Int. J. Industrial Electronics and Drives, Vol. 1, No. 1, 2009
1
Generalised space vector PWM for sinusoidal output
voltage generation with multiphase voltage source
inverters
Drazen Dujic, Martin Jones and Emil Levi*
School of Engineering,
Liverpool John Moores University,
Byrom St, Liverpool L3 3AF, UK
E-mail: Drazen.dujic@ieee.org
E-mail: m.jones2@ljmu.ac.uk
E-mail: e.levi@ljmu.ac.uk
*Corresponding author
Abstract: The paper presents a generalised approach towards the development of the space
vector pulse width modulation (SVPWM) for sinusoidal output voltage generation with two-level
multiphase voltage source inverters (VSIs), where the number of phases is an odd number. The
generalisation can greatly speed up the implementation since the tedious analysis of 2n voltage
space vectors of an n-phase inverter in the corresponding (n – 1)/2 2-D planes can be completely
avoided. Required dwell times for active space vectors are correlated with trigonometric
properties of multiphase systems, which account for the actual phase number. Feasibility of the
developed approach has been verified experimentally using the five-phase and seven-phase
SVPWM schemes as examples.
Keywords: pulse width modulation; PWM; space vector; multiphase; voltage source inverters;
VSIs.
Reference to this paper should be made as follows: Dujic, D., Jones, M. and Levi, E. (2009)
‘Generalised space vector PWM for sinusoidal output voltage generation with multiphase voltage
source inverters’, Int. J. Industrial Electronics and Drives, Vol. 1, No. 1, pp.1–13.
Biographical notes: Drazen Dujic received his Dipl. Ing. and MSc from the University of Novi
Sad, Serbia, in 2002 and 2005, respectively, and his PhD from the Liverpool John Moores
University, UK in 2008. From 2002 to 2006, he was with the Department of Electrical
Engineering, University of Novi Sad as a Research Assistant, and from 2006 till 2009, with
Liverpool John Moores University as a Research Associate. Currently, he is with ABB Corporate
Research Centre, Baden-Dättwil, Switzerland. His main research interests are in the areas of
design and control of advanced power electronics systems and high performance drives.
Martin Jones received his BEng (First Class Honours) and PhD from the Liverpool John Moores
University, UK in 2001 and 2005, respectively. He was a recipient of the IEE Robinson Research
Scholarship for his PhD studies and is currently with Liverpool John Moores University as a
Senior Lecturer. His current research focuses on modelling and control of electric motor drives.
Emil Levi received his MSc and PhD in Electrical Engineering from the University of Belgrade,
Serbia in 1986 and 1990, respectively. From 1982 till 1992, he was with the Dept. of Electrical
Engineering, University of Novi Sad. He joined Liverpool John Moores University, UK in May
1992 and is, since September 2000, Professor of Electric Machines and Drives. He serves as an
Editor of the IEEE Trans. on Energy Conversion, a Co-Editor-in-Chief of the IEEE Trans. on
Industrial Electronics and as a member of the Editorial Boards of the IET Electric Power
Applications and Int. J. of Industrial Electronics and Drives.
1
Introduction
Multiphase machines are nowadays considered as serious
contenders for various industrial applications (Singh, 2002).
These are predominately related to high power levels where,
due to the limitations of present power semiconductors,
ability of multiphase system to spread the power
across higher number of inverter legs is of great advantage.
Copyright © 2009 Inderscience Enterprises Ltd.
Another benefit offered by multiphase drives is an inherent
improvement in fault tolerance, when compared to the
three-phase drives (Levi et al., 2007; Levi, 2008). Finally,
higher number of phases yields smoother torque due to the
simultaneous increase of the frequency of the torque
pulsation and reduction of the torque ripple magnitude
(Apsley et al., 2006).
2
D. Dujic et al.
With regard to the stator winding design, multiphase
machines are more versatile than their three-phase
counterparts. The winding can be of distributed type, in
which case near-sinusoidal magneto-motive force (mmf)
distribution is achieved. Alternatively, the winding can be of
the concentrated type (a single slot per phase per pole), in
which case quasi-trapezoidal mmf distribution results
(Levi, 2008). The type of the multiphase machine stator
winding has a great impact on the nature of the output
voltage, required from the two-level multiphase voltage
source inverter (VSI). If the machine is with the distributed
winding, then VSI needs to provide purely sinusoidal output
voltage, without any low-order harmonics, since harmonic
currents for such voltage harmonics are restricted only by
the stator leakage impedance. On the other hand, when
machines are with concentrated windings, injection of
low-order harmonics is preferable since it leads to an
enhancement of the torque-per-ampere characteristic. Thus,
development of multiphase pulse width modulation (PWM)
schemes has to account for the type of the multiphase
machine’s winding.
The PWM schemes for multiphase VSIs have been
developed in the recent past using both carrier-based PWM
and space vector pulse width modulation (SVPWM)
approach. By and large, the emphasis, however, has been
placed on the SVPWM methods, although the carrier-based
approach is significantly simpler for implementation. This is
so since many of the properties of multiphase systems are
not immediately obvious when carrier-based PWM schemes
are used. Therefore SVPWM schemes were predominately
under investigation and such an approach is also followed in
this paper.
Since an n-phase system corresponds to an
(n – 1)-dimensional space (n is further on assumed to be an
odd number and the star-connected load is with a single and
isolated neutral point), the customary approach towards
designing a SVPWM scheme consists in decomposing the
(n – 1)-dimensional space into (n – 1)/2 2-D sub-spaces
(planes), using either real decoupling transformation or
symmetrical component approach (Zhao and Lipo, 1995;
Grandi et al., 2006a). Each of the available 2n voltage space
vectors of an n-phase VSI appears simultaneously in all
such 2-D planes (d1-q1, d2-q2, d3-q3, etc.). However, the
increase in the number of voltage space vectors that
accompanies the increase in the number of phases makes the
analysis and implementation of SVPWM schemes more and
more difficult.
Development of SVPWM scheme for a sinusoidal
output voltage generation with a five-phase VSI has been
presented in de Silva et al. (2004) and Iqbal and Levi
(2006), where four active space vectors were used per
switching period. Similarly, SVPWM for a seven-phase VSI
(Grandi et al., 2006b; Dujic et al., 2007a) and for a
nine-phase VSI (Dujic et al., 2007b; Grandi et al., 2007)
have been developed, utilising six and eight active space
vectors per switching period, respectively. In all these
SVPWM schemes for an n-phase VSI, n − 1 active space
vectors are selected in order to generate sinusoidal output
voltages. These are the active space vectors neighbouring
the reference space vector in the first (d1-q1) plane, while
zero reference space vectors are imposed in other planes.
Since each of the (n – 1)/2 planes contains certain low-order
harmonics, only fundamental component from the d1-q1
plane is synthesised, while low-order harmonics are kept at
zero average values. These SVPWM schemes are suitable
for multiphase machines with distributed winding.
