Page 307 - The Mechatronics Handbook

0066_frame_C14.fm Page 21 Wednesday, January 9, 2002 1:50 PM

The self-inductances of the stator windings are found to be

L ss = L asas = L bsbs = L ls + L m

The stator windings are displaced by 90 electrical degrees, and hence, the mutual inductances between
the stator windings are L asbs = L bsas = 0. Thus, we have

ψ as = L ss i as + ψ m sin θ rm and ψ bs = L ss i bs ψ m cos θ rm
–
Therefore, one finds

dL ss i as + ψ m sin θ rm )
(
u as = r s i as + ------------------------------------------------- = r s i as + L ss -------- + ψ m ω rm cos θ rm
di as
dt dt
(
–
dL ss i bs ψ m cos θ rm ) di bs
u bs = r s i bs + ------------------------------------------------- = r s i bs + L ss -------- ψ m ω rm sin– θ rm
-
dt dt
Using Newton’s second law
2
d θ rm
T e – B m ω rm – T L = J -------------
2
dt
we have
-- T e –(
------------ = 1 B m ω rm – T L )
dω rm
dt J
----------- = ω rm
dθ rm
dt
The expression for the electromagnetic torque developed by permanent-magnet motors can be obtained
by using the coenergy

2
-- L ss i as +(
W c = 1 2 L ss i bs ) + ψ m i as sin θ rm – ψ m i bs cos θ rm + W PM
2
Then, one has

T e = ----------- = ----------- i as cos( θ rm + i sin θ rm )
Pψ m
∂W c
bs
∂θ rm 2
Augmenting the circuitry transients with the torsional-mechanical dynamics, one finds the mathemat-
ical model of two-phase permanent-magnet micromotors in the following form:

1
-------- = – -----i as – ψ m θ rm + -----u as
r s
di as
-------ω rm cos
dt L ss L ss L ss
1
-------- = – -----i bs + ψ m θ rm + -----u bs
r s
di bs
-------ω rm sin
dt L ss L ss L ss
------------ = ----------- i as cos( θ rm + i bs sin θ rm ) – B m 1
dω rm
Pψ m
------ω rm –
--T L
dt 2J J J
----------- = ω rm
dθ rm
dt
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