JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA [619190]

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 45 Chapter 3.
SPACE-VECTOR PWM
3.1. Space-Vector Modulation
3.1.1. Three-Dimensional Vector Representation
A multilevel converter can synthesize output voltages from many discrete voltage
levels. Therefore, the functional diagram of an n-level diode-clamped converter can
be represented as shown in Fig. 3.1.
n-1
i2
C sc1sa2
C sa1
i1
a b c 1 1
0
ib ic ia sa0 VDC C
1 2 sb2
sb1sc2
sb0 sc0sa(n-1) sb(n-1) sc(n-1) 1)1(−=−nVvDC
nC
12−=nVvDC
C
11−=
nVvDC
C

Fig. 3.1. Functional diagram of an n-level diode-clamped converter.

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CHAPTER 3: SPACE-VECTOR PWM Page 46 Each switching state, or combination of phase-leg switches, produces a defined
set of three-phase voltages, which can be represented as vectors in the three-
dimensional Euclidean diagram (Fig. 3.2) [A26].
-1
-0.5
0
0.5
1 -1
-0.5
0
0.5
1 -1 -0.5 0 0.5 1
vab vbcvca
202 020 002 022
200 220 210 012
021 102
201 120121
010 122
011 112
001
212
101
211
100 221
110 DCca
Vv
DCab
Vv
DCbc
Vv 222
111
000

Fig. 3.2. Three-dimensional SV diagram.
The variables represented in Fig. 3.2 are the line-to-line voltages from a three-
level converter, as follows:
[] []T T
1or, ikkjjinVv v vDC
ca bc ab − − −−= = V Vr r
, (3.1)
where i, j, k ∈ [0, … , n-1], which define the position of the single-pole n-throw
switches of phases a, b and c, respectively. The vectors are labeled as ( i, j, k) in
order to simplify their notation.
Because of Kirchof’s Law, the sum of the line-to-line voltages is always zero; this
is really an equation of the plane in the line-to-line coordinate system. This means
that all of the vectors of a multilevel converter lie in a plane, and that is how they are usually represented.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 47 When the phase voltages are represented in the three-dimensional diagram, they
do not lie in a plane. However, they can be projected into a plane, thereby
representing an equivalent two-dimensional diagram.
Coming back to the three-dimensional representation, a voltage reference vector
(mr) that must be synthesized by PWM-averaged approximation can also be
represented in vector form, as follows:




+ ++ −+
=
)32cos()32cos() cos(
ˆooo
LL
ttt
Vm
θπωθπωθ ω
r, (3.2)
where LLVˆ is the amplitude of the line-to-line voltages. Since this vector has only two
degrees of freedom, it also lies on the same plane as the switching vectors. Using the
definition of vector norm, the length of the reference vector is
LLV m m m ˆ
23 2 2= + =rrr, (3.3)
while by the same definition, the length of the longest switching vector is
DCV V 2max=r
. (3.4)
The maximum length of the reference vector (3.3) that can be synthesized in
steady-state conditions equals the radius of the largest circle that can be inscribed in
the outer hexagon. Therefore, the maximum length of the reference vector is
max max23V mr r= . (3.5)
By substituting (3.4) into (3.5) and comparing the resulting equation with (3.3),
the maximum amplitude of the undistorted line-to-line voltage that can be synthesized
is
DC LL V V =maxˆ . (3.6)

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CHAPTER 3: SPACE-VECTOR PWM Page 48 3.1.2. Two-Dimensional Vector Representation
The Clarke’s Transformation [B12, B13] allows the three-dimensional vector
representation to be displayed in a two-dimensional diagram. Given three output
voltages of the converter ( va0, vb0 and vc0), the projection in a plane αβ (vα, vβ) of a
three-dimensional vector is
2
01
00
0 + avavav vj vVc b arrr r
+ + = =β α , (3.7)
where 32πjea=r.
Fig. 3.3(a) shows the three unitary director vectors of this transformation, while Fig.
3.3(b) shows an example for the case in which va0=200 V, vb0=300 V, and vc0= -100 V.

1ar
2ar0arvb0
va0
vc0

(a) 1
0avbrVr vb0
va0
vc02
0avcr
0
0avcr

(b)
Fig. 3.3. Clark’s Transformation: (a) director vectors, and
(b) example of spatial vector for v a0=200 V, vb0=300 V, and vc0=-100 V.
The aim of the SVM is to generate a reference vector ( mr
) in the same plane for
each modulation cycle. As the reference vector may not be the same as any vector
produced by the converter, its average value can be generated using more than one
vector per modulation cycle by PWM-averaged approximation. Selecting proper
vectors and applying them in a suitable order helps the devices achieve low switching
frequencies.
In steady-state conditions, the reference vector rotates at a constant angular
speed ( ω), which defines the frequency of the output voltages. The amplitude of the
fundamentals of those voltages is proportional to the length of the reference vector.

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CHAPTER 3: SPACE-VECTOR PWM Page 49 There are eight possible states for the two-level converter ( n3=23=8), which
produce the voltage vectors shown in Fig. 3.4. Six of these vectors have equal lengths
and are located every sixty degrees (100, 110, 010, 011, 001, 101). The other two
vectors are in the origin because of their null lengths (000, 111).

va0 vb0
vc0 001 010
011 100
101 110
000 111

Fig.3.4. SV diagram of the two-level converter.
The SV diagrams of the three-level and four-level converters have twenty-seven
and sixty-four vectors, respectively (Fig. 3.5).

va0 vb0
vc0 000
001
002 010
011
012 020
021
022 100
101
102 110
111
112 120
121
122
200
201
202 210
211
212 220
221
222
(a) va0vb0
vc0 000
001
002
003 010
011
012
013 020
021
022
023 030
031
032
033 100
101
102
103 110
111
112
113 120
121
122
123 130
131
132
133
200
201
202
203 210
211
212
213 220
221
222
223 230
231
232
233
300
301
302
303 310
311
312
313 320
321
322
323 330
331
332
333
(b)
Fig.3.5. SV diagrams of (a) the three-level converter, and (b) the four-level converter.
The redundant vectors in the diagram produce the same line-to-line voltages.
The three-level converter has six double vectors and one triple vector in the origin.
The four-level converter has twelve double vectors, six triple vectors and one tetra

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CHAPTER 3: SPACE-VECTOR PWM Page 50 vector in the origin. Proper utilization of these vectors will help the voltages of the
capacitors to achieve balance.
3.1.3. Limiting Area
Any set of three vectors 1vr, 2vr and 3vr in a plane ( αβ in Fig. 3.6) can generate any
reference vector mr in the same plane using PWM-averaged approximation, if the
reference vector lies in the triangle connecting the tips of 1vr, 2vr and 3vr.
β
1vr2vr3vr
mr
3θ1θθ2θ
α
Fig. 3.6. Limiting area to generate the reference vector )m(r
by using three vectors.
The average reference vector can be obtained by sequentially applying these
vectors in a modulation period in accordance with
∫∫∫∫+ + =m m T
+TT m+TT
T mT
mT
mdtvTdtvTdtvTdtmT
2 12 1
11
3 2
01
01 1 1 1 r r r r, (3.8)
where Tm is the modulation period, and T1+T2 ≤Tm.
Assuming that mr
remains approximately constant during a modulation period,
which is acceptable if Tm is much smaller than the line period ( T), then (3.8) can be
approximated as:
33 22 11 vdvdvdmrrrr+ + = , (3.9)
in which d1, d2 and d3 are the duty cycles of vectors 1vr, 2vr and 3vr, respectively. They
must satisfy the following condition:
13 2 1 = ++ ddd . (3.10)
The boundaries of the area that allows the reference vector to be generated can
be determined assigning zero value to one of the duty cycles. For example, imposing
d3=0, and by means of (3.9) and (3.10), the vector mr can be expressed as follows:

