Viscous Damping Effect on the CMUT Device in Air

Transcription

1 Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011, pp Viscous Damping Effect on the CMUT Device in Air Seung-Mok Lee Micromachined Sensing Laboratory, Ingen MSL Inc., Ayumino 2-7-1, Izumi, Osaka , Japan Bu-sang Cha and Masanori Okuyama Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka , Japan (Received 10 January 2011, in final form 9 March 2011) Acoustic damping of a capacitive micromachined ultrasonic transducer (CMUT) having a perforated circular membrane in air causes mechanical impedance to the plate vibration. We observed the effect of acoustic damping on the operation of the perforated membrane plate of a CMUT under various geometrical conditions. The damping force of the perforated plate was computed for each area ratio of the acoustic holes (AR) condition, and the results were introduced into calculations of the frequency response function (FRF) using the finite element method (FEM) to take the squeezefilm damping effect into account. The Q factor and the damping ratio of the CMUT under various AR and air-gap height (h) conditions of the device were studied; the acoustic damping effect is discussed in terms of design optimization. PACS numbers: n, Fg, Gm Keywords: Damping, Impedance, CMUT DOI: /jkps I. INTRODUCTION Recently, capacitive micromachined ultrasonic transducers (CMUTs) have attracted much attention for applications in small-size microphones and ultrasonic sensors. The main advantages of CMUTs compared to piezoelectric-type transducers include better acoustic matching to the propagation medium, higher efficiency, and broader bandwidth performance [1]. CMUTs are constructed with parallel plate capacitors similar in form to a condenser, with a vibrating plate and fixed substrate. The small space between these parallel plates is filled with air. This microstructure has an air gap of the order of micrometers; the air-damping effect principally occurs when air (fluid) is squeezed under or around a vibrating micromechanical structure in an extremely narrow gap and causes mechanical resistance to the vibration of the plate. For example, if the membrane plate vibrates due to an incoming acoustic wave, the membrane displacement is generally decreased by the resistance caused by the horizontal or the vertical motion of the squeezed air layer between the plates. On the other hand, proper air damping can extend the frequency bandwidth of a device [2], which is another merit of CMUTs relative to traditional piezoelectric transducers. The vibrating plate in this microstructure is usually smlee@ingen.co.jp; Fax: perforated to reduce the air-damping effect. These perforations (acoustic holes) also facilitate the etching process of the sacrificial layer in surface micromachining fabrication. However, membrane structures with a perforated plate have a reduced membrane area. The reduction of the membrane area implies a decrease in electrical energy due to a reduction of the basic capacitance of the device. The total impedance generated in a CMUT is due to acoustic and mechanical impedances, which generally occur because of the damping effect from the fluid (air) film and the mechanical stress of membrane, respectively. Impedance plays an important role in the Q factor of the CMUT device. These impedances vary widely depending on the geometric properties of the membrane structure, such as the perforation ratio of the vibrating plate, the height of the air-gap layer, and the thickness of the vibrating plate. Many studies have been conducted on the theoretical damping effect of microstructures [3] while the influence of the air-damping effect on a practical CMUT design has barely been reported. In this study, we considered the acoustic impedance that occurs in CMUTs, which is mainly influenced by the squeezed air-film damping of membrane structures with a narrow air gap and a perforated vibrating plate. The damping force and the damping ratio under each structural condition were calculated, and the Q factors of the perforated plate at a constant acoustic pressure were estimated using the finite element

