The MEMS Handbook MEMS Applications (2nd Ed) - M. Gad el Hak Episode 1 Part 6 ppt

This chapter will adopt a classification of microactuators based on the form of the input energy and will deal with only some of the microactuators listed in Table 5.4. In particular, piezoelectric, electro- magnetic, shape memory alloys, and electrostatic microactuators will be discussed. Some information will also be provided on polymeric, electrorheological, SMA polymeric, and chemical microactuators. 5.2 Piezoelectric Actuators The discovery of the piezoelectric phenomena is due to Pierre and Jacque Curie, who experimentally demonstrated the connection between crystallographic structure and macroscopic piezoelectric phe- nomena and published their results in 1880. Their first results were only on the direct piezoelectric effect (from mechanical energy to electric energy). The next year, Lippman theoretically demonstrated the exis- tence of an inverse piezoelectric effect (from electric energy to mechanical energy). The Curie brothers then gave value to Lippman’s theory with new experimental data, opening the way to piezoelectric actuators. After some tenacious theoretical and experimental work in the scientific community, Voigt synthesized all the knowledge in the field using a properly tensorial approach, and in 1910 published a comprehensive study on piezoelectricity. The first application of a piezoelectric system was a sensor (direct effect), a submarine ultrasonic detector, developed by Lengevin and French in 1917. Between the first and second World Wars many applications of natural piezoelectric crystals appeared, the most important being ultrasonic trans- ducers, microphones, accelerometers, bender element actuators, signal filters, and phonograph pick-ups. During World War II, the research was stimulated in the United States, Japan, and Soviet Union, resulting in the discovery of piezoelectric properties of piezoceramic materials exhibiting dielectric constants up to 100 times higher than common cut crystals. The research on new piezoelectric materials continued during the second half of the twentieth century with the development of barium titanate and lead zirconate titanate piezoceramics. Knowledge was also gained on the mechanisms of piezoelectricity and on the doping pos- sibilities of piezoceramics to improve their properties. These new results allowed high performance and low cost applications and the exploitation of a new design approach (piezocomposite structures, polymeric materials, new geometries, etc.) to develop new classes of sensors and, especially new classes of actuators. 5-8 MEMS: Applications TABLE 5.4 Classification of Microactuators Based on the Input Energy Input Energy Physical Class Actuator Electrical Electric and magnetic field Electrostatic Electromagnetic Molecular forces Piezoelectric Piezoceramic Piezopolymeric Magnetostrictive Electrostrictive Magnetorheological Electrorheological Fluidic Pneumatic High pressure Low pressure Hydraulic Hydraulic Thermal Thermal expansion Bimetallic Thermal Polymer gels Shape memory effect Shape memory alloys Shape memory polymers Chemical Electrolytic Electrochemical Explosive Pyrotechnical Optical Photomechanical Photomechanical Polymer gels Acoustic Induced vibration Vibrating © 2006 by Taylor & Francis Group, LLC 5.2.1 Properties of Piezoelectric Materials A piezoelectric material is characterized by the ability to convert electrical power to mechanical power (inverse piezoelectric effect) by a crystallographic deformation. When piezoelectric crystals are polarized by an electric tension on two opposite surfaces, they change their structure causing an elongation or a shortening, according to the electric field polarity. The electric charge is converted to a mechanical strain, enabling a relative movement between two material points on the actuator. If an external force or moment is applied to one of the two selected points, opposing a resistance to the movement, this “con- ceptual actuator” is able to win the force or moment, resulting in a mechanical power generation (Figure 5.13). The