Chapter three: Structural design, modeling, and simulation 191Figure 18... 192 Chapter three: Structural design, modeling, and simulation... Time-varying current radiates electromagnetic
Trang 1Chapter three: Structural design, modeling, and simulation 191
Figure 18 Timing Diagram
Trang 2192 Chapter three: Structural design, modeling, and simulation
Trang 3Chapter three: Structural design, modeling, and simulation 193
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Trang 6CHAPTER 4 CONTROL OF NANO- AND MICROELECTROMECHANICAL
SYSTEMS 4.1 FUNDAMENTALS OF ELECTROMAGNETIC RADIATION AND
ANTENNAS IN NANO- AND MICROSCALE
ELECTROMECHANICAL SYSTEMS
The electromagnetic power is generated and radiated by antennas Time-varying current radiates electromagnetic waves (radiated electromagnetic fields) Radiation pattern, beam width, directivity, and other major characteristics can be studied using Maxwell’s equations, see Section 2.2 We use the vectors of the electric field intensity E, electric flux density D,
magnetic field intensity H, and magnetic flux density B The constitutive
equations are
E
D = ε and B = µ H
where ε is the permittivity; µ is the permiability
It was shown in Section 2.2 that in the static (time-invariant) fields, electric and magnetic field vectors form separate and independent pairs That
is, E and D are not related to H and B, and vice versa However, for
time-varying electric and magnetic fields, we have the following fundamental electromagnetic equations
t
t z y x t
z y
x
∂
∂
µ ( , , , )
) , ,
,
×
) , , , ( ) , , , ( ) , , , ( ) , ,
,
t
t z y x t
z y x t
z y
×
∇
∂
∂ ε
ε
ρ ( , , , ) )
, ,
,
t z
y
⋅
0 ) , ,
,
⋅
∇ H x y z t ,
where J is the current density, and using the conductivity σ, we have
E
J = σ ; ρv is the volume charge density
The total current density is the sum of the source current JS and the conduction current density σ E (due to the field created by the source JS) Thus,
E J
JΣ= S+ σ
The equation of conservation of charge (continuity equation) is
∫
∫ ⋅ = −
v
v s
dv dt
d
d s ρ
and in the point form one obtains
t
t z y x t
z
y
∂
∂
−
=
⋅
∇ J ( , , , ) ρ ( , , , )
Trang 7
Therefore, the net outflow of current from a closed surface results in decrease of the charge enclosed by the surface
The electromagnetic waves transfer the electromagnetic power That is, the energy is delivered by means of electromagnetic waves Using equations
t
∂
∂
µ H
E = −
×
∇ and ∇ × H = E + J
t
∂
∂
we have
⋅
−
⋅
−
=
×
∇
⋅
−
×
∇
⋅
=
×
⋅
t
∂ ε
∂
∂ µ
) ( ) ( )
In a media, where the constitute parameters are constant (time-invariant),
we have the so-called point-function relationship
2 1 2 2 1
)
∂
−
=
×
⋅
In integral form one obtains
E H
E
s H
E
of presence
in the power dissipated ohmic
2
field magnetic and field electric the rate of change of energy stored in time
2 2 1 2 2
)
s
dv E dv
H E
t
∂
∂
The right side of the equation derived gives the rate of decrease of the electric and magnetic energies stored minus the ohmic power dissipated as heat
in the volume v The pointing vector, which is a power density vector,
represents the power flows per unit area, and
H
E
Furthermore,
∫
∫
v v
H E s
s
dv dv
w w t d
∂
∂
volume enclosed the
leaving
power
)
where wE=21ε E2 and wH=21µ H2 are the electric and magnetic energy densities; E2 1 J2
σ σ
ρσ = = is the ohmic power density
The important conclusion is that the total power transferred into a closed
surface s at any instant equals the sum of the rate of increase of the stored
electric and magnetic energies and the ohmic power dissipated within the
enclosed volume v.