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CHAPTER 3: SPACE-VECTOR PWM Page 51 ()2 11 2 vvdvmrrrr− + = . (3.11)
As all the duty cycles could potentially utilize values in the interval []10,, the tip of
the reference vector is on the line that joins the extremes of the vectors 2 1 and v vrr
(Fig. 3.7), which can be verified by simply making d1 vary within that interval.
Therefore, this segment is one boundary of the limiting area. The other two remaining
segments of the triangular region in Fig. 3.6 can be determined by assigning zero to d1
and d2 separately.
β

1vr2vr3vr
mr) (2 11 vvdrr−
α
Fig. 3.7. Boundary of the area, determined when d 3=0.
Any reference vector outside of this area requires that one or more duty cycles
be negative. This fact does not make physical sense; thus, it cannot be generated by
this set of three vectors.
3.1.4. Calculation of Duty Cycles
Equation (3.9) can be expressed by the following exponential notation:
3 2 1
33 22 11θ θ θ j j j jθevd evd evd emm + + = =r; (3.12)
therefore, the duty cycles of the vectors can be calculated according to either of the
following equation systems:








=



−
1cos
1 1 1sin sin sincos cos cos1
3 3 2 2 1 13 3 2 2 1 1
321
θθ
θ θ θθ θ θ
senmm
v v vv v v
ddd
, (3.13)
or

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CHAPTER 3: SPACE-VECTOR PWM Page 52 







=



−
1) Im() Re(
1 1 1) Im() Im() Im() Re() Re() Re(1
3 2 13 2 1
321
mm
v v vv v v
ddd
rr
r rrr rr
. (3.14)
In general terms, given any bi-dimensional stationary base frame, orthogonal or
not, the expression would be:









=



−
1 1 1 11
3 2 13 2 1
321
yx
y y yx x x
mm
v v vv v v
ddd
rr
rrrrrr
. (3.15)
The calculation process for any of these equation systems requires inverting a
matrix, which complicates the application of this method to a real-time processor
system. Additionally, the equation system may need to be solved more than once per modulation period (
Tm), since the region where the reference vector lies is previously
unknown. A simplified mathematical process should be found.
3.1.5. Calculation of Duty Cycles by Projections
A general method for calculating duty cycles of vectors is explained in this section. This method is based on determining some projections of the reference
vector, and it will be applied in order to simplify the modulation process later.
The vectors
1pr and 2pr in Fig. 3.8 are the projections from the reference vector mr
onto the segments that join the extreme of 3vr to 1vr and to 2vr, respectively.
β

1vr2vr
1pr2pr
α3vrmr

Fig. 3.8. Projections of the reference vector mr (1pr and 2pr).
Therefore, the reference vector can be expressed as follows:
3 2 1 vppmrrrr+ + = (3.16)

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CHAPTER 3: SPACE-VECTOR PWM Page 53 or
3
23 2
2
13 1
1 vlvvplvvpmrrrrrr+−+−= , (3.17)
where 1l and 2l are the lengths of the vectors 3 1vvrr− and 3 2vvrr−, respectively.
Finally, the reference vector mr can be expressed as:
3
22
11
2
22
1
111 vlp
lpvlpvlpmr rrr

− −+ + = . (3.18)
From (3.18), the duty cycles of the vectors can be directly deduced as follows:
22
11
3
22
2
11
1 1 and, ,lp
lpdlpdlpd − −= = = . (3.19)
If the balanced SV diagram is normalized to have triangular regions with unity
lengths ( 12 1 = =ll ), the calculation of those duty cycles is simplified as:
2 1 3 2 2 1 1 1 and, , pp d p d p d −−= = = . (3.20)
Calculation of duty cycles using this method is actually very functional. However,
the former condition that all the areas must be equilateral triangles is only possible if
the voltages of the DC-link capacitors are balanced. Thus, when dealing with the
unbalanced case (Chapter 5), these lengths can no longer be considered to be unity
because they change according to the present imbalance. In that case, (3.19) must
be applied.

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CHAPTER 3: SPACE-VECTOR PWM Page 54 3.2. The Three-Level Converter
3.2.1. SVM Under Voltage-Balanced Conditions
Suitable vectors from the SV diagram should be chosen for each modulation
cycle in order to generate the reference vector ( mr). The vectors nearest to mr are
the most appropriate selections in terms of their ability to minimize the switching
frequencies of the power devices, improve the quality of the output voltage spectra,
and the electromagnetic interference (EMI).
In Fig. 3.9, the SV diagram of the three-level converter is divided into sextants,
and each sextant is then divided into four triangular regions in order to show the
vectors nearest to the reference.

va0vb0
vc0000
001
002 010
011
012 020
021
022 100
101
102 110
111 120
121
122
200
201
202 210
211
212220
221
222 mr1st Sextan t2ndSextan t
3rd Sextan t
4th Sextan t
5th Sextan t6th Sextan t1 3
2
4
112

Fig. 3.9. Three-level vector diagram divided into sextants and regions .
Four groups of vectors can be distinguished in this diagram, as described in the
following.
(1) The “large vectors” (200, 220, 020, 022, 002 and 202) assign the output
voltages of the converter to either the highest or the lowest DC voltage levels. As
they do not connect any output to the NP, they do not affect the voltage balance of
the capacitors. These vectors can generate the highest AC voltage amplitudes

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CHAPTER 3: SPACE-VECTOR PWM Page 55 because they have the greatest lengths. In fact, these six vectors are equivalent to
the active ones of the two-level converter.
(2) The “medium vectors” (210, 120, 021, 012, 102 and 201) connect each output
to a different DC-link voltage level. Under balanced conditions, their tip end in the
middle of the segments that join two consecutive large vectors. The length of the
medium vectors defines the maximum amplitude of the reference vector for linear
modulation and steady-state conditions, which is 23 the length of the large vectors.
Since one output is always connected to the NP, the corresponding output current
will define the NP current ( i1). This connection produces voltage imbalances in the
capacitors, and these must be compensated.
(3) The “short vectors” (100-211, 110-221, 010-121, 011-122, 001-112 and 101-
212) connect the AC outputs to two consecutive DC-link voltage levels. Their length
is half the length of the large vectors. They are double vectors, which means that two
states of the converter can generate the same voltage vector. As they affect the NP
current in opposite ways, proper utilization of these vectors will help the NP voltage to
achieve balance.
(4) The “zero vectors” (000, 111 and 222) are in the origin of the diagram. They
connect all of the outputs of the converter to the same DC-link voltage level, and
therefore, they do not produce any current in the DC side.
3.2.2. Simplified Calculation of Duty Cycles
Taking into account the symmetry of all the sextants, it is interesting to reflect the
reference vector into the first sextant in order to reduce the number of relevant
regions (Appendix B). Also, the amplitude of the reference vector must be normalized
to fit into a diagram in which the triangular regions have unity lengths.
The theoretical maximum length of the normalized reference vector (nmr) is the
two-unity value. However, in steady-state conditions, its length is limited to 3 due to
the fact that longer lengths of this vector will be outside of the vector-diagram
hexagon (Fig. 3.10), and thus cannot be generated by modulation. Overmodulation is
produced if the normalized reference vector assumes lengths longer than 3 f o r
some positions of this vector, but it can never be outside of the hexagon
(overmodulation is covered in Section 6.5.)

