1968 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO.

6, NOVEMBER/DECEMBER 2012

Sensorless Vector Control of PM Synchronous

Motors During Single-Phase Open-Circuit
Faulted Conditions
Alberto Gaeta, Student Member, IEEE, Giacomo Scelba, Member, IEEE, and Alfio Consoli, Fellow, IEEE

Abstract—This paper deals with the study and development of Although it is expected that sensorless vector control strate-
new control solutions of permanent-magnet synchronous motor gies adopted with a healthy drive must be able to operate even
(PMSM) drives specifically aimed to operate under sensorless during fault conditions, in practice, they fail in the presence of
vector control even during fault conditions. The phase imbalance
produced by an open-phase fault leads to the failure of several such faults as single-phase open circuit. This abnormal working
sensorless estimation algorithms based on either machine models operation occurs because of the failure of one phase connection
or high-frequency signal injection. Exploiting a recently proposed or because of inverter faults mitigated by using fuses or the
machine model for three-phase faulted PMSM drives, the practical integrated protection of the gate drivers. Most of the several
implementation of sensorless vector controls for such drives under solutions proposed in the past to alleviate different drive and/or
asymmetric conditions due to a single-phase open-circuit fault is
performed. The effects of the magnetic/geometrical asymmetries machine faults [8]–[12] need at least the accessibility to the
during the fault are studied, and the techniques able to mitigate neutral point of the stator windings. Moreover, additional el-
such effects on the rotor position information are provided. The ements and power devices are required to modify the hardware
rotor position estimation is achieved in a wide operating range configuration due to the fault, and this clearly leads to an
during the fault as it is confirmed by experimental tests. increased cost for the entire drive system.
Index Terms—Fault-tolerant systems, models of electrical In this paper, a simple drive topology is considered, including
drives, permanent-magnet machines, sensorless control, variable- an inverter configuration with the neutral point of the machine
speed drives. connected either to the middle point of the dc bus or at the
output of a fourth inverter leg. The converter includes short-
I. I NTRODUCTION circuit protections such as high-speed fuses on each inverter leg
that, by means of hardware-based solutions, are forced to blow
A S permanent-magnet synchronous motors (PMSMs) ap-
pear the best effective and efficient option to favor the
adoption of electrical drives in several emerging applications,
whenever a short or open circuit of the power devices occurs
[10], [11]. Therefore, the faulted inverter leg can be isolated,
leaving the electrical machine to operate in a two-phase mode,
suitable control algorithms are needed to ensure proper op-
similarly to the case of disconnection of a motor phase.
eration of the drive both in healthy and faulted conditions.
The mathematical representation of the healthy machine,
This applies to any controlled configuration of the drive, in-
based on the assumption of a symmetrical and balanced
cluding sensorless vector control. In fact, to suitably fulfill
three-phase system, is no longer appropriate to describe the
the increasing demand for reduced drive cost and improved
unbalanced faulted conditions due to the open-phase circuit.
system reliability, most PMSM drive manufacturer proposes
Consequently, the equations used by model-based rotor position
for medium–high-performance applications to adopt sensorless
estimation techniques during healthy operation of the machine
vector control techniques during the drive healthy conditions
yield wrong estimates. Similarly, carrier-based estimation al-
[1]–[7].
gorithms that rely on the injection and demodulation of high-
frequency (HF) signals also fail after the fault as the additional
Manuscript received December 31, 2011; revised April 10, 2012; accepted voltages cannot be applied to the stator phases unless the
April 23, 2012. Date of publication October 23, 2012; date of current version voltages applied to the healthy phases are modified. Therefore,
December 31, 2012. Paper 2011-IDC-576.R1, presented at the 2011 IEEE
Energy Conversion Congress and Exposition, Phoenix, AZ, September 17–22,
although several papers have focused on the modeling and
and approved for publication in the IEEE T RANSACTIONS ON I NDUSTRY control of faulted PMSMs [8]–[12] and, particularly, a new
A PPLICATIONS by the Industrial Drives Committee of the IEEE Industry model for unbalanced single-phase open-circuit operation has
Applications Society. This work was performed in the frame of the project
“Azionamenti elettrici ad elevata robustezza di funzionamento,” funded by the
been presented to perform an effective field-oriented vector
Italian Ministero dell’Istruzione, dell’Università e della Ricerca - Progetti di control (FOVC) [13], very limited efforts have been made in
Ricerca di Interesse Nazionale 2008. order to achieve sensorless estimation during faulted conditions
A. Gaeta and G. Scelba are with the University of Catania, 95125 Catania,
Italy (e-mail: alberto.gaeta@diees.unict.it; gscelba@diees.unict.it). [14]–[18].
A. Consoli, deceased, was with the University of Catania, 95125 Catania, In order to overcome any limitation to practical implementa-
Italy. tion of sensorless techniques during faults, this paper will show
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org. that suitable modifications of the standard sensorless rotor posi-
Digital Object Identifier 10.1109/TIA.2012.2226192 tion algorithms are necessary during faulted operation and they

