Optimized Space Vector Modulation and Reg
PWM: A Reexamination
Jian Sun,
Horst Grotstollen
Institute for Power Electronics and Electrical Drives
Department of Electrical Engineering, University of Paderborn
Warburger Strasse 100,33098 Paderbom, Gemzany
Tel.: 5251-603881; Fax: 5251-603443; E-mail: sun@pblea.uni-paderborn.de
Abstract - The relationship between space vector modulation
and regular-sampled PWM techniques for hard-switching PWM
inverters and rectifiers are reexamined in this paper. Space
vector modulation with optimized apportioning of null vector
time that results in minimum current THD is investigated and is
compared with previously known methods. The conventional
optimal regular-sampled PWM method and its relation to the
optimized space vector modulation are discussed. Optimal
discontinuous modulation for minimizing both the switching
losses and the current THD will also be studied.
I.' INTRODUCTION
Pulse-width modulation (PWM) has been one of the most
intensively studied areas of power electronics for over three
decades now. For high-frequency voltage-source inverters as
depicted in Fig. 1, which can be used as a PWM rectifier as
well, two widely applied PWM methods are the suboscillation
method [l] and the space vector modulation [12]. Both
methods have moderate requirements on realization hardware
and feature good transient responses.
-7
I
L
I
-
I
i
I
Fig. 1. Three-phase PWM converter.
In the suboscillation method, pulse patterns are generated
by comparing reference voltages (also called modulation
waves) with a triangular wave. For practical implementation
using digital technologies, it is advantageous first to sample
the modulation wave once (or twice) every cycle of the triangular wave at regularly spaced intervals to produce a sampledhold modulation wave, and then to compare it (digitally) with
0-7803-3544-9196 $5.00 0 1996 IEEE
the triangular wave to define the switching instants. This
technique has been called regular-sampled PWM [2]. By
adding proper zero-sequence harmonics (triplen harmonics of
order 3, 6, ...) [3] to the modulation waves, the performance
of regular-sampled PWM can be improved in terms of
maximized output voltage range [4-61, minimum harmonic
distortions [2,7-81, or reduced switching losses [9-111.
Space vector modulation [12] is based on transforming
three-phase quantities into the a-P plane in which possible
output states of the converter as shown in Fig. 1 correspond to
eight voltage vectors: six active vectors that form a hexagon
and two null (zero) vectors located at the origin, see Fig. 3.
The desired three-phase sinusoidal output voltages correspond
to a circle in the a-p plane. In each switching period, the
required (sampled) reference voltage vector is realized by
using two directly adjacent active vectors; their duty ratios are
determined in such a way that the real output voltage vector,
when averaged over one switching cycle, coincides with the
reference vector. The sum of the two duty ratios is smaller
than one as long as the reference vector lies inside the circle
shown in Fig. 3, implying that the two active vectors occupy
only part of a switching cycle; the rest has to be filled out by
using null vectors and is called null-vector time.
The comparison between regular-sampled PWM and space
vector modulation has been the subject of many papers, e.g.
[ l l , 13-20]. However, the general relationship between them
has yet not been widely understood. On the other hand, the
apportioning of null-vector time between two null vectors
represents a degree of freedom that can be used to improve the
performance of space vector modulation. Although this has
been known for some years and has been addressed in some
papers [14-15, 211, a real optimal apportioning factor has
seldom been studied. In particular, the relation of such
optimized space vector modulation to regular-sampled P W
with different non-sinusoidal modulation waves has not been
sufficiently covered in the literature. Considering these, a
systematic study on space vector modulation and regularsampled PWM, especially the relationship between them and
their optimized forms, are presented in this paper. The paper is
organized as follows.
956
First, in Section 11, the gene:ral relation between space
vector modulation and regular-sampled PWM is reexamined,
and a one-to-one correspondence between them is established.
A direct optimization approach is; then taken in Section III to
determine ithe optimal apportioning of null-vector time in
space vector modulation. Using the general relation established, the corresponding optimal regular-sampled PWM is
then developed. In Section IV, the objective function for
optimization is extended to include the switching losses of the
converter, and the corresponding optimization results are
presented.
