Page 299 - The Mechatronics Handbook

0066_frame_C14.fm Page 13 Wednesday, January 9, 2002 1:49 PM

For a microstructure with outside diameter D r , the magnet strength is Q. Hence, the magnetic moment
is m = QD r , and the force is found as F = QB.
The electromagnetic torque is
1
T = 2F--D r sin α = QD r Bsin α = mBsin α
2
Using the unit vector in the magnetic moment direction a m, one obtains

T = m × B = a mm × B = QD r a m × B

For a current loop with the area A, the torque is found as

T = m × B = a mm × B = iAa m × B
For a solenoid with N turns, one obtains

T = m × B = a mm × B = iANa m × B

As the electromagnetic torque is found, using Newton’s second law, one has

--------- = 1 -- ∑ T Σ = 1 ( – -------- = ω r
dω r
dθ r
-- TT L),
dt J J dt
where T L is the load torque.
The electromotive (emf ) and magnetomotive (mmf ) forces can be used in the model development.
We have

⋅
⋅
emf = ∫ l ° Edl = ∫ l ° ( v × B) dl – ∫ s ∂B
------ds
∂t
motional induction transformer induction
generation
and
⋅
mmf = ∫ Hdl = ∫ s ° Jds + ∫ s ° ∂D
⋅
-------ds
l ∂t
For preliminary design, it is sufficiently accurate to apply Faraday’s or Lenz’s laws, which give the
electromotive force in term of the time-varying magnetic field changes. In particular,
dψ ∂ψ ∂ψ ∂ψ
emf = – ------- = – ------- – -------- -------- = – ------- – --------ω r
∂ψ dθ r
dt ∂t ∂θ r dt ∂t ∂θ r
∂ψ
where ------- is the transformer term.
∂t
The total flux linkages are
ψ = 1
--πN S Φ p
4
where N S is the number of turns and Φ p is the flux per pole.
For radial topology micromachines, we have

Φ p = µiN S
------------R in st L
2
P g e

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