NORMALIZED ABACUS FOR THE GLOBAL BEHAVIOR OF DIAPHRAGMS : PNEUMATIC, ELECTROSTATIC, PIEZOELECTRIC OR ELECTROMAGNETIC ACTUATION

Transcription

1 Journal of Modeling and Simulation of Microsystems, Vol., No. 2, Pages 49-6, 999. NORMALIZED ABACUS FOR THE GLOBAL BEHAVIOR OF DIAPHRAGMS : PNEUMATIC, ELECTROSTATIC, PIEZOELECTRIC OR ELECTROMAGNETIC ACTUATION O. Français ELMI, Groupe ESIEE, Cité Descartes, BP 99, 9362 Noisy Le Grand, France. I. Dufour Laboratoire d'electricité Signaux et Robotique (LESiR) - UPRESA CNRS Ecole Normale Supérieure de Cachan, 6 avenue du président Wilson, Cachan Cedex, France. Abstract The aim of this paper is to propose a complete normalized study of a diaphragm's behavior under classical excitation encountered in microsystem: pneumatic, electrostatic, piezoelectric and magnetic. The principle of modeling is widely described in order to be easily applied to other structure. First, a traditional analytical solution is recalled for the diaphragm's deflection based on energy minimization, through the use of a polynomial solution technique. Regarding the geometrical and physical parameters, three cases can be distinguished. They concern the diaphragm: acting as a pure plate, acting as a pure membrane, or exhibiting the effects of both elastic rigidity and initial internal stress. For these three cases, a standardization of the diaphragm's behavior under pneumatic actuation makes it possible to obtain an abacus that can then be used for any geometrical or physical parameter. The most interest of this paper is to establish simple abacuses for the diaphragm deflection under piezoelectric, magnetic or electrostatic actuation. The initial analytical formulae are obtained from the study of the physical phenomena. The abacuses for the piezoelectric and electromagnetic actuation can be deduced from the well known pneumatic actuation presented at the beginning of this paper. For the electrostatic actuation, the classical energy-based approach has been extended and allows to obtain abacuses for the center deflection, for the volume stroke and for the electrical capacitance. These abacuses have been compared measurements made on silicon plates under electrostatic actuation. Although the results presented are obtained by making certain approximations, this study provides order of magnitude results for the deflection, and in particular it can be used to find the effect of each parameter (current, voltage, thickness, length, material properties, etc.). The use of these abacuses is easy and numerous potential applications exist: electrostatic or piezoelectric micropump, accelerometers, microphones, microspeaker, etc. Keywords: modeling, membrane, plate, diaphragm, electrostatic, piezoelectric, electromagnetic. Introduction D h 3 w( xy, ) T h w( xy, ) pxy (, ) () In this paper, we consider a built-in diaphragm (Fig. ) characterized by its thickness h, its width a, its length b (the surface noted S is equal to ab), its elastic flexural rigidity D E 2( ν 2 ), where E is Young s modulus, v Poisson s ratio, and its initial internal stress T. The deflection under pressure p(x,y) has been denoted by w(x,y). The study presented below is limited to small deflections (w < h). The classical law governing the diaphragm s behavior under such assumptions and notations is []: where 2 2 x 2 + y 2 There is no simple analytical solution to equation (). The solution can thus be obtained using FEM simulation or approximate solution methods. In order to derive a parametric solution, we have opted for the classical approximate solution method based on energy minimization [2-4]. francaio@esiee.fr Computational Publications, ISSN

