Anais do XIX Congresso Brasileiro de Automática, CBA 2012.
ANALYSIS AND MODELING OF SIX-PHASE INDUCTION MOTOR UNDER OPEN PHASE FAULT
CONDITION
REGINALDO S. MIRANDA , EVANDRO C. GOMES – IEEE MEMBER
Electrical and Electronics Department, Federal Institute of Education, Science and Technology - IFMA
São Luis, Maranhão, Brasil
E-mail: rmiranda@ifma.edu.br/evandrogomes@ifma.edu.br
Abstract This paper presents the modeling of six-phase induction motor under open-phase fault condition. The analyses of
three different ways to model the fault condition are presented. The models are suitable for faults occurring in a single phase or
more than one. These models illustrate the existence of a pulsating torque when phases are opened. Simulation and experimental
results are provided to confirm the ability of these models to represent the fault condition.
Keywords Fault condition, six-phase machine, modeling.
Resumo Este artigo trata da modelagem de um motor de indução de seis fases sob condição de falta de fase aberta. São apresentadas três diferentes maneiras de modelar a condição de falta. Os modelos são adequados para representar a falta de uma ou
mais fases. Estes modelos permitem ilustrar a existência de oscilações no conjugado quando as fases estão abertas. Resultados
de simulação e experimental são fornecidos para confirmar a habilidade dos modelos em representar a condição de falta.
Palavras-chave Falta de fase, máquina de seis fases, modelagem.
1
Introduction
A higher degree of freedom can be achieved in an
electric drive system when the machine is changed
for another with a number of phases greater than
three. For applications where reliability is very important, the use of multi-phase systems has emerged
as a very feasible option. In particular, with loss of
one or more stator phases, a multiphase induction
machine can continue to be operated with an appropriate post-fault strategy.
Historically, the interest in multi-phase machines
was motivated by the possibility of dividing the output power. Then, reducing the torque oscillations
presented in drives with three-phase machines powered by six-step inverters. Among other reasons, the
fault tolerance is the most important. Currently, the
multi-phase machines can be used in high and low
power applications and the problem with torque oscillations can be solved with the PWM inverter. The
fault tolerance is still quite relevant.
During last decade is evident a large number of
publications on multi-phase machines. Many papers
reviewing the state of art in this type of drive system
show the interest in this issue, (Levi, 2008), (Singh,
2002), (Jones and Levi, 2002), (Bojoi, at al., 2006)
and (Levi, at al., 2007).
Among different configurations of multi-phase, a
six-phase induction machine (IM6P), whose stator
comprises two sets of three-phase coils displaced
from each other in an angle , is widely discussed in
the literature.
The six-phase machine modeling under normal
and fault conditions was reported in several papers.
The modeling is presented didactically in (White and
ISBN: 978-85-8001-069-5
Woodson, 1959). The analysis can be done similarly
to the three-phase case, i.e. the phase variable model
is transformed into another reference with new variables using a real or complex matrix. Nelson and
Krause presented the six-phase machine model with
arbitrary displacement between multiple winding sets
(Nelson and Krause, 1974).
A dqo model for six-phase induction machine including leakage inductance coupling was presented
by Lipo in (Lipo, 1980). (Zhao and Lipo,1995) used
the vector space decomposition approach to model a
six-phase induction machine. This same approach
was used by (Hadiouche, at al., 2000). The technique
turns the six-dimensional space of the machine into
three orthogonal subspaces of two dimensions. (Zhao
and Lipo, 1996), also presented another model to
represent the dynamic behavior of the machine under
unbalanced conditions. In (Pant, at al., 1999) is proposed a generalized model for analyzing the sixphase machine under balanced and unbalanced conditions (open circuit and short circuit). (Kianinezhad, at
al., 2008) proposed a general model to represent the
machine under faults.
Many papers have focused on the development of
fault-tolerant AC multi-phase drive systems. The
available degrees of freedom that exist in multiphase
machines are effectively utilized for an appropriate
post-fault operating strategy (Xu and Tolyat, 2002),
(Fu and Lipo, 1993), (Miranda, at. al., 2005) and
(Janhs, 1980)
Any proposed control technique with fault tolerance must be preceded by the behavior analysis of
system during fault. This analysis is usually performed by simulation using appropriate models.
Thus, this paper analyzes the effect of faults on dynamic behavior of the machine using models under
1408
Anais do XIX Congresso Brasileiro de Automática, CBA 2012.
fault condition. Three methods of open phase fault
representation are presented.
2 Six-phase motor modeling under balanced and
fault conditions.
mutual inductance between the stator and rotor and P
is the number of pair of poles.
Note that only dq plane is related to electromechanical energy conversion of the machine, while the
variables in other subspaces do not contribute to
torque production and they are only related to resistance and leakage inductance of the stator.
2.1 Model for balanced operation.
The motor considered in this study is depicted in
Figure 1. The stator has six phases divided into two
sets of symmetrical three-phase winding (with phase
shift between phases of 120º) separated by an angle
. This angle can take values such as 0º, 30º and 60º.
The rotor is a squirrel cage type and it can be modeled with six or three phases. In this study, the rotor
is considered equivalent to a three-phase winding.
The dynamic analysis of the motor is based on
space vector notation (Holtz and Springob, 1996) in
order to simplify the representation of multivariable
model reducing the order of the system in a half. A
vector decomposition is applied to the original sixdimensional system which is decomposed in three
orthogonal subspaces called (d,q), (x,y) e (o,o’).
Assuming a fixed reference frame of the stator, the
mathematical model that describes the dynamic behavior of the six-phase induction motor can be written as (White and Woodson, 1959).
v sdq  rs i sdq 
d
λ sdq
dt
Figure 1. Structure of a six-phase induction machine.
The dqxyo o stator variables in the model, depicted in Figure 1, can be determined from phase
variables, 123456, using the transformation equation:
(1)
w s135246  P6 w sdqxyoo '
(9)
v rdq  rr i rdq 
(2)
λ sdq
(3)
w s135246  ws1 ws3
ws5
ws 2
ws 4
λ rdq  lsr i sdq  lr i rdq
(4)
w sdqxyoo  wsd
wsq
wsx
wsy
wso wso
d
i sxy
dt
d
v soo '  rs i soo '  Lls i soo '
dt
d
v ro  rr iro  Llr iro
dt
Te  Plsr (isq ird  isd irq )
(5)
and
(6)
1

