Smart Material Systems and MEMS - Vijay K. Varadan Part 3 pot

First, we will discuss an accelerometer consisting of a proof mass suspended over an FET, with the gate electrode of the device attached to the suspended structure. The anchors of the ‘meander’ support are elevated to suspend the beam above the gate region (Figure 3.12). This arrangement provides a gap between the gate and the insulator layer, thus keeping the threshold voltage for the FET constant [16]. The meander beams attached to this system are configured such that the electrode moves in the direction shown in Figure 3.12. This motion of the gate electrode changes the transis- tor drain current without affecting the current density through the channel. The sensitivity S of this device is given by the following: S ¼ dI D dW ðA=mÞð3:18Þ where dI D is the change in drain current and dW is the change in the depth to which the gate is overlapping the channel. Vacuum cover enclosur Drive line (a) (b) (c) (d) Contacts for sense lines Resonant microbeam Silicon diaphragm Silicon Proof mass Silicon flexure P 1 P 2 V in V a V + V – V T Beam To p Substrate I DC Sense resistor Detector AGC amplifier Counter out V out Beam drive Voltage-controlled attenuator Differental amplifier Figure 3.11 Resonant microbeam system (a) showing cross-sectional views of the polysilicon beam attached to a silicon diaphragm (b) or silicon flexure (c), along with (d) a schematic of the related microbeam test circuit [15]. Reprinted from Sensors Actuators A, 35, Zook J D, Burns D W, Guckel H, Sniegowski J J, Engelstad R L and Feng Z, Characteristics of polysilicon resonant microbeams, pp. 51–59, Copyright 1992, with permission from Elsevier 54 Smart Material Systems and MEMS For a typical n-channel FET, the drain current is given by: I D ¼ C g mW 2L ½2ðV GS ÀV T ÞV DS ÀV 2 DS  for V DS <V GS ÀV T ð3:19Þ and: I D ¼ C g mW 2L ðV GS À V T Þ 2 for V DS ! V GS À V T ð3:20Þ where V GS and V DS are the gate-to-source and drain-to- source voltages, V T is the threshold voltage at which the channel begins to conduct, C g is the gate capacitance per unit gate area, m is the majority carrier mobility for the channel and W and L are the width and length of the channel, respectively. These equations show a linear relationship between the drain current and the channel width W. The threshold voltage for the FET is as follows: V T ¼ V FB À Q D C i À ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2qeN D ðV bi À V BS Þ p C i ð3:21Þ where Q D is the dose of the n-type impurity, N D is the doping concentration, e is the dielectric constant of the semiconductor and V FB , V bi and V BS are the flat-band voltage, built-in potential of the channel junction and substrate bias, respectively; C i is the capacitance of the gate, which is a series combination of the capacitance due to the air gap and that due to the insulator layer. For very thin insulator layers, this capacitance can be approximated to that due to air alone. Thus, the config- uration presented here results in a linear relationship for the device current to the mechanical motion. Further- more, the lateral motion permits larger amplitudes of variation. The inertial force of the mass causes the lateral movement when the device can be used as an accelerometer. Substrate Structural layer Gate electrode Insulator Channel Source Drain (b) (a) Meander beam Insulator Drain Source Anchor Direction o f vibration Polymer Structural layer Gate electrode Channel Source Drain Gate L W Gate/anchor Figure 3.12 Schematics of a movable-gate field effect transistor: (a) top view; (b) cross-sectional view. Sensors for Smart Systems 55 Table 3.1 Structures of Love, SAW, SH–SAW, SH–APM and FPW devices and comparison of their operation [17,18]. M. Hoummady, A. Campitelli and W. Wlodarski, ‘‘Acoustic wave sensors: design, sensing mechanisms and applications,’’ Smart Mater. Struct. 