IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 38, NO. 5, SEPTEMBER/OCTOBER 2002 1351
Torque Density Improvement in a Six-Phase Induction
Motor With Third Harmonic Current Injection
Renato O. C. Lyra, Member, IEEE, and Thomas A. Lipo, Fellow, IEEE
Abstract—The use of six-phase induction motor for indus-
trial drives presents several advantages over the conventional
three-phase drive such as improved reliability, magnetic flux har-
monic reduction, torque pulsations minimization, and reduction
on the power ratings for the static converter. For these reasons,
six-phase induction motors are beginning to be a widely acceptable
alternative in high power applications. A typical construction of
such drives includes an induction machine with a dual three-phase
connection, where two three-phase groups are spatially shifted
30 electrical degrees, a six-leg inverter, and a control circuit. By
controlling the machine’s phase currents, harmonic elimination
and torque-ripple reduction techniques could be implemented.
This paper describes a technique of injecting third harmonic zero
sequence current components in the phase currents, which greatly
improves the machine torque density. Analytical, finite-element,
and experimental results are presented to show the system opera-
tion and to demonstrate the improvement on the torque density.
Index Terms—Induction machine analysis and design, industrial
drives, six-phase drives.
I. INTRODUCTION
T
HREE-PHASE induction machines are today a standard
for industrial electrical drives. Cost, reliability, robustness,
and maintenance-free operation are among the reasons these
machines are replacing dc drive systems. The development of
power electronics and signal processing systems has eliminated
one of the greatest disadvantages of such ac systems, that is,
the issue of control. With modern techniques of field-oriented
vector control, the task of variable-speed control of induction
machines is no longer a disadvantage.
The need to increase system performance, particularly when
facing limits on the power ratings of power supplies and semi-
conductors, motivates the use of phase number other than three,
and encourages new pulsewidth modulation (PWM) techniques,
new machine design criteria, and the use of harmonic current
and flux components.
Paper IPCSD 02–027, presented at the 2001 Industry Applications Society
Annual Meeting, Chicago, IL, September 30–October 5, and approved for pub-
lication in the IEEE T
RANSACTIONS ON INDUSTRY APPLICATIONS by the Indus-
trial Drives Committee of the IEEE Industry Applications Society. Manuscript
submitted for review October 15, 2001 and released for publication May 29,
2002. This work was supported by the WEMPEC Consortium at the University
of Wisconsin, Madison, the Universidade Federal de Minas Gerais, Brazil, and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Brazil.
R. O.C. Lyra was with the Department of Electric and Computer Engineering,
University of Wisconsin, Madison, WI 53706 USA. He is now with the Depart-
ment of Electric and Computer Engineering, Universidade Federal de Minas
Gerais, Belo Horizonte, Brazil (e-mail: lyra@ieee.org).
T. A. Lipo is with the Department of Electric and Computer Engineering, Uni-
versity of Wisconsin, Madison, WI 53706 USA (e-mail: lipo@engr.wisc.edu).
Publisher Item Identifier 10.1109/TIA.2002.802938.
In a multiphase system, here assumed to be a system that
comprises more than the conventionalthree phases, the machine
output power can be divided into two or more solid-state in-
verters that could each be kept within prescribed power limits.
Also, having additional phases to control means additional de-
grees of freedom available for further improvements in the drive
system.
With split-phase induction machines, and appropriate drive
system, the sixth harmonic torque pulsation, typical in a six-step
three-phase drive, can be eliminated [1], [2]. Also, air-gap flux
created by fifth and seventh harmonic currents in a high-power
six-step converter-fed system is dramatically reduced with
the penalty of increased converter harmonic currents [3].
PWM techniques are employed to overcome this problem by
eliminating the harmonic currents in the modulation process
when the power ratings are not prohibitive.
Dual-stator machines are similar to split-phase machines with
the difference that the stator groups are not necessarily equal.
A dual-stator machine with different numbers of poles in each
three-phase group has been proposed in [4] to obtain controlla-
bility at low speeds. Two independent stator windings are used
in [5] for an induction generator system. One set of windings is
responsible for the electromechanical power conversion while
the second one is used for excitation purposes. A PWM con-
verter is connected to the excitation windings and the load is
connected directly to the power windings.