most frequently used piezoelectric materials are piezoceramics such as PZT, a polycrystalline ferroelectric material with a tetragonal-rhombahedral structure. These materials are generally composed of large divalent metal ions; such as lead; tetravalent metal ions, such as titanium or zirconium (Figure 5.14); and oxygen ions. Under the Curie temperature, these materials exhibit a structure without a cen- ter of symmetry. When the piezoceramics are exposed to temperatures higher than Curie point, they transform their structure, becoming symmetric and loosing their piezoelectric ability. Common piezoelectric materials are piezoceramics such as lead zirconate titanate (PZT) and piezoelec- tric polymers such as polyvinylidene fluoride (PVDF). To improve the performance of piezoceramics, the Microactuators 5-9 F F F d + − E (a) (b) (c) FIGURE 5.13 Inverse piezoelectric effect. An external force F is applied to a piezoelectric parallelepiped as in con- figuration (a). When an electric tension generator gives power to the actuator, it results in a displacement d, as in con- figuration (b). If the electric power is disconnected, the piezoelectric parallelepiped returns to its initial condition, as in configuration (c). FIGURE 5.14 Structure of a PZT cell under the Curie temperature. White particles are large divalent metal ions, gray particles are oxygen ions, and the black particle is a tetravalent metal ion. © 2006 by Taylor & Francis Group, LLC research proposed new formulations, PZN–PT and PMN–PT. These new formulations extend the strain from the 0.1%–0.2% (for PZT) to 1% (for the new formulations) and are able to generate a power den- sity five times higher than that of PZT. Piezoelectric polymers are usually configured in film structures and exhibit high voltage limits, but have low stiffness and electromechanical coupling coefficients. Piezoceramics are much stiffer and have larger electromechanical coupling coefficients; therefore poly- mers are not usually chosen as actuators. We can use the constitutive equations to describe the relationship between the electric field and the mechanical strain in the piezoelectric media: S ij ϭ s E ijkl T kl ϩ d ijk E k , i, j, k, l ∈ 1, 2, 3 (5.3) D i ϭ d ijk T jk ϩ ε ij T E j where D is the three-dimensional vector of the electric displacement, E is the three-dimensional vector of the electric field density, S is the second order tensor of mechanical strain, T is the second order tensor of mechanical stress, s E is the fourth order tensor of the elastic compliance, ε T is the second order tensor of the permeability, and d is the third order tensor of the piezoelectric strain. Note, through a transposition, d is related to d as that can be observed in the commonly used matrix form of the constitutive equation: S I ϭ s E I,J T J ϩ d I,j E j , i, j ∈ 1, 2, 3; I, J ∈ 1, 2 … 6 (5.4) D i ϭ d i,J T J ϩ ε T i,j E j The first version of the constitutive equation is useful to explicate the dimensions of the tensors, while the second is more concise. There are many ways of writing these equations: another interesting version (Equation 5.5) gives the strain in terms of stress and electric displacement, introducing the voltage matrix g and the β matrix: S I ϭ s I,J D T J ϩ g I,j E j , i, j ∈ 1, 2, 3; I, J ∈ 1, 2 … 6 (5.5) E i ϭ Ϫg i,j T J ϩ β T i,j D j If the tensors/vectors D, S, E, and T are rearranged into nine-dimensional column vectors, the constitu- tive equation can then take the form of Equation 5.6, Equation 5.7, Equation 5.8, or Equation 5.9, accord- ing to the selection of dependent and independent variables: Ά D ϭ ε T E ϩ d: T S ϭ d t E ϩ s E : T (5.6) Ά (5.7) Ά (5.8) Ά (5.9) The above constitutive equations exhibit linear relationships between the applied field and the result- ing strain. As an example, we can consider the tensor d. The experimental values of its components are obtained by an approximation, as it depends upon the strain and the applied electric field. This approx- imation consists of the hypothesis of low variation of applied voltage and the resulting strain. If the con- sidered region is