6 1997, # IOP Device type Substrate Typical Structure Particle displacement Transverse component Sensing frequency relative to wave relative to sensing medium/ propagation surface quantity Love ST-quartz 95–130 MHz Transverse Parallel Ice, liquid Rayleigh SAW ST-quartz 80 MHz–1 GHz Transverse parallel Normal Strain, gas SH–SAW LiTaO 3 90–150 MHz Transverse Parallel Gas, liquid SH–APM ST-quartz 160 MHz Transverse Parallel Gas, liquid, chemical Lamb/FPW Si x N y =ZnO 1–6 MHz Transverse parallel Normal Gas, liquid 3.10 ACOUSTIC SENSORS Acoustic sensors operate by converting electrical energy in to acoustic waves, the propagation characteristics of which could be influenced by the physical parameter being measured, and then converting this back to electrical energy for further processing. Various config- urations of acoustic wave devices are possible for sensor applications. The important characteristics of some of these devices are summarized in Table 3.1. The type of acoustic wave generated in a piezoelectric material depends mainly on the substrate material properties, the crystal cut and the structure of the electrodes utilized to transform the electrical energy into mechanical energy. A Rayleigh wave has both a surface-normal compo- nent and a surface-parallel component in the direction of propagation. The wave velocity is determined by the substrate material and the crystal cut. Most surface acoustic wave (SAW) devices operate under this mode and will be discussed further below. The energies of the SAW are confined to a zone close to the surface a few wavelengths thick [19]. Love waves are guided acoustic modes which propagate in a thin layer deposited on a substrate. The acoustic energy is concentrated in this guiding layer and results in a high-mass sensitivity. This wave mode is typically employed in gases, biochemical or viscosity sensors. The selection of a different crystal cut can yield shear horizontal (SH) surface waves instead of Rayleigh waves. The particle displacements of this wave are transverse to the wave propagation direction and parallel to the plane of the surface. The frequency of operation is determined by the inter-digitated transducer (IDT) finger spacing and the shear horizontal wave velocity for the particular substrate material. These have shown consid- erable promise in applications such as sensors in liquid media and biosensors [20–22]. In general, SH–SAWs are sensitive to mass loading, viscosity, conductivity and permittivity of the adjacent liquid. The configuration of SH–APM devices is similar to the Rayleigh SAW devices, but the wafer is thinner, typically a few acoustic wavelengths. SH waves excited by the transducer propagate in the bulk of the substrate, at an angle to the surface. These waves reflect between the plate surfaces as they travel in the plate between the input and output transducers. The frequency of operation is determined by the thickness of the plate and the design of the transducer. SH–APM devices are mainly used in liquid sensing and offer the advantage of using the back surface of the plate as the sensing active area. Lamb waves, also known as flexural plate waves (FPWs), are elastic waves that propagate in plates of finite thickness and are used for the health monitoring of structures and for flow sensors: as the fluid passes through a channel above the acoustic path, it affects the properties of the acoustic waves propagating on the substrate. Surface acoustic wave (SAW)-based sensors form an important part of the sensor family and in recent years have seen diverse applications ranging from gas and vapor detection to strain measurement [19]. SAW devices were first used in radar and communication equipment as filters and delay lines and were recently found to have several applications in sensors for various physical variables, including temperature, pressure, force, electric field and magnetic