           
                                             K 
                                             34 
                                           
                                         
                                       
                                       
                                         
                                           
                                             K 
                                             41 
                                           
                                         
                                         
                                           
                                             K 
                                             42 
                                           
                                         
                                         
                                           
                                             K 
                                             43 
                                           
                                         
                                         
                                           
                                             K 
                                             44 
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                                 ⁡ 
                                 
                                   [ 
                                   T 
                                   ] 
                                 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     θ 
                                   
                                 
                                 
                                   
                                     x 
                                   
                                 
                                 
                                   
                                     
                                       X 
                                       3 
                                     
                                   
                                 
                                 
                                   
                                     
                                       X 
                                       4 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           
                               
                           
                           ⇒ 
                           
                             ( 
                             
                               
                                 
                                   M 
                                 
                               
                               
                                 
                                   F 
                                 
                               
                               
                                 
                                   
                                     F 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     F 
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             - 
                             
                               [ 
                               
                                 K 
                                 ′ 
                               
                               ] 
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   θ 
                                 
                               
                               
                                 
                                   x 
                                 
                               
                               
                                 
                                   
                                     X 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     X 
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Hence, the system of equations: 
     
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               I 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               m 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 m 
                                 a 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 m 
                                 a 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 θ 
                                 ¨ 
                               
                             
                           
                           
                             
                               
                                 x 
                                 ¨ 
                               
                             
                           
                           
                             
                               
                                 
                                   X 
                                   ¨ 
                                 
                                 3 
                               
                             
                           
                           
                             
                               
                                 
                                   X 
                                   ¨ 
                                 
                                 4 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 k 
                                 t 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               k 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               θ 
                             
                           
                           
                             
                               x 
                             
                           
                           
                             
                               
                                 X 
                                 3 
                               
                             
                           
                           
                             
                               
                                 X 
                                 4 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               
                                 M 
                               
                             
                             
                               
                                 F 
                               
                             
                             
                               
                                 
                                   F 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   F 
                                   4 
                                 
                               
                             
                           
                           ) 
                         
                         - 
                         
                           
                             [ 
                             
                               K 
                               ′ 
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   θ 
                                 
                               
                               
                                 
                                   x 
                                 
                               
                               
                                 
                                   
                                     X 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     X 
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
 
                       
                       ⇒ 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   I 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   m 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     m 
                                     a 
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   0 
                                 
                                 
                                   
                                     m 
                                     a 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   
                                     θ 
                                     ¨ 
                                   
                                 
                               
                               
                                 
                                   
                                     x 
                                     ¨ 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       X 
                                       ¨ 
                                     
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       X 
                                       ¨ 
                                     
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             { 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         k 
                                         t 
                                       
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       k 
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                 
                                 ] 
                               
                               + 
                               
                                 [ 
                                 
                                   K 
                                   ′ 
                                 
                                 ] 
                               
                             
                             } 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 
                                   θ 
                                 
                               
                               
                                 
                                   x 
                                 
                               
                               
                                 
                                   
                                     X 
                                     3 
                                   
                                 
                               
                               
                                 
                                   
                                     X 
                                     4 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             M 
                           
                         
                         
                           
                             F 
                           
                         
                         
                           
                             
                               F 
                               3 
                             
                           
                         
                         
                           
                             
                               F 
                               4 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     It is important to note that the coupling between the coordinates in Equation (18) is due to the matrix [K′]. Evaluating the elements of [K′] from equations (4) and (17), the governing equation for the rotation, θ, of the diaphragm is: 
                       I   ⁢           ⁢     θ   ¨       +       (       k   t     +         (     d   2     )     2     ⁢     (       k   11     -     k   12     -     k   21     +     k   22       )         )     ⁢   θ     +       (     d   2     )     ⁢     (       k   11     +     k   12     -     k   21     -     k   22       )     ⁢   x     +       (     d   2     )     ⁢     (       k   13     -     k   23       )     ⁢     X   3       +       (     d   2     )     ⁢     (       k   14     -     k   24       )     ⁢     X   4         =   M           (   19   )               
where:
 
     
       
         
           
             
               K 
               ij 
             
             = 
             
               
                 - 
                 
                   
                     
                       ρ 
                       0 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                   V 
                 
               
               ⁢ 
               
                 A 
                 i 
               
               ⁢ 
               
                 
                   A 
                   j 
                 
                 . 
               
