Bichop R.H. (Ed.) Mechatronic Systems, Sensors, and Actuators: Fundamentals and Modeling

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15-4 Mechatronic Systems, Sensors, and Actuators

TABLE 15.1 Classification of Electromagnetic Microdevices Using the Electromagnetic

System–Geometry Classifier

FIGURE 15.1 Permanent-magnet synchronous micromachines with different geometry.

Geometry

Plate, P Spherical, S Torroidal, T Conical, N Cylindrical, C

Endless (closed), E

Open-Ended

(open),

Asymmetrical, A

Integrated,

Electromagnetic system

Endless electromagnetic system

and cylindrical geometry

Endless electromagnetic system

and conical geometry

Endless electromagnetic system

and spherical geometry

Stator

Rotor

Stator

Rotor

Stator

Rotor

Stator

Rotor

Stator

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Rotational and Translational Microelectromechanical Systems 15-5

into three horizontal and six vertical strips, and contains 18 sections, each identified by ordered pairs of

characters, such as (E, P) or (O, C).

In each ordered pair, the first entry is a letter chosen from the bounded electromagnetic system set

The second entry is a letter chosen from the geometric set

That is, for electromagnetic microdevices, the electromagnetic system–geometric set is

In general, we have

Other categorization can be applied. For example, single-, two-, three-, and multi-phase microdevices

are classified using a phase classifier

Therefore, Y × M × G × H = {(y, m, g, h) : y ∈ Y, m ∈ M, g ∈ G and h ∈ H}

Topology (radial or axial), permanent magnets shaping (strip, arc, disk, rectangular, triangular, or

other shapes), permanent magnet characteristics (BH demagnetization curve, energy product, hysterisis

minor loop), commutation, emf distribution, cooling, power, torque, size, torque-speed characteristics,

as well as other distinct features of microdevices can be easily classified.

That is, the devised electromagnetic microdevices can be classified by an N-tuple as

{microdevice type, electromagnetic system, geometry, topology, phase, winding, connection, cooling}.

Using the classifier, which is given in Table 15.1 in terms of electromagnetic system–geometry, the

designer can classify the existing motion microdevices as well as synthesize novel high-performance

microdevices. As an example, the spherical, conical, and cylindrical geometries of a two-phase permanent-

magnet synchronous microdevice are illustrated in Figure 15.2.

This section documents new results in structural synthesis which can be used to optimize the microde-

vice performance. The conical (existing) and spherical-conical (devised) microdevice geometries are

illustrated in Figure 15.2. Using the innovative spherical-conical geometry, which is different compared

to the existing conical geometry, one increases the active length L

and average diameter D

. For radial

flux microdevices, the electromagnetic torque T

is proportional to the squared rotor diameter and axial

length. In particular, where k

is the constant. From the above relationship, it is evident

FIGURE 15.2 Two-phase permanent-magnet synchronous microdevice (micromachine) geometry.

ME, O, I{}=

GP, S, T, N, C, A{}=

MG× E, F(), E, S(), E, T(),…, I, N(), I, C(), I, A(){}=

MG× m, g():mM and ∈ gG∈{}=

Hh: hH∈{}=

Cylindrical geometry

Rotor

Stator

Rotor

Stator

Rotor

Stator

Rotor

Stator

Conical geometry

Spherical geometry

Stator

Rotor

Stato

Rotor

Stator

Stato

Rotor

Stator

Spherical-conical

geometry

Endless electromagnetic system

Assymetrical geometr

Stator

Roto

Rotor

Stator

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15-6 Mechatronic Systems, Sensors, and Actuators

that the spherical-conical micromotors develop higher electromagnetic torque compared with the con-

ventional design. In addition, improved cooling, reduced undesirable torques components, as well as

increased ruggedness and robustness contribute to the viability of the proposed solution. Thus, using

the classifier paradigm, novel microdevices with superior performance can be devised.