If the source charge densityρv( x , y , z , t ) and the source current density )
,
,
,
( x y z t
J vary sinusoidally, the electromagnetic field also vary sinusoidally Hence, we have deal with the so-called time-harmonic electromagnetic fields The sinusoidal time-varying electromagnetic fields will
be studied Hence, the phasor analysis is applied For example,
z z y y x
E r a r a r a
r
Trang 8The electric field intensity components are the complex functions In particular,
Im Re
)
E r = + , Ey( r ) = EyRe+ jEyIm, Ez( r ) = EzRe+ jEzIm For the real electromagnetic field, we have
t E
t E
t
Ex( r , ) = xRe( r ) cos ω − xIm( r ) sin ω
One obtains the time-harmonic electromagnetic field equations In particular,
• Faraday’s law ∇ × E = − j ωµ H,
+
= + +
=
×
ω
σ ω ωε
σ
j j
ε ω
σ ρ +
=
⋅
∇
j
v
• Continuity of magnetic flux ∇ ⋅ H = 0,
• Continuity law ∇ ⋅ J = − jωρv, (4.1.1)
where
+ ε
ω
σ
j is the complex permittivity However, for simplicity we will
use ε keeping in mind that the expression for the complex permittivity
ε
ω
σ +
j must be applied.
The electric field intensity E, electric flux density D, magnetic field
intensity H, magnetic flux density B, and current density J are complex-valued
functions of spatial coordinates
From the equation (4.1.1) taking the curl of ∇ × E = − j ωµ H, which is rewritten as ∇ × E = − j ω B, and using ∇ × H = j ω D + J, one obtains
J E
E
E = ω µε = kv = − j ωµ
×
∇
×
,
where k v is the wave constant kv = ω µε , and in free space
c
kv = ω µ ε = ω
0
0
0 because the speed of light is
0 0
1
ε µ
=
sec
m
8
10
3×
=
The wavelength is found as
µε ω
π π
λ = 2 = 2
v
and in free space
ω
π π
kv
v
2 2
0
Trang 9Using the magnetic vector potential A, we have B = ∇ × A.
Hence,
0 )
×
and thus
Ë A
E + j ω = −∇ ,
where Ë is the scalar potential
To guarantee that ∇ × H = j ω D + J holds, it is required that
J E A
A A
µ = ∇ × ∇ × = ∇∇ ⋅ − ∇ = +
×
Therefore, one finally finds the equation needed to be solved
J Ë A
A
Taking note of the Lorentz condition ∇ ⋅ A = − j ωµε Ë, one obtains
J A
v
Thus, the equation for Ë is found In particular,
ε
ρv
v
k = −
+
∇2Ë 2Ë
The equation for the magnetic vector potential is found solving the following inhomogeneous Helmholtz equation
J A
v
The expression for the electromagnetic field intensity, in terms of the vector potential, is
ωµε
ω
j
To derive E, one must have A The Laplacian for A in different coordinate
systems can be found For example, we have
x x
v
A + = − µ
,
y y
v
A + = − µ
,
z z
v
A + = − µ
It was shown that the magnetic vector potential and the scalar potential obey the time-dependent inhomogeneous wave equation
) , ( ) , (
2
2
t
k Ω r = − r
∂
∂
−
The solution of this equation is found using Green’s function as
−
=
Ω ( r , t ) F ( r ' , t ' ) G ( r r ' ; t t ' ) dt ' d τ ',
' 4
1 )
'
;
'
r r r
−
−
=
−
The so-called retarded solution is
Trang 10∫∫∫ − − −
−
=
'
) ' '
, ' ( )
,
r r
r r r
For sinusoidal electromagnetic fields, we apply the Fourier analysis to obtain
∫∫∫ −
−
=
' 4
1
)
(
'
τ
e jk v
r r r r
r r
Thus, we have the expressions for the phasor retarded potentials
dv e
v
jk v
∫ −
' 4
)
(
'
r J r r A
r r
r
π
µ
,
dv e
v
jk v
∫ −
' 4
1
)
(
'
r r r Ë
r r
Example 4.1.1.
Consider a short (dl) thin filament of current located in the origin, see
electromagnetic field intensities
Figure 4.1.1 Current filament in the spherical coordinate system
Solution.
