Nr 64
Prace Naukowe Instytutu Maszyn, Napędów i Pomiarów Elektrycznych
Politechniki Wrocławskiej
Nr 64
Studia i Materiały
Nr 30
2010
Space vector modulation,
voltage source inverter, overmodulation
Khanh NGUYEN THAC*
A SIMPLE WIDE RANGE SPACE VECTOR PWM
CONTROLLER ALGORITHM FOR VOLTAGE-FED INVERTER
INDUCTION MOTOR DRIVE INCLUDING SIX-STEP MODE
In this paper a simple algorithm of space vector pulse width modulation (SVM) for two-level
voltage-fed inverter is proposed. The idea of algorithm is that a single algorithm covers the undermodulation and overmodulation range including six-step mode. The algorithm unifies equations to
determine angle of reference voltage vector Uc in undermodulation and overmodulation mode I,
therefore simplifies calculation program. The open loop control of the induction motor, a smooth operation during transition from the linear control to the six-step mode is demonstrated through experimental results using Matlab/Simulink simulation.
1. INTRODUCTION
Three-phase voltage source pulse width modulation (PWM) inverter have been
widely used for DC/AC power conversion, since they can produce a variable voltage
and variable frequency power [6], [9], [11], [12], [14]. However they require a dead time
to avoid the arm-short and snubber circuits to suppress the switching spike. Apart from
these aspects, the PWM inverters present an essential problem that they cannot produce
voltages as large as the six-step inverter can. That is, the DC bus voltage cannot be utilized to the maximum.
Many approaches have been reported in the literature to increase the range of the PWM
voltage source inverter (VSI) [1-5], [7], [8], [10], [13], [15-18]. Some of them are proposed
as extensions of the Sinusoidal PWM (SPWM) method [2], [4], [7], [8], [13], and others as
extensions for the Space Vector PWM (SVM) method [1-3], [5], [10], [15-18]. In the
__________
*
Wroclaw University of Technology, Institute of Electrical Machines, Drives and Measurements,
ul. Smoluchowskiego 19, 50-372 Wroclaw, Poland, khanh.nguyen.thac@pwr.wroc.pl
2
SPVM, by incrementing the reference voltage beyond the amplitude of the carrier triangular signal, some switching cycles are skipped and the voltage of each phase remains that of
one of the DC bus. This method shows a high non-linearity between voltage command and
output amplitude, and it requires an infinite amplitude command in order to reach a square
output voltage. In order to correct this non-linearity, a modification has been proposed to
the reference voltage used in SVM [11], which extends the range of the modulation index
to between 0.785 and 0.907, i.e. 15% greater than that obtained with the standard version
of SPWM. The over-modulation range starts from this point (modulation index M =
0.907), until the six-step mode (M = 1) is reached.
In this paper, a simple algorithm for space vector PWM to produce fundamental voltage
versus all range of the modulation index (from zero to unity) is proposed. When the
scheme is applied to the V/f control of induction motor (IM) drive, a wide range speed
results included field weakening mode can be obtained, what was show by simulations
performed using the MATLAB/SIMULINK.
2. SPACE VECTOR MODULATION
2.1. GENERAL SCHEME OF THE TWO-LEVEL INVERTER
The basic topology of the three-phase two-level voltage-fed inverter is expressed in Fig.
1(a), while Fig. 1 (b) shows its eight switching states of which six [U1 (100) – U6 (101)] are
active state vectors to from a hexagon, and two [U0 (000) and U7 (111)] are zero state
vectors that lie at the origin, that is noted by the command voltage vector Uc in sector 1.
(a)
DC side
U dc
U dc
2
(b)
 
