Improved Rotor Position Estimation
Of Salient-Pole PMSM Using High Frequency
Carrier Signal Injection
O. Mansouri-Toudert, H. Zeroug, F. Auger and A. Chibah

Abstract -- This paper presents a new rotor position
estimator for salient-pole permanent magnet synchronous
machines (PMSM) based on carrier signal injection. This
method first uses a new way to extract the negative frequency
component of the current, which contains the rotor position
information. Compared to the classical approach, this method
uses only one filter. It also performs an on line compensation
of the phase shift of the filter used for this extraction. This
method is used for sensorless field-oriented control,
confirming then its effectiveness under various operating
conditions.
Index Terms-- sensorless control, carrier signal injection,
surface-mounted permanent magnet synchronous machines,
demodulation.
I.
NOMENCLATURE
Ld, Lq : d-axis and q-axis stator inductance (H).
Rs : stator resistance (Ω).
vα, vβ, iα, iβ: voltages and currents in the Clark reference
frame.
va, vb, vc, ia, ib, ic: voltages and currents in the abc reference
frame.
vd, vq, id, iq: voltages and currents in the Park reference
frame.
s : derivative operator d/dt (Laplace variable).
m : rotor flux due to the permanent magnets (V.s)

θ r , θ r : real and estimated electrical angles between the
stator q-axis and the rotor q-axis (rad).

r and  r : real and estimated angular frequency of the
fundamental electrical excitation (rad/s).
c : carrier signal angular frequency (rad/s).
fc : carrier signal frequency (Hz).
m : mechanical rotor speed (rad/s).
 m* : desired mechanical rotor speed (rad/s).
Vc : injected carrier voltage (volt).
p : number of pole pairs.
J : total moment of inertia (kg m2).
f : viscuous friction coefficient (Nm s)
Tr : load torque (Nm).
Te : electromagnetic torque (Nm).
II.
P

ERMANENT
INTRODUCTION
magnet synchronous motors have recently
O. Mansouri-Toudert is with the « département Electrotechnique de la
Faculté de Génie Electrique et Informatique », Mouloud Mammeri
University, Tizi-Ouzou, Algeria (e-mail: toudert_ouiza@yahoo.fr). H.
Zeroug is with the Industrial Electrical Systems Laboratory, Faculty of
Electronics and Computing, Houari Boumediene University of Sciences
and Technology, Bab-Ezzouar, Algiers, Algeria. F. Auger and A. Chibah
are with LUNAM University and with the « Institut de Recherches sur
l’Energie Electrique » (IREENA), France (e-mail: Francois.Auger@univnantes.fr, arezki.chibah@etu.univ-nantes.fr).
come into wide use because of their high efficiency, large
power density and control simplicity [1], [2]. The control of
these machines involves the knowledge of the exact rotor
position, which is generally provided by a feedback sensor.
This position sensor (or rotational transducer) not only
increases cost, maintenance, and complexity, but also
impairs robustness and reliability of the drive system [1],
[3].
Therefore, many research studies are being conducted in
order to eliminate the mechanical sensor. Conventional
sensorless control schemes of the PMSM can be divided in
two categories. Methods from the first category perform an
estimation of the back-EMF [4], [5] using either voltage
models, state observers or Kalman filters [6], [7]. Methods
using a voltage model generally present a high sensitivity to
parameter variations, which affects performance and may
significantly degrade the rotor position estimation at low
speed [7]. Methods using state observers or Kalman filters
are less sensitive to parameter variations, but the common
problem of all the mentioned methods is that for lower
speeds, the back EMF has a very small amplitude, resulting
in an inaccurate rotor position estimation [4], [7] and in an
unstable speed control [8]. Methods of the second category
try to solve this problem thanks to the injection of a high
frequency voltage component used as a carrier signal [8],
[9], [10]. These methods can be used when a motor
saliency exists, even for surface-mounted permanent
magnet synchronous motors [11]. Their advantage is that
they are more devoted to accurately estimate the rotor
position at low speeds, and even during standstill. In this
category, however, several processings such as signal
heterodyning and filtering [12] are necessary, in order to
extract the position information contained in the negative
frequency component of the current. Therefore, phase shifts
are added to the rotor position, resulting in biased position
estimation. The most commonly used method to extract the
negative frequency component of the current involves three
continuous-time filters (a band-pass filter, a high-pass filter
and a low-pass filter) and two frequency shifts [13], [14],
[15]. In this article, a simple method is introduced, which
consists of only one continuous-time low-pass filter and
one frequency shift. Moreover, the phase shift of the lowpass filter is compensated, in order to nullify the position
estimation bias. This method has shown its effectiveness in
providing accurate rotor position estimations under various
operating conditions, and particularly in the low speed
range. The aim of the approach presented here may seem
close to a previous work published several years ago [16].
But in that paper, unbiased position estimation of an
interior PMSM was derived from a state estimator of the
saliency-based back-EMF. In our study, the position
estimation is derived from a high frequency current
injection, and the position estimation bias is cancelled
thanks to a phase compensation scheme. This paper is
organized as follows: In section II, a simplified model of a
PMSM is developed, and a high-frequency model is
presented in section III. In section IV, two methods for the
extraction of the negative frequency component of the
complex current signal are presented: the most often used
method and a new one. In section V, the rotor position
estimator is presented and a new scheme for the
compensation of the phase shift of the filters is given.
Simulation results are shown in section VI.
III. PMSM MODEL
The d-q model for a PM synchronous machine is very
adequate to present the analytical basis of the proposed
estimation method. Under the assumptions that the machine
contains only one single sinusoidally distributed saliency
and that only the inductances are considered in the high
frequency model, the stator voltage model in the rotor
reference frame is given by
vds  rs
v   
 qs   0
0  ids   s
 
