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

The next year, Lippman theoretically demonstrated the exis-tence of an inverse piezoelectric effect from electric energy to mechanical energy.. TABLE 5.4 Classification of Microactuators

This chapter will adopt a classification of microactuators based on the form of the input energy andwill 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 informationwill also be provided on polymeric, electrorheological, SMA polymeric, and chemical microactuators.

The discovery of the piezoelectric phenomena is due to Pierre and Jacque Curie, who experimentallydemonstrated 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 thengave 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 allthe 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 submarineultrasonic detector, developed by Lengevin and French in 1917 Between the first and second World Warsmany 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 duringthe second half of the twentieth century with the development of barium titanate and lead zirconate titanatepiezoceramics 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 lowcost applications and the exploitation of a new design approach (piezocomposite structures, polymericmaterials, new geometries, etc.) to develop new classes of sensors and, especially new classes of actuators

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

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 ashortening, 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 ormoment 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 (Figure5.13) The most frequently used piezoelectric materials are piezoceramics such as PZT, a polycrystallineferroelectric 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 (Figure5.14); 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, theytransform their structure, becoming symmetric and loosing their piezoelectric ability

Common piezoelectric materials are piezoceramics such as lead zirconate titanate (PZT) and tric polymers such as polyvinylidene fluoride (PVDF) To improve the performance of piezoceramics, the

research proposed new formulations, PZN–PT and PMN–PT These new formulations extend the strainfrom 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 structuresand 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 themechanical strain in the piezoelectric media:

S ij  s E ijkl T kl  d ijk E k

, i, j, k, l ∈ 1, 2, 3 (5.3)

D i  d ijk T jk εTijE 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 Eis the fourth order tensor of the elastic compliance, εTis 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  s E I,J T J  d I,j E j

g and the β matrix:

S I  s I,J D T J  g I,j E j

, i, j ∈ 1, 2, 3; I, J ∈ 1, 2 … 6 (5.5)

E i  g i,j T J βT

i,jDj

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, ing to the selection of dependent and independent variables:

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 imation consists of the hypothesis of low variation of applied voltage and the resulting strain If the con-

approx-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, 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

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)

where ρ is the density of the material and F is the resulting internal force reduced to a surface force (with

the divergence theorem)

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 oftrigonometric 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)

“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 piezoelectricmatrix of the “real” piezoceramic to take into account some unconsidered phenomena

∂S



∂t

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 Thissection 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 characterized by greater 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—fromsimple forms, such as rectangular patches or thin disks, to custom very complex shapes Because of thereduced displacements, piezoelectric materials are not usually used directly to generate a motion, but areconnected to the user by a transmission element (Figure 5.2) Therefore, the piezoelectric actuators arenot 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, theprimary design parameters of a piezoelectric actuator (referring to the entire actuating machine, not onlythe 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 offbetween these three parameters; the actuator architecture is devoted to increment one or two of these

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.

parameters at the cost of the other parameters Piezoelectric actuators are characterized by noticeableexerted forces and high frequencies, but also significantly reduced strokes; therefore the architecturedesigns 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, soanother 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 thesame 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 force exerted by a single mentary layer The leverage effect on the stroke is linearly proportional to the ratio between the elemen-tary piezoelectric length and the actuator length

ele-The most common stack architectures can gain some microns stroke, exerting some kilonewtonsforces, with about ten microseconds time responses

Bender actuators consist of two or more layers of PZT materials subjected to electric potential ences, which induce opposite strain on the layers (Figure 5.17) The opposite strains cause a flexion of thebender, due to the induced internal moment in the structure This architecture is able to generate anamplification of the stroke as a quadratic function of the length of the actuator, resulting in a stroke of

more than one millimeter Different configurations of bender actuators are available, such as end ported, cantilever, and many other configurations with different design tricks to improve stability orhomogeneity of the movement.

sup-Unimorph actuators are a special class of bender actuators, which are composed of a PZT layer and anon-active host Two common unimorph architectures are: Rainbow, developed by Heartling (1994) andThunder, 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 ability to combine theelementary blocks in series or parallel configurations to form an arrayed actuation system with improvedstroke 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 Leverarm actuators are machines composed of an elementary actuator and a transmission able to amplify thestroke and reduce the generated force The transmission utilized is a leverage mechanism or a multistageleverage system To reduce design complexity, the leverage system is generally composed of two simpleelements: a thin and flexible member (the fulcrum) and a thicker, more rigid, long element (the leveragearm) 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 a reduced sec-tion The result is a very high stroke amplification (approximately 100 times); however, this amplificationinvolves some problems due to the presence of fluids, microfluidic phenomena, and high frequencymechanical waves that are transmitted to the fluids The third class of externally amplified actuators isflextensional actuators This class is characterized by the presence of a flexible component with a propershape, able to amplify displacement It differs from the lever arm actuators approach, because of itsclosed-loop configuration, resulting in a higher stiffness but reduced amplification power To incrementthe stroke amplification, this class of actuators can be used in a building-block architecture A typicalexample is a stack of Moonie actuators The research on actuation architecture is very dynamic Newdesign solutions emerge in literature and on the market frequently; therefore it would be improper togeneralize 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 arebasically reducible to inchworm systems (Figure 5.18) In general, they are composed of three or moreactuators, alternatively contracting, to simulate an inchworm movement The resulting system is a veryprecise actuator, with very high stroke (more than 10 mm), but with a reduced natural frequency.The behavior of PZT actuators can be affected by undesired physical phenomena such as hysteresis Infact, 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 precisionand 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 isgained, a too-large potential is imposed to the actuation system, or a too-high mechanical stress isapplied To avoid undesired phenomena, the actuator should be maintained within a proper range oftemperatures, mechanical stresses, and electrical potential A design able to counteract such undesiredeffects could be studied and a control system implemented The piezoelectric effect could be implemented