11. RELATIONSHIP BETWEEN THE TWO
MODULATION METHODS
A. Structures of Pulse Pattems
Consider first the regular-sampled PWM where the
modulation wave is sampled twic;e every cycle of the triangular wave. This is usually called double-edge modulation [2],
and is scheimatically illustrated in Fig. 2. For simplicity, all
voltages are normalized by half of the DC-link voltage, vdc.
Hence the amplitude of the triangular wave, c(t), is one, and
the magnituide of each reference wave may not exceed one in
order to avoid the presence of low-frequency harmonics. The
switching instants in each switching period is determined by
comparing the sampled-hold modulation waves with the triangular wave, as illustrated in Fig. 2. With simple geometric
calculations.,the switching instants can be determined to be
T, = T; ( 1 -e,)/2
i
T b = T; (1-f?b)/2
T, = T , . (1 - e , ) /2
Fig. 2. Regular-sampled PWM with double-edge modulation.
VlOO
(1)
where T, is the switching period. The corresponding pulse
patterns are shown in Fig. 2, where it is assumed that
e , 2 eb 2 e , .
Consider now the pulse patterns generated by space vector
modulation. Notice that the transformation from three phase
quantities to the a-p coordinate system is defined to be
Fig. 3. Output voltage vectors of a three-phase PWM converter in the a-Pplane.
The reference voltage vector correspondingto e,, eb, and e,,
as used for regular-sampledPWM illustrated in Fig. 2, is
--1
2
h
2
Referring to Fig. 1, each phase of the converter can output two
different (normalized) voltages: + 1 when the upper transistor
is on, and -1 when the lower is on. Thus, there exist totally 23
= 8 possible combinations of output voltages, which, when
transformed using (2), form a hexagon in the a-(3 plane, as
shown in Fig. 3. The binary indexes used there correspond to
the conduction state of each phase for producing the voltage
vector. For example, the vector vlo0 corresponds to v, = VdJ2
and V b = V, 3 -vd&?.
Since it is assumed that e , 2 eb 2 e,, it is easily to check that
e,20,
ep>0;
eP<,/5.e,.
The reference vector thus lies inside the triangle formed by
vectors vloo and vllo (Sector I) and can be realized by using
these two adjacent vectors as well as the two null vectors. To
reduce switching frequency, it is a common practice [13] to
output these vectors in the sequence of
yo00 + V l o o
957
-+
V l l O -+
Vlll-+
V l l O + V l o o + yo00
T I = T b - T , = T,(e,-eb)/2,
T3 = T,-Tb = T s ( e b - e , ) / 2 ,
as can be easily verified, the two sets of pulse patterns are
identical if To = T,, which is equivalent to
1 +e,
y=- T, - Tc (4)
To+Tl
T,+T,-T,
2-ea+e,'
. 4.
Pulse patterns generated by space vector modulation.
in each pair of switching periods. This results in the pulse
patterns as illustrated in Fig. 4, where the determination of
durations To, T1,T3, and T7 will be discussed later.
Comparing Fig. 2 and Fig. 4, it is clear that the pulse
patterns produced by using regular-sampled PWM and space
vector modulation have the same structure. The discussion has
been based on the assumption that e,2eb2e,, but the
conclusion is valid in general, as can be easily verified. For
regular-sampled PWM with single-edge modulation, that is,
the modulation wave is sampled only once every cycle of the
triangular wave, there also exists a corresponding space vector
modulation scheme in which each phase is switched twice
every switching cycle at instants symmetrical about the center
of the switching period, so that the produced pulse patterns
still have the same structure. Considering this, further discussions will be concentrated on double-edge modulation and its
corresponding space vector modulation.
This expression is valid for reference vectors lying in Sector I
( e , 2 eb 2 ec),but can be easily generalized to other sectors as
1 + emin
Y=
2 - emax + emin
where emin= min{e,, eb, e,}, e, = "{e,,
(5)
eb, e,}.