rs  iqs   r
 r  ds 
 
s  qs 
(1)
And the magnetic flux is given by
ds   Lds
    0
 qs  
0  ids  m 

Lqs  iqs   0 
(2)
When transformed into a stationary reference frame αβ, the
above equations can be rewritten as [17]
vs 
is  d s 
v   rs i    
 s 
 s  dt  s 
with
(3)
s 
is 
   L ( r ) i   m
 s 
 s 
cos( r )
 sin( ) 
r 

(4)
and
 L  L cos(2 r )  L sin(2 r ) 
L ( r )  

  L sin(2 r ) L  L cos(2 r )
where L  Ld  Lq  / 2 and L  Lq  Ld  / 2 are
respectively the average inductance and the zero-to-peak
differential inductance. Of course, the machine is not
powered
to
produce
heat,
but
to
generate
an
electromagnetic torque that can be modelized as


Te  p m  Ld  Lq  I ds I qs
(5)
This torque allows us to act on the rotor speed as shown by
the mechanical equation
Jp
dr
 f p r  Te  Tr
dt
(6)
Therefore, this electromagnetic torque produced by the
stator currents may be used either to compensate the load
torque and the friction losses or to produce a speed
variation.
IV. PRINCIPLE OF THE CARRIER SIGNAL INJECTION BASED
of the PMSM motor terminals a balanced set of high
frequency low magnitude sinusoidal voltages [9], [18] :


j  c t  
v _ c   Vc sin ct 
2

(7)



V
e
 
c

v _ c   Vc cosct  

v _ c
with ωc=fc. The frequency fc (around 1 kHz) and the
magnitude Vc (around 10 V) of this carrier signal are
chosen to ensure an acceptable increase of the total
harmonic distorsion (THD) of the phase current [10], [19]
and an insignificant torque ripple resulting from Eq. (5). At
a high frequency, the PMSM model equations (3)-(4) can
be simplified to [3]:
dL
 v _ c  
i 
 i 
 r    _ c   L  r  d   _ c 
v    rs I 2 
dt
dt i  _ c 
 i  _ c 
  _c  
d i _ c 
 L  r 
(8)