2 JMSM, Vol., No. 2, Pages 49-6, 999. Bottom view y Section 2.2. Coefficient Calculation Given the expression for the total energy, the n equations (3) can be combined in matrix form, as follows: b x h Fig. : Geometry of the diaphragm. 2. Energy Minimization The deflection w(x,y) of a diaphragm under pressure p(x,y) corresponds to the diaphragm position that has the minimum total energy. We have chosen to approximate this deflection by a linear combination of polynomial functions f i : n wxy (, ) a i f i ( x, y) (2) i Let U tot represent the total energy of the diaphragm; the deflection is then obtained the n coefficients a i, which are the solutions of the n equations: U tot (3) a i 2.. Energy Calculation a The total energy of the diaphragm is composed of two terms [2,5]: the potential energy due to pressure acting on the diaphragm: b 2 a 2 w U ext pxy (, ) dw dxdy (4) b 2 a 2 and the potential internal energy due to both the elastic flexural rigidity and the initial internal stress : U int U D + U T b 2 a 2 U D -- D (6) 2 h 3 ( w( xy, ))wxy (, ) dxdy b 2 a 2 and b 2 a 2 U T -- T. (7) 2 h( w( xy, ))wxy (, ) dxdy b 2 a 2 Therefore, the total energy can be expressed as: (5) U tot U ext + U int U ext + U D + U T. (8) z x du int du a (9) da i ext i da i where: du int da i is a ( n n) matrix that contains, for each row i - column j pair, the second partial derivative of the U int energy respect to the ith and jth coefficients. It should be noted that this matrix is not dependent upon the type of excitation. Instead, it depends solely upon the approximate functions f i. a i is a ( n ) vector that contains the n coefficients a i. du ext da i is a ( n ) vector that contains the derivatives of U ext respect to each of the coefficients a i. The coefficients a i can then be derived as follows: a i () In this equation, the two matrices are dependent upon the approximate functions f i. In the following section, we will show that the number n and the choice of the n functions both serve to determine the accuracy of the results Choice of Polynomial Functions Simple functions du int da i du ext da i In order to avoid the constraints of the diaphragm s dimensions [6], the following variable substitution is introduced: X 2x a; Y 2y b, the origin being located at the center of the diaphragm: X [, ] and Y [, ]. The expression of the deflection can then be written the new variables: n W( X, A i F i ( X, () i where W( X, wax ( 2, by 2) and F i ( X, f i ( ax 2, by 2). The polynomial functions F i must be consistent the built-in diaphragm s boundary conditions: F i ( ±, Y ) F i ( X, ± ) and F i ( ±, Y ) F i ( X, ± ) X Y. Function F i has therefore been chosen the term ( X 2 ) 2 ( Y 2 ) 2 as a factor. Moreover, since only symmetrical actuation is being considered, the approximate functions must be even. The simplest functions corresponding 5

3 Normalized Abacus for the Global Behavior of Diaphragms : Pneumatic, Electrostatic, Piezoelectric or Electromagnetic Actuato these criteria are [6]: first type of functions have been indicated by solid lines and the second type of functions by dashed lines. ( X 2 ) 2 ( Y 2 ) 2 X j Y k (2) j and k being even numbers Supplementary functions.6 If the numbers j and k were to increase, the polynomial.4 functions would be shifted towards the edges of the diaphragm. Thus, from Fig. 2 ( all of the function.2 maxima being taken as equal to ), we can observe that it is not necessary to use many functions of this type, in that the functions corresponding to high polynomial orders will be of -.2 little significance in the deflection s estimation. -.4 Lack of functions F -.6 i j8 -.8 j j Fig. 3: Contour of the functions (limit.98). F i.5 j2 j4 j6.5 Fig. 2: Section of the simple polynomial functions: Y. In order to compensate for the lack of functions in the intermediate zone between the edge and the center, we propose employing other polynomial functions: ( X 2 ) 2 + nl ( Y 2 ) 2 + nm X 2l Y 2m ( l, m) {(, ),(, ), (, )} number Used functions (3) and n being an even For the following simulations, we have considered 34 F i functions: 25 of the first type (Eq. (2)), j { 2468 ; ; ; ; } and k { 2468 ; ; ; ; }; and 9 of the second type (Eq. (3)), n { 246 ; ; } and ( l, m) {(, ),(, ), (, )}. In order to visualize the area corresponding to the localization of the functions, the contour of the various F i functions has been plotted in the [X,Y] plane, the maxima taken as equal to for each function and a contour value equal to.98 (see Fig. 3). The gray areas correspond to values less than.98 and the white areas to values greater than.98. The X Fig. 3 demonstrates the advantage of using functions of the second type (in the intermediate zone) instead of a greater number of functions of the first type (placed along the edges). 3. Pneumatic Actuation In the previous section, a traditional methodology for solving Eq. () was recalled. The result respect to the deflection under pneumatic pressure ( pxy (, ) constant P ) depends on the diaphragm s geometrical characteristics (h, a and b) as well as on its physical parameters (D and T ). Since the electromagnetic and the piezoelectric actuation (sections 4 and 5) can be deduced from the simpler case of the pneumatic actuation, we have developed in this section the classical case of the pneumatic actuation respect to the physical parameters D and T. In fact, in this section, the case of the plate (the opposite force due to T is negligible) and the case of the membrane (the opposite force due to D is negligible) are to be examined first. Once these two cases have been taken into consideration, the deflection can then be standardized. Afterwards, we will show that, for some determinate diaphragm configurations, these two simpler cases cannot actually be applied. For such configurations, another standardization procedure has thereby been proposed. 3.. Plate Behavior In this case, Eq. () becomes: This equation is equivalent to: D h 3 w( xy, ) P r ND ND ND X X 2 Y r Y 4 (4) (5) 5

