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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 6, NOVEMBER 2005
Analysis of Multiphase Space Vector Pulse-Width
Modulation Based on Multiple d–q Spaces Concept
Hyung-Min Ryu, Member, IEEE, Jang-Hwan Kim, Student Member, IEEE, and Seung-Ki Sul, Fellow, IEEE
Abstract—Multiphase motors are usually designed to have the
concentrated winding and nonsinusoidal airgap flux density distribution in order to maximize the torque per ampere. This means
that the phase voltage of a multiphase motor has the nonsinusoidal
waveform. Accordingly, the conventional analysis on a multiphase
space vector pulse-width modulation (SVPWM), which is confined
to a sinusoidal phase voltage, should be extended to a nonsinusoidal phase voltage. In this paper, based on a multiple – spaces
concept a novel analysis on a multiphase SVPWM to synthesize
an arbitrary nonsinusoidal phase voltage is proposed. Throughout
this paper, a five-phase inverter is used as a practical example. The
basic concepts can be easily extended to an -phase inverter.
multiphase SVPWM scheme should be extended to synthesize
a nonsinusoidal phase voltage.
In this paper, a new analysis on a multiphase SVPWM is presented. Using a multiple – spaces concept, we explain how to
realize an arbitrary nonsinusoidal phase voltage in terms of the
applying times of available switching vectors. Based on the proposed SVPWM scheme, the analysis on the maximum modulation index of a multiphase inverter is also presented. Throughout
this paper, a five-phase inverter is used as a practical example. The
basic concepts can be easily extended to an -phase inverter.
Index Terms—Multiphase motors, multiphase space vector
pulse-width modulation (SVPWM), multiple – spaces concept,
nonsinusoidal voltage.
II. MULTIPLE – SPACES CONCEPT
I. INTRODUCTION
K
ELLY et al. [1] mentioned an -phase unified pulse-width
modulation (PWM) scheme by the extension of a
three-phase unified PWM. In an -phase system, the th harmonic voltage offset can be calculated by averaging between the
maximum and minimum reference phase voltages. This voltage
offset equally divides the applying times of two zero switching
vectors within the sampling interval so that the dc-link voltage
utilization can be increased compared to a sine-triangle PWM.
In [1], Kelly et al. also verified that an -phase space vector
PWM (SVPWM) scheme can be described in terms of the
applying times of available switching vectors on the basis of
the space vector concept. However, the paper only focuses on
how to realize a sinusoidal phase voltage.
As is widely known, most multiphase motors are designed to
have the nonsinusoidal back-EMF voltage in order to increase
the torque per ampere [2]–[8]. For example, multiphase permanent magnet motors are usually designed to have the nearly
rectangular air-gap flux density distribution and the concentrated winding with a full-pitch in order to maximize the torque
per ampere [2]. Toliyat et al. [3] verified that the injection of
the third harmonic current in a five-phase induction motor with
the concentrated winding enables the air-gap flux density to be
nearly trapezoidal so that the torque per ampere can be increased
by 10%. In both cases, the back-EMF voltages have the nonsinusoidal waveforms. Therefore, the conventional analysis on a
Manuscript received August 25, 2004; revised February 25, 2005. Recommended by Associate Editor J. H. R. Enslin.
H.-M. Ryu is with the Automation Laboratory, INTECH Factory Automation
Company, Kyonggi-do 449-910, Korea (e-mail: hmryu@intech-fa.co.kr).
J.-H. Kim and S.-K. Sul are with the School of Electrical Engineering and
Computer Science, Seoul National University, Seoul 151-742, Korea (e-mail:
ghks95@eepel.snu.ac.kr; sulsk@plaza.snu.ac.kr).
Digital Object Identifier 10.1109/TPEL.2005.857551
In a three-phase motor, all the (6
1)th harmonics of
3)th
three-phase variables, with the exception of the (6
harmonics corresponding to zero-sequence components, can
be equivalently represented by the following two-dimensional
(2-D) complex space vector (hereafter referred to as space
vector) on a stationary reference frame [9]
(1)
where
2 3 . From (1), the space vector on a synchronously rotating reference frame can be defined as follows:
(2)
where
and denotes the fundamental frequency, or
the angular velocity of reference field. Using the space vector of
(2), the fundamental harmonic of three-phase variables can be
regarded as a dc component. This idea is of key importance to
the field-oriented vector control and synchronous frame current
control of a three-phase motor.
