1144
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998
A Novel Overmodulation Technique for
Space-Vector PWM Inverters
Dong-Choon Lee, Member, IEEE, and G-Myoung Lee
Abstract— In this paper, a novel overmodulation technique
for space-vector pulsewidth modulation (PWM) inverters is proposed. The overmodulation range is divided into two modes
depending on the modulation index (MI). In mode I, the reference
angles are derived from the Fourier series expansion of the
reference voltage which corresponds to the MI. In mode II, the
holding angles are also derived in the same way. The strategy,
which is easier to understand graphically, produces a linear
relationship between the output voltage and the MI up to sixstep operation. The relationship between those angles and the MI
can be written in lookup tables or, for real-time implementation,
can be piecewise linearized. In addition, harmonic components
and total harmonic distortion (THD) of the output voltage are
analyzed. When the method is applied to the V/f control of the
induction motor, a smooth operation during transition from the
linear control range to the six-step mode is demonstrated through
experimental results.
Index Terms—Fourier series, inverter utilization, overmodulation, space-vector PWM.
I. INTRODUCTION
T
HREE-PHASE voltage-source pulsewidth modulation
(PWM) inverters have been widely used for dc/ac power
conversion since they can produce a variable voltage and
variable frequency power. However, they require a dead time
to avoid the arm-short and snubber circuits to suppress the
switching spike. Apart from these ancillary aspects, the PWM
inverters have an essential problem that they cannot produce
voltages as large as the six-step inverters can. That is, the dc
bus voltage cannot be utilized to the maximum.
To increase the voltage utilization of the sinusoidal PWM
inverter, a method of the addition of the third harmonics to
the reference voltage was proposed by which the fundamental
component can be increased by 15.5% [1]. In a space-vector
PWM inverter, which is widely used, the voltage utilization
factor can be increased to 0.906, normalized to that of the sixstep operation [2]. On the other hand, different discontinuous
PWM strategies were analyzed in [3], where the modulation
waveform of a phase has at least one segment of 60 which
is clamped to the positive and/or negative dc bus for, at most,
a total of 120 in a fundamental period during which no
switching in either inverter arm occurs. Recently, it is shown
that discontinuous PWM schemes and the space-vector PWM
Manuscript received August 20, 1997; revised January 28, 1998. This work
was supported by the Electrical Engineering and Science Research Institute
(EESRI), Korea, under Project 95-67. Recommended by Associate Editor,
O. Ojo.
The authors are with the School of Electrical and Electronic Engineering,
Yeungnam University, Kyungbuk 712-749, Korea.
Publisher Item Identifier S 0885-8993(98)08236-2.
Fig. 1. Diagram of space voltage vectors.
can be obtained by properly adding a zero-sequence voltage to
the original modulation waveform [4]. By injecting the zerosequence voltage, the modulation index can be increased up
to 0.906.
On the other hand, a few off-line PWM methods were
proposed to optimize the performance index. With those
strategies, not only either particular harmonic components can
be eliminated [5] or total harmonics may be minimized [6], but
also the maximum utilization of the inverter can be obtained.
However, since their transient responses are slow, it is difficult
for them to be applied to high-performance motor drives.
It had not been a great interest to increase the inverter
utilization until a few recent overmodulation methods were
proposed [7]–[11]. Kerkman modeled the inverter gain as a
function of the modulation index (MI) using a describing
function from which a compensated modulation index to give
the desired fundamental voltage component was approximately
derived for practical implementation [7]. However, the approximate inverter model gives a nonlinear inverter gain. In [8]
and [9], this nonlinear characteristic was eliminated by using
a simple lookup table. The result is a linear input to output
voltage transfer function from PWM to six-step operation of
the inverter.
Holtz proposed a continuous control of PWM inverters
in the overmodulation range [10]. In this scheme, there are
two modes of overmodulation depending on the modulation
index. In mode I, however, the fundamental voltage cannot
be generated as exactly equal to the reference voltage since
the contribution of the voltage increment around each corner
of the hexagon to the fundamental component differs from
that of the voltage decrement around the center of each side
0885–8993/98$10.00  1998 IEEE
LEE AND LEE: NOVEL OVERMODULATION TECHNIQUE FOR SPACE-VECTOR PWM INVERTERS
1145
Fig. 2. Trajectory of reference voltage vector and phase voltage waveform in mode I.
of the hexagon since it is dealt with in an average meaning.