4 JMSM, Vol., No. 2, Pages 49-6, 999. N D ( X, W( X, C D P S 2 C D D h Vol N D Bouwstra Ansys r b a.5 Thus, for a given rectangularity (r being fixed), the solution to Eq. (5) the method presented in section 2 on "energy minimization" is sufficient to determine the deflection for each geometrical and physical parameter. The product of the solution N D (X, to Eq. (5) and the value of C D is in fact the deflection W(X,. In Fig. 4, the solution N D (,) of Eq. (5) has been plotted versus the value of the rectangularity r Fig. 5: Variation of Vol N D versus r Timoshenko Ansys N (,) D 3.2. Membrane Behavior In this case, Eq. () becomes: T h w( xy, ) P. (8) The same standardization as that used in the plate case can be performed for the membrane; Eq. (8) is then equivalent to: 2 2 NT NT r (9) X r Y 2 N T ( X, W( X, C T C T P S T h Fig. 4: Variation of N D (, ) versus r. The entire normalized deflection N D (X, can be used to obtain the stroke volume (Vol): Vol N D Vol S --C 4 D Vol N D (6) N D ( X, dxdy. (7) Then the stroke volume can also be normalized and a single abacus (Fig. 5) is sufficient to find the stroke volume for each plate under any pneumatic pressure. Moreover, in Figs. 4 and 5, we have plotted some of the results published by other authors [3,7,8] as well as some FEM simulations performed the software ANSYS. A very high level of agreement has been observed between the results of the energy based method and those based on other methods. r b a Thus, like for the plate case, the solution to Eq. (9) by the method presented in section 2 on "energy minimization", together the calculation of the C T value, are sufficient to determine the total deflection of the membrane. The standardized result for the center deflection, N T (,), has been presented in Fig. 6. It is obvious that, in this case, the solution method presented here is in agreement other results [9]. In the same way as for the plate behavior, it is possible to obtain the stroke volume (Vol) using the normalized deflection N T (X, S Vol --C (2) 4 T Vol N T Vol N T N T ( X, dxdy (2). Then the stroke volume can also be normalized and a single abacus (Fig. 7) is sufficient to know the stroke volume for each membrane under any pneumatic pressure. 52

5 Normalized Abacus for the Global Behavior of Diaphragms : Pneumatic, Electrostatic, Piezoelectric or Electromagnetic Actua ANSYS and [9] - Membrane Plate Pure plate case Diaphragm Pure membrane case Fig. 6: Variation of N T (, ) versus r Bouwstra [8] Vol N T Fig. 8: Variation of N CDT (, ) versus C DT for a square-shaped diaphragm. In Fig. 9, we have plotted the same curve for the normalized volume Vol N CDT. Using Fig. 9, it is possible to determine the volume (Vol):.35.3 Vol C D Vol N CDT (23) Fig. 7: Variation of Vol N T versus r General Case As for the two previous cases, Eq. () can be written as follows, to obtain a standardized result: pure plate case Diaphragm Membrane Plate Pure membrane case N CDT ( X, C DT N CDT ( X, N CDT ( X, W( X, C D (22) Fig. 9: Variation of Vol N CDT versus C DT for a square-shaped diaphragm. C DT C D C T ST D h 2 In Fig. 8, the result obtained in the square-shaped diaphragm case (r ) has been presented for the center point. As opposed to the two simplified cases (plate and membrane), the solution depends upon both the rectangularity r and the parameter C DT, in the general case. Figs. 8 and 9 demonstrate that for certain values of C DT, the diaphragm can be considered either as a pure plate ( C DT <.25 ) or as a pure membrane ( C DT > 4 ). For intermediate value of C DT (.25 < C DT < 4 ); the behavior of the diaphragm can be found from the general solution of Eq. (22). The abacuses of Figures 8 and 9 must then be used. 53