A particular case of split-phase or dual-stator machine,
the six-phase machine can be built by splitting a three-phase
winding into two groups. Usually these three-phase groups
are displaced by 30 electrical degrees from each other. This
arrangement composes an asymmetrical six-phase machine
since the angular distance between adjacent phases is not all
the same [6]. The analysis of an induction machine for multiple
phases and arbitrary displacement between them is presented
in [2] where the six-phase induction machine is used as an
example and an equivalent circuit has been derived. The
model for a six-phase machine was developed in [7].
Reliability is one of the advantages in using six-phase sys-
tems. In the case of failure of one of the phases, either in the
machine or in the powerconverter, the system can still operate at
a lower power rating since each three-phase group can be made
independent from each other. In the case of losing one phase, the
six-phase machine can continue to be operated as a five-phase
machine as described in [8].
The inherent third harmonic component in the winding func-
tions of the machine [9], [10] suggests the use of third harmonic
currents to improve its performance. Torque improvement can
be obtained by using multiphase windings with injection of
0093-9994/02$17.00 © 2002 IEEE
1352 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 38, NO. 5, SEPTEMBER/OCTOBER 2002
third harmonic currents. Such a nine-phase induction machine
drive was investigated in [11]. The complexity of the power
system, which includes series and parallel transformers, in-
creases the systemcost and must be evaluated for each particular
application. The use of a voltage-controlled system does not
guarantee the phase alignment between fundamental and third
harmonic currents, especially at low speeds, and a poor low
speed and dynamic behavior can be expected for this system.
This paper describes a technique of injecting third harmonic,
zero-sequence current components in the phase currents that
improve the machine torque density. Analytical, finite-element,
and experimental results are presented to show the system oper-
ation and to demonstrate the improvement on the torque density.
II. T
ORQUE IMPROVEMENT WITH THIRD HARMONIC
CURRENT INJECTION
The asymmetric six-phase machine is derived from a
conventional three-phase induction machine. For a two-pole
concentrated-winding three-phase machine, the magnetomotive
force (MMF) acting across the air gap associated with the stator
currents is
(1)
where
, , and are the machine’s winding functions
[12] and compose a set of 120
phase-shifted square waves with
amplitude
(each coil has turns). Using Fourier anal-
ysis decomposition, and assuming balanced three-phase cur-
rents with amplitude
, it can be shown that the even and all
triple harmonics are zero and the MMF is given by
(2)
where
(3)
(4)
(5)
and the wave contains a 0% third harmonic, 20% fifth harmonic,
and 14% seventh harmonic, plus smaller portions of higher har-
monics. Setting the sine function argument equal to a constant,
to establish a fixed point in the waveform, and differentiating
with respect to time, the rotational speed can be calculated.
The peak fundamental component rotates in the direction of
increasing
with angular speed , the fifth harmonic rotates
in the direction of decreasing
at 1/5 the speed of the fun-
damental component and the seventh harmonic rotates at the
same direction of the fundamental with 1/7 of its speed. The
fifth harmonic produces a negative-sequence component of flux
that produces negative or braking torque. The seventh produces
positive torque but it is only useful between 0–1/7 of the syn-
chronous speed.
If a neutral connection is provided, zero-sequence current
components can flow in the machine. Considering a zero-
sequence current
the MMF due to this current can be
calculated to be
(6)
where the triple harmonic of the square-wave winding functions
are clearly represented. This suggests the use of triplens of the
fundamental current frequency to produce torque corresponding
to the zero sequence winding functions. Since most of the ma-
chines have a discrete distributed winding, the zero-sequence
winding function is nonzero and can be explored to produce
extra torque. Injecting a third harmonic current component
and neglecting higher harmonics, the MMF is
(7)
This quantity represents a standing or pulsating wave in the air
gap and not a rotating wave. This component is undesirable
since it produces braking and pulsating torques.
Distributing the windings over more slots can reduce the har-
monic content of the MMF. For a distribution over two slots, or
two slots per pole per phase, the new fundamental MMF is
(8)
This result corresponds to a three-phase machine with dis-
tributed windings or to a six-phase concentrated winding ma-
chine. Two three-phase winding groups, spatially phase shifted
by 30
, compose the six-phase machine.