Likewise, this space vector concept can be extended to a
five-phase motor. Since five-phase variables have four degrees
of freedom with the exception of zero-sequence components,
two space vectors should be used in order to define the equivalent transformation of five-phase variables. These two space
vectors can be reasoned out because a five-phase motor has the
following two kinds of phase displacements according to the
harmonics of winding density distribution.
1)th harmonics of winding density distribu1) The (10
tion have the displacement of , , , , and phases
counterclockwise.
3)th harmonics of winding density distribu2) The (10
tion has the displacement of , , , , and phases counterclockwise.
0885-8993/$20.00 © 2005 IEEE
RYU et al.: ANALYSIS OF MULTIPHASE SPACE VECTOR PULSE-WIDTH MODULATION
1365
Fig. 2. Circuit diagram of a five-phase PWM inverter.
Fig. 1. Two orthogonal 2-D spaces of a five-phase motor. (a) d –q space. (b)
d –q space.
Therefore, two space vectors, which can equivalently express
, , , , and phase variables except zero-sequence components, can be defined as follows:
Fig. 3. Two equivalent load configurations of a five-phase PWM inverter. (a)
1–4. (b) 2–3.
respectively, can be defined as follows:
(5)
(6)
(3)
(4)
where
2 5.
The space vectors of (3) and (4) have the following properties.
1)th harmonics
1) Using the space vector of (3), the (10
of five-phase variables can be equivalently expressed as
1) in the
ac components with the frequency of (10
stationary – space depicted in Fig. 1(a).
3)th harmonics
2) Using the space vector of (4), the (10
of five-phase variable can be equivalently expressed as
3) in the
ac components with the frequency of (10
stationary – space depicted in Fig. 1(b).
The number of subscripts used to differentiate two – spaces
means the lowest order among the harmonics equivalently transformed to each – space.
From (3) and (4), the space vectors in the – and –
spaces synchronously rotating with the frequency of and 3 ,
Using the space vectors of (5) and (6), the fundamental and
third harmonics of five-phase variables can be regarded as dc
components.
1 -phase motor, it
By generalizing this basic concept to a 2
can be easily seen that there exist -orthogonal – spaces, where
1 of phase
all the harmonics with the order of less than 2
variables can be equivalently represented as dc components.
III. FIVE-PHASE SPACE VECTOR PWM
A. Definition of Switching Vectors
In a five-phase PWM inverter, as shown in Fig. 2, two
different equivalent load circuits can be generated depending
on the switching pattern [1]. For example, when the switching
pattern is (10000) the equivalent load circuit consists of one
impedance in series with a group of four parallel impedances,
as shown in Fig. 3(a). Likewise, the switching pattern (11000)
generates the two to three equivalent load configuration, as
shown in Fig. 3(b). In the switching pattern depicted by a
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 6, NOVEMBER 2005
five-digit binary number, each digit means the switching func, , and
in order, where “1” indicates the
tion , ,
upper switch of the corresponding leg is on, while “0” indicates
the lower switch is on.
,
,
,
, and
By (3) and (4), five phase voltages
, as decided by the switching patterns, can be expressed as
the following switching vectors in – and – spaces:
(7)
(8)
By utilizing the zero-sequence voltage, the switching vectors
can be calculated directly from the switching functions
(9)
(10)
denotes dc-link voltage.
where
Using (9) and (10), thirty nonzero switching vectors, with
the exception of the two zero switching vectors (00000) and
(11111), can be depicted on – and – planes, as shown
in Fig. 4(a) and (b), respectively.
B. Selection of Switching Vectors
According to the equivalent load configuration and the magnitude, thirty nonzero switching vectors can be classified into
the following three sets.
— {2–3,1} switching vectors
(11001), (11000), (11100), (01100), (01110),
(00110), (00111), (00011), (10011), (10001).
— {1–4} switching vectors
(10000), (11101), (01000), (11110), (00100),
(01111), (00010), (10111), (00001), (11011).
— {2–3,2} switching vectors
(01001), (11010), (10100), (01101), (01010),
(10110), (00101), (01011), (10010), (10101).
In the name of the switching set, e.g., {2–3,1}, the number
following a comma is additionally used in order to differentiate
the switching sets with the same equivalent load configuration.
And the number “1” means that the magnitude of switching vectors in the corresponding switching set is the greatest among the
switching vectors with the same equivalent load configuration
on the – plane.
From the average vector concept during one sampling period
[10], a reference voltage vector on the – plane can be
realized by adjusting the applying times of the nearest two
{1–4} switching vectors and two {2–3,1} switching vectors.
