Numerical fluid structural interface analysis in condenser microphone design

Transcription

1 Journal of Mechanical Science and Technology 5 (3) () 65~6 DOI.7/s Numerical fluid structural interface analysis in condenser microphone design Akbar Ranjbar, Mohammad-Ali Saeimi-Sadigh and Bashir Behjat * Acoustic Research Center, Electrical Engineering Department, Shahed University, Tehran, Iran (Manuscript Received June, ; Revised Ocotober 3, ; Accepted November, ) Abstract Condenser microphones are widely used in electronic and acoustic applications. Although, various mechanical and electro-mechanical methods have been developed to design and analyze these sensors. However, due to the difficulty of fluid-structural-electrical couplings, none of them can introduce a method that consider all parameters of the design together. This research concerns the effects of four main parameters: a) Air gap size; b) Number of holes on the back-plate; c) holes radius size and d) location of the holes in back plate, on the response of the microphones. This analysis have been carried out based on coupled finite element and finite volume method using ANSYS-CFX software to simulate fluid-structure interaction between the diaphragm and air in the air-gap region. By using this method, the effects of the geometric parameters on the response of the microphone have been investigated. Results show that, increasing air gap size, holes radius, and holes number decrease the damping effects of the air between diaphragm and back plate. On the other hand, increasing the distance between the holes has the opposite effect. In addition, results reveal that among these four parameters, increasing the number of holes on back plate is the most efficient method in reducing air-gap damping effects. Keywords: Condenser microphone; ANSYS-CFX; Finite element methods; Fluid structure interaction Introduction Up to now, there has been much research on theoretical modeling and numerical analysis of condenser microphones. These models are mainly based on transforming the microphone structure into equivalent mechanical or electrical components and these equivalent models are analyzed by basic mechanical equations []. In the next paragraph some recent numerical and analytical works have been reviewed. For the first time Zucherwar [] presented an analytical response for the microphone. His results were based on the model which includes all microphone geometrical characteristics. He solved the formulas which resulted from the fluidstructure analysis of the model, however he did not consider the initial voltage and pretension in his analysis. In 98, Zhon [3] determined the microphone sensitivity. He used microphone average diaphragm displacement in his work and included all microphone geometrical characteristics, air viscosity and the effect of the different parameters on the microphone response such as air gap and hole dimensions. In 994, Donk et al. [4] obtained the microphone static response from an analytical model. Although they just modeled the air gap This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim * Corresponding author. Tel.: , Fax.: address: bashir.behjat@gmail.com KSME & Springer and excluded other geometrical parameters, but including the initial voltage and tension in derived equations and studying of its effects on the microphone response were the advantages of their research. In 997 Mutschlecner [5] used fluid approximation for gases to present theoretical analyses for the microphone. He took into acount the interaction between fluid and structure while he ignored the diaphragm initial voltage and tension in their dynamic analysis. Pedersen [6] presented a complete discrete model for microphone. In his work, all geometrical characteristic of the system have been modeled by an equivalent electric circuit. Circuit element values were calculated considering the fact that energies that stored in both systems have the same value. Pedersen took into account the initial diaphragm tension, but he did not look over the initial voltage. A finite element microphone analysis was presented by Ying in 998 [7]. He modeled the microphone diaphragm by ANSYS software and ignored the fluid- structure coupling, but he considered the diaphragm initial tension. Rajalingham [8] took into consideration the non-linear electrical force and diaphragm analytical equations. He solved the nonlinear equations by Newton-Raphson method. Kainz [9] worked on acoustical dispersion from diaphragm and he used matrix transformation to solve the coupling equations. Dadic [] modeled condenser microphone using finite element method. This method was used to determine microphone nonlinear characteristics and the effect of nonlinear electrical field on