2 -748- Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011 assumed to be operating in a vacuum condition, there is no loading on the membrane by any medium; the force acting on the membrane (F men ) consists of a mechanical spring force (F mech ) and electrostatic force (F elec ) given by [4] F men = F mech + F elec. (1) Fig. 1. (a) Cross-sectional schematic diagram of a CMUT device. It is usually made of a thin film membrane, upper and bottom electrodes and Si substrate. (b) Electrical circuit model used to detect the signal of a CMUT in the receive mode. method. The conditions for the air-gap height (h) and area-ratio of acoustic holes (AR) of the CMUT, which are the most influential factors for the device damping effect, were considered, and the influences of the acoustic impedance on the CMUT operations were discussed comparing it with other factors, such as the mechanical impedance and the fixed capacitance. II. EQUIVALENT CIRCUIT MODEL AND TRANSFORMATION EFFICIENCY A CMUT device is made of thin membranes that are essentially parallel plate capacitors with gaps between the plates. Figure 1(a) is a cross-sectional schematic diagram of a typical CMUT. The thin-film electrode on a silicon substrate wafer on which the membrane is fabricated makes up one of the plates of the capacitor. The other plate is the metal electrode on top of the membrane. The membrane is generally made of a dielectric material, most commonly low-stress silicon nitride (Si 3 N 4 ) or poly-si thin film. CMUTs can be used both as acoustic transmitters and receivers. In this study, we only considered the receiving mode as an ultrasonic sensor for the gas (air) regime. This sensor can be used in the frequency range from audible to ultrasound. In the receive mode, the harmonic vibrations of the membrane, which are caused by an incident acoustic wave, are detected by the capacitance variation; this produces a current flow in the external electric circuit, which is amplified for further processing. Figure 1(b) shows a schematic of the electrical circuit used to detect acoustic waves with CMUTs. When a DC bias is applied to the membrane, it is pulled down by the electrostatic force, which is balanced by the mechanical restoring force of the membrane. If the DC bias is greater than the collapse voltage, the electrostatic force overwhelms the mechanical restoring force, and the membrane falls onto the substrate. Therefore, the bias should be lower than the collapse voltage. The collapse voltage can be approximated by using a first-order model of the membrane behavior; this neglects all the electrical fringing fields and the membrane curvature. If the CMUT is The electrostatic force exerted by the capacitor can be obtained by differentiating the potential energy of the capacitor with respect to the position of the membrane: F elec = d dx ( 1 2 CV 2 ) = εsv 2 2(h 0 x) 2. (2) Here, V is the voltage across the capacitor, C is capacitance, ε is the electric permittivity, S is the area of the electrode, x is the displacement of the membrane, and h 0 is the gap between the capacitor plates. The mechanical spring force can be represented as F mech = kx, (3) where k is the spring constant of the membrane plate. If Eqs. (2) and (3) are introduced into Eq. (1), the total force relation acting on the membrane is given by m d2 x(t) εs[v (t)]2 dt 2 + kx(t) = 0, (4) 2[h 0 x(t)] 2 where x(t) is gap as a function of time due to membrane vibration. If only a DC bias is acting on the device, the time dependence of the displacement becomes zero, and the first term of Eq. (4) can be neglected. As the DC bias increases in this condition, at some point, the electrostatic force overwhelms the spring s restoring force, and the membrane collapses. This point can be found when there is a double real root at x > h 0. The collapse voltage is, consequently, given by V coll = 8kh ε 0 S. (5) In Eq. (5), V coll is a function of the distance between the electrodes, the area of the electrode, the dielectric constant, and the dimensions of the device. If V coll is increased, the operating DC bias can be raised, which improves the electrostatic force expressed by Eq. (2). The energy transformation or loss that occurs in CMUTs is usually analyzed using the equivalent circuit model introduced by Mason for an electrostatic device [1,4]. A CMUT equivalent circuit model is shown in Fig. 2. The capacitance C 0 in the electrical domain denotes the base capacitance of the CMUT, and a negative capacitance accounts for the spring softening effect due to the electromechanical interaction [4]. The transformation ratio from mechanical energy into electrical energy of the device is given by n = V DC ε 0 ε 2 S (ε 0 t m + εh(t)) 2, (6)

