Page 295 - The Mechatronics Handbook

0066_frame_C14.fm Page 9 Wednesday, January 9, 2002 1:39 PM

and two magnetic field vectors are used. The electric field vectors are the electric field intensity, , the
E
D
electric flux density, , and the current density, . The magnetic field vectors are the magnetic field
J
intensity H and the magnetic field density . The differential equations for microelectromechanical
B
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
E
H
(
∂Hx, yzt)
,,
∇ × Ex, yzt) = – µ-------------------------------
(
,,
∂t
∂Ex, yzt,,(
)
∂Ex, yzt,,(
)
(
(
(
∇ × Hx, yzt) = ε------------------------------ + Jx, yzt) = ε------------------------------ + σEx, yzt)
,,
,,
,,
∂t ∂t
ρ v x, yzt)
(
,,
⋅
(
∇ Ex, yzt) = -----------------------------
,,
ε
∇ Hx, yzt) = 0
⋅
(
,,
where ε is the permittivity, µ is the permeability, σ is the conductivity, and ρ v is the volume charge density.
The constitutive (auxiliary) equations are given using the permittivity ε, permeability tensor µ, and
conductivity σ. In particular, one has
D = εE or D = εE + P
(
B = µH or B = µ H + M)
J = σ E or J = ρ νv
The Maxwell’s equations can be solved using the boundary conditions on the field vectors. In two-
region media, we have
⋅
a N × ( E 2 – E 1 ) = 0, a N × ( H 2 – H 1) = J s, a N ( D 2 – D 1 ) = ρ s , a N ( B 2 – B 1 ) = 0
⋅
where is the surface current density vector, a N is the surface normal unit vector at the boundary from
J s
is the surface charge density.
region 2 into region 1, and ρ s
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
E and to model electromagnetic fields in MEMS, one has
H
2 2
∂E
∂J
∂ E
∂ D
∇ × ( ∇ × E) = ∇ ∇ E) ∇ E = −µ------ µ---------- = – µσ------ µε---------
(
⋅
2
–
–
–
∂t ∂t 2 ∂t ∂t 2
2
∂H
∂ H
⋅
∇ × ( ∇ × H) = ∇ ∇ H) ∇ H = – µσ------- µε----------
(
2
–
–
∂t ∂t 2
The following pair of homogeneous and inhomogeneous wave equations
2

∂ E
∂E
2
∇ E µσ ------ µε--------- = ∇ -----
ρ v
–
–
ε
∂t ∂t 2 
2
∂H
∂ H
2
∇ H µσ------- µε---------- = 0
–
–
∂t ∂t 2
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