Converter
T1
SA
T3
D4
SB
D3
SC
D6
SC
U 2 110 
D5
B
U AB
IB
C
IC
AC side
LB
UB
LC
SB
SC
SC
EA
EB
EC
UC
Induction Motor
Gate drive
UA
RC
U1 100 
c
U

 tR / Ts  Uc
4
or
LA
RB
Uc
ct
Se
SB
RA
U 4  011
U 7  000 
SA
SA
U 7 111
0t  
  
or
6
IA
D2
Se
ct
A
T2
Ts
T6
 tL /
SA
Se
ct
T4
U dc
2
or
3
U A0
0
1
or
SB
Sector 2
ct
Se
D1
U 3  010 
T5
U 5  001
Sector 5
U 6 101
N
Fig. 1. Two level voltage-fed inverter (a), switching states of the inverter (b)
3
The operation in undermodulation range is determined by the modulation index M,
which is defined as the ratio between the magnitude of the command or reference
voltage vector and the peak value of the fundamental component of the square-wave
voltage. The modulation index (M) varies between 0 and 1. In the undermodulation
region [0  M  0.907) , shown in Fig. 2(a), the reference voltage vector U c remains
within the hexagon. The overmodulation region is subdivided into two modes: mode I
[0.907  M  0.952) and mode II [0.952  M  1.00] . Fig. 2(b) shows a reference
vector for mode I and the lower and upper trajectory limits for this region. Fig. 2(c)
shows operation in overmodulation mode II and the corresponding trajectory limits.
U 2 110 
U 2 110 
D
M = 0.907
Uc

A

*
h

U1 100 
a 
Uc
h
*
U1 100 
UR
B
*
c


U*c

 /6
U
M=1
h
Uc
C

M = 0.952
M = 0.952
M = 0.907
UL
U 2 110 
h
U1 100 
 b
c
Fig. 2. Regions of the inverter operation: undermodulation [0  M  0.907] (a), Overmodulation mode I
[0.907  M  0.952] (b), Overmodulation mode II [0.952  M  1.00] (c)
2.2. UNDERMODULATION REGION
For calculation effective times (tR and tL) in undermodulation region (M ≤ 0.907)
of sector N ( N  1  6 ), we use vector diagram shown in the Fig. 3.
U2
i
U i 1
U c

N 2
Uc
U

U3
U R
UR
N
U L
U4

3
UL
1
3
N

*
N

3
 N  1

U L

U c U R
U1

Fig. 3. Vector diagram of the reference vector voltage Uc in the sector N
4
The effective times of the inverter switching states in undermodulation region are
obtained from simple trigonometrically relationships, according to the vector diagram
in Fig. 3, and can be expressed by the following equation:
U c  U R  U L  Ui
tR 
3Ts
U dc
tR
t
 Ui 1 L
Ts
Ts

 
  
U c sin  3 N   U c cos  3 N  










 U c sin  3  N  1   U c cos  3  N  1  





t0  t7   Ts  tR  tL  / 2
tL 
(1)
3Ts
U dc
(2)
where
tR , tL , t0 – effective time for the right, left and zero switching vectors, respectively,
Ts  1 / f s – sampling time (fs – switching frequency),
U c , U c –  components of the reference voltage vector Uc,
U dc – DC bus voltage.
After some modifications, we obtain:
tR 
2 3



MTs sin  N   
3





MTs sin     N  1 

3


t0  t7  Ts  t R  t L  / 2
tL 
2 3
(3)
where  is angle of U c (see Fig.3).
In the case of M is a modulation index, the space vector modulation is defined as:
M
where:
Uc
U1 six  step 