3 Viscous Damping Effect on the CMUT Device in Air Seung-Mok Lee et al Fig. 2. Electrical equivalent circuit model of a CMUT. where t m is the thickness of the membrane and h(t) is the gap height as a function of time. This equation notes that the transformation efficiency is a function of the DC bias, the electrode area, the permittivity of the dielectric material, and the membrane displacement. For design optimization of a CMUT, the transformation efficiency should be maximized and the mechanical and acoustic impedances occurring in the device should be minimized. On the mechanical side, the mechanical impedance Z m is defined as the ratio of the pressure applied to the membrane to the average velocity of the membrane in a vacuum condition. Thus, the mechanical impedance denotes the resistance on the membrane due to vibration. The mechanical impedance of the membrane is calculated by solving the fourth-order differential equation of motion on the membrane [1]: [ ] ak 1 k 2 (k 2 J 0 (k 1 a)j 1 (k 2 a) + k 1 J 1 (k 1 a)j 0 (k 2 a)) Z m = jωρt m 2(k1 2 + k2 2 )J, (7) 1(k 1 a)j 2 (k 2 a) ak 1 k 2 (k 2 J 0 (k 1 a)j 1 (k 2 a) + k 1 J 1 (k 1 a)j 0 (k 2 a)) where J 0, J 1, and J 2 are Bessel function, ω is the angular frequency, and ρ is the density of the membrane plate. k 1 and k 2 are given by the following equations: d2 d2 + 4cω k 1 = 2 d + 4cω, k 2 = j 2 + d, 2c 2c c = (Y 0 + T )l 2 t 12(1 σ 2 ), d = T ρ. (8) In these equations, Y 0 is Young s Modulus, T is the residual stress, and σ is Poisson s ratio. Ladabaum et al. [5] reported the optimization of the mechanical impedance for CMUTs. Z m is derived from an energy formulation, and the critical assumption is that the tension generated by a displacement is small compared to the total tension of the membrane. The dimensions of the membrane (thickness, radius), frequency, and tension are decisive factors for lowering the mechanical impedance. However, since these factors are usually determined by the resonant frequency of the device and the fabrication process, the scope for regulating them to improve the efficiency is limited. On the other hand, the acoustic impedance Z a denotes the resistance due to loading by the surrounding medium. This mostly originates from the thin air film of the air gap. CMUTs have a thin air film of about µm in thickness under the membrane; the film causes resistance (damping) to the membrane vibration. Since many holes are perforated in the membrane to lower the damping force, the hole ratio to the whole membrane area subsequently influences the fluid behavior and the damping effect of the device. The impedance elements are placed in series to maintain consistency with definitions, and the parasitic capacitance terms of the device are not considered in this model. The acoustic impedance is generally controlled by using the AR and h conditions, which are almost independent to the resonant frequency. III. SQUEEZE FILM DAMPING CMUT structures have a thin air layer between the vibrating plate and the fixed substrate. The thin air layer presents a counter-reactive force on the moving plate, and the fluid behavior in the microstructure has been explained by the squeeze film damping theory, which is based on the Navier-Stokes equation. 1. Fluid Film Behavior in a Microstructure The total forces acting on the control volume and the application of Newton s law of motion to the time rate of change of momentum can be computed to obtain an equation of motion governing the fluid behavior. There are three components of the total force: (1) the net pressure acting normal to the bounding surface, (2) the net shear force acting tangential to the surface, and (3) any force, such as gravity, acting throughout the volume. The isothermal motion of the fluid in the presence of external and internal viscous forces is described by the Navier-Stokes equation, in which all forces acting on the control volume are considered as an equation of continuity [6] ρ + (ρu) (9) and a momentum equation [ ] u ρ + (u )u = ρg p+µ 2 u+(µ+λ) ( ν). (10)