In a multiphase drive with concentrated stator winding it
is desirable to utilise higher stator current harmonic
injection for the purpose of the torque enhancement. The
odd harmonics below the phase number n can be used and
one output voltage harmonic per additional plane
(d2-q2, d3-q3, …, d(n – 1)/2 – q(n – 1)/2) can be injected. Thus the
reference voltage space vectors now have non-zero values in
all the planes. Ryu et al. (2005) have presented development
of such a SVPWM scheme for five-phase drives, where
simultaneous synthesis of both fundamental and the third
harmonic is performed. Active space vectors are selected on
the basis of the fundamental reference in the d1-q1 plane
(four neighbouring active space vectors). This limits the
achievable third harmonic voltage in the d2-q2 plane, but it is
not a problem since the third harmonic voltage reference in
the d2-q2 plane is considerably smaller than the fundamental
reference in the d1-q1 plane. Another PWM scheme with the
third harmonic injection for a five-phase VSI, based on
summation of device turn-on times for creating the desired
references in the two planes and the initial vector selection
still based only on the neighbouring vectors in the d1-q1
plane, has been developed by Ojo et al. (2006). In a
seven-phase system, the third and the fifth harmonic can be
injected (Locment et al., 2006), while a nine-phase system
allows injection of the third, the fifth and the seventh
harmonic (Coates et al., 2001).
While the implementation steps of various SVPWM
schemes may vary, they all produce desired reference space
vector(s) at the output of an n-phase VSI. In this paper,
some common characteristics, observed in the structure of
multiphase SVPWM schemes, are presented, which can
greatly speed up the implementation stage of multiphase
SVPWM schemes. Although the focus is primarily on the
sinusoidal output voltage generation with multiphase VSIs,
some remarks regarding harmonic injection SVPWM
schemes are also given.
This paper is organised as follows. Section 2
introduces basic notation, which is used throughout the
paper. In Section 3 multiphase SVPWM for sinusoidal
output voltage generation is presented, using the
five-phase, seven-phase and nine-phase VSIs as examples.
The generalisation, which follows from these PWM
schemes, is summarised in Section 4. Validity of the
generalisation is verified on the three-phase SVPWM in
Section 5. In addition, the effects of the application of
three-phase based SVPWM (two active vectors only) to
other multiphase systems are analysed. DC bus utilisation is
considered in Section 6. Experimental results, which verify
theoretical development, are presented in Section 7, while
Section 8 concludes the paper.
Generalised space vector PWM for sinusoidal output voltage generation with multiphase voltage source inverters
The analysis in this paper is restricted to continuous PWM
schemes and the operation in the linear region of
modulation. Thus, the modulation index is defined as the
ratio of the peak value of the fundamental phase voltage and
one half of the DC bus voltage:
V
M = m
Vdc 2
(1)
The generalised decoupling (Clarke’s) transformation, using
power-variant form, is given for an n-phase system with
(α = 2π n) :
cos(α )
⎡1
⎢0
sin(α )
⎢
⎢1
cos(2α )
⎢
sin(2α )
⎢0
2⎢ M
M
Cn = ⎢
n⎢
n −1
⎢ 1 cos( 2 )α
⎢
⎢ 0 sin( n − 1)α
⎢
2
⎢1 2
12
⎣
L
L
L
L
M
L
L
L
cos(n − 1)α
sin(n − 1)α
⎤
⎥
⎥
cos 2(n − 1)α ⎥
⎥
sin 2(n − 1)α ⎥
⎥
M
⎥
n −1 ⎥
cos(n − 1)(
)α ⎥
2
⎥
n −1 ⎥
sin(n − 1)(
)α
⎥
2
⎥
12
⎦
(2)
By combining successive pairs of rows into corresponding
complex quantities, complex space vectors (symmetrical
components) are obtained (White and Woodson, 1959).
Transformation (2) decomposes an n-dimensional vector
space into (n – 1)/2 2-D and mutually orthogonal planes,
which are defined with the first (n – 1)/2 pairs of rows
in (2), d1-q1, d2-q2, d3-q3, etc. The last row defines a
single-dimensional zero-sequence sub-space that can not be
excited in the case of an isolated neutral point of a
star-connected n-phase load.
For the sake of simplicity, the following trigonometric
constants are used in subsequent analysis:
π
K p = sin( p );
n
π
L p = cos( p )
n
(3)
Here p = 1, 2, 3,… (n – 1)/2 (up to the number of 2-D planes
for a given n). In the particular case when p = 1, the use of
subscript is omitted. The same symbols K and L are used for
all considered phase numbers, although the values are
different for any given n, in accordance with (3).
All the voltage space vector magnitudes are normalised
with respect to one half of the value of the DC bus voltage.
There is only one reference space vector in the d1-q1 plane,
which, after normalisation, can be expressed using the
previously defined modulation index as:
v * = vd*1 + jvq*1 = Me jϑ
Finally, instead of dealing with the times of application
(T) of various space vectors, a notion of duty cycle (relative
on-time over the switching period Ts) is introduced as:
δ=
T
Ts
(5)
Additional subscripts are used for δ in order to clearly link
duty cycles with certain space vectors or inverter legs.
3
Multiphase SVPWM
In order to identify common characteristics of multiphase
SVPWM schemes for sinusoidal output voltage generation,
three multiphase topologies are further briefly addressed. In
particular, five-phase, seven-phase and nine-phase SVPWM
schemes are elaborated next.
3.1 Five-phase SVPWM
If decoupling transformation (2) is applied to a five-phase
system, all 25 = 32 voltage space vectors are obtained in the
d1-q1 and d2-q2 planes, as shown in Figure 1 (only tips of
space vectors are shown). Space vectors are labelled with
decimal numbers, which, when converted into binary
representation, reveal the values of the switching functions
(ml) of each of the inverter leg (l = A, B, C,…). Thus ‘1’
corresponds to the upper/lower switch in particular leg
being in the ‘on’/‘off’ state, while ‘0’ has the opposite
meaning. There are ten sectors that can be identified in each
plane and each sector is with the angular span of π 5 .
Let the reference space vector be in the first sector
(s = 1). The four active space vectors are selected in the
d1-q1 plane (de Silva et al., 2004; Iqbal and Levi, 2006), as
illustrated in Figure 2. The active space vectors are
designated, with respect to the d1-q1 plane, as ‘a’ and ‘b’
space vectors (positioned along the lines that separate the
sectors), while the additional numbers are used to
distinguish between different active vector magnitudes.
Figure 1
Voltage space vectors of a five-phase VSI, (a) in d1-q1
plane (b) in d2-q2 plane
12
0.6Vdc
0.4Vdc
14
0
30
8
13
0 31
9
5
16
25
21
2
11
7
29
26
22
15
24
20
10
6
-0.2Vdc
-0.4Vdc
28
4
0.2Vdc
q1
Preliminary remarks
v
2
23
18
27
1
17
(4)
Angle ϑ defines the instantaneous reference space vector
position in the d1-q1 plane.
