Page 296 - The Mechatronics Handbook

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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
E
H
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
A
∂A
∇ × A = B = µH and E = – ------- – ∇V
∂t
The electromagnetic field is derivative from the potentials. Using the Lorentz equation

∇ A = – ∂V
⋅
-------
∂t
the inhomogeneous vector potential wave equation to be solved is
2
∂A
∂ A
2
– ∇ A + µσ ------- + µε--------- = – µσ∇V
∂t ∂t 2
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 ρ v , the Lorenz force, which relates the electromagnetic and mechanical
phenomena, is found as
F = ρ v E + v × B) = ρ v E + J × B
(

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

F = ∫ ( ρ ν E + × B) v = 1 ∫ ↔
--- T αβ ds⋅
µ s °
J
d
v
The electromagnetic stress energy tensor (the second Maxwell stress tensor) is
0 E x E y E z
↔
T αβ = – E x 0 B z – B y
– E y – B z 0 B x

– E z B y – B x 0
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
T s = T s + T s M
E

E 1 D 1 – 1 --E j D j E 1 D 2 E 1 D 3 B 1 H 1 – 1 --B j H j B 1 H 2 B 1 H 3
2 2
= 1 + 1
E 2 D 1 E 2 D 2 – --E j D j E 2 D 3 B 2 H 1 B 2 H 2 – --B j H j B 2 H 3
2 2
E 3 D 3 – 1 B 3 H 3 – 1
E 3 D 1 E 3 D 2 --E j D j B 3 H 1 B 3 H 2 --B j H j
2 2
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