Multiphase machines are characterized by high power density, enhanced faulttolerant capacity, and low torque pulsation. For a voltage source inverter supplied multiphase machine, the probability of load imbalances becomes greater and unwanted loworder stator voltage harmonics occur. This paper deals with the PWM control of multiphase inverters under unbalanced load conditions and it proposes a novel nearfivevector SVPWM algorithm based on the fivephase sixleg inverter. The proposed algorithm can output symmetrical phase voltages under unbalanced load conditions, which is not possible for the conventional SVPWM algorithms based on the fivephase fiveleg inverters. The cause of extra harmonics in the phase voltages is analyzed, and an
xy
coordinate system orthogonal to the
αβz
coordinate system is introduced to eliminate loworder harmonics in the output phase voltages. Moreover, the digital implementation of the nearfivevector SVPWM algorithm is discussed, and the optimal approach with reduced complexity and low execution time is elaborated. A comparison of the proposed algorithm and other existing PWM algorithms is provided, and the pros and cons of the proposed algorithm are concluded. Simulation and experimental results are also given. It is shown that the proposed algorithm works well under unbalanced load conditions. However, its maximum modulation index is reduced by 5.15% in the linear modulation region, and its algorithm complexity and memory requirement increase. The basic principle in this paper can be easily extended to other inverters with different phase numbers.
I. INTRODUCTION
In the late 1960s, multiphase machines were proposed to reduce the low frequency torque ripple in machine drives. Multiphase machines are characterized by enhanced fault tolerance, reduced perphase converter rating, and low torque pulsation. Hence, significant efforts have been put into the development of multiphase drives since the late 1960s. In the 1990s, the inverter ac drives facilitated the utilization of multiphase machines. The research of multiphase drives started accelerating due to the developments of ship propulsion in the second half of the 1990s. In the last decade, multiphase machines have been attracting more and more attention in various applications including electric vehicles, “more electric” aircraft and ship propulsion
[1]

[8]
.
Taking the costs, the faulttolerant capacity, and the occurrence probability of faults into account, the optimal phase number of a multiphase machine is five or six. In a fivephase machine supplied by a fivephase fiveleg inverter, there are four control degrees of freedom, and the zero sequence component of the phase voltages is nonzero under unbalanced load conditions. By applying the decoupling (Clarke’s) transformation, the original fourdimensional (4D) stator system can be decomposed onto two twodimensional (2D) uncoupled planes and one zero sequence component. The
α

β
plane is identical to the corresponding plane for a threephase machine. For a fivephase machine with distributed windings, the MMF distribution is nearsinusoidal. The other plane, named
x

y
, is not related to electromechanical energy conversion. The impedance for the
x

y
stator current components is actually the stator winding leakage impedance. Thus, the applied voltages should contain minimum amplitude
x

y
components in order to avoid large
x

y
stator currents. For a fivephase machine with concentrated windings, the MMF distribution is quasirectangular. The current harmonics that map onto the
x

y
plane can be utilized to couple with the corresponding spatial magnetic field harmonics to obtain additional average torque components
[1]
,
[8]

[11]
.
The conventional vector control and the direct torque control (DTC) which are applicable in threephase machine drives, are also available in the multiphase drives. However, it is important to note that in multiphase drives, the loworder stator voltage harmonics that map onto the voltage
x