  12

1   12
P6 

3  c1

 c2
c
 3
0
1
0
2
2
3
2
 12
3
2
2
2
d
λ rdq  jr λ rdq
dt
 ls i sdq  lsr i rdq
v sxy  rs i sxy  Lls
with
(7)
(8)
where vsdq = vsd + jvsq, isdq = isd + jisq and sdq = sd +
jsq are the voltage, current and stator flux vectors in
dq, respectively; vsxy = vsx + jvsy, isxy = isx + jisy and
sxy = sx + jsy are the voltage, current and stator flux
vectors in xy, respectively, vsoo’ = vso + jvso’, isoo’ = iso
+ jiso’ and soo’ = so + jso’ are the voltage, current
and stator flux vectors in oo´, respectively (the variables of the rotor are identified with the subscript r);
Te is the electromagnetic torque; r is the angular
velocity of the rotor, rs and rr are the resistances of
stator and rotor respectively; ls = Lls +3Lms, and lr =
Llr +3Lms are the self-inductances of the stator and
rotor respectively, Lls and Llr are the leakage inductance of the stator and rotor respectively; lsr is the
ISBN: 978-85-8001-069-5


 12

3
2
2
2
s1
c1
 s1
2
2
s2
c2
 s2
2
2
s3
c3
 s3
2
2
3
2
ws 6 T ,
T


2

2

2
2 
 (10)
 22 

 22 
 22 
2
2
and
c1 = cos(), c2 = cos(2/3 + ), c3 = cos(4/3 + ),
s1 = sin(), s2 = sin(2/3 + ) e s3 = sin(4/3 + ).
The mechanical equation completes the dynamic
behavior of the six-phase motor:
J
dm
 Bm  Te  TL
dt
(11)
1409
Anais do XIX Congresso Brasileiro de Automática, CBA 2012.
flux components due to currents in others stator and
rotor windings.
If the switch S1 is opened (i1 = 0), the voltage equation becomes:
vsf1 
M
k
Figure 2. Block diagram of a six-phase induction motor.
Equations (1) to (8) and the mechanical equation
can be represented graphically by block diagrams as
depicted in Figure 2. The dashed line identifies the
electrical equations. The neutral line of both threephase sets is assumed to be separated. In this condition, the plane variables oo’ are naturally zero and
they are not presented in Figure 2. Note that dq model is identical to the three-phase motor. The difference is the branch defined by xy axes. This branch
produces very high currents excited by a vxy voltage.
The diagram depicted in Figure 2 is used here as a
reference to develop the phase fault operation models.
2.2 Fault Analysis
dik
dt
(14)
Equation (14) defines the floating point potential
in the opened phase s1. Note that in normal operating
condition (12), the voltage vs1 is a known input signal.
Thus, a solution to flux differential equation is obtained. However, this situation is no longer valid in
the fault condition in (14), because vsf1 is unknown.
In this way, it is necessary to develop models that
take the fault condition in account.
In the next sections, three models for simulation of
phase fault condition will be analyzed. In following
discussions, only one opened phase will be considered and the phase s1 is chosen in all cases. But the
event can be extended to more than one opened
phase.
2.2 Case I (Model using variable resistance)
The first case presents the fault condition in a
simple way. A variable resistance is inserted in series
with the phase where the fault occurs. In this case, the
resistor is the opening element in the phase. The
resistor R replaces the switch S1 in fault condition.
The procedure is illustrated in Figure 4.
The absence of phase can be represented generically by an open switch in series with the phase. This
situation is illustrated in Figure 3.