Similarly, given a reference vector e, + jep in the a-p
plane, an apportioning factor y, and the corresponding pulse
patterns generated by space vector modulation, it is possible to
determine three voltages e,, eb, and e, with which regularsampled PWM will generate the same pulse patterns. To see
this, notice first that two pulse patterns as shown in Fig. 2 or
Fig. 4 are the same if they have equal average value over one
switching cycle. Also, with regular-sampled PWM, each
average phase voltage (over one switching cycle) is equal to
the sampled value of the modulation wave. Thus, to produce
the same pulse patterns as from space vector modulation, the
sampled-hold modulation wave e,, eb, and e, for regularsampled PWM must satisfy
B. Apportioning of Null-Vector Erne
I
.....................
I
. Ip . I
=
Generally, with space vector modulation, the reference
L- CJ
vector is realized by two directly adjacent active vectors, with
durations determined in such a way that the resulting average where Vo is the average zero-sequence component contained
output voltage vector coincides with the reference vector [12]. in the outputs generated by space vector modulation, that is a
Using this principle, the durations T1 and T3 in Fig. 4 are function of e,, ep, and the apportioning factor yt:
determined as follows where T, is the switching period:
1
= 2 y - 1 + - 2[ e a ( 1 - 3 y ) + J S e p ( l - y ) ]
(7)
v0
Since (6) has a unique solution for each given set of (ea, ep, y),
it can be concluded that for space vector modulation using a
These determine also the total null-vector time (To + T7 = T, - certain apportioning factor, there exists an equivalent regularTl - T3), but not yet its apportioning between the two null sampled PWM that will generate the same pulse patterns.
1 , is, each individual time To and T7 is
vectors vooo and ~ 1 ~that
still not known. Thus, to complete the definition of pulse
patterns, an apportioningfactor y defined as
t. Function (7) is valid when the reference Yector is inside Sector I.
For other cases, similar functions can be obtained using
7T
Go = 2y- 1 + h m y c o s ( 0 - ( 2 k - 1 ) -) +
' 6
kx
mcos(0--)
k = 1, 3, 5
3
(3)
needs to be specified using some criteria. In the conventional
space vector modulation [ 131, y is simply set to 0.5 such that
the null-vector time is equally divided between vm and ~ 1 1 1 .
5T
mcos(e- ( k - i ) - )
3
Consider now the problem of choosing a proper apportioning factor such that the pulse patterns shown in Fig. 4 are
identical to those in Fig. 2. Since
A-,
k = 2, 4, 6
where m =
and 0 is the position angle of the reference
vector measured from vector vim [17].
958
C. Modijications of Modulation Waves
The previous discussions have focused on details of pulse
patterns within every switching cycle, and demonstrated the
equivalence between regular-sampled PWM and space vector
modulation. As can be seen from Fig. 2, all three switching
instants will be shifted by a same amount if a common value is
added to each modulation wave. Evidently, this will result in
increased (or reduced) average output phase voltage, but the
average line voltage remain unaffected. In fact, the added
common component affects only the positions of line voltage
pulses, but their widths remain the same. Considering this,
different zero-sequence harmonics (triplen harmonics of order
3, 6, 9, 12, ...) [3] can be purposely added to the three-phase
sinusoidal reference waves to improve the performance of
regular-sampled PWM.
-1t
. .
.
.
.
i
50
100 150 200
250 300
350
50
100 150 200 250 300 350
The same effect can be achieved with space vector
modulation by properly choosing the apportioning factor.
Thus, for each three modified, non-sinusoidal modulation
waves ea(t),eb(t), and e,(t), (5) can be used to define a corresponding lfunction of apportioning factor for an equivalent
space vector modulation that produce the same modulated
- 1 L ...
50 100 150 200 250 300 350
three-phase voltages. Inversely, for space vector modulation
with the apportioning factor defined by a certain function, the
corresponding modulation waves for the equivalent regularsampled PWM can be determined from (6). To demonstrate
this, Fig. 5 shows the corresponding apportioning factors for
two regular-sampled PWM methods. In Fig. 6, the zero-0.51
sequence component, Vo(e), and the modulation wave, e,@),
-11 . .
U
1
corresponding to several simple apportioning methods known
50
100
150
200
250
300
350
in the literature are illustrated. The modulation index m is
defined as the amplitude of the required output phase voltage Fig. 6. Zero-sequence component v @ ) and modulation wave
e,(€)) for m = 1 and corresponding to space vector modulation
as normalized by half of the DC-link voltage.
with a) y = 0.5; b) y = 1; c) y = 0; d) y = 0.5 [l - sgn(cos38)].