y
plane should be avoided
[12]
. Therefore, a qualified method for inverter PWM control is required.
The current research related to the PWM control of multiphase inverters mainly focus on twolevel inverters because they are simple and lowcost. This paper deals with the PWM control of multiphase twolevel inverters, and the modulation strategies for multilevel inverters will not be discussed. The same as that for the threephase machines, there are mainly two PWM methods: carrierbased PWM (CBPWM) and space vector PWM (SVPWM). The SVPWM algorithm facilitates digital implementation, and is characterized by a high modulation index and a constant switching frequency. Therefore, it is the most popular method
[6]
,
[7]
,
[12]
. Neglecting the resistor of the stator, the flux linkage space vector
ψ
_{s}
synthesized by the stator can be expressed as:
ψ
_{s}
≈
∫U_{S}
dt
, where
U_{S}
is the voltage space vector synthesized by the stator. Consequently, the symmetrical phase voltages are required in order to synthesize the circular rotating
ψ
_{s}
. The above is the basic theory of the SVPWM algorithm.
The probability of load imbalances among phases becomes greater in multiphase machines as the number of phases increases. When an
n
phase machine with a single neutral point is supplied by an
n
phase nleg inverter, the sum of the phase currents is constrained to be zero. It is not possible for the inverter to output
n
symmetrical phase voltages under unbalanced load conditions because the potential of the neutral point of the machine cannot be controlled independently. Thus, the trajectory of the synthesized
ψ
_{s}
is no longer circular under unbalanced load conditions.
A novel nearfivevector SVPWM (NFVSVPWM) algorithm for fivephase machine drives is presented in this paper, which can output five symmetrical phase voltages under unbalanced load conditions. The proposed NFVSVPWM algorithm applies to a fivephase sixleg inverter. The midpoint of the additional leg connects to the neutral point of the machine in the fivephase sixleg inverter. The effect of the switching vectors on the
x

y
plane has been taken into account in the NFVSVPWM algorithm and there are no loworder harmonics such as the 3rd, 7th, etc. in its output phase voltages. Furthermore, the digital implementation of the NFVSVPWM algorithm is investigated, and the optimal approach with reduced complexity and low execution time is elaborated. A comparison of the PWM techniques is given. The pros and cons of the NFVSVPWM algorithm are analyzed. Finally, simulation and experimental results validate the effectiveness of the proposed NFVSVPWM algorithm. The basic principle in this paper can be easily extended to other inverters with different phase numbers.
II. TOPOLOGY OF FIVEPHASE SIXLEG INVERTER
SVPWM control under unbalanced load conditions is based on the fivephase sixleg inverter in this paper.
Fig. 1
shows the topology of the fivephase sixleg inverter. The midpoint of the additional leg, F, connects to the neutral point of the loads in this topology. For the conventional fivephase fiveleg inverter, the constraint that the sum of the phase currents need to be zero has to be satisfied and it is not possible for this topology to output five symmetrical phase voltages under unbalanced load conditions. In the fivephase sixleg inverter, the constraint releases, and the potential of the neutral point of the loads can be controlled independently. Consequently, with the help of the proposed topology, the zero sequence component of the output phase voltages can be controlled independently. There are five control degrees of freedom in the fivephase sixleg topology. The incremental control degree of freedom also diversifies the postfault control strategies of the fivephase machines. Moreover, the fivephase sixleg topology does not require an access to the midpoint of the DC bus, and is able to accommodate certain opencircuit faults in the switches or machines
[13]

[15]
. These two advantages are not investigated in this paper.
Topology of fivephase sixleg inverter.
III. NEARFIVEVECTORS SVPWM ALGORITHM
In this section, a fivephase machine that adopts the distributed windings and symmetrical winding arrangement is taken as a practical example to explicate the principle of the NFVSVPWM algorithm. The flow chart of the NFVSVPWM algorithm is shown in
Fig. 2
.
 A. Definition of Switching Vectors
As shown in
Fig. 1
, n is the neutral point of the loads, z is the midpoint of the DC bus, and A, B, C, D, E, F are the midpoints of each leg, respectively. Note that, point z is defined to facilitate calculation, which does not exist in applications. The switching function is defined as
S_{i}
. When the upper switch of the corresponding leg is on,
S_{i}
equals to 1. And
S_{i}
equals to 0 when the lower switch is on. The leg voltage (the voltage between the midpoint of a leg and point z) of each phase can be calculated by
where
i
=
A,B,C,D,E,F
, and
V_{d}
denotes the voltage of the DC bus.
Flow chart of NFVSVPWM algorithm.
The phase voltage (the voltage between the midpoint of a leg and point n) is calculated by
There are 2
^{6}
= 64 switching patterns for a fivephase sixleg inverter. Every switching pattern can be referred to as a sixdigit binary number, each digit of which corresponds to
S_{F}
,
S_{A}
,
S_{B},
,
S_{C}
,
S_{D}
,
S_{E}
in order.
The switching vectors of the fivephase sixleg inverter are defined in equation (3). Assuming that ω is the fundamental frequency of the output phase voltages, through the transform of (3), the (10k±1)th harmonics of the phase voltages can be equivalently expressed as circular rotating space vectors with the frequency of (10k±1)ω in the stationary
αβ
coordinate system, while the (10k±3)th harmonics of the phase voltages can be equivalently expressed as circular rotating space vectors with the frequency of (10k±3)ω in the stationary
xy
coordinate system
[16]

