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From ψ as = (L ls + L m – L ∆m cos2θ r )i as , one obtains a set of three first-order nonlinear differential
equations. In particular, we have

1
-------- = -------------------------------------------------i as – -------------------------------------------------i as ω r sin2θ r + -------------------------------------------------u as
di as
2L ∆m
r s
dt L ls + L m – L ∆m cos2θ r L ls + L m – L ∆m cos2θ r L ls + L m – L ∆m cos2θ r
--------- = 1 -- L ∆m i as sin( 2 2θ r – B m ω r – T L )
dω r
dt J
-------- = ω r
dθ r
dt
Example 14.5.3: Mathematical Model of Two-Phase Permanent-Magnet
Stepper Micromotors
For two-phase permanent-magnet stepper micromotors, we have

u as = r s i as + dψ as
-----------
dt
u bs = r s i bs + dψ bs
-----------
dt

where the flux linkages are ψ as = L asas i as + L asbs i bs + ψ asm and ψ bs = L bsas i as + L bsbs i bs + ψ bsm .
Here, u as and u bs are the phase voltages in the stator microwindings as and bs; i as and i bs are the phase
currents in the stator microwindings; ψ as and ψ bs are the stator flux linkages; r s are the resistances of the
stator microwindings; L asas , L asbs , L bsas , and L bsbs are the mutual inductances.
The electrical angular velocity and displacement are found using the number of rotor tooth RT,

ω r = RTω rm
θ r = RTθ rm

where ω r and ω rm are the electrical and rotor angular velocities, and θ r and θ rm are the electrical and rotor
angular displacements.
The flux linkages are functions of the number of the rotor tooth RT, and the magnitude of the flux
linkages produced by the permanent magnets ψ m . In particular,

ψ asm = ψ m cos RTθ rm ) and ψ bsm = ψ m sin RTθ rm )
(
(
The self-inductance of the stator windings is

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

The stator microwindings are displaced by 90 electrical degrees. Hence, the mutual inductances between
the stator microwindings are zero, L asbs = L bsas = 0.
Then, we have

(
ψ as = L ss i as + ψ m cos RTθ rm ) and ψ bs = L ss i bs + ψ m sin RTθ rm )
(

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