0093-9994/$31.00 © 2012 IEEE

GAETA et al.: VECTOR CONTROL OF PM SYNCHRONOUS MOTORS DURING OPEN-CIRCUIT FAULTED CONDITIONS 1969

can be introduced with limited theoretical and implementation Such a model can be rewritten in orthogonal reference
efforts in the FOVC under open-phase conditions performed frames, stationary or synchronous with the rotor flux position,
through the model presented in [13]. Different sensorless esti- by applying respectively the traditional abc/qd0s Kss and the
mation methods, either model-based or carrier-signal-injection- qd0s /qd0r s Krs matrix transformations [19]
based, will be considered in the following study, highlighting ⎡2 ⎤ ⎡ ⎤
3 − 21 − 12 cos(θr ) − sin(θr ) 0
the main differences of implementation of each technique in
Kss = ⎣ 0 − √3 √3 ⎦ s Krs = ⎣ sin(θr ) cos(θr ) 0 ⎦ .
1 1
case of healthy and faulted operations through experimental 1 1 1 0 0 1
tests performed on a laboratory system. It is important to point 3 3 3
(3)
out that, even if the proposed methodology will be applied in
this paper only to a limited number of sensorless techniques, In qdos coordinates, the machine equations are
it can be straightforwardly extended to other encoderless
s d s
algorithms. vqdo = Kss vabc
s
= rss isqdo + Λ (4)
dt qdo
Λsqdo = Kss Λsabc = Lsqdo isqdo + Msqdo λpm (5)
II. P ERMANENT-M AGNET S YNCHRONOUS
M ACHINE M ODEL where
A. Healthy Operation rss = diag(rs ) (6)
Excluding saturation and other nonlinearities, the model Msqdo = [ sin(θr ) cos(θr ) 0 ] T
(7)
of a PMSM in balanced and symmetrical conditions can be ⎡ Ld +Lq ⎤
represented in an abc stationary reference frame by 0 0
2 Lq − Ld
Lsqdo =⎣ 0 Ld +Lq
0 ⎦+
d 2 2
vsabc = rs isabc+ Λsabc 0 0 Lls
dt ⎡ ⎤
Λsabc = Labc isabc + Mabc λpm (1) cos [2(θr )] − sin [2(θr )] 0
× ⎣ − sin [2(θr )] − cos [2(θr )] 0 ⎦ . (8)
where isabc , vabc
s
, and Λsabc are the stator currents, voltages, and 0 0 0
fluxes, respectively, rs = diag(rs ) is the stator resistance, the
The electromagnetic torque includes two contributions, indi-
terms of the matrix Labc are
  cated as Ter and Tem , due respectively to the reluctance effect
Ld + Lq − 2Lls and the permanent-magnet excitation
Lhh = Lls +
3 3p T
  Ter = (Ld − Lq ) isqdo
Ld − Lq 24
− cos [2(ϑr + α)] ⎡ ⎤
3 sin(2θr ) cos(2θr ) 0 
  × ⎣ cos(2θr ) − sin(2θr ) 0 ⎦ isqdo (9)
1 Ld + Lq − 2Lls
Lkh = Lhk = − 0 0 0
2 3 ⎡ ⎤
  3p T cos(θr )
Ld − L q Tem = λpm isqdo ⎣ − sin(θr ) ⎦ . (10)
− cos [2(ϑr + γ)] 22
3 0
⎧ ⎧ π
⎨ 0, h=a ⎨ − 3 , (h, k) = (a, b) By applying s KrS , the model of the machine in the reference
α = − 23 π, h = b γ = π, (h, k) = (b, c) (2)
⎩ 2 π, h = c ⎩π frame synchronous with the rotor flux is obtained
3 3 , (h, k) = (a, c)
r
vqdo = s Krs vqdo
s
and the matrix Mabc is ⎡ ⎤
    T 0 1 0
d
2
Mabc = sin(2θr ) sin 2θr − π
2
sin 2θr + π . = rrs irqdo + ωre ⎣ −1 0 0 ⎦ Λrqdo + Λrqdo (11)
3 3 dt
0 0 0
Note that Lls is the leakage inductance, Lq and Ld are the Λrqdo = s Krs Λsqdo = Lrqdo irqdo + Mrqdo λpm (12)
q- and d-axis synchronous inductances, λpm is the permanent-
where the matrices are
magnet flux, and θr is the rotor position; the pole numbers will
be indicated with p, the rotor speed will be indicated with ωr , rrs = diag(rs ) (13)
and ωre = (p/2)ωr .
The angular positions α and γ indicate the spatial dis- Mrqdo = [ 0 1 0 ]T (14)
placement of the stator-phase inductances, while the sub- ⎡ Ld +Lq ⎤ ⎡ ⎤
0 0 1 0 0
script letters h and k represent the phase indexes. In the 2 Lq −Ld ⎣
Lrqdo = ⎣ 0
Ld +Lq
0 ⎦+ 0 −1 0 ⎦ .
case of surface-mounted permanent-magnet synchronous ma- 2 2
0 0 Lls 0 0 0
chines (SMPMSMs), the equations are modified assuming that
L d = Lq . (15)
1970 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 6, NOVEMBER/DECEMBER 2012

Torque expressions are given by new reference frame, but the components on the two axes have
different amplitudes.
3p 3P
Tem = λpm irq Ter = (Ld − Lq )irq ird . (16) According to [13], by using the following two matri-
22 22 ces Rv and RI , an additional reference frame transforma-
tion synchronous with the rotor flux position is applied to
(19) and (20)
B. Open-Phase Fault Operation  
cos(θr + β2 ) − sin(θr + β2 ) T
As an open-phase fault occurs, assuming that the neutral RV = = (RV )−1
point of the stator windings is accessible, the model of the sin(θr + β2 ) cos(θr + β2 )
machine can be rewritten as proposed in [13], where a set of  
1 cos(θr + β2 ) −3 sin(θr + β2 )
new reference frame transformations has been used in order to (RI ) =
3 sin(θr + β2 ) 3 cos(θr + β2 )
provide an analytical representation of the machine based in a
 