15.3 MEMS Fabrication

Microelectromechanics, which integrates micromechanics and microelectronics, requires affordable, low-

cost, high-yield fabrication technologies which allow one to fabricate 3-D microscale structures and

devices. Micromachining is a key fabrication technology for microscale structures, devices, and MEMS.

Microelectromechanical systems fabrication technologies fall into three broad categories: bulk machining,

surface machining, and LIGA (LIGA-like) techniques [1–3].

15.3.1 Bulk Micromachining

Bulk and surface micromachining are based on the modified CMOS and specifically designed microma-

chining processes. Bulk micromachining of silicon uses wet and dry etching techniques in conjunction

with etch masks and etch-stop-layers to develop microstructures from the silicon substrate. Microstruc-

tures are fabricated by etching areas of the silicon substrate to release the desired 3-D microstructures.

The anisotropic and isotropic wet etching processes, as well as concentration dependent etching techniques,

are widely used in bulk micromachining. The microstructures are formed by etching away the bulk of

the silicon wafer to fabricate the desired 3-D structures. Bulk machining with its crystallographic and

dopant-dependent etch processes, when combined with wafer-to-wafer bonding, produces complex 3-D

microstructures with the desired geometry. Through bulk micromachining, one fabricates microstruc-

tures by etching deeply into the silicon wafer. There are several ways to etch the silicon wafer. The

anisotropic etching uses etchants that etch different crystallographic directions at different rates. Through

anisotropic etching, 3-D structures (cons, pyramids, cubes, and channels into the surface of the silicon

wafer) are fabricated. In contrast, the isotropic etching etches all directions in the silicon wafer at same

(or close) rate, and, therefore, hemisphere and cylinder structures can be made. Deep reactive ion etching

uses plasma to etch straight walled structures (cubes, rectangular, triangular, etc.).

15.3.2 Surface Micromachining

Surface micromachining has become the major fabrication technology in recent years because complex

3-D microscale structures and devices can be fabricated. Surface micromachining with single-crystal

silicon, polysilicon, silicon nitride, silicon oxide, and silicon dioxide (as structural and sacrificial materials

which deposited and etched) is widely used to fabricate microscale structures and devices on the surface

of a silicon wafer. This affordable low-cost high-yield technology is integrated with IC fabrication

processes guaranteeing the needed microstructures-IC fabrication compatibility. The techniques for

depositing and patterning thin films are used to produce complex microstructures and microdevices on

the surface of silicon wafers (surface silicon micromachining) or on the surface of other substrates. Surface

micromachining technology allows one to fabricate the structure as layers of thin films. This technology

guarantees the fabrication of 3-D microdevices with high accuracy, and the surface micromachining can

be called a thin film process. Each thin film is usually limited to thickness up to 5

m, which leads to

fabrication of high-performance planar-type microscale structures and devices. The advantage of surface

micromachining is the use of standard CMOS fabrication processes and facilities, as well as compliance

with ICs. Therefore, this technology is widely used to manufacture microscale actuators and sensors

(microdevices).

Surface micromachining is based on the application of sacrificial (temporary) layers that are used to

maintain subsequent layers and are removed to reveal (release) fabricated (released or suspended) micro-

structures. This technology was first demonstrated for ICs and applied to fabricate microstructures in

the 80s. On the surface of a silicon wafer, thin layers of structural and sacrificial materials are deposited

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Rotational and Translational Microelectromechanical Systems 15-7

and patterned. Then, the sacrificial material is removed, and a micromechanical structure or device is

fabricated. Figure 15.3 illustrates a typical process sequence of the surface micromachining fabrication

technology.