             
           
         
       
     
     Note that if the diaphragm is symmetric, A 1 =A 2 , and A 3 =A 4 . As a result, the coefficients of x, X 3 , and X 4  in equation (19) become zero. This causes the governing equation for rotation to be independent of the other coordinates as well as independent of the volume, V (i.e., I{umlaut over (θ)}+k t θ=M). The rotation is also independent of the area of the slits  114 , because of the assumption that the pressure created within the back volume  108  is spatially uniform and therefore does not create any net moment on the diaphragm  102 . 
     In the foregoing analysis, it has been assumed that the microphone diaphragm  102  is symmetric about the central pivot point  106 . As mentioned above, in this case, the diaphragm  102  behaves like a differential microphone diaphragm and has a first-order directional response. If, however, the diaphragm  102  is designed to be asymmetrical with respect to pivot point  106 , then the directionality departs from that of a differential microphone and tends toward that of a nondirectional microphone. The effect of the back volume  108  on the rotation of the diaphragm  102  can be determined by extending the foregoing analysis to this non-symmetric case. 
     In the following, expressions are derived for the forces and moment that are applied to the microphone diaphragm  102  due to an acoustic plane wave. For plane waves, the pressure acting on the diaphragm  102  is assumed to be of the form p=Pe îωt e (−îk     x     x−îk     y     y) , where 
                 k   x     =       ω   c     ⁢   sin   ⁢           ⁢   ϕ   ⁢           ⁢   sin   ⁢           ⁢   θ       ,     
     ⁢       k   y     =       ω   c     ⁢   sin   ⁢           ⁢   ϕ   ⁢           ⁢   cos   ⁢           ⁢   θ                 and                 k   z     =       ω   c     ⁢   cos   ⁢           ⁢   ϕ       ,         
where the angles are defined in  FIG. 5 . The net moment due to the incident sound is given by
 
             M   =       ∫       -     L   x       /   2         L   x     /   2       ⁢       ∫       -     L   y       /   2         L   y     /   2       ⁢       Pⅇ       ⅈ   ^     ⁢           ⁢   ω   ⁢           ⁢   t       ⁢     ⅇ     (         -     ⅈ   ^       ⁢     k   x     ⁢   x     -       ⅈ   ^     ⁢     k   y     ⁢   y       )       ⁢   x   ⁢           ⁢     ⅆ   x     ⁢           ⁢     ⅆ   y                 
where L x  and L y  are the lengths in the x and y directions, respectively.
 
     The expression for the moment can be integrated separately over the x and y directions to give 
               ⇒   M     =       Pⅇ       ⅈ   ^     ⁢           ⁢   ω   ⁢           ⁢   t       ⁢       ∫       -     L   x       /   2         L   x     /   2       ⁢       ⅇ       -     ⅈ   ^       ⁢     k   x     ⁢   x       ⁢   x   ⁢           ⁢     ⅆ   x     ⁢       ∫       -     L   y       /   2         L   y     /   2       ⁢       ⅇ       -     ⅈ   ^       ⁢     k   y     ⁢   y       ⁢           ⁢       ⅆ   y     .                     
Integrating over the y coordinate
 
     
       
         
           
             
               becomes 
               ⇒ 
               M 
             
             = 
             
               
                 
                   P 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ⅇ 
                     
                       
                         ⅈ 
                         ^ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           ⅇ 
                           
                             ⅈ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               k 
                               y 
                             
                             ⁢ 
                             
                               
                                 L 
                                 y 
                               
                               / 
                               2 
                             
                           
                         
                         - 
                         
                           ⅇ 
                           
                             ⅈ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               k 
                               y 
                             
                             ⁢ 
                             
                               
                                 L 
                                 y 
                               
                               / 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         - 
                         ⅈ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         k 
                         y 
                       
