Smart Material Systems and MEMS - Vijay K. Varadan Part 4 pot

Electromechanical Systems: MEMS’ 2000, IEEE, Piscat- away, NJ, USA, pp. 142–147 (2000). 13. W. Riethmuller and W. Benecke, ‘Thermally excited silicon microactuators’, IEEE Transactions: Electron Devices, 35, 758–763 (1988). 14. Q.A. Huang and N.K.S. Lee, ‘Analysis and design of polysilicon thermal flexure actuator’, Journal of Microme- chanical and Microengineering, 9, 64–70, (1999). 15. G. Sun and C.T. Sun, ‘Bending of shape-memory alloy- reinforced composite beam’, Journal of Materials Science, 30, 5750–5754 (1995). 16. D. Wood, J.S. Burdess and A.J. Harris, ‘Actuators and their mechanisms in microengineering’, in Proceedings of the Colloquium on Actuator Technology: Current Practice and New Developments, No. 110, IEE, London, UK, pp. 7/1–7/3 (1996). 84 Smart Material Systems and MEMS 5 Design Examples for Sensors and Actuators 5.1 INTRODUCTION The principles of several sensors and actuators have been discussed in Chapters 3 and 4. Several of these devices are employed in numerous applications in civil, military, aerospace and biological areas, as will be demonstrated in Part 4 of this text. This chapter is intended to provide the basic understanding of the design of some of these sensors and actuators. Examples of sensors presented here include the piezoelectric and piezoresistive types. A chemical sensor based on the surface accoustic wave (SAW) principles is described. A fiber-optic gyroscope represents the optical segment of sensors in this chapter. In addition, the design of microvalves and pumps required in several biomedical applications is also included here. 5.2 PIEZOELECTRIC SENSORS Lead zirconate titanate (commonly known by the acro- nym PZT) is arguably the most widely used component in smart systems. The importance of this material comes from the fact that it exhibits significant piezoelectric properties. Piezoelectricity refers to the phenomenon in which forces applied to a slab of a material result in the generation of electrical charges on the surfaces of the slab. This is due to the distribution of electric charges in the unit cell of a crystal when force is applied. In these crystals, the force applied along one axis of the crystal leads to the appearance of positive and negative charges on opposite sides of the crystal along another axis. The strain induced by the force leads to a physical dis- placement of the charge within the unit cell. This polariza- tion of the crystal leads to an accumulation of charge: Q ¼ dF ð5:1Þ In the above equation, the piezoelectric coefficient d is a 3 Â3 matrix. In general, forces in the x,y,z directions contribute to charges produced in any of the x,y,z direc- tions. Values of the piezoelectric coefficients of these materials are usually made available by the manufacturer. Typical values of the piezoelectric charge coefficients are 1–100 pC/N. Some of the other properties of PZT are listed in Table 5.1. Once the charge is known, the voltage across the plate of the piezoelectric material can be determined by: V ¼ Q=C ð5:2Þ where the parallel plate capacitance of this configuration is: C ¼ e 0 e r A d ð5:3Þ Thus, in order to produce a larger voltage one can resort to reducing the area of the sensor. However, it must be cautioned that piezoelectrics are not generally good dielectrics. These materials have substantial leakage losses. In other words, the charge across a pair of electrodes may vanish over time. Therefore, there is a time constant for retention of voltage on the piezoelectric after the application of a force. This time constant depends on the capacitance of the element, and the leakage resistance. Typical time constants are of order of 1 s. Because of this effect, piezoelectrics are not used for static measurements such as weight. The reversible effect is used in piezoelectric actua- tors. Application of a voltage across such a material results in dimensional changes in the crystal. The coefficients involved are exactly the same as in Smart Material Systems and MEMS: Design and Development Methodologies V. K. Varadan, K. J. Vinoy and S. Gopalakrishnan # 2006 John Wiley & Sons, Ltd. ISBN: 0-470-09361-7 Equation (5.1). The change in length per unit applied voltage is given by: dL V ¼ FL EA  d 11 FL e 0 e r A  ¼ e 0 e r Ed 11 ð5:4Þ The strain in the above expression depends only on the piezoelectric coefficient, the dielectric constant and Young’s modulus. Therefore, it may be inferred that objects of a given piezoelectric material, irrespective of their shape, would undergo the same fractional change in length upon the application of a given voltage. Most sensors using the piezoelectric effect require a charge-amplifying preamplifier. A simple circuit