Usually, the sacrificial layer is made of silicon dioxide (SiO

), phosphorous-doped silicon dioxide, or

silicon nitride (Si

). The structural layers are then typically formed with polysilicon, and the sacrificial

layer is removed. In particular, after fabrication of the surface microstructures and microdevices (micro-

machines), the silicon wafer can be wet bulk etched to form cavities below the surface components, which

allows a wider range of desired motion for the device. The wet etching can be done using hydrofluoric

and buffered hydrofluoric acids, potassium hydroxide, ethylene-diamene-pyrocatecol, tetramethylam-

monium hydroxide, or sodium hydroxide. Surface micromachining technology was used to fabricate

rotational micromachines [6]. For example, heavily-phosphorous-doped polysilicon can be used to

fabricate rotors and stators, and silicon nitride can be applied as the structural material to attain electrical

insulation. The cross-section of the slotless micromotor fabricated on the silicon substrate with polysilicon

stator with deposited windings, polysilicon rotor with deposited permanent-magnets, and bearing is

illustrated in Figure 15.4. The micromotor is controlled by the driving/sensing and controlling/processing

ICs. To fabricate micromotor and ICs on a single- or double-sided chip (which significantly enhances

the performance), similar fabrication technologies and processes are used, and the compatibility issues

are addressed and resolved. The surface micromachining processes were integrated with the CMOS

technology (e.g., similar materials, lithography, etching, and other techniques). To fabricate the integrated

MEMS, post-, mixed-, and pre-CMOS/micromachining techniques can be applied [1–3].

15.3.3 LIGA and LIGA-Like Technologies

There is a critical need to develop the fabrication technologies allowing one to fabricate high-aspect-

ratio microstructures. The LIGA process, which denotes Lithography–Galvanoforming–Molding (in

German words, Lithografie–Galvanik–Abformung), is capable of producing 3-D microstructures of up

to centimeter high with the aspect ratio (depth versus lateral dimension) more than 100 [2,7,8]. The

LIGA technology is based upon X-ray lithography, which guarantees shorter wavelength (in order from

FIGURE 15.3 Surface micromachining.

FIGURE 15.4 Cross-section schematics for slotless permanent-magnet brushless micromotor with ICs.

1. Deposition and patterning

of the sacrificial layer

Silicon substrate

Sacrificial layer

2. Deposition and patterning

of the structural layer

Silicon substrate

Sacrificial layerSacrificial layer

Structural layer

3. Etching of the sacrificial

layer

Structural layer

Silicon substrate

Micromachined structure

Silicon substrate

Stator

Rotor

Bearing post

Bearing flange

Permanent

magnet

Stator

windings

Rotor

Stator

Insulating

ICs

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15-8 Mechatronic Systems, Sensors, and Actuators

few to 10 Å, which leads to negligible diffraction effects) and larger depth of focus compared with optical

lithography. The ability to fabricate microstructures and microdevices in the centimeter range is partic-

ularly important in the actuators and drives applications since the specifications are imposed on the

rated force and torque developed by the microdevices, and due to the limited force and torque densities,

the designer faces the need to increase the actuator dimensions.

15.4 MEMS Electromagnetic Fundamentals and Modeling

The MEMS classifier, structural synthesis, and optimization were reported in Section 15.2. The classifica-

tion and optimization are based on the consideration and synthesis of the electromagnetic system, analysis

of the magnetomotive force, design of the MEMS geometry and topology, and optimization of other

quantities. Different rotational (radial and axial) and translational motion microdevices are classified

using endless (closed), open-ended (open), and integrated electromagnetic systems.

Our goal is to approach and solve a wide range of practical problems encountered in nonlinear design,

modeling, analysis, control, and optimization of motion microstructures and microdevices with driving/

sensing circuitry controlled by ICs for high-performance MEMS. Studying MEMS, the emphases are

placed on:

•

Design of high-performance MEMS through devising innovative motion microdevices with radi-

ating energy microdevices, microscale driving/sensing circuitry, and controlling/signal processing

ICs

•

Optimization and analysis of rotational and translation motion microdevices

•

Development of high-performance signal processing and controlling ICs for microdevices devised