The magnetic vector potential has only a z component, and thus, from
J A
v
we have
ds
i J
A k
Az+ v z = − µ z = − µ
,
where ds is the cross-sectional area of the filament.
z
idl
θ φ
r
φ
a
r
a
θ
a
Trang 11Taking note of the spherical symmetry, we conclude that the magnetic
vector potential A z is not a function of the polar and azimuth angles θandφ In particular, the following equation results
0
∂
∂
∂
∂
z v
z k A r
A
r
r
It is well-known that the solution of equation 2 2 0
2
=
ψ
ψ
v
k d
d
has two
components In particular, ejk v r (outward propagation) and e−jk v r (inward propagation) The inward propagation is not a part of solution for the filament located in the origin Thus, we have
r jk t
j v
ae
r
t = ω−
ψ ( , ) (outward propagating spherical wave)
In free space, we have
) / (
)
,
( t r = aej ω t−r c
Substituting
r
Az = ψ
, one obtains c
r j
r
a r A
ω
−
= )
To find the constant a, we use the volume integral
∫
∫
∫
v
z v
z s
d r z v
c d d r A dv
2 2
2
where the differential spherical volume is dv = rd2sin θ d θ d φ dr; r d is the differential radius
Making use of
r c j z
r
r
a r c
j r
A A
ω
+
−
=
∂
∂
=
⋅
we have
idl a
d d ae
r c
d
r c j d
2
0 0
+
−
one has
π
µ
4
0idl
a =
Thus, the following expression results
c r j
r
idl r
A
ω
π
=
4
)
Therefore, the final equation for the magnetic vector potential (outward propagating spherical wave) is
z c r j
e r
idl
A
ω
π
=
4
)
Trang 12From az = arcos θ − aθsin θ, we have
) sin cos
( 4
)
π
µ
θ
ω
a a
r j
e r
idl
r
The magnetic and electric field intensities are found using
A
B = ∇ × and
ωµε
ω j
Then, one finds
φ
ω
ω π
θ
r j
e r cr
j r
idl r
=
×
∇
0
1 4
sin )
(
1
)
.
1 sin
4
1 cos
4
)
(
3 2 2
2 0
0
3 2 0
0
θ ω
ω
ω ω θ πω
ε
µ
ω θ πω
ε µ
a
a E
c r j
r c r j
e r cr
j r c
cidl
j
e r cr j cidl
j
r
−
−
+ +
−
−
=
The intrinsic impedance is given as
0
0
0
ε
µ
=
Z , and
0 0 0 0
1
µ
ε
=
=
Z
Near-field and far-field electromagnetic radiation fields can be found,
simplifying the expressions for H(r) and E(r).
For near-field, we have
φ
ω
π
θω
r j
e cr
idl j r
sin )
( 1
)
(
4 )
0
θ
θ π
ω ε
µ
a E
r c
cidl j
r =
The complex Pointing vector can be found as
) ( )
2
1E r × H r
The following expression for the complex power flowing out of a sphere
of radius r results
π
ωµ π
ε µ µ ω
12 12
) ( )
(
2 2 0 2 2 0 0 0 2
* 2
d r
s
=
=
⋅
×
The real quality is found, and the power dissipated in the sense that it
travels away from source and cannot be recovered
Trang 13Example 4.1.2.
Derive the expressions for the magnetic vector potential and electromagnetic field intensities for a magnetic dipole (small current loop) which is shown in Figure 4.1.2
Figure 4.1.2 Current loop in the xy plane
Solution.