                     
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         
                           - 
                           
                             L 
                             x 
                           
                         
                         / 
                         2 
                       
                       
                         
                           L 
                           x 
                         
                         / 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           
                             - 
                             
                               ⅈ 
                               ^ 
                             
                           
                           ⁢ 
                           
                             k 
                             x 
                           
                           ⁢ 
                           x 
                         
                       
                       ⁢ 
                       x 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⇒ 
                 M 
               
               = 
               
                 P 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ⅇ 
                   
                     
                       ⅈ 
                       ^ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ω 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     t 
                   
                 
                 ⁢ 
                 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       sin 
                       ( 
                       
                         
                           
                             k 
                             y 
                           
                           ⁢ 
                           
                             L 
                             y 
                           
                         
                         2 
                       
                     
                   
                   
                       
                   
                 
                 ⁢ 
                 
                   
                     ∫ 
                     
                       
                         - 
                         
                           L 
                           x 
                         
                       
                       / 
                       2 
                     
                     
                       
                         L 
                         x 
                       
                       / 
                       2 
                     
                   
                   ⁢ 
                   
                     
                       ⅇ 
                       
                         
                           - 
                           
                             ⅈ 
                             ^ 
                           
                         
                         ⁢ 
                         
                           k 
                           x 
                         
                         ⁢ 
                         x 
                       
                     
                     ⁢ 
                     x 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         ⅆ 
                         x 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     Integrating by parts for the x-component gives: 
     
       
         
           
             
               ⇒ 
               M 
             
             = 
             
               P 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ⅇ 
                 
                   
                     ⅈ 
                     ^ 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   ω 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   t 
                 
               
               ⁢ 
               
                 
                   
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 k 
                                 y 
                               
                               ⁢ 
                               
                                 L 
                                 y 
                               
                             
                             2 
                           
                           ) 
                         
                       
                     
                     
                       
                         k 
                         y 
                       
                       ⁢ 
                       
                           
                       
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             L 
                             x 
                           
                           2 
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ⅇ 
                                 
                                   
                                     - 
                                     
                                       ⅈ 
                                       ^ 
                                     
                                   
                                   ⁢ 
                                   
                                     k 
                                     x 
                                   
                                   ⁢ 
                                   
                                     
                                       L 
                                       x 
                                     
                                     / 
                                     2 
                                   
                                 
                               
                               + 
                               
                                 ⅇ 
                                 
                                   
                                     ⅈ 
                                     ^ 
                                   
                                   ⁢ 
                                   
                                     k 
                                     x 
                                   
                                   ⁢ 
                                   
                                     
                                       L 
                                       x 
                                     
                                     / 
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                           
                             - 
                             
                               ⅈk 
                               x 
                             
                           
                         
                       
                       + 
                       
                         
                           1 
                           
                             k 
                             x 
                             2 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               ⅇ 
                               
                                 
                                   ⅈ 
                                   ^ 
                                 
                                 ⁢ 
                                 
                                   k 
                                   x 
                                 
                                 ⁢ 
                                 
                                   
                                     L 
                                     x 
                                   
                                   / 
                                   2 
                                 
                               
                             
                             - 
                             
                               ⅇ 
                               
                                 
                                   - 
                                   
                                     ⅈ 
                                     ^ 
                                   
                                 
                                 ⁢ 
                                 
                                   k 
                                   x 
                                 
                                 ⁢ 
                                 
                                   
                                     L 
                                     x 
                                   
                                   / 
                                   2 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 . 
               