•

Development of mathematical models with minimum level of simplifications and assumptions in

the time domain

•

Design of optimal robust control algorithms

•

Design of intelligent systems through self-adaptation, self-organization, evolutionary learning,

decision-making, and intelligence

•

Development of advanced software and hardware to attain the highest degree of intelligence,

integration, efficiency, and performance

In this section, our goal is to perform nonlinear modeling, analysis, and simulation. To attain these

objectives, we apply the MEMS synthesis paradigm, develop nonlinear mathematical models to model

complex electromagnetic-mechanical dynamics, perform optimization, design closed-loop control sys-

tems, and perform data-intensive analysis in the time domain.

To model electromagnetic motion microdevices, using the magnetic vector and electric scalar potentials

and V, respectively, one usually solves the partial differential equations

using finite element analysis. Here,

, and

are the permeability, conductivity, and permittivity.

However, to design electromagnetic MEMS as well as to perform electromagnetic–mechanical analysis

and optimization, differential equations must be solved in the time domain. In fact, basic phenomena

cannot be comprehensively modeled, analyzed, and assessed applying traditional finite element analysis,

which gives the steady-state solutions and models. There is a critical need to develop the modeling tools

that will allow one to augment nonlinear electromagnetics and mechanics in a single electromagnetic–

mechanical modeling core to attain high-fidelity analysis with performance assessment and outcome

prediction.

Operating principles of MEMS are based upon electromagnetic principles. A complete electromagnetic

model is derived in terms of five electromagnetic field vectors. In particular, three electric field vectors

∇

– A

µσ

∂

-------

µε

∂

---------

µσ

∇V–=

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Rotational and Translational Microelectromechanical Systems 15-9

and two magnetic field vectors are used. The electric field vectors are the electric field intensity, , the

electric flux density, , and the current density, . The magnetic field vectors are the magnetic field

intensity and the magnetic field density . The differential equations for microelectromechanical

motion device are found using Maxwell’s equations, constitutive (auxiliary) equations, and classical

mechanics.

Maxwell’s partial differential equations in the - and -domain in the point form are

where

is the permittivity,

is the permeability,

is the conductivity, and

is the volume charge density.

The constitutive (auxiliary) equations are given using the permittivity

, permeability tensor

, and

conductivity

. In particular, one has

The Maxwell’s equations can be solved using the boundary conditions on the field vectors. In two-

region media, we have

where is the surface current density vector, is the surface normal unit vector at the boundary from

region 2 into region 1, and is the surface charge density.

The constitutive relations that describe media can be integrated with Maxwell’s equations, which relate

the fields in order to find two partial differential equations. Using the electric and magnetic field intensities

and to model electromagnetic fields in MEMS, one has

The following pair of homogeneous and inhomogeneous wave equations

∇ Ex, yzt,,()×

∂

Hx, yzt,,()

∂

-------------------------------

–=

∇ Hx, yzt,,()×

∂

Ex, yzt,,()

∂

------------------------------

Jx, yzt,,()+

∂

Ex, yzt,,()

∂

------------------------------

Ex, yzt,,()+==

∇ Ex, yzt,,()⋅

x, yzt,,()

-----------------------------

∇ Hx, yzt,,()⋅ 0=

E or D

EP+==

H or B

HM+()==

E or J

–()× 0, = a

–()× J

= a

–()⋅

,= a

–()⋅ 0=

∇∇E×()×∇∇E⋅()∇

– E −

∂

------

∂

----------

–

µσ

∂

------

–

µε

∂

---------

–===

∇∇H×()×∇∇H⋅()∇

– H

µσ

∂

-------

–

µε

∂

----------

–==

∇

µσ

∂

------

–

µε

∂

---------

– ∇

-----

∇

µσ

∂

-------

–

µε

∂

----------

– 0=

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15-10 Mechatronic Systems, Sensors, and Actuators

is equivalent to four Maxwell’s equations and constitutive relations. For some cases, these two equations

can be solved independently. It must be emphasized that it is not always possible to use the boundary

conditions using only and , and thus, the problem not always can be simplified to two electromagnetic

field vectors. Therefore, the electric scalar and magnetic vector potentials are used. Denoting the magnetic

vector potential as and the electric scalar potential as V, we have

The electromagnetic field is derivative from the potentials. Using the Lorentz equation

the inhomogeneous vector potential wave equation to be solved is

To model motion microdevices, the mechanical equations must be used, and Newton’s second law is

usually applied to derive the equations of motion.

