A Novel SVPWM Overmodulation Technique Based On Voltage Correcting Function
Ilioudis C. Vasilios,
Student Member, IEEE
Dept. of Electrical and Computer Engineering Aristotle University of Thessaloniki Thessaloniki, Greece email: ilioudis@teithe.gr
Margaris I. Nikolaos,
Member, IEEE
Dept. of Electrical and Computer Engineering Aristotle University of Thessaloniki Thessaloniki, Greece email: margaris@eng.auth.gr
Abstract
—This paper presents a new Space Vector Pulse Width Modulation (SVPWM) strategy for voltage regulation of a Voltage Source Inverter (VSI) enabling the continuous transition from the linear modulation to the sixstep mode. Overmodulation operation is based on a correcting function method modifying properly the amplitude and the phase angle of the VSI output voltage. The evaluation of the method is explored via frequency analysis of the inverter voltage and motor currents in order to be able to evaluate the possible impact of the drive system. Simulation results demonstrate the effectiveness of the proposed SVPWM Overmodulation algorithm applied on sensorless speed control of a salientpole Synchronous Machine (SM) via Matlab/Simulink utility.
Keywords
Synchronous Machine (SM); SVPWM Voltage Correcting Function (VCF); SVPWM Overmodulation; SixStep Operation.
N
OTATION
V
dc
= dclink voltage supplied to inverter
u
max
= maximum output voltage of inverter
T
s
= PWM switching time
T
0
= zero state switching time
T
1
= first state switching time of the sector
T
2
= second state switching time of the sector
u
*
= reference voltage vector
u
c
= compensated voltage vector
= angle of the reference voltage vector
0
=angle of the inverter input voltage vector
m
= SVPWM sector number
u
*
,
u
*
=

axis reference voltages
u
= inverter output voltage
u
, u
=

axis voltages
i
, i
=

axis currents I. I
NTRODUCTION
In vector control, the controlling variables of the electrical machine expressed in dq synchronous rotating frame need to be converted back to threephase PWM signals before being applied to the inverter. The desired three phase voltages and currents of the machine are applied by means of a Voltage Source Inverter (VSI) or Current Source Inverter (CSI) using an effective PWM scheme. There two main methods to generate these signals in order to feed with power the motor: the Sinusoidal Pulse Width Modulation (SPWM) and the Space Vector Pulse Width Modulation (SVPWM). Sinusoidal PWM has been the most widely used technique in ac motor control. It is a simple carrierbased PWM, where a triangular carrier wave is modulated by a sine wave and their intersection points determine the switching times of the inverter power devices.
On the other hand Space Vector PWM (SVPWM) is a more sophisticated technique that provides better dc bus utilization and lower total harmonic distortion (THD) of the inverter voltage signal compared with SPWM. The appropriate PWM timer values needed in SVPWM can be calculated from the corresponding magnitude and angle parameters (i.e.
u
and
) using the dq voltage or current coordinates. Although SVPWM technique has maximum modulation index (MI) greater compared to SPWM nevertheless it is also unable to make full use of the inverter’s supply voltage. Operating in linear area, the maximum MI for SPWM and SVPWM are 0.5 and 0.866 respectively. In order to generate sinusoidal voltages the vector
u
*
must be maintained within the SVPWM hexagon (see Fig.1), inside the inscribed cycle representing the linear area of SVPWM. In other words, the modulation index, should not exceed
3/2
0.866. Several techniques have been proposed to extend the modulation index range above 0.866 for SVPWM [1][8]. These techniques are referred to as overmodulation. Such an extension of the inverter voltage range may cause undesired effects even serious problems especially in the ac motor traction applications (
u
*
u
c
) [1]–[3]. Operation of these modulation schemes in the overmodulation area as well as continuous transition into the sixstep mode is still a challenging research area [6], [10]. However overmodulation methods can be useful where the voltage supply needs to be varied (e.g. in automotive applications), since they do provide a higher voltage to the motor improving the dc bus utilization. In this paper, according to the operational principle of SVPWM, a novel overmodulation technique is proposed implementing a simple magnitudeangle algorithm. Implementing this unified auxiliary function, the output voltage of three phases could increase continuously up to that of sixstep operation. In addition, the total harmonic distortion (THD) of the output voltage using the proposed novel overmodulation
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682
technique is analyzed. Simulation results are presented at the end of this paper, which prove that the proposed overmodulation technique could be practicable. II. S
VPWM
O
VERMODULATION
B
ASED ON
A
N
A
UXILIARY
C
ORRECTING
F
UNCTION
In this section, an analysis of the novel overmodulation strategy for the spacevector PWM is presented. The derived algorithm is based on a compensated voltage function suitable to extend inverter operation from linear modulation up to sixstep mode. For analysis simplicity, the deadtime effect is neglected. When developing overmodulation algorithm the modulation index (MI) is used as a main characteristic of spacevector PWM. This is effectively the ratio of the magnitude of the voltage vector
u
to the maximum amplitude of the voltage vector provided by the inverter (
2/3
)
V
dc
. Here the modulation index for SVPWM inverters is defined as
max
23
ccdc
uu MI Vu
(1)
where
u
max
is the maximum amplitude of voltage reference vector
u
*
operating in the linear area (
u
max
=(2/3)
3) and
u
c
is the amplitude of the inverter input voltage referred as compensated voltage. According to the modulation index, the SVPWM range is divided into two regions,
Linear Modulation Area
and
Overmodulation Area
described in the following sections.