for this purpose is shown in Figure 5.1. Another recently devel- oped material with sizeable piezoelectric properties is poly(vinylidene fluoride) (PVDF). This can usually be treated during fabrication to have a good piezoelectric coefficient in the direction of interest. Being polymeric, films of this material can be made at low cost. A related copolymer P(VDF–TrFE) also shows significant piezo- electric properties. The properties of PVDF and P(VDF– TrFE) are given in Table 5.2. Both of these materials are used in acoustic sensors because of their strong piezo- electricity, low acoustic impedance (useful in underwater applications, since there are only small mismatches with those of water) and flexibility (which permits applica- tions on curved surfaces). Therefore, transducers with wide operating bandwidths can be easily designed using PVDF. This also results in improvements in the overall performance of sensors such as hydrophones used for sensing acoustic fields. As the sensor size decreases, it becomes necessary to provide an amplifier or buffer in close proximity to overcome the sensitivity loss due to interconnected capacitances. This calls for the concept of sensors integrated with electronics. The discussion below shows the integration of a sensor where an on-chip MOSFET is implemented in which the sensor is placed over the extended gate metal electrode of the MOSFET. The MOSFET amplifier takes care of the loss of the sensor signal due to the finite capacitances of the cables that are used to drive the signal to the signal processing unit. A schematic of the device structure is shown in Figure 5.2 [3]. This is fabricated using six levels of photo masks. The device consists of a sensing part and an amplifying part. A PVDF film is used as the sensing material and an n-channel MOSFET with an extended aluminum gate is used as the electronic interface to the PVDF sensor. The basic structure is fabricated using a standard NMOS process. Transistors with large W/L Table 5.1 Electromechanical properties of PZT. Property Value Density (g/cm 3 ) 7.7–8.1 Maximum energy density (J/m 3 ) 102 Young’s modulus (GPa) 60–120 Tensile strength (MPa) 25 (dynamic); 75 (static) Compressive strength (MPa) 520 Curie temperature (  C) 160–350 Operational temperature range (  C) À273 to 80 Inducible strain (1–2 kV/m) at (mm/m) 1–2 Response time Very fast (typically kHz, up to GHz) Charge generating sensor Q C V out r – + Figure 5.1 Schematic of a piezoelectric sensor which uses a charge preamplifier. Table 5.2 Properties of PZT, PVDF and P(VDF–TrFE) [1,2]. F. S. Foster, K. A. Harasiewicz and M. D. Sherar, ‘‘A History of Medical and Biological Imaging with Polyvinylidene Fluoride (PVDF) Transducers,’’ IEEE Trans. Ultrasonics Ferroelectrics Freq. Control, UFFC-47, # 2000 IEEE Property PZT-5A PVDF P(VDF –TrFE) Thickness mode 0.49 0.14 0.25–0.29 coupling coefficient Relative permittivity, e r 1200 12–13 7–8 Density (g/cm 3 ) 7.75 1.78 1.88 Acoustic impedance, 33.7 3.92 4.37 Z (MRa) Maximum 365 80 115–145 temperature (  C) 86 Smart Material Systems and MEMS ratios are preferred in order to obtain a large transcon- ductance, g m , and low noise. Hence, MOSFETs with different W/L ratios are preferred for this application. The operating principle of this device can be explained as follows. The incident acoustic signal initiates the charge redistribution on the surfaces of the PVDF film that, in turn, changes the charge on the gate of an n-type MOSFET. The shift in gate voltage is used to modulate the drain current in a common source configuration. Since the FET is an important component in these sensors, its electrical characteristics are important in determining the behavior of these sensors. They also help to determine the operating point for the integrated sensors. In a MOSFET, the drain current I D is produced when electrons flow from source to drain. So, the existence of the channel is the cause of current flow. If V GS is the gate- source voltage of the MOSFET and V T the threshold voltage, then the condition for a channel to exist is that V GS > V T . With source and substrate terminals at ground potential, the threshold voltage V T is given by [4]: V T ¼ V T mos þ V FB ð5:5Þ where V T mos is the threshold voltage of the MOS capacitor and V FB is the flat band voltage: The threshold voltage of the MOS capacitor V T mos is given by: V T mos ¼ 2fðbÞþ Q b C ox ð5:6Þ where F(b) is the bulk potential, Q b is the maximum space charge density per unit area of the depletion region and C ox is the gate oxide capacitance. From these relations, it is evident that V T is a function of the material properties of the gate conductor and insulation, the thickness of the gate insulator, the channel doping and the