Page 634 - The Mechatronics Handbook

0066_Frame_C20.fm Page 104 Wednesday, January 9, 2002 1:44 PM

The expression for the electromagnetic torque was derived in [8]. In particular,

T e = L ∆m i as sin 2q r
2

where L ∆m is the half-magnitude of the sinusoidal magnetizing inductance L m variations,
()
L m q r = L m – L ∆m cos 2q r
Thus, to develop the electromagnetic torque, the current i as must be fed as a function of the rotor
angular displacement θ r . For example, if i as = i M Re( sin 2q r ), then

1
T eaverage = --- ∫ p L ∆m i as sin 2q r q r = 1 2
2
d
--L ∆m i M
p 0 4
The micromotor under our consideration is the synchronous micromachine, and the obtained expres-
sion for the phase current is very important to control the microtransducer. In particular, the Hall-effect
position sensor should be used to measure the rotor displacement, and the ICs must feed the phase
current as a nonlinear function of θ r . Furthermore, the electromagnetic torque is controlled by changing
the current magnitude i M .
The mathematical model of the single-phase reluctance micromotor is found using Kirchhoff’s and
Newton’s second laws. In particular, we have

u as = r s i as + dy as (circuitry equation—Kirchhoff’s law)
----------
dt
2
d q r
T e – B m w r – T L = J---------- (torsional-mechanical equation—Newton’s law)
2
dt
Here, the electrical angular velocity ω r and displacement θ r are used as the mechanical system variables.
dy as
From u as = r s i as + ---------- and the flux linkage equation ψ as = (L ls + L m − L ∆m cos2θ r ) as , using the torsional-
dt
mechanical dynamics, one obtains a set of three first-order nonlinear differential equations which models
single-phase reluctance micromotors. In particular, we have
1
-------- = – --------------------------------------------------i as – --------------------------------------------------i as w r sin 2q r + --------------------------------------------------u as
2L ∆m
r s
di as
dt L ls + L m – L ∆m cos 2q r L ls + L m – L ∆m cos 2q r L ls + L m – L ∆m cos 2q r
-- L ∆m i as sin(
--------- = 1 2 2q r – B m w r – T L )
dw r
dt J
-------- = w r
dq r
dt
As the mathematical model is found and the micromotor parameters are measured, nonlinear simu-
lation and analysis can be straightforwardly performed to study the dynamic responses and analyze the
micromotor efficiency. In particular, the resistance, inductances, moment of inertia, viscous friction
coefficient, and other parameters can be directly measured or identified based upon micromotor testing.
The steady-state and dynamic analysis based upon the lumped-parameter mathematical model is straight-
forward. However, the lumped-parameter mathematical models simplify the analysis, and thus, these
models must be compared with the experimental data to validate the results.
The disadvantage of single-phase reluctance micromotors are high torque ripple, vibration, noise, low
reliability, etc. Therefore, let us study three-phase synchronous reluctance micromotors.

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