dt i  _ c 
where I2 is the two-dimensional identity matrix. This
equation shows that a current i _ c  i _ c  j i _ c is
induced by this carrier voltage injection. The stator current
can hence be decomposed into three components [10]: the
first one is a positive frequency component that rotates in
the same direction as the injected voltage, the second one is
a negative frequency component that rotates in the opposite
direction as the injected voltage, and the third one is the
low frequency excitation component, resulting from the
low frequency controlling voltage and from the stator flux
and from the rotor flux:
i  icp e


j  c t  
2

 icn e


j    c t  2 r  
2

 i _ s , (9)
where the amplitude of the positive and negative frequency
components are respectively given by
icp 
L . Vc
L . Vc
, icn  2
(10)
2
( L  L )  c
( L  L2 )(c  2r )
2
From (10), one can see that only the negative frequency
component
icn e


j   c t  2 r  
2

contains the information
about the rotor positionr. As a consequence, so as to
perform rotor position estimation, it is necessary to use
some appropriate signal processings to extract this
component [2], [10].
V.
STATOR CURRENT SIGNAL PROCESSING
Several methods using analog filters and frequency
shifts may be used to isolate the rotor position information
r contained in the second term of (9) and to get rid of the
other undesirable components. In this paper, only two
methods will be presented and compared. An emphasis is
made on the second one, which is original, more simple and
more effective.
Fig. 1. Most frequently used method for rotor position estimation
ESTIMATION METHODS
If Ld differs from Lq, it is possible to estimate the rotor
position by superimposing to the low frequency excitation
A. Previously proposed method
One of the most widely used demodulation method [3],
[14], [15], [18], [19] first consists (see Fig. 1) of an analog
band-pass filter (BPF) centered on the carrier frequency fc.
This filter aims at preserving the two low amplitude high
frequency components of the measured stator current and at
canceling the low frequency excitation current. The
resulting complex valued current is multiplied by
e  jct (this operation is sometimes called a heterodyning
process), to perform a frequency translation of
c
. As a
approximately the same bandwidth (±40 Hz around -1000
Hz). The classical demodulation process provides a perfect
rejection around DC and around +1000 Hz, but the
proposed method provides a higher attenuation anywhere
else. Fig. 3 also shows that the slope of the phase around
-1000 Hz is higher for the proposed demodulator than for
the classical method, justifying therefore the need for a
phase compensation scheme.
result, the negative frequency component is moved
around  2 c , and the positive frequency component
0
modulus (dB)
-20
becomes a DC component, which can then be removed by a
high-pass filter. The output of this high-pass filter is
 j 2ct
multiplied by e
, so as to translate the negative
frequency component around zero and to provide a signal
-40
-60
-80
-100
-120
-140
-1500
j 2
proportional to e r . This signal is cleaned by a low-pass
filter, and a phase extraction is finally used to provide the
rotor position estimation [13].
-1000
-500
-1000
-500
0
500
1000
1500
0
500
1000
1500
400
200
phase (deg)
0
-200
-400
-600
-800
-1000
-1500
frequency (Hz)
Fig. 2. Simple method proposed for rotor position estimation
B. Proposed processing
The review of the previous method raises questions about
the real need for as many steps to achieve the final result.
Therefore, we tried to find a simpler method. This new
method (see Fig. 2) first consists of a multiplication of the
measured stator current i
frequency translation of
 c
by e
 j c t
, to perform a
. As a result, the negative
frequency component is moved around zero, and the
remaining two components are located at frequencies
higher than the carrier frequency:
i e j ct   icn e