6 JMSM, Vol., No. 2, Pages 49-6, Piezoelectric Actuation 4.. Principle of the actuation the expression for the force F x is: F x T h dy + F x (24) The piezoelectric actuation of a diaphragm is possible by depositing a piezoelectric layer on it and by applying a potential difference between the two piezoelectric faces (Fig. ). The thickness of the piezoelectric layer is given by h 2, Young s modulus E 2, Poisson s ratio v 2 and the elastic 2 flexural rigidity D 2 E 2 2( ν 2 ). The thickness, Young s modulus, Poisson s ratio and elastic flexural rigidity of the diaphragm are noted h, E, v and D. where F x is the internal force due to the deflection. Balancing the forces and moments on the sections of the piezoelectric and the diaphragm elements, we obtain the four following equations: F x + F 2x F y + F 2y (25) (26) h + h 2 F x M 2 x + M 2x h + h 2 F y M 2 y + M 2y (27) (28) Fig. : Piezoelectric actuation. where the expressions for the bending moments versus the curvature radii are [3]: 3 ν M x D h dy (29) r x r y The elongation of the piezoelectric layer causes the deflection of the diaphragm. In the following section we present the equation for the onset of the physical phenomena Equation setting Let us consider an element of the system contained between four sections: (x,y), (x,y+dy), (x+dx,y) and (x+dx,y+dy). The internal forces on the piezoelectric material section can be reduced to the forces F 2x and F 2y and to the bending moments M 2x and M 2y. Similarly, the internal forces on the diaphragm are reduced to the forces F x and F y and to the bending moments M x and M y (Fig. ). 3 ν 2 M 2x D 2 h dy r x r y 3 ν M y D h dx r y r x 3 ν 2 M 2y D 2 h dx r y r x Combining these equations, we obtain: F x F 2x (3) (3) (32) F2x M2x M2x F2x D h D 2 h 2 ν D h + ν 2 D 2 h dy h + h 2 r x r y F y F 2y (33) Fx Mx F2y Fy M2y My Mx Fig. : Forces and moments on an elementary volume. Fx The initial internal stress is included in the internal forces. For example, if the initial stress of the diaphragm is given by T, (34) The relative elongation of the piezoelectric material and diaphragm must be the same at the interface: and D h D 2 h 2 ν D h + ν 2 D 2 h dx h + h 2 r y r x d 3 U F 2x ν 2 F 2y h h 2 E 2 h 2 dy E 2 h 2 dx 2r x F x E h dy ν F y E h dx h r y (35) 54