impurities at the silicon–insulator interface. If N a is the acceptor atom concentration and N i is the intrinsic concentration, then the bulk potential F(b) is given by: fðbÞ¼ KT q ln N a N i  ð5:7Þ If e si is the permittivity of the silicon substrate, then the maximum space charge density Q b is given by: Q b ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4e si qN a fðbÞ p ð5:8Þ The flat band voltage V FB is given by: V FB ¼ f ms À Q fc C ox ð5:9Þ where F ms is the work function at the metal–semicon- ductor interface, Q fc is the surface charge state and C ox is the gate oxide capacitance. If E g is the band gap energy and F(b) the bulk potential, then the work function F ms is given by: f ms ¼ ÀE g 2q þ fðbÞð5:10Þ To operate the MOSFET as an amplifier, it must be biased at a point in the saturation region where the transconductance is proportional to the applied gate voltage but is independent of the drain voltage. For the MOSFET to operate in the linear region, the drain source voltage V DS < ðV GS À V T Þ. Then the drain source current I ds is given by: I ds ¼ Wm n C ox 2L V GS À V T ðÞ 2 ð5:11Þ where L is the length, W the width and m n the surface mobility of the carriers in the channel of the MOSFET. Hence the length L, the width W and the gate insulator thickness of the MOSFET are decided based on the above equations. The channel length of these devices is + + PVDF SiO 2 p-Si Source Gate Drain Figure 5.2 Schematic of the cross-section of PVDF–MOSFET hydrophone device [3]. Design Examples for Sensors and Actuators 87 designed as 10 mm. From the I–V characteristic curves, the carrier mobility m n was obtained as about 600– 700 cm 2 /(Vs). The threshold voltages for both the devices are within À0.5 to 0.3 V, which means the devices are ‘depletion-mode’ n-channel MOSFETs. Since the resistance of the MOSFET is quite small and the equivalent output impedance of the PVDF transducer is simply a capacitor (C 0 ), the ideal PVDF–MOSFET structure may be modeled as shown in Figure 5.3. In the equivalent circuit, V g is the signal voltage reaching the gate of the MOSFET (induced gate voltage), C 0 is the ‘clamped’ capacitance of the PVDF film, C sub is the extended gate electrode-to-substrate capacitance, C gs and C gd are the gate-to-source and gate-to-drain capacitances, respectively, g m is the transconductance and R D is the resistance of the resistor connected to the drain of the MOSFET. From the equivalent circuit, it is easy to get: V g V PVDF ¼ C 0 C 0 þ C sub þ C gs þ C gd 1 þ g m R D ðÞ ð5:12Þ where C 0 is related to the thickness of the PVDF film. In the design shown here, the PVDF film has a thickness of 110 mm. The values of C sub , C gs and C gd can be calcu- lated from the structural and geometrical parameters of the MOSFET (Table 5.3). Equation (5.12) shows that the induced gate voltage can be improved by minimizing C sub , C gs and C gd . When an acoustic signal reaches the PVDF transducer, a small voltage is generated and partially transmitted to the gate of the MOSFET. The small variation of the gate voltage in turn induces the voltage variation across R D . The voltage gain is: V 0 V g ¼Àg m R D ¼À Wm n C g L V GS À V T ðÞR D ð5:13Þ The sensitivity of the sensor with the electronics built in can be obtained as: V 0 P 1 ¼ V 0 V g  V g V PVDF  V PVDF P 1  ð5:14Þ 5.3 MEMS IDT-BASED ACCELEROMETERS The concept and design principles underlying an MEMS–IDT (inter-digitated transducer)-based acceler- ometer are based on the use of surface acoustic waves (SAWs). This unique concept is a departure from the conventional comb-driven MEMS accelerometer design. By designing the seismic mass of the accelerometer to float just above a high-frequency Rayleigh surface acous- tic wave sensor, it is possible to realize the accuracy and versatility required for the measurement of a wide range of accelerations. Another unique feature of this device is that because the SAW device operates at radiofrequen- cies (RFs), it is easier to be able to connect the 1DT device to a planar antenna and read the acceleration remotely by wireless transmission and reception. This unique combination of technologies results in a novel accelerometer that can be remotely sensed by an RF communication system, with the advantage of no power requirements at the sensor site. In the device described here, a conductive seismic mass is placed close to the substrate (at a distance of less than one acoustic wavelength). This serves to alter the electrical boundary condition as discussed above. Pro- gramable tapped delay lines have used the principle of air gap coupling between the SAW substrate and a silicon ‘superstrate’ to form individual MOS capacitors. These capacitors are then used to control the