3
-0.6Vdc
3
-0.6Vdc -0.4Vdc -0.2Vdc
19
0
v
d1
(a)
0.2Vdc 0.4Vdc 0.6Vdc
4
D. Dujic et al.
Figure 1
Voltage space vectors of a five-phase VSI, (a) in d1-q1
plane (b) in d2-q2 plane (continued)
10
0.6Vdc
0.4Vdc
11
8
v
q2
0.2Vdc
2
14
9
15
0
25
31
6
7
13
17
29
-0.6Vdc
16
22
5
-0.6Vdc -0.4Vdc -0.2Vdc
20
21
0
v
δ 2E = δ1 + δ a1
δ 3O = δ 2O + δ a 2 ;
δ 3E = δ 2E + δ b 2
δ 4O
δ 4E
= δ 3O
Table 1
(b)
Principle of calculation of times of application of the
active space vectors, (a) first plane (b) second plane
vb 2
q1
vb1
q2
vb 2
π 5
v*
d1
va 2
va1
d2
va1
va 2
a)
b)
vb1
In order to average the reference space vector over the
switching period (δ = 1) and simultaneously neutralise
low-order harmonics in the d2-q2 plane, the following
conditions must be satisfied in each of the two planes,
respectively:
v * = va1δ a1 + va 2δ a 2 + vb1δ b1 + vb 2δ b 2
0 = va1δ a1 + va 2δ a 2 + vb1δ b1 + vb 2δ b 2
(6)
It can be seen from (6) that zero reference space vector is
imposed in the d2-q2 plane. After some manipulation, the
solution, applicable to every sector (s = 1 to 10), expressed
by means of duty cycles and introduced trigonometric
constants [(3) with n = 5], is of the form:
δ a1 = KM sin( s
π
5
δ a 2 = K 2 M sin( s
δ 0 = δ 31
− ϑ );
π
+ δ a2
Per-leg duty cycle disposition through sectors
1
2
3
4
5
A
δ5
δ 4E
δ 3O
δ 2E
δ1
B
δ 4O
δ5
δ5
δ 4E
δ 3O
C
δ 2O
δ 3E
δ 4O
δ5
δ5
D
δ1
δ1
δ 2O
δ 3E
δ 4O
E
δ 3O
δ 2E
δ1
δ1
δ 2O
6
7
8
9
10
A
δ1
δ 2O
δ 3E
δ 4O
δ5
B
δ 2E
δ1
δ1
δ 2O
δ 3E
C
δ 4E
δ 3O
δ 2E
δ1
δ1
D
δ5
δ5
δ 4E
δ 3O
δ 2E
E
δ 3E
δ 4O
δ5
δ5
δ 4E
2π 5
ϑ
(8)
0.2Vdc 0.4Vdc 0.6Vdc
d2
Figure 2
= δ 3E
Here superscripts ‘O’ and ‘E’ stand for odd and even sector,
respectively. Distribution of these per-leg duty cycles
through the sectors is summarised in Table 1.
23
4
δ 2O = δ1 + δ b1 ;
+ δ b2 ;
δ 5 = δ1 + δ a1 + δ a 2 + δ b1 + δ b 2
28
1
-0.4Vdc
30
19
12
-0.2Vdc
18
24
3
0
δ1 = δ O 2
26
27
(7) in accordance with the predetermined switching pattern
and are given with:
Figure 3
π
δ b1 = KM sin(ϑ − ( s − 1) )
5
π
− ϑ ); δ b 2 = K 2 M sin(ϑ − ( s − 1) )
5
5
π
1
1
= δ O = [1 − K 2 M cos((2s − 1) − ϑ )]
2
2
10
Switching pattern for the first sector is shown in Figure 3,
from where the per-leg duty cycles, considering (8) and
contents of Table 1, can be easily identified. In general, the
sequence of space vectors in all odd sectors is given with 0,
a1, b2, a2, b1, 31, b1, a2, b2, a1, 0. In even sectors it is
governed with 0, b1, a2, b2, a1, 31, a1, b2, a2, b1, 0. Thus
symmetrical switching pattern is obtained, with two
commutations per inverter leg, which is convenient for
practical implementation using digital signal processor
(DSP).
δ O δ a1 δ b 2 δ a 2 δ b1 δ O δ b1 δ a 2 δ b 2 δ a1 δ O
(7)
An equal distribution of the total duty cycle of the zero
space vectors (δO) among two zero space vectors (δ0 and δ31)
is assumed in (7). This degree of freedom can be utilised in
a different manner, resulting in various continuous and
discontinuous PWM schemes (Holmes and Lipo, 2003).
Duty cycles for each inverter leg, necessary for the final
implementation, are obtained by summation of the results of
Switching pattern of a five-phase SVPWM in Sector 1
(see online version for colours)
4
2
2
2
2
2
2
2
2
2
4
mA
mB
mC
mD
mE
v0 v16 v24 v25 v29
v31 v29 v25 v24 v16 v0
Generalised space vector PWM for sinusoidal output voltage generation with multiphase voltage source inverters
Figure 4
Figure 4
56
60
0.4Vdc
24
58
28
92
62
0.2Vdc
61
44
20
26
8
v
q1
29
0
30
12
94
46
22
-0.2Vdc
31
54 42
63 76
10
4
45
14
95
2
6
11
15
82
91
37
111
5
75
39
79
-0.6Vdc
123
117
96
105
33
115
97
98
73
119
107
101
83
65
35
99
69
-0.2Vdc
111
99
56
9
73
108
44
32
62
15 38 3
57
121
127
51
109 63 97
12
0
39
103
25
60
48
13
1
77 31
36
100
7
61
49
125
5
27
96
52
98
72
14
50
79
102
115
76
119
17
93
69
23
117
0
v
Figure 5
67
0.2Vdc
22
0
v
q1
q2
v
51
18
-0.2Vdc
59
19
48
83
17
11
27
86
37
60
66
q2
110
70
44
83
92
80
22
86
87
84
0.2Vdc 0.4Vdc 0.6Vdc
vb3
π 7
v*
4
106
101
121
29
75
9
d2
d1
vb1
68
2π 7
13
76
88
93
73
0.2Vdc 0.4Vdc 0.6Vdc
vb 2
d2
va 2
va 3
vb3
vb1
77
89
0
v
va 3
q3
vb 2
14
117
65
(a)
a)
va 2
108
124
111
78
84
71
104
40
94
97 45 64 12
109
5
69
120
107
74
80 28
125
92
15
79
85
41
8
105
72
24
81
100
96
95
25
(b)
116
126
91
-0.6Vdc -0.4Vdc -0.2Vdc
46
103
1
57
-0.4Vdc
-0.6Vdc
98
6
53
31
va1
36
32
119
42
2
118
d3
vb 2
102
112
99 47
20
7 122
58
127
30
115 63 82
33
0
23
87
43 56 10
49
16
61
113
3
67
21
26
123
90
35
0
52
114 62
55
82
85
vb1
118
50
94
70
Calculation of times of application of the active space
vectors, (a) first plane (b) second plane (c) third plane
0.2Vdc 0.4Vdc 0.6Vdc
38
39
18
81
ϑ
34
114
Six active space vectors are selected per switching period, in
order to average the reference space vector in the d1-q1
plane and simultaneously achieve zero average values in the
d2-q2 and d3-q3 planes (zero reference space vectors) (Dujic
et al., 2007a). Situation illustrated in Figure 5 shows the
selected neighbouring active space vectors in the first sector
of the d1-q1 plane, as well as their activated pairs in the other
two planes.