             
           
         
       
     
     Simplifying the above gives: 
     
       
         
           
             
               
                 
                   
                     ⇒ 
                     M 
                   
                   = 
                   
                     P 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           ⅇ 
                           
                             
                               ⅈ 
                               ^ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                         
                         [ 
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     
                                       k 
                                       y 
                                     
                                     ⁢ 
                                     
                                       L 
                                       y 
                                     
                                   
                                   2 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               k 
                               y 
                             
                             ⁢ 
                             
                                 
                             
                           
                         
                         ] 
                       
                       [ 
                       
                         
                           
                             - 
                             
                               
                                 L 
                                 x 
                               
                               
                                 
                                   ⅈ 
                                   ^ 
                                 
                                 ⁢ 
                                 
                                   k 
                                   x 
                                 
                               
                             
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     k 
                                     x 
                                   
                                   ⁢ 
                                   
                                     L 
                                     x 
                                   
                                 
                                 2 
                               
                               ) 
                             
                           
                         
                         - 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅈ 
                                 ^ 
                               
                             
                             
                               k 
                               x 
                               2 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     k 
                                     x 
                                   
                                   ⁢ 
                                   
                                     L 
                                     x 
                                   
                                 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Because the dimensions of the diaphragm are very small relative to the wavelength of sound, the arguments of the sin and cosine functions are very small, which results in 
               sin   ⁡     (         k   y     ⁢     L   y       2     )       ≈           k   y     ⁢     L   y       2     .           
The second term in brackets in Equation (20) is expanded to second order using Taylor&#39;s series. Using
 
               cos   ⁢           ⁢   θ     ≈     1   -       θ   2     2                 and                 sin   ⁢           ⁢   θ     ≈     θ   -       θ   2     6         ,         
in Equation (16),
 
     
       
         
           
             M 
             ≈ 
             
               P 
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   
                     
                       ⅇ 
                       
                         
                           ⅈ 
                           ^ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         2 
                         ⁢ 
                         
                           ( 
                           
                             
                               L 
                               y 
                             
                             2 
                           
                           ) 
                         
                       
                       ] 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                           
                             - 
                             
                               L 
                               x 
                             
                           
                           
                             
                               ⅈ 
                               ^ 
                             
                             ⁢ 
                             
                               k 
                               x 
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 
                                   k 
                                   x 
                                   2 
                                 
                                 ⁢ 
                                 
                                   L 
                                   x 
                                   2 
                                 
                               
                               8 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅈ 
                               ^ 
                             
                           
                           
                             k 
                             x 
                             2 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 
                                   k 
                                   x 
                                 
                                 ⁢ 
                                 
                                   L 
                                   x 
                                 
                               
                               2 
                             
                             - 
                             
                               
                                 
                                   k 
                                   x 
                                   3 
                                 
                                 ⁢ 
                                 
                                   L 
                                   x 
                                   3 
                                 
                               
                               48 
                             
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
                 . 
               
             
           
         
       
     
     Simplifying gives: 
     
       
         
           
             
               
                 
                   M 
                   ≈ 
                   
                     P 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         
                           ⅈ 
                           ^ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ω 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     ⁢ 
                     
                       L 
                       y 
                     
                     ⁢ 
                     
                       
                         
                           k 
                           x 
                         
                         ⁢ 
                         
                           L 
                           x 
                           3 
                         
                       
                       
                         12 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅈ 
                           ^ 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     The net force is given by a surface integral of the acoustic pressure, 
             F   =     -       ∫       -     L   x       /   2         L   x     /   2       ⁢       ∫       -     L   y       /   2         L   y     /   2       ⁢     P   ⁢           ⁢     ⅇ       ⅈ   ^     ⁢           ⁢   ω   ⁢           ⁢   t       ⁢     ⅇ         -     ⅈ   ^       ⁢     k   x     ⁢   x     -       ⅈ   ^     ⁢     k   y     ⁢   y         ⁢           ⁢     ⅆ   x     ⁢           ⁢       ⅆ   y     .                   
Carrying out the integration gives:
 
     
       
         
           
             
               F 
               -- 
             
             ⁢ 
             P 
             ⁢ 
             
                 
             
             ⁢ 
             
               ⅇ 
               
                 
                   ⅈ 
                   ^ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ω 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 t 
               
             
             ⁢ 
             
               
                 2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   sin 
                   ⁡ 
                   
                     ( 
                     
                       
                         
                           k 
                           x 
                         
                         ⁢ 
                         
                           L 
                           x 
                         
                       
                       2 
                     
                     ) 
                   
                 
               
               
                 k 
                 x 
               
             
             ⁢ 
             
               
                 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     sin 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             k 
                             y 
                           
                           ⁢ 
                           
                             L 
                             y 
                           
                         
                         2 
                       
                       ) 
                     
                   
                 
                 
                   k 
                   y 
                 
               
               . 
             
