Page 303 - The Mechatronics Handbook

0066_frame_C14.fm Page 17 Wednesday, January 9, 2002 1:49 PM

L m
L m max
L m
L m
L m
L m min

0 3 2 r
2
FIGURE 14.8 Magnetizing inductance L m (θ r ).

Assume that this variation is a sinusoidal function of the rotor angular displacement. Then,

L m θ r = L m – L ∆m cos 2θ r
()
is the average value of the magnetizing inductance and L ∆m is half of the amplitude of the
where L m
sinusoidal variation of the magnetizing inductance.
The plot for L m (θ r ) is documented in Fig. 14.8.
The electromagnetic torque, developed by single-phase reluctance motors is found using the expression
for the coenergy W c (i as , θ r ). From W c (i as , θ r ) = --(L ls + L m – L ∆m cos 2θ r )i as , one finds
1
2
2
[
(
1 2
∂W c i as ,θ r ) ∂ --i as L ls + L m – L ∆m cos2θ r )]
(
2
T e = --------------------------- = ---------------------------------------------------------------------- = L ∆m i as sin 2θ r
2
∂θ r ∂θ r
The electromagnetic torque is not developed by synchronous reluctance motors if IC feeds the
dc current or voltage to the motor winding because T e = L ∆m i as sin2θ r . Hence, conventional control
2
algorithms cannot be applied, and new methods, which are based upon electromagnetic features must
be researched. The average value of T e is not equal to zero if the current is a function of θ r . As an
illustration, let us assume that the following current is fed to the motor winding:
i as = i M Re( sin 2θ r )
Then, the electromagnetic torque is

2
(
T e = L ∆m i as sin 2θ r = L ∆m i M Re sin 2θ r ) sin2θ r ≠ 0
2
2
and
T eav = 1 ∫ π L ∆m i as sin2θ r dθ r = 1 --L ∆m i M
2
2
---
π 0 4
The mathematical model of the microscale single-phase reluctance motor is found by using Kirchhoff’s
and Newton’s second laws

u as = r s i as + dψ as (circuitry equation)
-----------
dt
2
d θ r
T e – B m ω r – T L = J---------- ( torsional-mechanical equation)
2
dt

©2002 CRC Press LLC