Uc
2
U dc

(4)
5
U c  U c2  U c2 – phase peak value,
U1( six  step ) – fundamental peak value of the square-phase voltage wave.
The modulation index M varies from 0 to 1 at the square-wave output. The length of the
U c vector, in the whole range of  is equal to U dc / 3 . This is a radius of the circle
inscribed in the hexagon in Fig. 1(b). At this condition the modulation index is equal:
M 
U dc / 3
2 * U dc / 
 0.907
(5)
This means that 90.7% of the fundamental voltage at the square wave can be obtained.
2.3. OVERMODULATION REGION
In the algorithm where overmodulation region is considered, two operation modes
depending on the reference voltage value are defined. In the mode I the algorithm
modifies only the voltage vector amplitude, in the mode II both the amplitude and
angle of the voltage vector are influenced [10],[15].
The overmodulation mode I is shown in Fig. 2(b). In this mode voltage vector U c
crosses the hexagon boundary at two points (B and C) in each sector. There is a loss
of fundamental voltage in the region where reference vector exceeds the hexagon
boundary. To compensate for this loss, the reference vector amplitude is increased in
the region where the reference vector is in the hexagon boundary. The magnitude of
the reference vector is changed from Uc to U*c , while the angle is transmitted without
any changes (  *   ). A modified reference voltage trajectory proceeds partly on the
hexagon (line length BC) and partly on the circle (curve length AB and CD). This
trajectory is shown with solid line in Fig. 2(b). This mode extends the range of the
modulation index up to 0.952.
In the hexagon trajectory part only active vector are used. The duration of these
vectors tR and tL are obtained from trigonometrical relationships and can be expressed by the following equation [10]:
3 cos  *  sin  *
3 cos  *  sin  *
tL  Ts  tR
tR  Ts
(6)
t0  t7  0
In the general case, we can calculate the switching time using following equations:
6




3 cos     N  1   sin     N  1 
3
3


tR  Ts




3 cos     N  1   sin     N  1 
3
3


tL  Ts  tR
(7)
t 0  t7  0
Region II starts from M = 0.952 and reaches six-step mode M = 1.00 (Fig. 2c). The
trajectory U c is maintained on the hexagon, which defines amplitude of the reference
voltage vector. The angle is calculated from the following equation [10]:
0

   h 
*
 
 6  h 6
 3
0    h
for
h     3  h
(8)
 3  h     3
where  h is the hold-angle. This angle uniquely controls the fundamental voltage. It
is a nonlinear function of the modulation index [10], [15].
The modified vector is held at a vertex of the hexagon for holding angle  h over
particular time and then partly tracks the hexagon sides in every sector for the rest of
the switching period. The holding angle  h is nonlinear function of the modulation
index, which can be piecewise linearized, what is presented in [15]:
6.40M  6.09

 h  11.75M  11.34
48.96M  48.43

0.9517  M  0.980
for
0.980  M  0.9975
(9)
0.9975  M  1.0000
In Fig. 2(c) the reference vector trajectory generated for the first sector is shown.
This trajectory is obtained in three steps. First part, if angle  is less than the respective
value of  h , the algorithm hold the vector at the vertex U 1 . Next part is for  from  h
to  3   h . In this angle range, the reference vector moves along the hexagon. In the
last range, from  3   h to  3 , the vector U c held until the next vertex U 2 . The ondurations are calculated by substituting  * for  in (6).
The overmodulation mode II works up to the six-step mode for  h equal to zero.
The six-step mode is characterized by selection of the switching vector for one-sixth
7
of the fundamental period. In this case, the maximum possible inverter output voltage
is generated.
3. SIMPLE SVM ALGORITHM
Fig. 4 shows the flowchart for SVM algorithm, where U c and U c are  components (real and imaginary) of the reference voltage vector U c , respectively of U c , 
is the angular distance from the Re axis in Fig. 3 ( Uc  U c  ).
U c ,U c
Calculate the angle  (11)
Determine the sector N (12)
Calculate the
modulation index M (4)
0  M  0.907
0.952  M  1.00
M
Calculate  h  9 
0.907  M  0.952
Calculate  * 8
Calculate tR , tL  7 
Calculate tR , tL , t0  3
Calculate tR , tL by  *  6 
Caculate TA_ON , TB_ON , TC_ON
Switch gates
Fig. 4. Simple algorithm of SVM
The angle  is given by the following equation:
0  arcsin Uc / Uc 
 0

  2   0
  

0
(10)
U c  0 and U c  0
for
U c  0 and U c  0
U c  0
The sector number N is calculated from following equation:
(11)
8
 0     / 3
 / 3    2 / 3
 2 / 3     
    4 / 3
 4 / 3    5 / 3
 5 / 3    2 
1

2
3

N 
4
5

6
(12)
In the Fig. 5 the switching times for sector 1 are shown for the illustration.