4 -750- Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011 Here, u, ρ, µ, λ, g, and p are the fluid velocity (u = iu + jv + kw), density, shear viscosity, and bulk viscosity, the acceleration of gravity, and the gas pressure, respectively. In typical CMUT device structures, air fills the cavity between the substrate and the movable membrane plate. If the membrane plate moves downward, the inside gas pressure increases due to the plate motion, which squeezes the gas out from the edges of the plate. If the membrane moves up, the pressure between the plates drops, and the gas is sucked back into the small cavity. The viscous drag of air during the membrane vibration creates a dissipative mechanical force on the plate that opposes the motion. This dissipative force is called squeeze-film damping. To analyze this damping behavior, several assumptions were considered: 1) the gap height h is much smaller than the lateral extent of the movable plate, 2) there is no pressure gradient in the normal direction, 3) the gas obeys the ideal gas law, and 4) the system is isothermal. Under these assumptions, the Navier-Stokes equation and the ideal gas law can be combined to yield the Reynolds equation: (P h) 12η = [(1 + 6K n )h 3 P P ]. (11) Here, the pressure P depends on x and y but not on z, h is the time- and position-dependent gap height, and K n is the Knudsen number, which is the ratio of the mean free path of the gas molecules to the gap. At ambient conditions, the mean free path of air is about µm. Therefore, if the air gap is in the range of several microns, then 0.01 < K n < 0.1, which corresponds to slip flow. Because this value of K n is relatively small, it can be treated as negligible in the Reynolds equation. In addition, the relative movement of the fluid film in the lateral direction is not considered. Thus, for the squeeze film damping problems encountered in microelectromechanical system (MEMS) devices, the Reynolds equation can be reduced to x ( P h 3 µ P x ) + ( ) P h 3 P y µ y = 12 (hp ), (12) where h is the thickness of the fluid film. This is the nonlinear Reynolds equation for isothermal squeeze-film damping of a compressible gas. For the normal motion of parallel plates, h and µ are not functions of position, and Eq. (11) can be simplified as 2 x 2 P y 2 P 2 = 24µ (hp ) h 3. (13) If the pressure is normalized by the ambient pressure P a, where P = P/P a, by using the normalized variables x = x/l, ỹ = y/l, h = h/h 0 c and τ = ωt, the nonlinear Reynolds equation for small amplitudes around its balanced position can be written in non-dimensional form as 2 x P ỹ P 2 2 = 2σ ( h P ). (14) τ Here, σ is the squeeze number: σ = 12µωl2 P a h 2, (15) 0 where, ω is the angular frequency, l is the typical length of the plate or radius of a circular plate and h 0 is gap height at steady state. l also corresponds to the radius of an equivalent circular pressure cell in the structure of circular plate. The squeeze number represents the compressibility of the fluid film. Equation (14) shows that the squeeze number is proportional to the square of the lateral dimension to air-gap ratio (l/h 0 ) and the oscillation frequency (ω) and that it is inversely proportional to the ambient pressure P a. If σ is much lower than unity (σ 1), the compressibility can be neglected, and the flow is treated as incompressible. For higher values of σ, the compressibility leads to a significant air-spring effect, which is undesirable because it can adversely affect the dynamic behavior of the device. In CMUT structures with perforated membranes, the squeeze number can be exploited by controlling the size and number of air holes. In MEMS structures with small amplitudes, the assumption of a rigid structure simplifies the model to a one-degree-of-freedom spring-mass-damper. For small amplitude displacements of the plate, the compressible Reynolds equation can be linearized using the perturbation parameters ˆp (in pressure) and ĥ (in gap) [7]. We can linearize the Reynolds equation for an assumed operating point and neglect the higher-order terms as 2 ˆp x ˆp y 2 = 12µ P a h 3 a [ ] ˆp h a + P ĥ a. (16) Here, p = P a + ˆp and h = h a + ĥ. If air is assumed to be incompressible (it i.e., dp/dt = 0), Eq. (15) can be reduced to the incompressible Reynolds equation 2 ˆp x ˆp y 2 = 12µ h 3 a h. (17) The squeeze number of the CMUT is so small that the fluid flow corresponds to the incompressible region. 2. Damping Force Acting on a Vibrating Plate In the CMUT structure, the damping force acting on the vibrating plate consists of two main components: the viscous flow of air when air is squeezed out of (or sucked into) the plate region and the force compressing the air film. We refer to the force component related to the viscous flow as the viscous damping force and to the air compression component as the elastic damping force. The viscous damping force dominates the total damping force in relatively low-frequency regions while