2 66 A. Ranjbar et al. / Journal of Mechanical Science and Technology 5 (3) () 65~6 the microphone mechanical components. Also an appropriate electrical analysis for electrical parameters has been performed. Chen et al. [] have obtained the complete diaphragm equations under electrical charge by using the theory of plates and shells. Then they transformed these equations into mechanical equivalent circuit and performed an electrical analysis. In this paper the microphone fluid-structure coupling problem has been investigated. The microphone has been modeled with all of the geometrical characteristics in ANSYS software, then using CFX software (one of the ANSYS Products); coupled fluid-structural analysis has been performed. It is worth to note that all the geometrical characteristics and initial tension of diaphragm have been taken into account in his analysis.. Microphone theoretical response Generally, fluid structure coupling issue emerges in the diaphragm air layer coupling equations in condenser microphone and includes equation of motion, Navier-Stokes equation, continuity equation, and equation of state. Considering these equations and the momentum transfer equation, the diaphragm displacement could be finalized as []: ( θ ) p,, (, ) (, ) i p r z η r θ + k η r θ = +. () T T This equation is written in the cylindrical coordinate system (r,θ,z) and in the time domain. p i is the incident sound pressure on the diaphragm, p is the reaction pressure on the diaphragm, η is the displacement of the diaphragm, T is the pretension in the diaphragm and k is the number of sound wave in the diaphragm which is defined as: k σ = πω M T where σ M is the diaphragm mass surface density and ω is the frequency of sound. Eq. (l) is a mixed integral-differential equation. Because the reaction pressure depends on the integral of the diaphragm displacement η, on the diaphragm surface. We assumed a uniform distribution of sound pressure on the surface of the diaphragm. The air velocity can be stated as scalar potential and vector potential function: V = gradφ + A (3) where A is the vector potential and is the scalar potential. The following condition should hold for the vector function A: divα=. (4) () While k=w/c and L=iw/ν are number of scalar and vector wave numbers, c is the sound speed in air and ν is the kinematic viscosity of air. The above equations include the microphone air diaphragm coupling, although the non linear electric force is not considered. 3. Microphone modeling The presented model in this section (Fig. ) includes diaphragm, back plate, air gap, back chamber, and the holes. Fluid structure coupling occurs between the diaphragm and the air available in the air gap. The coupling should be defined for the software between the diaphragm and the air in the air gap. Initial diaphragm tension is also modeled, so the effect of this tension on the bending reaction of the diaphragm is equivalent to the Young s Modulus increasing. The equation of the plate that includes initial tension is defined [3]: dr r dr a r D z d φ dφ k q + + φ + = where a, is the radius of the diaphragm, D is the flexural rigidity of plate, the is related to displacement by φ=dw/dr and q is the applied pressure on the diaphragm. The final solution of the above equation with clamped boundary condition: 4 kr qa J J ( k ) a qa a r w = kj k D 4kD ( ) ( ) 3 W is the displacement for the diaphragm under tension T. Also J and J are first and second Bessel type functions respectively. For a regular diaphragm with the same boundary condition without initial tension, the displacement is: 4 qa r w =. 64D a By equating the Eqs. (7) and (8), it will result the Young s Modulus of the diaphragm under specific initial tension 4 Ta D =. Ta 6D J D The resulted nonlinear equation is solved simultaneously in each steps using ANSYS and CFX software solution. (6) (7) (8) (9) Substituting Eqs. (3) and (4) into the Eq. (), yields: φ + k φ = + = o, A LA o, (5) 4. Results In this section, results of the fluid-structure analysis including diaphragm deflection and the velocity and the pressure of

3 A. Ranjbar et al. / Journal of Mechanical Science and Technology 5 (3) () 65~6 67 Fig.. The model used for simulation. Fig. 3. Air gap mesh. Fig.. The model of the Diaphragm s mesh. Fig. 4. Back plate holes and outside ring mesh model. the air have been presented. In addition, effects of different parameters such as air gap distance, number of the holes, and location of the holes on the microphone general operation and sensitivity have been investigated. 