           
         
       
     
     Again, for small angles this becomes
 
 F=−Pe   îωt ( L   x   L   y )  (22)
 
     Using Equations (15), (18) and (19): 
     
       
         
           
             
               
                 
                   [ 
                   
                     
                       
                         I 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         m 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           m 
                           a 
                         
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           m 
                           a 
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         
                           θ 
                           ¨ 
                         
                       
                     
                     
                       
                         
                           x 
                           ¨ 
                         
                       
                     
                     
                       
                         
                           
                             X 
                             ¨ 
                           
                           3 
                         
                       
                     
                     
                       
                         
                           
                             X 
                             ¨ 
                           
                           4 
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   { 
                   
                     
                       [ 
                       
                         
                           
                             
                               k 
                               t 
                             
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             k 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                     + 
                     
                       [ 
                       
                         K 
                         ′ 
                       
                       ] 
                     
                   
                   } 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         θ 
                       
                     
                     
                       
                         x 
                       
                     
                     
                       
                         
                           X 
                           3 
                         
                       
                     
                     
                       
                         
                           X 
                           4 
                         
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               ( 
               
                 
                   
                     
                       
                         Pⅇ 
                         
                           
                             ⅈ 
                             ^ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ω 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                       ⁢ 
                       
                         L 
                         y 
                       
                       ⁢ 
                       
                         
                           
                             k 
                             x 
                           
                           ⁢ 
                           
                             L 
                             x 
                             3 
                           
                         
                         
                           12 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅈ 
                             ^ 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       - 
                       
                         
                           Pⅇ 
                           
                             
                               ⅈ 
                               ^ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             t 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               L 
                               x 
                             
                             ⁢ 
                             
                               L 
                               y 
                             
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       - 
                       
                         PA 
                         a 
                       
                     
                   
                 
                 
                   
                     
                       - 
                       
                         PA 
                         a 
                       
                     
                   
                 
               
               ) 
             
           
         
       
     
     Let 
               K   eq     =     {       [           k   t         0       0       0           0       k       0       0           0       0       0       0           0       0       0       0         ]     +     [     K   ′     ]       }           
and assume θ=Θe îωt ,x=Xe îωt ,X 3 =X 3 e îωt  and X 4 =X 4 e îωt             .

     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   K 
                                   eq 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     1 
                                     , 
                                     1 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 I 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   3 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   , 
                                   4 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   K 
                                   eq 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     , 
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   , 
                                   3 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   , 
                                   4 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   3 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   3 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   K 
                                   eq 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     3 
                                     , 
                                     3 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   m 
                                   a 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ω 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   3 
                                   , 
                                   4 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   4 
                                   , 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   4 
                                   , 
                                   2 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 K 
                                 eq 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   4 
                                   , 
                                   3 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 
                                   K 
                                   eq 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     4 
                                     , 
                                     4 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   m 
                                   a 
                                 
                                 ⁢ 
                                 
                                   ω 
                                   2 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               Θ 
                               / 
                               P 
                             
                           
                         
                         
                           
                             
                               X 
                               / 
                               P 
                             
                           
                         
                         
                           
                             
                               
                                 X 
                                 3 
                               
                               / 
                               P 
                             
                           
                         
                         
                           
                             
                               
                                 X 
                                 4 
                               
                               / 
                               P 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               L 
                               y 
                             
                             ⁢ 
                             
                               
                                 
                                   k 
                                   x 
                                 
                                 ⁢ 
                                 
                                   L 
                                   x 
                                   3 
                                 
                               
                               
                                 12 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅈ 
                                   ^ 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               ( 
                               