S

FIGURE 5.17 Bender actuator, where S is the total stroke.

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

5.3.1 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 site poles In 1750, John Michell (1751) and then in 1785, Charles Coulomb (1785–1789) developed aquantitative 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

oppo-a moppo-agnet In 1821, Foppo-aroppo-adoppo-ay (1821) discovered the moment of the moppo-agnetic force Ampere (1820) observed

a magnetic equivalence of an electric circuit In 1876, Rowland (1876) demonstrated that the magnetic effectsdue to moving electric charges are equivalent to the effects due to electric currents In 1831, MichaelFaraday (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

Though not considered so by his contemporaries, his theory was revolutionary, as it did not require thepresence 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 weredeveloped 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 motorswere later developed and research in this field remained very active Research in the field of electricmicroactuators started in 1960 with W McLellan, who developed a 1/64th inch cubed micromotor inanswer to a challenge by R Feynman Since then an indefinite number of inventions and prototypes havebeen presented to the scientific community, patented, and marketed Therefore, outlining the history ofmicroelectromagnetic actuators is an almost impossible task; however, by observing the new technologiesproduced, we are able to trace the key inventions and ideas to the formation of actual components Forexample, the isotropic and anisotropic etching techniques that were developed in the 1960s generatedbulk micromachining in 1982; sacrificial layer techniques also developed in the 1960s generated surfacemicromachining 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, andtype of electromagnetic phenomena (Tables 5.5–5.8)

We will use the Lagrange equations of motion or Newtonian equations of motions to derive the els of each single type of actuator:

Cylindrical Spherical Toroidal Conical Complex shape

TABLE 5.6 Classification of Electromagnetic Microactuators Based

on Movement Movement of the Actuator Prismatic

Rotative Complex movement

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 iand

Q irepresent, 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 jis a

vecto-rial force applied to the system, r is the vector representing the position and the geometrical tion of the system, and m is the mass of the system The presence of rotary movement can imply the use

configura-of an analogous rotary version configura-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 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

inter-a Linter-agrinter-angiinter-an inter-approinter-ach, we cinter-an propose the synthetic Tinter-able 5.9, to inter-associinter-ate einter-ach elementinter-ary pinter-arinter-ameter witheach 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

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

to backlash between mechanical components can be considered introducing stiffness function, which aregoverned by the distance between two consecutive components of the kinematical chain The resultingelastic force can be evaluated with the product of the stiffness function and the distance between the twocomponents We consider this function as constant or polynomial function of the distance between thetwo components; however, it assumes the value of zero when the absolute distance between the two con-sidered consecutive components is less than a half of the backlash:

where k12is the modified stiffness function, x is the generalized position of a mechanical component of the microactuator, y is the generalized position of the consecutive mechanical component, g is the back- lash, and k12is the real stiffness of the junction between the two adjacent components (Figure 5.20)

In a similar way, the backlash can also be considered in the definition of the dissipative forces betweentwo adjacent components The definition of the friction is a more complex problem, because of the pres-ence of different type of frictions Three important friction phenomena should be considered: Coulombfriction, viscous friction, and static friction:

F⬅冢f v  f sexp冢ϕ冨ξ.

where f v , f s , f c and ϕ are optimal parameters, while ξ ⬅ x  y is the relative position of two adjacent

mechanical components The presence of backlash introduces a discontinuity in the frictional parameters(Equation 5.19), and the situation can radically increment the complexity of the frictional model

F⬅冦f v(l  ∆1δ(ξ))  f s(l  ∆2δ(ξ))exp冤ϕ(1  ∆3δ(ξ))冨ξ.冨冥 f c(1  ∆4δ(ξ))冨ξ.冨冧 ,

where ∆1, ∆2, ∆3, and ∆4are optimal parameters, which take into account the variation of frictional effects

in the “backlash-disconnected” joint Many other observations can also be introduced to model a ular phenomenon such as the hysteretic behavior of general mechanical systems, but these remarks will

partic-be considered only for shape memory microactuators An example for electrical parameters is thedependence of resistivity based on temperature (Equation 5.20) In fact, a fluctuation in the temperaturerange due to unexpected phenomena and changes in internal temperature caused by electrical powerfeeding and Joule effect affect the resistivity of a microelectromechanical system, and its related resist-ance, thus altering its functionality