5 Viscous Damping Effect on the CMUT Device in Air Seung-Mok Lee et al Table 1. Geometry and mechanical properties of the membrane plate. Radius of membrane plate 225 µm Thickness 1.5 µm Young s modulus 166 GPa Poisson s ratio Density 5,410 kg/m 3 the elastic damping force dominates in relatively highfrequency regions. Griffin et al. [7] discussed the compressible squeeze-film damping problem for a strip plate. They determined firstly the film s response (step damping force: F (s)) to a step change in the film thickness ( z(s)) and then used the principle of superposition combined with a convolution integral formulation to determine the film s response to any displacement function. The step response function for small amplitude damping of an arbitrary source waveform can be obtained through the Laplace transform given by F (s) = 96µLW 3 π 4 h 3 0 n odd 1 n s ω c s z(s). (18) For small amplitude excitations, terms after the second term in Eq. (17) are negligible, so the infinite series solution was truncated only the first term of the sum: F (s) = 96µLW 3 π 4 h 3 0 s 1 + s/ω c z(s), (19) where the approximate cutoff frequency ω c squeeze-film air damping is given by of the ω c = π2 h 2 0P 0 12µW 2. (20) Here, W can be replaced by l, which is the radius of the circular plate. The cutoff frequency corresponds to a specific cutoff squeeze number, and the elastic force becomes equal to the viscous force at the cutoff frequency (or cutoff squeeze number). The squeeze number shown in Eq. (14) is equal to πω/ω c. Blech et al. [8] solved each damping force for a rectangular plate as an infinite series expansion. The solutions are given by f d (σ) = 64σ π 6 f e (σ) = 64σ π 8 m,n(odd) m,n(odd) m 2 + (n/η) 2 (nm) 2 {(m 2 + m/η 2 ) 2 + σ 2 /π 4 }, 1 (nm) 2 {(m 2 + m/η 2 ) 2 + σ 2 /π 4 }. (21) Here, f d, f e, σ, and η are the viscous damping force, the elastic damping force, the squeeze number, and aspect ratio of the rectangle, respectively. By equating f d (σ) Fig. 3. Schematic structure of a perforated membrane plate. Each cell has a hexagon shape with an acoustic hole, and the circular membrane plate is constructed with many unit cells. r 1 and r 2 are the radii of the acoustic hole and the unit cell, respectively. and f e (σ), the quantitative value of cutoff squeeze number can be calculated. We computed each damping force of a CMUT device by using Eq. (20), and the results are discussed later in section V. IV. MODELING AND SIMULATIONS 1. Modeling a Perforated Membrane Plate To simulate the damping effect using the finite element method, we first designed a perforated membrane plate. We considered the geometry of the acoustic holes and the mechanical properties of the plate. The mechanical properties and dimensions of the designed membrane plate are shown in Table 1. The membrane is a circular perforated plate with numerous small acoustic (air) holes, and the holes with radius r 1 are distributed in a hexagonal array, as shown in Fig. 3. The membrane plate is constructed with many unit cells of circular plates with acoustic holes, and the cells are arranged regularly all over the plate. r 2 means the radius of a unit cell in Fig. 3. The AR condition of the plates was established as 10 50%, and the design details of each element are shown in Table 2. The plate is consisted of silicon-nitride and platinum electrode films, and the radius and the thickness were 225 µm and 1.5 µm, respectively. The resonant frequency of the plate was designed to be about 70 KHz while it changed a little depending on the AR condition. On the other hand, the air gap height h was designed to be in the range of 1.5, 2.0, 2.5, and 3.0 µm. A perforated membrane plate model fabricated for the FEM analysis is shown in Fig. 4.