[20]
.
where
denote the
α
,
β
,
x
,
y
and zero sequence component of
V
_{k}
, respectively.
By the derivations above, a switching vector can be correlated with a space vector in the
αβz
coordinate system and a space vector in the
xy
coordinate system. The distribution of switching vectors in the
xy
coordinate system is shown in
Fig. 3
, and that in the
αβz
coordinate system is shown in
Fig. 4
. When the reference voltage space vector is known, particular switching vectors can be applied to synthesize it.
Switching vectors in the xy coordinate system.
Switching vectors in the αβz coordinate system.
Fig. 5
shows the distribution of all the switching vectors in the 2D stationary
αβ
coordinate system with the exception of the zero sequence component. The switching vectors locate in ten sectors. Referring to the distribution of the switching vectors in the
αβ
coordinate system, the space composed of the 64 switching vectors in the
αβz
coordinate system can be divided into ten triangular prisms. Furthermore, according to the magnitude of the switching vectors in the
αβ
coordinate system, they can be classified into three patterns: large vectors, medium vectors, and small vectors. When the small vectors are selected to synthesize the reference voltage space vector, the onoff frequency of the inverter during a PWM period doubles as the legs of the same switching state are not adjacent. Thus, the small vectors should not be selected in SVPWM control.
Switching vectors in the αβ coordinate system.
 B. Selection of Switching Vectors in NFVSVPWM Algorithm
The SVPWM algorithm is based on the principle of vector composition. The applying time of each switching vector must be no less than zero, and the sum of the applying times of the selected switching vectors must be no greater than a PWM period.
The NFVSVPWM algorithm selects five nonzero switching vectors and two zero switching vectors as applying vectors in each PWM period. Once the combination of five nonzero switching vectors is fixed, the reference voltage vector
V_{ref}
that can be synthesized is confined to the polyhedron composed of the five selected nonzero switching vectors. The following equation shows the relationship between
V_{ref}
and the applying vectors:
where
T_{S}
is the PWM period, and
T
_{k+2}
,⋯,
T
_{k2}
are the applying times of
V
_{k+2}
,⋯,
V
_{k2}
, respectively.
The combination of the selected nonzero switching vectors varies with the location of
V_{ref}
in the
αβz
coordinate system, by which both the minimum onoff frequency and the maximum modulation index should be achieved. Therefore, the selected nonzero switching vectors should be adjacent.
The triangular prism in which
V_{ref}
locates and the polyhedron which
V_{ref}
maps into determine the combination of switching vectors. The triangular prism is judged by the
V_{α}
,
V_{β}
components of
V_{ref}
, and the polyhedron is determined by the polarities of the phase voltages corresponding to
V_{ref}
. There are four combinations of nonzero applying vectors in a triangular prism, each of which is composed of three large vectors and two medium vectors. The phase voltages that correspond to
V_{ref}
are calculated by equation (5). The digital implementation of the determination of the location of
V_{ref}
will be discussed in Section IV.
The judging conditions in triangular prism 1 are listed in
Table I
. The four combinations of nonzero switching vectors in triangular prism 1 are shown in
Fig. 6
. The judging conditions in other triangular prisms can be obtained by the same means.
Four combinations of nonzero switching vectors in triangular prism 1.
JUDGING CONDITIONS OF NFVSVPWM ALGORITHM IN TRIANGULAR PRISM 1
JUDGING CONDITIONS OF NFVSVPWM ALGORITHM IN TRIANGULAR PRISM 1
For a fivephase machine with distributed windings, the expected
x