out of the field of linearity, then new values should be used to estimate the tensor d (the constitutive equations are linear, but the value of d is different in each small considered region). Or, a unique nonlinear constitutive equation could be used (d is no more constant but is a function of S and E ϭ β S D Ϫ h : S T ϭ Ϫh t D ϩ c D : S E ϭ β T D Ϫ g : T S ϭ g t D ϩ s D : T D ϭ ε S E ϩ e : S T ϭ Ϫe t E ϩ c E : S 5-10 MEMS: Applications © 2006 by Taylor & Francis Group, LLC E, resulting in a theoretically correct but really complex approach). Another consideration is based on the “aging effect” of piezoceramic materials represented by a logarithmic decay of their properties with time. Therefore, over time, a new value of d should be estimated to obtain a correct model for the piezoceramic material. Considering linear constitutive equations (for each small region of the considered variables) in com- bination with the hypothesis of low strain, we can write: ∇ s v ϭ , with ∇ s (o) ϵ ΂ ∇ ΂ o ΃ ϩ ∇ ΂ o T ΃΃ (5.10) where v is the speed of a basic element of piezoceramic. Using Equation 5.10, the equation of motion (Equation 5.11), the Maxwell equations (Equation 5.12), and the previously mentioned constitutive Equations 5.7 and 5.8 (the latter is used only to find the time derivative of D), we can obtain the general Christoffel equations of motion (Equation 5.13). ∇oT ϭ ρ Ϫ F (5.11) where ρ is the density of the material and F is the resulting internal force reduced to a surface force (with the divergence theorem). Ά Ϫ∇ ϫ E ϭ µ 0 ∇H ϭ ϩ J (5.12) where µ 0 is the permeability constant, H is the electromagnetic induction, and J is current density. Ά ∇oc E : ∇ S v ϭ ρ Ϫ ϩ ∇oe Ϫ∇ ϫ ∇ ϫ E ϭ µ 0 ε S ϩ µ 0 e:∇ S ϩ µ 0 (5.13) In order to obtain a simple set of equations from Equation 5.13, we can neglect the presence of force, cur- rent density, and the rotational term of E. The Fourier theorem allows us to transform, under reasonable hypotheses, a periodical function (or in general a function defined into a finite time frame) into a sum of trigonometric functions such that we can consider only a single wave propagating through the media. The usual geometries of piezoelectric actuators are planar, therefore we will consider only planar waves (Equation 5.14). Under these simplifications, we can obtain a simplified set of Christoffel equations (Equation 5.15). f(r,t) ϭ e j( ω иtϪkIor) (5.14) where ω is the angular pulsation, I is the direction of the wave, and the constant k should be determined. Ά (5.15) where V is the electric potential, while l i , l j , l i,K and l Lj are the matrices of the directional cosines. The resulting equation can be solved to calculate the value of the potential energy to have a desired speed (under the limitations of the used technology). Knowing the model for the actuating principle, this equa- tion can be implemented in a more complex model of a complete actuator or it can be used as a “metaphor” for the behavior of the actuator. The second approach allows the definition of a “virtual” piezoelectric object implementing, through a proper calibration, some correction of the piezoelectric matrix of the “real” piezoceramic to take into account some unconsidered phenomena. Ϫk 2 (l iK c E KL l Lj ) и v j ϩ ρω 2 v i ϭ Ϫj ω и k 2 (l iK e Kj l j ) и V ω 2 k 2 (l i ε S ij l j ) и V ϭ Ϫj ω и k 2 (l i e iL l Lj ) и v j ∂J ᎏ ∂t ∂v ᎏ ∂t ∂ 2 E ᎏ ∂t 2 ∂E ᎏ ∂t ∂F ᎏ ∂t ∂ 2 v ᎏ ∂t 2 ∂D ᎏ ∂t ∂H ᎏ ∂t ∂v ᎏ ∂t 1 ᎏ 2 ∂S ᎏ ∂t Microactuators 5-11 © 2006 by Taylor & Francis Group, LLC 5.2.2 Properties of Piezoelectric Actuators In 2001, Niezrecki et al. in 2001 proposed a review of the state of the art of piezoelectric actuation. This section will use this scheme and provide some explanations of the most common actuation systems. Piezoelectric actuators are composed of elementary PZT parts that can be divided into three categories (depending on the used piezoelectric relation) of axial actuators, transversal actuators, and flexural actu- ators (Figure 5.15). Axial and transversal actuators are characterizedbygreater stiffness, reduced stroke, and higher