The magnetic dipole moment is equal to the current loop are times current That is,
z
z M i
r a a
0
For the short current filament, it was derived in Example 4.1.1 that
z c r j
e r
idl
A
ω
π
=
4
)
In contrast, we have
∫
=
l
dl r
i
'
1
4
0
π
µ
The distance between the source element dl and point O ( r , θ ,π2) is
denoted as r’ It should be emphasized that the current filament is lies in the xy
plane, and
φ φ φ
φ
dl = a 0 = ( − axsin + aycos )0
Thus, due to the symmetry
∫
−
=
2 /
2 /
0 0
'
sin 2
π
π
π
r
ir
a
where using the trigonometric identities one finds
φ
θ sin
sin 2 '2 r2 r02 rr0
Assuming that r2 >> r02, we have
+
≈ 1 1 sin θ sin φ
'
r
r r
z
θ
r
'
r
0
r
φ d
ir0 φ
) , , ( r θ π2 O
dl
Trang 14sin 4 sin
sin sin 1 2
'
sin 2
2
2 0 0 2
/
2 /
0 0
0
2 /
2 /
0 0
θ µ
φ φ φ θ π
µ
φ
φ π
µ
φ π
π φ
π
π φ
r
ir d
r
r r
ir
d r ir
a a
a
A
=
+
=
=
∫
∫
−
−
Having obtained the explicit expression for the vector potential, the
magnetic flux density is found In particular,
) sin cos
2 ( 4
sin
2 0 0 2
2 0
µ
θ
a A
r
ir r
ir
Taking note of the expression for the magnetic dipole moment
z
i
r a
0
π
= , one has
r
r r
a
2
2 0 0
4
sin
µ θ
µ
It was shown that using = ∫
l
dl r
i
'
1 4
0
π
µ
A , the desired results are obtained
Let us apply ∫
−
=
l
r c j
dl r
e i
' 4
' 0
ω
π
µ
From j c r r r e j c r
c j e
ω
−
'
, we have
π
µ π
φ
ω ω
sin 1
4 '
)]
' ( 1 [
0
c j c l
r c j
r
M dl
r
e r r j
+
=
−
−
Therefore, one finds
θ π
ω
ω ω
3 0
2 2
r c j
c c
e r r j c
M j
−
θ π
ε µ
ω
ω
4
2
3 2
0
0 2
3 0
3 3 2
2
r c j
c c
r j r c
M j
+
θ π
ε µ
ω
ω ω
ω
4
3 2
0
0 2
3 0
3 3 2
2
r c j
c c
c
e r j r r j c
M j
−
−
−
Trang 15The electromagnetic fields in near- and far-fields can be straightforwardly derived, and thus, the corresponding approximations for theEφ, Hr and Hθ
Let the current density distribution in the volume is given as J ( r0), and for far-field from Figure 4.1.3 one has r ≈ r ' − r0
Figure 4.1.3 Radiation from volume current distribution
The formula to calculate far-field magnetic vector potential is
dv e e
jk r
−
4
)
A r
π
µ
, and the electric and magnetic field intensities are found using
ωµε
ω
j
and B = ∇ × A
We have
e r
Z jk
v
jk r
r r jk v
4
)
E r
) ( )
H r = Yv r×
Example 4.1.3.
Consider the half-wave dipole antenna fed from a two-wire transmission line, as shown in Figure 4.1.4 The antenna is one-quarter wavelength; that is,
v
λ 41
4
1 ≤ ≤
− The current distribution is i ( z ) = i0cos kvz Obtain the
equations for electromagnetic field intensities and radiated power
x
z
0
r
r
r
'
r J
Source
Trang 16and integrating the derived expression over the surface
φ θ θ θ
θ π π
ππ
d d Z
i
sin sin
) cos ( cos
8
2
0 0
2 2 1 2
2
0
2
0
0
6
If the current density distribution is known, the radiation field can be found Using Maxwell’s equations, using the electric and magnetic vector
potentials AE and AH, we have the following equations
( ∇2+ kv2) AE = − µ JE
,
( ∇2+ kv2) AH = − ε JH
,
H E
E j
ε ωµε
E H
H
j
µ ωµε
The solutions are
r r J
jk r jk
4
)
(
π
µ
,
r r J
jk r jk
4
)
(
π
ε
Example 4.1.4.