field, as well as che- mical compounds. A SAW device consists of a piezo- electric wafer, IDTs and reflectors on its surface. The IDT is the ‘cornerstone’ of SAW technology, converting the electrical energy into mechanical energy, and vice versa, and hence are used for exciting as well as detecting the SAW. An IDT consists of two metal comb-shaped electrodes placed on a piezoelectric substrate (Figure 3.13). An electric field, created by the voltage applied to the electrodes, induces dynamic strains in the piezoelectric substrate, which in turn launches elastic waves. These waves contain, among others, the Rayleigh waves which run perpendicular to the electrodes with velocity V R . If a harmonic voltage, v ¼ v 0 exp ( jot), is applied to the electrodes, the stress induced by a finger pair travels along the surface of the crystal in both directions. To ensure constructive interference and in-phase stress, the distance between two neighboring fingers should be equal to half the elastic wavelength, l R . d ¼ l R =2 ð3:22Þ v d Figure 3.13 Finger spacings and (d) and their role in determi- nation of the acoustic wavelength (n) in an inter-digitated transducer [23]. Sensors for Smart Systems 57 The associated frequency is known as the synchronous frequency and is given by the following: f 0 ¼ V R =l R ð3:23Þ At this frequency, the transducer efficiency in converting electrical energy to acoustical, or vice versa, is max- imized. The width of each electrode finger is generally chosen as half the period. Its length determines the acoustic beamwidth and hence is not as significant in this preliminary design. The number of pairs of fingers are however critical in choosing the device bandwidth. The impulse response of the basic IDT is a rectangle. The Fourier transform of a rectangle is a sinc function whose bandwidth in the frequency domain is propor- tional to the length of the rectangular window in the space domain. As a result, a narrow bandwidth requires the IDT to have a large number of fingers. A schematic of a SAW device with IDTs ´ metallized onto the surface is shown in Figure 3.14 [23]. The exact calculation of the piezoelectric field driven by the inter-digital transducer is rather elaborate [19]. For simplicity, analysis of the IDT is carried out by means of numerical models. The frequency response of a single IDT can be simplified by the delta-function model [19]. The SAW velocity on the substrate depends on its density and elastic and piezoelectric constants. The principle of SAW sensors is based on the fact that the SAW traveling time between the IDTs changes with variation in the physical variables. Acoustic sensors offer a rugged and relatively inex- pensive platform for the development of wide-ranging sensing applications. A unique feature of acoustic sen- sors is their direct response to a number of physical and chemical parameters, such as surface mass, stress, strain, liquid density, viscosity, dielectric and conductivity pro- perties [24]. Furthermore, the anisotropic nature of piezo- electric crystals allows for various angles of cut, with each cut having unique properties. Applications, such as, for example, a SAW-based accelerometer utilize a quartz crystal with an ST-cut, which has an effective zero temperature coefficient [25], with a negligible frequency shift through changes in temperature. Again, depending on the orientation of the crystal cut, various SAW sensors with different acoustic modes may be constructed, with a mode ideally suited towards a particular application. Other attributes include very low internal loss, uniform material density and elastic con- stants and advantageous mechanical properties [26]. The principal means of detection of the physical property change involves the transduction mechanism of a SAW acoustic transducer, which involves transfer of signals from the mechanical (acoustic wave) to the electrical