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

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21-122 Mechatronic Systems, Sensors, and Actuators

21.5.8.1 Electromagnetic System Modeling in Microactuators with Permanent

Magnets: High-Fidelity Modeling and Analysis

In this section we focus our efforts to derive the expanded equations for the electromagnetic torque and

force on cylindrical permanent-magnet thin films, see Figure 21.135. The permanent-magnet thin film is

assumed to be uniformly magnetized and the equations are developed for two orientations of the mag-

netization vector (the orientation is parallel to the axis of symmetry, and the orientation is perpendicular

to this axis). Electromagnetic fields and gradients produced by the planar windings should be found at a

point in inertial space, which coincides with the origin of the permanent-magnet axis system in its initial

alignment. Our ultimate goal is to control microactuators, and thus, high-fidelity mathematical models

(which will result in viable analysis, control, and optimization) must be derived. To attain our objective,

the complete equations for the electromagnetic torque and force on a cylindrical permanent-magnet thin

films are found.

The following notations are used: A, R, and l are the area, radius, and length of the cylindrical permanent

magnet; B is the magnetic flux density vector; B

is the expanded magnetic flux density vector; [

∂

B] is

the matrix of field gradients [T/m]; [

∂

] is the matrix of expanded field gradients [T/m]; F and T are

the total force and torque vectors on the permanent-magnet thin film; i is the current in the planar

microwinding; m is the magnetic moment vector [A m

]; M is the magnetization vector [A/m]; r is the

position vector (x, y, z are the coordinates in the Cartesian system),

is the inertial coordinate vector-transformation matrix; W and Π are the work and potential energy;

is the Euler orientation for the 3-2-1 rotation sequence; ∇ is the gradient operator; subscript ij represents

partial derivative of i component in j-direction; subscript (ij)k represents partial derivative of ij partial

derivative in k-direction;

–

(bar over a variable) indicates that it is referenced to the microactuator

coordinates.

21.5.8.2 Electromagnetic Torques and Forces: Preliminaries

The equations for the electromagnetic torque and force on a cylindrical permanent-magnet thin film are

found by integrating the equations for torques and forces on an incremental volume of the permanent-

magnet thin film with magnetic moment Mdv over the volume. Figure 21.135 illustrates the microactuator

with the cylindrical permanent-magnet thin film in the coordinate system, which consists of a set of

orthogonal body-fixed axes that are initially aligned with a set of orthogonal x-, y-, z-axes fixed in the

inertial space.

The equations for the electromagnetic torque and force on an infinitesimal current can be derived

using the fundamental relationship for the force on a current-carrying-conductor element in a uniform

magnetic field. In particular, for a planar current loop (planar microwinding) with constant current i in

the uniform magnetic field B (vector B gives the magnitude and direction of the flux density of the

external field), the force on an element dl of the conductor is found using the Lorentz force law

Assuming that the magnetic flux is constant through the magnetic loop, the torque on a planar current

loop of any size and shape in the uniform magnetic field is

F idlB×

T i r dlB×()×

rB⋅()dlBrdl⋅

–

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Actuators 21-123

Using Stokes’s theorem, one has

The electromagnetic torque T acts on the infinitesimal current loop in a direction to align the magnetic

moment m with the external field B, and if m and B are misaligned by the angle

, we have

The incremental potential energy and work are found as

Using the electromagnetic force, we have

and

21.5.8.3 Coordinate Systems and Electromagnetic Field

The transformation from the inertial coordinates to the permanent-magnet coordinates is

We use the transformation matrix

If the deflections are small, we have

T i

dA ∇ rB⋅()B ∇ r×()dA⋅

–×

idAB×

T iAB× mB×==

TmB

sin=

dW dΠ T d

and Wsin Π mB

cos– mB⋅–== = == =

dW dΠ– F dr⋅∇Πdr⋅–== =

F ∇Π– ∇ mB⋅()m ∇⋅()B== =

r T

r =

coscos

sincos

sin–

sincos–cossinsin

sincos+sinsinsin

cossin

sin

sin+cossincos

cossin–sinsincos

coscos

, r

coscos

sincos

sin–

sincos–cossinsin

sincos+sinsinsin

cossin

sinsin+cossincos

cossin–sinsincos

coscos

−

–

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21-124 Mechatronic Systems, Sensors, and Actuators

It should be emphasized that we use the 3-2-1 orthogonal transformation matrix for the z-y-x Euler

rotation sequence, and

are the rotation Euler angle about the x, y, and z axes.