q–d-axis reference frame synchronous with the rotor flux. In −1 3 cos(θr + β2 ) 3 sin(θr + β2 )
particular, a mathematical model of PMSM in a qdsu orthog- (RI ) = . (22)
− sin(θr + β2 ) cos(θr + β2 )
onal stationary reference frame during an open-phase fault is
first obtained by exploiting the matrix transformations A and The machine model in this new reference frame is
B, indicated in the following:  
  1 1  0 1 d
1 1
r r r
vqd = rs iqd + ωre Λrqd + Λrqd (23)
Λ= (A)−1 = 12 2
(17) −1 0 dt
1 −1 − 21
2
⎡  −1  ⎤ Λrqd = Lrqd irqd + Mrqd λpm (24)
β1 −β3
2 cos 0
⎢ 2 ⎥ 3p 
B=⎢  ⎥ Te = Ter + Tem = λpm irq + (Ld − Lq )irq · ird (25)
⎣  −1  ⎦ 22
β1 −β3
0 2 sin 2
rrs = 2 · diag(rs ) + rs E(θr ) Mrqd = [ 0 1 ]T (26)
⎡   ⎤  
2 cos β1 −β 2
3
0 Lq + Lls 0
(B)−1 = ⎣  ⎦ .
r
  (18) Lqd = + Lls E(θr ) (27)
0 Ld + Lls
0 2 sin β1 −β 3
2  
cos [2(θr + β2 )] sin [2(θr + β2 )]
The machine model in qdsu reference frame is E(θr ) = . (28)
sin [2(θr + β2 )] − cos [2(θr + β2 )]
d su
su
vqd = BAvsij = rsu su
s iqd + Λ (19) Compared to the equivalent model valid for healthy opera-
dt qd
tion, additional terms appear in the voltage and flux equations
Λsu
qd = BAΛsij = Lsu su su
qd iqd + Mqd λpm (20) that include the matrix E(θr ) and are proportional to the stator
resistance and leakage inductance, respectively [15]. Those
where
terms make the open-phase model, represented in the reference
 
sin(θr + β2 ) frame synchronous with the rotor position, dependent on the
rsu
s = diag(rs ) Msu
qd = rotor flux position.
cos(θr + β2 )
⎡ 1  Ld +Lq  ⎤ Differently than (23)–(28), previously presented models for
3 2 + 2Lls 0 faulty operation such as [11] assume that the machine is op-
Lsu
qd =
⎣  ⎦ erating at steady state; they also neglect the stator resistance
Ld +Lq
0 2 voltage drop and adopt a different set of reference frame
transformations. The final model proposed in [11] is in the
1 Ld − L q qdr coordinates and includes additional terms in the voltage
+−
2 3 equations that depend on the rotor position and are related to
 
cos [2(θr + β2 )] −3 sin [2(θr + β2 )] the speed, permanent-magnet flux linkage, and the magnetizing
× and leakage inductances of the machine.
− sin [2(θr + β2 )] −3 cos [2(θr + β2 )]
⎧ 2 
⎪ − π, 0, 23 π → open phase ‘a’ fault
⎨ 3 
(β1 , β2 , β3 ) = 0, π3 , 23 π → open phase ‘b’ fault (21) III. S ENSORLESS FOVC U NDER O PEN -P HASE FAULT
⎪
⎩ 
0, − 3 , − 3 π → open phase ‘c’ fault
π 2 The machine model representing the PMSM under open-
phase fault allows us to perform a medium–high-performance
s
where vij and Λsij are the stator voltage and flux vectors of the field-oriented control even during the fault with no modifi-
two healthy phases. cations of the current control structure. Moreover, the same
The coefficients β1 , β2 , and β3 are chosen according to the model can be used to improve the performance of rotor flux
open-phase fault condition, as indicated in (21). Note that qdsu position estimation algorithms, which fail during this fault
electrical stator quantities are still sinusoidal waveforms in this situation.
GAETA et al.: VECTOR CONTROL OF PM SYNCHRONOUS MOTORS DURING OPEN-CIRCUIT FAULTED CONDITIONS 1971

Fig. 1. Block diagram of the fault-tolerant vector control.

Basically, the FOVC tolerant to open-phase fault can be

implemented as shown in Fig. 1 [13], where an additional feed-
forward voltage drop compensation algorithm is included in
case of fault. Obviously, also under faults, the implementation
of FOVC requires the knowledge of the rotor flux position
θr [20]. However, θr must be determined by taking into account
the effects of the fault.
Specifically, as the stator-phase open circuit occurs, the
current of the faulted phase decays to zero, and as it has been
widely described in [8], in order to maintain the same load
condition, the stator-phase currents must be displaced √ by 60
electrical degrees with their amplitudes increased 3 times of
those before fault.
Based on the similarities between the proposed model, valid
for open-phase operation and expressed in rotor qdr coordi-
nates, and the model valid for balanced operation, expressed in
rotor qd0r coordinates, one can expect that the same sensorless
techniques can be used with minor modifications even after
an open-phase fault. Although this is almost completely true,
the particular structure of the proposed transformation matrices Fig. 2. Three-phase PMSM drive tolerant to open-phase fault.
and the presence in the novel model of additional coupling
inverter output voltage
terms must be taken into account, particularly when dealing  
⎧ s 2 s 2 
with HF injection-based sensorless techniques. Moreover, the ⎪
⎪ s 2
proposed model assumes only the existence of the fundamental ⎨ min viq + vjq + vnq
s
viq − vnq
s
= vins (29)
stator current harmonic during the fault, excluding any other ⎪
⎪
⎩ s
additional components. In practice, due to the unbalanced op- vjq − vnq
s s
= vjn
erating conditions, a third harmonic appears in the two healthy ⎡ s ⎤ ⎡ 2 ⎤
stator currents. This current harmonic is produced by a third- viq 3 − 13  s 
⎢ s ⎥ ⎢ 1 2 ⎥
vin
order harmonic of the back electromotive force (EMF), and its ⎣ vjq ⎦ = ⎣ − 3 3 ⎦ s
. (30)
effects can be mitigated in the vector control implementation, vjn
s
vnq −3 −3
1 1
as indicated in [18].
Another important issue deals with the reduction of the The relationship (30), coupled to the inverter topology mak-
vector voltage capacity under fault [10]. This drawback can be ing use of a fourth inverter leg, allows one to re-establish the
overcome by connecting the neutral point of the stator windings space vector voltage capacity of the healthy drive.
to an additional inverter leg (Fig. 2) and transforming the
s s
reference voltage signals vin and vjn into three new reference
s s s A. Estimation Strategies Based on a Machine Model
signals viq , vjq , and vnq representing the stator-phase voltages
Synchronous With the Rotor Position
referred to the middle point of the additional leg. The control
s
voltage signal of the additional leg vnq is the voltage potential The first sensorless technique (Method I) is based on an
variation of the neutral point of stator windings referred to the estimation algorithm that performs the tracking of the rotor
middle point of the dc bus. flux position by leading to zero the error eqd between the
The new reference signals (30) are determined by solving the calculations of the back EMFs in the correct and estimated
system (29) that assumes as a constraint the minimization of the reference frames [21]. This standard model-based sensorless
1972 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 6, NOVEMBER/DECEMBER 2012