domain [19]. Small perturbations affecting the acoustic wave would manifest themselves as large changes when converted to the electromagnetic (EM) domain because of the difference in velocity between the two waves. Given that the velocity of propagation of the SAW on a piezoelectric substrate is 3488 m=s and the AC voltage is applied to the IDT at a synchronous frequency of 1 MHz, the SAW wavelength is given by l ¼ v=f ¼ 3:488  10 À3 m. The EM wavelength in this case is lc ¼ c=f , where c ¼ð3 Â10 8 m=sÞ is the velocity of light. Thus, lc ¼ 30 m, and the ratio of the wave- lengths ðl=lcÞ¼1:1  10 À5 . 3.11 POLYMERIC SENSORS Several well-known sensing mechanisms have been dis- cussed so far in this chapter. This and the next section will dwell on two material systems that have not been explored to their fullest potential. The advancement of silicon-based micro systems is intimately intertwined with developments in silicon semiconductor processing technology. Accordingly, var- ious processing approaches have been established for the integration of silicon-based micro systems with standard complimentary metal oxide semiconductor (CMOS) pro- cessing. For precision devices, and for devices requiring To source To detecto r Uniform fin g er s p acin g IDTs’ center-to-center separation M W λ R Constant finger overlap Figure 3.14 Schematic of a SAW device with IDTs metallized onto the surface [23]. 58 Smart Material Systems and MEMS integrated electronics, silicon is presently unrivaled. However, it is not necessarily the best material for all applications. For example, structures fabricated on this is limited to 2-D or very limited 3-D systems, unpackaged silicon devices are incompatible with many chemical and biological substances and fabrication requires sophisti- cated, expensive equipment operated in a clean-room environment. These often limit the low-cost potential of silicon-based micro systems. Polymer-based micro sys- tems are rapidly gaining momentum due to their potential for conformability and other special characteristics not available with silicon. In general, polymer-based devices may not be as small or as complex as those with silicon. However, polymers are often flexible, chemically and biologically compatible, available in many varieties and can be fabricated in truly 3-D shapes. Most of these materials and their fabrication processes are inexpensive. Perhaps one of the most important advantages of sensors using polymeric materials, in the context of smart systems, is their potential for being distributed over a large area. Polymer sensors are particularly advantageous in ‘moderate-performance’ devices which are low cost or disposable [27]. Unlike many silicon devices that are often packaged inside polymers, sensors built with poly- mers can even be ‘self-packaged’. Active polymer com- ponents can take advantage of several functional polymers to increase their functionality. Polymer sensors may be divided into two categories. The first uses the piezoelectric properties observed in some functional polymers while the second uses the change in conduc- tivity of some other polymers when exposed to changing environmental conditions. Since the discovery of strong piezoelectricity in poly (vinylidene fluoride) (PVDF) in 1969, piezoelectric poly- mers have been extensively investigated for various applications [28]. There are some unique features of piezoelectric polymers that make them attractive for use as sensing elements, including their relatively low acoustic impedance, broadband acoustic performance, flexible form and availability in large area films, and ability to be dissolved and coated onto various substrates. In the successful applications of piezoelectric polymer technology, these characteristics have prevailed over their inherent disadvantages of relatively weak piezo- electric properties, large dielectric and elastic losses, and