exertable forces, while flexural actuators can achieve larger strokes but exhibit lower stiffness. Although we have shown in Figure 5.15 piezoelectric elementary parts with a parallelepiped shape, piezoelectric materials are produced in a wide range of forms using different production techniques—from simple forms, such as rectangular patches or thin disks, to custom very complex shapes. Because of the reduced displacements, piezoelectric materials are not usually used directly to generate a motion, but are connected to the user by a transmission element (Figure 5.2). Therefore, the piezoelectric actuators are not just “simple actuators,” they are complete machines with an actuation system (the PZT element) and a transmission allowing the transformation of mechanical generated power in a desired form. In fact, the primary design parameters of a piezoelectric actuator (referring to the entire actuating machine, not only the elementary PZT part) include – the functional parameters — displacement, force, and frequency — and – the design parameters — size, weight, and electrical input power. Underlining only the functional parameters, the generated mechanical power is essentially a trade off between these three parameters; the actuator architecture is devoted to increment one or two of these 5-12 MEMS: Applications V V V F F F (a) (b) (c) FIGURE 5.15 Elementary PZT part, where F is the exerted force and V is the electric potential difference between two faces. (a) is an axial actuator, (b) is a transversal actuator, and (c) is a flexural actuator. © 2006 by Taylor & Francis Group, LLC parameters at the cost of the other parameters. Piezoelectric actuators are characterized by noticeable exerted forces and high frequencies, but also significantly reduced strokes; therefore the architecture designs aim to improve the stroke, reducing force or frequency. A distinction can then be made between –force-leveraged actuators and –frequency-leveraged actuators. The leverage effect can be gained with an integrated architecture or with external mechanisms, so another distinction can be made between – internally leveraged actuators and – externally leveraged actuators. The most common internally force-leveraged actuators include: (1) Stack actuators (2) Bender actuators (3) Unimorph actuators and (4) Building-block actuators. The externally force-leveraged actuators can be subdivided as: (1) Lever arm actuators (2) Hydraulic amplified actuators (3) Flextensional actuators and (4) Special kinematics actuators. The frequency-leveraged actuators can be, in general, led back to inchworm architecture. Stack actuators consist of multiple layers of piezoceramics (Figure 5.16). Each layer is subjected to the same electrical potential difference (electrical parallel configuration), so the total stroke results is the sum of the stroke of each elementary layer, while the total exertable force is the forceexertedby a single ele- mentary layer. The leverage effect on the stroke is linearly proportional to the ratio between the elemen- tary piezoelectric length and the actuator length. The most common stack architectures can gain some microns stroke, exerting some kilonewtons forces, with about ten microseconds time responses. Bender actuators consist of two or more layers of PZT materials subjected to electric potential differ- ences, which induce opposite strain on the layers (Figure 5.17). The opposite strains cause a flexion of the bender, due to the induced internal moment in the structure. This architecture is able to generate an amplification of the stroke as a quadratic function of the length of the actuator, resulting in a stroke of Microactuators 5-13 V F s S FIGURE 5.16 Stack actuator. V is the electrical potential difference applied to each piezoelectric element, F is the total exertable force. The stroke s of a single piezoelectric element is proportional to s, while the total stroke S is pro- portional to S. © 2006 by Taylor & Francis Group, LLC more than one millimeter. Different configurations of bender actuators are available, such as end sup- ported, cantilever, and