technique has to be modified to take into account the differences Matrix products can be simplified as follows:
between the motor model valid under open-phase operation    
and the model for balanced operation. In fact, as for the 0 1 −1 0 1
R R =
model of open-phase operation, by substituting (24) in (23), we −1 0 −1 0
obtain    
Eext cos δ
  R = Eext
r r r 0 1 0 sin δ
vqd = 2rs iqd + rs E(θr )iqd + ωre Lls E(θr )irqd  
−1 0 d −1 rest  d 0 1 rest
   R R iqd = irest + ω i
0 1 Lq + Lls 0 dt dt qd
δ
−1 0 qd
+ ωre ir  
−1 0 0 Ld + Lls qd 0 1
−1
    RE(θr )R = E(θrest )
0 Lq + Lls 0 d r −1 0
+ ωre + i
λpm 0 Ld + Lls dt qd RE(ϑr )R−1 = E(θrest )
d 
d  d
+ Lls E(θr )irqd . (31) R E(θr )R−1 irest = E(θrest ) irest + (ωδ − 2ωrest )
dt dt qd
dt qd
 
Adding to and subtracting from the q-axis equation the 0 1
expression × E(θrest )irest
−1 0 qd

dirq
(Ld + Lls ) + ωre (Lq + Lls )ird (32) where θrest = θr + δ and ωδ = (d/dt)δ.
dt
The final model, useful for rotor position estimation, can be
an equivalent qdr model synchronous with the rotor position is written as
obtained  
  Eext
cos δ rest
= vqd − 2rs irest
r r 0 1 r sin δ qd
vqd = 2rs iqd + ωre (Lq + Lls ) i
−1 0 qd  
  − ωre (Lq + Lls )
0 1 rest
i
d Eext −1 0 qd
+ (Ld + Lls ) irqd + + rs E(θr )irqd
dt 0
  d rest
0 1 d  + −(Ld + Lls ) i
+ ωre Lls E(θr )irqd + Lls E(θr )irqd dt qd
−1 0 dt  
0 1 rest
− (ωrest − ωre )(Ld + Lls ) i
(33) −1 0 qd

where d 
+ −rs E(θrest )irest
qd − Lls E(θrest )irest
qd
dt
d r  
Eext = −(Ld − Lq ) i + ωre [λpm + (Ld − Lq )ird ] 0 1
dt q + ωre Lls E(θrest )irest
qd . (36)
−1 0
is indicated as the extended EMF.
By using the reference frame transformation (34), (33) can be Assuming that the terms including Lls are negligible and
represented in a qdrest reference frame rotating at the estimated ωrest ∼
= ωre , the previous relationship can be rewritten as
rotor speed ωrest and lagging an error angle δ from the qdr    
reference frame eq cos δ
= Eext
  ed sin δ
cos(δ) − sin(δ) T  
R= = (R)−1 . (34) ∼ 0 1 rest
sin δ cos(δ) rest
= vqd − 2rs irest − +ω L i
qd re q
−1 0 qd
The result of this operation is d
− Ld irest − rs E(θrest )irest
qd . (37)
  dt qd
0 1
rest
vqd = 2rs RR−1 irest+ ωre (Lq + Lls )R R−1 irqd
qd
−1 0 Note from (37) that a mismatching between the real stator
  resistance and that used in (37) produces an additional ripple
d −1 rest  Eext
+ (Ld + Lls )R R iqd + R into the estimated rotor position, at double pulsation than the
dt 0
rotor speed, due to the term rs E(θrest )irest
qd . As proposed in
+ rs RE(θr )R−1 irest
qd
[21], from (37), it is possible to estimate the rotor flux position
  through a suitable tracking algorithm that maintains the error
0 1
+ ωre Lls R E(θr )R−1 irest δ to zero. This goal is obtained by driving to zero the error
−1 0 qd
eqd between calculations of the back EMFs in the correct and
d  estimated reference frames, according to the block diagram
+ Lls R E(θr )R−1 irest
qd . (35) in Fig. 3.
dt
GAETA et al.: VECTOR CONTROL OF PM SYNCHRONOUS MOTORS DURING OPEN-CIRCUIT FAULTED CONDITIONS 1973