y
components of
V_{ref}
are zero. For a fivephase machine with concentrated windings, it is possible to obtain additional average torque components by taking advantage of the harmonics that map onto the
x

y
plane. Furthermore, the zero component of
V_{ref}
is required to be zero in the vector control for multiphase drives under normal conditions, and it is possible that the expected zero component of
V_{ref}
is nonzero under postfault conditions.
A fivephase machine that adopts the distributed windings and symmetrical winding arrangement is used as a practical example throughout this paper, and therefore the expected
x

y
components of
V_{ref}
are zero. The basic concepts in this paper can also be applied to the machines with different structures.
 C. Applying Times of Switching Vectors in NFVSVPWM Algorithm
In this part, it is assumed that
V_{ref}
maps into the polyhedron I of the triangular prism 1 in the
αβz
coordinate system to make a concrete illustration of the NFVSVPWM algorithm. In that case,
V_{25}
,
V_{24}
,
V_{31}
,
V_{16}
,
V_{29}
are selected as nonzero switching vectors in the NFVSVPWM algorithm.
The distribution of the preferred switching vectors in the
xy
coordinate system is shown in
Fig. 7
.
V_{31}
corresponds to a zero space vector in the
xy
coordinate system. The orientation of
V_{25}
is opposite to that of
V_{16}
, and the orientation of
V_{24}
is opposite to that of
V_{29}
. For a machine with distributed windings, the expected xy components of
V_{ref}
are zero. In the
xy
coordinate system, the distribution of the selected switching vectors is utilized to achieve that the synthesis of the applying vectors is zero.
Distribution of the preferred switching vectors in the xy coordinate system.
If the following constraint conditions are satisfied, the synthesis of the applying vectors in the
xy
coordinate system will be zero, and there will be no (10k±3)th harmonics in the output phase voltages.
where │
V_{S}
│ and │
V_{M}
│ denote the magnitudes of the small vector and medium vector, respectively. Both │
V_{S}
│ and │
V_{M}
│ are constants.
Based on the analysis above, the applying times of the switching vectors in the NFVSVPWM algorithm are obtained by solving equations
Additionally, two zero switching vectors
V_{0}
,
V_{63}
are also applied in each PWM period to avoid a discontinuous modulation.
Assuming that
T_{sum}
=
T
_{25}
+
T
_{24}
+
T
_{31}
+
T
_{29}
+
T
_{16}
, an overmodulation occurs when
T_{sum}
>
T_{s}
. In that case, the applying times of
V_{0}
,
V_{63}
equal to zero, and in order to reduce the distortion of the output phase voltages to its least extent, the synthesis of the switching vectors should maintain the orientation of the reference voltage vector. Therefore, the solution of equation (7) should be normalized before it can be utilized to generate switching signals. The normalization of the solution is shown as follows:
When
T_{sum}
≤
T_{s}
, the solution of equation (7) can be utilized to generate the switching signals without any modification. The applying times of
V_{0}
,
V_{63}
are calculated by
In order to reduce the total harmonic distribution (THD) of the phase voltage, a symmetrical switching sequence is adopted to generate the switching signals in the NFVSVPWM algorithm. In polyhedron I of triangular prism 1, the switching sequence is arranged as
V_{0}