where ρ is the resistivity, ρ0is the reference resistivity, αi is the temperature dependence coefficient, T is the temperature of the resistor, and T 0is the reference temperature Other examples of non-linearity inthe 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 tions between magnetic, electric, and mechanical fields, and referring to the Maxwell’s equations, theycould eventually apply a finite element approach to improve the model’s power:

rela-∇oE  0

∇oB  0

∇ B  µε

5.3.3 DC Mini- and Microactuators

Because of thermal problems related with brush friction, DC microactuators are not the most viable choicewithin the class of electromagnetic microactuators To develop a complete model of the actuator, one canselect a cylindrical geometry, having a rotational movement and an unlimited stroke One can also con-sider a single conductive coil, which is turned on a cylindrical rotor, able to rotate around its axis The coil

is also embedded in a magnetic field B and it is fed with an electric current as in Figure 5.21 Due to theaction of the magnetic field on the electric current, the coil is affected by the mechanical couple:

where τ is the mechanically generated couple, A is the area vector perpendicular to the coil, and B is the

magnetic field intensity We can now consider a set of n coils equally spaced on the rotor, which are

affected by a set of p magnetic fields perpendicular to the rotor axis The same current i is in all the coils,

and the electric potential of the rotor is applied to the coils by brushes, which are used to change thedirection of the current every half turn of the rotor The rotor then, is subject to the force computed by:

When the number of coils, and/or the number of magnetic fields, is sufficiently high, the mechanicaltorque expression can be further simplified:

Therefore, under the assumption that the magnetic field is generated by the stator’s current, the magnetictorque can be expressed as the product of a constant term and two electric variables: the rotor’s and thestator’s currents:

To obtain the value of the angular speed of the rotor, we assume the temporary disregard of tions and other “parasitic phenomena” due to their effect, only before and after the transformation fromelectric to mechanical energy For this reason, all the input electrical energy from the rotor is convertedinto mechanical energy and the angular speed of the rotor can be expressed:

dissipa-ω

where e is the rotor’s electrical potential With the aid of the transformation equations (Equation

5.25–5.26), the microactuator behavior can be described from a standpoint of circuit’s approach as inFigure 5.23

2p

π

FIGURE 5.22 Scheme of a DC microactuator, where B is the magnetic field intensity, i is the rotor’s current, and i s

is the stator’s current used to generate the magnetic field.

The symbols mentioned in Figure 5.23 are summarized as follows:

(s) Stator’s circuit

v s is the stator’s electric potential

i s is the stator’s current

R s and L sare the resistance and the inductance of the stator

(r) Rotor’s circuit

v r is the rotor’s electric potential

i r is the rotor’s current

R r and L rare the resistance and the inductance of the rotor

e r is the rotor’s potential that linked with the rotor’s current, is transformed in mechanical power whichvalue is related with the stator’s current and with the theoretical motion speed:

e is the theoretic position of the rotor

J m is the motor’s inertia

J L is the load’s inertia

f is the reduced friction coefficient of the system as described in Equation 5.19

k is the reduced stiffness coefficient of the system as described in Equation 5.17

g is the reduced backlash coefficient of the system

where r (1) is the mechanical reduction ratio With a simple circuit analysis, a lumped parameter model

of the microactuator can be obtained (Equation 5.30):

parameters of the model are not able to reach the desired predictive precision, more complex modelscould be considered in order to take into account neglected phenomena Considering the introduction offictitious mechanical compliances or electrical parasitic phenomena could be followed by a calibrationprocedure ruled directly by real experimental results or by a finite element model Should a model not beimplemented in a controlling algorithm, a finite element model can be directly used to provide good pre-dictive result Some useful finite element software are available on the market to achieve a proper model,thus creating new finite element software is not necessary.

5.3.4 Induction Mini- and Microactuators

Induction actuators consist of a mobile and a static part and the transformation from electric to ical energy due to the inductance of each part of the microactuator This section will address a cylindricalrotary actuator with unlimited stroke and, to simplify the analysis, only a common configuration is con-sidered: a three phase, two pole actuation system Figure 5.24 shows a rotary induction motor The mobilepart (the rotor) has a cylindrical shape and is able to rotate around its axis; the static part (the stator) hasthe same axis as the rotor and is separated from it by an air gap Both are composed of ferromagneticmaterial and incorporate lengthwise holes carrying conductive wires that are close to the air gap

mechan-To develop a mathematical model of the dynamic behavior of the considered microactuator, it is essary to define a reference system and an angular variable, which identify the position of the rotor

respect to the first stator’s winding (s 1):

θ

where θ is indicated in Figure 5.25, as the angle between s 1 and r 1 , and f is the number of phases The

dynamic model can be generalized as a multi-pole micromachine for which the ideal actuator speed can

be calculated by:

θ

where p is the (even) number of poles, θ is indicated in Figure 5.25, and θ a is the ideal position of theactuator Therefore, the dynamic electromagnetic behavior of the system can be described by the matrixequation:

FIGURE 5.24 Physical scheme of rotary induction actuator s, r, and g represent, respectively, the stator, the rotor,

and the gap.