low dielectric constants. In addition to its piezo- electric properties, PVDF also offers pyroelectric proper- ties [17]. PVDF is a semicrystalline high-molecular-weight poly- mer formed by the linking together of simple 1,1-difluor- oethylene (VDF) molecules. Under precisely controlled reaction conditions, a molecular structure of PVDF with a 90 % head-to-tail arrangement (i.e. CH 2 –CF 2 –(CH 2 – CF 2 ) n –CH 2 –CF 2 ) [29] can be obtained. PVDF is approxi- mately half crystalline and half amorphous. The most common polymorph form of PVDF, the a-phase, is pro- duced by crystallization from the melt or solution. The a-phase can be transformed into the polar form, the b- phase, by mechanically stretching or rolling at elevated temperatures. Since all of the dipole moments become perpendicular to the chain axes, microscopically, each crystallite has a net dipole moment and is piezoelectric. However, on the macroscopic scale, there is no polariza- tion within the polymer due to the random orientation of the dipole moments of the crystallites. In order to render the PVDF film piezoelectric, poling is required, which involves the application of an electric field. This step preferentially aligns the dipoles of the crystallites in the direction of the applied electric field and thus produces a net polarization. In the copolymer (P(VDF–TrFE)), the increased number of the relatively large fluorine atoms prevents the formation the of tg þtg-conformation. This extends the polymer chains to crystallize directly into the b-phase. The copolymer also needs a final poling step to make it fully piezoelectric. The two main poling techni- ques are conventional two-electrode poling (also referred to as thermal poling) and corona poling. A listing of the properties of poled PVDF and its copolymer P(VDF– TrFE) is provided in Table 3.2 [30]. Several standard processes are available for the deposi- tion of polymer thin films. Some films which are used for gas sensing employing SAW devices are listed in Table 3.3. These could be deposited on a substrate by deposition methods such as spin coating, dip coating and in situ polymerization. 3.12 CARBON NANOTUBE SENSORS After carbon nanotubes (CNTs) were first discovered by Iijima in 1991 [31], several researchers have reported excellent mechanical, electrical and thermal properties for these materials, both theoretically and experimen- tally. In recent years, such nanotubes have been intro- duced into microelectronics and micro electromechanical systems (MEMS). These nanotubes are also regarded as promising materials for nanotechnology and nano elec- tromechanical systems (NEMS). Fundamentally, CNTs can be considered as rolled-up cylinders of graphite sheets of sp 2 -bonded carbon atoms with diameters less than 100 nm. The length of an individual carbon nanotube could typically vary from Sensors for Smart Systems 59 tens of nanometers to several microns. Caps have always been observed at both ends of these cylinders, which could be hemispheres of a fullerene, such as C 60 . Carbon nanotubes can be divided into two categories, i.e. single- walled nanotubes (SWNTs) and multi-walled nanotubes (MWNTs), according to the number of grahene layers. Some properties of CNTs, such as conductivity varia- tion and the electrostrictive effect, have been used in implementing sensors using them. The design of such sensors follow the principles discussed earlier in this chapter. In the following, we present a somewhat differ- ent approach that makes use of the variation in electro- magnetic properties of a transmission line coated with a layer of a CNT [32]. Based upon the change in this electrical property in composite thin films of carbon naotubes (as the vapor concentration varies), monitoring of the reflection phase at radio frequencies has been proposed for real-time wireless sensing applications. The reflection phase of electromagnetic