B. Estimation Strategies Based on Stationary Reference

Frame Machine Model
In case the adopted sensorless estimation method exploits a
machine model calculated in the orthogonal stationary refer-
ence frame [22], [23], the model for faulty operation described
by (19)–(21) can be used. Usually, stationary reference frame
models are aimed to achieve the rotor flux position estimation in
nonsalient PMSMs, where Lq = Ld . In that case, from (4)–(8)
appears that the back-EMF terms into the voltage equations
are due only to the permanent-magnet flux, as Lsqd includes
constant terms. Hence, estimation of the rotor position can be
Fig. 3. Model-based sensorless estimation—Method I. easily achieved by integration of the voltage equations.
During an open-phase fault, the model of the machine again
needs to be improved and substituted by a model represented
in the qdsu reference frame that can be adapted to most EMF-
based methods. As in such a model, the q- and d-axis electrical
quantities have different amplitudes, this could refrain from
applying it in those sensorless algorithms that exploit both
model and carrier injection to achieve the rotor position [24].
However, this restriction can be overcome by defining an
analytical model of the faulted machine that is symmetrical
even in a stationary reference frame. Such a goal can be
achieved considering that the proposed model in rotor qdr coor-
dinates has been obtained by using two different transformation
Fig. 4. ωr = 60 rad/s 40% Te_rated : CH1: Estimation error [7.2◦ /div]. matrices for voltages/fluxes and currents in order to remove in
CH2: ωr [40 rad/s/div]. Time: [0.10 s/div]. the model the asymmetry associated to the fault and obtain a
model structure similar to that of the machine during healthy
operation. In order to preserve the model symmetry in the sta-
tionary qds reference frame, all variables must be obtained by
applying the inverse of the transformation Rv to the electrical
quantities in the rotor reference frame qdr . As for the voltages
     s
vqs vqr v
= R−1 −1
· r = RV · RV · B · A · is
vds V vd vj
 s
v
= B · A · is
vj
⎡  −1   −1 ⎤
Fig. 5. ωr = 60 rad/s 40% Te_rated : CH1: Estimation error [7.2◦ /div]. 2 cos β1 −β 3
2 cos β1 −β3
= ⎣ −1 ⎦
CH2: ωr [40 rad/s/div]. Time: [0.10 s/div]. 2 2
 −1  
β1 −β3 β1 −β3
2 sin 2 − 2 sin 2
In practice, the differences with the standard healthy imple-  s
mentation consist in the novel reference frame transformation v
· is . (38)
and in the analytical expression (37) of back-EMF quantities, vj
including the term rs E(θrest )irest
qd and the two-times resistance
voltage drop 2rs irest
qd . Similarly, for the currents
The dynamic behavior of the estimator is depending on the      
proportional–integral compensator gains and the bandwidth of isq ir is
= R−1 · rq = R−1 · RI · B · A · is
the low-pass filter [21]. isd V id V ij
As previously mentioned, the error between the stator re-  
∗ isi
sistance adopted in the model and the real value produces =R · B · A · s
additional ripples in the estimated rotor position, in the speed, ij
and, thus, in the torque during sensorless control implementa- ⎡  −1   −1 ⎤
2 β1 −β3 2 β1 −β3
cos cos
tion. Figs. 4 and 5 highlight this drawback, depicting the rotor =⎣ −1 ⎦
3 2 3 2
 −1  
β1 −β3
speed ωr and the rotor position estimation error during faulted 2 sin 2 − 2 sin β1 −β
2
3

operation; the additional stator resistance voltage drop term  s

related to E(θrest ) is neglected in Fig. 4 while it is included i
· is (39)
in Fig. 5. ij
1974 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 6, NOVEMBER/DECEMBER 2012

where vi , vj and ii , ij are the voltages and currents of the two

remaining working phases after fault, where
1 
−1 0
R∗ = RV · RI = 3 . (40)
0 1
s su
As from (38)vqd = vqd , the model of the machine in qds
coordinates can be obtained directly from the unbalanced qdsu
reference frame by applying the transformation R∗ only to isu
qd

d s Fig. 6. Model-based sensorless estimation—Method II.

s
vqd su
= vqd = rs (R∗ )−1 isqd + Λ (41)
dt qd
∗ −1 su
Λsqd = Λsu su s
qd = Lqd (R ) iqd + Mqd λpm (42)

where
 
3rs 0
rss = rs (R∗ )−1 = Msqd = Msu
qd
0 rs
∗ −1
Lsqd = Lsu
qd (R )
Fig. 7. HF injection technique for sensorless fault-tolerant drive.
⎡  ⎤
Ld +Lq
+ 2Lls 0 Ld − Lq C. Estimation Strategies Based on Carrier Signal Injection
=⎣  ⎦ + −
2
Ld +Lq 2
0 2 Essentially, sensorless algorithms based on injection of addi-
⎡   ⎤ tional sinusoidal HF (300 Hz ÷ 5 kHz) signals consist of two
cos[2(θr + β2 )] −sin 2 θr + β1 +β3
fundamental parts: injection and demodulation algorithms.
×⎣    ⎦.
2

−sin 2 θr + β1 +β 3
−cos[2(θr + β2 )] During healthy operation of the drive, voltage injection can
2
be equivalently performed in abc, in qd0s stationary reference
frames, or in the estimated d-axis direction. In the first case, a
Note that the transformations used in (38) and (39) and
balanced set of three sinusoidal HF voltages is injected, while
required to calculate the electrical quantities from the reference
in the second case, two-equal-amplitude 90◦ phase-shifted si-
frame ij s to qds do not depend on the rotor flux position.
nusoidal voltages are injected. In the last mentioned injection
Assuming that the effect of the leakage inductance is negligible,
technique, a single sinusoidal voltage is superimposed to the
the asymmetries of the model are reduced only to the stator
d-axis component of the fundamental voltage in the reference
resistance voltage drop.
frame synchronous with the rotor flux position.
The electromagnetic torque contributions in this new refer-
Under open-phase fault, the same voltage injection technique
ence frame are given by
cannot be performed in the abc stationary reference frame
 
3p s T cos(θr + β2 ) without suitably modifying the voltage amplitudes applied to
Tem = λpm iqd (43) the healthy phases, in order to maintain the same HF magnetic
22 − sin(θr + β2 )
field. On the other hand, if additional voltage injection before
3p T
Ter = (Ld − Lq ) isqd the fault is performed in the qd0s reference frame, a new
24 reference frame must be adopted for the injection algorithm
 