The other combinations of switching vectors increase the
number of switching or decrease the maximum magnitude
of the realizable voltage vector. It should be noted that only
the switching vectors with the maximum magnitude in each
equivalent load configuration are normally used.
Fig. 4. Thirty nonzero switching vectors on (a) d –q plane and (b) d –q
plane.
C. Calculation of the Applying Times of Switching Vectors
For example, a – axes reference voltage vector located
in sector 1, as shown in Fig. 5, can be realized by solving the
following complex equation:
(11)
(12)
RYU et al.: ANALYSIS OF MULTIPHASE SPACE VECTOR PULSE-WIDTH MODULATION
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Fig. 5. Realization of a reference voltage vector located in sector 1 on the
d –q plane.
where the unknown , , , and denote the applying times
of the switching vectors of (10000), (11000), (11001), and
(11101), respectively. The subscript of indicates the number
“1” in the corresponding switching pattern, or the applying
means
order of nonzero switching vectors in ON sequence.
the sampling period, or half of the switching period. indicates
the applying time of zero voltage due to (00000) and (11111).
and
represent the magnitudes of {1–4} and {2–3,1}
switching vectors, respectively, which can be calculated using
(9) as follows:
Fig. 6. Realization of a reference voltage vector located on the d –q plane.
From (11) and (15), the following matrix equation between
the given – – – axes voltages and unknown , , ,
and can be obtained as follows:
(17)
(13)
(18)
(14)
Four switching vectors on the – plane corresponding to
four switching patterns, which are selected to realize a reference voltage vector located in sector 1 on the – plane, are
depicted as shown in Fig. 6. Note that the magnitude of {2–3,1}
switching vectors is the greatest on the – plane but the
smallest on the – plane. In the same manner as that of (11),
a – axes reference voltage vector can be realized by solving
the following complex equation:
(15)
where
on the
lows:
denotes the magnitude of {2–3,1} switching vectors
– plane, which can be calculated using (10) as fol-
(16)
where the subscript of matrix means sector number.
of (18) has a full rank, the applying times
Since the matrix
of nonzero switching vectors are uniquely decided by the following matrix equation:
(19)
The generalized matrix
according to the sector number
can be derived like (20), shown at the bottom of the next page.
The resultant switching signals during one switching period
can be drawn, as shown in Fig. 7.
IV. MAXIMUM MODULATION INDEX
In this section, the analysis on the maximum realizable output
voltage of a five-phase inverter in the linear modulation region
is addressed.
The modulation index can be defined as the ratio of the amplitude of the fundamental phase voltage to the dc-link voltage
like [1]
(21)
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 6, NOVEMBER 2005
Fig. 7. Switching signals when d –q axes reference voltage vector is located
in sector 1.
Fig. 8. Trajectory of the d –q axes voltage vector with the maximum
modulation index when the phase voltage is sinusoidal.
When the phase voltage is sinusoidal, or the magnitude of the
– axes voltage vector is zero, the following relation between
the applying times of nonzero switching vectors can be derived
from (15) and (16):
Fig. 9. Trajectories of the voltage vectors with the maximum modulation index
when the magnitude ratio and phase between the d –q and d –q axes
voltage vectors are 0.236 and [rad], respectively. (a) d –q plane. (b) d –q
plane.
can be drawn as the dotted circle in Fig. 8. Then, the maximum
modulation index can be calculated as
(22)
In this case, the trajectory of the – axes voltage vector
with the maximum magnitude in the linear modulation region
(23)
(20)
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1369
Fig. 10. Waveforms of the phase voltage, offset voltage, and pole voltage
generated from the d –q and d –q axes voltage vectors of Fig. 9. The Y -axis
is normalized by dc-link voltage.
From (23), it can be seen that the maximum modulation index
of a five-phase inverter is reduced by 9% compared with a threephase inverter with the maximum modulation index of 0.5774.
However, when the – axes voltage vector rotates along the
dotted trajectory of Fig. 9(a) and
atan
atan
rad
(24)
– axes voltage vector can rotate along the dotted
the
inscribed circle of the decagon, at the vertexes of which the
{2–3,1} switching vectors are located, in Fig. 9(b). Then, the
modulation index can be calculated as
10
and the magnitude of the
culated as
–
0.6155
(25)
axes voltage vector can be cal0.1453
(26)
In other words, if the – axes voltage vector has the magnitude of 23.6% compared to the – axes voltage vector and
defined in (24) is rad , the maximum moduthe phase
lation index of a five-phase inverter can be increased by 6.6%
compared to that of a three-phase inverter. In this case, the phase
voltage has the flat-top waveform, as shown in Fig. 10, and thus
in the viewpoint of the triangular comparison PWM scheme
using the offset voltage [1], the increase of the maximum modulation index can be easily understood.
