SENSORLESS FIELD ORIENTED CONTROL OF BRUSHLESS ...

Figure 2.26 – Rotor-stator flux linkage for a sinusoidal and arbitrary trapezoidal motor.
The expressions for torque and bEMF were derived previously as Equations (2.31)-(2.32) and
Equations (2.36)-(2.37).
(2.31): ( ) N
D Y
B sin(
) i(
t)
T r
p r
(2.32): ( ) K
sin( ) i(
t)
T r
t r
(2.36): e( ) NDYB sin( ) (
t)
r p r
(2.37): e( ) K sin( ) (
t)
r e r
Comparing Equations (2.43)-(2.44), (2.31)-(2.32), and (2.36)-(2.37) makes clear the equivalence
shown in Equation (2.48).
R Ke Kt NDY Bp
(2.48)
Equation (2.48) shows that the units of e K and t
K are the units of flux linkage, Wb( turns)
.
Further, the derivative of Equation (2.44) with respect to rotor position is Equation (2.49).
d
R( r) Rsin( r)
(2.49)
d r
Comparing Equation (2.49) with Equations (2.32) and (2.37) makes clear the relationships shown
in Equations (2.50) and (2.51).
d
Tt () R( r)
it ()
(2.50)
d r
d
et () R( r) ()
t
(2.51)
d r
Similar to Equation (2.48), these can be written as Equation (2.52).
44