The field B and gradients of B produced by the microcoils fixed in the inertial frame and expressed

assuming that the electromagnetic fields can be described by the second-order Taylor series. Expanding

B about the origin of the x, y, z system as a Taylor series, we have [18]

where

We den ote

Then,

Hence, the first-order gradients are given as

The expanded field is expressed in the permanent-magnet coordinates as

where .

Br∇⋅()B

r ∇⋅()

B++=

∂

--------

∂

----------

r++=

∂

--------

∂

--------

∂

--------

∂

--------

and

∂

----------

∂

------

∂

---------

∂

------

∂

---------

∂

------

∂

---------

∂

------

∂

---------

∂

------

∂

---------

∂

------

∂

---------

∂

------

∂

---------

∂

------

∂

---------

∂

------

∂

---------

∂

--------

and B

ij()k

∂

------

∂

----------

∂

--------

[]and

∂

----------

ix()x

ix()y

ix()z

iy()x

iy()y

iy()z

iz()x

iz()y

iz()z

eij

∂

------

∂

----------

r+ B

ij [B

ij()x

ij()y

ij()z

]

r+==

Br∇⋅()B

r ∇⋅()

B++=

B T

B and ∇ T

∇==

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Actuators 21-125

Using r = , one has

and

21.5.8.4 Electromagnetic Torques and Forces

Now let us derive the fields and gradients at any point in the permanent magnet using the second-order

Taylor series approximation. To eliminate the transformations between the inertial and permanent

magnet coordinate systems and simplify the second-order negligible small components, we assume that

the relative motion between the magnet and the reference inertial coordinate is zero and the T

trans-

formation matrix is used (otherwise, the second-order gradient terms will lead to cumbersome results).

The magnetization (the magnetic moment per unit volume) is constant over the volume of the

permanent-magnet thin films, and m = Mv.

Assuming that the magnetic flux is constant, the total electromagnetic torque and force on a planar

current loop (microwinding) in the uniform magnetic field is

where

Case 1: Magnetization Along the Axis of Symmetry

For orientation of the magnetization vector along the axis of symmetry (x-axis) of the permanent-magnet

thin films, we have

Thus, in the expression ,

∂

--------

∂

----------

r++=

∂

-------

∂

---------

r++

∂

-------

∂

---------

r++

∂

-------

∂

---------

r++

TMB

r+× M ∇⋅()B

×()vd

FM∇⋅()B

M ∇⋅()B

∂

[]M

exx

exy

exz

eyx

eyy

eyz

ezx

ezy

ezz

M ∇⋅()B

∂

[]M M

exx

exy

exz

(MB

r (M ∇⋅ )B

)×+× vd=

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21-126 Mechatronic Systems, Sensors, and Actuators

the terms are

Therefore,

and

The terms in the derived equations must be evaluated.

Let us find the analytic expression for the electromagnetic torque . In particular, we have

where

and

Therefore,

Furthermore,

Thus, for , one has

rM∇⋅()B

× M

B–

exy

exz

exx

exz

x–

−B

exx

exy

and MB

× M

–

exz

exy

z–()vd

exx

exz

x–()vd

–=

exy

exx

y–()vd

exz

y vd

y vB

xx()z

xy vB

xy()z

xz()z

zy() vd

y vd

0, xy vd

0, zy vd

0===

ydzdxd

––

–

−R

l–

===

exz

yvd

xy()z

exy

zvd

xy()z

exz

exy

z–()vd

xy()z

– 0== =

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Actuators 21-127

Then, for , we obtain

Finally, we obtain the expression for T

Thus, the following electromagnetic torque equations result:

The electromagnetic forces are found as well. In particular, from

and

using the expressions for the expanded magnetic fluxes, e.g.,

and performing the integration, one has the following expressions for the electromagnetic forces as the

function of the magnetic field:

dv M

exx

exz

x–()dv

–=

zx()x

-----

zy()y

zz()z

+++v– B

xx()z

-----

– v+=

xx()z

– B

yy()z

zz()z

+++–=

dv M

exy

exx

y–()dv

xx()y

– B

yy()y

–

yz()z

–

xx()z

– B

yy()z

zz()z

+++–=

xx()y

– B

yy()y

yz()z

––+=

exx

exy

exz

exx

xx()x

xx()y

xx()z

z+++()vd

, F

===

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21-128 Mechatronic Systems, Sensors, and Actuators