So, it gives somewhat nonlinear transfer characteristics of
the inverter in overmodulation mode I. For mode II, there
is also no adequate explanation of the method of controlling
the fundamental component of the output voltage.
Another digital continuous control for the space-vector
PWM inverter was proposed in [11], where two modes of
the overmodulation in [10] are incorporated in single mode,
by which the implementation becomes simpler, but the linear
transfer characteristic of the inverter is lost in theory and much
higher harmonics are generated.
In this paper, a novel overmodulation strategy for the spacevector PWM to produce the exact fundamental voltage versus
the modulation index is proposed, where reference angles and
holding angles based on Fourier series expansion of the desired
output voltage are derived. The principle is most simple to
understand graphically. With this scheme, a linear control of
the inverter output voltage can be obtained over the whole
overmodulation range. For the dc-link voltage disturbance, the
proposed method is shown to be effective as well. In addition,
harmonic components of the output voltage and the total
harmonic distortion (THD) are analyzed. When the scheme
is applied to the V/f control of induction motor drives, it is
demonstrated that a smooth transient operation can be obtained
in overmodulation range by experimental results.
II. A NOVEL OVERMODULATION STRATEGY
(1)
is the phase voltage reference and
is the inverter
where
input voltage.
According to the modulation index, the PWM range is
divided into three regions as follows.
MI
At first, a principle of the space-vector modulation is described briefly. The space voltage vectors involve six effective
vectors and two zero vectors as shown in Fig. 1. A voltage
reference vector is composed of time-average components of
two effective vectors adjacent to it and one zero vector. That is,
(2)
is the sampling period of the PWM and
and
where
are time intervals of applying
and
vectors, respectively.
and
for zero-voltage vectors
The time intervals of
are calculated as
(3)
(4)
(5)
where is a phase angle of the reference voltage vector.
, the space-vector modulation generates
Below MI
sinusoidal output voltages. The trajectory of output voltages
traces a circle inscribed to the hexagon. Above
at MI
it, the voltage waveform of the inverter is distorted, where
magnitude becomes smaller than that of the reference voltage.
B. Overmodulation Mode I
In this section, a novel overmodulation strategy for the
space-vector PWM is derived from developing Fourier series
expansion of the waveform of the phase voltage reference
which gives the desired fundamental component. For simple
analysis, a dead-time effect is neglected. The modulation index
for PWM inverters is defined here as
MI
A. Linear Modulation
MI
The overmodulation mode I is operated when the magnitude
which is boosted
of a compensated voltage reference vector
is between
to produce a desired fundamental voltage of
two radii of an inscribed circle and a circumscribed circle
of the hexagon. Fig. 2 shows the trajectory of three voltage
vectors rotating in a complex plane (left part) and the phase
voltage waveform of an actual voltage reference vector
(bold line) transformed in a time domain (right part) [12],
which is modulated actually by the inverter. Here, the
denotes a reference angle measured from the vertex to the
intersection of the compensated voltage vector trajectory with
the side.
For a given voltage reference, the phase voltage waveform
is divided into four segments. The voltage equations in each
1146
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998
limited up to the side of the hexagon. Then, the switching
intervals through (3)–(5) are corrected as [13]
(12)
(13)
(14)
As known from Fig. 2, the upper limit in mode I is when
Then, the modulation index is 0.952, which is known
from (10) and (11). When the MI is higher than 0.952, another
overmodulation algorithm is needed.
Fig. 3. Reference angle with regard to modulation index (solid line: numerical, dashed line: piecewise linearized).
segment are expressed as
for
(6)
for
(7)
for
(8)
for
MI
C. Overmodulation Mode II
In mode I, the angular velocity of the compensated and
actual voltage reference vectors is both the same and constant
for each fundamental period. Under such a condition, output
cannot be generated since
voltages higher than MI
there exists no more surplus area to compensate for the voltage
loss even if the modulation index is increased above that.
In modulation ranges higher than 0.952, the actual voltage
reference vector is held at a vertex for particular time and
then moves along the side of the hexagon for the rest of
controls the time
the switching period. The holding angle
interval the active switching state remains at the vertices,
which uniquely controls the fundamental voltage. A basic
concept of the mode II is similar to [10], where it lacks an
explicit explanation about how to derive the algorithm.
