Environmental Engineering Reference
In-Depth Information
i
ds
l
dr
+
+
L
m
/
L
r
w
Σ
sL
s
L
m
/
L
r
+
+
u
ds
+
+
i
dr
Σ
Σ
Σ
L
r
/(
pL
r
+
r
r
)
1/(
psL
s
+
r
s
)
L
m
r
r
/
L
r
Σ
+
-
+
+
L
m
/
L
r
1/
L
r
X
sw
e
+
+
3
P
2 2
Σ
L
m
/
L
r
T
em
w
Σ
-
+
X
sw
e
sL
s
u
qs
-
-
+
+
L
r
/(
pL
r
+
r
r
)
1/(
psL
s
+
r
s
)
L
m
r
r
/
L
r
Σ
Σ
Σ
+
-
1/
L
r
+
L
m
/
L
r
i
qr
Σ
-
L
m
/
L
r
i
qs
l
qr
Figure 7.3 Block diagram model of the induction machine in synchronous d-q
frame
The classical model for an induction machine is next obtained by substituting
the expressions for flux linkage into the expressions for stator and rotor voltage in
(7.1), obtaining
u
qs
¼ð
r
s
þ
L
s
p
Þ
i
qs
þ
w
e
L
s
i
ds
þ
L
m
pi
qr
þ
w
e
L
m
i
dr
u
ds
¼ð
r
s
þ
L
s
p
Þ
i
ds
w
e
L
s
i
qs
þ
L
m
pi
dr
w
e
L
m
i
qr
0
¼ð
r
r
þ
L
r
p
Þ
i
qr
þ
s
w
e
L
r
i
dr
þ
L
m
pi
qr
þ
s
w
e
L
m
i
ds
0
¼ð
r
r
þ
L
r
p
Þ
i
dr
s
w
e
L
r
i
qr
þ
L
m
pi
ds
s
w
e
L
m
i
qs
l
qr
¼
L
r
i
qr
þ
L
m
i
qs
l
dr
¼
L
r
i
dr
þ
L
m
i
ds
m
em
¼
ð
7
:
3
Þ
3
2
P
2
ð
l
qr
i
dr
l
dr
i
qr
Þ
The equations in (7.3) can easily be put into matrix form for a clearer under-
standing of the relationships between stator currents, rotor currents and the terminal
voltages in the synchronous reference frame. In most control scenarios it would be
appropriate as a next step to solve (7.3) for the currents in differential form and then
to simulate the system for their response. In this derivation, a block diagram
approach is taken to more clearly illustrate the cause and effect relations amongst
the machine variables and parameters involved. Figure 7.3 is the resultant model
for the cage rotor induction machine using the solutions for currents in (7.3).
Under rotor flux FOC it is necessary that l
qr
= 0 so that only
d
-axis rotor flux
exists. For this condition to prevail, the output of the block driving
q
-axis rotor flux