amount of RF coupling from the input IDT on the SAW substrate to the output terminal on the silicon chip [5]. This principle has also been successfully implemented in the realization of SAW ‘convolvers’ [6]. The seismic mass consists of a micromachined silicon structure which incorporates reflectors and flexible beams. The working of the device is as follows. The V PVDF C sub C gs R D C gd g m V g C 0 V g GD S Figure 5.3 Equivalent small-signal model of a PVDF–MOS- FET device. Table 5.3 Structural and geometrical parameters of a fabricated MOSFET (all values in mm). Parameter Value Field oxide thickness 1 Gate oxide thickness 30.0 Metal thickness 0.25 n þ junction depth 0.5 p-Type substrate thickness 250 SU-8 thickness 11 88 Smart Material Systems and MEMS IDT generates a Rayleigh wave, and the array of reflec- tors reflect this wave back to the IDT. The phase of the reflected wave is dependent on the position of the reflectors. If the positions of the reflectors are altered, then the phase of the reflected wave is also changed. The reflectors are part of the seismic mass. In response to acceleration, the beam flexes, so causing the reflectors to move. This can be measured as a phase shift of the reflected wave. By calibrating the phase shift measured with respect to the acceleration, the device can be used as an acceleration sensor. Alternatively, the measurement can be done in the time-domain, in which case the delay time of the reflection from the reflectors is used to sense the acceleration. A schematic of an MEMS–IDT-based accelerometer is shown in Figure 5.4. For waves propa- gating in the piezoelectric medium, there are two sets of equations, namely the mechanical equation of motion and Maxwell’s equation for the electrical behavior. The equation of motion is as follows: r @ 2 u i @t 2 ¼ X 3 j ¼1 @T ij @x i ð5:15Þ where r is the density of the material, u i is the wave displacement in the ith direction and T ij is the stress. This equation is intercoupled by the constitutive relation: T ij ¼ X k X l c E ijkl S kl À X k e kij E k ð5:16Þ where c E ijkl is the stiffness tensor for a constant electric field, i.e. if the electric field (E) is held constant, this tensor relates changes in T ij to changes in S kl . The electric displacement (D) is determined by the field E and the permittivity tensor e ij . In a piezoelectric material, the electric displacement is also related to the strain: D i ¼ X j e S ij E j þ X j X k e kij S jk ð5:17Þ where e S ij is the permittivity tensor for constant strain and e kij is the coupling constant between the elastic and electric fields. The constitutive equations for piezoelectric materials relating the stress T, strain S, electric field E and electric displacement D are given by Equations (5.15) and (5.16). It can be seen that the electric field and the electric field displacement are coupled in this set of equations. For a non-piezoelectric material, e kij ¼ 0 and there is no cou- pling between the elastic and electric fields. The sym- metric strain tensor is given by: S ij ¼ 1 2 @u i @x j þ @u j @x i  ð5:18Þ where u is the wave displacement. The electromagnetic quasi-static approximation: E i ¼À @f @x i ð5:19Þ rD ¼ 0 ð5:20Þ for an electric potential f can be used here to make further simplifications. The rotational part of the electric field due to the existence of a moving magnetic field is neglected. This approximation (Equation (5.19)) is valid as the acoustic velocity is small when compared to that of the electromagnetic wave. Contacts Absorber Beam Polysilicon seismic mass IDT LiNbO 3 crystal Figure 5.4 Schematic of an MEMS–IDT-based accelerometer. Design Examples for Sensors and Actuators 89 By incorporating Equations (5.17)–(5.19) in Equation (5.20) results in a system of four coupled equations relating the electric potential with three components of displacement in a piezoelectric crystal: r @ 2 u i @t 2 ¼ X j X k e kij @ 2 f @x j @x k þ X l c E ijkl @ 2 u k @x j @x l ! ð5:21Þ X i X j e S ij @ 2 f @x i @x j À X l e ijk @ 2 u j @x i @x k ! ¼ 0: ð5:22Þ where i, j and k vary from 1 to 3. The problem of wave propagation on anisotropic substrates can be solved by the method of partial waves. Plane wave solutions of the form: u m j ¼ a m j e ikb m x 3 e ikðx 1 ÀvtÞ ð5:23Þ f m ¼ a m 4 e ikb m x 3 e ikðx 1 ÀvtÞ ð5:24Þ are considered where j ¼ 1–3andm ¼1–4. The coordi- nate system is aligned to the substrate such that the propagation is along x 1 and the surface normal is in the x 3 direction. Therefore, the surface wave decays along the x 3 direction. In Equations (5.23) and (5.24), k is the wave number, b is the decay factor and v is the phase velocity. The partial wave solutions are substituted into Equations (5.21) and (5.22). The weighing