Page 250 - Rashid, Power Electronics Handbook

14 Inverters 239

the eight states of the VSI normalized with respect to v (Table switching frequency. Although there is not a systematic
i
14.3), which generates the eight space vectors (V , approach to generate a SV sequence, a graphical representa-
i
i ¼ 1; 2; ... ; 8) in Fig. 14.18. As expected, V to V are tion shows that the sequence V , V iþ1 , V (where V is
6
z
i
z
1
nonnull line voltage vectors and V and V are null line- alternately chosen among V and V ) provides high perfor-
8
7
7
8
voltage vectors. mance in terms of minimizing unwanted harmonics and
The objective of the SV technique is to approximate the reducing the switching frequency.
line-modulating signal space V with the eight space vectors
c
(V , i ¼ 1; 2; .. . ; 8) available in VSIs. However, if the modu- 14.3.4.3 The Normalized Sampling Frequency
i
lating signal V is laying between the arbitrary vectors V and The normalized carrier frequency m in three-phase carrier-
i
c
V , only the nearest two nonzero vectors (V and V ) and f
iþ1 i iþ1 based PWM techniques is chosen to be an odd integer number
one zero SV (V ¼ V or V ) should be used. Thus, the
z 7 8 multiple of 3 (m ¼ 3 n; n ¼ 1; 3; 5; ...). Thus, it is possible
f
maximum load line voltage is maximized and the switching
to minimize parasitic or nonintrinsic harmonics in the PWM
frequency is minimized. To ensure that the generated voltage
waveforms. A similar approach can be used in the SVM
in one sampling period T (made up of the voltages provided
s technique to minimize uncharacteristic harmonics. Hence, it
by the vectors V , V , and V used during times T , T , and
i iþ1 z i iþ1 is found that the normalized sampling frequency f should be
sn
T ) is on average equal to the vector V , the following
z c an integer multiple of 6. This is due to the fact that in order to
expression should hold:
produce symmetrical line voltages, all the sectors (a total of 6)
should be used equally in one period. As an example, Fig.
V T ¼ V T þ V iþ1 T iþ1 þ V T z ð14:37Þ 14.19 shows the relevant waveforms of a VSI SVM for f ¼ 18
i
c
z
s
i
sn
and ^ v ¼ 0:8. Figure 14.19 con®rms that the ®rst set of
c
The solution of the real and imaginary parts of Eq. (14.37) relevant harmonics in the load line voltage are at f , which
sn
for a line-load voltage that features an amplitude restricted to is also the switching frequency.
0 ^ v 1 gives
c
T ¼ T ^ v sinðp=3 ÿ yÞ ð14:38Þ 14.3.5 DC Link Current in Three-Phase VSIs
s
c
i
T iþ1 ¼ T ^ v sinðyÞ ð14:39Þ Due to the fact that the inverter is assumed to be lossless and
s
c
T ¼ T ÿ T ÿ T iþ1 ð14:40Þ constructed without storage energy components, the instanta-
i
s
z
neous power balance indicates that
The preceding expressions indicate that the maximum
fundamental line-voltage amplitude is unity as 0 y p=3. v ðtÞ i ðtÞ¼ v ðtÞ i ðtÞþ v ðtÞ i ðtÞþ v ðtÞ i ðtÞ
ab
b
a
i
bc
i
ca
c
This is an advantage over the SPWM technique which achieves ð14:41Þ
p
a 3=2 maximum fundamental line-voltage amplitude in the
linear operating region. Although, the SVM technique selects where i ðtÞ, i ðtÞ, and i ðtÞ are the phase-load currents as
a b c
the vectors to be used and their respective on-times, the shown in Fig. 14.20. If the load is balanced and inductive, and
sequence in which they are used, the selection of the zero a relatively high switching frequency is used, the load currents
space vector, and the normalized sampled frequency remain become nearly sinusoidal balanced waveforms. On the other
undetermined. hand, if the ac output voltages are considered sinusoidal and
For instance, if the modulating line-voltage vector is in the dc link voltage is assumed constant v ðtÞ¼ V , Eq. (14.41)
i
i
sector 1 (Fig. 14.18), the vectors V , V , and V should be can be simpli®ed to
2
1
z
used within a sampling period by intervals given by T , T ,
2
1
8 p  p  9
and T , respectively. The question that remains is whether > 2V sinðotÞ 2I sinðot ÿ fÞ >
z
o1
o
p
the sequence (i) V ÿ V ÿ V , (ii) V ÿ V ÿ V ÿ V , i ðtÞ¼ 1 < þ 2V sinðot ÿ 120 Þ p  =

z
2

z
1
2
1
z
2I sinðot ÿ 120 ÿ fÞ
(iii) V ÿ V ÿ V ÿ V ÿ V , (iv) V ÿ V ÿ V ÿ V ÿ i V > p  o1 p  o >
2
1
z
1
1
z
z
z
2
i:
;
o
o1
V ÿ V ÿ V , or any other sequence should actually be þ 2V sinðot ÿ 240 Þ 2I sinðot ÿ 240 ÿ fÞ
2 1 z
used. Finally, the technique does not indicate whether V ð14:42Þ
z
should be V , V , or a combination of both.
7 8
where V is the fundamental rms ac output line voltage, I is
o1 o
14.3.4.2 Space-Vector Sequences and Zero the rms load-phase current, and f is an arbitrary inductive
Space-Vector Selection load power factor. Hence, the dc link current expression can be
further simpli®ed to
The sequence to be used should ensure load line voltages that
feature quarter-wave symmetry in order to reduce unwanted
harmonics in their spectra (even harmonics). Additionally, the i ðtÞ¼ 3 V o1 I cosðfÞ¼ p  V o1 I cosðfÞ ð14:43Þ
3
o
i
l
zero SV selection should be done in order to reduce the V i V i