V
5
V
5
5
V
5
7
7
V 7
7
V , V
,
V , V
V , V
V , V
1
16
1
V
1
16
V
1
16
V
1
16
V
V
14
V
14
V
14
V
14
3
V
V
3
V
12
3
12
3
V
12
12
V
V
V
V
10
10
10
10
V
V
V
V
V
V
V V
15
, V
V 13
15
13
15
13
15
13
1
16
V
V
V
V
V
V
V 11
11
9
11
9
11
9
V 9
Fig. 6. The possible space including the voltage space vector (the dodecahedron).
RP
1
V
2
V
3
V
RP
1
V
2
V
3
V
1
9
V
10
V
12
V
41
9
V
13
V
14
V
5
2
V
10
V
12
V
42
5
V
13
V
14
V
7
2
V
4
V
12
V
46
5
V
6
V
14
V
8
2
V
4
V
8
V
48
5
V
6
V
8
V
9
9
V
10
V
14
V
49
9
V
11
V
15
V
13
2
V
10
V
14
V
51
3
V
11
V
15
V
14
2
V
6
V
14
V
52
3
V
7
V
15
V
16
2
V
6
V
8
V
56
3
V
7
V
8
V
17
9
V
11
V
12
V
57
9
V
13
V
15
V
19
3
V
11
V
12
V
58
5
V
13
V
15
V
23
3
V
4
V
12
V
60
5
V
7
V
15
V
24
3
V
4
V
8
V
64
5
V
7
V
8
V
Table 2. The active vector of different tetrahedrons
Each tetrahedron is formed by three NZVs (non-zero vectors) confounded with the edges
and two ZVs (zero vectors) ( 1
V ,
16
V ). The NZVs are presenting the active vectors
nominated by 1
V , 2
V and 3
V Tab. 2. The selection of the active vectors order depends on
several parameters, such as the polarity change, the zero vectors ZVs used and on the
sequencing scheme. 1
V , 2
V and 3
V have to ensure during each sampling time the equality
of the average value presented as follows:
⋅
=
⋅
+
⋅
+
⋅
+
⋅
+
⋅
=
+
+
+
+
re
V f z
T
1
V
1
T
2
V
2
T
3
V
3
T
0
V 1 01
T
016
V
016
T
z
T
1
T
2
T
3
T
0
T 1
0
T 16 (9)
The last thing in this algorithm is the calculation of the duty times. From the equation given
in (9) the following equation can be deducted:
The Space Vector Modulation PWM Control Methods Applied on Four Leg Inverters
239
⎡
⎤
ar
V ef
⎡ 1
V a
2
V a
3
V a ⎤
⎡ 1
T ⎤
⎡ 1
T ⎤
⎢
⎥ ⎢
⎥ 1 ⎢ ⎥
1
⎢ ⎥
⎢
⎥ =
⋅
⋅
=
⋅
⋅
bre
V f
⎢ 1
V b
2
V b
3
V b ⎥
⎢ 2
T
M
⎥
⎢ 2
T ⎥
(10)
⎢
⎥
z
T
z
T
V
⎢
⎥
⎢ ⎥
⎢ ⎥
⎢⎣
⎥ ⎣ 1
V c
2
V c
3
V c ⎦
⎣ 3
T ⎦
⎣ 3
T
cref
⎦
⎦
M
Then the duty times:
T ⎤
V
⎡
⎤
⎡ 1
aref
⎢
⎥
⎢ ⎥
1
−
=
⋅
⋅
2
T
z
T M
⎢ b
V ref ⎥
⎢ ⎥
(11)
⎢
⎥
⎢ ⎥
⎣ 3
T ⎦
⎢⎣ cr
V ef ⎥⎦
4. 3D-SVM in α − β − γ coordinates for four leg inverter
This algorithm is based on the representation of the natural coordinates a , b and c in a new 3-D orthogonal frame, called α − β − γ frame [72-80], this can be achieved by the use of the
Edit Clark transformation, where the voltage/current can be presented by a vector V :
V
⎡ α ⎤
⎡
⎡ ⎤
a
V ⎤
Iα
⎡ I ⎤
⎢
⎥
a
⎢ ⎥
V
V
⎢ ⎥
=
⎢ ⎥
β = C ⋅
⎢
⎥
⎢
=
= ⋅
b
V ⎥ I I⎢β
C I
⎥
⎢ b ⎥
(12)
⎢ V ⎥
⎢ ⎥
⎢ ⎥
⎣
⎢ ⎥
γ ⎦
⎣ c
V ⎦
I
⎣ γ ⎦
⎣ Ic ⎦
C represents the matrix transformation:
⎡ 1
−1 2
−1 2 ⎤
2 ⎢
⎥
C = ⋅ 0
3 2 −
⎢
3 2⎥
(13)
3 ⎢1 2 1 2
1 2 ⎥
⎣
⎦
When the reference voltages are balanced and without the same harmonics components in
the three phases, the representations of the switching vectors have only eight possibilities
which can be represented in the α − β plane. Otherwise in the general case of unbalance
and different harmonics components the number of the switching vectors becomes sixteen,
where each vector is defined by a set of four elements ⎡ S , S , S ,
⎤
⎣ a b c Sf ⎦ and their positions in
the α − β − γ frame depend on the values contained in these sets Tab. 3.
