Environmental Engineering Reference
In-Depth Information
this unaligned position. The inductance is low, so the rise time of the current is very
rapid until the current amplitude reaches the inverter current limit,
I
. At this point, the
inverter current regulator begins to PWM the current, so that it remains at its
commanded value while the rotor moves from unaligned to aligned position. After
sufficient dwell, the current is switched OFF and allowed to decay along the flux
linkage-current trajectory for fully aligned position back to the origin. The coun-
terclockwise motion about the
l
-
I
diagram encloses an area,
W
, representing the net
energy exchanged to mechanical energy at the shaft. The area beneath the unaligned
position line represents phase A leakage inductance. The area from the fully aligned
curve to the
l
-axis represents energy stored in the winding and returned to the supply,
the reactive kVA in other words. The spacing between the
l
I
curves in Figure 5.53
is a function of rotor angle,
q
. Bear in mind that as the inverter sequences between
the phases in the order A-B-C, the rotor indexes clockwise as noted by
W
.
Torque production in the VRM is given by (5.32) for average and instanta-
neous values:
mN
r
W
2
p
T
¼
@
W
ð
i
,
q
Þ
@q
T
avg
¼
ð
Nm
Þ
ð
5
:
32
Þ
where
W
equals the energy converted per working stroke of the machine. That is the
energy converted per phase excitation. It requires excitation of all
m
-phases to move
the rotor by one rotor tooth pitch - hence, the quantity in the numerator for average
torque of the number of 'working' strokes per revolution times the work performed
per stroke. The flux linkage in the VRM is given by (5.33) and the expression for
induced voltage. The variable
L
(
q
) is the inductance variation with rotor position:
l ¼
L
ðqÞ
i
d
l
dt
¼
L
di
dt
þ
i
dL
d
q
dt
E
¼
ð
5
:
33
Þ
d
q
d
q
dt
¼ w
From these expressions, it is easy to compute the electrical power as the pro-
duct of back-emf,
E
, and current and obtain
2
Li
2
d
dt
1
1
2
i
2
dL
P
e
¼
þ
d
q
w
ð
5
:
34
Þ
1
2
i
2
dL
T
¼
d
q
where the first term in the expression for electric power represents the reactive volt-
amps, that is, derivative of the stored field energy, and the second term is the
mechanical output power. Instantaneous torque is rewritten in (5.34) to highlight
the fact that it is a function of current squared. This latter point can be interpreted to
mean that part of the input current is used to excite, for example, magnetize, the
machine and part is used to develop mechanical work.