Here, detailed expressions based on Fourier series expansion
just in the same way as in mode I will be developed. From
Fig. 4, the voltage equations in four segments are expressed as
(9)
for
and is an angular velocity of the fundamental
where
voltage reference vector.
Expanding (6)–(9) in a Fourier series and taking the fundamental component of it, the resultant equation can be expressed
as
for
for
(15)
(16)
(17)
for
(18)
(10)
where
and denote integral ranges of each voltage
where
function as shown in Fig. 2. Integrating (10) numerically,
with regard to the
we can obtain the value of
Since
represents the peak value of the fundamental
component, from the definition of the modulation index of (1)
MI
(19)
(20)
(11)
which gives a
Thus, a relationship between the MI and the
linearity of the output voltage is determined, which is plotted
in a solid line in Fig. 3.
For the voltage reference vector exceeding the side of the
hexagon, the inverter cannot generate the output voltage as
large as the voltage reference since the maximum output is
and
are phase angles of the actual voltage reference
The
and
vector rotating for
, respectively, as shown in Fig. 5.
and
are derived as follows. The
The two angles of
to
at
actual voltage reference vector rotates from
a little higher speed while the fundamental one is rotating at
to
Equation (19)
constant speed from
LEE AND LEE: NOVEL OVERMODULATION TECHNIQUE FOR SPACE-VECTOR PWM INVERTERS
1147
Fig. 4. Trajectory of reference voltage vector and phase voltage waveform in mode II.
Fig. 6. Holding angle with regard to modulation index (solid line: numerical,
dashed line: piecewise linearized).
Fig. 5. Angular displacement of reference and actual voltage vectors.
expansion as
is simply derived from a proportional relationship for angular
displacements of these two vectors as
(22)
(21)
Thereafter, the actual voltage reference vector is held at a
vertex while the fundamental one is continuously rotating
to
For
, the
from
situation is reversed. The actual voltage reference vector is
held at a vertex while the fundamental one is rotating from
to
At
, the actual
voltage reference vector starts to rotate and is aligned with
The same analogy as the
the fundamental one at
gives the expression
above for
of (20), which is also applied for
Substituting (15)–(18) into (10) and matching the result of
its integral with (11), a relationship between the modulation
index and the holding angle is obtained, which is plotted in
a solid line in Fig. 6.
III. HARMONIC ANALYSIS
In Section II, the reference angle
and the holding angle
were derived which give a linear inverter gain in the
complete overmodulation range. Here, harmonic components
of the output voltage are analyzed using the Fourier series
is given by (6)–(9) in mode I and (15)–(18) in
where
mode II. A numerical integration of (22) shows that even-order
harmonics and triplen harmonics are eliminated in the output
voltage. The four lowest harmonic components (5th, 7th,
11th, and 13th) versus the MI are illustrated in Fig. 7. Some
harmonic components are absent at the particular modulation
index. Fig. 8 shows voltage harmonic spectra through fast
Fourier transform (FFT). The magnitude of each harmonic
component coincides well with the result of (22).
The THD factor is defined as
THD
(23)
where and are the rms value and fundamental component
of the phase voltage, respectively. Fig. 9 shows THD factor
of the output voltage. As the modulation index increases,
especially in mode II, the THD is deteriorated steeply and
The THD for [8] and [10]
it culminates to 0.311 at MI
is similar to that in this method. However, the THD in [11] is
much higher, as shown in Fig. 9, since the voltage waveform
has jumps.
1148
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998
Fig. 7. Harmonic voltage components.
Fig. 10. Experiment system.
(a)
(b)
(c)
Fig. 8. Harmonic spectra by FFT, normalized to fundamental component at
1:0 (simulation).
MI
=
(d)
Fig. 11. Phase voltage waveforms. (a) MI = 0:78; (b) MI = 0:938; (c)
MI = 0:968; and (d) MI = 1:0:
Fig. 9. Total harmonic distortion.
IV. EXPERIMENTS
AND
DISCUSSIONS
To confirm the validity of the proposed scheme, experiments
were performed for the V/f control of induction motor drive
fed by an insulated gate bipolar transistor (IGBT) PWM
inverter. Fig. 10 shows the experiment system with a digital signal processor (DSP) board. In practice, such a highperformance DSP is not required for implementation of the
V/f control of the induction motor. Also, the dc-link voltage
is usually measured for the space-vector modulation and
overvoltage protection, and the current is measured only for
monitoring. The inverter switching frequency is 3.5 kHz, and
the dc-link voltage is 287 V, which is set a little lower than
at nominal operation in order to show distinctly the effect of
the overmodulation algorithm. The induction motor used for
experiments is rated at 3 Hp, 220 V, and 60 Hz.