The amplitude of the fundamental component is, then,
(9)
and the constant
, as expected, is the usual dis-
tribution factor for two slots per pole per phase. The funda-
mental component istherefore reduced by4.1% when compared
to the no-phase-shift full-pitch case. Doing similar analysis for
the fifth and seventh harmonics, it can be shown that with the
distribution over two slots, the fifth harmonic component is re-
duced from 20% to 5% and the seventh harmonic goes from
14% to 3.7%. For this case, if a third harmonic zero-sequence
component is injected, it would again produce a standing wave
and is of no practical value again.
If, however, two sets of three-phase currents phase are shifted
in time by 30
to comprise the currentsin the six-phase machine,
the fundamental component of the MMF can be found to be the
same as in the concentrated winding machine (3) but now both
fifth and seventh harmonics are reduced to zero. With injection
of third harmonic zero-sequence current components as
and (10)
where
and correspond to the two three-phase winding
groups, the zero-sequence MMF is now
(11)
LYRA AND LIPO: TORQUE DENSITY IMPROVEMENT IN A SIX-PHASE INDUCTION MOTOR 1353
Fig. 1. Approximated equivalent circuit of a three-phase induction motor
where the rotor leakage inductance is neglected.
This result corresponds to a rotating field with angular
speed equal to the fundamental angular speed. Hence, the
zero-sequence component can be now used to produce a
second positively rotating flux component synchronized with
the fundamental component.
A. Flux Distribution With Third Harmonic Current Injection
The possibility of injecting third harmonic current compo-
nents into the machine without producing pulsating torques en-
ables the ability to reshape the machine’s flux distribution in a
similar manner to the technique applied in PWM modulators. In
PWM modulators, a third harmonic voltage reference is added
to the fundamental component to increase the modulation index
beyond the unity modulation index without distortion produced
by dropping pulses. For the equivalent approach applied to the
modulating machine flux, it is desired to increase the funda-
mental component of flux without saturating the machine.
The appropriate target reference flux waveform, containing
the third harmonic contribution, is defined as
(12)
Using an optimization process, the relation between funda-
mental and third harmonic components can be determined for
the best iron utilization and the air-gap flux is defined as a func-
tion of the maximum allowed flux distribution
as
(13)
From (13), it is clear that for keeping the same peak value for
the flux density in the air gap, with injection of the third har-
monic component the peak of the fundamental flux component
is higher.
B. Torque Improvement With Third Harmonic Current
Injection
The benefit of using the third harmonic component can now
be investigated. Using an approximate equivalent circuit for the
induction machine, as shown in Fig. 1, where the rotor leakage
inductance is neglected, and assuming peak values for the vari-
ables, the rotor current is
(14)
and the torque is defined by
(15)
The peak voltage
is proportional to the peak air-gap flux
density [12] by
(16)
where
is the area of one magnetic pole and is the
number of series connected turns. The factor
expresses the
average value of
in terms of its peak value.
In the case of the asymmetric six-phase machine, the max-
imum allowable fundamental flux density can be increased by
. Since all other parameters in (15) remain the same for
this machine, the increase in torque obtained by raising the fun-
damental component of flux density, while keeping the same
peak tooth and air-gap flux density is
-
(17)
where
corresponds to the torque production in a three-
phase induction machine used as baseline for the evaluation.
There is an additional 33% in the torque production for the
six-phase machine with third harmonic injection due to the in-
crease in the fundamental flux. In addition to that, the contribu-
tion of the third harmonic component must be considered. The
torque produced by the third harmonic currents in the six-phase
machine is
(18)
and the third harmonic voltage can be computed as
(19)
where
.
The slip for the third harmonic is
(20)
and the rotor resistance can be determined [13] to be
(21)
where
is the number of series connected turns of one of
the three pairs of poles of the third harmonic,
is the number
of rotor slots, and
is the resistance of a rotor bar taking into
account the effect of the end ring. The factor of 3 is used since
the three pole pairs of the third harmonic are connected in series.
Inserting these expressions in the torque equation for the third
harmonic (18)
-
(22)
1354 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 38, NO. 5, SEPTEMBER/OCTOBER 2002
Fig. 2. Torque enhancement considering reduction in the air-gap flux and
increase in the surface current density.
Using a similar derivation, an equivalent expression can be
found for the baseline machine torque
(23)
Taking the ratio the contribution of the third harmonic can be
found
-
(24)
Also, the contribution of the third harmonic is 7.4% of the
value produced by the baseline machine. The total torque
improvement is, then,
- -
% % (25)
It must be noted, however, that the peak value stator core flux
has not been maintained constant so that the amount of improve-
ment that can actually be realized depends upon the saturation
level permitted in the stator core.