6 -752- Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011 Table 2. Dimension details of the acoustic holes in the perforated membrane plate. Area ratio of acoustic holes (AR) 10% 20% 30% 40% 50% Radius of holes r 1 (µm) Radius of equivalent cell r 2 (µm) Distance between the centers of two neighboring holes (µm) Fig. 4. Perforated membrane plate model for FEM simulations. Membrane plates having each AR condition were constructed with 30,000 50,000 meshes. 2. Damping Coefficient of a Perforated Plate To consider the damping effect on the displacement simulation, we computed the damping coefficient of the plate for each condition. The dynamic performances of the membrane structure can be approximated as an equivalent spring-mass-damper vibrating system. In mechanical vibrations, the differential equation for the system is given by mẍ + cẋ + kx = f, (22) where m is the mass, c is the damping coefficient, k is the spring constant, and f is the applied force. The damping coefficient is defined as the ratio of the damping force to the velocity of the plate. If m is the equivalent mass (for a single-degree-of-freedom model) of an oscillating structure and ω n is the fundamental angular frequency, the damping ratio ζ is then given by [8] ξ = C 2mω n = C actual C critical. (23) Under the given mass and frequency conditions, the damping ratio ξ of the vibrating system is defined as the ratio of the actual damping coefficient (C actual ) to the critical damping coefficient (C critical ) at which no oscillations take place. If the density of the holes in the perforated membrane is n, the area of the cell containing holes corresponds to A C = 1/n. The damping force of the whole plate is the summation of that for each cell. The base approximations in squeeze-film damping are that (1) the whole plate is much larger than a cell and all cells are identical, so the air flow between the cells is negligible (infinite plate approximation); and (2) the plate is thinner than the radius of hole, so the pressure difference causing the air flow through the hole is negligible (thin plate approximation). In a circular plate, the damping coefficient of a cell (C c ) can be written as C c = 3π 2h 3 µr4 1 = 3 2πh 3 µa2 C. (24) The holes of the plate were assumed to be distributed in a hexagonal array, as shown in Fig. 3. The total damping force (F T ) and coefficient of the perforated plate (C T ) are given by F T C T = A T F C = 3µA T ḣk(η), (25) A C 2πnh3 = 3µA2 T k(η), (26) 2πh3 where A T is the whole cell area of the membrane and ḣ means the time derivative of gap height due to membrane vibration. k(η) is given by k(η) = 4η 2 η 4 4 ln η 3. (27) Equation (29) shows that the damping coefficient is principally dominated by the area of the perforated plate and by the air-gap height. 3. FRF and Q Factor Simulations The FRF of each CMUT model was simulated using the FEM, and ANSYS, commercial software for FEM analysis, was used for these analyses. A modal analysis was used to interpret the resonant mode of the membrane plates; useful characteristics include a) showing the displacement maxima in a vibration and b) revealing the frequencies of natural vibrations. First, we computed the damping ratio and the damping coefficient of the perforated plate model; the FRF and Q factor of each AR and h condition under constant pressure were consequently simulated using the FEM. The damping coefficient results for each condition were used to simulate the membrane behavior. Based on the Q factor results, we approximated the influence of the acoustic impedance for each plate model. The simulation process is shown as a flow chart in Fig. 5.