Let us consider a case using lookup tables for data angle.
First, the reference and holding angles are calculated off line
and stored in the memory with regard to the increment of
to
. If a desired reference voltage is
0.001 from MI
given, a modulation index is calculated by (1) and the reference
angle or holding angle corresponding to it is read out from the
lookup table. In the case of the mode I, the magnitude of
a compensated reference voltage vector is easily calculated
using the reference angle as
from which the switching interval is calculated. In mode II,
the holding angle according to the MI is first determined, and
then the phase angle of the actual voltage reference vector is
determined by considering (19) and (20) with regard to the
Then, of course, its magnitude reaches the side of the hexagon.
Fig. 11 shows the waveform of the output phase voltage
at different modulation indices, which represents the voltage
LEE AND LEE: NOVEL OVERMODULATION TECHNIQUE FOR SPACE-VECTOR PWM INVERTERS
1149
(a)
(b)
(c)
(d)
Fig. 14. Transient responses for dc-link voltage disturbance (experimental).
Fig. 12. Phase current waveforms. (a) MI
MI = 0:968; (d) MI = 1:0:
= 0 78
:
;
(b) MI = 0:938; (c)
(a)
Fig. 15. Transient responses for dc-link voltage disturbance (simulation).
(b)
!
Fig. 13. Transient responses for motor frequency change. (a) Linear
I and (b) mode I
mode II.
! mode
value averaged over each switching period for easy monitoring. The phase currents corresponding to each voltage in
Fig. 11 are illustrated in Fig. 12. According to the increase of
the modulation index, the phase currents are more distorted.
Fig. 13 shows transient responses of the voltage and current
for the change of the motor frequency. Since a linearity of
the voltage modulation is guaranteed, the motor current is not
changed abruptly, but smoothly.
When a disturbance in the dc-link voltage occurs, the
inverter is often operated in overmodulation range. Fig. 14
shows the transient responses in case of the decrease of 10%
from the operating dc-link voltage. Since the inverter input
voltage is decreased, the modulation index is boosted so that
the fundamental component of the output voltage can be kept
the same. In Fig. 15, at a similar condition to that in Fig. 14,
torque ripples due to current harmonics are generated, but the
average torque is kept constant. Since the torque ripples are
filtered by the motor inertia, the motor speed is little changed.
Fig. 16 shows FFT spectra of the phase voltage analyzed
by a digital oscilloscope of which the results are the same as
those in Fig. 8.
If the hardware memory cannot allow lookup tables for the
reference angle and the holding angle, they can be calculated
in real time by piecewise-linear approximation as shown in
dashed lines in Figs. 3 and 5. Then, a transfer characteristic of
the output to the modulation index is shown in Fig. 17, from
which it is known that the nonlinearity is sufficiently tolerable
1150
IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 13, NO. 6, NOVEMBER 1998
It is expected that the overmodulation algorithm is very
effective to the PWM inverter controls in a frequent variation
of the utility voltage or battery-fed inverter system.
APPENDIX
The equations of the reference and holding angle piecewise
linearized as a function of the MI are as follows.
A. Mode I
MI
Fig. 16. Harmonic spectra by FFT, normalized to fundamental component
1:0 (experimental).
at MI
=
MI
MI
MI
MI
MI
B. Mode II
MI
MI
MI
MI
MI
MI
REFERENCES
Fig. 17.
Output voltage versus modulation index in piecewise linearization.
for practical purposes. The piecewise-linearized equations for
two overmodulation modes are given in the Appendix.
V. CONCLUSIONS
In the space-vector modulation, a linear control of the
by
inverter output voltage was obtained up to MI
a novel overmodulation strategy. The method is based on
Fourier series representation of the reference voltage, where
a graphical transformation between complex voltage vectors
and the phase voltage in time domain is used implicitly.
The reference angles in mode I and the holding angles in
mode II were derived as a function of the modulation index by numerical analysis. These data can be written in
lookup tables or, for real-time implementation, piecewise
linearized.
