Page 302 - The Mechatronics Handbook

0066_frame_C14.fm Page 16 Wednesday, January 9, 2002 1:49 PM

three nonlinear differential equations for the studied translation microdevise are found as

(
(
di rA g l f +[ 2A f µ f x + 2d)] 2µ f A f A g l f + 2A f µ f x + 2d)
----- = – --------------------------------------------------------i + -------------------------------------------------iv + -------------------------------------------------u a
(
dt 2 A g l f + 2A f µ f x + 2d) 2
N µ f µ 0 A f A g N µ f µ 0 A f A g
dx
------ = v
dt
2 2 2
dv --------------------------------------------------------------i – ---- k s1 x +( k s2 x ) – B v
1
----- =
2
2
N µ f µ 0 A f A g
-----v
(
[
dt mA g l f + 2A f µ f x + 2d)] 2 m m
Example 14.5.2: Mathematical Model of an Elementary Synchronous
Reluctance Micromotor
Consider a single-phase reluctance micromotor, which can be straightforwardly fabricated using con-
ventional CMOS, LIGA, and LIGA-like technologies. Ferromagnetic materials are used to fabricate
microscale stator and rotor, and windings can be deposited on the stator, see Fig. 14.7.
The quadrature and direct magnetic axes are fixed with the microrotor, which rotates with angular
velocity ω r . These magnetic axes rotate with the angular velocity ω. Assume that the initial conditions
are zero. Hence, the angular displacements of the rotor θ r and the angular displacement of the quadrature
magnetic axis θ are equal, and
θ r = θ = ∫ t t 0 ω r τ() τ = ∫ t t 0 ωτ() τ.
d
d
The magnetizing reluctance ℜ m is a function of the rotor angular displacement θ r . Using the number
of turns N S , the magnetizing inductance is

2
()
L m θ r = ----------------- .
N S
()
ℜ m θ r
This magnetizing inductance varies twice per one revolution of the rotor and has minimum and maxi-
mum values, and
2 2
N S
L m min = ------------------------- , L m max = -------------------------
N S
()
()
θ =0,π,2π,… 1 3 5
ℜ m max θ r ℜ m min θ r
r θ =--π,--π,--π,…
r 2 2 2
Stator
ICs
Quadrature Magnetic Axis Direct Magnetic Axis
i as r T, e
N
u as (t) s
r , L Rotor
s
s
t
r r ( )d r 0
r t
0
as
FIGURE 14.7 Microscale single-phase reluctance motor with rotational motion microstructure (microrotor).

©2002 CRC Press LLC