coefficients of these plane waves are chosen to satisfy the mechanical and electrical boundary conditions at the surface of the crystal. In equations of motion, the material parameters are expressed in terms of axes which are selected for con- venient boundary conditions and excitation requirements. The tabulated values of these material parameters are expressed according to the crystalline axes. It is neces- sary to transform the material parameters to match the coordinate system of the problem. In certain cases, this is a mere interchange of the coordinate axes (as in YZ lithium niobate). For more complex situations (128  YZ lithium niobate) the parameters are transformed using an appropriate transformation matrix. The elements of this matrix are the direction cosines between the crystalline axis and the ‘problem’ axis. A YZ lithium niobate crystal is usually the material of choice in the design of devices of this type as it has the highest electromechanical coupling efficiency. The basic principle of the device depends on the strength of the piezoelectric coupling. ‘YZ lithium niobate’ indicates that the x 3 axis is parallel to the crystal axis Y, and x 1 is parallel to the crystal axis Z. The orientation of x 3 is called the ‘cut’ of the crystal. For YZ lithium niobate, the crystal is Y-cut and Z-propagating. Since the material tensors, permittivity and piezoelectric tensors are speci- fied with reference to the crystal axes, they need to be transformed into a frame defined by x 1 , x 2 , x 3 . The rotated material parameters are: C ¼ c 33 c 13 c 13 000 c 13 c 11 c 12 0 c 14 0 c 13 c 12 c 11 0 Àc 14 0 00 0c 66 0 c 14 0 c 14 Àc 14 0 c 44 0 00 0c 14 0 c 44 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ð5:25Þ e ¼ e 33 e 31 e 31 000 000Àe 22 0 e 15 0 Àe 22 e 22 0 e 15 0 2 4 3 5 ð5:26Þ E ¼ E 33 00 0 E 11 0 00E 11 2 6 6 4 3 7 7 5 ð5:27Þ The partial wave solutions (Equations (5.23) and (5.24)) are substituted into Equations (5.21) and (5.22), after the material parameters have been rotated to match the coordinate system defined for the problem, to get: m 11 À rV 2 m 12 m 13 m 14 m 12 m 22 À rV 2 m 23 m 24 m 13 m 23 m 33 À rV 2 m 34 m 14 m 24 m 34 m 44 À rV 2 2 6 6 4 3 7 7 5  a 1 a 2 a 3 a 4 2 6 6 4 3 7 7 5 ¼ 0 ð5:28Þ where: m 11 ¼ c 55 b 2 þ 2c 15 b þ c 11 m 12 ¼ c 45 b 2 þ c 14 þ c 56 ðÞb þ c 16 m 13 ¼ c 35 b 2 þ c 13 þ c 55 ðÞb þ c 15 m 14 ¼ e 35 b 2 þ e 15 þ e 31 ðÞb þ e 11 m 22 ¼ c 44 b 2 þ 2c 46 b þ c 66 m 23 ¼ c 34 b 2 þ c 36 þ c 45 ðÞb þ c 56 m 24 ¼ e 34 b 2 þ e 14 þ e 36 ðÞb þ e 16 m 33 ¼ c 33 b 2 þ 2c 35 b þ c 55 m 34 ¼ e 33 b 2 þ e 13 þ e 35 ðÞb þ e 15 m 44 ¼ÀE 33 b 2 þ 2E 13 b þ E 11  : 90 Smart Material Systems and MEMS For YZ lithium niobate, m 12 , m 23 and m 24 ¼ 0. For non- trivial solutions, the determinant of the coefficients of a must vanish. For a given value of the phase velocity, setting this determinant equal to zero results in an eighth- order equation in the decay constant (b). These roots of b are purely real or conjugate pairs. Only values with negative imaginary parts are admissible as these roots lead to waves that decay with depth (i.e. surface waves). There exist four such roots of b. For each of the four roots, the corresponding eigenvalues and the eigenvectors are determined. A linear combination of the partial waves is then formed: U j ¼ X m C m a m j e ikb m x 3 ! e ikðx 1 ÀvtÞ ð5:29Þ f ¼ X m C m a m 4 e ikb m x 3 ! e ikðx 1 ÀvtÞ ð5:30Þ These are then substituted into the boundary conditions in the mechanical and electrical domains of the problem. The mechanical boundary condition states that the sur- face of the crystal is ‘mechanically free’. There is no component of force in the x 3 direction on the surface (x 3 ¼ 0). This further implies that T 31 , T 32 and T 33 ¼ 0. In the electrical domain, since a conductive plate is placed at a height h above the substrate, the potential goes to zero at x 3 ¼ h. The potential above the surface satisfies Laplace’s equation. The potential and the elec- tric displacement (D) in the direction normal to the substrate are continuous at x 3 ¼ 0. This boundary con- dition is represented by the following equations. The potential in the air gap is given by: f 1 ðx 3 Þ¼ Be kx 3 þ Ce Àkx 3  e ikðx 1 ÀvtÞ ð5:31Þ The potential at x 3 ¼ h is zero. Therefore: f 1 ðhÞ¼ Be kh þ Ce Àkh  e ikðx 1 ÀvtÞ C ¼ÀBe 2kh ð5:32Þ The potential given by Equation (5.27) is equal to the potential given by the plane wave solution at the surface (x 3 ¼ 0). Equating these results in an expression for the unknown constant B: f 1 ð0Þ¼B 1 Àe 2kh  e ikðx 1 ÀvtÞ ¼ fð0Þð5:33Þ fð0Þ¼ X m c m a m 4 ! e ikðx 1 ÀvtÞ ð5:34Þ B ¼ P m c m a m 4 1 À e 2kh ð5:35Þ The electric displacement in the air gap is given by: D 3 ¼Àe 0 @f 1 ðx 3 Þ @x 3 