waves reflected from a load was determined by load impedance. For this purpose, composite thin films with funtionalized carbon nanotubes (f-CNTs) were coated onto an interdigital coplanar waveguide, as shown in Figure 3.15, and the phase change of the reflected waves due to the presence of an organic gas was evaluated. Table 3.2 Comparison of typical properties of PVDF and P(VDF–TrFE) [30]. Property PVDF P(VDF–TrFE) Coupling coefficient k 31 0.12 0.20 k t 0.14 0.25–0.29 Piezoelectric strain constant (10 À12 m=V or C/N) d 31 23 11 d 33 À33 À38 Piezoelectric stress constant (10 À3 Vm=N) g 31 216 162 g 33 À330 À542 Pyroelectric coefficient, P (10 À6 C=(m 2 K) 30 40 Young’s modulus, Y (10 9 N=m 2 )2–43–5 Relative permittivity, e=e 0 12–13 7–8 Mass density, r (10 3 kg=m) 1.78 1.82 Speed of sound, c (10 3 m=s) 2.2 2.4 Acoustic impedance, Z (MRa) 3.92 4.37 Loss tangent, tan d e (at 1 kHz) 0.02 0.015 Temperature range (  C) À40 to 80 À40 to 115 Table 3.3 Typical examples of polymer thin films used in gas sensors. Measurand Coating Hydrogen Palladium SO 2 Triethanolamine NO 2 Lead phthalocyanine Toluene Polydimethylsiloxane Water vapor/humidity Polymide, SiO 2 , cellulose acetate H 2 SWO 3 CO Metal phthalocyanine CO 2 Polyethyleneimine CH 4 Metal phthalocyanine NH 3 Platinum Power divider Gas sensor (CNT/PMMA) Reference load (NiCr thin film) RF signal Figure 3.15 Schematic of a sensor based on the phase changes in a transmission line coated with a carbon nanotube composite. 60 Smart Material Systems and MEMS When a reflected wave exists on a ‘lossless’ transmis- sion line terminated with a load impedance, Z L ¼ a þ jb, the voltage across T the line is given by the following: V ¼ V þ e Àjbz þ V À e jbz ð3:24Þ where V þ and V À are the amplitude constants of the incident and reflected waves, respectively, and b is the phase constant for the ‘lossless’ line. The voltage reflec- tion coefficient, G L , is described by the ratio of V À to V þ as follows [33]. G L ¼ V À V þ ¼ Z L À Z C Z L þ Z C ð3:25Þ and the voltage at any point on the transmission line (z < 0) is given by the following: V ¼ V þ  e Àjbz þjG L je jðyþbzÞ  ð3:26Þ where: G L ¼jG L je iy ð3:27Þ and: jG L j¼ ½ða 2 À Z 2 c Þþb 2 þ4b 2 Z 2 c ½ða þZ 2 c Þþb 2  2 () ½ ð3:28Þ plus: y ¼ tan À1 2bZ c ða 2 À Z c 2 Þþb 2 ! ð3:29Þ where Z c is the characteristic impedance of the transmis- sion line. According to Equation (3.29), the phase of the reflected waves in a transmission line is determined by load impedance. Typical changes in the phase of the reflected waves with respect to the load impedance of a transmission line are illustrated in Figure 3.16. As long as the imaginary part of the load impedance (b)islow, the reflected wave phase exhibits a large phase shift with a small change in the real part of the load impedance (a) near the characteristic impedance. The basic sche- matic of phase monitoring in this newly designed sensor employs a variable resistor with a small imaginary impedance as a load terminating a coplanar waveguide (Figure 3.15). REFERENCES 1. S. Fatikow and U. Rembold, Microsystem Technology and Microrobobics, Springer-Verlag, Berlin, Germany (1997). 2. P. Rai-Choudhury (Ed.), Handbook of Microlithography, Micromachining and Microfabrication, Vol. 2, Microma- chining and Microfabrication, SPIE Optical Engineering Press, Bellingham, WA, USA (1997). z 1 z 2 0 20 40 60 80 100 120 140 160 180 200 6055504540 Real part of impedance (ohm) S 11 Phase (degrees) 1 2 Figure 3.16 Relationship of the reflection (S 11 ) phase to the real and imaginary parts of the load impedance (Z L ¼ a þ jb), with z 1 ¼ a þj1 and z 2 ¼ a þj2. Sensors for Smart Systems 61 3. W.P. Eaton and J.H. Smith, ‘Micromachined pressure sensors – review and recent developments’, Smart Materials and Structures, 6, 530–539 (1997). 4. C. Lee, T. Itoh and T. 