,
111. MINIMIZATION OF HARMONIC
CURRENT DISTORTION
0.4
0.2
0‘
350
m = 0.8
Fig. 5. Apportioning factors corresponding to regular-sampled P W I with a) e, = mcos8 - ( m / 6 ) cos38 [4]; b) pure
sinusoidal modulation waves.
As mentioned before, adding a zero-sequence component to
the modulation waves for regular-sampled PWM or changing
the apportioning factor in space vector modulation leads to the
shift of pulse positions in output line voltages within a
switching period, thus will also affect the converter output
current waveform. Therefore, harmonic current distortions can
be reduced by properly selecting the zero-sequence
component or the apportioning factor. This optimization possibility has been studied in several papers, but most previous
works have focused on comparing different apportioning
factors or modulation waves in use. A direct optimization
approach is taken here to find the real optimal apportioning
factors that result in minimum harmonic current distortiont.
t. When preparing this paper, the authors found out that a direct optimization approach has also been taken in [22] where similar
results have been reported.
959
Substitute 8 in (8) with 0, and solve (8).
Finally, the local harmonic current rms value, iTHD,is
determined as a function of m, 0k, N,and y:
A. Optimal Apportioning Factor
The optimization is based on the following assumptions:
The converter is connected to a symmetrical three-phase
inductive load, each phase of which consists of an inductor
L and a sinusoidal counter voltage source. This represents
two typical applications: as an inverter supplying an AC
motor or as a rectifier with power factor correction.
The switching frequency is high compared to the fundamental frequency such that the load counter voltage can be
treated as constant within each switching cycle.
The converter operates in steady state, and the deviation of
each phase current from its sinusoidal reference is zero at
the beginning of each switching cycle.
To simplify the notation, voltages are normalized by Vd&
and currents by Vdc/(27cflL),fi being the fundamental
frequency of converter output. Based on these, the model
describing current deviations from the sinusoidal reference
waves in the a-p coordinate system is found to be
Z/N
&
im
(,-,
N
%,,N,y) = 7c
5
[Aii+Ai;]dB
(10)
0
With help of the computer algebra system Mathematica, a
closed-form express is found for i,,
and is given in the
appendix. Using that expression, optimal apportioning factor
that minimizes ,i
is calculated and is given below:
yop = 1/2+
n
8 + 3m2- 8 h s i n ( 0 c -)
3
+ 3m2sin ( 2 8 + n-)6
B. Evaluation of Optimization Results
Notice that (11) is valid for 8 E [0, 60'1, but can be easily
extended to other intervals based on the symmetry of three
phases. The results are illustrated in Fig. 7. For m larger than
1 2 f i l 4 9 the function yields values that are outside the range
[0, I] and need to be limited. The flat top in Fig. 7 is due to
where T i s the transformation matrix defined in (2); sa,sb, and this limitation. As can be seen, the optimal apportioning factor
s, are equal to +I or -1, depending on whether the upper or varies around 0.5. For each m, the curve is symmetrical about
lower transistor of the corresponding phase leg is on; 8 = 1x6Oo,I = 1, 2, ..., 6 . It is very close to the line y = 0.5
8 = 27cf1t, and m is the modulation index as defined in the when m is small, indicating that conventional space vector
last section. This model can be interpreted as following: The modulation with y = 0.5 is optimal in low modulation index
output harmonic currents are due to harmonic voltages across region. In contrast, the curve is better approximated by
the load inductors that are the differences between converter function y = [l + sgn(cos38)]12 in the high modulation index
output voltages and their fundamental components (the first region, which would suggest that it might be advantageous to
use flat-top modulation in that range (refer to Fig. 6).
and second term in the right-hand side of (8), respectively).
r--7
LSd
As can be seen, the model (8) does not involve any load
parameter, which allows the optimal pulse patterns to be determined as a function of the modulation index. The objective is
to minimize the total (global) harmonic current distortion
1
0.8
2n
2
1
[Ai, + Ai;] de.