On the contrary, when the – axes voltage vector rotates
is zero, the
along the dotted trajectory of Fig. 11(a) and
– axes voltage vector will rotate along the dotted inscribed
circle of the decagon, at the vertexes of which {1–4} switching
vectors are located, in Fig. 11(b). Then, the modulation index
can be calculated as
0.3804
and the magnitude of the
culated as
–
(27)
axes voltage vector can be cal0.2351
(28)
Fig. 11. Trajectories of the voltage vectors with the maximum modulation
index when the magnitude ratio and phase between the d –q and d –q
axes voltage vectors are 0.618 and 0 [rad], respectively. (a) d –q plane. (b)
d –q plane.
In this case, the phase voltage has the peak-top waveform as
shown in Fig. 12 and thus in the viewpoint of the triangular
comparison PWM scheme using the offset voltage the decrease
of the maximum modulation index can be easily understood.
From the previous results, it is clear that the maximum modulation index of a five-phase PWM inverter depends on the phase
voltage waveform, which is decided by the magnitude ratio and
between the – and – axes voltage vectors.
phase
This means that the power density of a five-phase inverter depends on the phase voltage waveform as well as power factor.
For example, a five-phase surface-mounted permanent magnet
synchronous motor (SMPMM) [2] typically has the maximum
modulation index of 0.57 0.59 because the phase voltage has
is close to rad .
a flat-top waveform, and so the phase
On the other hand, a five-phase synchronous reluctance motor
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IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 20, NO. 6, NOVEMBER 2005
Fig. 12. Waveforms of the phase voltage, offset voltage, and pole voltage
generated from the d –q and d –q axes voltage vectors of Fig. 11. Y -axis is
normalized by dc-link voltage.
TABLE I
ELECTRICAL SPECIFICATIONS OF A FIVE-PHASE IPMSM
(SRM) with a salient-pole rotor [8] has the maximum modulation index of less than 0.45 because the phase voltage has
is close to zero.
a peak-top waveform, and so the phase
Therefore, in the viewpoint of the power density of motor drive
system, a five-phase SRM with a salient-pole rotor has a disadvantage compared to a five-phase SMPMM.
V. EXPERIMENTAL RESULT
The experiment on synchronous frame current control [11],
the output voltage of which is synthesized by using the proposed
SVPWM, has been carried out.
As a tested machine, a five-phase interior permanent magnet
synchronous motor (IPMSM) is used and a three-phase spindle
induction motor operating a speed control mode is used as a
load machine. Table I shows the electrical specifications of a
five-phase IPMSM.
Two intelligent power modules (IPMs) are used as the
switching devices of a five-phase PWM inverter and the proposed SVPWM algorithm is fully digitally implemented by
the control board based on digital signal processor (DSP)
TMS320VC33. A serial digital-to-analog converter with four
channels is employed in order to observe the control variables
stored in memory on an oscilloscope. The switching frequency
of SVPWM is 5 kHz and then the current control bandwidth is
set to 2000 rad/s.
Fig. 13. Experimental result of the proposed SVPWM method when the current
of a five-phase motor is regulated at the synchronously rotating d –q and d –q
axes. (a) Waveforms of u-phase reference current, u-phase sampled current,
and u-phase current error. (b) Trajectories of the sampled current vectors and
reference voltage vectors on the stationary d –q and d –q axes.
Fig. 13 shows the experimental result when the – – –
axes currents on synchronously rotating reference frames are
regulated to be 20 [A], 20 [A], 10 [A], and 10 [A]. The
motor speed is regulated to be a constant 1000 [r/min] by the
load machine. Fig. 13(a) shows the -phase reference current,
-phase sampled current, and -phase current error from top to
bottom. Fig. 13(b) shows the trajectories of the sampled current
vectors and reference voltage vectors on the stationary –
and – axes. This experimental result reveals that by employing the proposed SVPWM method the nonsinusoidal phase
current including the third harmonic component can be regulated without steady-state error.
VI. CONCLUSION
In this paper, a novel analysis on a multiphase SVPWM to
realize a nonsinusoidal phase voltage has been proposed. The
multiple – spaces concept, which can equivalently analyze
a multiphase ac motor like a dc motor, is presented. Based on
this concept, it is explained that an arbitrary reference voltage
vector of a five-phase ac motor can be synthesized in terms of
the applying times of available switching vectors in a five-phase
PWM inverter.