                                 
                                   L 
                                   x 
                                 
                                 ⁢ 
                                 
                                   L 
                                   y 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               A 
                               a 
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               A 
                               a 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Using Equation (23), the displacement and rotation relative to the amplitude of the pressure, X/P and θ/P, as a function of the excitation frequency, ω may be computed. 
     Based on the foregoing analysis, it may be observed that if the air in the back volume  108  is considered to be in viscid, the performance of the differential microphone diaphragm  102  is not degraded if the depth of the backing cavity  110  is reduced significantly. Thus the microphone  100  can be fabricated without the need for a backside hole behind the diaphragm  102 . The fabrication process for the surface micromachined microphone diaphragm is shown in  FIGS. 6   a - 6   d.    
     Referring now to  FIG. 6   a , there is shown a bare silicon wafer  200  before fabrication is begun. Such silicon wafers are known to those skilled in the art and are not further described herein. 
     As may be seen in  FIG. 6   b , a sacrificial layer (e.g., silicon dioxide)  202  is deposited on an upper surface of wafer  200 . While silicon dioxide has been found suitable for forming sacrificial layer  202 , many other suitable material are know to those of skill in the art. For example, low temperature oxide (LTO), phosphosilicate glass (PSG), aluminum are known to be suitable. Likewise, photoresist material may be used. In still other embodiments, polymeric materials may be used to form sacrificial layer  202 . It will be recognized that other suitable material may exist. The choice and use of such material is considered to be known to those of skill in the art and is not further described herein. Consequently, the invention is not considered limited to a specific sacrificial layer material. Rather, the invention covers any suitable material used to form a sacrificial layer in accordance with the inventive method. 
     Over sacrificial layer  202 , a layer of structural material (for example polysilicon) is also deposited. While polysilicon has been found suitable for the formation of layer  204 , it will be recognized that layer  204  may be formed from other materials. For example, silicon nitride, gold, aluminum, copper or other material having similar characteristic may be used. Consequently, the invention is not limited to the specific material chosen for purposes of disclosure but covers any and all similar, suitable material. Layer  204  will ultimately form diaphragm  102  ( FIG. 2 ). 
     As is shown in  FIG. 6   c , the diaphragm material, layer  204  is next patterned and etched to form the diaphragm  102 , leaving slits  114 . 
     Finally, as may be seen in  FIG. 6   d , the sacrificial layer  202  under diaphragm  102  is removed leaving cavity  110 . After the removal of the sacrificial layer, the microphone diaphragm  102  has a back volume  108  with a depth equal to the thickness of the sacrificial layer  202 . The microphone is shown schematically in  FIG. 7 . 
     To convert motion of diaphragm  102  into an electronic signal, comb fingers incorporated at  208  ( FIG. 7 ) may be integrated with the diaphragm. Such comb or interdigitated fingers are described in detail in copending U.S. patent application Ser. No. 11/198,370 for COMB SENSE MICROPHONE, filed Aug. 5, 2005. 
     As an alternative sensing scheme, the fundamental microphone structure of  FIG. 7  may be modified slightly to include two conductive layers  206  disposed between silicon chip  200  and additional conductive layer  204  to form back plates forming fixed electrodes of capacitors. These back plates are electrically separated from each other in order to allow differential capacitive sensing of the diaphragm motion. 
     It should be noted that one could employ both the comb fingers  208  and the back plate  206  to perform capacitive sensing. In this case, in addition to serving as an element of a capacitive sensing arrangement, a voltage applied to comb sense fingers  208  may be used to stabilize diaphragm  102 . The voltage applied between the comb fingers and the diaphragm can be used to reduce the effect of the collapse voltage, which is a common design issue in conventional back plate-based capacitive sensing schemes. 
     It will be recognized that many other sensing arrangements may be used to convert motion of diaphragm  102  to an electrical signal. Consequently, the invention is not limited to any particular diaphragm motion sensing arrangement. 
     Since other modifications and changes varied to fit particular operating requirements and environments will be apparent to those skilled in the art, the invention is not considered limited to the example chosen for purposes of disclosure, and covers all changes and modifications which do not constitute departures from the true spirit and scope of this invention. 
     Having thus described the invention, what is desired to be protected by Letters Patent is presented in the subsequently appended claims.