4. Finite elements model In this section the fluid and structure mesh model is presented. The diaphragm is very thin stainless steel having young modulus of 7 GPa and mass density of 78 kg/m 3. Table depicts the geometric parameters of the microphone which have been used in each case study. For the diaphragm modeling, the Shell 63 element has been used. The boundary conditions are applied by defining zero values for the displacement and the rotation in the diaphragm outer nodes. To reduce the computational time, the meshes are finer where there is high pressure gradient and are larger elsewhere. The number of elements that have been used in this model, includes 36 for air gap, 6579 for back plate holes, and 3959 for the back chamber. The models are analyzed by ANSYS and CFX software using Intel (R) Core (TM) Duo with 4GB RAM. Fig. shows the mesh of the diaphragm. There is an air gap between the diaphragm and the back plate which plays a damper role for the diaphragm displacement due to the pressure response. Mesh model for the air gap is shown in Fig. 3 which element 3D Flotran 4 for fluid mesh has been used. Fig. 4 shows the back plate holes mesh with outside ring region between the back plate and the microphone body. Back chamber mesh model is shown in Fig. 5. Due to the very thin air gap thickness, there was not a uniform mesh model possibility, hence the air gap mesh was done by three different sections and then these sections were assembled in CFX software. 4. Method of analysis To design a condenser microphone, a fluid- structure numerical analysis has been performed to obtain deflection of the diaphragm in each case study. Modal and harmonic analysis are also necessary to obtain mode shape and system response in different frequencies. Differential equation for transient analysis is:... () M X+ CX+ KX = f t. () In the above equation, M is the system mass matrix, C is the damping matrix, K is the stiffness matrix and f(t) is the load vector. In this section the results of structure dynamic response computation under time varying load has been presented. The results are shown as the contour of the displacement; Von-Mises effective stress and fluid pressure diagram.

4 68 A. Ranjbar et al. / Journal of Mechanical Science and Technology 5 (3) () 65~ gap micron gap 3micron gap 5 micron pressure(pas) line Fig. 5. Isometric diagram of back chamber mesh model. Pressure(pas) pressure.e+.e- 4.E- 6.E- 8.E-.E+.E+ Time(sec) Fig. 6. Time varying loading vector. Displacemnt(m).E-6 5.E-7.E+ -5.E-7 -.E-6 -.5E-6.E+.E- 4.E- 6.E- 8.E-.E+.E+ Time(sec) Fig. 7. Diaphragm transient response without structure fluid coupling. Fig. 7 shows the transient dynamic response for the diaphragm without fluid-structure coupling with zero damping coefficients under time varying load as shown in Fig. 6. Loading process is implemented in a specific time interval, which could be obtained from working frequency after harmonic analysis. During the process, the pressure was reached to its maximum from zero and then removed from the diaphragm in a very short time interval. Fig. 7 indicates that after removing the pressure, the diaphragm is oscillating within.36 micrometer amplitude due to the inertia forces and no damping effect. However, in condenser microphone, the air flow between the diaphragm and back plate will result in a damping force which causes the U_Z Fig. 8. Air gap effects on the pressure produced on the back plate and on its radius. oscillating amplitude to decay. In this section variable loading vector is defined up to the maximum pressure exertion moment since the goal is to determine the design parameters effect on the diaphragm deflection. Applied pressure value on the models under consideration is 5 Pa. To consider air gap increase effects on diaphragm deflection, the model was analyzed with three different air gap distances of, 3, and 5 microns. Parameters effect analysis was performed through preparing plate pressure contours and also diaphragm deflection counter. Diaphragm deflection will increase with pressure gradient decrease due to increase of air gap, so it was concluded that increasing the air gap is an effective tool to decrease the fluid damping effect. The diagram of pressure on the back plate is shown in Fig. 8. Since the produced pressure has a direct effect on the damping quantity, 5% decrease in produced pressure presents 5% increase in the air gap, so this method is an effective one in damping reduction. The diaphragm deflection diagram in Fig. 9 is presented to indicate the effect of air gap between the diaphragm and back plate. Increasing air gap will result in diaphragm deflection increase. In other word it is concluded that air gap increase causes the diaphragm to have a better response in the lower pressure. Fig. shows the diaphragm Von-Mises stress diagram. It is necessary to check Von-Mises stress of the plate against yield stress to avoid elasticity property reduction due to hardening phenomenon. The results indicated an increase in air gap between the diaphragm and back plate that will have a considerable effect on system response especially on reactive pressure on the back plate which directly results in diaphragm damping and deflection. To observe the back plate holes distance effect on creation pressure on the back plate and also diaphragm deflection, the diaphragm model has been analyzed in three different cases with 5 mm, 37mm, and 4mm, holes distance from the back plate center. With increasing the back plate holes distance, it is expected to have a pressure increase on the back plate due to the outgoing air flow reduction rate.