RYU et al.: ANALYSIS OF MULTIPHASE SPACE VECTOR PULSE-WIDTH MODULATION
In addition, through the analysis on the maximum modulation index it is shown that unlike a three-phase inverter the
maximum modulation index of a five-phase inverter is dependent on the phase voltage waveform according to the third harmonic component. This result means that the power density of
a five-phase inverter depends on the phase voltage waveform
as well as power factor according to the motor type. The proposed SVPWM method is proved through an experiment using
a five-phase IPMSM.
1371
Hyung-Min Ryu (S’00–M’05) was born in
Kwangju, Korea, in 1975. He received the B.S.,
M.S., and Ph.D. degrees in electrical engineering
from Seoul National University, Seoul, Korea, in
1997, 2000, and 2004, respectively.
He is currently a Research Engineer with INTECH
Factory Automation Company, Kyonggi-do, Korea.
His current research interests are power electronic
control of electric machines and power converter
circuits.
REFERENCES
[1] J. W. Kelly, E. G. Strangas, and J. M. Miller, “Multiphase inverter
analysis,” in Proc. IEEE Int. Electric Machines Drives Conf. (IEMDC),
2001, pp. 147–155.
[2] L. Parsa and H. M. Toliyat, “Multiphase permanent magnet motor
drives,” in Proc. Industrial Application Soc. Annu. Meeting, Oct. 2003,
pp. 401–408.
[3] H. Xu, H. A. Toliyat, and L. J. Petersen, “Five-phase induction motor
drives with DSP-based control system,” IEEE Trans. Power Electron.,
vol. 17, no. 4, pp. 524–533, Jul. 2002.
[4] Y. Kats, “Adjustable-speed drives with multiphase motors,” in Proc.
IEEE Int. Electric Machines Drives Conf. (IEMDC), May 1997, pp.
TC2/4.1–TC2/4.3.
[5] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated
winding induction machine for adjustable speed drive applications part 1
(motor analysis),” IEEE Trans. Energy Conv., vol. 6, no. 4, pp. 679–683,
Dec. 1991.
[6] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated
winding induction machine for adjustable speed drive applications part
2 (motor design and performance),” IEEE Trans. Energy Conv., vol. 6,
no. 4, pp. 684–692, Dec. 1991.
[7] H. A. Toliyat, T. A. Lipo, and J. C. White, “Analysis of a concentrated
winding induction machine for adjustable speed drive applications-experimental results,” IEEE Trans. Energy Conv., vol. 9, no. 4, pp.
695–700, Dec. 1994.
[8] H. A. Toliyat, L. Y. Xue, and T. A. Lipo, “A five phase reluctance motor
with high specific torque,” IEEE Trans. Ind. Appl., vol. 28, no. 3, pp.
659–667, May/Jun. 1992.
[9] D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of AC
Drives. Oxford, UK: Oxford Univ. Press, 1996.
[10] H. W. Van der Broeck and H. C. Skudelny, “Analysis and realization of
a pulse width modulator based on voltage space vectors,” IEEE Trans.
Ind. Appl., vol. 24, no. 1, pp. 142–150, Jan./Feb. 1988.
[11] H.-M. Ryu, J.-W. Kim, and S.-K. Sul, “Synchronous frame current control of multiphase synchronous motor—part I. Modeling and current
control based on multiple d–q spaces concept under balanced condition,” in Proc. Industrial Application Soc. Annu. Meeting, Oct. 2004,
pp. 63–70.
Jang-Hwan Kim (S’02) was born in Kwangju,
Korea, in 1975. He received the B.S. and M.S.
degrees in electrical engineering from Seoul National University, Seoul, Korea, in 1999 and 2001,
respectively, where he is currently pursuing the
Ph.D. degree.
His current research interests are power electronic
control of electric machines, power quality systems,
and power converter circuits.
Seung-Ki Sul (S’78–M’87–SM’98–F’00) was born
in Korea in 1958. He received the B.S., M.S., and
Ph.D. degrees in electrical engineering from Seoul
National University, Seoul, Korea, in 1980, 1983, and
1986, respectively.
He was with the Department of Electrical and
Computer Engineering, University of Wisconsin,
Madison, as an Associate Researcher from 1986 to
1988. He was then with Gold-Star Industrial Systems
Company as a Principal Research Engineer from
1988 to 1990. Since 1991, he has been a member
of the faculty of the School of Electrical Engineering and Computer Science,
Seoul National University, where he is currently a Professor. His current
research interests are power electronic control of electric machines, electric
vehicle drives, and power converter circuits.