sin [2(θr + β2 )] cos [2(θr + β2 )] s  after the fault.
× iqd . (44)
cos [2(θr + β2 )] − sin [2(θr + β2 )] A convenient reference frame is qds , where the additional
voltages are still 90◦ shifted with regard to each other and with
Note that (43) and (44) confirm that the torque expressions the same amplitude. Fig. 7 shows the block control scheme of
have the same structure of (9) and (10) calculated in case of this HF injection solution.
healthy motor. As for the demodulation algorithms, a suitable modification
An example of straightforward implementation of the pro- of the standard signal manipulation is required to extract the
posed estimation method (Method II) is shown in Fig. 6, where correct rotor position information.
the proposed model written in the stationary reference frame A sensorless technique is selected in this paper to practically
qds is exploited to achieve the rotor position estimation in evaluate differences and similarities between the rotor flux
SMPMSMs. Similar to sensorless algorithms applied to healthy position estimation algorithms used in case of healthy system
drives, the estimation of the rotor flux position is obtained by and the algorithms that should be used in case of open-phase
manipulating the electrical stator quantities. Note that a phase- fault. The chosen algorithm is based on the injection of an
shift correction β2 , depending on the faulted phase, is required additional HF (500 Hz–3 kHz) rotating voltage vector Vshf
in order to correctly calculate the rotor position. and on demodulation of the HF stator current vector Ishf ,
As it is well known, the ideal integrator is substituted by a generated by the signal injection and modulated by the machine
low-pass filter to limit offset and drift effects. saliency [3].
GAETA et al.: VECTOR CONTROL OF PM SYNCHRONOUS MOTORS DURING OPEN-CIRCUIT FAULTED CONDITIONS 1975

Fig. 8. Carrier-signal-injection-based sensorless technique—Method III.

The amplitude of the HF current vector Ishf depends on the

rotor position and can be related to the angular position of the
HF additional voltage vector.
Assuming a healthy PMSM with a single dominant saliency
Fig. 9. Experimental harmonic content and time waveform of |Ishf |2 in
in the d, q rotor reference frame, supplied only with an HF healthy and faulted drives.
voltage Vshf , neglecting the resistive terms, and operating at
low rotor speed, the squared amplitude |Ishf |2 can be ex- TABLE I
M OTOR I NAMEPLATE AND PARAMETERS
pressed as
 
2
Vshf 1 1
|Ishf | =
2
− 2 cos2 (θhf − θr )
(ωhf − ωr )2 L2dshf Lqshf
2
Vshf
+ (45)
(ωhf − ωr )2
t t are shown in Fig. 9 for the motor with the technical data of
where θr = 0 ωr dt + θro and θhf = 0 ωhf dt + θhf o , both Table I, highlighting different harmonic contents at the same
expressed in electrical radians, are the angular positions of operating conditions (40% Te_rated ).
the rotor q-axis and the HF injected voltage, measured from In particular, the harmonic at 2(ωhf − ωre ) that is exploited
the q s -axis of a stationary reference frame, respectively. It is to carry out the rotor position estimation with the healthy
possible to identify the absolute position of the rotor d, q-axis drive is reduced during the fault; in addition, other harmonics
through the detection of the maximum and minimum points appear around the injection frequency (830 Hz), and the most
of |Ishf |2 . In fact, the minimum points of |Ishf |2 occur at significant are at 2ωhf and 2(ωhf + ωre ).
θhf = π/2 + kπ + θr (k = 0, 1, 2, . . . n) and maximum points An explanation of this result can be offered with the help of
at θhf = kπ + θr (k = 0, 1, 2, . . . n). Consequently, the posi- the proposed open-phase fault model. In particular, by carefully
tion θr can be easily obtained from θhf , which is known. examining in (23)–(28) the effects on the rotor position esti-
In case of open-phase fault, as the neutral point of the stator mation of the disturbance terms related to E(θr ), although the
windings is accessible, it is still possible to create an HF rotat- stator resistance voltage drop can be considered negligible, at
ing field to properly map the anisotropies of the machine in this high frequencies, the same simplification cannot be applied to
fault situation. This can be obtained by applying an additional the disturbance terms proportional to Lls . The proposed model,
HF voltage ellipse to the stator terminals of the machine able valid during open-phase fault, can be rewritten for HF operation
to impose an average 60 electrical degrees phase shift also to by eliminating the low-frequency terms
the HF phase stator currents [13]. This condition is achieved  
by imposing a phase shift between the two HF voltages vijhf s r Lqhf + Llshf 0 d r
vqdhf = i
equal to 120 electrical degrees as for the healthy operation. 0 Ldhf + Llshf dt qdhf
While the injection technique is the same adopted before the d r
fault, the demodulation strategy must be modified; in fact, dif- + Llhf E(θr ) i . (46)
dt qdhf
ferently than during healthy conditions, calculation of |Ishf |2
requires that the reference frame transformations given by A, If (46) would be used for rotor position estimation, its last
B, RI , and (RV)−1 must be used to calculate the q − d stator term would lead to a severe degradation of the rotor position
currents in a stationary reference frame, similarly to healthy estimation. In fact, using at HF the same notation adopted for
operation. Fig. 8 shows the block diagram of the modified low-frequency modeling, it can be assumed that the terms Lqhf
sensorless estimation strategy (Method III). and Ldhf consist of two contributions due to the magnetizing
Adoption of the new transformation matrices modifies the and leakage HF fluxes, respectively; contribution of the leakage
original implementation of the sensorless method that was used term Llhf is more relevant at the carrier frequencies as the
for healthy operation of the machine. Experimental measure- leakage flux is higher compared to the case of low-frequency
ments of the electrical signal |Ishf |2 before and after the fault flux [25]. Carrier-injection-based sensorless methods do not
1976 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 6, NOVEMBER/DECEMBER 2012

Fig. 10. Simulated harmonic content of |Ishf |2 in healthy and faulted drives.