ð5:36Þ The electric displacement on the surface of the crystal is given by: D 3 ðx 3 ¼ 0Þ¼Àke 0 P m c m a m 4 1 À e 2kh 1 þ e 2kh  e ikðx 1 ÀvtÞ ð5:37Þ This equation is obtained from the potential equation in Equation (5.25). The expression for the electric field displacement from the plane wave solution of potential is similarly obtained and is given by: D 3 ðx 3 ¼ 0Þ¼ X j e E 3j E j þ X j X k e 3jk S jk ð5:38Þ From the above equations, the relevant electrical bound- ary conditions can be obtained. The choice of the suspension for the seismic mass determines the linearity of motion and the sensitivity to residual strain. A single support is the simplest and lowest spring constant design, but allows substantial offline motion and rotation in the suspended mass. Two parallel supports remove the rotation component of the motion, but still introduce offline motion due to curvature of the beams (arclength is preserved, while vertical distance is not). A two-sided support removes that problem but greatly increases the sensitivity to residual stress. In addition to launching surface waves, the IDT can also generate bulk acoustic waves. These waves can propagate in any direction within the body of the sub- strate material. In this design, the principal effect of the generation of bulk waves is reduction of the power available for the generation of surface waves. The fol- lowing strategies may be useful to minimize the genera- tion of bulk waves: (a) The bottom surface of the piezoelectric substrate is roughened and coated with a soft conductor like silver epoxy. (b) Use substrate geometries that are not rectangular in shape. (c) Choice of the right number of IDT fingers. The input power is converted into bulk wave energy and sur- face wave energy as P ¼ P S þ P B , where P S repre- sents the power in the excited SAW wave and P B the component that is radiated as bulk waves. The ratio Design Examples for Sensors and Actuators 91 of P S to P B decreases drastically as the number of finger pairs in the exciting IDT is reduced. For YZ lithium niobate, the amount of input power converted into transverse bulk waves increases almost expo- nentially as the number of IDT fingers is reduced below five. 5.4 FIBER-OPTIC GYROSCOPES Fiber-optic gyroscopes are miniature solid-state optical devices for the precise measurements of mechanical rotation in inertial space. Conventionally used mechan- ical gyroscopes involve a spinning mass and ‘gimbaled’ mountings. Optical gyroscopes are free of such moving parts and may be used for a wide range of applications, for example, navigation, exploration and in the manu- facturing and defence industries. The basic theory of rotation sensing by optical means is known as the Sagnac effect, since this possibility was first demonstrated by G. Sagnac in 1913. The type of interferometer used to measure rotation is known as the Sagnac interferometer (Figure 5.5). Two identical light beams traveling in opposite directions around a closed path experience a phase difference when the loop is rotated about its axis, and this phase difference is proportional to the rotation rate [7]. Consider the inter- ferometer shown in Figure 5.5. Here, a light beam is split by using a beam splitter and the two beams (B 1 and B 2 ) are made to travel in a circular path. When the inter- ferometer is at rest in an inertial frame of reference, the pathlength of the counter-propagating waves are equal since light travels at the same speed in both directions around the loop. The time taken by the beam B 1 to complete the circular path is: t 1 ¼ 2pr c ð5:39Þ where r is the radius of the circular path. Similarly, the time taken by the beam B 2 is also of the same value. Therefore: t 1 ¼ t 2 ¼ t ð5:40Þ If the interferometer is rotating at a speed of m/s in the clockwise direction and the observer is motionless in the original inertial frame, the time taken by B 1 to complete the circular path is less than B 2 . In this case, the time taken by B 1 to complete the circular path is given by: t 1 ¼ 2pr c þ tr c ð5:41Þ Similarly, the time taken by B 2 to complete the circular path is: t 2 ¼ 2pr c À tr c ð5:42Þ Therefore, the difference between the propagation times of the two waves is: Át ¼ t 1 À t 2 ¼ 2tr c ¼ 4pr 2  c 2 ð5:43Þ Obviously, B 2 will reach its destination before B 1 . For a continuous wave of frequency o, this corresponds to a phase shift: Áf ¼ oÁt ¼ 4pr 2  c 2 o ¼ 4o c 2 A ð5:44Þ where A is the area of the circular path. It may also be noted that this result would remain unchanged even when the interferometer is filled with a medium of refractive index n because of the Fresnel–Fiezeau drag effect due to the movement of the medium compensating for the increased optical pathlengths. The advantage of using an optical-fiber coil to form the interferometer is that the Sagnac phase difference increases with the number of turns or length of the fiber. In this special case, Equation (5.44) can be rewrit- ten as: Áf ¼ 2p LD lc A ð5:45Þ where L is the length of the fiber and D is the diameter of the coil. Fiber-optic gyroscopes are broadly classified into two- types. The first type is an open-loop fiber-optic gyroscope r B 1 B 2 B Figure 5.5 Schematic of a Sagnac interferometer. 