2
(9)
0.6
However, since Ai, and Aip are assumed to be zero at the
beginning of each switching cycle, this objective is equivalent
to minimizing the local harmonic current rms value (over
each switching cycle). Besides, due to the symmetry of output
voltages, only the interval fk[0, 60'1 needs to be considered.
0.4
ZTHD =
0
With reference to Fig. 4, the harmonic current distortion
over the kth switching cycle beginning at 8, = M N , N being
the ratio of switching frequency to the fundamental frequency,
is calculated in following steps:
j0,
a) Calculate T I and T3 (with me as the reference vector).
b) Determine the null-vector times TOand T7 as functions of
the apportioning factor as well as the modulation index.
0.2
8"
I
50
100
150
200Fig. 7. Optimal apportioning factor for variable m.
To see the effect of optimization, total harmonic current
distortion corresponding to using different y, including the
optimal one derived above, are calculated for modulation
index varying from 0 to 2/& by integrating (13) (in the
Appendix) over one fundamental cycle. The results are illustrated in Fig. 8 by the variable
960
3.5
0.3
3
0.2
0.1
0
0 1.5
-0.1
1
0.5
-0.2
0.2
0.4
0.6
0.8
1
m
Fig. 8. Total harmonic distortion1 as a function of the modulation index m. a) y = 0, 1; b) 'y = 0.5 [l - sgn(cos38)I; c)
e, = mcos8 - (m/6) cos38; d) with sinusoidal modulation
waves; e) y = 0.5; f) with optimized y.
IV. REDUCTION OF SWITCHING LOSSES
8 Jill?
0=--.
71.
-0.3
0
io
20
30
40
50 8 60°
Fig. 9. Qptimal zero-sequence voltage, Vo(8), for m = 21fi
(solid line), as compared to the pure sinusoidal function
defined by (12) (dashed line).
Consider now the modulation wave e,@) shown in Fig. 6.b)
ITHD
It might be quite
to see that
e (corresponding
to y = Oe5) andf(with Optimized are 'lose to each Other for
all modulation index, despite of the fact that the apportioning
factors are very different. Also surprising is the results with
flat-top m'Ddulation (curve b):
the Optima' aPPortioning fac:tor for large is better approximated
by y = [1 sgn(cos3e)1127 as mentioned be'fore, the
harmonic
distortion is still higher than that with y = 0.5. Nevertheless,
the
that the
space vector
modulatioin with y = 0.5 is a faiirly good choice for practical
applications and can achieve almost the best results in terms of
harmonic current distortion.
C. Equivalent Optimal Regular-SampledPWM
Using (6) and (7), modulation waves for the equivalent
optimal regular-sampled PWM can be established. To this end,
the resultiing average zero-sequence voltage Vo(0) is first
determined by substituting (11) into (7). Since
e, = mcos8, ep = msin8,
- d) for phase A. As can be seen, each modulation wave is
clamped to the positive or negative DC-like voltage (*VdJ2)
time interval. Consequently,either the upper or the
in a
lower transistor in phase A remains conducting, eliminating
thus the switching losses in that
The Samehappens to
phase B and phase C as well (but in different intervals). The
modulation in this
has been givendifferent
in the
literature, such
nyo-phase
[SI, bus-clamping
modulation [151,six-step modulation [26], or discontinuous
modulation [11, 231. The
mo-phasemodulation is
adopted in this paper. with space vector modulation, twophase modulation corresponds to setting the apportioning
factor to either 1 or 0 in a certain interval. In either case, only
two phases are switched once in each switching cycle, and the
conducting state of the other phase remains unchanged (refer
to Fig. 4).
Since with two-phase modulation, switching losses in each
switching period are reduced, the switching frequency can be
increased without increasing the total switching losses, which
in turn results in reduced harmonic current distortion. The
V0(8) becomes a function of m and 8 which can be reduced to
increasein switching frequency depends on the
the following simple form (by using again Mathematica):
characteristics of power devices. With power MOSFET's, the
major switching losses occur during switching on and are
vo(e) = --m4 cos30
(12) dependent of the DC-link voltage. Therefore, the switching
This coinc;ideswith the result reported in 121 where it has been losses Will be reduced by 1/3 in each switching Cycle in which
Or 1, independent Of the load current* Hence the
considered to be approximate because of the use of numerical y =
switching
frequencyt can be increased by a factor 312. In
optimization methods for deriving it.
contrast, when insulated-gate bipolar transistors (IGBT's) are
Since y defined by (11) needs to be limited in .
