⎡ 1
a
a 4
7
a ⎤ U
⎡ a⎤
T
⎢ ⎥
p ⎢
⎥ ⎢
⎥
=
⋅
⎢ 2
T
a
⎥
⎢ 2 a 5
8
a
U
⎥ ⎢ b
(20)
U
⎥
dc
⎢ ⎥
⎢
⎥ ⎢
⎥
⎣ 3
T ⎦
⎣ 3
a
6
a
9
a ⎦ U
⎣ c ⎦
This method can be applied in both frames a − b − c and α − β − γ in the same way. The switching states x , y and z or the voltage vectors x
V , y
V and z
V are independent of the
coordinates and are determined only from the relative values of Ua , Ub and Uc . All matrix elements ia take the values 0, 1 or -1. Therefore, the calculations need only the addition and
subtraction of Ua , Ub and Uc except the coefficient p
T Udc . The ia values are determined
from the following relations where they can be presented as a function of elementary
relative voltages:
⎧ 1 U =
⎧
=
⎧
=
a
U 1
1 U
U
1 U
U
⎪
a
2
a
3
=
⎪
⎪
⎨−
=
= ⎨−
=
= ⎨−
=
1
a
1 Ua U 2 , a 2
1 Ua U 3 , a 3
1 Ua U 4
⎪
⎪
⎪
⎩ 0 otherwise
⎩ 0 otherwise
⎩ 0 otherwise
⎧ 1 U =
⎧
=
⎧
=
b
U 1
1 U
U
1 U
U
⎪
b
2
⎪
b
3
⎪
a = ⎨−
=
= ⎨−
=
= ⎨−
=
4
1 Ub U 2 , a 5
1 Ub U 3 , 6
a
1 Ub U 4
(21)
⎪
⎪
⎪
⎩ 0 otherwise
⎩ 0 otherwise
⎩ 0 otherwise
⎧ 1 U =
⎧
=
⎧
=
c
U 1
1 U
U
1 U
U
⎪
c
2
c
3
=
⎪
⎪
⎨−
=
= ⎨−
=
= ⎨−
=
7
a
1 Uc U 2 , 8
a
1 Uc U 3 , 9
a
1 Uc U 4
⎪
⎪
⎪
⎩ 0 otherwise
⎩ 0 otherwise
⎩ 0 otherwise
The Space Vector Modulation PWM Control Methods Applied on Four Leg Inverters
247
If we substitute these values in (20) and according to the given definitions of i , j , k and r , the application duration of adjacent vectors, can be expressed in (22), it shows that they
are only depending on the relative voltage vectors U 1 , U 2 , U 3 and U 4 .
⎡ ⎤
⎡
−
1
T
U 1 U 2 ⎤
T
⎢ ⎥
p
⎢
⎥
=
⋅
−
⎢ 2
T
U
⎥
⎢ 2 U 3
(22)
U
⎥
dc
⎢ ⎥
⎢
−
⎥
⎣ 3
T ⎦
U
⎣ 3 U 4 ⎦
6. 3D-SVM new algorithm for four leg inverters
A new algorithm of tetrahedron determination applied to the SVPWM control of four leg
inverters was presented by the authors in [82]. In this algorithm, a new method was
proposed for the determination of the three phase system reference vector location in the
space; even the three phase system presents unbalance, harmonics or both of them. As it was
presented in the previous works the reference vector was replaced by three active vectors
and two zero vectors following to their duty times [34-35], [69-80]. These active vectors are
representing the vectors which are defining the special tetrahedron in which the reference
vector is located.
In the actual algorithm the numeration of the tetrahedron is different from the last works,
the number of the active tetrahedron is determined by new process which seems to be more
simplifiers, faster and can be implemented easily. Form (5) and (12) the voltages in the αβγ
frame can be presented by:
V
⎡
⎤
⎡ S − ⎤
α
a
Sf
⎢
⎥
⎢
⎥
V
⎢ β = C ⋅
⎥
⎢ −
⎥ ⋅
b
S
Sf
g
V
(23)
⎢ V ⎥
⎢
⎥
⎣ γ ⎦
−
⎢⎣ c
S
Sf ⎥⎦
It is clear that there is no effect of the fourth leg behaviours on the values of the components
in the α − β plane. The effect of the fourth leg switching is remarked in the γ component.
The representation of these vectors is shown in Fig. 12 –c-.
