Page 301 - The Mechatronics Handbook

0066_frame_C14.fm Page 15 Wednesday, January 9, 2002 1:49 PM

where x is the displacement of a translational microstructure (plunger), m is the mass of a movable
plunger, B v is the viscous friction coefficient, k s1 and k s2 are the spring constants (the spring can be made
∂W c i, x)
(
from polysilicon), and F e (t) is the magnetic force which is found using the coenergy W c , F e (i, x) = ----------------------- .
∂x
The stretch and restoring forces are not directly proportional to the displacement, and these forces
are different on either side of the equilibrium position. The restoring/stretching force exerted by the
2
polysilicon spring is expressed as (k s1 x + k s2 x ).
Assuming that the magnetic system is linear, the coenergy is expressed as

(
W c i, x) = 1 2
--Lx()i
2
Then

F e i, x( ) = 1 2dL x()
--i --------------
2 dx

The inductance is found as

2
2
N
Lx() = ------------------ = --------------------------------------------------
N µ f µ 0 A f A g
ℜ f + ℜ g A g l f + 2A f µ f x + 2d)
(
where ℜ f and ℜ g are the reluctances of the ferromagnetic material and air gap, A f and A g are the associated
cross section areas, and l f and (x + 2d) are the lengths of the magnetic material and the air gap.
Hence
2 2 2
dL – ---------------------------------------------------------
------ =
2N m f m 0 A f A g
(
dx [ A g l f + 2A f µ f x + 2d)] 2
Using Kirchhoff’s law, the voltage equation for the phase microcircuitry is
dψ
u a = ri + -------
dt

where the flux linkage ψ is expressed as ψ = L(x)i.
One obtains

di
dL x()dx
u a = ri + L x()----- + i--------------------
dt dx dt
and thus

2 2 2
di r 2N µ f µ 0 A f A g 1
----- = – -----------i + --------------------------------------------------------------------iv + -----------u a
(
[
dt Lx() Lx() A g l f + 2A f µ f x + 2d)] 2 Lx()
Augmenting this equation with differential equation
2
d x
Ft() = m-------- + B v dx ( k s1 x + k s2 x ) + F e t()
------ +
2
dt 2 dt
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