many other configurations with different design tricks to improve stability or homogeneity of the movement. Unimorph actuators are a special class of bender actuators, which are composed of a PZT layer and a non-active host. Two co mmon unimorph architectures are: Rainbow, developed by Heartling (1994) and Thunder, developed at NASA Langley Research Center (Wise, 1998). These are characterized by a pre- stressed configuration. Being stackable, they are able to gain important strokes (some millimeters). Building block actuators consist of various configurations characterized by the abilitytocombine the elementary blocks in series or parallel configurations to form an arrayed actuation system with improved stroke by series arrays and improved force by parallel arrays. There are various state-of-the-art elemen- tary blocks available such as C-blocks, recurve actuators, and telescopic actuators. The first class of externally leveraged actuators to be examined is the lever arm actuator class. Lever arm actuators are machines composed of an elementary actuator and a transmission able to amplify the stroke and reduce the generated force. The transmission utilized is a leverage mechanism or a multistage leverage system. To r e duce design complexity, the leverage system is generally composed of two simple elements: a thin and flexible member (the fulcrum) and a thicker, more rigid, long element (the leverage arm). Another externally leveraged architecture is hydraulic amplification. In this configuration, a piezo- electric actuator moves a piston, which pumps a fluid into a tube moving another piston of areduced sec- tion. The result is a very high stroke amplification (approximately 100 times); however, this amplification involves some problems due to the presence of fluids, microfluidic phenomena, and high frequency mechanical waves that are transmitted to the fluids. The third class of externally amplified actuators is flextensional actuators. This class is characterized by the presence of a flexible component with a proper shape, able to amplify displacement. It differs from the lever arm actuators approach, because of its closed-loop configuration, resulting in a higher stiffness but reduced amplification power. To increment the stroke amplification, this class of actuators can be used in a building-block architecture. Atypical example is a stack of Moonie actuators. The research on actuation architecture is very dynamic. New design solutions emerge in literature and on the market frequently; therefore it would be improper to generalize these classifications based on only the three described classes of externally leveraged actuators: lever arm, hydraulic, and flextensional. The final class of externally leveraged actuators uses the frequency leverage effect. These actuators are basically reducible to inchworm systems (Figure 5.18). In general, they are composed of three or more actuators, alternatively contracting, to simulate an inchworm movement. The resulting system is a very precise actuator, with very high stroke (more than 10mm), but with a reduced natural frequency. The behavior of PZT actuators can be affected by undesired physical phenomena such as hysteresis. In fact, hysteresis can account for as much as 30% of the full stroke of the actuator (Figure 5.19). An addi- tional problem is the occurrence of spurious additional resonance frequencies under the natural fre- quencies. These additional frequencies introduce undesired vibrations, reducing positioning precision and the overall performance of the actuator. Furthermore, the depoling effect, which results in an unde- sired depolarization of artificially polarized materials, occurs when a too-high temperature of the PZT is gained, atoo-large potential is imposed to the actuation system, or a too-high mechanical stress is applied. To avoid undesired phenomena, the actuator should be maintained within a proper range of temperatures, mechanical stresses, and electrical potential. A design able to counteract such undesired effects could be studied and a control system implemented. The piezoelectric effect could be implemented 5-14 MEMS: Applications S FIGURE 5.17 Bender actuator, where S is the total stroke. © 2006 by Taylor & Francis Group, LLC to sense mechanical deformations.With a very compact design, a controlled electromechanical system can be developed. This is one of the many reasons piezoelectric actuators have become so successful. 