5 A. Ranjbar et al. / Journal of Mechanical Science and Technology 5 (3) () 65~6 69 deflection(m).e+ -.E-6 -.E-6-3.E-6-4.E-6 gap micron gap 3 gap 5 Pressure(Pas) rhole.6 rhole.5-5.e e radius Fig. 9. Air gap effects on diaphragm displacement Radius Fig.. the effect of number of holes on Diaphragm displacement..e-6.e+ -.E-6 Fig.. Air gap effects on Von-Mises tension in diaphragm. (a) (b) Fig.. Diaphragm displacement due to the different Back plate distance (a) 5mm; (b) 4mm. deflection(m) -.E-6-3.E-6-4.E-6-5.E-6-6.E-6 5.E-.E-.5E-.E-.5E- 3.E- 3.5E- 4.E- 4.5E- 5.E- radius 6 hole 8 hole 5.5E- 6.E- 6.5E- 7.E- 7.5E- 8.E- 8.5E- 9.E- 9.5E-.E+ Fig. 3. The curves for the pressure on the back plate for the 5mm and 6mm back plate holes radiuses. To show the diaphragm deflection in above three cases, the displacement contour is shown in Fig. (a), and Fig. (b) for the 5mm and 4mm hole distance models. The effect of fluid pressure increase on the diaphragm deflection reduction is clearly observed. It is worth to mention that on the above models, a little more pretension was applied to the diaphragm to reduce the diaphragm deflection due to the electrical charges. To consider the effect of increasing number of holes, a model with 8 holes arranging in two rows that each row have 4 holes with the 37mm distance from the diaphragm center for the first row and 47mm for the second row was analyzed. As shown in Fig., the displacement diagrams indicate a considerable diaphragm deflection effect for an increase in the number of holes. To determine the back plate holes radius size effect on the pressure gradient created on the back plate, the holes radius on the model with 4mm holes distance with respect to the diaphragm center was increased %. As mentioned before, this model showed a 5% increase in maximum pressure on the back plate with respect to the base model. Thus the holes radius was increased from 5mm to 6mm which resulted in 44% increase in each area of the holes. Fig. 3 shows the pressure curve on the back plate. Comparing the areas under the curves for the two cases, the damping due to the fluid pressure crea-

6 6 A. Ranjbar et al. / Journal of Mechanical Science and Technology 5 (3) () 65~6 Table. Base model dimensions. Parameters Symbols Parameters Quantities Diaphragm Radius R.75 (m) Back Plate Radius Rb.87 x Ro (m) Back plate hole radius r hole.5 (m) Back plate holes location rh.4 x Ro (m) Air gap thickness gap (micro meter) Initial tension σ.7 x 6 (pas) Number of holes N 6 percent increase in maximum pressure is resulted form 5- mm increase in back plate holes distance. Maximum pressure range experience considerable increase with increasing holes distance. Adding two more holes reduces maximum pressure from 3 Pa. to.66 Pa. in addition to considerable decreasing the maximum pressure range. Increasing the number of holes is the most effective method in reduction of fluid damping between the diaphragm and back plate. By increasing percent of back plate holes radius, the maximum pressure created on the back plate will decrease up to 5percent. Acknowledgment The authors would like to thank Dr. H. R. Hassani, the head of Electrical Engineering College of Shahed University, Dr. H. R. Massah, the head of the Acoustic Research Institute, and Dr. F. Mohseni sponsor of Acoustic Research Institute for their support and enthusiastic efforts to help our research goals. References