(a)
0.4Vdc
66
88
30 64
68
116
91
20
21
d1
54
126
67
6 89
124
112
65 19
95
54
71
28
16
4
90
78
71
-0.6Vdc -0.4Vdc -0.2Vdc
0.6Vdc
26
2
120
24
113
101 55
75
123
35
37
74
122
103
3
7
33
110
8
(c)
81
51 64
66
87
70
100
34
-0.6Vdc -0.4Vdc -0.2Vdc
113
114
102
1
19
45
11
53
49
34
85
47
0.2Vdc
10
104 58
105 59
121
32
109
41
104
89
106
17
55 68
74 43
77
47
23
50
0
38
78
-0.4Vdc
127
18
116
80
84 53
90 59 72 41
36
118
93
46
112
125
108
110
9
27 21
86
13
122
16
126
25
57
40
0.4Vdc
0
106
107
40
-0.6Vdc
48
88
52
43
-0.4Vdc
120
124
42
29
Voltage space vectors of a seven-phase VSI in,
(a) d1-q1 plane (b) d2-q2 plane (c) d3-q3 plane
0.6Vdc
Voltage space vectors of a seven-phase VSI in (a) d1-q1
plane (b) d2-q2 plane (c) d3-q3 plane (continued)
0.6Vdc
q3
During the synthesis of the seven-phase SVPWM one has to
deal with 27 = 128 space vectors of a seven-phase VSI. In
order to generate sinusoidal output voltage, all three planes
(d1-q1, d2-q2 and d3-q3) need to be simultaneously
considered (Grandi et al., 2006b; Dujic et al., 2007a). Space
vectors are designated as in the previous case and their
positions (tips) in all three planes are shown in Figure 4.
Similar to the five-phase case, there is no position in any
plane that is occupied by two (or more) different space
vectors (except for two zero space vectors in the origin of
each plane). There are now 14 sectors that can be identified,
each spanning the angle π/7.
v
3.2 Seven-phase SVPWM
5
b)
(b)
va1
va 3
va 2
3π 7
d3
va1
vb3
(c) )
The system of space vector equations that needs to be
solved in order to determine duty cycles for active space
vectors is given with (each row corresponds to one plane):
v * = va1δ a1 + va 2δ a 2 + va 3δ a 3 + vb1δ b1 + vb 2δ b 2 + vb3δ b3
0 = va1δ a1 + va 2δ a 2 + va 3δ a 3 + vb1δ b1 + vb 2δ b 2 + vb3δ b3 (9)
0 = va1δ a1 + va 2δ a 2 + va 3δ a 3 + vb1δ b1 + vb 2δ b 2 + vb3δ b3
6
D. Dujic et al.
General solution of (9), valid in every sector (s = 1 to 14), is
of the form:
δ a1 = KM sin( s
π
π
7
δ a 2 = K 2 M sin( s
− ϑ ); δ b1 = KM sin(ϑ − ( s − 1) )
7
π
π
− ϑ ); δ b 2 = K 2 M sin(ϑ − ( s − 1) )
7
7 (10)
π
π
δ a 3 = K3 M sin( s − ϑ ); δ b3 = K3 M sin(ϑ − ( s − 1) )
7
7
π
1
1
δ 0 = δ127 = δ O = [1 − K3 M cos((2s − 1) − ϑ )]
2
2
14
Trigonometric constants of (3) are used in (10) and this time
n = 7. The same as in the five-phase case, an equal
distribution of the total zero space vector duty cycle among
two zero space vectors is assumed.
Final per-leg duty cycles, necessary for DSP
implementation, are obtained as:
δ1 = δ O 2
and it is equal to 29 = 512. There are now four planes to
consider and each one can be divided into 18 sectors. Due to
the huge number of space vectors, their presentation is
omitted and only final expressions for duty cycles are given
in order to illustrate similarities with the previous two
multiphase SVPWM schemes.
To generate sinusoidal output voltage, eight
neighbouring active space vectors are selected per switching
period (Dujic et al., 2007b). These are illustrated in Figure 7
for the situation when the reference space vector is
positioned in the first sector of the d1-q1 plane. Setting the
zero reference values in other three planes, a system of
equations similar to (6) and (9) can be written, which after
some manipulations yields solution of the form:
δ a1 = KM sin( s
π
δ 2E = δ1 + δ a1
δ a 3 = K3 M sin( s
δ 3O = δ 2O + δ a 2 ;
δ 3E = δ 2E + δ b 2
δ 4O
δ 5O
δ 6O
δ 4E
δ 5E
δ 6E
δ a 4 = K 4 M sin( s
+ δ b3 ;
+ δ a3 ;
= δ 3E
= δ 4E
= δ 5E
+ δ a3
(11)
+ δ b3
+ δ b2 ;
+ δ a2
δ 7 = δ1 + δ a1 + δ a 2 + δ a 3 + δ b1 + δ b 2 + δ b3
Symmetrical switching pattern is obtained again, which is
illustrated in Figure 6 for the first sector. The sequence of
the applied space vectors in all odd sectors is 0, a1, b2, a3,
b3, a2, b1, 127, b1, a2, b3, a3, b2, a1, 0, while in even
sectors it is 0, b1, a2, b3, a3, b2, a1, 127, a1, b2, a3, b3, a2,
b1, 0.
Figure 6
δ 0 = δ 511
Figure 7
δ O δ a1δ b 2 δ a 3 δ b3 δ a 2 δ b1 δ O
4 2 2 2 2 2 2
2
π
π
− ϑ ); δ b3 = K3 M sin(ϑ − ( s − 1) ) (12)
9
9
π
π
− ϑ ); δ b 4 = K 4 M sin(ϑ − ( s − 1) )
9
1
1
π
= δ O = [1 − K 4 M cos((2 s − 1) − ϑ )]
2
2
18
9
Principle of calculation of times of application of the
active space vectors, (a) first plane (b) second plane
(c) third plane (d) fourth plane
vb3
vb 2
q2
vb 4
vb 2
π 9
*
d1
ϑ v
va 2 v a 3 v a 4
va1
Switching pattern of a seven-phase SVPWM in
Sector 1 (see online version for colours)
π
− ϑ ); δ b 2 = K 2 M sin(ϑ − ( s − 1) )
9
9
q1
vb1
− ϑ ); δ b1 = KM sin(ϑ − ( s − 1) )
9
π
δ a 2 = K 2 M sin( s
δ 2O = δ1 + δ b1 ;
= δ 3O
= δ 4O
= δ 5O
π
9
)
δ b1 δ a 2 δ b3 δ a 3δ b 2 δ a1 δ O
mB
vb3
va 4
vb1
(a)
(b)
q4
vb 3
vb1
3π 9
mC
d2
va1 va 3
vb3
vb 2
mA
va 3
2π 9
va 2
q3
2 2 2 2 2 2 4
vb 4
va 4
va 2 d 3
va1
4π 9
vb 2
d4
va 4 va1
a2
vb 4
mD
vb1
vb 4
mE
(c)
mF
mG
v0 v64 v96 v97 v113v115v123 v127 v123v115v113 v97 v96 v64 v0
3.3 Nine-phase SVPWM
Following the same procedure as before, a nine-phase
SVPWM for sinusoidal output voltage can be developed.
The number of space vectors now increases significantly
(d)
Solution given with (12) is valid for all sectors (s = 1 to 18)
and trigonometric constants (3) are determined with n = 9.
Similar to the previous two cases, per-leg duty cycles can be
obtained by summation of duty cycles of (12) in accordance
with the predetermined switching pattern per every sector.
Sequence of applied space vectors in all odd sectors is 0, a1,
b2, a3, b4, a4, b3, a2, b1, 511, b1, a2, b3, a4, b4, a3, b2, a1,
0; it is again different for all even sectors and is 0, b1, a2,
b3, a4, b4, a3, b2, a1, 511, a1, b2, a3, b4, a4, b3, a2, b1, 0.