Case 2: Magnetization Perpendicular to the Axis of Symmetry

For orientation of the magnetization vector perpendicular to the axis of symmetry, the following equation

is used to find the electromagnetic torque:

where

and

Thus,

Expressing the fluxes and performing the integration, we have the following expressions for the torque

components as the function of the magnetic field:

The electromagnetic forces are found to be

T (MB

rM∇⋅()B

)×+× vd

M ∇⋅()B

∂

[]M M

exz

eyz

ezz

rM∇⋅()B

× M

B–

eyz

ezz

exz

ezz

x–

exz

eyz

===

× M

–

exz

eyz

z–()vd

–=

ezz

x–()vd

eyz

exz

y–()vd

xx()y

-----

yy()y

yz()z

+++–=

xz()y

-----

– B

xx()x

-----

xy()y

+++=

xy()y

-----

–=

exz

eyz

ezz

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Actuators 21-129

Thus, the expressions for the electromagnetic force and torque components are derived. These equa-

tions provide one with a clear perspective on how to model, analyze, and control the electromagnetic

forces and torques changing the applied magnetic field because the terms

can be viewed as the control variables. It must be emphasized that the electromagnetic field (B

and B

(ij)k

)

is controlled by regulating the current in the planar microwindings and designing the microwindings

(or other radiating energy microdevices). As was discussed, the derived forces and torques must be used

in the torsional-mechanical equations of motion for the microactuator, and, in general, the six-degree-

of-freedom microactuator mechanical dynamics results. These mechanical equations of motion are easily

integrated with the derived electromagnetic equations, and closed-loop systems can be designed to attain

the desired microactuator performance. These equations guide us to the importance of electromagnetic

features in the modeling, analysis, and design of microactuators.

21.5.8.5 Some Other Aspects of Microactuator Design and Optimization

In addition to the electromagnetic-mechanical (electromechanical) analysis and design, other design and

optimization problems are involved. As an example, let us focus our attention on the planar windings.

The ideal planar microwindings must produce the maximum electromagnetic field, minimizing the

footprint area, taking into consideration the material characteristics, operating conditions, applications,

power requirements, and many other factors. Many planar winding parameters and characteristics can

be optimized, for example, the dc resistance must be minimized to improve the efficiency, increase the

flux, decrease the losses, etc. To attain good performance, in general, microwindings have the concentric

circular current path and no interconnect resistances. For N-turn winding, the total dc resistance r

found to be

where

is the winding material resistivity, t

is the winding thickness, r

and r

are the outer and inner

radii of the k-turn winding, respectively.

To achieve the lowest resistance, the planar winding radii can be optimized by minimizing the resis-

tance, and the minimum resistance is denoted as r

tmin

. In particular, making use of first- and second-

order necessary conditions for minimization, one has

where r

is the inner or outer radius of an arbitrary turn of the optimized planar windings from the

standpoint of minimizing the resistance.

Then, the minimum value of the microcoil resistance is given by

where r

and r

are the outer and inner radii of the windings (i.e., r

O Nth microcoil

and r

I 1st microcoil

respectively.

∂

--------

and B

ij()k

∂

------

∂

---------

πρ

----------

()ln

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

k=1

-------- 0 and

--------

> 0=

tmin

πρ

----------

()

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

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21-130 Mechatronic Systems, Sensors, and Actuators

Thus, using the number of turns and turn-to-turn spacing, the outer and inner radii of the k-turn

winding are found as

For spiral windings, the averaging (equivalency) concept should be used because the outer and inner

radii are the functions of the planar angle, see Figure 21.138. Finally, it should be emphasized that the

width of the Nth microcoil is specified by the rated voltage current density versus maximum current

density needed, fabrication technologies used, material characteristics, etc.

21.5.9 Micromachined Polycrystalline Silicon Carbide Micromotors

Articles [19,20] report the silicon-based fabrication of reluctance micromotors. This section is focused

on a new enabling technology to fabricate microtransducers. Multilayer fabrication processes at low

temperature and micromolding techniques were developed to fabricate SiC microstructures and salient-

pole micromotors, which can be used at a very high temperature (400°C and higher) [21,22]. This was

done through the SiC surface micromachining. Advantages of the SiC micromachining and SiC technol-

ogies (high temperature and ruggedness) should be weighted against fabrication drawbacks because new

processes must be designed and optimized. Reactive ion etching is used to pattern SiC thin films; however,

many problems, such as masking, low etch rates, and poor etch selectivity, must be addressed and resolved.