TABLE II
M OTOR II NAMEPLATE AND PARAMETERS

TABLE III
L OAD S YSTEM C HARACTERISTICS

Fig. 11. Rotor position estimation algorithm including saliency decoupling

terms.

need any model to estimate the rotor position, and we here

use (46) only to calculate the current |Ishf |2 before and after
the fault. This is shown with the simulated results in Fig. 10
and confirms that, in faulted conditions, the additional term
proportional to E(θr ) and to the leakage inductance produces
the increased harmonic content of the HF currents.
In order to reduce the effect of the additional harmonic
content, saliency decoupling algorithms or filtering techniques
need to be specifically included in the demodulation algo-
rithm [26].
Fig. 11 shows a possible technical solution. As the presence
of the additional harmonics is due to the magnetic asymmetries
of the machine, compensation of the harmonic content of the
signal |Ishf |2 is performed by means of a saliency decou-
pling network. The amplitudes I1 , I2 and phases ϕ1 , ϕ2 of the
harmonic components at 2ωhf and 2(ωhf + ωre ) exploited to
remove the additional harmonics from the signal |Ishf |2 are Fig. 12. Method I—Motor I: 90 rad/s—50% Te_rated : CH1: Measured θr
[180◦ /div]. CH2: Estimation error [7.2◦ /div]. CH3: Estimated ωr [40 rad/s/
obtained at different load conditions by means of an offline div]. Time: [0.01 s/div].
procedure. Phases ϕ1 and ϕ2 are measured with respect to the
signal 2θhf that is calculated starting from the phase of the HF motor drive whose specifications are shown in Table III is used
voltage vector Vhf . as load.
The previously described sensorless estimation techniques
have been exploited to perform a sensorless FOVC with the
IV. E XPERIMENTAL R ESULTS
system steadily operating under open-phase fault (phase c
In order to evaluate the capability of the three modified opened). The experimental results in Figs. 12 and 13 show the
sensorless algorithms, experimental tests have been executed sensorless control performed at steady state when the estima-
at steady state and transient, by utilizing an experimental tion algorithms indicated in Figs. 5 and 8 are used under faulted
setup consisting of two electrical drives mechanically coupled condition. Note that the maximum rotor position errors from
through a torque meter and controlled through a DSP-based the measured rotor angles are limited in both cases below five
control board. The technical specifications of the two PMSM electrical degrees.
tested motors are indicated in Tables I and II. The neutral point Figs. 14 and 15 show the torque-producing component of the
of the stator windings of the motors is connected to the middle stator current, the rotor position estimation, and the measured
point of the dc bus of the same drive by means of a triac. A dc speed during a speed transient at no load applied to both
GAETA et al.: VECTOR CONTROL OF PM SYNCHRONOUS MOTORS DURING OPEN-CIRCUIT FAULTED CONDITIONS 1977

Fig. 13. Method II—Motor II: ωr = 90 rad/s—50% Te_rated : CH1: Esti- Fig. 16. Method II—Motor II transient healthy–faulty operations: CH1: θrest
mated θr [180◦ /div]. CH2: Estimation error [7.2◦ /div]. CH3: Estimated ωr [360◦ /div]. CH2: ωr [100 rad/s/div]. CH3: ifaulty_phase [10 A/div]. CH4: inc
[40 rad/s/div]. Time: [0.01 s/div]. [10 A/div]. Time: [0.005 s/div].

Fig. 14. Method I—Motor I: ωr = 60–120 rad/s: CH1: Estimated θr Fig. 17. Method I—Motor I: ωr = 50 rad/s—50% Te_rated : CH1: ωr
[180◦ /div]. CH2: iqs [10 A/div]. CH3: ωr [40 rad/s/div]. Time: [0.1 s/div]. [40 rad/s/div]. CH2: ΔVnc [50 V/div]. CH3: ia [10 A/div]. CH4: inc
[10 A/div]. Time: [0.02 s/div].

Fig. 15. Method II—Motor II: ωr = 60 rad/s–120 rad/s: CH1: Estimated θr

Fig. 18. Method I—Motor I transient healthy–faulty operations, no load:
[180◦ /div]. CH2: iqs [10 A/div]. CH3: ωr [40 rad/s/div]. Time: [0.1 s/div].
CH2: ωr [100 rad/s/div]. CH3 ifaulty_phase [5 A/div]. CH4: inc [5 A/div].
Time: [0.2 s/div].
sensorless implementations, again in faulted conditions. Of
course, the dynamic behavior is strongly affected by the elec- Fig. 17 shows the ripple superimposed to the voltage of the dc
tromagnetic torque availability due to the fault. bus middle point ΔVnc , measured with respect to the negative
Fig. 16 shows the transient between sensorless healthy and dc bus rail. This ripple is produced by the current inc circulating
faulty operations at 50% of rated torque and ωr = 120 rad/s; between the neutral point of the stator windings and the middle
the total mechanical inertia of the system mitigates the speed point of the dc bus. Note that the harmonic compensation
variation during the time interval (7 ms) that has been intention- technique [13] mitigates the effects of the third harmonic of
ally included in this test to emulate the time required to identify the back EMF on the currents of the healthy stator phase. This
the fault, isolate the faulted inverter leg, and activate the triac in does not happen for inc , consisting of the fundamental and third
order to switch to the proposed sensorless FOVC. harmonics.
1978 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 48, NO. 6, NOVEMBER/DECEMBER 2012

the fault-tolerant proposed solution is performed in the presence

of the harmonics produced by the magnetic asymmetries of
the machine. This test has been performed with a sensored
vector control. Improvement of the rotor position estimation
due to the elimination of the additional harmonic components
is clearly visible in Fig. 20(b), where a noticeable reduction of
the estimation error is achieved applying the proposed saliency
decoupling procedure.
Note that position control has not been tested as it is expected
that this control loop shows the same limits and drawbacks of
the speed control loop. We also expect that the transient be-
tween healthy and faulty operations during the position control
is more critical.
Fig. 19. Open-phase fault, 13 rad/s—40% Te_rated : CH1: Estimated θr
[180◦ /div]. CH1: Measured θr [180◦ /div]. CH3: Estimation error [32◦ /div].
Time: [0.04 s/div]. V. C ONCLUSION
A novel machine model for three-phase faulted PMSMs has
been exploited in this work to perform rotor position estimation
in a wide operating range during the open-phase fault. Applica-
tions of this study to some model- and carrier-based sensorless
methods have been described and experimentally evaluated
through several speed and torque transients, confirming that
fault operation is critical for any sensorless technique and
that specific modifications must be introduced to allow their
practical implementation in such abnormal situations.