Generalised space vector PWM for sinusoidal output voltage generation with multiphase voltage source inverters 7
This is illustrated in Figure 8 for the case of the first
sector and can easily be extrapolated to all the other sectors.
Figure 8
Switching pattern of a nine-phase SVPWM in Sector 1
(see online version for colours)
δ O δ a1 δ b 2 δ a 3 δ b 4 δ a 4 δ b3 δ a 2 δ b1 δ O
4 2 2 2 2 2 2 2 2
2
δ b1 δ a 2 δ b3 δ a 4 δ b 4 δ a 3 δ b 2 δ a1 δ O
2 2 2 2 2 2 2 2 4
mA
mB
mC
mD
mE
mF
mG
mH
mI
v0
v256
v384
v385
v449
v451
v483
v487
v503
v511
v503
v487
v483
v451
v449
v385
v384
v256
v0
4
Generalised multiphase SVPWM
There is a great similarity between sets of equations (7),
(10) and (12), used in the implementation of the SVPWM
for five-phase, seven-phase and nine-phase VSIs. This
offers a possibility to develop a generic SVPWM algorithm,
applicable to all odd phase number VSIs, which can
simplify and speed up implementation of the SVPWM
schemes aimed at sinusoidal output voltage generation.
Common for all n-phase sinusoidal SVPWM schemes
presented so far is the need to use n – 1 active space vectors
per switching period. As the phase number n increases, the
number of available space vectors of an n-phase VSI also
increases as 2n. While in the case of a five-phase VSI it is
still relatively easy to deal with 32 space vectors, this
becomes tedious for higher phase numbers. Thus, selection
of the set of n – 1 active space vectors is not always a
straightforward task, while in carrier-based PWM schemes
this happens naturally after comparison with carrier signal.
Once when the set of n – 1 active space vectors is
identified and the reference space vector magnitude and
position are known, it is necessary to solve an algebraic set
of n – 1 equations, in order to obtain duty cycles of each
active space vector. These equations [for example, (6) and
(9)] are generated considering positions and magnitudes of
the selected active space vectors in all (n – 1)/2 2-D planes.
The aim is to generate the fundamental in the first plane,
while zero average value is imposed as the restriction in all
the other planes.
However, regardless of the phase number, the pattern of
appearance of solutions is almost identical, provided that
these are expressed by means of trigonometric constants
defined with (3). This can be observed by comparing the
solutions for five-phase, seven-phase and nine-phase
SVPWM schemes that are given with (7), (10) and (12),
respectively. Obviously, these solutions have been obtained
using the knowledge of the selected set of active space
vectors and attributes of each of the space vectors in every
plane. Yet, the form of solutions allows a generalisation,
since the only important variables that change with the
change of the phase number are the trigonometric constants.
Clearly, the number of used active space vectors also
changes, as well as their attributes in different planes which
are characteristic for a particular phase number.
Based on (7), (10) and (12), a generic solution for duty
cycles of the active space vectors used for an n-phase
SVPWM for sinusoidal output voltage generation can be
written as:
⎡ δ a1
⎢ δ
⎢ a2
⎢ M
⎢
⎣⎢δ a ( n −1)
⎤ ⎡ K
⎥ ⎢ K
2
⎥=⎢
⎥ ⎢ M
⎥ ⎢
⎥ ⎣⎢ K ( n −1)
2⎦
⎤
⎥
⎥ M sin( s π − ϑ )
⎥
n
⎥
⎥
2⎦
⎡ δ b1
⎢ δ
⎢ b2
⎢ M
⎢
⎣⎢δ b ( n −1)
⎤ ⎡ K
⎥ ⎢ K
2
⎥=⎢
⎥ ⎢ M
⎥ ⎢
⎥ ⎣⎢ K ( n −1)
2⎦
⎤
⎥
⎥ M sin(ϑ − ( s − 1) π )
⎥
n
⎥
⎥
2⎦
(13)
Duty cycles of two zero space vectors, assuming equal
distribution of the total duty cycle of the zero space vectors,
are in the form:
1
2
δ 0 = δ 2n −1 = [1 − K ( n −1) 2 M cos((2 s − 1)
π
2n
− ϑ )]
(14)
Once calculated, duty cycles need to be summed properly in
order to obtain per-leg duty cycles that will be distributed in
accordance with the current sector. To perform this,
knowledge of the sequence of space vectors within the
switching period is necessary. In all SVPWM methods
presented, symmetrical switching pattern is generated by
having zero space vector (0) at the beginning and at the end
of the switching period and zero space vector 2n – 1 in the
middle of the switching period. The ordering of active space
vectors, between two zero space vectors, during the first
half of the switching period in the first sector, is illustrated
in Figure 9, for all three analysed topologies. It can be seen
that the ordering of active space vectors follows the rule that
is easy to generalise for any phase number.
The sequence starts with the smallest ‘a’ space vector
and changes in an alternating manner, through the selected
set of active space vectors, ending with the smallest ‘b’
space vector. The sequence is then reversed during the
second half of the switching period between zero space
vectors 2n – 1 and 0, thus being the mirror image of the
sequence from the first half of the switching period.
Although illustrations in Figure 9 are for the first sector, the
same applies to all the other odd sectors. With regard to the
ordering of the active space vectors in the even sectors, it is
easy to notice that the sequences of active space vectors in
all even sectors are obtained by swapping the sequences of
the first and the second half of the odd sector sequences. In
all even sectors, during the first half of the switching period,
D. Dujic et al.
8
the sequence is identical to the sequence in odd sectors
applied during the second half of the switching period and
vice versa.
Based on these considerations, it is possible to
generalise the set of expressions that yield the values of the
final per-leg duty cycles. Thus, in all odd sectors of an
n-phase SVPWM, per-leg duty cycles (n of them), in
increasing order, can be calculated as:
δ1 =
It is visible from (15) and (16) that the smallest per-leg duty
cycle (δ1) corresponds to the application of the zero space
vector 2n – 1 in the middle of the switching period, while
the largest duty cycle (δn) is the result of summation of δ1
and all the active space vector duty cycles. Obviously,
different placements of the zero space vectors can be
selected, but this issue is beyond the scope of this paper.
Finally, once when per-leg duty cycles are calculated,
they need to be distributed properly among inverter legs
depending on the current sector. Based on the distribution
given in Table 1 for a five-phase system, it is possible to
establish general pattern for per-leg duty cycle distribution.
It is enough to consider situation with respect to the first
inverter leg A. It can be observed that the first n sectors
receive duty cycles in the order from the largest one to the
smallest one, respecting the odd/even sector relations. Then,
in the remaining n sectors, the order of duty cycles is
reversed and they are applied from the smallest one to the
largest one, again respecting the odd/even sector relations.
For all the other inverter legs, duty cycle disposition is
obtained by simple shifting of the sequence of the previous
leg by two sectors. Since sectors span π/n, this is effectively
a shifting of 2π/n degrees, in accordance with the spatial
shift of the machine’s phase windings. Thus, disposition for
leg B is obtained by shifting duty cycles of leg A by two
sectors, leg C receives resulting duty cycles of leg B shifted
again by two sectors and so on. In this way, duty cycle
disposition for all n inverter legs in all 2n sectors is
obtained. This is summarised for the general case in
Table 2. It is easy to verify that Table 2 can be reduced to
the previously given Table 1, by replacing the phase number
n with the appropriate value (n = 5 in the case of Table 1).