Articles [21,22] report two single-layer reactive ion etching-based polycrystalline SiC surface microma-

chining processes using polysilicon or SiO

as the sacrificial layer. In addition, the micromolding process,

used to fabricated polysilicon molds in conjunction with polycrystalline SiC film deposition and mechan-

ical polishing to pattern polycrystaline SiC films, are introduced. The micromolding process can be used

for single- and multilayer SiC surface micromachining.

The micromotor fabrication processes are illustrated in Figure 21.139. A 5–10

m thick sacrificial

molding polysilicon is deposited through the LPCVD on a 3–5

m sacrificial thermal oxide. The rotor-

stator mold formation can be made on the polished (chemical-mechanical polishing) polysilicon surface,

enabling the 2

m fabrication features using standard lithography and reactive ion etching. After the

mold formation and delineation, the SiC is deposited on the wafer using atmospheric pressure chemical

vapor deposition reactor. In particular, the phosphorus-doped (n-type) polycrystalline SiC films are

deposited on the SiO

sacrificial layers at 1050°C with 0.5–1

m/h rate (deposition is not selective, and

SiC will be deposited on the surfaces of the polysilicon molds as well). Mechanical polishing of SiC is

needed to expose the polysilicon and planarize the wafer surface (in [21,22], the polishing was done with

m diameter diamond suspension, 360 N normal force, and 15 rad/sec pad rotation—the removal rate

of SiC was reported to be 100 nm/min). The wafers are polished until the top surface of the polysilicon

mold is exposed (polishing must be stopped at once due to the fast polishing rate). The flange mold is

fabricated through the polysilicon and the sacrificial oxide etching (using the KOH and BHF, respectively).

The 0.5

m bearing clearance low-temperature oxide is deposited and annealed at 1000°C. Then, the

m polycrystalline SiC film is deposited and patterned by reactive ion etching to make the bearing.

FIGURE 21.138 Planar spiral microwinding.

r ( )

-------

1/N

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Actuators 21-131

The release begins with the etching (BHF solution) to strip the left-over bearing clearance oxide. The

sacrificial mold is removed by etching (KOH system) the polysilicon. It should be emphasized that the

SiC and SiO are not etched during the mold removal step. Then, the moving parts of the micromotor

were released. The micromotor is rinsed in water and methanol, and dried with the air jet.

Using this fabrication process, the micromotor with the 100–150

m rotor diameter, 2

m airgap, and

m bearing radius, was fabricated and tested in [21, 22]. The rated voltage was 100 V and the maximum

angular velocity was 30 rad/s. For silicon and polysilicon micromotors, two of the most critical problems

are the bearing and ruggedness. The application of SiC reduces the friction and improves the ruggedness.

These contribute to the reliability of the SiC-based fabricated micromachines.

21.5.10 Axial Electromagnetic Micromotors

The major problem is to devise novel microtransducers in order to eliminate fabrication difficulties and

guarantee affordability, efficiency, reliability, and controllability of MEMS. In fact, the electrostatic and

planar micromotor fabricated and tested to date are found to be inadequate for a wide range of appli-

cations due to difficulties associated and the cost. Therefore, this section is devoted to devising novel

affordable rotational micromotors.

Figure 21.140 illustrates the devised axial topology micromotor, which has the closed-ended electromag-

netic system. The stator is made on the substrate with deposited microwindings (printed copper coils can

be made using the fabrication processes described as well as using a double-sided substrate with one-sided

deposited copper thin films through conventional photolithography processes). The bearing post is fab-

ricated on the stator substrate and the bearing hold is a part of the rotor microstructure. The rotor with

permanent-magnet thin films rotates due to the electromagnetic torque developed. It is important to

emphasize that the stator and rotor are made using conventional well-developed processes and materials.

It is evident that conventional silicon and SiC technologies can be used. The documented micromotor

has a great number of advantages. The most critical benefit is the fabrication simplicity. In fact, axial

FIGURE 21.139 Fabrication of the SiC micromotors: cross-sectional schematics.

FIGURE 21.140 Slotless axial electromagnetic micromotor (cross-sectional schematics) with controlling ICs.

Substrate

Oxide Oxide

SiC

Substrate

Oxide

SiC

Substrate

Oxide

SiC

Polysilicon

Substrate

Oxide

Polysilicon

Stator

Rotor

Substrate

ICs

Rotor

Permanent magnet

Planar windings

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