V_{16}

V_{24}

V_{25}

V_{29}

V_{31}

V_{63}

V_{63}

V_{31}

V_{29}

V_{25}

V_{24}

V_{16}

V_{0}
in order. The resultant switching signals are shown in
Fig. 8
.
Switching signals when V_{ref} locates in polyhedron I of triangular prism 1.
IV. DIGITAL IMPLEMENTATION OF NFVSVPWM ALGORITHM
Two of the steps in the NFVSVPWM algorithm are timeconsuming: the determination of the location of
V_{ref}
and the calculation of the applying times of the switching vectors. The intensive computations required in these two steps hinder the online application of the NFVSVPWM algorithm. In this section, the maximal offline calculation of the NFVSVPWM algorithm has been investigated.
 A. Determination of the location of Vref
Firstly, the triangular prism in which
V_{ref}
locates is determined. Then the polyhedron which
V_{ref}
maps into is judged. The first procedure is realized according to the
V_{α}
,
V_{β}
components of
V_{ref}
. The second procedure is determined according to the polarities of the phase voltages corresponding to
V_{ref}
.
Five temporary variables are defined in equation (10). In the
αβ
coordinate system, the sector which
V_{ref}
maps into is determined by the value of the sectordetermination function. The sectordetermination function is shown as follows:
where sign(·) is the sign function. The value of sign(·) equals to 1 when the independent variable is greater than zero, otherwise it is zero.
A sector in the
αβ
coordinate system corresponds to the triangular prism of the same number in the
αβz
coordinate system. Thus, the triangular prism in which
V_{ref}
locates can be judged by the value of the function S. The corresponding relationships between the triangular prism number and function
S
are shown in
Table II
.
CORRESPONDING RELATIONSHIPS BETWEEN THE TRIANGULAR PRISM NUMBER AND FUNCTIONS
CORRESPONDING RELATIONSHIPS BETWEEN THE TRIANGULAR PRISM NUMBER AND FUNCTION S
The polyhedron which
V_{ref}
maps into is judged by the polarities of the phase voltages corresponding to
V_{ref}
. There are four polyhedrons in a triangular prism. As shown in
Table I
, only three of the phase voltages are required to determine the polyhedron. In triangular prism 1, the three phase voltages for the polyhedron determination are calculated by (12), and unnecessary calculation is avoided.
 B. Calculation of the applying times of switching vectors
In this part, it is also assumed that
V_{ref}
maps into polyhedron I of triangular prism 1 in the
αβz
coordinate system to explain the reduction of the arithmetic operations in the NFVSVPWM algorithm. The applying times of the nonzero switching vectors are calculated by
In equation (13), the calculation of the inverse matrix of the fiveorder matrix requires a large amount of arithmetic operations. Calculating it offline and then applying the results online provides a possible means to employ the NFVSVPWM algorithm online. However, it is difficult to obtain the analytical solution of the inverse matrix of a fiveorder matrix. For a fivephase machine with distributed windings, the expected
x

y
components of
V_{ref}
are zero. As shown in
Fig. 7
, the distribution of the preferred switching vectors in the
xy
coordinate system is utilized to reduce the calculation of equation (13). If the constraint conditions in (6) are satisfied, the synthesis of applying vectors in the
xy
coordinate system will be zero. Then, the applying times of
V_{25}
,
V_{24}
,
V_{31}
are calculated by
where
denotes the
αβz
components of
V_{k}
.
The applying times of the other two nonzero switching vectors can be obtained through equations in (6). The order of the inverse matrix is reduced to three. Compared with equation (13), the calculation in equation (14) has been significantly cut down. Moreover, it is relatively easy to obtain the analytical solution of the inverse matrix.
Assuming that
A
denotes the inverse matrix in equation (14), matrix
A
varies with the location of
V_{ref}
. There are forty combinations of nonzero applying vectors, each of which corresponds to a certain matrix
A
. A detailed analysis shows that all these coefficients of matrix
A
are made from the reduced set of eight constants. The eight constants are listed in
Table III
, and the coefficients of matrix
A
in triangular prism 1 are shown in
Table IV
.
EIGHT CONSTANTS CONSISTED IN MATRIXA
EIGHT CONSTANTS CONSISTED IN MATRIX A
COEFFICIENTS OF MATRIXAIN TRIANGULAR PRISM 1
COEFFICIENTS OF MATRIX A IN TRIANGULAR PRISM 1
This analysis cuts down the arithmetic operations of the NFVSVPWM algorithm and facilitates its digital implementation.
V. COMPARISON WITH EXISTING PWM ALGORITHMS
As mentioned in Section I, the CBPWM and SVPWM algorithms are the main approaches in the PWM control of multiphase inverters. According to
[1]
, the CBPWM with zerosequence injection and the SVPWM are exact equivalents in the PWM of multiphase voltage source inverters. With an optimization of the space vector sequence, full dc bus utilization and stator current ripple minimization can be achieved by both algorithms. The published SVPWM strategies are mainly based on (4L) and (2L+2M) methods. The (4L) SVPWM algorithm adopts four large nonzero switching vectors per switching period to synthesize
V_{ref}
while the (2L+2M) SVPWM algorithm adopts two large and two medium nonzero switching vectors per switching period to synthesize
V_{ref}
. They provide the same utilization of the dc bus voltage in the linear modulation region. The (4L) SVPWM algorithm leads to a smaller phase voltage THD on the
x