U 2 110 
A
TB ON
B
UL
Uc
C

U1 100 
UR
Sector 1
TB OFF
U 0 U1
t0 tR
U 2 U7 U7 U 2
tL t7 t7 tL
Ts
U1 U 0
tR t0
Ts
Tp
Fig. 5. Switching sequences for the first sector
The switching times for the first sector are given by following equations:
TAON  t0
TB ON  t0  t R
(13)
TC ON  t0  tR  tL
and TOFF  TP  TON  2Ts  TON because of symmetry.
In the other sectors, the switching times calculation is similar to those of the first
sector and in the overmodulation region only difference is that t0  0 .
4. SIMULATION RESULTS
In order to validate the proposed algorithm, a V/f open–loop controlled induction
motor drive with 5kHz switching frequency was simulated. The parameters of the
induction motor are given in the Table 1.
9
Table 1. Parameters of the induction motor
Number of phase windings ( ms)
Power (PN)
Voltage (UN)
Current (IN)
Frequency (fN)
Number of pole pairs (pb)
Base speed (nN)
Nominal torque (TN)
3
5500W
220V
14.2A
50Hz
3
910rpm
0.1972Nm
Based on the algorithm analyses, some simulation results using
MATLAB/SIMULINK are shown below, in Fig. 6 – Fig. 9.
In Fig. 6 transients of the electromagnetic torque ( me ) and motor speed are shown,
for nominal load torque of the motor, switched on with the delay  =0s. Also smooth
changes of the modulation index M is illustrated in this figure, in the whole range of
the SVM modulation ( 0  M  1 ). It can be seen that, in undermodulation region
(t<0.55s), after starting the me oscillates around the load torque (mL) value with small
amplitude; in this case the current and flux have sine wave because the voltage is in
the linear PWM region (see Fig. 7). In the next range – in overmodulation PWM region (t>0.55s), the me oscillates around mL with bigger amplitude (for
0.907≤M<0.925), while the current and stator flux shapes are not longer sine wave
and their amplitudes increase (Fig. 7).
Fig. 6. Electromagnetic torque ( me ), rotor speed ( m ), load torque ( mL ) and modulation index ( M )
when maximum reference frequency equal to nominal frequency (50 Hz)
10
Fig. 7. Voltage, current and flux waveforms when VSI transition from PWM to six-step mode
with nominal load torque and reference frequency
Fig. 8 and 9 show simulation results when reference speed ref takes three levels
(1.0, 1.5 and 2.0 [p.u.]), while the load torque is zero (mL=0). In Fig. 8, the relationship between the electromagnetic torque (me), speed and modulation index (M) are
shown, when ref =1 and M=0.969, as well as ref =1.5 and ref =2 and the M=1, respectively.
Fig. 8. Electromagnetic torque (me), rotor speed (m), load torque (mL) and modulation index (M) when
reference speed (ref) changed from zero to 2.
In Fig. 9 trajectories of the stator and rotor flux vector are demonstrated, correspond-
11
ing to three levels of speeds and modulation indexes. When ref >1, the IM operates in
the field weakening region, with the six-step mode of VSI, so trajectories of stator
flux look like hexagon.
Fig. 9. Stator flux and rotor flux in no load mode when ref changed:
(1) – ref =1, (2) – ref =1.5, (3) – ref = 2
5. CONCLUSIONS
This paper proposes a simple space vector PWM algorithm. The algorithm has
been described in detail and shows the continuous control of SVM inverter in the
whole range of modulation index, from zero to unity. The simulation results show that
the algorithm works very well in under and overmodulation regions. The algorithm
can be applied to the IM motor control in a wide range speed, including field weakening mode.
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