Finally, general layout of an n-phase SVPWM
modulator, which is in accordance with the DSP
implementation used during the subsequent experimental
testing of the presented SVPWM schemes, is given in
Figure 10.
δO
2
⎡ δ2O ⎤
⎢
⎥ ⎡0
⎢ δ3O ⎥ ⎢ 0
⎢
⎥ ⎢
O
⎢ δ4 ⎥ ⎢ 0
⎢ M
⎥ ⎢
⎢
⎥ ⎢L
O
⎢δ(n−1) 2+1 ⎥ ⎢ 0
⎢
⎥=⎢
⎢δ(On−1) 2+2 ⎥ ⎢ 0
⎢
⎥ ⎢L
⎢ M
⎥ ⎢
⎢ O ⎥ ⎢0
δ
⎢ n−2 ⎥ ⎢
⎢ O ⎥ ⎢0
⎢ δn−1 ⎥ ⎢⎣ 1
⎢⎣ δn ⎥⎦
0
0
0 L 0
0 L 0
0 L 0
0 L 0
0
1
0
0 L 0
0 L 1
1
L L L L L L L L
0
0 L 0
1 L 1
1
0 0 L 1 1 L 1 1
L L L L L L L L
0
1 L 1
1 L 1
1
1 L 1
1 L 1
1
1
1
1 L 1
1 L 1
1
1 ⎤ ⎡ δb1 ⎤
⎢
⎥
1 ⎥⎥ ⎢ δa2 ⎥
1 ⎥ ⎢ δb3 ⎥
⎥
⎥ ⎢
L⎥ ⎢ M ⎥
1 ⎥ ⎢δa(n−1) 2 ⎥
⎥ + δ1
⎥⋅ ⎢
1 ⎥ ⎢δb(n−1) 2 ⎥
⎢
⎥
L⎥ ⎢ M ⎥
⎥
1 ⎥ ⎢ δa3 ⎥
⎥
⎥ ⎢
1 ⎥ ⎢ δb2 ⎥
1 ⎦⎥ ⎣⎢ δa1 ⎦⎥
(15)
For even sectors, basically the same calculations can be
applied, but this time with the reversed sequences of the
duty cycles of active space vectors, in accordance with the
reversed sequence of active space vectors. Therefore:
δ1 =
δO
2
⎡ δ2E ⎤
⎢
⎥ ⎡0
⎢ δ3E ⎥ ⎢ 0
⎢
⎥ ⎢
E
⎢ δ4 ⎥ ⎢ 0
⎢ M ⎥ ⎢
⎢
⎥ ⎢L
⎢δ(En−1) 2+1 ⎥ ⎢ 0
⎢
⎥=⎢
⎢δ(En−1) 2+2 ⎥ ⎢ 0
⎢
⎥ ⎢L
⎢ M ⎥ ⎢
⎢ E ⎥ ⎢0
⎢ δn−2 ⎥ ⎢
⎢ E ⎥ ⎢0
⎢ δn−1 ⎥ ⎢⎣ 1
⎢⎣ δn ⎥⎦
Figure 9
0
0
0 L 0
0 L 0
0 L 0
0 L 0
0
1
0
0 L 0
0 L 1
1
L L L L L L L L
0
0 L 0
1 L 1
1
0 0 L 1 1 L 1 1
L L L L L L L L
0
1 L 1
1 L 1
1
1 L 1
1 L 1
1
1
1
1 L 1
1 L 1
1
1 ⎤ ⎡ δa1 ⎤
⎢
⎥
1 ⎥⎥ ⎢ δb2 ⎥
1 ⎥ ⎢ δa3 ⎥
⎥
⎥ ⎢
L⎥ ⎢ M ⎥
⎢
1 ⎥ δb(n−1) 2 ⎥
⎥ + δ1
⎥⋅ ⎢
1 ⎥ ⎢δa(n−1) 2 ⎥
⎢
⎥
L⎥ ⎢ M ⎥
⎥
1 ⎥ ⎢ δb3 ⎥
⎥
⎥ ⎢
1 ⎥ ⎢ δ a2 ⎥
1 ⎦⎥ ⎣⎢ δb1 ⎦⎥
(16)
Sequences of active space vectors during the first half of the switching period in the first sector for, (a) five-phase SVPWM
(b) seven-phase SVPWM (c) nine-phase SVPWM
vb 2
q1
q1
vb1
q1
vb3
vb 2
vb1
va1
(a)
vb1
d1
va 2
vb3
vb 2
vb 4
d1
d1
va1
va 2
(b)
va 3
va1
va 2
(c)
va 3 va 4
Generalised space vector PWM for sinusoidal output voltage generation with multiphase voltage source inverters 9
Table 2
Per-leg duty cycle disposition through sectors
Leg | sector
1
2
3
A
δn
δ nE−1
δ nO− 2
…
2n – 2
2n – 1
2n
δ1
…
δ nE− 2
δ nO−1
…
δ1
δn
B
δ nO−1
δn
δn
…
δ 3O
δ 2E
…
δ nE− 4
δ nO−3
δ nE− 2
C
δ nO−3
δ nE− 2
δ nO−1
…
δ 5O
δ 4E
…
δ nE− 6
δ nO−5
δ nE− 4
M
M
M
M
…
M
M
…
M
M
M
δ 2O
δ 3E
δ 4O
…
δn
δ nE−1
…
δ 2E
δ1
δ1
M
M
M
…
M
M
…
M
M
M
n–2
δ nO− 6
δ nE− 7
δ nO−8
…
δ 6O
δ 7E
…
δ nE−3
δ nO− 4
δ nE−5
n–1
δ nO− 4
δ nE−5
δ nO− 6
…
δ 4O
δ 5E
…
δ nE−1
δ nO− 2
δ nE−3
n
δ nO− 2
δ nE−3
δ nO− 4
…
δ 2O
δ 3E
…
δn
δn
δ nE−1
(n + 1)/2
M
…
n
n+1
Figure 10 General layout of an n-phase SVPWM modulator for sinusoidal output voltage generation (see online version for colours)
v
*
Equations
(13)
Sector
determination
δ1
δ a1 , δ a 2 ,...δ a ( n −1) 2
M ,ϑ
(14)
δ 0 ,δ 2
n
Equations
(15)
−1
(16)
δ b1 , δ b 2 ,...δ b ( n −1) 2
s
In order to calculate per-leg duty cycles for the next
switching period, during the current switching period, for a
given reference space vector and the particular n-phase
topology under consideration, at first sector s must be
determined depending on the current position of the
reference space vector. Once when the sector is determined,
using current attributes of the reference space vector, duty
cycles of n – 1 active space vectors can be calculated based
on (13). Duty cycles of the zero space vectors can be
determined based on (14) and it is assumed here that the
total zero vector duty cycle is equally shared between two
zero space vectors. Although, as noted already, beyond the
scope of this paper, this distribution can be done in some
other manner and it represents the degree of freedom that
can be used to alter characteristics of the SVPWM scheme
(Holmes and Lipo, 2003).