Using the volume charge density

, the Lorenz force, which relates the electromagnetic and mechanical

phenomena, is found as

The electromagnetic force can be found by applying the Maxwell stress tensor method. This concept

employs a volume integral to obtain the stored energy, and stress at all points of a bounding surface can

be determined. The sum of local stresses gives the net force. In particular, the electromagnetic stress is

The electromagnetic stress energy tensor (the second Maxwell stress tensor) is

In general, the electromagnetic torque developed by motion microstructures is found using the elec-

tromagnetic field. In particular, the electromagnetic stress tensor is given as

∇ A× B

H== and E

∂

-------– ∇V–=

∇ A⋅

∂

-------–=

∇

– A

µσ

∂

-------

µε

∂

---------

µσ

∇V–=

Ev+ B×()

EJB×+==

EJ+ B×()vd

---

αβ

↔

ds⋅

αβ

↔

0 E

– 0 B

–

– B

– 0 B

– B

– 0

– E

–

– B

–

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Rotational and Translational Microelectromechanical Systems 15-11

For the Cartesian, cylindrical, and spherical coordinate systems, which can be used to develop the

mathematical model, we have

Maxwell’s equations can be solved using the M

ATLAB environment.

In motion microdevices, the designer analyzes the torque or force production mechanisms.

Newton’s second law for rotational and translational motions is

where

and

are the angular velocity and displacement, v and x are the linear velocity and displacement,

is the net torque, is the net force, J is the equivalent moment of inertia, and m is the mass.

15.5 MEMS Mathematical Models

The problems of modeling and control of MEMS are very important in many applications. A mathe-

matical model is a mathematical description (in the form of functions or equations) of MEMS, which

integrate motion microdevices (microscale actuators and sensors), radiating energy microdevices, micro-

scale driving/sensing circuitry, and controlling/signal processing ICs. The purpose of the model devel-

opment is to understand and comprehend the phenomena, as well as to analyze the end-to-end behavior.

To model MEMS, advanced analysis methods are required to accurately cope with the involved highly

complex physical phenomena, effects, and processes. The need for high-fidelity analysis, computationally-

efficient algorithms, and simulation time reduction increases significantly for complex microdevices, restrict-

ing the application of Maxwell’s equations to problems possible to solve. As was illustrated in the previous

section, nonlinear electromagnetic and energy conversion phenomena are described by the partial differ-

ential equations. The application of Maxwell’s equations fulfills the need for data-intensive analysis capa-

bilities with outcome prediction within overall modeling domains as particularly necessary for simulation

and analysis of high-performance MEMS. In addition, other modeling and analysis methods are applied.

The lumped mathematical models, described by ordinary differential equations, can be used. The process

of mathematical modeling and model development is given below.

The first step is to formulate the modeling problem:

•

Examine and analyze MEMS using a multilevel hierarchy concept, develop multivariable input-

output subsystem pairs, for example, motion microstructures (microscale actuators and sensors),

radiating energy microdevices, microscale circuitry, ICs, controller, input/output devices.

•

Understand and comprehend the MEMS structure and system configuration.

•

Gather the data and information.

•

Develop input-output variable pairs, identify the independent and dependent control, disturbance,

output, reference (command), state and performance variables, as well as events.