Figure 4. Representation of a phase fault using resistor.
The voltage phase in normal and fault condition
can be represented by
Figure 3. Representation of a phase fault through open switch.
The voltage equation in the phase with the closed
switch S1 can be represented by:
vs1  vsf1  R1is1 
d s1
dt
(12)
where vsf1 , is the phase voltage, is1 is the phase current, R1 is the resistance of the winding and,
s1  L1is1 
 Mi
k
(13)
k
is the linkage flux of the winding. L1 is the selfinductance of the winding and Mik represents the
ISBN: 978-85-8001-069-5
vsf1  vs1  Ris1
(12)
The resistor R has zero value in normal operation
condition and this value tends to infinity in fault
condition. Therefore, the current will be zero during
fault. The increase in resistance R represents an increase in phase resistance and this change of the
phase resistance has a direct effect on the system time
constant, i.e., the time constant becomes too small,
increasing the simulation time inevitably.
The block diagram, in Figure 5, represents the
fault condition described in case I. The block IM6P
corresponds to the block enclosed by the dashed line
in Figure 2.
1410
Anais do XIX Congresso Brasileiro de Automática, CBA 2012.
for the case in which the phase s1 is opened in the sixphase machine with  = 60°.
The application of this transformation in the phase
equations of the machine yields a model with new
variables defined by the same voltages equations (1)
and (2), related to dq stator and rotor planes respectively. The voltage equation in xyo plane is defined
by:
Figure 5. Model for operation with open phase using a resistor as
an opening element.
v sxyo  rs i sxyo  Lls
2.2.2 Case II (Model using a new transformation
matrix)
The second case was proposed by Lipo and Zhao
(Lipo and Zhao, 1996). The authors applied this
methodology for a six-phase motor with  = 30°
considering only one opened phase. Later,
Kianinezhad (Kianinezhad et. al, 2008) extended this
analysis to all fault possibilities with different values
of . The idea is obtain a new motor model to represent the fault condition. It is not difficult to show that
the voltage equations can still be represented by the
complex space vector. However, flux linkage equations are not symmetrical. The procedure is described
below.
Initially, only five phases are considered in operation during fault, yielding a five-dimension system. A
new transformation matrix, P5, is defined to stator
variables in order to obtain a new model. This matrix
splits the system into two mutual orthogonal subspaces called (d,q) and (x,y,o). Analogously to the original case, the subspace dq will be related to electromechanical energy conversion, while xyo will be only
related to the loss, i.e., the five remaining phases
produce the MMF equivalent to that produced by two
windings centered on the d and q axis with a 90º
phase shift.
The decomposition matrix for rotor variables of
the machine remains the same as the transformation
for balanced operation because the rotor still maintains a balanced winding structure.
A new variable transformation is defined as
(Kianinezhad, 2008)
w s 35246  P5 w sdqxyo
(13)
with
T
w s35246  ws3 ws5
ws 2
ws 4
ws6  ,
w sdqxyo   wsd
and
wsx
wsy
wso 
wsq
T
 0,3536 0,5 0, 4102 0, 2265 0, 6367 
 0,3536 0,5 0, 0898 0, 7735 0,1367 