Each vector can be expressed by three components following the three orthogonal axes as
follows:
i
V
⎡
⎤
α
⎢
⎥
i
i
V = V
⎢ β ⎥
⎢
⎥
i
⎢ Vγ ⎥
⎣
⎦
Where:
i = 1,16
(12)
240
Electric Machines and Drives
It is clear that the projection of these vectors onto the αβ plane gives six NZVs and two
ZVs; these vectors present exactly the 2D presentation of the three leg inverters, it is
explained by the nil value of the γ component where there is no need to the fourth leg.
On the other side Fig. 7 represents the general case of the four legs inverter switching
vectors. The different possibilities of the switching vectors in the α − β − γ frame are shown
clearly, seven vectors are localised in the positive part of the γ axis, while seven other
vectors are found in the negative part, the two other vectors are just pointed in the
(0,0,0) coordinates, this two vectors are very important during the calculation of the
switching times.
Vector
Sa b
S c
S Sf
Vγ
Vα
Vβ
9
V
0001
1
−
0 0
10
V
0011
− 1
− 1
3
3
2
11
V
0101
−
− 1
+ 1
3
3
3
13
V
1001
+ 2 3
0
12
V
0111
−2 3
0
14
1
V
1011
−
+ 1
− 1
3
3
3
15
V
1101
+ 1
+ 1
3
3
16
V
0000 0
0
0
Vector
Sa b
S c
S Sf
Vγ
Vα
Vβ
1
V
1111
0 0 0
2
V
0010
− 1
− 1
3
3
1
3
V
0100
+
− 1
+ 1
3
3
3
5
V
1000
2 3
0
4
V
0110
− 2 3
0
6
2
V
1010
+
+ 1
− 1
3
3
3
7
V
1100
+ 1
+ 1
3
3
9
V
1110
1
+
0 0
Table 3. Switching vectors in the αβγ frame
The Space Vector Modulation PWM Control Methods Applied on Four Leg Inverters
241
γ − axis
γ
V
β − axis
1⋅ Vg
V 8
V
2
7
+ ⋅ Vg
3
V 4 V 3
V 6
1
V 5
+ ⋅ Vg
3
V 2
α − axis
0 ⋅ Vg
V V
1
16
V 15
V 12
− 1 ⋅ Vg
V
3
14
V 11
2
V 13
− ⋅ Vg
3
V 10
V 9
−1⋅ Vg
Fig. 7. Presentation of the switching vector in the αβγ frame
Varef
a, b, c
Vα ref
Vbref
V
α , β
Vβ ref
cref
+
V
⋅ V
α ref
β ref
+
+
Vβ
V
ref
β ref
True
True
True
True
V
≥ 3 ⋅ V
α ref
β ref
V
≥ 3 ⋅ V
V
≥ 3 ⋅ V
V
≥ 3 ⋅ V
α ref
β ref
α ref
β ref
α ref
β ref
1
P
P 2 P 4
P 5 P 3
P 2 P 6
P 5
Fig. 8. Determination of the prisms
The position of the reference space vector can be determined in two steps.