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Collin, Foundations for Microwave Engineering, McGraw-Hill, New York, NY, USA (1992). 62 Smart Material Systems and MEMS 4 Actuators for Smart Systems 4.1 INTRODUCTION In this chapter, the basic principles of common electro- mechanical actuators are briefly discussed. The energy conversion schemes presented here include piezoelectric, electrostrictive, magnetostrictive, electrostatic, electro- magnetic, electrodynamic and electrothermal. Most of the schemes are reciprocal and hence these devices are generally referred to as transducers. Although some of these schemes are not quite amenable for smart micro- mechanical systems, they do have the potential for being used in such systems in the foreseeable future. One important step in the design of these mechanical systems is obtaining their electrical equivalent circuits from analytical models. This remains the main focus of this chapter. However, relevant examples of fabricated prototypes from the published literature are also included wherever necessary. In what follows we extensively make use of electromechanical analogies to arrive at electrical equivalent circuits of transducers. These equivalent circuits are neither unique nor exact, but would serve as an easily understood tool in trasnducer design. The use of these electrical equivalent circuits would also facilitate use of the vast resources available for modern optimization programs for electrical circuit design into transducer designs. A list of useful electromechanical analogies is given in Table 4.1 [1]. These are known as mobility analogies. These analogies become useful when one needs to replace mechanical components with electrical compo- nents which behave similarly, forming the equivalent circuit. As a simple example, the development of an electrical equivalent circuit of a mechanical transmission line component is discussed here [1]. The variables in such a system are force and velocity. The input and output variables of a section of a ‘lossless’ transmission line can be conveniently related by an ABCD matrix form as follows: _ x 1 F 1  ¼ cos bxjZ 0 sin bx j Z 0 sin bx cos bx "# _ x 2 F 2  ð4:1Þ where: Z 0 ¼ 1 A ffiffiffiffiffiffi rE p ffiffiffiffiffiffi C 1 M 1 r ð4:2Þ and: b ¼ o v p ð4:3Þ and: v p ¼ ffiffiffiffi E r s ¼ 1 ffiffiffiffiffiffiffiffiffiffi C l M l p ð4:4Þ In these equations, A is the cross-sectional area of the mechanical transmission line, E its Young’s modulus and r the density; C l and M l are the compliance and mass per unit length of the line, respectively. Now, looking at the electromechanical analogies in Johnson [1], the expres- sion for an equivalent electrical circuit can be obtained in the same form as Equation (4.1) above: V 1 I 1  ¼ cos bxjZ 0 sin bx j Z 0 sin bx cos bx "# V 2 I 2  ð4:5Þ In Equation (4.5), the quantities in the components of the matrix are also represented by equivalent electrical parameters as follows: Z 0 ¼ ffiffiffi m e r ¼ ffiffiffiffiffiffi L 1 C 1 r ð4:6Þ 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 [...]... À 2Q 33 T3 D3 À 2Q 13 T1 D3 For non-zero stress in the thickness direction of the slab: E3 ¼ a1 D3 þ a2 D2 þ a3 D3 T3 3 D3 À P0 E3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À 4Q44 T5 D1 ðe0 eT Þ2 À aðeT Þ2 w 33 33 ð4:96Þ À 2Q 13 T2 D3 ð4:102Þ s1 ¼ sD T1 þ sD T2 þ sD T2 þ Q 33 D2 þ Q 13 D2 11 12 13 1 2 þ Q 13 D2 3 ð4:1 03 s2 ¼ sD T1 þ sD T2 þ sD T3 þ Q 13 D2 þ Q 33 D2 12 11 13 1 2 and: s3 ¼ b1 T3 þ b2 D2 3 þ... ð4:104Þ sD T 2 13 þ Q 33 D2 3 ð4:99Þ Solving for E: sD T 1 13 D1 ¼ D2 ¼ 0 76 Smart Material Systems and MEMS we get: The voltage due to the fixed charge Q0 is: D3 À P0 E3 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À 2b2 T3 D3 2 E2 À aðD À P Þ2 e0 3 0 ð4:110Þ   Àx 2b2 Q0 d À V0 dne b1 A v2 ¼ ð4:114Þ A capacitive term C0 is defined as follows: and: s3 ¼ b1 T3 þ b2 D2 3 ð4:111Þ C0 ¼ sD 33 and b2 ¼ Q 33 The equivalent... B2 5 ¼ 4 0 d31 d31 d 33 0 B3 2 3 T1 6T 