Consider the slot (one-half wavelength long slot is dual to the half-wave dipole antenna studied in Example 4.1.3), which is exited from the coaxial line, see Figure 4.1.5 The electric field intensity in the z-direction is
( l z )
k
E
E = 0sin v − Derive the expressions for the magnetic vector potential and electromagnetic field intensities
Figure 4.1.5 Slot antenna
x
y
z
Slot
r
(l z)
k
E sin v − 2
: Field
0
l
Slot
Trang 17Using the magnetic current density JH, from
∫
∫
∫
∫ ∇ × ⋅ = ⋅ = − ⋅ − ⋅
s
H s
l s
d d
j dl
the boundary conditions for the magnetic current sheet are found as
H n
The slot antenna is exited by the magnetic current with strength
( l z )
k
E sin v −
2 0 in the z axis For half-wave slot we have
( l z )
k
i
( ∇2+ kv2) AH = − ε JH
,
θ
θ π
θ π ωµε
r
Y i j j
−
=
sin 2
) cos cos(21 0 0
,
sin
2
) cos cos(21 0
φ
θ π
θ π
A
r
i
−
=
×
∇
−
s r J
s
jk r jk
H = − v ∫ v ( )
16
)
(
π
ε
The boundary condition an× E = −21JH = an× axE0sin kv( l − z ) is satisfied by the radiated electromagnetic field
The radiation pattern of the slot antenna is the same as for the dipole
References
1 Hayt W H., Engineering Electromagnetics, McGraw-Hill, New York,
1989
2 Collin R E., Antennas and Radiowave propagation,” McGraw-Hill, New
York, 1985
3 Paul C R., Whites K W., and Nasar S A., Introduction to Electromagnetic Fields, McGraw-Hill, New York, 1998.
Trang 184.2 DESIGN OF CLOSED-LOOP NANO- AND
MICROELECTROMECHANICAL SYSTEMS USING THE
LYAPUNOV STABILITY THEORY
The solution of a spectrum of problems in nonlinear analysis, structural synthesis, modeling, and optimization of NEMS and MEMS lead to the development of superior high-performance NEMS and MEMS In this section,
we address introductory control issues Mathematical models of NEMS and MEMS were derived, and the application of the Lyapunov theory is studied as applied to solve the motion control problem.
It was illustrated that NEMS and MEMS must be controlled Nano- and microelectromechanical systems augment a great number of subsystems, and to control microscale electric motors, as discussed in previous chapters, power amplifiers (ICs) regulate the voltage or current fed to the motor windings These power amplifiers are controlled based upon the reference (command), output, decision making, and other variables Studying the end-to-end NEMS and MEMS behavior, usually the output is the nano- or microactuator linear and angular displacements There exist infinite number of possible NEMS and MEMS configurations, and it is impossible to cover all possible scenarios Therefore, our efforts will be concentrated on the generic results which can be obtained describing NEMS and MEMS by differential equations That is, using the mathematical model, as given by differential equations, our goal is develop control algorithms to guarantee the desired performance characteristics addressing the motion control problem (settling time, accuracy, overshoot, controllability, stability, disturbance attenuation, et cetera)
Several methods have been developed to address and solve nonlinear design and motion control problems for multi-input/multi-output dynamic systems In particular, the Hamilton-Jacobi and Lyapunov theories are found to
be the most straightforward in the design of control laws
The NEMS and MEMS dynamics is described as
u x B d r x
F
t
x & ( ) = ( , , ) + ( ) ,y = H ( x ),umin ≤ ≤ u umax, ( ) x t0 = x0,
(4.2.1) where x∈X⊂c is the state vector; u∈U⊂m is the bounded control vector;
r∈R⊂band y∈Y⊂bare the measured reference and output vectors; d∈D⊂s
is the disturbance vector; F(⋅):c×b×s→c and B(⋅):c→ c×m are jointly continuous and Lipschitz; H(⋅):c→b is the smooth map defined in the neighborhood of the origin, H(0) = 0.
Before engaged in the design of closed-loop systems, which will be based upon the Lyapunov stability theory, let us study stability of time-varying nonlinear dynamic systems described by
) , (
)
( t F t x
x & = , x ( t0) = x0, t ≥ 0
The following Theorem is formulated