A. Linear Modulation Area (0
u
*
V
dc
/
3, Linear MI).
Based on the principle of the spacevector modulation, the space voltage vectors involve six effective vectors and two zero vectors as shown in Fig. 2. Here
is the phase angle of the reference voltage vector,
m
is the sector number (m=1,2…6) and
is the angle between the voltage vectors
u
*
and
V
m
given by
=[
(m1)
/3
] (see Fig. 1 and Fig. 2). Considering for simplicity that
u
*
lies in first sector (
m=1,
=
), then the voltage reference vector
u
*
is composed of timeaverage components of two effective vectors adjacent to it and one zero vector
120*120120,7
sss
TTT uuuuVVV TTT
(2) where
T
s
is the sampling period of the PWM and
T
1
,
T
2
and T
0
are time intervals of applying voltage vectors
V
1
,
V
2
and
V
0, 7
respectively (
V
0,
7
implies
the zerovoltage vectors V
0
and
V
7
). These time intervals of
T
1
,
T
2
and
T
0
are calculated as follows
*1
3sin3
sdc
uTT V
(3)
*2
3sin
sdc
uTT V
(4)
012
s
TTTT
Figure 1. Linear and overmodulation area in SVPWM. Figure 2. Operation in linear area and sector numbering.
*
13sinsin3
ssdc
uTT V
*
31cos6
sdc
uT V
(5) In this mode of operation, the angle
, the amplitude of the compensated and actual voltage reference vectors (
u
c
and
u
*
respectively) are the same for each fundamental period. Therefore both voltage vectors exactly coincide, that is
u=u
*
, when
0
u
*
V
dc
/
3
.
3rd IEEE International Symposium on Power Electronics for Distributed Generation Systems (PEDG) 2012
683
B. Overmodulation Area (V
dc
/
3 <
u
*
(2/3)V
dc
, Non Linear MI).
Operation in overmodulation mode occurs when the magnitude of the reference space vector,
u
*
, is greater than
V
dc
/
3
causing the end of the vector to be located outside of the SVPWM hexagon (
0.866<MI
1.0
). Therefore, the trajectory of the reference voltage vector intersects the hexagon at two points for each sector (see Fig. 4). The intersection angle
is given by
*
1cos63
dc
V au
(6) Operating in overmodulation area, the inverter cannot generate the output voltage as large as the voltage reference since the provided maximum output is limited up to the sides of the hexagon. Therefore the amplitude of compensated reference voltage
u
c
might be different than the actual one of input reference voltage
u
*
applied to SVPWM algorithm. As a result the produced voltage waveform is not exactly sinusoidal including terms of higher harmonics. To extend inverter output voltage avoiding largely harmonic distortion of its waveform, it is proposed a simple modification for the reference voltage before its space vector modulation. The magnitude of the compensated voltage 
u
c
 is given by
0000
,3
dcc
V uug
(8) Here
g
0
(
0
,
0
)
is a function of auxiliary variables
0
and
0
defined analytically by
000000
6cos116,cos6
g
(9) where
0000
66,1sgn166
(10) and
610
c
e
(11) Here
c
is a small positive number (
0<c
0.05
) used to improve the smoothness of function
u
c
. This function
g
0
(
0
,
0
)
is called
Auxiliary Correcting Function
(ACF) and it includes a simple adjustment of the compensated voltage applied on SVPWM algorithm operating in the overmodulation area. Finally using (8) and (9), the compensated voltage
u
c
could be expressed as a function of angular position
0
,
0
and angle
0
in complex form as follows
0
00
ja
uue
Figure 3. Block diagram of SVPWM connected to VSI. Figure 4. Overmodulation using Auxiliary Correcting Function (red line).
