Page 668 - Mechatronics with Experiments

654 MECHATRONICS

i + i + i = 0
2 1 2 3
i i + i + i = i
3 1 2 3 s
1
i
s
i 2 i 1 i 3
3 i 1 i 2 FIGURE 8.33: Brushless motor current vector
and phase current components (three-phase Y
wound).

The current in each of the three phases is controlled by 120 degrees phase shift
relative to each other. The rotor position is tracked by a position sensor. The vector sum of
phase currents are commutated relative to the rotor position such that two magnetic fields
are always perpendicular to each other. The algebraic sum of the currents in three phases
is zero, but not the vector sum (Figure 8.33). A current feedback loop is used in the PWM
circuit to regulate the current in each phase with sufficient dynamic bandwidth. Notice that
the magnitude of the magnetic field generated by the permanent magnets is constant and not
actively controlled and it rotates with the rotor. Whereas the stator field (a vector quantity
which has magnitude and direction) is controlled actively by the drive current control loop.
In digital implementation, the current command, the commutation, and current control
algorithms can reside either in the drive or in a higher level controller. In the latter case, the
drive performs only the PWM modulation and is called “the power block.” In most cases,
current commutation and regulation algorithms are implemented in the drive.
The stator being static by definition, the angle of each stator phase is fixed on the
diagram (Figure 8.33). The magnitude of each phase current (i , i , i ) for a given total
a b c
⃗
current vector (i ) is the projection of the current vector onto the phase (Figure 8.33).
s
⃗
i = i ⋅ ⃗ u n (8.181)
n
s
⃗ u is the unit vector for the phase, ⃗ u for phase a, ⃗ u for phase b, ⃗ u for phase c.
n a b c
The derivation below shows that the sinusoidal commutation algorithm for a three-
phase brushless motor (which is designed to have a sinusoidal back EMF) produces the
same torque–current relationship as that of a brush-type DC motor. Let us assume that
the brushless motor has three phase winding and each phase has a sinusoidal back EMF
as a function of rotor position. As a result, the current-to-torque gain for each individual
phase has the same sinusoidal function. For each phase, they are displaced from each other
by a 120 degrees angle as a result of the physical distribution of the windings around the
periphery of the stator.
Consider that rotor is at angular position, , and each phase of the stator has current
values i , i , i . The torque generated by each winding is T , T , and T .
a b c
a
c
b
∗
T = i ⋅ K ⋅ sin( ) (8.182)
a a T
◦
∗
T = i ⋅ K ⋅ sin( + 120 ) (8.183)
b b T
◦
∗
T = i ⋅ K ⋅ sin( + 240 ) (8.184)
c c T
Let us control the current (with commutation and current feedback control algorithms)
such that each phase is 120 degrees apart from each other and sinusoidally modulated as a
function of the rotor position.
i = i ⋅ sin( ) (8.185)
a
◦
i = i ⋅ sin( + 120 ) (8.186)
b
◦
i = i ⋅ sin( + 240 ) (8.187)
c