92 Smart Material Systems and MEMS with a dynamic range of the order of 1000 to 5000, with a scale-factor accuracy (inclusive of non-linearity and hysterisis effects) of about 0.5 %, and sensitivities that vary from less than 0.01 degrees/h to 100 degrees/h and higher. These fiber-optic gyroscopes are generally used for low-cost applications where dynamic range and linearity are not crucial. The second type is the closed- loop fiber-optic gyroscope that may have a dynamic range of 10 6 and a scale-factor linearity of 10 ppm or better. These types of fiber-optic gyroscopes are primar- ily targeted at medium- to high-accuracy navigation applications that have high turning rates and require high linearity and large dynamic ranges. Figure 5.6 illustrates the open-loop configuration. This consists of a fiber coil, two directional couplers, a polarizer, an optical source and a detector. A piezo- electric (PZT) device wound with a small length at one end of the fiber coil applies a non-reciprocal phase modulation. Light from the laser traverses the first directional coupler, polarizer and then the second direc- tional coupler where it is split into two signals of equal intensity that travel around the coil in opposite direc- tions. The light recombines at the coupler, returning through the polarizer, and half of the light is directed by the first coupler into a photo detector. This configura- tion permits measuring the difference in phase between the two signals to one part in 10 16 . This is possible due to the principle of reciprocity. Light passing from the laser through the polarizer is restricted to a single state of polarization, and the directional couplers and coil are made of special polarization-maintaining fibers to ensure a single-mode path. Beams of light in both directions travel through the same pathlength. Almost all environ- mental conditions (except rotation) have the same effects on both beams and are canceled out. Hence, this gyroscope is sensitive only to rotation about the axis perpendicular to the plane of the coil. The light intensity returning from the coil to the polarizer is a raised cosine function, having a maximum value when there is no rotation and a minimum when the optical phase difference is Æp (half an optical wavelength). This effect can be shown to be independent of the shape of the optical path and of the propagation medium. Modulating PZT with a sinusoidal voltage impresses a differential optical phase shift between the two light beams at the modulating frequency. The inter- ferometer output when there is no rotation of the coil exhibits the periodic behavior shown in Figure 5.7, whose frequency spectrum comprises Bessel harmonics of the modulation frequency. Since the phase modulation is symmetrical, only even harmonics are present; the ratio of the harmonic amplitudes depends on the extent of phase modulation. When the coil is rotated, the modulation occurs about the shifted position of the interferometer response. The modulation is unbalanced, and the funda- mental and odd harmonics will also be present (Figure 5.8). The amplitudes of the fundamental and odd harmonics are proportional to the sine of the angular rotation rate, while the even harmonics have a cosine relationship. The simplest demodulation scheme synchronously detects the signal at the fundamental frequency. Further improvements in dynamic range and linearity can be obtained by using a closed-loop configuration where the phase shift induced by rotation is compen- sated by an equal and opposite artificially imposed phase shift. One way to accomplish this is to introduce a frequency shifter into the loop, as shown in Figure 5.9. The relative frequency difference of the light beams propagating in the fiber loop can be controlled, resulting in a net phase difference that is proportional to the length of the fiber coil and the frequency shift. In Laser First coupler Polarizer second coupler Detector Amplifier LD drive Demodulator Analog rate output x-Splice location PZT phase modulator Sensing Coil Oscillator Figure 5.6 Schematic of a fiber-optic gyroscope in the open-loop configuration [8]. Reproduced with permisison of KVH Industries Inc Design Examples for Sensors and Actuators 93 [...]