j  2 r  
2

 icp e


j  2ct  
2

 i _ s e jct
These last two components can be removed with a low-pass
filter, providing again a signal i _ dem (see Fig. 2)
proportional to e
j 2 r
.
Fig. 3. Frequency responses of the two demodulators: proposed method
(blue dashed lines) and classical demodulation process (red solid lines).
VI. UNBIASED SPEED AND POSITION ESTIMATION
A.
Speed and position estimator
The demodulation of the extracted negative frequency
component provides the signals Icn cos(2r) and Icn sin(2r).
The rotor position could be estimated from these signals
with an inverse tangent function, but a more noise
insensitive estimation is obtained by a non-linear secondorder angle tracking observer [9], [10] suited to this case, as
shown in Fig. 4. This observer provides an estimation of
both the position and the speed of the rotor. The setting of
this observer can be done as proposed in [21]. From the
maximal electromagnetic torque of the machine Temax, the
highest angular acceleration can be computed as
  Tem ax J
(11)
The first coefficient Kb (see Fig. 4) can then be chosen as
In both cases, Bessel filters should be used, because of
their maximally-flat group delay (or maximally linear phase
response), leading to a position estimation error
proportional to the rotor speed. Their coefficients are
computed as shown in [20], [22]. After numerous trials, we
set the cutoff frequency of all the low-pass filters to 40 Hz.
The band-pass filter is centered at fc and its bandwidth has
been set to 400 Hz. The cutoff frequency of the highpass
filter has been set to 10 Hz. In the previously proposed
demodulation method, second-order filters can be used
successfully. By cons, a fourth-order low-pass-filter is used
in the second method, to sufficiently attenuate the low
frequency excitation component. This filter is realized with
two second-order low pass-filters in series.
Fig. 3 shows the frequency response of the two
demodulators. This figure shows that they have
Kb   ~
 r_max
where
~
 r_max
,
(12)
is the maximum permissible position
estimation error resulting from such an angular
acceleration. The second coefficient Ka can be computed as
K a  2m Kb ,
(13)
where m is a damping ratio chosen by the user. This
parameter can be chosen so as to obtain a desired peak
overshoot when the actual rotor position abruptly changes.
For example, choosing an overshoot of 5% leads to
m=1.945 [21].
VII.
Fig. 4. Block diagram of the nonlinear second-order angle tracking
observer
B. Compensation of the position estimation error
An extensive use of this estimator revealed that a biased
position estimation is obtained when the angle tracking
observer (Fig. 4) follows the negative frequency component
extractor (Fig. 1 or 2). This bias comes from the phase shift
() of the filters used during the demodulation process.
Since the expressions of these phase shifts are known and
since the angle tracking observer also estimates the rotor
speed, one may attempt to remove this estimation bias by
subtracting   c  2ˆ r  to   c  2 r  . This
compensation scheme is more precisely described by the
block diagram shown in Fig. 5. In this figure, BP(),
HP(), LP2() and LP4() are respectively the arguments
of the band-pass filter, high-pass filter, second-order lowpass filter and fourth-order low-pass filter. The angle  is
the sum of the phase shifts of the filters used for the
negative
frequency
component
extraction.
This
compensation scheme can hence be used for any of the
demodulation methods presented in section IV : when the
“classical” method is used, K1=K2=K3=1 and K4=0,
whereas when the proposed method is used, K1=K2=K3=0
and K4=1.
Fig. 5. The proposed estimation error compensation scheme.
SIMULATION RESULTS
To evaluate the performances of the described methods, a
set of experiments have been performed on a MATLAB/
SIMULINK simulation. The numerical values of the
parameters of the simulated machine come from the
identification of an industrial 4.4 kW SM-PMSM (Yaskawa
SGMGH-44DCA6F) and are given in the appendix. The
motor is fed by a power converter which receives three
control signals from a classical vector control ensured by
three PI controllers. The speed and position estimations are
performed by a negative frequency component extractor
and demodulator, an angle tracking observer and a phase
shifts compensator.
Fig. 7 and Fig. 8 show the estimation error of the rotor
position obtained with the “classical” method when the
reference of the speed controller is constant and
respectively equal to 10 r/min and 50 r/min. These results
show that without compensation, the estimation error
increases with speed. This may lead to an unstable speed
control, since the estimated position is used in the abc/dq
and dq/αβ transformations of the vector control, as shown
in Fig. 6. When using a compensation of the phase shifts,