5.3 Electromagnetic Actuators 5.3.1 Electromagnetic Phenomena The research on electromagnetic phenomena and their ability to generate mechanical interactions is ancient. The first scientific results were from William Gilbert (1600), who, in 1600, published De Magnete, a treatise on the principal properties of a magnet — the presence of two poles and the attraction of oppo- site poles. In 1750, John Michell (1751) and then in 1785, Charles Coulomb (1785–1789) developed a quantitative model for these attraction forces discovered by Gilbert. In 1820, Oersted (1820) and inde- pendently, Biot and Savart (1820), discovered the mechanical interaction between an electric current and a magnet. In 1821, Faraday (1821) discovered the moment of the magnetic force. Ampere (1820) observed a magnetic equivalence of an electric circuit. In 1876, Rowland (1876) demonstrated that the magnetic effects due to moving electric charges are equivalent to the effects due to electric currents. In 1831, Michael Faraday (1832) and, independently, Joseph Henry (1831) discovered the possibility of generating an elec- tric current with a variable magnetic field. In 1865, James Clerk Maxwell (1865) developed the first com- prehensive theory on electromagnetic field, introducing the modern concepts of electromagnetic waves. Microactuators 5-15 (a) (b) (c) (d) (e) (f) (g) FIGURE 5.18 Sequential inchworm movement. Strain Field FIGURE 5.19 Hysteresis in PZT actuators. © 2006 by Taylor & Francis Group, LLC Though not considered so by his contemporaries, his theory was revolutionary, as it did not require the presence of a media, the ether, to propagate the electromagnetic field. Later, in 1887, Hertz (1887) exper- imentally demonstrated the existence of electromagnetic waves. This opened up the possibility of neglect- ing the ether and, in 1905, aided the formulation of the theory of relativity by Albert Einstein (1905). While electromagnetic physics was being studied, new advances in electromagnetic motion systems were developed. The earliest experiments were undertaken by M.H. Jacobi (1835) in 1834 (moving a boat). Though the first complete electric motor was built by Antonio Pacinotti (1865) in (1860). The first induc- tion motor was invented and analysed by G. Ferraris (1888) in 1885 and later, independently, by N. Tesla (1888), who registered a patent in the United States in 1888. Many other macro electromagnetic motors were later developed and research in this field remained very active. Research in the field of electric microactuators started in 1960 with W. McLellan, who developed a 1/64th inch cubed micromotor in answer to a challenge by R. Feynman. Since then an indefinite number of inventions and prototypes have been presented to the scientific community, patented, and marketed. Therefore, outlining the history of microelectromagnetic actuators is an almost impossible task; however, by observing the new technologies produced, we are able to trace the key inventions and ideas to the formation of actual components For example, the isotropic and anisotropic etching techniques that were developed in the 1960s generated bulk micromachining in 1982; sacrificial layer techniques also developed in the 1960s generated surface micromachining in 1985. Some more recent technologies include silicon fusion bonding, LIGA technol- ogy, micro electro, and discharge machining. 