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CHAPTER 3: SPACE-VECTOR PWM Page 56
1 (Maximum Amplitude)
nmr3=nm
1 200 210220
211
100221
110
222
111
000 3
2
1 4

Fig. 3.10. Maximum length of the normalized reference vector in steady-state conditions .
Defining a modulation index m that potentially uses values in the interval
[]10, m∈ for linear modulation, the length of the normalized reference vector would
be:
( )3 0 3 ≤ ≤ =n n m m m . (3.21)
In Fig. 3.11, the normalized reference vector is decomposed into the axes
located at zero and sixty degrees, obtaining projections m1 and m2, respectively.

1 nmr
1200 210 220
211
100221
110
222
111
000 3
2
1 4
2mr
1mrθn

Fig. 3.11. Projections of the normalized reference vector in the first sextant.
The lengths of the new vectors are determined as follows:
32 and
32 1n
nn
n nsinm msincosm mθ θθ =


− = . (3.22)
In accordance with the general method revealed in Section 3.1.5, these values
are the direct duty ratios of the vectors, as in the following:
2 1 111 2 221 110 1 211 100 1 and m m d m d,m d/ / − −= = = . (3.23)

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CHAPTER 3: SPACE-VECTOR PWM Page 57 Even though there are three redundant vectors in the origin, only the combination
“111” is considered because it achieves the best modulation sequences in terms of
switching frequency.
The cases for which the normalized reference vector is located in Regions 1, 2
and 3 are shown in Fig. 3.12.

1 nmr
1 200 210 220
211
100 221
110
222
111
000 3
2
1 4
2mr
1mrθn
m1-1

Region 1
1 nmr
1 200 210220
211
100 221
110
222
111
0003
2
1 4 2mr
1mrθn 1-m1
1-m2

Region 2
1 nmr
1 200210 220
211
100 221
110
222
111
0003
2
1 4 2mr
1mrθn m2-1

Region 3
Fig. 3.12. Projections for Regions 1, 2 and 3.
Table 3.1 summarizes the information needed to ascertain the region where the
reference vector lies and the duty cycles of the nearest vectors in the first sextant.
Table 3.1. Summary of information for the SVM.
Case Region Duty Cycles
m1>1
1 d200=m1-1
d210=m2
d100/211 =2-m1-m2
m1≤1
m2≤1
m1+m2>12 d100/211 =1-m2
d110/221 =1-m1
d210=m1+m2-1
m2>1
3 d210=m1
d220=m2-1
d110/221 =2-m1-m2
m1≤1
m2≤1
m1+m2≤1 4 d100/211 =m1
d110/221 =m2
d111=1-m1-m2
For all cases, it is assumed that the sum of m1 and m2 is not greater than 2;
otherwise, the reference vector would be outside of the hexagon, and thus could not
be reproduced by modulation.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 58 3.2.3. The dq-gh Transformation
It is quite common for the control stage to provide the reference vector in terms of
dq coordinates. Transformation (3.24) is proposed to translate these dq components
directly into some very useful variables for the modulation:
with


=

−
qd
ghdq
hg
mm
mmT (3.24)
() (),cos sin32 cos 32 sin
)1(2

 + +
−=−
r rr r
DCghdqθ θπθ πθ
n Vk T
where θr is the coordinate reference angle.
Using this transformation, the control signals given in the orthogonal rotating
coordinate frame (the dq components) are directly translated into a non-orthogonal
base in the stationary coordinate frame (as gh components). In fact, the new
reference axes are the ones that limit the first sextant, and which are separated by
60ș.

gh
d q
θr ∝ mh
mq
∝ mg md mr

Fig. 3.13. Graphical representation of the dq-gh transformation.
The value of the coefficient k in (3.24) must be defined so that in the new axes,
the SV diagram will fit into a two-unit-per-side hexagon. This coefficient depends on which sort of variables (phase or line-to-line variables) are processed for the control,
and also on the coefficient used for the dq transformation. For example, when using a
power-conservative dq transformation (coefficient
32 ) and dealing with phase
variables, the parameter k must be unity. When the line-to-line variables are used for
the control, k must be 31 .

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CHAPTER 3: SPACE-VECTOR PWM Page 59 From the values mg and mh, the sextant where the reference vector lies can be
directly found, as can the components m1 and m2 of the equivalent vector in the first
sextant (Appendix B). These relationships are illustrated in Fig. 3.14 and Table 3.2.

va0 vb0
vc0 2 h
g2
-2
-2(m1,m2)(-m1,m1+m2)
m1 m2
(-m2,-m1) (-m1-m2,m1)
(m2,-m1-m2)(m1+m2,-m2)

Fig. 3.14. Components gh from different reference vectors. Equivalences in the first sextant .
Table 3.2. Determination of the sextant and the equivalent components m 1 and m 2 in the first
sextant .
gh Components
(mg, mh)
Sextant Equivalent Components
in the First Sextant
(m1, m2)
mg≥0 mh≥0 1st m1=mg m2=mh
mg<0 mh≥0 mg+mh≥0 2nd m1=-mg m2=mg+mh
mg<0 mh≥0 mg+mh<0 3rd m1=mh m2=-mg-mh
mg<0 mh<0 4th m1=-mh m2=-mg
mg≥0 mh<0 mg+mh<0 5th m1=-mg-mh m2=mg
mg≥0 mh<0 mg+mh ≥0 6th m1= mg+mh m2=-mh
Using a DSP to process the duty cycles of the vectors will be very fast if the dq-
gh transformation is applied and Tables 3.1 and 3.2 are used.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 60 3.2.4. Defining Real Vectors
When the most suitable sequence for the vectors in the first sextant to achieve
low switching frequency has been defined, the next and final step is to apply the
calculated duty cycles to the corresponding vectors. This task requires the knowledge
of the real sextant in which the reference vector lies (Table 3.2), and it can be
performed by simply interchanging the states of the output phases in accordance with
the equivalences given in Table 3.3. The sequences of vectors obtained in the
corresponding sextant will preserve the same switching frequencies that the original
sequence defined in the first sextant.
Table 3.3. Interchanges of the output states depending on the sextant in which the reference
vector lies (after making calculations in the first sextant).
1st Sextant 2nd Sextant 3rd Sextant 4th Sextant 5th Sextant 6th Sextant
a a → b a → b a → c a → c a
b b → a b → c b b → a b → c
c c c → a c → a c → b c → b
3.2.5. Modulation Techniques
So far, the duty cycles of the vectors nearest to the reference have been
calculated. However, the short vectors used in each modulation period are not yet
defined. Two techniques are discussed in this dissertation: NTV modulation and
symmetric modulation.
3.2.5.1. NTV Modulation
The NTV modulation technique uses only three of the closest vectors per
modulation cycle. Thus, a single short vector will be selected from each pair. The
choice is made according to the objective of maintaining balanced voltages in the
DC-link capacitors; therefore, the present voltage imbalance and the direction of the instantaneous output currents must be known. The NP current ( i
1) must be positive in
order to discharge the lower capacitor, and must be negative to charge it. For
example, if ia is positive, vector 100 will discharge the lower capacitor ( i1=ia>0), and
vector 211 will charge it ( i1=ib+ic= -ia<0).