the estimation error becomes lower than one degree in both
cases. Fig. 8, 9 and 10 show the results obtained under the
same conditions with the proposed method for the
extraction and demodulation of the negative frequency
component. These results show that without compensation,
the estimation bias is higher than with the “classical”
method. But once the phase shifts are compensated, the
estimation error still remains below one degree. This means
that with the proposed method, a highly stable speed
control is obtained at the price of a demodulator with a
lower complexity.
Fig. 11 shows the true speed obtained when the
reference of the speed control is quickly varying. This
figure shows that the PI controllers of the vector control
have been set for the actual speed to closely follow the
desired speed. Fig. 12 and 13 respectively show the speed
estimation error and the position estimation error obtained
with both the “classical” and the proposed demodulator,
both with phase shifts compensation. These figures show
that when the phase shifts are compensated, the “classical”
demodulator provides a slightly more accurate estimation
of speed and position, but at the price of a more complex
process. It should be underlined that the classical
demodulator is generally used without phase shifts
compensation.
VIII.
Fig. 6. Block diagram of the sensorless vector control of a salient PMSM
using high frequency signal injection.
CONCLUSION
This paper has presented a new position estimator of the
rotor position of a salient permanent magnet synchronous
machine (PMSM) using carrier signal injection. The
commonly used method generally contains three analog
filters. Since the proposed one uses only one analog filter, it
is simple and effective, and leads to a lower implementation
processing time. It was shown that when the compensation
scheme takes into account the operating speed, filter phase
shift is greatly reduced, thus leading to an accurate rotor
position estimation. Simulation results presented in this
paper demonstrate the effectiveness of the proposed
method. Further researches include the validation of this
method on an experimental testbed, and a comparison with
the approach presented in [23]. As mentioned in [5], carrier
signal injection based methods provide an estimation of
twice the electrical angle. As a result, an ambiguity of 180
degrees affects the determination of the rotor position, and
this ambiguity requires to be properly managed.
compensation (blue dashed lines) and with phase compensation (red solid
line). The speed reference is constantly equal to 50 r/min.
5
Position error with simple method (deg)
IX. APPENDIX
TABLE I
MOTOR MODEL PARAMETERS
Parameter
Rated flux
Rated output power
Rated torque
Rated speed
Rated voltage
Rated current
Load torque
rs
Ld
Lq
P
Value
0.32 Wb
4.4 kW
28.4 Nm
1500 r/min
400 V
16.5 A
0 Nm
0.25 
4.8 mH
4.1 mH
4
0
-5
-10
0
0.05
0.1
0.15
time (s)
Fig. 9. Position estimation error using the proposed method without phase
compensation (blue dashed lines) and with phase compensation (red solid
line). The speed reference is constantly equal to 10 r/min.
5
10
Position error with simple method (deg)
Position error with the classical method (deg)
15
0
-5
5
0
-5
-10
-10
-15
0
0.05
0.1
0.15
0
0.05
0.1
Fig. 7. Position estimation error using the classical method without phase
compensation (blue dashed lines) and with phase compensation (red solid
line). The speed reference is constantly equal to 10 r/min.
0.15
time (s)
time (s)
Fig. 10. Position estimation error using the proposed method without
phase compensation (blue dashed line) and with phase compensation (red
solid line). The speed reference is constantly equal to 50 r/min.
15
10
8
desired and measured speed(r/min)
Position error with classical method (deg)
10
6
4
2
0
-2
-4
5
0
-5
-10
-6
-15
-8
-10
0
0.05
0.1
0.15
time (s)
-20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time (s)
Fig. 8. Position estimation error using the classical method without phase
Fig. 11. Desired (red solid line) and true (blue dotted line) rotor speed.
[5]
4
[6]
3
speed estimation error (r/min)
2
[7]
1
0
[8]
-1
[9]
-2
-3
-4
[10]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time (s)
Fig. 12. Speed estimation error obtained with the classical method (blue
dotted lines) and proposed method (red solid line).
[11]
[12]
5
4
[13]
Position estimation error (deg)
3
2
[14]
1
[15]
0
-1
[16]
-2
-3
[17]
-4
-5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
time (s)
[18]
Fig. 13. Position estimation error obtained with the classical method (blue
dotted lines) and simple method (red solid line), both with phase
compensation.
[19]
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[20]
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[2]
[3]
[4]
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[21]
[22]
[23]
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