7 Normalized Abacus for the Global Behavior of Diaphragms : Pneumatic, Electrostatic, Piezoelectric or Electromagnetic Actua- d 3 U F 2y ν 2 F 2x h h 2 E 2 h 2 dx E 2 h 2 dy 2r y F y E h dx ν F x E h dy h r y (36) Where U is the applied voltage and d 3 the piezoelectric constant. The first term of Eqs represents the elongation due to the piezoelectric properties of material. The second and third terms represent the elongation of the center of the piezoelectric section due to the internal forces F 2x and F 2y. The fourth term represents the variation of the elongation between the center and the lower edge of the piezoelectric section. The two first terms of the second part of Eqs represent the elongation of the center of the diaphragm section due to the internal forces F x and F y. The last term is due to the variation of the elongation between center and upper edge of the diaphragm section. Combining Eqs , we obtain: ---- r x ---- r y 6d 3 Uh ( h + h 2 )( + v )( + v 2 )D D D h ( + ν ) D 2h2 ( + ν2 ) D D 2 ( + ν )( + ν 2 )h h 2 ( 4h + 6h h 2 + 4h ) (37) These equations are quasi-similar to the equations presented in [], where the authors presented equations for an equivalent flexural rigidity and a bending moment due to the piezoelectric actuation. The difference between our result and the result given by [] comes from the coupling between the x and y axis that we have taken into account in Eqs and For small diaphragm deflections, the expression for the radius of curvature versus the deflection w can be approximated by and ---- r x d 2 w dx d2 w ( + ( d 2 w dx 2 )) dx 2 (38) d 2 w dy d2 w (39) r y ( + ( d 2 w dy 2 )) dy 2 According to Eqs , the differential equation governing the diaphragm deflection is: w 2d 3 Uh ( h + h 2 )( + v )( + v 2 )D D D h ( + ν ) D 2h2 ( + ν2 ) D D 2 ( + ν )( + ν 2 )h h 2 ( 4h + 6h h 2 + 4h ) (4) 4.3. Analysis and Solution Eq. (4) proves that the piezoelectric actuation of a diaphragm is equivalent to the pneumatic actuation of a pure membrane. The solution presented in the section 3.2 can be used to obtain the deflection of each diaphragm point and the abacus is used to obtain the center deflection and the stroke volume rapidly. In order to use the equivalence between the piezoelectric actuated diaphragm and the pneumatic actuated membrane, we need to consider the analytical equivalence: 3d 3 Uh ( h + h 2 )( + v )( + v 2 )D D 2 S C T D h ( + ν ) D 2h2 ( + ν2 ) D D 2 ( + ν )( + ν 2 )h h 2 ( 4h + 6h h 2 + 4h ) (4) where S is the surface of the diaphragm. 5. Electromagnetic Actuation NI B I I I I I I I I I I I Fig. 2: Magnetic actuation (bottom and cross-section views). B 5.. Principle of the actuation The electromagnetic actuation of a diaphragm is possible by placing the system in a area where there is a magnetic field and while causing a current to flow in N conductors deposited on the diaphragm (Fig. 2). The deflection can be controlled by varying both the current I and the magnetic field B Equation setting Let us consider a rectangular surface area length dy and width a/n. This diaphragm element is then subjected to a force df: df IdyB. (42) 55

8 JMSM, Vol., No. 2, Pages 49-6, 999. Each elementary element is then subjected to a constant pressure P: IdyB P dya N 5.3. Analysis and solution NIB a (43) Eq. (43) proves that the electromagnetic actuation of a diaphragm is equivalent to the pneumatic actuation of a diaphragm. The solution presented in the section 3 can be used to obtain the deflection of each point on the diaphragm and the abacuses can be used to obtain rapidly the center deflection and the stroke volume. In order to use the equivalence between magnetic actuated diaphragm and pneumatic actuated diaphragm, we have to consider the analytical equivalence: P NIB a (44) W ( X, ew n ( X, C D (47) For a given rectangularity (r being fixed), simulations several C D values must be run. C D lies between and the value corresponding to the plate center s pull-in. To obtain the variation of W n respect to C D, the energy minimization method has once again been used. Furthermore, we have developed an iterative calculation procedure to determine the coefficients a i. At each increment of the calculation, the new position is integrated into the pressure in order to converge towards the solution. Fig. 3 presents the standardization of the center deflection, W n (, ). Other authors [] have classically presented the limit prior to the pull-in effect as a displacement corresponding to one-third of the e value. In this instance, using the energy-based method, we have obtained a limit displacement that corresponds to a larger value (the displacement for which C D is equal to.8). 6. Electrostatic Actuation In this case, the pressure acting on the diaphragm is not constant, but rather depends on the deflection. An electrode is placed in front of the diaphragm at a distance e, and a voltage U is applied between the diaphragm and the electrode. An electrostatic pressure P e (X, is then created: P e ( X, P ( W( X, e) 2 (45) Electrostatic Pneumatic Po εu 2 P , 2e 2 and ε the permeability of the space between the electrode and the diaphragm. In this section, we will concentrate on a square diaphragm. This particular approach can also be applied to a rectangular diaphragm. 6.. Plate Behavior In this case, Eq. () is equivalent to: W n ( X, W( X, e P S 2 C D D h 3 e 56 W n ( X, C D ( W n ( X, ) 2 (46) For a fixed value of C DW therefore, only one solution for W n actually exists. Once n ( X, has been identified as a function of C D, deducing the value of W(X, for all plate characteristics is thus made possible: Fig. 3: Variation of W n (, ) versus C D for a square-shaped plate. A comparison more classical methods has been performed in Fig. 3: the method considers the pressure to be constant and equal to P. It appears that for very small deflections, the pressure can indeed be considered to be constant. However for larger deflections, the dependence of the pressure on the deflection changes the system s behavior, and, the pneumatic model is no longer sufficient to produce an accurate simulation. A normalization of the volume and of the capacitance is also possible and then the abacuses presented in Figs. 4-5 are sufficient for all cases. The real volume (Vol) and capacitance (C) are obtained via the abacus and the two following formulas: Vol es -----Vol 4 N D εs C -----C 4e N D (48) (49)