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CHAPTER 3: SPACE-VECTOR PWM Page 61
C C
sa1 sb1 sc1 i1
vC1
a vC2
b c 2
1 1
0 ia sa2
sa0 sb2
sb0 sc0 sc2 100
ib ic

i1=ia
vC1vC2
i1
C C
sa1 sb1 sc1
a b c 2
1 1
0 iasa2
sa0 sb2
sb0 sc0 sc2 211
ib ic

i1=ib+ic=-ia
Fig. 3.15. Example of control of the NP current by proper selection of the double vectors.
Vector 100 produces i 1=ia, whereas vector 211 produces i 1=ib+ic=-ia.
Since all of the modulation will be calculated in the first sextant, this criterion can
be expressed as shown in Table 3.4(a). When the reference vector lies in the first
sextant, currents 'iaand 'ic in these tables will be ai and ci, respectively, but they
must be changed when it lies in another sextant. These equivalences are given in
Table 3.4(b).
Table 3.4. (a) Criteria for selection between vectors 100 and 211, and vectors 110 and 221;
and (b) equivalences of currents to process calculations in the first sextant.
(a)
Selection between 100 and 211 Selection between 110 and 221
vC1>vC2 'ai>0 y1 vC1>vC2 'ci>0 y2
0 0 0 (100) 0 0 1 (221)
0 1 1 (211) 0 1 0 (110)
1 0 1 (211) 1 0 0 (110)
1 1 0 (100) 1 1 1 (221)
)0'() (2 1 1 > ⊕ > =a C C i v v y ) 0'() (2 1 2 > ⊕ > =c C C i v v y
(b)
Equivalences 1st Sext. 2nd Sext. 3rd Sext. 4th Sext. 5th Sext. 6th Sext.
'ai ia ib ib ic ic ia
'ci ic ic ia ia ib ib
Table 3.5 shows the sequences of the vectors in the first sextant that are more
capable of minimizing the switching frequencies of the devices. These sequences
depend on the short vectors that are selected according to voltage-balance
requirements. The number of changes or steps between consecutive vectors

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CHAPTER 3: SPACE-VECTOR PWM Page 62 associated with each sequence is also indicated in this table. The worst cases are
Region 2 (vectors 100-221) and Region 4 (vectors 100-221), both of which require
four switching steps, which is twice the number required by any other sequences.
When a sequence is repeated in a subsequent modulation period, the sequence is
flipped in order to minimize the number of steps from one cycle to the next.
Table 3.5. Sequences of vectors in the first sextant by NTV modulation.
Region Short Vectors Sequences Steps
100 100-200-210 // 210-200-100 2 // 2 1 211 200-210-211 // 211-210-200 2 // 2
100-110 100-110-210 // 210-110-100 2 // 2
100-221 100-210-221 // 221-210-100 4 // 4
211-110 110-210-211 // 211-210-110 2 // 2 2
211-221 210-211-221 // 221-211-210 2 // 2
110 110-210-220 // 220-210-110 2 // 2 3 221 210-220-221 // 221-220-210 2 // 2
100-110 100-110-111 // 111-110-100 2 // 2
100-221 100-111-221 // 221-111-100 4 // 4
211-110 110-111-211 // 211-111-110 2 // 2 4
211-221 111-211-221 // 221-211-111 2 // 2
Since the selection of the double vectors in the NTV modulation is based on
comparators and logical functions, a nonlinear control is performed. Although it is not
possible to achieve a value that is precisely zero for the average NP current over a
modulation period, if the short vectors are properly chosen, the sign of this average
current tends to balance those voltages, and the objective is nonetheless generally
achieved (this is not true for all cases, as discussed in Section 3.2.6.1).
Despite the simplicity of using only three vectors per modulation cycle, the
following drawbacks exist:
– there are significant switching-frequency ripples in the voltages of the
capacitors; and
– when changing sequences due to a new region or different selection of short
vectors, two switching steps can be produced (two legs must switch one level). Due
to this fact, and that there are some sequences that require four steps (Table 3.5),
the switching frequencies will not be constant.
The symmetric modulation approach can overcome these disadvantages and
keep constant the switching frequency.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 63 3.2.5.2. Symmetric Modulation
Symmetric modulation is characterized by using four vectors per modulation
sequence. Dealing with another variable in the calculation of the duty cycles allows
the equation of the NP current to be included in the equation system:
44 33 22 11 vdvdvdvdmrrrrr+ + + = , (3.25a)
14 3 2 1 =+ ++ dddd , and (3.25b)
() () ()[]
{}0
2,0,,11 1 11 1 11 1 1 = − + − + − =∑
∈kjick ij b j ki a i jk i d did di d d i . (3.25c)
Equation (3.25a) is, in fact, a pair of equations. Equation (3.25c) shows the
relationship between the local averaged value of the NP current, the duty cycles of
the vectors, and the AC currents. Mathematically speaking, this new equation allows
the NP current to equal zero each modulation period.
Symmetric modulation is very similar to NTV in some respects. The new vector
added to the sequence is one of the short vectors that was not selected using NTV.
Thanks to this, the duty cycles are basically calculated by the same process, with the only difference being that the duty cycle applied in NTV to only one of the dual
vectors is now shared between both of them. For instance, if the reference vector lies
in Region 1, the sequence will be 100-200-210-211. To satisfy the zero-NP-current
condition, the duty cycle calculated for 100/211 should now be properly distributed
between them.

000
100110
111 200210
211220
221
222 2H
2L3
4L4H1

Fig. 3.16. New regions for symmetric modulation. Example of vector sequence for Region 1.
For this modulation, Regions 2 and 4 are now split into 2L, 2H, 4L and 4H. The
line that divides such regions, which coincides with vector 210, has been drawn
according to a geometric-symmetry criterion. The best vector sequences in the first

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CHAPTER 3: SPACE-VECTOR PWM Page 64 sextant are given in Table 3.6; these sequences best minimize the switching
frequencies of the devices.
Table 3.6. Sequences of vectors in the first sextant by symmetric modulation.
Regions Sequences Steps
1 100-200-210-211 // 211-210-200-100 3 // 3
2L 100-110-210-211 // 211-210-110-100 3 // 3
2H 110-210-211-221 // 221-211-210-110 3 // 3
3 110-210-220-221 // 221-220-210-110 3 // 3
4L 100-110-111-211 // 211-111-110-100 3 // 3
4H 110-111-211-221 // 221-211-111-110 3 // 3
In general terms, the local averaged NP current must be zero to achieve constant
voltage in the NP. However, the exact NP current required must be calculated for
each modulation period in order to compensate for errors that occur due to
tolerances and some assumptions. A modulation period delay is taken into account to
obtain the reference NP current, so that the value calculated during the present
period will be applied the next one. The voltages of the capacitors at the beginning of the next period ( k+1) are:

C vC2
vC1C 2
1
0 i1
icm
()



+ − =+ + =
∫∫
+
++
+
. )2 (1and, 21
)(1)1(
1)1 (1)(2)1(
1)1 (2
kCTk
kTcmkCkCTk
kTcmkC
vdti iCvvdti iCv
m
mm
m
(3.26)
The component icm in (3.26) is the common current through both DC-link
capacitors. On the other hand, the NP current ( i1) splits into 50% for each capacitor
according to the general expression given in (4.4).
These voltages at the end of the next period can be expressed as:
()
()


+ − =+ + =
++
++++
++
∫∫
. 21and, 21
)1 (1)2(
)1(1)2 (1)1 (2)2(
)1(1)2 (2
kCT k
TkcmkCkCT k
TkcmkC
vdti iCvvdti iCv
m
mm
m
(3.27)

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 65 Substituting (3.26) into (3.27) and imposing the balance condition
)2 (1)2 (2+ += kCkC v v reveals that
)(1)1(
1)2(
)1(1)(21 1
kCTk
kTT k
TkkC v dtiCdtiCvm
mm
m+ − −=∫∫+ +
+. (3.28)
The common current cmi has disappeared in (3.28) because this current does not
affect the NP voltage balance. Finally, this equation can be rewritten as:
[] )(1)(2)(1)1 (1k kCkC
mk i v vTCi − − = + , (3.29)
where )1 (1+ki and )(1ki are the discrete local averaged values of 1i over each
respective modulation period, such that
∫ ∫+
+++
= =m
mm
mT k
Tk mkTk
kT mk dtiTi dtiTi)2(
)1(1)1 (1)1(
1)(1 .1and,1 (3.30)
Therefore, (3.29) defines the averaged NP current required to achieve voltage
balance in the capacitors at the end of the next modulation period. Notice that all the
voltages in this equation are sensed at the beginning of the present period k.

