Environmental Engineering Reference
In-Depth Information
Equations (5.11) can best convey their meaning through a vector diagram in
the
d
-
q
plane according to the convention for
d
- and
q
-axes given for the stator
current,
i
s
.
U
s
jX
d
i
ds
u
ds
jX
s
i
s
q
i
ds
i
s
d
q
-axis
g
i
qs
u
qs
E
s
l
dr
d
-axis
Figure 5.18 Vector diagram for the SPM machine
The electrical power associated with the SPM machine is calculated in the
d
-
q
frame as shown in (5.12) and (5.13). In the latter relationship, (5.11) for
u
qs
is
substituted into (5.12):
P
e
pu
¼
Re
f
u
qds
i
qds
g
P
e
pu
¼
u
q
i
q
þ
u
d
i
d
ð
W
Þ
ð
5
:
12
Þ
P
e
pu
¼
n
pu
½
E
0
i
q
þð
X
d
X
q
Þ
i
d
i
q
W
Þ
ð
5
:
13
Þ
where * is the conjugate.
The currents given in (5.13) can be converted back to the stator current by
using the definition of current angles shown in Figure 5.18 relative to the
q
-axis,
and in doing so, we obtain the more common expressions for electrical power in a
synchronous machine. Equation (5.14) can form the basis of the M/G sizing
operation necessary to design for a specific power level, for example, peak regen-
erating power. To proceed from this point, it is necessary to have an understanding
of what constitutes the back-emf,
E
0
and the expressions for
d
- and
q
-axis reac-
tances (inductances).
In a practical machine, the rotor magnets are separated from the stator bore by
a physical airgap,
g
, in which the electromagnetic interaction takes place. For a
permanent magnet of remanence,
B
r
, the airgap flux density,
B
g
is given as
B
r
ð
Wb
=
m
2
B
g
¼
Þ
ð
5
:
14
Þ
1
þð
m
r
g
=
L
m
Þ