5.3.2 Properties of Electromagnetic Actuators Electromagnetic actuators can be classified according to four attributes: geometry, movement, stroke, and type of electromagnetic phenomena (Tables 5.5–5.8). We will use the Lagrange equations of motion or Newtonian equations of motions to derive the mod- els of each single type of actuator: ΄ ΅ Ϫ ϭ Q i , i ϭ 1,2… n (5.16) Α m jϭl F j (t, r) ϭ m d 2 r ᎏ dt 2 ∂(⌫ Ϫ ⌸) ᎏᎏ ∂q i ∂(⌫ ϩ D) ᎏᎏ ∂q . i d ᎏ dt 5-16 MEMS: Applications TABLE 5.5 Classification of Electromagnetic Microactuators Based on Geometry Geometry of the Actuator Planar Cylindrical Spherical Toroidal Conical Complex shape TABLE 5.6 Classification of Electromagnetic Microactuators Based on Movement Movement of the Actuator Prismatic Rotative Complex movement © 2006 by Taylor & Francis Group, LLC where the first and second equation represents, respectively, Lagrangian and Newtonian approach; the used symbols Γ, D, and Π denote, respectively, the kinetic, potential, and dissipated energies; while q i and Q i represent, respectively, the generalized coordinates and the generalized forces applied to the system. The Newtonian approach corresponds to the use of Newton’s second law of motion, where F j is a vecto- rial force applied to the system, r is the vector representing the position and the geometrical configura- tion of the system, and m is the mass of the system. The presence of rotary movement can imply the use of an analogous rotary version of Newton’s second law. Multibody systems can be also considered with the use of a simple and concise matrix method. To develop MEMS models, Γ, D, and Π, in the Lagrangian approach, and F, in the Newtonian approach, should take account of mechanical and electrical terms. We will first consider the Lagrangian approach as it is able to simply mix many forms of physical inter- actions without separating the model into many parts. If we consider elementary mechanical movement (prismatic and rotational) and elementary electric circuits, and then apply a lumped parameters model with a Lagrangian approach, we can propose the synthetic Table 5.9, to associate each elementary parameter with each kind of energy mentioned in (Equation 5.16) and select the correct generalized coordinates and forces. The proposed table is mainly outlining many other forms of energy deemed worthy of consideration, as well as other basic or complex models to be taken into account. For instance, the effect of impacts due Microactuators 5-17 TABLE 5.7 Classification of Electromagnetic Microactuators Based on the Stroke Stroke of the Actuator Limited stroke Unlimited stroke TABLE 5.8 Classification of Electromagnetic Microactuators Based on Electromagnetic Phenomena Electromagnetic Phenomena of the Actuator Direct current microactuators Induction microactuators Syncronous microactuators Stepper microactuators TABLE 5.9 Selection of Terms for Lagrange Equations of Motion Generalized Elementary Movement Terms of Lagrange Equation Elementary Lumped Parameter Prismatic mechanical movement Kinetic energy m: mass (prismatic inertial term) Potential energy k: linear stiffness Dissipative energy f: linear friction Generalized coordinate x: linear coordinate along the trajectory Generalized force F: applied force Rotational mechanical movement Kinetic energy J: moment of inertia Potential energy k: rotational stiffness Dissipative energy f: rotational friction Generalized coordinate t: rotational coordinate along the trajectory Generalized force T: applied torque Electric circuit Kinetic energy L: selfinductance M: mutual inductance Potential energy C: capacitance Thermal Dissipative energy R: resistance Generalized coordinate q: electrical charge Generalized force u: applied voltage © 2006 by Taylor & Francis Group, LLC [...]... mechanical stage, τL is the torque of the load; Jm and JL are, respectively, the inertia of the motor and the inertia of the load; r ( 1) is the mechanical reduction ratio; k and f are, respectively the stiffness (Equation 5 .17 ) and the friction (Equation 5 .19 ) coefficient of the reduction stage; θ, θR and θL are, respectively, the theoretic position of the rotor, the theoretic position after the reduction stage... (Equation 5.40) and the position of the load θ ϭ θR и r (5.40) The computation of the electric torque τe is then related to the calculation of the generated power: p ϭ eTi ϭ (Ri ϩ ϕ)Ti (5. 