y
plane, but its total phase voltage THD is higher than the corresponding one of the (2L+2M) SVPWM algorithm. The (4L) SVPWM algorithm will always generate higher current THD and squared rms current ripple at the equal average switching frequency than the (2L+2M) SVPWM algorithm, and its implementation is more difficult
[18]
,
[19]
.
The existing SVPWM approaches mostly apply to the twolevel fivephase fiveleg inverters. It is not possible for them to output five symmetrical phase voltages under unbalanced load conditions since the zero sequence component of the phase voltages cannot be controlled independently. By contrast, the NFVSVPWM algorithm applied to the fivephase sixleg inverters can control the zero sequence component of the phase voltages independently and it can output five symmetrical phase voltages under unbalanced load conditions. The NFVSVPWM algorithm is built on the (2L+2M) SVPWM algorithm. Thus, their switching characteristics are identical. The performances of (2L+2M) SVPWM algorithm are the same as those of the CBPWM with zerosequence injection. In addition, it is superior in in terms of switching characteristics when compared with the (4L) SVPWM algorithm. Accordingly, the (2L+2M) SVPWM algorithm is chosen for comparison with the NFVSVPWM algorithm in this section. Note that, the (2L+2M) SVPWM algorithm is denoted further as 4VSVPWM for conciseness.
Table V
provides a comparison of the major characteristics of the 4VSVPWM and NFVSVPWM algorithms.
The maximum modulation index
M
is defined as
M
=
V
_{1}
/ (
V_{d}
/ 2) , where
V
_{1}
is the maximum fundamental peak phase voltage. The value of
M
in
table V
is calculated in the linear modulation region. For the NFVSVPWM algorithm,
M
relates to the zero sequence component of
V_{ref}
. However, it is possible to obtain the value of
M
when the zero sequence component of
V_{ref}
equals to zero. The calculation procedure is given in Appendix A.
The THD of phase voltage
i
is defined as
where
F_{k}
stands for the kth component in the spectrum of the phase voltage
i
, and h represents the harmonic order closest to the uppermost frequency in calculation. The phase voltage THD in
Table V
has been calculated up to 3 kHz on the basis of the experimental results (given in Section VII). The PWM switching frequency is set to 13.2 kHz in the experiments, and the current ripple is negligible at such a high switching frequency. Moreover, the switching characteristics of the NFVSVPWM and 4VSVPWM algorithms are the same. Thus, the switching characteristics are not listed in
Table V
.
The complexity of the algorithms has been analyzed by comparing the number of program statements executed in a switching period, and the memory requirement has been evaluated by the number of constants and variables stored in the memory. The data in
Table V
depend on how the algorithms have been carried out. Nevertheless, it can be inferred that the NFVSVPWM algorithm is a little more complex than the 4VSVPWM algorithm.
MAIN CHARACTERISTICS OF THE COMPARED SVPWM STRATEGIES
MAIN CHARACTERISTICS OF THE COMPARED SVPWM STRATEGIES
To sum up, the switching characteristics and voltage THD of the NFVSVPWM algorithm are as good as those of 4VSVPWM algorithm and the NFVSVPWM algorithm is able to output five symmetrical phase voltages under unbalanced load conditions. However, the maximum modulation index of the NFVSVPWM algorithm is reduced by 5.15% in the linear modulation region, and its algorithm complexity and memory requirement increase.
VI. SIMULATION RESULTS
A Matlab/Simulink simulation environment has been designed to validate the effectiveness of the proposed NFVSVPWM algorithm. The performances of two algorithms under unbalanced load conditions are compared: the 4VSVPWM algorithm based on a fivephase fiveleg inverter and the NFVSVPWM algorithm.
The
V_{ref}
of both algorithms on the
αβ
plane are space vectors with a magnitude of 8.5 V, rotating at 518.1 rad/s, and their expected
x