... (undeformed) configuration and uti its position after some time t ¼ t In terms of the unit vectors ei , they can be expressed as follows: Undeformed position :u0 ¼ u0 ei i Deformed position :u ¼ uti ei Smart Material Systems and MEMS: Design and Development Methodologies V K Varadan, K J Vinoy and S Gopalakrishnan # 2006 John Wiley & Sons, Ltd ISBN: 0 -4 7 0-0 936 1-7 106 Smart Material Systems and MEMS Due to the... N Croce, B Magnani and P Dario, ‘A piezoelectric-driven stereolithography-fabricated micropump’, Journal of Micromechanical and Microengineering, 5, 177–179 (1995) Part 3 Modeling Techniques Smart Material Systems and MEMS: Design and Development Methodologies V K Varadan, K J Vinoy and S Gopalakrishnan # 2006 John Wiley & Sons, Ltd ISBN: 0 -4 7 0-0 936 1-7 6 Introductory Concepts in Modeling One of the... transfer’, in Proceedings of IEEE: MEMS 98, IEEE, Piscataway, NJ, USA, pp 361–366 (1998) 10 V.K Varadan and V.V Varadan, ‘Microsensors, microelectromechanical systems (MEMS) and electronics for smart structures and systems , Smart Materials and Structures, 9, 953–972 (2000) 11 M.J Vellekoop, ‘Acoustic wave sensors and their technology’, Ultrasonics, 36, 7– 14 (1998) 12 J.A Chilton and M.T Goosey (Eds), Special...  ! Áv c1 þ 4c3 G0 ffi Àoh rf ðc1 þ c2 þ c3 Þ À v0 v0 ð5:76Þ Áa oh ¼ ðc1 þ 4c3 Þ 2 G00 k0 v0 ð5:77Þ 100 Smart Material Systems and MEMS where ci represents the substrate specific material constants, h is the film thickness, rf is the film density, v0 is the unperturbed Rayleigh wave velocity, G0 and G00 are the real and imaginary parts of the shear modulus of the polymer film, respectively, and o is the... gauges have the same nominal resistance so that the Wheatstone bridge is in the ‘balance state’ and there is no voltage output The radial- and tangential-resistance changes due to the radial 96 Smart Material Systems and MEMS and tangential strains, respectively, are obtained by considering the temperature-shift-induced strain: ÁRr e ¼ kgf ð"r þ etemp Þ R ð5:55Þ ÁRt ¼ kgf ð"t þ etemp Þ e R ð5:56Þ The voltage... Piscataway, NJ, USA, pp 47 1 48 6 (1998) 26 M Ritcher, R Linnemann and P Woias, ‘Robust design of gas and liquid micropumps’, Sensors and Actuators: Physical, A68, 48 0 48 6 (1998) 27 X.N Jiang, Z.Y Zhou, X.Y Huang, Y Li, Y Yang and C.Y Liu, ‘Micronozzle/diffuser flow and its application in micro valveless pumps’, Sensors and Actuators: Physical, A70, 81–87 (1998) 28 M.C Carrozza, N Croce, B Magnani and P Dario, ‘A... neighboring material particles Consider two material particles having coordinates ðx0 ; y0 ; z0 Þ and ðx0 þ dx0 ; y0 þ dy0 ; z0 þ dz0 Þ After the motion, these particles will have the coordinates ðx; y; zÞ and ðx þ dx; y þ dy; z þ dzÞ The initial and final distances between these neighboring particles are given by: ð6:17Þ ð6:16Þ The above measure gives the relative displacements between the two material particles,... 102 Smart Material Systems and MEMS Pumping Piozoelectrc disk chamber Ball valves Figure 5.15 Schematic (cross-sectional view) of a polymer micropump fabricated using the microstereolithography (MSL) process [28] M C Carrozza, N Croce, B Magnani, and P Dario, A piezoelectric-driven stereolithographyfabricated micropump, J Micromech Microeng 5, 1995, # IOP REFERENCES 1 F.S Foster, K.A Harasiewicz and. .. Ferroelectrics and Frequency Control, 39, 657–659 (1992) 23 R Rossberg and H Sandmaier, ‘Portable micro liquid dosing system’, in Proceedings of IEEE: MEMS 98, IEEE, Piscataway NJ, USA, pp 526–531 (1998) 24 X Yang, C Grosjean, Y.C Tai and C.M Ho, ‘A MEMS thermopneumatic silicone rubber membrane valve’, Sensors and Actuators: Physical, A 64, 101–108 (1998) 25 A.K Henning, ‘Microfluidic MEMS , in Proceedings... medical and biological imaging with polyvinylidene fluoride (PVDF) transducers’, IEEE Transactions: Ultrasonics, Ferroelectrics and Frequency Control, 47 , 1363–1371 (2000) 2 Website: [http://www.msiusa.com/piezo_download_listing htm #PART1 -INT.pdf] 3 B Zhu, ‘Design and development of PVDF based MEMS micro acoustic sensor and accelerometer’, Ph.D Dissertation, Department of Engineering Science and Mechanics, . 1995, # IOP 102 Smart Material Systems and MEMS Part 3 Modeling Techniques Smart Material Systems and MEMS: Design and Development Methodologies V. K. Varadan, K. J. Vinoy and S. Gopalakrishnan #. c 14 0 c 13 c 12 c 11 0 Àc 14 0 00 0c 66 0 c 14 0 c 14 Àc 14 0 c 44 0 00 0c 14 0 c 44 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ð5:25Þ e ¼ e 33 e 31 e 31 000 000Àe 22 0 e 15 0 Àe 22 e 22 0 e 15 0 2 4 3 5 ð5:26Þ E. problem, to get: m 11 À rV 2 m 12 m 13 m 14 m 12 m 22 À rV 2 m 23 m 24 m 13 m 23 m 33 À rV 2 m 34 m 14 m 24 m 34 m 44 À rV 2 2 6 6 4 3 7 7 5  a 1 a 2 a 3 a 4 2 6 6 4 3 7 7 5 ¼ 0 ð5:28Þ where: m 11 ¼