0
00130
6cos1163cos6
jmdc
V e
(12) where
00
(1)/3
m
(13) In the proposed method the whole SVPWM area (linear and overmodulation area) is considered as unique area and it is also supposed that the compensated voltage vector
u
c
is exactly the same with the reference one
u
*
in the linear area. On the
3rd IEEE International Symposium on Power Electronics for Distributed Generation Systems (PEDG) 2012
684
contrary the compensated voltage vector
u
c
is different from the reference one
u
*
in the overmodulation area with amplitude
u
c
and phase angle displacement
0
given by (8), (11) and (13) respectively. A unified general form referred to SVPWM compensated voltage is expressed by means of the following formula
**
11sgn32
dcc
V uuu
*0
1sgn3
dc
V uu
**
11sgn32
jdc
V uue
0
*0
1sgn3
jdc
V uue
(14) In the following section important properties of the unified function
u
c
(u
*
,
0
,
0
)
are discussed such as is its periodicity, integrability and continuity over the entire space vector modulation interval [
0
,
(2/3)V
dc
] and angle interval [
0
,
2
] considering the variables
0
and
0
. III. A
NALYSIS OF
A
UXILLIARY
C
ORRECTING
F
UNCTION
For the compensated voltage analysis, it is useful to explore the continuity of the function
u
c
over the entire interval of
u
c
, [
0
,
2V
dc
/3
] and how its phase angle
0
is affected. Another important characteristic of this function to be investigated is its integrability on the interval [
0
,
2
]. Also it is useful to explore some important properties of the compensated voltage function with respect to the intersection angle
.
A. Continuity of
u
c
over the interval [0, (2V
dc
/3)] (entire modulation area). 1) SVPWM Linear Modulation Area (0
u
*
<V
dc
/
3).
When the reference voltage vector lies into the linear area (
u
*
[
0,V
dc
/
3
]), then from (14) it results that
u
c
satisfies the following relation
13**
jm jc
uueue
(15a) since
u
*
[
0,V
dc
/
3
], it should be
*
11sgn312
dc
Vu
and
*
11sgn302
dc
uV
Obviously the compensated voltage vector is exactly the same as the reference voltage (
u
c
=
u
*
).
2) SVPWM Limit Cycle at
u
*
=V
dc
/
3.
t
u
*
=V
dc
/
3
the angle
becomes equal to
/6
, since any hexagon side is tangential to inscribed circle. As a result from (10), (11) and (13), it will be
0
=
/6
,
0
=
and
0
=
, that is from (11), (10)
610
6
c
e
and
006
6,lim
006
66lim1sgn166
This implies that the phase angle
0
of the compensated vector
u
c
is continuous on the interval [
0
,
2
] during the transition from linear to overmodulation area. In the same manner it could be proved as well that the compensated voltage amplitude
u
c
is a continuous function at
u
*
=V
dc
/
3
, since lefthand limit and righthand limit are equal, that is
*
3
lim
dc
cV u
u
*
**3
1lim1sgn323
dc
jjdcdcV u
V V uuee
(15b) and
*
3
lim
dc
cV u
u
0*
*03
1lim1sgn323
dc
j jdcdcV u
V V uuee
(15c) In addition for
u
*
=V
dc
/
3
it will be
00
**00
123
jj jjjdcc
V uueueueuee
(15d) Therefore considering (15a)(15d) it can be seen that the unified function
u
c
(u
*
,
0
,
0
)
is a piecewise continuous function of
u
*
over the entire space vector modulation interval [
0
,
(2/3)V
dc
].
3) SVPWM Overmodulation Area (V
dc
/
3 <
u
*
2V
dc
/3).
Once the reference voltage vector lies outside the linear area (
u
*
[
V
dc
/
3, 2V
dc
/3
]), then from (14) it results that
u
c
satisfies the following relation
00
1300
jm jc
uueue
(15e)
3rd IEEE International Symposium on Power Electronics for Distributed Generation Systems (PEDG) 2012
685