7 Viscous Damping Effect on the CMUT Device in Air Seung-Mok Lee et al Fig. 5. Estimation process for the acoustic damping effect on CMUTs having perforated membranes. Fig. 7. Dependences of the viscous and the elastic damping forces of CMUTs on the squeeze number. The membrane plate is assumed to be a square. Fig. 6. Squeeze number variation of each membrane plate model with changing ratio of the unit cell radius to the airgap height (r 2/h). V. RESULTS AND DISCUSSION 1. Squeeze Number and Damping Force of CMUT To consider the damping mechanism of a CMUT device, we calculated the squeeze number of the CMUT structure. The dimensions of the designed perforated plate (CMUT) model are shown in Table 2; the air-gap height and the radius of a cell were established as and µm, respectively. The squeeze number of the designed CMUT model was computed first using Eq. (14), and the results are shown in Fig. 6. The squeeze numbers were relatively low ( 0.03), and the air-gap ratio (l/h 0 ) and the resonant frequency (ω) were the main variables influencing them. From these results, the vis- cous and the elastic damping forces of the CMUT were computed using Eq. (20); the membrane plate was assumed to be a square (η = 1). The results are shown in Fig. 7; the viscous damping force linearly increased with increasing squeeze number, and the lower the AR condition, the higher the viscous damping force. The elastic damping force was subsequently lower than the viscous force, which means that the viscous damping dominated the total damping force in the relatively low-squeezenumber region while the elastic damping force dominated in the high-squeeze-number region. The cutoff squeeze number (σ C ), the point of equilibrium of the elastic and the viscous damping force, was 21. Since the squeeze number for the CMUT structure was usually very small, the compressibility of the device was subsequently low. Thus, the viscous damping force dominated the squeeze film damping of the device, and the fluid film was incompressible. In this region, the damping force was thus influenced by the AR and the h conditions of the device structure. Reasonable design goals for the CMUT are maximizing the compliance ( C m A/tm) 3 for higher mechanical sensitivity and a high base capacitance (C b A/h) for better electrical efficiency. This clearly requires a sufficiently wide surface area (A) and a small membrane thickness (t m ) and air-gap height (h) for the device structure. Since incompressible viscous damping dominates the fluid film behavior, a wide area and a small air gap lead to large squeeze-film damping. In contrast, a small area and a large air gap that lower the damping force deteriorates the electrical efficiency of the device. To minimize the damping and to maximize the capacitance, a trade-off analysis should be carried out when designing the perforation geometry and air-gap conditions.

8 -754- Journal of the Korean Physical Society, Vol. 58, No. 4, April 2011 Fig. 8. Damping ratio (ζ) of CMUTs at each area ratio (AR) and air-gap height (h). 2. Damping Ratio The damping ratio is the ratio of the actual damping coefficient to the critical damping coefficient. We computed the damping ratios (ζ) for each device condition as a function of the air-gap height (h) and the area ratio of acoustic holes (AR); the results are shown in Fig. 8. Because the damping coefficient of the perforated membrane is generally proportional to k(η)/h 3, as shown in Eq. (29), it decreases with increasing AR. On the other hand, it increases with decreasing h. Because an increase in AR leads to a decrease in membrane area, the resistance of the air layer is generally decreased. The largest variation was shown at h = 1.5 µm, which implies that the ratio of the displacement to the gap height was largest at that condition. Since these results only considered a thin hole-plate structure, the resistance of the air flow through the holes was not considered. If the thickness of the plate is larger than the radius of a hole, the damping effect of the holes and the border effect should be considered. 