k k+1 k+2 Sensing
vC1(k) and v C2(k)
mT Time)(1ki )1(1+ki

Fig. 3.17. Sequence of modulation cycles.
Since the local averaged NP current reference )1 (1+ki is given by (3.29), the next
step is to achieve this value by properly distributing the duty cycle of the short
vectors. The instantaneous NP current 1i can be expressed in terms of the control
functions of these vectors as:
c b a is sisis s i ) ( ) (110 221 210 100 211 1 − + + − −= . (3.31)
Assuming constant AC output currents during a modulation period, the discrete
local averaged operator transforms this expression into:
[] [] )()(110)(221)()(210)()(100)(211)(1kck k kbk kak k k i d d i d i d d i − + + − −= . (3.32)

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 66 For the next modulation period k+1
[]
[] .)1 ()1 (110)1 (221)1 ()1 (210)1 ()1 (100)1 (211)1 (1
+ + ++ + + + + +
− ++ + − −=
kck kkbk kak k k
i d di d i d d i (3.33)
Two cases can be distinguished depending on the position of the reference
vector. If the tip of the reference vector lies in Regions 1, 2L or 4L
) orș30 (2 1m mn ≥ ≤θ , vectors 100-211 define the beginning and ending of each
modulation sequence. These vectors can be distributed according to a variable 1x:
). 1(2and,) 1(21211/100
211 1211/100
100 xdd xdd + = − = (3.34)
Since the duty cycles must be positive, this distribution variable considers values
in the interval []1,11−∈x . Taking into account the vectors used in these regions,
)1 (1+ki can be expressed as:
.)1 ()1 (110)1 ()1 (210)1 ()1 (1)1 (211/100)1 (1+ + + + + + + + − + −= kck kbk kak k k i d i d i x d i (3.35)
Isolating the variable )1 (1+kx obtains the following:
)1 ()1 (211/100)1 (1)1 ()1 (110)1 ()1 (210)1 (1+ ++ + + + +
+− −=
kakk kck kbk
ki di i d i dx . (3.36)
Due to the intrinsic delay of the modulation process, most of the variables in
(3.36) are known during the present period k. The duty cycles are calculated during
this k period, and the value of )1 (1+ki is the exact averaged current required to reach
voltage balance, as defined by (3.29). However, the AC currents during the next
period k+1 are unknown, since they are sensed at the beginning of the period k. Yet,
these currents can be extrapolated to the next period by using the first-order Taylor’s Series, as follows:
)1(,,)(,,)1(,,)(,,)(,,)1 (,, 2 −−
+ − =−+ ≈ kcbakcba m
mkcbakcbakcbakcba i i TTi ii i (3.37)
On the other hand, if the reference vector lies in Regions 3, 2H or 4H
) orș30 (2 1m mn < >θ , vectors 110-221 will be the beginnings and endings of all of
the modulation sequences. Similarly, the distribution variable x2 can be calculated as
)1 ()1 (221/110)1 (1)1 ()1 (210)1 ()1 (211)1 (2+ ++ + + + +
++ −=
kckk kbk kak
ki di i d i dx , (3.38)

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CHAPTER 3: SPACE-VECTOR PWM Page 67 and the duty cycles of vectors 110 and 221 will be
), 1(2and), 1(22221/110
221 2221/110
110 xdd xdd + = − = (3.39)
where []1,12−∈x .
If the distribution variables x1 and x2 have values outside the interval []1,1− , they
remain saturated to the closest limit.
Equations (3.36) and (3.38) provide the values of the distribution variables in the
first sextant. If the reference vector is not in this sextant, the AC currents will be
interchanged according to the real sextant. These equivalences are given in Table
3.7.
Table. 3.7. Equivalences of currents.
1st Sextant 2nd Sextant 3rd Sextant 4th Sextant 5th Sextant 6th Sextant
ai b ai i→ b ai i→ c ai i→ c ai i→ ai
bi a bi i→ c bi i→ bi a bi i→ c bi i→
ci ci a ci i→ a ci i→ b ci i→ b ci i→
3.2.5.3. Simulated Results
NTV and symmetric modulation are tested by simulation in the following. For all
of the examples, the converter is supplied by a DC voltage source VDC=1800 V and
operates with an R-L star-connected load with parameters R=1 Ω and L=2 mH. The
DC-link capacitors are C=1000 µF and the fundamental frequency of the AC voltages
is f=50 Hz. Both modulation techniques have been checked for different modulation
indices ( m) and modulation periods ( Tm).
The values of the total harmonic distortion (THD) given in the figures consider the
following definition:
122
ˆˆ
100 (%)VV
THDhh∑∞
== , (3.40)
in which hVˆ is the amplitude of the h-order harmonic. This expression can be better
handled with the following simplification:

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 68 1ˆ21001ˆˆ
100 (%)2
12
2
112
− =− =∑∞
=
VV
VV
THDRMS hh
(3.41)
in which VRMS is the RMS value of the waveform.
Fig. 3.18 and Fig. 3.19 show the voltages of the DC-link capacitors ( vC1 and vC2),
a line-to-line voltage ( vab) and the output currents ( ia, ib and ic) when the converter
operates with modulation period Tm= 50 µs (or modulation frequency fm=20 kHz).
NTV is tested in Fig. 3.18 and symmetric modulation in Fig. 3.19.
Symmetric modulation generates better output voltage spectra, since the first
group of harmonics have frequencies around half of the modulation frequency, while
the spectra of the NTV contain lower frequency components. Additionally, symmetric
modulation generates less THD. Another advantage is the smaller amplitude of the
high-frequency ripple in the voltages of the capacitors. Nevertheless, two important
disadvantages of symmetric modulation must be remarked on. On the one hand,
there exists a low-frequency ripple in the voltages of the capacitors when the
converter operates with low modulation indices. This fact is a consequence of the
lesser degree of freedom in the utilization of redundant vectors, since one of the dual
vectors cannot be chosen in regions 2L, 2H, 4L and 4H. On the other hand, the
switching frequencies of the devices are higher with symmetric modulation. The
mean switching frequencies (meansf ) given in the figures are calculated for complete
turn-on and turn-off cycles in all the switches of the bridge and divided by 12.
In Fig. 3.20 the converter operates with a modulation period of Tm= 500 µs (or
modulation frequency fm=2 kHz). NTV is tested under two conditions, the “normal”
NTV presented in Fig. 3.20(a), which does not consider the intrinsic one-period delay
that exists between sensing variables and application of the modulation, and the NTV
shown in Fig. 3.20(b), in which the values of the sensed voltages and currents are
extrapolated to the next modulation period. In the second case, less NP voltage
ripple is achieved but at the price of higher switching frequencies of the devices.
Symmetric modulation is applied in Fig. 3.20(c). In accordance with Section
3.2.5.2, this modulation includes compensation for one-period delay. Since the
objective of this modulation is to achieve equal voltages in the capacitors at the end
of any modulation period, the NP voltage ripple produced by the switching frequency
has less amplitude. A magnified detail of these voltages is presented in Fig. 3.21.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 69

0 5 10 15 20 25 30 35 40
Time (ms) (V),(A)
vC1, vC2
ia i c ib
-800 -600 -400 -200 0 200 400 600 800 1000
vab/4

0 5 10 15 20 25 0 50010001500
THD = 77.57 %
fs mean = 2256 Hz (V)V
habˆ
Frequency (kHz)
(a)

0 5 10 15 20 25 30 35 40vC1, vC2
ia i c ib
-800 -600 -400 -200 0 200 400 600 800 1000
vab /4
Time (ms) (V),(A)

0 5 10 15 20 25 0 50010001500
Frequency (kHz)(V)V
habˆ
THD = 44.56 %
fs mean = 2186 Hz
(b)

0 5 10 15 20 25 30 35 40vC1, vC2
ia i c ib
vab /4
-800 -600 -400 -200 0 200 400 600 800 1000
Time (ms) (V),(A)

0 5 10 15 20 25 0 50010001500
Frequency (kHz)(V)V
habˆ
THD = 38.17 %
fs mean = 1812 Hz
(c)
Fig. 3.18. NTV modulation with modulation period T m=50 µs:
(a) m=0.4; (b) m=0.6; and (c) m=0.8.