9 Normalized Abacus for the Global Behavior of Diaphragms : Pneumatic, Electrostatic, Piezoelectric or Electromagnetic Actua Electrostatic Pneumatic Po Electrostatic Pneumatic Fig. 4: Variation of Vol N D versus C D for a square-shaped plate Fig. 6: Variation of W n (, ) versus C T for a square-shaped membrane Electrostatic Pneumatic Po Fig. 5: Variation of versus C D for a square-shaped plate. C N D 6.2. Membrane behavior The same approach can be used as that employed in the plate case and we only present the abacuses obtained for the center deflection, volume and capacitance versus C T (Figs. 6-8) and the equations required to use these abacuses: P S C T T he W (, ) ew n (, ) (5) (5) Vol C es -----Vol 4 N T εs -----C 4e N T (52) (53) Fig. 7: Variation of Vol N T versus C T for a square-shaped membrane Fig. 8: Variation of C N T versus C T for a square-shaped membrane. 57

10 JMSM, Vol., No. 2, Pages 49-6, General case As for the two previous cases, in order to obtained normalized results (abacuses), the same solution must be found for the following equation:.8 : Measurements : M odeling and 2 W n ( X, C DT W n ( X, C D ( W n ( X, ) 2 T S C DT D h 2 P S 2 C D D h 3 e (54) C' pull-in It is then possible to obtain a data base C DT and C D as inputs and the deflection, volume or capacitance as outputs. 7. Verification of Certain Electrostatic Results In order to verify the modeling approach of the electrostatic actuation used in this paper, we produced, at the ESIEE laboratory, built-in plates of various surfaces on an S.O.I. (Silicon On Insulator) silicon plate which were then welded onto a plate of glass containing metallic electrodes. The geometry of the entire deflected Si plates has been measured by a technique of microscopic interferometry developed at the IEF laboratory. The digitized interferograms (Fig. 9) were obtained by using an optical microscope fitted a Michelson interferometric objective, along a sodium discharge illumination (λ 589 nm) and a CCD camera connected to a frame grabber ( ): model- Fig. 2: Standardization of the center deflection ing and measurements. Various measurements were carried out in order to verify the pull-in voltage value (Fig. 2). The characteristics of the systems used were: e 5 µm; h 2 µm; D 5.6 GPa and S is variable. 24 (V) 2 6 : Modeling W n, : Measurements (mm ) Fig. 2: Pull-in voltage versus plate surface area: modeling and measurements. In this section the abacuses of the plate behavior under electrostatic actuation have been subjected to comparative measurements, and the results have proved most satisfactory. Fig. 9: Sample interferograms square-shaped plate under electrostatic actuation. In this section, we will only present the case of the plate (out large initial stresses): simulations are compared to measurements conducted on the systems developed. Our attention is turned to a comparison of the center plate deflection. The measurements have been standardized in order to be placed in the plane ( C D, W n (, )) (Fig. 2). 8. Conclusion We have recalled herein the energy-based solution method and its implementation for obtaining the deflection at each point of a diaphragm. Both plate and membrane behavioral patterns can be easily derived and standardized. Therefore, in each of these two cases and a fixed rectangularity, only one solution is needed in order to determine the deflection for every geometrical (size, thickness) and physical (applied 58