3. Influence of Acoustic Impedance on the Q Factors For a membrane vibration at a certain acoustic pressure, the total impedance generated in the plate can be represented as the sum of the membrane impedance (Z m ) originating from the plate deformation and the acoustical impedance (Z a ) originating from the squeezefilm s air-damping; the relations among these mechanical impedances and energy transformation during CMUT operation are shown in the equivalent circuit model in Fig. 2. If there is no electrical bias in the system, the total mechanical impedance (Z T ) can be, consequently, Fig. 9. Dependence of the frequency response function (FRF) of perforated membrane plate on the AR conditions of 10 50% for (a) h (air-gap height) = 1.5 µm and (b) h = 2.0 µm. represented as Z T = Z m + Z a. (28) To consider the influences of acoustic impedance on the CMUT efficiency, we examined the FRF of the plate models. At first, the FRFs of the perforated plate model having h 1.5 µm and 2.0 µm were examined at a pressure of 1 Pa, and the results are shown in Figs. 9(a) and 9(b). The mechanical and the geometrical properties of membrane plate are shown in Tables 1 and 2. The amplitude of the plate is known to increase with increasing AR and h conditions. The resonant frequency (f r ) somewhat decreased with increasing AR due to the decreasing plate volume, as shown in Fig. 10(a); fr was khz. The Q factors for each AR and h conditions were also computed under an acoustic pressure of 1 Pa., and the results are shown in Fig. 10(b). The Q factor increased linearly with the AR, which is inversely proportional to the damping ratio at relatively low air gap heights (h = µm). However, there was a

9 Viscous Damping Effect on the CMUT Device in Air Seung-Mok Lee et al of the holes. The holes of the plate usually reduce the mechanical stress originating from the vibration. The Q factor is mainly influenced by the AR and h conditions at 1 Pa pressure condition, which originates from the damping effect of CMUT device structure. Thus, the damping force is mainly influenced by the AR and the h conditions, and those conditions dominate the Q factor variation of the plate. The membrane impedance also affects the Q factor variation at relatively high-h conditions, which is mainly due to a reduced damping effect at that condition. The hole geometries of the plate are thought to influence the mechanical stress of the plate and to consequently affects the membrane impedance in the vibration. VI. SUMMARY Fig. 10. (a) Dependence f r (resonant frequency) of the perforated membrane model on AR variation and (b) dependence of the Q factor on the air-gap height and the AR obtained from the frequency response of the perforated membrane. little irregularity at h = 3 µm. This result shows that because the damping effect on the membrane is not very large at high h, typical linearity does not exist, and the membrane impedance originating from the plate structure in vibration is more influential. In general, the Q factor is inversely proportional to the damping ratio and linearity depends on the AR variation. Since the thin air-gap condition ( 3 µm) leads to a relatively large airdamping effect, Q factors are not so high, in principle. We think that the acoustic damping force dominates the total impedance at h = 2.5 µm, which leads to a linearity in the Q factor variations. In contrast, at h = 3.0 µm, there are no dominant factors because the influence of the damping effect becomes small; the membrane impedance originating from the plate structure has a relatively large influence on the plate operation. We think that the membrane impedance originating from the plate vibration is subsequently influenced from the geometries The influence of the acoustic impedance on the Q factor of CMUT devices in air is examined. The total impedance of the vibration of a CMUT device can be represented as the sum of the mechanical and the acoustic impedances in air. We modeled perforated membrane plates having various AR (10% - 50%) conditions; we then analyzed the displacements and the Q factors of those plates by using the FEM. First, we computed the damping ratio for each AR and air-gap height (h); then, we introduced the computation results to the simulation of plate vibration. We observed that the damping ratio decreased with decreasing membrane area. At low values of h, the Q factor was considerably influenced by the acoustic impedance originating from the damping force; at high values of h, the membrane s impedance has a greater influence on the Q factor. REFERENCES [1] W. P. Mason, Electromechanical Transducers and Wave Filters (Van Nostrand, London, 1948). [2] R. Pratap, S. Mohite and A. K. Pandey, J. Indian Inst. Sci. 87, 75 (2007). [3] M. Bao and H. Yang, Sens. Actuators, A 136, 3 (2007). [4] A. Ergun, G. Yaralioglu and B. T. Khuri-Yakub, J. Aerosp. Eng. 16, 76 (2003). [5] I. Ladabaum, X. Jin, H. T. Soh and B. T. Khuri-Yakub, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45, 678 (1998). [6] R. Panton, Incompressible Flow, 2nd ed. (John Wiley & Sons, New York, 1995). [7] W. Griffin, H. Richardson and S. Yamanami, ASME J. Basic Eng. 88, 451 (1966). [8] J. Blech, ASME J. Lubric. Technol. 105, 615 (1983).