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CHAPTER 3: SPACE-VECTOR PWM Page 70

0 5 10 15 20 25 30 35 40vC1, vC2
ia i c ib
-800 -600 -400 -200 0 200 400 600 800 1000
vab/4
Time (ms) (V),(A)

0 5 10 15 20 250 50010001500
Frequency (kHz)(V)V
habˆ
THD = 76.94 %
fs mean = 2291 Hz
(a)

0 5 10 15 20 25 30 35 40 vC1, vC2
ia i c ib
-800 -600 -400 -200 0 200 400 600 800 1000
vab /4
Time (ms) (V),(A)

0 5 10 15 20 250 50010001500
Frequency (kHz)(V)V
habˆ
THD = 44.41 %
fs mean = 2295 Hz
(b)

0 5 10 15 20 25 30 35 40vC1, vC2
ia i c ib
-800 -600 -400 -200 0 200 400 600 800 1000
vab /4
Time (ms) (V),(A)

0 5 10 15 20 250 50010001500
Frequency (kHz)(V)V
habˆ
THD = 38.06 %
fs mean = 2164 Hz
(c)
Fig. 3.19. Symmetric modulation with modulation period T m=50 µs:
(a) m=0.4; (b) m=0.6; and (c) m=0.8.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 71

-600 -400 -200 0 200 400 600 800 1000
0 5 10 15 20 25 30 35 40
Time (ms) (V),(A)
ia i c ib vC1,vC2
vab /4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0200 400 600 800 1000 1200
Frequency (kHz)(V)V
habˆ
THD = 46.02 %
fs mean = 245 Hz
(a)

-600 -400 -200 0 200 400 600 800 1000
0 5 10 15 20 25 30 35 40
Time (ms) (V),(A)
ia i c ib vC1, vC2
vab /4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0200 400 600 800 1000 1200
Frequency (kHz)(V)V
habˆ
THD = 45.53 %
fs mean = 296 Hz
(b)

-600 -400 -200 0 200 400 600 800 1000
0 5 10 15 20 25 30 35 40
Time (ms) (V),(A)
ia i c ib vC1, vC2
vab /4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0200 400 600 800 1000 1200
Frequency (kHz)(V)V
habˆ
THD = 45.19 %
fs mean = 248 Hz
(c)
Fig. 3.20. NTV and symmetric modulation with modulation period T m=500 µs and m=0.6:
(a) NTV; (b) NTV with compensation for one-period delay; and (c) symmetric modulation.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 72
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 840 860 880 900 920 940 960
Time (ms) (V)
vC1 v C2

Fig. 3.21. Voltages in the DC-link capacitors with symmetric modulation. Magnification from
Fig. 3.20(c).
In conclusion, NTV presents some interesting advantages when the converter
operates with small modulation periods. These advantages are the greater control of
the low-frequency NP voltage ripple, the lower switching frequencies in the devices,
as well as the easier implementation because of fewer calculations required for NP
voltage control. Symmetric modulation presents advantage when the converter
operates with large modulation periods. In such conditions, the NP voltage ripple
produced by the switching frequency becomes significant and this modulation
technique can handle it better than NTV.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 73 3.2.6. Limits of Control of the Voltage Balance
In this section, the limits of the ability to control the NP balance for both
modulation techniques are revealed.
When the reference vector is located in the first sextant, the local averaged NP
current can be expressed as:
ph i iD=1 , (3.40)
with () ()[]110 221 210 211 100 d d d d d − − =D
and []T
c b a ph iii=i .
To validate this expression for the entire vector diagram, the equivalent AC
currents for each sextant must be taken into account. This correction should be done
according to Table 3.7.
A new transformation matrix S is introduced for the purpose of interchanging the
currents depending on which sextant the reference vector occupies, as follows:
ph i iSD=1 , with




+ + ++ + ++ + +
=
2 1 6 5 4 36 3 4 1 5 25 4 3 2 6 1
ssssssssssssssssss
S , (3.41)
where si defines the sextant in which the reference vector lies, such that
{} .,,,,, i,i msi 654321otherwise.0sextant theinlies if1=
=r
(3.42)
As in steady-state conditions, the term phi in (3.41) will be time-dependent; thus,
a rotating coordinate transformation can be included to handle the constant values
for those variables, as follows:
dqT i iSD=1 , (3.43)
where T T 1ST TS TS Sdq dq dq T = = =−, and ph dq dq iT i= .
Equation (3.43) is general for the three-level diode-clamped converter, since it
allows the local averaged NP current to be analyzed for any SVM technique.
The model can be extended to converters with higher numbers of levels. For the
general case of an n-level diode-clamped converter with multiple mid points (MPs):

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 74 dqT MP iSD i= , (3.44)
where []T
1 2 2… ii in MP −=i , and
[] [] []
[] [] []
[] [] []




− − −− − −− − −
=∑ ∑ ∑∑ ∑ ∑∑ ∑ ∑
≥≥≠
≥≥≠
≥≥≠≥≥≠
≥≥≠
≥≥≠≥≥−≠− − −
≥≥−≠− − −
≥≥−≠− − −
kjikjik ij
kjikjij ki
kjikjii jkkjikjik ij
kjikjij ki
kjikjii jkkjinkjik n n nij
kjinkjinj n k ni
kjinkjin ni jk n
d d d d d dd d d d d dd d d d d d
1,,11 1
1,,11 1
1,,11 12,,22 2
2,,22 2
2,,22 22 ,,)2)(2( )2(
2 ,,)2()2( )2(
2 ,,)2)(2( )2(
M M M
D

{} 1,…,2,1,0 ,, − ∈ ∀ n kji .
3.2.6.1. Limits of NTV Modulation
In this section, the limits of the ability to control the NP balance in the NTV
modulation technique are analyzed. Such limits will be determined by selecting the
dual vectors such that they reach either maximum or minimum NP current. For instance, for the maximum case, vector 100 will be selected if
ia is positive, and
vector 211 if ia is negative. The situation is similar for vectors 110 and 221, taking into
account the direction of ic. The minimum NP current is found by the opposite
reasoning.
In Fig. 3.22, the waveforms that limit the maximum and minimum local averaged
NP current are shown for some examples. These waveforms are normalized by the
amplitude of the AC phase currents ( Iˆ), which are in steady-state conditions.
Different lengths of the reference vector have been considered for a zero-degree
current phase angle ( ϕ=0o).

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 75

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(a)
0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(b)

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(c)
0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees)Iiˆ/1
mini1 avgi1

(d)

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(e)
0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(f)
Fig. 3.22. Normalized maximum (solid line) and minimum (dashed line) local averaged NP
current. Examples given for purely resistive load ( ϕ=0o):
(a) m=1; (b) m=0.9541; (c) m=0.9; (d) m=0.7; (e) m=0.5; and (f) m=0.3.