 (14)
P5   0,3536 0,5 0,5704 0,5469 0, 0235


 0, 7071 0, 0 0, 6602 0, 2265 0,1133 
 0,3536 0,5
0, 25
0, 0
0, 75 

ISBN: 978-85-8001-069-5
(15)
d
i sxyo
dt
The difference with the original model is presented in the flux equations. The model for balanced
operation is symmetrical and this symmetry is lost in
a fault condition, i.e., the d and q axis inductances are
not the same. The flux equations are represented by
new variables such as
sd  lsd isd  M d ird
(16)
sq  lsq isq  M q irq
(17)
rd  M d isd  lr ird
(18)
rq  M q isq  lr irq
(19)
where the difference in the inductance values is expressed by
lsd  Lls  2 Lms ; lsq  Lls  3Lms
Md 
2, 447
2
Lms ; M q 
3
2
Lms
The self inductance of the rotor, lr, for dq axes
remains symmetrical, since the rotor model is the
same as the pre-fault condition. The equations (16)(19) suggest that the machine model in dq plane is a
two phase machine model with asymmetrical dq
windings.
The torque can be now represented by new variables such as (Lipo and Zhao, 1996)
Te  P(M q isqird  M d isd irq )
(20)
This equation can be used to predict torque oscillations under faulty conditions such as open phases.
An analytical development is presented in
(Kianinezhad, at al., 2008). The developed equation
sets in this section can be represented graphically
using a block diagram for simulation of this model.
This block diagram is depicted in Figure 6 to emphasize the difference in the model without fault.
1411
Anais do XIX Congresso Brasileiro de Automática, CBA 2012.
vs1 
vs1 
1
3
1
3
[(ls  lls )
(vsd  vsx )
(23)
d
d
isd  lsr ird )
dt
dt
(24)
Applying (24) in the motor model, ensure is1 = 0
in the steady state condition. The new topology to
simulate fault condition is depicted in Figure 8.
Figure 6. Block diagram of a six-phase induction motor with an
open phase.
The model topology used for simulation in fault
condition is depicted in Figure 7.
Figure 7. Model for operation with open phase using new variables to fault condition.
Figure 8. Model for operation with open phase using additional
voltage.
In this procedure is not necessary to design the
motor model again because the block IM6P, in Figure 8, continues to be defined as same as in Figure 2.
But, the way how the motor is supplied has changed.
Furthermore, this model does not increase the constant time like the variable resistance model.
3 Simulation and Experimental Results
2.2.3 Case III (Model using a new voltage phase)
In this section, a new voltage phase is defined under fault condition in order to simulate the phase
opening in a six-phase motor. The simulation of the
machine with an opened phase is performed assuming
a constraint of zero current phase. The opened phase
equation is defined using the machine equations for
this condition. This voltage is applied to the motor
and it must maintain a zero current during a fault.
This procedure is presented below considering the
machine model with double neutral defined in the
reference frame stator. Consider the phase 1 is open
while the other phases are connected to the motor.
The restriction for this condition is represented by
is1 = 0. The application of the transformation equation (9) defines the relationships between the d and x
axis currents like
isx  isd
(21)
Applying (21) in (5) and using only the real part
of resultant equation, the x axis equation can be written as (22) due to condition (21).
vsx  rs isd  Lsl
disd
dt
(22)
Considering the fault, a new voltage phase can be
obtained applying (22) in the transformation (9). This