= , E

, E

,==D

= , D

, D

,==

= , H

, H

,==B

= , B

, B

= , E

, E

,==D

= , D

, D

,==

= , H

, H

,==B

= , B

, B

= , E

, E

,==D

= , D

, D

,==

= , H

, H

,==B

= , B

, B

---------

--------

-----

----

------

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15-12 Mechatronic Systems, Sensors, and Actuators

•

Making accurate assumptions, simplify the problem to make the studied MEMS mathematically

tractable (mathematical models, which are the idealization of physical phenomena, are never

absolutely accurate, and comprehensive mathematical models simplify the reality to allow the

designer to perform a thorough analysis and make accurate predictions of the system performance).

The second step is to derive equations that relate the variables and events:

•

Define and specify the basic laws (Kirchhoff, Lagrange, Maxwell, Newton, and others) to be used

to obtain the equations of motion. Mathematical models of electromagnetic, electronic, and

mechanical microscale subsystems can be found and augmented to derive mathematical models

of MEMS using defined variables and events.

•

Derive mathematical models.

The third step is the simulation, analysis, and validation:

•

Identify the numerical and analytic methods to be used in analysis and simulations.

•

Analytically and/or numerically solve the mathematical equations (e.g., differential or difference

equations, nonlinear equations, etc.).

•

Using information variables (measured or observed) and events, synthesize the fitting and mis-

match functionals.

•

Verify the results through the comprehensive comparison of the solution (model input-state-

output-event mapping sets) with the experimental data (experimental input-state-output-event

mapping sets).

•

Calculate the fitting and mismatch functionals.

•

Examine the analytical and numerical data against new experimental data and evidence.

If the matching with the desired accuracy is not guaranteed, the mathematical model of MEMS must be

refined, and the designer must start the cycle again.

Electromagnetic theory and classical mechanics form the basis for the development of mathematical

models of MEMS. It was illustrated that MEMS can be modeled using Maxwell’s equations and torsional-

mechanical equations of motion. However, from modeling, analysis, design, control, and simulation

perspectives, the mathematical models as given by ordinary differential equations can be derived and used.

Consider the rotational microstructure (bar magnet, current loop, and microsolenoid) in a uniform

magnetic field, see Figure 15.5. The microstructure rotates if the electromagnetic torque is developed.

The electromagnetic field must be studied to find the electromagnetic torque.

The torque tends to align the magnetic moment with and

FIGURE 15.5 Clockwise rotation of the motion microstructure.

m B,

TmB×=

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Rotational and Translational Microelectromechanical Systems 15-13

For a microstructure with outside diameter D

, the magnet strength is Q. Hence, the magnetic moment

is m = QD

, and the force is found as F = QB.

The electromagnetic torque is

Using the unit vector in the magnetic moment direction one obtains

For a current loop with the area A, the torque is found as

For a solenoid with N turns, one obtains

As the electromagnetic torque is found, using Newton’s second law, one has

where is the load torque.

The electromotive (emf ) and magnetomotive (mmf ) forces can be used in the model development.

We have

and

For preliminary design, it is sufficiently accurate to apply Faraday’s or Lenz’s laws, which give the

electromotive force in term of the time-varying magnetic field changes. In particular,

where is the transformer term.

The total flux linkages are

where N

is the number of turns and Φ

is the flux per pole.

For radial topology micromachines, we have

T 2F

sin QD

sin mB

sin===

TmB× a

mB× QD

B×== =

TmB× a

mB× iAa

B×== =

TmB× a

mB× iANa

B×== =

---------

–(),==

--------

emf Edl⋅

B×()dl⋅

∂

------

–==

motional induction

generation

transformer induction

mmf Hdl⋅

Jds⋅

∂

-------

ds+==

emf

-------–

∂ψ

∂

-------–

∂ψ

∂θ

-------

–

--------

∂ψ

∂

-------–

∂ψ

∂θ

-------

–

== =

∂ψ

∂

---- ---

------------

in st

= L

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