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CHAPTER 3: SPACE-VECTOR PWM Page 76 For the case in which m=1 in Fig. 3.22, the upper waveform that limits the
maximum NP current dips into negative values. Therefore, it is not possible to
achieve positive NP current values for some positions of the reference vector when
this is required for voltage-balance control. Similarly, this stipulation also exists for
the lower waveform that limits the minimum NP current, since both waves are
symmetric. Under those conditions, the local averaged value of this current over a
modulation period cannot be confined to zero. As a result, a low-frequency harmonic
appears in the NP current, and this eventually becomes NP voltage oscillations.
In contrast, for the case in which m=0.9541, the upper local averaged NP current
waveform always consists of positive values (the lowest NP current waveforms will
not be considered henceforth, since they are symmetric). Therefore, the current is
always controlled and balance is achieved. In fact, this length of the reference vector
is the limiting case for the unity PF, because the minimum value of the upper NP
current waveform is zero. For smaller amplitudes of the reference vector, the NP
current is always controlled. Those waveforms also provide information as to the
spare NP current available for achieving balance.
In Fig. 3.23, the load is considered to be purely inductive. For this case, full NP
current control is not achieved for reference vector lengths greater than m=0.5774.

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CHAPTER 3: SPACE-VECTOR PWM Page 77

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(a) 0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(b)

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(c) 0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(d)

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(e) 0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(f)
Fig. 3.23. Normalized maximum (solid line) and minimum (dashed line) local averaged NP
current. Examples given for purely inductive load ( ϕ=-90o):
(a) m=1; (b) m=0.9; (c) m=0.7; (d) m=0.5774; (e) m=0.5; and (f) m=0.3.

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CHAPTER 3: SPACE-VECTOR PWM Page 78 The three-dimensional diagram in Fig. 3.24(a) shows the minimum values from
the highest NP currents’ waveforms (mini1). The area above the black (zero) plane is
where the minimum value is positive; therefore, full control for the NP current can
always be achieved. The area below this plane shows negative values for the NP
current waveforms, which means that a third-order harmonic will appear in the NP
current.
On the other hand, Fig. 3.24(b) shows the averaged value from those waveforms,
worked out over a whole line period (avgi1). Since these values are always positive,
the entire surface is above the zero plane. Hence, despite the low-frequency ripple,
the control always retains the ability to keep those oscillations at one half the level of
the DC-link voltage. This figure also provides information about how quickly the NP
can be re-balanced after an unbalanced transition.

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CHAPTER 3: SPACE-VECTOR PWM Page 79
Current Phase Angle
(Degrees) Iimin
ˆ1
Modulation Inde x
m
(a)
Current Phase Angle
(Degrees) Iiavg
ˆ1
Modulation Index
m
(b)
Fig. 3.24. (a) Minimum local averaged values and (b) whole-line-period averaged values of
the upper NP current waveforms.
The surfaces in Fig. 3.24 contain symmetries with respect to the current phase
angles 0o and 180o. For any modulation index,
) 180()( ,1 1 ϕ ϕ ± =±=o
avg min f f i i , (3.46)
or
ϕcos where), ( ,1 1 = ±= PF PFf i iavg min . (3.47)

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CHAPTER 3: SPACE-VECTOR PWM Page 80 Fig. 3.25 shows some sections from Fig. 3.24(a) where the curves are given for
different PFs. One conclusion is that the unity PF is the most favorable case,
because it can contribute more current to the NP balance. On the other hand, the
closer the modulation index is to 0.5, the better the NP current control. This is logical
since the duty cycles of the short vector are higher in those conditions, and, as a
consequence, they have more NP current control.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1-0.8 -0.6-0.4-0.200.20.4 0.60.81I iminˆ
1
 PF =1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Modulation Index, m
Fig. 3.25. NP current availability for different PF values.
The knowledge of the NP current availability will be useful for analyzing the case
in which two three-level converters are connected back-to-back, so that they can
share the NP-balancing task. This analysis is performed in Chapter 6.
Fig. 3.26 shows the maximum NP voltage ripple that could occur as a result of
the NP current. The normalized amplitude of the ripple ( 2NPnV∆ ) is defined as
follows:
Cf IV V
RMSNP NPn 2
2∆=∆. (3.48)
Figure 3.26(b) provides helpful information for determining the value of the
capacitors for a practical application. For example, assuming the worst operating

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CHAPTER 3: SPACE-VECTOR PWM Page 81 conditions of unity modulation index ( m=1) and current angle ϕ=-84o or ϕ=+96o, the
maximum normalized ripple is:
02973.02
2=∆=∆
Cf IV V
RMSNP NPn. (3.49)
If the values were IRMS=220 A, f=50 Hz, and C=550 µF, the amplitude of the NP
voltage ripple would be:
VCfI V VRMS NPn NP8.237
10 5505022002973.02 26=
⋅ ⋅=∆=∆
−. (3.50)
It can be observed that the value of the total DC-link voltage VDC does not affect
the NP voltage ripple. This statement is true under the assumption that the AC currents are not affected by the NP oscillation, which can be acceptable in the case
of relatively small NP-voltage amplitudes. Significant inductive loads contribute to make output currents less sensitive to NP voltage oscillations.

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CHAPTER 3: SPACE-VECTOR PWM Page 82

Current Phase Angle
(Degrees) 2NPnV∆
Modulation Index
m

(a)

2NPnV∆
-150 -100 -50 0 50 100 150 00.0050.01 0.0150.02 0.025 0.03
m=1
m=0.9
m=0.8
m=0.7
m=0.6
Current Phase Angle
(Degrees)
(b)
Fig. 3.26. Normalized NP voltage ripple for NTV.

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CHAPTER 3: SPACE-VECTOR PWM Page 83 3.2.6.2. Limits of Symmetric Modulation
The NP current limits of the symmetric modulation technique will be determined
in a manner similar to the process described for NTV modulation. To achieve maximum and minimum NP averaged current, the control variables
x1 and x2 will be
fixed to extreme values: {} .1,1 ,2 1 −=xx Thus, only three vectors will be used for the
modulation.
Waveforms of the local averaged NP current are given for different lengths of the
reference vector. For the examples in Figs. 3.27 and 3.28, the current angles have
been considered to be zero and an inductive ninety degrees, respectively.
The maximum length of the reference vector to achieve full control of the NP
voltage with a current angle of zero is the same as for the NTV modulation technique (
m=0.9541). However, for a purely reactive load angle, the maximum value is smaller
(m= 0.5) than for NTV modulation.
The symmetric modulation technique has less NP current control than NTV
modulation. Although symmetric modulation usually uses four vectors per modulation
cycle, only three vectors are used when the NP voltage imbalance is at the edge of or
beyond the control limits. Furthermore, in contrast with NTV modulation, only one
short vector per modulation cycle is selected according to NP voltage-balancing
requirements. The other short vector is assigned, depending on whether the
normalized reference vector is above or below thirty degrees. Thus, as compared
with NTV modulation, there is one fewer variable for the NP current control when the
reference vector lies in the inner regions.
The normalized amplitude of the low-frequency NP ripple is presented in Fig.
3.30. This figure shows the existence of this ripple for an extensive operating area.
Nevertheless, the maximum value of the ripple is the same than with NTV
( 02973.02/= ∆NPnV ) and it is produced under the same operating conditions ( m=1
and current angle ϕ=-84o or ϕ=+96o).

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CHAPTER 3: SPACE-VECTOR PWM Page 84
0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees)Iiˆ/1

(a) 0 50 100 150 200 250 300 350-1-0.8-0.6-0.4-0.200.20.40.60.81
Reference Vector Angle (Degrees) Iiˆ/1

(b)

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees)Iiˆ/1

(c) 0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(d)

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees)Iiˆ/1

(e) 0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(f)
Fig. 3.27. Normalized maximum (solid line) and minimum (dashed line) local averaged NP
current for symmetric modulation. Examples given for a purely resistive load ( ϕ=0o):
(a) m=1; (b) m=0.9541; (c) m=0.9; (d) m=0.7; (e) m=0.5; and (f) m=0.3.

