Page 305 - The Mechatronics Handbook

0066_frame_C14.fm Page 19 Wednesday, January 9, 2002 1:49 PM

Taking note of the circuitry equations, one has

[
(
dL ss i as + ψ m cos RTθ rm )]
u as = r s i as + -------------------------------------------------------------- = r s i as + L ss di as RTψ m ω rm sin RTθ rm )
(
-------- –
dt dt
dL ss i bs + ψ m sin RTθ rm )]
[
(
(
u bs = r s i bs + ------------------------------------------------------------- = r s i bs + L ss -------- + RTψ m ω rm cos RTθ rm )
di bs
dt dt
Therefore, we obtain
(
-------- = – -----i as + ---------------ω rm sin RTθ rm ) + -----u as
di as r s RTψ m 1
dt L ss L ss L ss
1
(
-------- = – -----i bs ---------------ω rm cos RTθ rm ) + -----u bs
di bs
r s
RTψ m
–
dt L ss L ss L ss
Using Newton’s second law, we have
dω rm 1
------------ = -- T e –( B m ω rm – T L )
dt J
----------- = ω rm
dθ rm
dt
The expression for the electromagnetic torque developed by permanent-magnet stepper micromotors
must be found. Taking note of the relationship for the coenergy

W c = 1 2 L ss i bs ) + ψ m i as cos ( RTθ rm ) + ψ m i bs sin ( RTθ rm ) + W PM
-- L ss i as +(
2
2
one finds the electromagnetic torque:

T e = ----------- = – RTψ m i as sin ( RTθ rm ) i bs cos ( RTθ rm )]
[
∂W c
–
∂θ rm
Hence, the transient evolution of the phase currents i as and i bs , rotor angular velocity ω rm , and dis-
placement θ rm , is modeled by the following differential equations:

1
-------- = – -----i as + RTψ m ( RTθ rm ) + -----u as
r s
di as
---------------ω rm sin
dt L ss L ss L ss
1
-------- = – -----i bs – RTψ m ( RTθ rm ) + -----u bs
r s
di bs
---------------ω rm cos
dt L ss L ss L ss
------------ = – --------------- i as sin[ ( RTθ rm ) i bs cos ( RTθ rm )] – B m 1
dω rm
RTψ m
–
------ω rm –
--T L
dt J J J
----------- = ω rm
dθ rm
dt
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