In addition, each harmonic component and the THD of the
output voltage were analyzed. The THD factor in this scheme
was shown to be lower than that of other method. In spite of
the dc-link voltage disturbance, the fundamental component of
the inverter output voltage can be kept constant by boosting
the modulation index. When the method is applied to the V/f
control of the induction motor, a smooth operation during
transition from the linear control range to the six-step mode
was demonstrated through experimental results.
[1] G. Buja and G. Indri, “Improvement of pulse width modulation techniques,” Arch. fr Elektrotech., vol. 57, pp. 281–289, 1975.
[2] H. W. van der Broek, H. C. Skudelny, and G. V. Stanke, “Analysis and
realization of PWM based on voltage space vectors,” IEEE Trans. Ind.
Applicat., vol. 24, no. 1, pp. 142–150, 1988.
[3] J. W. Kolar, H. Ertl, and F. C. Zach, “Influence of the modulation
method on the conduction and switching losses of a PWM converter
system,” IEEE Trans. Ind. Applicat., vol. 27, no. 6, pp. 1064–1075,
1991.
[4] V. Blasko, “A hybrid PWM strategy combining modified space vector
and triangle comparison methods,” in IEEE PESC Conf. Rec., 1996, pp.
1872–1878.
[5] H. S. Patel and R. G. Hoft, “Generalized techniques of harmonic
elimination and voltage control in thyristor inverters: Part I—Harmonic
elimination,” IEEE Trans. Ind. Applicat., vol. 9, pp. 310–317, May/June
1973.
[6] G. S. Buja, “Optimum output waveforms in PWM inverters,” IEEE
Trans. Ind. Applicat., vol. 16, no. 6, pp. 830–836, 1980.
[7] R. J. Kerkman, B. J. Seibel, D. M. Brod, T. M. Rowan, and D. Leggate,
“A simplified inverter model for on-line control and simulation,” IEEE
Trans. Ind. Applicat., vol. 27, no. 3, pp. 567–573, 1991.
[8] R. J. Kerkman, T. M. Rowan, D. Leggate, and B. J. Seibel, “Control
of PWM voltage inverters in the pulse dropping region,” IEEE Trans.
Power Electron., vol. 10, no. 5, pp. 559–565, 1995.
[9] R. J. Kerkman, D. Leggate, B. J. Seibel, and T. M. Rowan, “Operation
of PWM voltage source-inverters in the overmodulation region,” IEEE
Trans. Ind. Electron., vol. 43, no. 1, pp. 132–141, 1996.
[10] J. Holtz, W. Lotzkat, and A. M. Khambadkone, “On continuous control
of PWM inverters in the overmodulation range including the six-step
mode,” IEEE Trans. Power Electron., vol. 8, no. 4, pp. 546–553, 1993.
[11] S. Bolognani and M. Zigliotto, “Novel digital continuous control of
SVM inverters in the overmodulation range,” IEEE Trans. Ind. Applicat.,
vol. 33, no. 2, pp. 525–530, 1997.
[12] P. Vas, Electrical Machines and Drives: A Space-Vector Theory Approach. New-York: Oxford Science, 1992.
[13] T. G. Habetler, “A space vector-based rectifier regulator for ac/dc/ac
converters,” in Proc. EPE, vol. 2, 1991, pp. 101–107.
LEE AND LEE: NOVEL OVERMODULATION TECHNIQUE FOR SPACE-VECTOR PWM INVERTERS
Dong-Choon Lee (S’90–M’95) was born in Korea
in 1963. He received the B.S., M.S., and Ph.D.
degrees in electrical engineering, all from Seoul
National University, Seoul, Korea, in 1985, 1987,
and 1993, respectively.
He was a Research Engineer at Daewoo Heavy
Industry from 1987 to 1988. He also was at the
Research Institute of Science Engineering of Seoul
National University under a Post-Doctoral Fellowship for one year. He has been a Faculty Member of
the School of Electrical and Electronic Engineering,
Yeungnam University, Kyungbuk, Korea, since 1994. Also, he is currently a
Visiting Scholar at the Department of Electrical Engineering, Texas A&M
University, College Station. His research interests include ac machine drives,
static power converters, and DSP applications.
1151
G-Myoung Lee was born in Korea in 1970. He
received the B.S. degree from Kyungil University,
Korea, in 1995 and the M.S. degree from Yeungnam
University, Kyungbuk, Korea, in 1997, both in electrical engineering. He is currently working toward
the Ph.D. degree at Yeungnam University.
His research interests are motor drives and controls and PWM converters and inverters.