tion could be determined for further necessary analyses. Displacement contours for the above cases are also shown in Fig Conclusion (a) (b) Fig. 4. Diaphragm displacement contour for the different holes radiuses (a) 5mm; (b) 6mm. In this paper a new model for condenser microphones has been introduced. Finite element method is used to obtain the design parameters of the microphone. ANSYS and CFX software has been used to solve the nonlinear fluid structure interaction of the microphone. The influence of microphone parameters on the response is investigated. Some of the results of the work can be indicated as: Increasing the air gap results in decreasing the pressure on the back plate and consequently causes diaphragm deflection to increase. If the effective tension on the diaphragm goes beyond the submission limit, it will cause the diaphragm elasticity to reduce. About 5-6 [] L. L. Beranek, Acoustics, Acoustical Society of America, New York (996). [] A. J. Zucherwar, Theoretical response of condenser microphones, J. Acoust. Soc. Am. 64 (5) (978) [3] R. Zhon, Analysis of the Acoustic Response of Circular condenser electert microphones, J. Acoust. Soc. Am., 69 (4) (98) -3. [4] A. Donk, P. R. Scheeper, W. Olthus and P. Bergveld, Modeling of silicon condenser microphones, Sensors and Actuators A, 4 (994) 3-6. [5] J. P. Mutschlecner and R. W. Whitaker, Design and operation of infrasonic microphones, Los Alamos National Laboratory (997). [6] M. Pedersen, W. Olthuis and P. Bergveld, High-Performance Condenser Microphone with Fully Integrated CMOS Amplifier and DC DC Voltage Converter, J. Microelectromechanical Sys., 7 (4) (998) [7] M. Ying, Q. Zou and S. Yi, Finite-element analysis of silicon condenser microphones with corrugated diaphragms, Finite Elements in Analysis and Design, 3 (998) [8] C. Rajalingham and R. B. Bhat, Influence of an electric field on diaphragms stability and vibration in a condenser microphone, J. Sound and Vibration, (5) ( 998) [9] W. Kainz, Condenser microphone model. Application of T-matrix method of Waterman to acoustic scattering from an elastic obstacle, J. Acoust. Soc. Am., 4 () ( 998) [] M. Dadic, Numerical determination of electric forces and capacitance of pressure condenser microphone, IEEE Instrumentation and Measurement technology conference (). [] J. Chen, Y. Hsu, S. Lee, T. Mukherjeea and G. K. Fedder, Modeling and simulation of a condenser microphone, Sensors and Actuators A, (8) 4-3. [] G. S. K. Wong and T. F. W. Embleton, AIP Handbook of

7 A. Ranjbar et al. / Journal of Mechanical Science and Technology 5 (3) () 65~6 6 Condenser Microphones: Theory, Calibration and Measurements, American Institute of Physics edition (994). [3] S. Timoshenko and S. Woinowsky-Krieger, Theory of plates and shells, McGraw-Hill Companies, second edition (959). Bashir Behjat entered the Mechanical Engineering program at the Amirkabir University of Technology. He received his bachelor s degree, along with a major degree in Solid Mechanics in 6. After getting as an honored student among the students of the faculty, Bashir Behjat was accepted into the master s program in the Mechanical Engineering department at Amirkabir University of Technology in 6. After accepting as a PhD student in Tabriz University in Iran in September 9, he continued his PhD in this university as a researcher in the field of Nonlinear Finite element method and FGPM Materials.