Environmental Engineering Reference
In-Depth Information
to solve for the time increments given in (6.15) with the proviso that the reference
vector,
U
k
, remains constant during a switching cycle:
U
k
T
¼
U
k
T
k
þ
U
k
þ
1
T
k
þ
1
ð
6
:
15
Þ
where (6.15) must be solved for
T
k
and
T
k+
1
as fractions of the switching cycle
period,
T
. The null vector,
U
0
or
U
7
, must persist for a time increment given by (6.16):
T
0
¼
T
T
k
T
k
þ
1
ð
6
:
16
Þ
When the null vector on time,
T
0
= 0, SVPWM has reached its limit of applic-
ability and its modulation depth is maximized for full PWM. Beyond this limit,
pulse dropping must occur just as it does for sine-triangle PWM (see Figure 6.11).
Now, by referring again to Figure 6.13 and rewriting (6.15) and (6.16) as integrals it
can be seen that the reference vector
U
k
must satisfy
ð
ð
T
k
þ
1
ð
ð
T
T
k
T
U
k
dt
¼
U
k
dt
þ
U
k
þ
1
dt
þ
ð
6
:
17
Þ
U
0
;
7
dt
0
0
T
k
T
k
þ
T
k
þ
1
where
k
is index of vectors adjacent to the sector in which the reference vector is
located. Because the switching frequency is at least an order of magnitude greater
than the fundamental frequency, the transitions between states in (6.17) will occur
for essentially quasi-static behaviour of the commanded reference.
Refer again to Figure 6.13 and note the angle that the vectors
U
k
,
U
k+
1
and
U
s
make with the
a
-axis and then rewrite (6.15) as (6.18) in terms of actual magnitudes:
h
3
3
i
3
2
m
i
U
dc
½
cos
ð
0
Þþ
j
sin
ð
0
Þ
T
k
þ
3
2
m
i
U
dc
cos
p
p
þ
j
sin
3
2
m
i
U
dc
½
cos
ðgÞþ
j
sin
ðgÞ
¼
ð
6
:
18
Þ
where the modulation index has been defined previously as the ratio of the max-
imum sinusoidal synthesized wave amplitude to that of an equivalent square wave
magnitude, and
g
=
w
t
. Therefore, solving (6.18) results in expressions for
T
k
and
T
k
+1
as
T
k
¼
m
i
T
sin
ð
p
=
3
g
Þ
sin
ðp=
3
Þ
ð
6
:
19
Þ
sin
ðgÞ
sin
T
k
þ
1
¼
m
i
T
ð
p
=
3
Þ
T
0
¼
T
T
k
T
k
þ
1