y
components are zero. There is no zero sequence component of
V_{ref}
in the 4VSVPWM algorithm since the 4VSVPWM algorithm cannot control the zero sequence component of the output phase voltages independently. The expected zero sequence component of the NFVSVPWM algorithm is zero. Four 5 ohm resistors connected in a star connection constitute the loads of phase A, B, C, and E, while phase D is opencircuit in order to test the performances of the SVPWM algorithms under extreme unbalance load conditions.
Fig. 9
illustrates the simulation results. For the 4VSVPWM algorithm, the
x

y
components of the output phase voltages are zero, but the amplitudes of the output phase voltages differ drastically. It is obvious that the 4VSVPWM algorithm is unable to output symmetrical phase voltages under unbalanced load conditions because it cannot control the zero sequence component of the phase voltages independently. By contrast, the proposed NFVSVPWM algorithm can output symmetrical phase voltages under unbalanced load conditions. Both the
x

y
components and the zero sequence component of the output phase voltages are zero, and there are no 3rd, 7th, etc. harmonics in the output phase voltages. Note that, the phase voltages in
Fig. 9
are filtered by lowpass filters with the same cutoff frequency of 3 kHz to facilitate observation.
Simulation results for V_{ref}= 8.5e^{jwt}(V), ω = 518.1(rad/s) under unbalanced load conditions. From left to right: 4VSVPWM, NFVSVPWM. From top to bottom: output phase voltages, voltage trajectory in the αβz coordinate system and xy coordinate system, and voltage harmonic distribution of phase E.
VII. EXPERIMENTAL RESULTS
An experimental bench has been built in the laboratory. The experimental bench is comprised of four parts: the voltage source inverter (VSI), the control unit, the switching devices, and the loads. A threephase uncontrolled rectifier paralleling with filter capacitors works as the VSI of the system. The SVPWM algorithms are fully digitally implemented by a control board based on a fixedpoint digital signal processor (DSP) TMS320F2812. Two intelligent power modules (IPMs) act as the switching devices of the fivephase PWM inverter.
By connecting the output of leg F to the neutral point of the loads, the six legs build a fivephase sixleg inverter, which the NFVSVPWM algorithm is based on. By disconnecting the neutral point of the loads from the output of leg F, the remaining five legs build a fivephase fiveleg inverter, which the 4VSVPWM algorithm is based on. The load parameters in the experiments are identical to those in the simulations: the loads of phase A, B, C, and E are composed of four 5 ohm resistors connected in star connection, while the phase D is opencircuit.
The
V_{ref}
of both algorithms on the
αβ
plane is a space vector with a magnitude of 8.5 V, rotating at 518.1 rad/s, and the
x