Calculated duty cycles are further summed in order to
obtain the final per-leg duty cycles using (15) for odd
sectors and (16) for even sectors. Finally, calculated per-leg
duty cycles are distributed to inverter legs (Table 2), based
on the current sector value. These values are actually loaded
in the corresponding ‘compare’ registers of the DSP PWM
units and are compared against up/down running
counters/timers, before producing final PWM signals at the
s
mA
mB
δ 2O , δ 3O ,...δ nO−1
δ 2E , δ 3E ,...δ nE−1
Table 2
mn −1
mn
δn
s
DSP output pins, which are further provided to the inputs of
the IGBT drivers of the multiphase VSI.
It is important to note that, in order to implement
multiphase SVPWM schemes based on the proposed
general structure of the modulator, the actual analysis of the
space vectors of a multiphase system in all characteristic
planes can be omitted completely. The reason for this is
related to the form of the generic solutions for duty cycles
of the active space vectors, which are based on the use of
trigonometric constants. These constants are characteristic
for each particular topology and they embed proper dwell
times for each active space vector. Thus, although
developed SVPWM schemes are for different n-phase VSIs,
their form is very similar and it allows for generalisation
that can greatly speed up the implementation.
5
Three-phase SVPWM and its application to
multiphase systems
Following the guidelines given in Section 4, it is easy to
show that the three-phase SVPWM is encompassed by the
presented generalisation. The common feature of all n-phase
sinusoidal SVPWM schemes is the need to use n – 1 active
space vectors per switching period. Therefore, in a
10
D. Dujic et al.
three-phase system, there are two active space vectors per
switching period. In addition, two zero space vectors are
applied as well (zero and seven).
Based on (13), the generic solution for duty cycles of
two active space vectors (valid for any sector s = 1 to 6)
used for the three-phase SVPWM, for sinusoidal output
voltage generation, can be written as:
δ a = KM sin( s
π
3
3
1
2
δ 0 = δ 7 = δ O = [1 − KM cos((2 s − 1)
π
6
− ϑ )]
(18)
In the next step, per-leg duty cycles need to be calculated
using the knowledge of the switching pattern in every
sector. Similar to the cases illustrated in Figure 9, switching
pattern in all odd sectors is 0, a, b, 7, b, a, 0, while in even
sectors it is 0, b, a, 7, a, b, 0. Therefore, per-leg duty cycles
are determined using (15) and (16) as:
δ1 = δ O 2
δ 2O = δ1 + δ b ;
δ 2E = δ1 + δ a
δ 3 = δ1 + δ a + δ b
(19)
Once when per-leg duty cycles are calculated, they need to
be distributed properly through all six sectors. This is
summarised in Table 3. These equations, developed for
three-phase SVPWM, are well known (Holmes and Lipo,
2003) and they confirm the feasibility of the proposed
generalisation.
Table 3
M
π
sin( s − ϑ )
4
n
K ( n −1) 2
n
M
π
(20)
sin(ϑ − ( s − 1) )
δb =
4
n
K ( n −1) 2
n
nL( n −1) 2
1
1
π
δ 0 = δ 2n −1 = δ O = [1 − M
cos((2 s − 1) − ϑ )]
2
2
2 K ( n −1) 2
2n
δa =
π
δ b = KM sin(ϑ − ( s − 1) ) (17)
− ϑ );
Similar to the development of the SVPWM for other
multiphase VSIs, ‘a’ and ‘b’ active space vectors are those
that separate the sectors and use of subscript is omitted here
since two active space vectors are of the same magnitude.
Trigonometric constant K of (3) is now obtained by
inputting n = 3. Under the assumption of the equal
distribution of the total zero space vector duty cycle among
two zero space vectors, one has from (14):
1
2
average value over the switching period. Thus, this
approach can be used for multiphase machines with
concentrated winding. It is simple to establish that duty
cycles for two largest active space vectors, in the case of
application of this approach to multiphase SVPWM, are:
Per-leg duty cycle disposition through sectors
Leg | sector
1
2
3
4
5
6
A
δ3
δ 2E
δ1
δ1
δ 2O
δ3
B
δ 2O
δ3
δ3
δ 2E
δ1
δ1
C
δ1
δ1
δ 2O
δ3
δ3
δ 2E
It is interesting to note that the principle of three-phase
SVPWM (utilisation of two active vectors only) can be
easily applied to systems with higher phase numbers. This
has been discussed in de Silva et al. (2004), Iqbal and Levi
(2006) and Dujic et al. (2007a), where it has been shown
that this approach inherently introduces low-order
harmonics into the phase voltages. They appear as the result
of activation of the mapped active space vector pairs in the
planes other than the first one, which now do not yield zero
The remaining synthesis of the modulator is the same as the
one presented so far, with a difference that only two active
space vectors ‘a’ and ‘b’ with the largest magnitude are
involved in the switching pattern. Thus, considering, for
example, the five-phase case, the switching pattern in the
first sector is the same as the one shown in Figure 3, if only
active space vectors a2 and b2 are present. Similar applies
to all the other phase numbers.
6
DC bus voltage utilisation
Before presenting experimental results, it is interesting to
compare the DC bus utilisation of multiphase VSIs, for
these two rather different SVPWM strategies. The
maximum value of M, regardless of whether the SVPWM
based on use of n – 1 or just two active space vectors per
switching period is used, can be obtained from the total zero
space vector duty cycle. One needs to consider the condition
when the reference space vector is in the middle of the
sector (the first sector, for example) and determine for what
value of M the total duty cycle of the zero space vectors
becomes zero.
Thus, in the case of SVPWM scheme based on the use
of only two adjacent largest active space vectors per
switching period, one can find that the maximum level the
fundamental can reach is of the value:
n −1 π
sin(
)
2 K ( n −1) 2 2
2 n
=
M max − h (n) =
n L( n −1) 2 n cos( n − 1 π )
2 n
(21)
The additional sub-script ‘h’ is used to emphasise that this
value of the fundamental is obtained at the expense of low
order harmonic existence. The magnitudes of these
harmonic components are in a fixed relation to the
magnitude of the fundamental over the whole range of the
modulation index, as it will be shown shortly. On the other
hand, when purely sinusoidal output voltage is generated,
utilisation of the DC bus is determined as:
M max (n) =
1
K ( n −1)
=
2
1
n −1 π
sin(
)
2 n
(22)
Generalised space vector PWM for sinusoidal output voltage generation with multiphase voltage source inverters 11
M
max-h
max
1.22
Modulation indices
1.20
1.18
1.16
1.14
1.12
1.10
1.08
1.06
1.04
1.02
1
3
5
7
9
11
Number of phases
13
15
As can be seen, multiphase SVPWM schemes based on the
use of only two adjacent largest active space vectors offer
an increase in the DC bus utilisation as the number of
phases increases. The maximum value increases towards the
maximum value obtainable during the square-wave mode
(2n-mode) of operation M = 4 π ≈ 1.2731. However, with
an increase in the number of phases, n-phase SVPWM
schemes based on the use of n – 1 active space vectors per
switching period show a decrease in the value of the
maximum obtainable modulation index. It can be verified
that the trend of decrease is towards the value of the
modulation index equal to M = 1. This is the same value as
the one offered by the simplest sinusoidal modulation,
irrespectively of the number of phases.
