Page 669 - Mechatronics with Experiments

ELECTRIC ACTUATORS: MOTOR AND DRIVE TECHNOLOGY 655
The total torque developed as a result of the contribution from each phase is
T = T + T + T c (8.188)
a
m
b
◦
◦
2
2
2
∗
= K ⋅ i ⋅ (sin +sin ( + 120 ) +sin ( + 240 )) (8.189)
T
Note the trigonometric relation,
◦
◦ 2
◦ 2
(sin( + 120 )) = (cos sin 120 +sin cos 120 ) (8.190)
(√ ) 2
3 1
= cos − sin (8.191)
2 2
√
3 2 1 2 3
= cos + sin − sin cos (8.192)
4 4 2
√
3 2 1 2 3
◦ 2
(sin( + 240 )) = cos + sin + sin cos (8.193)
4 4 2
3
2
2
∗
T = K ⋅ i ⋅ (sin + cos ) (8.194)
m T 2
Hence, the torque is a linear function of the current, independent of the rotor angular
position, and the linearity constant torque gain (K ) is a function of the magnetic field
T
strength.
T = K ⋅ i (8.195)
m T
where,
∗ 3
K = K ⋅ (8.196)
T T
2
Therefore, a sinusoidally commutated brushless DC motor has the same linear rela-
tionship between current and torque as does the brush-type DC motor. Most implementa-
tions make use of the fact that the algebraic sum of three phase currents is zero. Therefore,
only two of the phase current commands and current feedback measurements are imple-
mented. Third phase information for both the command and feedback signal is obtained
from the algebraic relationship (Figures 8.32, 8.33).
Actual back EMF of a real motor is never perfectly sinusoidal nor trapezodial. The
ultimate goal in commutation is to maintain a current-torque gain that is independent
of the rotor position, that is a constant torque gain. In order to achieve that, the current
commutation algorithm must be matched to the back EMF function of a particular motor.
Clearly, if a trapezoidal current commutation algroithm is used with a motor which has a
sinusoidal back EMF function, the current-to-torque gain will not be constant. The resulting
motor is likely to have large torque ripple.
Let us consider the effect of commutation angle error on the current–torque rela-
tionship of the motor. Let be the measurement error in rotor angle. That is is the
e a
actual motor angle, and is the measured angle that is used for commutation algo-
m
rithm, = − . It can be shown that the current–torque relationship under non-zero
e a m
commutation angle error conditions is
T = K ⋅ i ⋅ cos( ) (8.197)
e
T
m
Notice that
1. when = 0, we have the ideal condition,
e
◦
2. when = 90 , then there is no torque generated by the current,
e
3. when −90 ≤ ≤ 90, effective torque constant is less than ideal,
e