voltage equation has the following form:
ISBN: 978-85-8001-069-5
The models presented in this paper were simulated using the package MATLAB/Simulink. The implementation is based on the block diagrams in Figures 2, 5, 6, 7 and 8 arranged directly in Simulink.
The motor used in the simulations is a six-phase
motor,  = 60o, 1 cv, 4 poles, 220V, 60Hz, whose
parameters are presented in Table 1. The simulations
have been performed at a low stator electrical frequency, at 20 Hz, in order to have a better visualization of the results.
Torque, velocity and phase currents of the motor
for case I are presented in Figure 9 (variable resistance). The electromagnetic torque presents oscillations when the motor operates with phase fault and
the frequency of these oscillations is twice the frequency of the currents. The speed also oscillates, but
with lower amplitude due to low-pass filter characteristic of the mechanical system.
The results for case II (fault model) are presented
in Figure 10. Only torque and speed are presented in
steady state and the similarity of the results can be
noted. The results for case III (new voltage) are presented in Figure 11. Again the results present in fact
the system behavior during a phase fault. Only torque
and velocity are presented for these results. There is a
slight difference in the fault start up. The other results
are all very similar to previous cases.
An experimental result is shown in Figure 12. The
result was obtained in the same working conditions of
simulations, i.e., the motor has the same parameters
and the same opened phase s1. However, the source
1412
Anais do XIX Congresso Brasileiro de Automática, CBA 2012.
used in experimental result is a six-phase inverter
while it was considered an ideal source in simulation.
The result of Figure 12 has to be compared with
those obtained by simulation of Figure 9 and they are
very similar. The experimental result confirmed the
validity of the presented analysis.
Table I. Motor Parameters
1 cv, 4 poles, 220V, 60Hz, six phase motor ( = 60o)
Parameters
Valor
12,5
rs []
8,9
rr []
Lsl [H]
0,031
Lrl [H]
0,031
lsr [H]
1,39
(a)
(b)
(a)
Figure 10. Simulation result with opened phase s1 (case II): (a)
Torque, (b) velocity.
(b)
(a)
(c)
Figure 9. Simulation result with opened phase s1 (case I): (a)
Torque, (b) velocity and (c) currents.
(b)
Figure 11. Simulation result with opened phase s1 (case III): (a)
Torque, (b) velocity.
ISBN: 978-85-8001-069-5
1413
Anais do XIX Congresso Brasileiro de Automática, CBA 2012.
Figure 12. Experimental result with opened phase.
4 Conclusion
In this paper, the operation of the six-phase induction motor with an opened phase was investigated.
Three ways to simulate the fault condition were presented. The results demonstrate the ability of the
models to represent the effect of the fault. The result
obtained from case I has a simulation time much
larger than other cases. The model of case II is quite
general, allowing the opening of other phases only
defining the new transformation matrix. However, the
model does not present the transition between the pre
and post-fault conditions. The model in case III has a
fast simulation and since it uses the same motor model in the pre-fault condition, it can present the transition between operation modes. Although the analysis
conducted in this study considers only one opened
phase, the models can be generalized to a larger
number of opened phases. An experimental result
have been presented to demonstrate the feasibility of