7 2 6 27 0 m T 6 7 6 T3 7 6 11 6 7þ4 0 m22 T 6T 7 6 47 0 0 6 7 4 T5 5 d15 0 0 0 3 7 05 0 32 3 H1 0 76 7 0 54 H2 5 ð4: 135 Þ H3 m 33 T T6 Assuming linear relationships between B and H and between S and H, the internal energy may be written as follows: 1 1 Si Ti þ Hm Bm 2 2 1 1 1 1 Ti sij Tj þ Ti dmi Hm þ Hm dmi Ti þ Hm mmk Hk ¼ 2 2 2 2 ¼ Ue þ 2Ume þ Um ð4: 136 Þ U¼ while... fabrication of small-sized magnets and current-carrying coils In this case, however, the coil is also movable This remains a fabrication challenge, as miniaturized components are required for MEMS applications v¼ bs h 33 q þ h 33 w bl ð4:88Þ F¼ D C 33 Qbl w þ h 33 q tan Qh ð4:89Þ D where C 33 is the elastic stiffness of the piezoelectric material at constant electric displacement, h 33 is the piezoelectric... and: ð4:101Þ ð4:106Þ þ 4Q44 D1 D3 T1 ¼ T2 ¼ T4 ¼ T5 ¼ T6 ¼ 0 À 2Q 33 T2 D2 À 2Q 13 T3 D2 À 2Q 13 T1 D2 À 4Q44 T4 D3 þ Q 13 D2 1 ð4:105Þ s5 ¼ ð1Þ P0 D1 À E1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À 2Q 33 T1 D1 À 2Q 13 T2 D1 ðe0 eT Þ2 À aðeT Þ2 w 11 11 þ sD T 3 33 s4 ¼ sD T4 þ 4Q44 D2 D3 44 sD ¼ 66 p with the restriction that D3 À P0 < ð0:99e0 e= aÞ Generalizing this one-dimensional nonlinear model to three... ¼ 1,2 and 3 The elastic compliances at constant H are:  @Si   ¼sH ij @Tj H ð4: 130 Þ ð4: 131 Þ For small variations in dT and dH, the constitutive relationships may be linearized as follows: Si ¼ SH Tj þ dki Hk ij Bm ¼ dmj Tt þ mT Hk mk ði ¼ 1; 6Þ ðm ¼ 1; 2; 3 0 B1 3 2 0 0 0 d31 0 0 d15 0 0 d31 72 3 7 7 H1 d 33 76 7 7 74 H2 5 0 7 7 H 3 7 0 5 0 0 3 3 0 7 0 7 7 0 7 7 7 0 7 7 7 0 5 s66 H ð4: 134 Þ 0... x0 Þ2 ð4 :30 Þ Actuators for Smart Systems 67 Fðv; xÞ ¼ ¼   q0 q2 0 x vþ kÀ d þ x0 e0 Ae ðd þ x0 Þ e0 Ae v0 ðd þ x0 Þ2 vþ kÀ e0 A e v 2 0 We start with rewriting the second part of Equation (4 .33 ) with q on the left-hand side and taking the time derivative for current: ! ðd þ x0 3 x ð4 :31 Þ Note that the system is in equilibrium as long as the second term on the right-hand side of Equation (4 .31 ) is... as: Ume k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi Ue Um ð4: 137 Þ Usually magnetostrictive materials operate in the longitudinal mode This reduces the stresses strains, and magnetic field components in the direction with subscript Actuators for Smart Systems 79 3 , while all others are zero These conditions simplify the magnetomechanical coupling coefficient to: 2 k 33 ¼ 2 d 33 mT sH 33 33 By Faraday’s law, the induction through... into the first part of Equation (4 .33 ): The matrix form of Equations (4.25) and (4 .31 ) is: 2 C0 kC0 FÀ U G G ð4 :34 Þ It may be noticed that a stable equilibrium state exists for 0 < K < 1 The typical values for K are between 0.05 and 0.25 Transduction may also be expressed in such a way as to connect between electrical variables (on the left-hand side) and mechanical variables (on the right-hand side) The... constant and Q is the phase constant By defining the static capacitance C0 , the transfer factor G and the spring constant k as follows: C0 ¼ (a) I bl ; bs h 33 G ¼ h 33 (b) bl ; bs h 33 v When subjected to mechanical stress, certain anisotropic crystalline materials generate charge This phenomenon, D C 33 bl h i 1: Γ v 4.5 PIEZOELECTRIC TRANSDUCERS k¼ 1/k* u F C0 jX Figure 4.10 Schematic (a) and equivalent . Physics, 32 , 237 6– 237 9 (19 93) . 23. V .K. Varadan and V.V. Varadan, ‘Microsensors, actuators, MEMS and electronics for smart structures’, in Handbook of Microlithography, Micromachining and Microfabri- cation,. follows: Z 0 ¼ ffiffiffi m e r ¼ ffiffiffiffiffiffi L 1 C 1 r ð4:6Þ 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 v p ¼ 1 ffiffiffiffiffi me p ¼ 1 ffiffiffiffiffiffiffiffiffiffi L 1 C 1 p ð4:7Þ In. SPIE, 57 63, 32 6 33 2 (2005). 33 . R.E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New York, NY, USA (1992). 62 Smart Material Systems and MEMS 4 Actuators for Smart Systems 4.1