It can be concluded that, in multiphase machines with
concentrated windings, better DC bus utilisation than for the
three-phase case can be achieved. Some of the low order
harmonics, generated in addition to the fundamental, are in
this case even desirable since they enable torque
enhancement. This has been demonstrated in the past for
five-phase drives (Ryu et al., 2005), seven-phase drives
(Locment et al., 2006) and nine-phase drives (Coates et al.,
2001). In real-world applications, this means that rated
operating voltage of the multiphase machine can be reached
from the lower DC bus voltage, thanks to harmonic
injection that simultaneously enhances developed torque.
On the other hand, multiphase machines with distributed
windings suffer from poorer DC bus utilisation, compared
to three-phase machines. As Figure 11 indicates, the
maximum value of the modulation index in the linear region
of modulation tends towards unity as the number of phases
increases.
7
Experimental results
The experimental results are collected from the
star-connected static ‘R-L’ load connected to a custom-built
multiphase VSI. SVPWM schemes are implemented in a
Figure 12 Five-phase SVPWM based on the use of, (a) two active
space vectors (b) four active space vectors per
switching period
M = 1.2311
300
Phase voltage (V)
M
1.24
200
100
0
-100
-200
-300
0.02
Phase voltage spectrum (V rms)
1.2731
1.26
0.03
0.04
0.05
100
200
300
0.06
Time (s)
0.07
0.08
0.09
0.1
400
500
Frequency (Hz)
600
700
800
0.07
0.08
0.09
0.1
400
500
Frequency (Hz)
600
700
800
250
200
150
100
50
0
0
(a)
M = 1.0515
300
Phase voltage (V)
Figure 11 Dc bus utilisation as the function of the number of
phases (see online version for colours)
TMS320F2812 DSP, which is used to control the
multiphase VSI. During measurements DC bus voltage was
around 600 V, while the switching frequency has been set to
5 kHz. One phase voltage of the multiphase load is directly
measured using the HP35665A dynamic signal analyser and
it is low pass filtered with a cut-off frequency of 1.6 kHz.
The results presented include phase voltage waveform and
its spectrum for the operation with the maximum achievable
value of M, which is a function of the phase number and
50 Hz reference. Experimental results for five-phase and
seven-phase cases are shown.
Phase voltages and their spectra, for the five-phase load,
obtained using the two SVPWM schemes, are shown in
Figure 12. It can be seen that when only two active space
vectors per switching period are used [Figure 12(a)], phase
voltage spectrum contains low-order harmonics.
200
100
0
-100
-200
-300
0.02
Phase voltage spectrum (V rms)
Expressions (21) and (22) are graphically illustrated in
Figure 11, for the phase numbers up to 15.
0.03
0.04
0.05
100
200
300
0.06
Time (s)
250
200
150
100
50
0
0
(b)
In addition to the fundamental, spectrum shows presence of
the third harmonic (around 23.8%) and the seventh
harmonic (around 2.5%). Thus, as already emphasised, a
simple extension of the three-phase SVPWM cannot
produce a sinusoidal output voltage. As a consequence
of the second plane not being considered, harmonics
characteristic for the d2-q2 plane are generated
12
D. Dujic et al.
(10k ± 3, k = 0, 1, 2…). Application of four active space
vectors per switching period [Figure 12(b)], with properly
determined duty cycles, generates purely sinusoidal output
phase voltages without any low-order harmonics from the
second plane. This is evidenced by the spectrum of the
phase voltage, from which it is clear that only the
fundamental component from the first plane is present.
Experimental results for the seven-phase case are shown
in Figure 13. Generated output phase voltage when only two
largest active space vectors are used [Figure 13(a)] is not
sinusoidal and contains low-order harmonic components.
This is the result of activated space vectors in the d2-q2 and
d3-q3 planes, which were not considered during the
development of the modulation scheme. Phase voltages
contain the fifth harmonic (around 10.36%) and the ninth
harmonic (around 2.02%) from the d2-q2 plane, as well as
the third harmonic (around 25.58%) and the 11th harmonic
(around 0.72%) from the d3-q3 plane.
Figure 13 Seven-phase SVPWM based on the use of, (a) two
active space vectors and (b) six active space vectors per
switching period
M = 1.2518
Phase voltage (V)
300
200
100
0
-100
-200
-300
Phase voltage spectrum (V rms)
0.02
0.03
0.04
0.05
100
200
300
0.06
Time (s)
0.07
0.08
0.09
0.1
400
500
Frequency (Hz)
600
700
800
0.07
0.08
0.09
0.1
400
500
Frequency (Hz)
600
700
800
250
200
150
100
50
0
0
(a)
M = 1.0257
Phase voltage (V)
300
200
100
0
-100
-200
-300
Phase voltage spectrum (V rms)
0.02
0.03
0.04
0.05
100
200
300
0.06
Time (s)
250
200
150
100
50
0
0
(b)
In general, in the case of a seven-phase system,
characteristic harmonics of the d2-q2 plane are 14k ± 5,
while in the d3-q3 plane one has 14k ± 3 (k = 0, 1, 2…)
harmonics. Similar to the five-phase case, the amount of
these harmonics stays in fixed relation to the fundamental
over the whole range of the achievable modulation index.
The application of six active space vectors per switching
period produces purely sinusoidal output voltages, as shown
in Figure 13(b). This is the consequence of properly
determined duty cycles for active space vectors, based on
consideration of all three planes of the seven-phase system.
Therefore, zero average values over the switching period are
achieved during the operation of a modulator.
8
Conclusions
Some common properties of n-phase SVPWM schemes for
sinusoidal output voltage generation are discussed in this
paper. Characteristic for all these schemes is the application
of n – 1 active space vectors over the switching period. This
enables averaging of the reference space vector in the
d1-q1 plane, with simultaneous zero average voltage in all
the other planes, so that the low-order harmonics are
neutralised in all the remaining planes.
It has been demonstrated that a great similarity exists in
the structure of these SVPWM schemes, which allows for
certain generalisation and unification. Presented generic
structure can be used during the implementation of any odd
multiphase SVPWM scheme directly, without tedious
analysis of the voltage space vectors in corresponding
planes. Proper dwell times are naturally embedded into duty
cycles by means of trigonometric constants and this
provides zero average values in all the planes other than the
first one. It was also demonstrated that the standard
three-phase SVPWM scheme is encompassed by this
generalisation and is just a special case of the generalised
SVPWM.
SVPWM schemes for generation of purely sinusoidal
output voltages are characterised with a decrease in the DC
bus utilisation as the number of phases increase. In contrast
to that, direct application of three-phase SVPWM principle
to multiphase systems (use of only two largest active space
vectors per switching period), offers an increase in DC bus
utilisation with an increase in the number of phases.
However, this is achieved at the expense of presence of
low-order harmonics in the phase voltages. While this is
unacceptable for multiphase machines with distributed
windings, it can be used in the case of multiphase machines
with concentrated windings. The amount of low-order
harmonic injection is not controllable and is in fixed relation
to the fundamental.
Feasibility of presented generalisation has been verified
experimentally using the five-phase and seven-phase
systems.
Acknowledgement
This work was supported in part by the Engineering and
Physical Sciences Research Council (EPSRC) under grant
EP/C007395, in part by Semikron, UK, in part by MOOG,
Italy and in part by Verteco, Finland.
Generalised space vector PWM for sinusoidal output voltage generation with multiphase voltage source inverters 13
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