Alternatively, rather than maintain the peak air-gap flux con-
stant after addition of the third harmonic, the peak flux den-
sity could be reduced by 1.732/2 or 0.866 and the fundamental
component kept constant. In this case, the slot area available for
copper could be increased permitting a 14% increase in current
and a 14% increase in torque for the same tooth and core flux
density.
It can be shown [15] that the total torque enhancement is a
function of the peak air-gap flux, therefore, a function of a flux
reduction factor
, and the initial tooth-to-slot aspect ratio ,
which determines the percentage increase allowed for the cur-
rent density in the machine. Fig. 2 shows the theoretical torque
enhancement in the induction machine as a function of these
two factors. For a typical 0.5 tooth-to-slot aspect ratio and 87%
(a)
(b)
Fig. 3. Phases
a
and
x
winding distribution. (a) Winding function. (b)
Harmonic composition.
flux reduction, a torque enhancement of approximately 14% is
obtained, as mentioned before.
III. M
ACHINE DESIGN
For verification, an asymmetric six-phase induction motor
was designed using a conventional three-phase motor as base-
line. The new winding distribution accommodates in the same
frame size as the baseline three-phase motor and both machines
have similar peak air gap fluxes.
From a single-layer three-phase stator, a double-layer six-
phase distribution is implemented by dividing the three phases
into two groups that are spatially shifted 30 electrical degrees.
Fig. 3 shows the winding functions, and their harmonic compo-
sition, for phases
and . The spatial phase shift between the
two windings and the presence of the third harmonic component
in the distribution necessary to interact to the injected third har-
monic currents can be seen.
For the baseline machine, from the nameplate and geomet-
rical data, the air-gap flux is calculated. The stator phase voltage
of the machine is calculated as [13]
(26)
where
(27)
LYRA AND LIPO: TORQUE DENSITY IMPROVEMENT IN A SIX-PHASE INDUCTION MOTOR 1355
Fig. 4. Finite-element mesh with 6063 nodes and 3006 surfaces.
where is the stator inner diameter, the stator length, the
number of poles, and
the peak fundamental air-gap flux.
The number of series connected turns per phase is defined as
turns/coil coil side/slot number of slots
number of phases circuits
(28)
Thus, for a single-layer three-phase machine, one has
(29)
For a 230-V connection with
conductors per slot,
circuits, and stator slots, the number of se-
ries-connected turns per phase
. Using the physical
dimensions of the machine, the peak fundamental air-gap flux
density is calculated to be
T when 60-Hz operation
and unit winding factor are considered.
For a six-phase machine with a double-layer winding,
(30)
With the same physical dimensions and same peak funda-
mental air-gap flux density, the flux per pole
is the same for
the six-phase machine when only the fundamental component
is considered. From (26), it is seen that the number of serial-
connected turns per phase must be the same for the six-phase
machine and, so, for keeping the same
, either has to be
doubled or
has to be reduced to its half. For this machine con-
figuration, the reduction of
guarantees the accommodation of
the conductors in the slots. Finally, the electric loading has to be
checked for the new winding configuration. The electric loading
for the machine is determined by
(31)
with all geometrical variables in (30) constant, the reduction in
the number of circuit by half causes the stator current to reduce
by half. This is reasonable since now there are two three-phase
windings instead of one in the baseline machine.
To measure theflux distribution in theair gap, afull pole pitch
search coil [14] was inserted in the machine stator and, with the
aid of an integrator, the flux can be determined.
Fig. 5. Air-gap flux density distribution without third harmonic current
injection. Peak phase current
I
=
0
:
5
A.
Fig. 6. Air-gap flux density distribution with third harmonic current injection.
Peak phase current
I
=0
:
5
A. Peak third harmonic phase current
I
=0
:
4
A.
Fig. 7. Air-gap flux density distribution with third harmonic current injection.
Peak phase current
I
=0
:
58
A. Peak third harmonic phase current
I
=
0
:
4
A.
IV. FINITE-ELEMENT ANALYSIS
Finite-element analysis is conductedin the six-phase machine
to calculate the air gap flux distribution and to demonstrate the