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CHAPTER 3: SPACE-VECTOR PWM Page 85
0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(a) 0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(b)

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(c) 0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(d)

0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(e) 0-1-0.8-0.6-0.4-0.200.20.40.60.81
50 100 150 200 250 300 350
Reference Vector Angle (Degrees) Iiˆ/1

(f)
Fig. 3.28. Normalized maximum (solid line) and minimum (dashed line) local averaged NP
current for symmetric modulation. Examples given for a purely inductive load ( ϕ=-90o):
(a) m=1; (b) m=0.9; (c) m=0.7; (d) m=0.5; (e) m=0.3; and (f) m=0.1.

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CHAPTER 3: SPACE-VECTOR PWM Page 86

Current Phase Angle
(Degrees) Iimin
ˆ1
Modulation Inde x
m

(a)
Current Phase Angle
(Degrees) Modulation Index
m Iiavg
ˆ1

(b)
Fig. 3.29. (a) Minimum local averaged values and (b) whole-line-period averaged values of
the upper NP current waveforms.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 87

Current Phase Angle
(Degrees) 2NPnV∆
Modulation Index
m

(a)

-150 -100 -50 0 50 100 1500 0.005 0.010.015 0.02 0.025 0.03 m=1
m=0.9
m=0.8
m=0.7
m=0.6
m=0.5
m=0.4
m=0.3
m=0.2
m=0.1
Current Phase Angle
(Degrees) 2NPnV∆
(b)
Fig. 3.30. Normalized NP voltage ripple for symmetric modulation.

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CHAPTER 3: SPACE-VECTOR PWM Page 88 3.2.7. Modulation Algorithm
Fig. 3.31 summarizes the steps required for the application of the proposed fast
SVM algorithm in a DSP. This diagram considers the NTV modulation technique,
however it can be easily adapted to symmetric modulation with slight changes.

dq-gh Transformation
Calculation of Duty Cycles
(Table 3.1)
Define Short Vectors and Best
Sequence (Tables 3.4 and 3.5) Projection into the 1st Sextant
(Table 3.2) Reference Vector (md, m q)
mg, m h
m1, m 2
d1, d2 , d3
Interchange Phases
(Table 3.3) dv1, dv2 , dv3 (1st Sextant )
dv1, dv2 , dv3 (Real Vectors ) vC1
vC2
ia
ib

Fig. 3.31. General diagram of the SVM algorithm.

The short time required for processing this modulation algorithm hinges on these
following points.
– The dq-gh transformation directly translates the control variables given in dq
coordinates into a non-stationary coordinate system, providing useful variables for
the modulation.
– All of the calculations are made in the first sextant; therefore the total number of
regions involved is divided by six.
– Most of the operations required are based on products and comparison
operations, which are quickly processed by a DSP.

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CHAPTER 3: SPACE-VECTOR PWM Page 89 3.2.8. Experimental Results
The proposed SV-PWM algorithm has been programmed into the 32-bit floating-
point digital processor (Sharc ADSP 21062) of the SMES system. Its instruction
processing time of 25 ns allows the algorithm to be processed in less than 4 µs. The
modulation period of the three-level converter is Tm=50 µs (fm=20 kHz). An
asynchronous motor is connected as a load.
In Fig. 3.32, the DC-link voltage is provided by a DC power supply adjusted to 60
V. This figure shows a line-to-line output voltage and two output currents for the modulation index
m=0.9. Fig. 3.33 shows the same variables for the case when
m=0.4. These experimental results have been also verified by simulation. An R-L
load is used for the simulation results, so the values have been adjusted to achieve
the same amplitude and phase angle for the output currents.
Fig. 3.34 shows the voltages of the two DC-link capacitors, a low-pass-filtered
line-to-line output voltage and an output phase current. The capacitors are forced to
have a permanent voltage imbalance by means of two DC power supplies. The upper
one is adjusted to 60 V and the lower one to 10 V. When the NP connection is
released, the modulation process itself controls the voltage balance. As the selection
of dual vectors is properly made, balance is achieved.

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CHAPTER 3: SPACE-VECTOR PWM Page 90

CvC2
vC1 VDC
60 V
C2
1
0AC
MOTOR
Three-Level
Converter a
b
c ibia

Ch 1: vab
Ch 3: ia
(10A/div)
Ch 4: ib
(10A/div )

0 5 10 15 20 25 30 35 40 45 50 -20 -10 0 10 20 -50 0 50 100
Time (ms)(V)
(A) -100 vab
ia i b
Fig. 3.32. Line-to-line voltage (v ab) and output currents (i a and i b), for modulation index
m=0.9.

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CHAPTER 3: SPACE-VECTOR PWM Page 91

Ch 1: vab
Ch 3: ia
(10A/div)
Ch 4: ib
(10A/div )

0 5 10 15 20 25 30 35 40 45 50 -20 -10 0 10 -50 0 50 100
Time (ms) (V)
(A)20 -100 vab
ia i b
Fig. 3.33. Line-to-line voltage (v ab) and output currents (i a and i b), for modulation index
m=0.4.

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CHAPTER 3: SPACE-VECTOR PWM Page 92

CvC2
vC1VDC2
60 V
C2
1
0 VDC1
10 V AC
MOTOR
Three-Level
Converter a
b
c ib

Ch 1: vC2
Ch 2: vC1
Ch 3: vab
Ch 4: ib
(20A/div )

Fig. 3.34. DC-link voltages (v C1 and v C2), filtered line-to-line voltage (v ab) and output phase
current (i b). The NP voltage is released to be controlled by the modulation itself.

JOSEP POU TECHNICAL UNIVERSITY OF CATA LONIA

CHAPTER 3: SPACE-VECTOR PWM Page 93 3.3. Conclusions of the Chapter
Some general modulation issues are presented in this chapter. The duty cycles are
calculated by projections of the reference vector, and by the dq-gh transformation,
which translates the control variables into a pair of components very useful for the
modulation. An equivalent reference vector is used for processing calculation in the
first sextant. This equivalent vector has the propriety that interchanging the final states
of the phase legs automatically generates the original reference vector.
For the case of the three-level converter, two modulation techniques are analyzed:
NTV and symmetric modulation. Assuming a modulation period much smaller than the line period (
Tm<<T), the first method presents certain advantages, such as a lower
switching frequency and an extended operation area without low-frequency NP voltage oscillations. Better performance than NTV is obtained with symmetric modulation when
dealing with large modulation periods, from the standpoint of output voltage spectra
and NP voltage ripple. Some practical graphics are presented to obtain the amplitude
of the low-frequency NP voltage ripple for both modulation strategies. This information
is very useful for the calculation of the values of the DC-link capacitors.
NTV modulation has been implemented in the DSP of the SMES system. This
processor has an instruction time of 25 ns, and the algorithm requires less than 4
µs to
be processed. As all of the calculations are made in the first sextant, only four regions
must be considered for the modulation, instead of the twenty-four that comprise the whole diagram. Therefore, most of the operations can be realized by just checking
some conditions of small tables. Optimal sequences of vectors are applied in order to
achieve minimal switching frequencies for the devices of the bridge. The criterion used
for the selection of the short vectors helps the NP voltage to achieve balance. As the
modulation process itself performs the voltage-balancing task, the control stage is
relieved of this duty.

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CHAPTER 3: SPACE-VECTOR PWM Page 94