"
intervals used, the major switching losses are due to turning off and are
when m is, larger than 12JZ149 = 1.12, as mentioned before, dependent of the current. nus,
the reduction in switching
the zero-siequencevoltage will also differ from that defined by
(12). To see this, ?,(e) corresponding to the optimal appor- t. This should be understood as the local average switching fietioning fac:tor as limited to the range [0, 11 is illustrated in Fig.
quency measured over a ceratin interval in which two-phase modu9 for m = 21& and is comparedl to that defined by (12).
lation is used.
96 1
losses depends on the load current and varies from switching
cycle to switching cycle. A detailed study can be found in [21]
and [23]. Here we consider only the use of MOSFET's to
further demonstrate the equivalence between space vector
modulation and regular-sampled PWM.
To achieve the least harmonic current distortion, the local
harmonic current rms value, imD, resulting from using the
optimal apportioning factor (1 1) should be compared with
two-phase modulations (y = 0 or l), taking into account that
the switching frequency can be increased by 50% with twophase modulation. If the two-phase modulation results in
lower ~THD,
that is, if
the apportioning factor will be set to 0 or 1, instead of yOp
defined by (11). Accordingly, the switching frequency will be
increased by 50%.
The optimal apportioning factor as modified in this way is
illustrated in Fig. 10 for different modulation indexes.
Comparing the results with those shown in Fig. 7, it can be
seen that the differences are in the high modulation index
region ( ~ ~ 0 . where
7)
two-phase modulation in conjunction
with increased switching frequency leads to smaller local
harmonic current rms value in some regularly spaced intervals
whose width varies with the modulation index. To see the
effect of the modified optimal apportioning factor, the
resulting total harmonic current distortion for different m is
illustrated in Fig. 11 and is compared with that using the
optimal apportioning factor defined by (11). As can be seen,
they are identical for m smaller than 0.7, as two-phase
modulation is only used in higher modulation index region.
2.5
\
2
1.5
0 1
0.5
n
" 0
0.2
0.4
0.6
0.8
1
m
Fig. 11. Total harmonic distortion as a function of the modulation index m. a) With optimized y defined by (11); b) With
modified optimal y as shown in Fig. 10.
0.4
0.2
0
-0.2
-0.4
o
50
100
150 200 250 300 350'8
Fig. 12.Zero-sequence components for the equivalent regularsampled PWM corresponding to the modified optimal apportioning factor shown in Fig. 10.
v. CONCLUS
The relationship between space vector modulation and
regular-sampled PWM is reexamined in this paper. It is
demonstrated that the apportioning of null-vector time
between two null vectors for space vector modulation and the
In Fig. 12, the corresponding zero-sequence components for zero-sequence components added to the modulation waves in
the equivalent optimal regular-sampled two-phase modulation regular-sampled PWM represent a degree of freedom that can
are illustrated.
be properly matched such that both modulation methods will
generate the same outputs. They can also be utilized to
optimize the performance of each modulation method in terms
of harmonic current distortion and/or switching losses. For
space vector modulation, an analytical expression is derived
for the optimal apportioning factor that results in minimum
THD. The corresponding optimal regular-sampled P W is
shown to be that with third harmonic injection. Two-phase
modulation with increased switching frequency is also studied
and is found to feature lower THD in the high modulation
index region.
I
50
100
150
200-
Fig. 10. Optimal apportioning factor for variable frequency
operation and constant switching losses.
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APPENDIX
The local harmonic current nns value is a function of the
modulation index, the switching frequency, the apportioning
factor, and the angle 8:
.2
~THD
n3m2
288~[go(m, y)
gl(my
y)
’
Y+g2(m9
y) ’ $1
where functions go, gl,and 82 are defined as following:
x
2
+9m sin(2x--) + 4 f i m s i n
6
x
R
- 144- 54m2+72fimsin
7
963
36mcos3x- 9 h m 2 s i n (4x+ -)
3
(13)