y
components of
V_{ref}
are zero. Additionally, the zero component of
V_{ref}
of the NFVSVPWM algorithm is zero. In order to reduce the computational effort and model industrial applications more analogously, the applying times of the switching vectors are calculated thirtytwo times in a period of
V_{ref}
. The PWM switching frequency is set to 13.2 kHz. The output phase voltages in the experiments are shown in
Fig. 10
, and the FFT analysis of phase voltage E is shown in
Fig. 11
. The results reveal that the proposed NFVSVPWM algorithm is able to output five symmetrical phase voltages under extreme unbalanced load conditions, and eliminate the (10k±3)th harmonics of the phase voltages. The amplitudes of the output phase voltages in the NFVSVPWM algorithm are smaller than those of
V_{ref}
. This is due to the existence of the collectoremitter saturation voltage of the IGBTs. The phase voltages in
Fig. 10
and
11
are filtered by lowpass filters with the same cutoff frequency of 3 kHz to facilitate observation.
Output phase voltages for ω = 518.1(rad/s) under unbalanced load conditions. From left to right: 4VSVPWM, NFVSVPWM.
Phase voltage E and FFT analysis in NFVSVPWM algorithm.
VIII. CONCLUSIONS
The proposed NFVSVPWM algorithm based on a fivephase sixleg inverter can output symmetrical phase voltages under unbalanced load conditions, which is not possible for the conventional SVPWM algorithms based on the fivephase fiveleg inverters. The cause of the extra harmonics in the phase voltages is analyzed, and an
xy
coordinate system orthogonal to the
αβz
coordinate system is introduced to eliminate the loworder harmonics in the output phase voltages. Thus, there are no loworder harmonics such as the 3rd, 7th, etc. in the output phase voltages for the NFVSVPWM algorithm. The digital implementation of the NFVSVPWM algorithm is investigated, and the optimal approach with reduced complexity and low execution time is given. A comparison of the proposed algorithm and other existing PWM algorithms is also provided. The switching characteristics and voltage THD of the NFVSVPWM algorithm are as good as those of the 4VSVPWM algorithm. However, the maximum modulation index of the NFVSVPWM algorithm is reduced by 5.15% in the linear modulation region, and algorithm complexity and memory requirement of it increase. The effectiveness of the proposed NFVSVPWM algorithm has been validated by the simulation and experimental results. The basic principle in this paper can be easily extended to other inverters with different phase numbers.
Acknowledgements
This work was supported in part by the 863 Plan of China under Project 2011AA11A261, in part by National Natural Science Foundation of China under Project 51077026 and 51377033.
BIO
Ping Zheng (M’04.SM’05) received her B.S., M.S., and Ph.D. degrees from the Harbin Institute of Technology, Harbin, China, in 1992, 1995, and 1999, respectively. Since 1995, she has been with Harbin Institute of Technology, where she has been a Professor since 2005. She is a Member of the IEEE IAS Electric Machines Committee, IEEE Industry Applications Society and International Compumag Society. She is the author or coauthor of more than 140 published refereed technical papers and four books. She also holds 30 Chinese invention patents. Prof. Zheng was the recipient of more than 20 technical awards including the “China Youth Science and Technology Award” by the Organization Department of the Communist Party of China, in 2009. Her current research interests include electric machines and control, hybrid electric vehicles, and unconventional electromagnetic devices.
Pengfei Wang received his B.S. and M.S. degrees in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2007 and 2009, respectively, where he is currently working toward his Ph.D. degree. He is the author or coauthor of 7 published papers. His current research interests include faulttolerant permanentmagnet synchronous machines, and the control strategies of multiphase machines under healthy and postfault conditions.
Yi Sui received his B.S. and M.S. degrees in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2009 and 2011, respectively, where he is currently working toward his Ph.D. degree. He is the author or coauthor of 15 published papers. His current research interests include the faulttolerant permanentmagnet synchronous machine systems used in pure electric vehicles, permanentmagnet linear machines, and the high performance servo machine systems used in unconventional areas.
Chengde Tong received his B.S., M.S., and Ph.D. degrees in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2007, 2009 and 2013, respectively. He is currently a Lecturer with Department of Electrical Engineering, Harbin Institute of Technology. He is the author or coauthor of more than 30 published papers. His current research interests include electric drives and the energy management of hybrid electric vehicles, freepiston Stirling engines and permanentmagnet linear machines.
Fan Wu received his B.S. and M.S. degrees in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2010 and 2012, respectively, where he is currently working toward his Ph.D. degree. He is the author or coauthor of 14 published papers. His current research interests include electromagnetic design for electrical machines, including analytical analysis, finite element analysis and prototyping.
Tiecai Li received his B.S. and M.S. degrees from the Harbin Institute of Technology, Harbin, China, in 1977 and 1990, respectively. He is currently a Professor with the Harbin Institute of Technology. His current research interests include motor drives and control, integrated motor systems, intelligent computer interface, and network information appliances.
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