41) ϭ (iTRTi )1 ϩ (iTLTi)2 ϩ (iTLTi)3 where the e and i are, respectively the vector of the electric potentials and the vector of the electric currents (Equation 5.34), R is the matrix of the electric resistance... Alloys: Thermomechanical Derivation with Non-Constant Material Functions,” J Intell Mater Syst Struct., 4(2), pp 229–242 Brinson, L.C., and Huang, M.S (19 96) “Simplifications and Comparisons of Shape Memory Alloy Constitutive Models,” J Intell Mater Syst Struct 16 , pp 10 8 11 4 Buehler, W.J., and Wiley, R.C (19 65 ) “Nickel-based Alloys Technical Report,” US Patent 3 ,17 4,8 51 Chang, L.C., and Read, T.A (19 51) ... Special Issue on Micro-Machine System, (19 91) J Robotics Mechatronics Tanaka, (19 86) “Thermomechanics of Transformation Pseudoelasticity and Shape Memory Effect in Alloys,” Int J Plast., 1, pp 59–72 Tesla, N (18 88) US Patents 3 81, 96 8-3 81, 96 9-3 82, 27 9-3 90, 41 5-3 90, 820 Wise, S.A (19 98) “Displacement Properties of RAINBOW and THUNDER Piezoelectric Actuators,” Sensors Actuators A, 69 , pp 33–38 Wu, X.D.,... ϕϭLиi Ά e ϭ [es1 es2 es3 ee 0 0]T, i ϭ [is1 is2 is3 ϕ ϭ [ϕs1 ϕs2 ϕs3 ϕe ϕD ϕQ]T ie iD iQ]T, (5.45) where the vector e, i and ϕ represent, respectively, the electric potential, the electric current and the magnetic flux, the matrix R of the electric resistance and the matrix L of the inductance are defined in Equation 5. 46 The indices s1, s2, and s3 associated with each element, refer to the stator’s... obtain the electric torque τe; only the evaluation of the time derivative of the transposed matrix of the inductances is required: 2π 0 L 21 , L ϭ [i ] ᎏ θ (5.42) LT ϭ L 12 i,j i 1 3, j 1 3, ii,j ϭ Ϫ и Msr sin θ ϩ 3 (j Ϫ i) 0 12 ΄ ΅ ΄ ΅ Because the time derivative of the transposed matrix of the inductances is linearly dependent by the time derivative of the angular position of the rotor, the electric... respectively, the stator and rotor winding resistances, while I(3) and 0(3) are, respectively, the three-dimensional identity and zero matrixes L1 ,1 L1,2 (5. 36) Lϭ L L2,2 2 ,1 ΄ ΅ where the submatrixes L1 ,1 and L2,2, respectively, of statoric and rotoric inductances are defined in Equation 5.37, while the submatrixes L1,2 and L2 ,1 of mutual inductances are defined in Equation 5.38 ΄ Ls L1 ,1 ϭ Ms Ms... winding The submatrixes L1 ,1, L2,2, L1,2 and L2 ,1 are defined in Equations 5.47–5.49 Ά ΄ ΅ 4π l0 ϩ L00 ϩ L2 cos 2θ Ϫ ᎏ (i Ϫ 1) , i ϭ j 3 L1 ,1 ϭ [li,j]i 1 3, j 1 3 li,j ϭ ’ ΄ ΅ L00 4π Ϫᎏ ϩ L2 cos 2θ Ϫ ᎏ (5 Ϫ i Ϫ j) , i 2 3 (5.47) j where l0 is the statoric dispersion selfinductance, L00 is the net statoric selfinductance, and L2 is the amplitude of the second Fourier term of each self-inductance of the. .. Other examples of non-linearity in the electrical parameters can be observed in non-linear components, such as diodes Regarding the general microelectromechanical model theory, a suitable approach allows the modeling of electro-mechanical systems using a lumped parameter approach If designers consider the general relations between magnetic, electric, and mechanical fields, and referring to the Maxwell’s... Metals The Gold-Cadmium Beta Phase” Trans AIME 18 9, pp 47–52 Coulomb, C.A (17 85 17 89) “Memoires sur l’electricite et le magnetisme,” Memoires de l’Academie Royale des Science de Paris Einstein, A (19 05) “Zur Elektrodynamik bewegter Korper,” Annalen der Physik 17 , pp 8 91 9 21 Faraday, M (18 21) “On some new Electro-Magnetical Motions, and the Theory of Magnetism,” Quarterly Journal of Science 12 , pp 75–96 . stage, 2 π ᎏ 3 M r M r L r M r L r M r L r M r M r M s M s L s M s L s M s L s M s M s L 1, 2 L 2,2 L 1, 1 L 2 ,1 0(3) I(3) и R r I(3) и R s 0(3) Microactuators 5-2 3 s 1 s 2 s 3 1 s 1 s Ϫ2 s 2 s Ϫ3 s 3 s r 1 r 2 r 3 1 r 3 r Ϫ2 r 1 r Ϫ3 r 2 r s FIGURE. generating an elec- tric current with a variable magnetic field. In 18 65 , James Clerk Maxwell (18 65 ) developed the first com- prehensive theory on electromagnetic field, introducing the modern concepts. and a magnet. In 18 21, Faraday (18 21) discovered the moment of the magnetic force. Ampere (18 20) observed a magnetic equivalence of an electric circuit. In 18 76, Rowland (18 76) demonstrated that the