11 Normalized Abacus for the Global Behavior of Diaphragms : Pneumatic, Electrostatic, Piezoelectric or Electromagnetic Actuapressure, flexural rigidity or initial internal stress) parameter. We have also presented the limit for these two cases and the solution technique for the general case using the same standardization principle. Then we have presented the differential equation for the deflection of diaphragms under piezoelectric or magnetic actuation. The analytical formulation of these two problems is similar to the case of pneumatic actuation. The analogy of the diaphragm piezoelectric actuation and the pneumatic membrane actuation is presented and can be solved the pneumatic solution. The case of magnetic actuation is comparable to the pneumatic case, and so the three cases (pure plate, pure membrane or general diaphragm) can be encountered and solved using the well known energy based solution. For the more complicated case of electrostatic actuation, the classical energy based method has been adapted. With this actuation, we developed a standardization of the deflection as well as an abacus that allows the deflection to be easily identify for both cases (plate or membrane). Some measurements made on plates under electrostatic actuation have been presented and are in good agreement the abacus. Therefore, in each of these three actuation principles it is possible to determine the deflection for all geometrical (size, thickness) and physical (flexural rigidity, initial internal stress, applied voltage, applied current and magnetic field) parameters. Acknowledgments The authors gratefully acknowledge Patrick Sangouard (ESIEE) for the production of the experimental silicon builtin plate and Alain Bosseboeuf (IEF) for the optical measurements. This work was partially presented at the International Conference on Modeling and Simulation of Microsystems, MSM 98 [2]. References [] L. Landau and E. Lifshitz, Theory of Elasticity, Mir, Moscow, 967. [2] R. Legtenberg, J. Gilbert, S.D. Senturia and M. Elwenspoek, Electrostatic curved electrode actuators, IEEE Journal of Microelectromechanical Systems, vol.6, n 3 (September 997) pp [3] S. Timoshenko and S. Woinowski-Krieger, Theory of plates and Shells, McGraw-Hill, New York, 2nd edn, 959. [4] R. Steinmann, H. Friemann, C. Prescher and R. Schellin, Mechanical behaviour of micromachined sensor membranes under uniform external pressure affected by in-plane stresses using a Ritz method and Hermite polynomials, Sensors and actuators A, 48 (995), pp [5] J. Courbon, Plaques minces élastiques, Editions Eyrolles. [6] G. Blasquez, Y. Naciri, P. Blondel, N. Ben Moussa and P. Pons, Static response of miniature capacitive pressure sensors square or rectangular silicon diaphragm, Revue Phys. Appl. Vol 22 (987), pp55-5 [7] J. Bay, O. Hansen and S. Bouwstra, Quasi-static analyses of diaphragms and capacitive microphones using finite element calculation, MME 97, Southampton, UK, September 997, pp [8] P. Boman and S. Bouwstra, Linear behaviour of differential pressure loaded square diaphragms elastic support, MME'97, Southampton, UK, September 997, PP [9] E. Bonnotte, Etude des proprietes mecaniques des films minces. Application au silicium monocristallin, PhD Thesis, Besançon, December 994. [] A. Klein and G. Gerlach, Modelling of piezoelectric bimorph structures using an analog hardware description language, Microsim 97, Lausanne, Suisse, September , pp [] J.I. Seeger, S.B. Crary, Analysis and simulation of MOS capacitor feedback for stabilizing electrostatically actuated mechanical devices, Microsim II, Computational mechanics publications, pp 99-29, 998. [2] O. Français, I. Dufour, Energy-Based Solution Method for the Global Behavior of Diaphragms under Pneumatic and Electrostatic Pressure, Technical Proceedings of the First International Conference on Modeling and Simulation of Microsystems, Semiconductors, Sensors and Actuators, MSM 98, Santa Clara, California, April 6-8, 998, pp Received in Cambridge, MA, USA, 22 nd October 999 Paper /

12 JMSM, Vol., No. 2, Pages 49-6,