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[13] A. Consoli, A. Gaeta, and G. Scelba, “Modeling and control of Giacomo Scelba (S’04–M’07) received the M.S.
PMSMs under open-phase fault,” in Proc. IEEE ECCE, 2011, pp. 1684– and Ph.D. degrees in electrical engineering from the
1691. University of Catania, Catania, Italy, in 2002 and
[14] S. Green, D. J. Atkinson, A. G. Jack, B. C. Mecrow, and A. King, “Sensor- 2005, respectively.
less operation of a fault tolerant PM drive,” Proc. Inst. Elect. Eng.—Elect. In 2004, he was a Visiting Student at Rockwell
Power Appl., vol. 150, no. 2, pp. 117–125, Mar. 2003. Automation Standard Drives Development, Mayfield
[15] F. De Belie and J. Melkebeek, “Sensorless two-phase current control of Heights, OH. He is currently an Assistant Profes-
a faulty three-phase salient-pole PMSM,” in Proc. IEEE ICEMS, 2009, sor with the Department of Electric, Electronic and
pp. 1–6. Computer Engineering, University of Catania. His
[16] O. Wallmark, L. Harnefors, and O. Carlson, “Post-fault operation of current research interests include sensorless con-
fault-tolerant inverters for PMSM drives,” in Proc. Int. Conf. EPE, 2005, trol, digital signal processing, and ac drive control
pp. 1–10. technologies.
[17] O. Wallmark, L. Harnefors, and O. Carlson, “Control algorithms for a
fault-tolerant PMSM drive,” IEEE Trans. Ind. Electron., vol. 54, no. 4,
pp. 1973–1980, Aug. 2007. Alfio Consoli (M’79–SM’88–F’00), deceased, re-
[18] A. Gaeta, G. Scelba, A. Consoli, and G. Scarcella, “Sensorless estimation ceived the Graduate degree in electrical engineering
in PMSMs under open-phase fault,” in Proc. IEEE Int. Conf. SLED, 2011, from the Politecnico di Torino, Turin, Italy, in 1972.
pp. 27–34. From 1973 to 1974, he was with Fabbrica Italiana
[19] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of Electric Ma- Automobili Torino, Turin. In 1975, he joined the
chinery and Drive Systems, 2nd ed. West Lafayette, IN: Purdue Univ., Department of Electrical, Electronic, and Systems
2002, pp. 109–125. Engineering, University of Catania, Catania, Italy,
[20] D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of AC where he became a Professor of electrical engineer-
Drives. London, U.K.: Oxford Univ. Press, 1996. ing in 1985, becoming the Head of the Center for
[21] S. Morimoto, K. Kawamoto, M. Sanada, and Y. Takeda, “Sensorless con- Promotion and Transfer of Innovation Technology,
trol strategy for salient-pole PMSM based on extended EMF in rotating which he founded in 1999. In 1985, he was a Visiting
reference frame,” IEEE Trans. Ind. Appl., vol. 38, no. 4, pp. 1054–1061, Professor at the University of Wisconsin, Madison. In 1987, he became the
Jul./Aug. 2002. Coordinator of scientific activities of the Ph.D. in electrical engineering and in
[22] M. Cacciato, G. Scarcella, G. Scelba, S. M. Bille, D. Costanzo, and energy at the University of Catania. He was involved in industry cooperation
A. Cucuccio, “Comparison of low-cost-implementation sensorless programs and in national and international research projects in industry in
schemes in vector controlled adjustable speed drives,” in Proc. IEEE Int. the fields of energy conversion systems, electrical drives, robotics, and power
Conf. SPEEDAM, 2008, pp. 1082–1087. electronics. He was a coauthor and a coeditor of the book Modern Electric
[23] Z. Chen, M. Tomita, S. Ichikawa, S. Doki, and S. Okuma, “Sensorless Drives (Kluwer, 2000). He was the author or coauthor of more than 250
control of interior permanent magnet synchronous motor by estimation of technical papers. He was the holder of four international patents.
an extended electromotive force,” in Conf. Rec. IEEE IAS Annu. Meeting, Prof. Consoli was a member of the Executive Council of the European
2000, pp. 1814–1819. Power Electronics Association as the Chairman of the Chapter on Electrical
[24] A. Piippo, M. Hinkkanen, and J. Luomi, “Analysis of an adaptive observer Drives. From 1997 to 2001, he was a member of the Executive Board of the
for sensorless control of interior permanent magnet synchronous motors,” IEEE Industry Applications Society (IAS). He was a member of the Executive
IEEE Trans. Ind. Electron., vol. 55, no. 2, pp. 570–576, Feb. 2008. Committee of the IEEE Power Electronics Society, where he also served as the
[25] J.-H. Jang, J.-I. Ha, M. Ohto, K. Ide, and S.-K. Sul, “Analysis Chairman of the Technical Committee on “Motor Drives and Actuators” and
of permanent-magnet machine for sensorless control based on high- an Associate Editor of the IEEE T RANSACTIONS ON P OWER E LECTRONICS.
frequency signal injection,” IEEE Trans. Ind. Electron., vol. 40, no. 6, In 1980, he was a recipient of a NATO grant from Purdue University, West
pp. 1595–1604, Nov./Dec. 2004. Lafayette, IN, the Third Prize Paper presented at the IEEE IAS Annual Meeting
[26] P. Garcia, F. Briz, D. Raca, and R. D. Lorenz, “Saliency-tracking-based in 1998, and the Best Paper published in the IEEE T RANSACTIONS ON
sensorless control of AC machines using structured neural networks,” P OWER E LECTRONICS in 2000. He was a Distinguished Lecturer of IEEE for
IEEE Trans. Ind. Appl., vol. 43, no. 1, pp. 77–86, Jan./Feb. 2007. 2002–2004.

Alberto Gaeta (S’08) received the M.S. and Ph.D.

degrees in electrical engineering from the Univer-
sity of Catania, Catania, Italy, in 2008 and 2011,
respectively.
He is currently with the University of Catania.
His research interests include electric machinery and
electric drives with particular attention to the re-
search and optimization of fault-tolerant and sensor-
less control techniques.