the models
References
Levi, E. N (2008). Multiphase Electric Machines for
Variable-Speed Applications. IEEE Transactions
on Industrial Electronics, Vol. 55, No. 5, pp.
1893 - 1909.
Singh, G. K. (2002), Multi-phase induction machine
drive research - A survey, Electr. Power Syst.
Res., vol. 61, no. 2, pp. 139–147.
Jones, M. and Levi, E. (2002) , A literature survey of
state-of-the-art in multiphase AC drives, in Proc.
UPEC, Stafford, U.K., pp. 505–510.
Bojoi R., Farina, Profumo, F. F. and Tenconi, A.
(2006), Dual-three phase induction machine
drives control - A survey, IEEJ Trans. Ind.
Appl., vol. 126, no. 4, pp. 420–429.
Levi, E., Bojoi, R., Profumo, F., Toliyat, H. A., and
Williamson, S. (2007), Multiphase induction motor drives - A technology status review, IET
Electr. Power Appl., vol. 1, no. 4, pp. 489–516.
ISBN: 978-85-8001-069-5
White, D.C., and Woodson, H.H. (1959), Electromechanical energy conversion. John Wiley and
Sons, New York, NY.
Nelson, R. H.; Krause, P. C. (1974), Induction machine analysis for arbitrary displacement between multiple windings sets. IEEE Trans. on
Power App. and Syst., v. 93, n. 3, p. 841–848.
Lipo, T. A. (1980), A d-q model for six phase induction machine. In: Rec. Int. Conf. Eletric Machine,Greece. pp. 860–867.
Zhao, Y.; Lipo, T. A. (1995), Space vector pwm
control of dual three-phase induction machine
using vector space decomposition. IEEE Trans.
Ind. Applicat., v. 31, n. 5, p. 1100–1109.
Hadiouche, D., Razik, H., and Rezzoug, A. (2000),
Modelling of a double-star induction motor with
an arbitrary shift angle between its three phase
windings. Proc. 9th Int. Conf. on Power Electronics and Motion Control PEMC, Kosice, Slovakia, pp. 5.125–5.130.
Zhao, Y. and Lipo, T. A. (1996), Modeling and control of a multi-phase induction machine with
structural unbalance. Part I: Machine modeling
and multidimensional current regulation, IEEE
Trans. Energy Convers., vol. 11, no. 3, pp. 570–
577.
Zhao, Y. and Lipo, T. A. (1996), Modeling and control of a multi-phase induction machine with
structural unbalance. Part II: Field-oriented control and experimental verification, IEEE Trans.
Energy Convers., vol. 11, no. 3, pp. 578–584.
Pant, V., Singh, G. K. and Singh, S. N. (1999), Modeling of a multi-phase induction machine under
fault condition. In: Conf. Rec. IEEE PEDS. v.
11, p. 92–97.
Kianinezhad, R., Nahid-Mobarakeh, B., Baghli, L.,
Betin, F., and Capolino, G. A. (2008), Modeling
and Control of Six-Phase Symmetrical Induction
Machine Under Fault Condition Due to Open
Phases, IEEE Transactions On Industrial Electronics, Vol. 55, No. 5, pp. 1966 – 1977.
Xu, H., and Toliyat, H.A. (2002), “Resilient Current
Control of Five-Phase Induction Motor under
Asymmetrical Fault Conditions”, Proceedings of
the 2002 Applied Power Electronics Conference
and Exposition (APEC 2002), March 10-14, Dallas, TX, pp. 64-71.
Fu, J. R. and Lipo, T. A. (1993), “Disturbance Free
Operation of a Multiphase Currente Regulated
Motor Drive with an Opened Phase”, IEEE
Transactions on Industry Applications, vol. 30,
no. 3, pp. 1267-1274.
Miranda, R. S., Jacobina, C. B., Lima, A. M. N., and
Rossiter, M. B. (2005), Operação de um sistema
de acionamento com motor de seis fases tolerante
a faltas”, Revista Brasileira de Eletrônica de Potência - SOBRAEP, v. 10, n. 1, pp. 15–22.
Jahns, T. M. (1980), Improved reliability in solidstate AC drives by means of multiple
independent phase-drive units, IEEE Trans. Ind.
Appl.,vol. IA-16, no. 3, pp. 321–331, May/Jun.
1414