6.1 Determination of the truncated triangular prisms
As it is shown in the previous sections, the three algorithms are based on the values of the
a − b − c frame reference voltage components. In this algorithm there is no need for the calculation of the zero (homopolar) sequence component of the reference voltage. Only the
values of the reference voltage in a − b − c frame are needed. The determination of the truncated triangular prism (TP) in which the reference voltage space vector is located is
based on four coefficients. These four coefficients are noted as C 0 , C 2 , C 3 and C 4 . Their values can be calculated via two variables x and y which are defined as follows:
V
x
α
=
(24)
V
248
Electric Machines and Drives
γ − axis
V 8
γ − axis
α − axis
V
V 8
7
V 4
α − axis
V
V
V 7
6
3
V
β − axis
4
V
V 6
V
3
5
V 2
V 5
V 2
TP =
V V
β − axis
2
1
16
V V
β − axis
1
16
V 15
TP = 1
α − axis
V
V
TP =
15
3
12
V 12
V 14
V 14
TP = 4
TP = 6
V 11
V 11
V
V
13
13
V
V
TP = 5
10
10
V
V
9
9
(a) (b) (c)
Fig. 12. Presentation of the possible switching vectors in a − b − c
V
y
β
=
(25)
V
Where:
2
2
V = Vα + Vβ
(26)
The coefficients can be calculated as follows:
⎡
1
⎤
C
⎢
⎥
⎡ 0 ⎤
⎛ 5
⎞
⎢
⎢ ⎥
INT ⎜
x ε ⎥
− − ⎟
⎢
⎝
⎠⎥
C
⎢ 1
2
⎥ = ⎢
⎥
(27)
C
⎢ ⎥
− − ε
2
INT
⎢
(1 y ) ⎥
⎢ ⎥
C
⎢
⎥
⎣ 3 ⎦
⎛ 5
⎞
⎢ INT
+ x + ε ⎥
⎜
⎣
⎝ 2
⎟
⎢
⎠⎥⎦
ε is used to avoid the confusion when the reference vector passes in the boundary between
two adjacent triangles in the αβ plane, the reference vector has to be included at each
sampling time only in one triangle Fig. 12-c-. On the other hand, as it was mentioned in the
first family works, the location of the reference vectors passes in six prism Fig. 12-b-, but
effectively this is not true as the reference vector passes only in six pentahedron or six
truncated triangular prism (TP) as the two bases are not presenting in parallel planes
following to the geometrical definition of the prism Fig. 12-a-. The number of the truncated
prism TP can be determined as follows:
2
TP = 3
i
C + ∑ −
(28)
2
( 1) CiCi+1
i=0
The Space Vector Modulation PWM Control Methods Applied on Four Leg Inverters
249
6.2 Determination of the tetrahedrons
In each TP there are six vectors, these vectors define four tetrahedrons. Each tetrahedron
contains three active vectors from the six vectors found in the TP. The way of selecting the
tetrahedron depends on the polarity changing of each switching components included in
one vector. The following formula permits the determination of the tetrahedron in which the
voltage space vector is located.
3
T = 4( TP − 1) + 1 +
h
∑ ia
(29)
1
Where:
a = 1 if V ≥ 0
else a = 0
i
i
i
i = a, b, c
To clarify the process of determination of the TP and Th for different three phase reference system voltages cases which may occurred. Figures 13 and 14 are presenting two general
cases, where:
•
Figures noted as ‘a’ present the reference three phase voltage system;
•
Figures noted as ‘b’ present the space vector trajectory of the reference three phase
voltage system ;
•
Figures noted as ‘c’ present the concerned TP each sampling time, where the reference
space vector is located;
•
Figures noted as ‘d’ present the concerned Th in which the reference space vector is
located.
Case I: unbalanced reference system voltages
300
Va
Vb
Vc
200
30
)
20
100
10
)V
itude (V
a (
0
0
agn
gam
M
-10
V
-20
-100
oltageV
-30
400
-200
200
400
200
0
Vbe
0
ta
-200
(
-300
V
-200
)
Valpha (V)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
-400
Time (s)
-400
(a) (b)
6
24
5.5
22
20
5
18
4.5
P
16
h
f T
4
f T
o
14
o
ber
3.5
ber
12
um
um
3
10
he N
he N
T
8
T
2.5
6
2
4
1.5
2
1
0
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Time (s)
Time (s)
(c) (d)
Fig. 13. Presentation of instantaneous three phase reference voltages, reference space vector,
TP and Th
250
Electric Machines and Drives
Case II Unbalanced reference system voltages with the presence of unbalanced harmonics
500
400
300
150
)
100
200
Ve (
50
)
100
Va (
0
0
agnitud
gam
-50
V
-100
tge M
-100
ola -200
V
-150
-300
400
200
400
-400
200
V
0
beta
0
(V -200
-500
)
-200
Valpha (v)
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
-400
Time (s)
-400
(a) (b)
6
25
5.5
5
20
4.5
P
H
4
f T 15
f the T
o
o 3.5
ber
ber
um
um
3
10
he N
he N
T
2.5
T
2
5
1.5
1
0
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
0
0.002 0.004 0.006 0.008
0.01
0.012 0.014 0.016 0.018
0.02
Time (s)
Time (s)
(c) (d)
Fig. 14. Presentation of instantaneous three phase reference voltages, reference space vector,
TP and Th
6.3 Calculation of duty times
To fulfill the principle of the SVPWM as it is mentioned in (9) which can be rewritten as
follows:
3
⋅
= ∑ ⋅