A Novel Sensorless Initial Rotor Position Estimation Method for Permanent Magnet Synchronous Motors

International Journal of Power Electronics and Drive System (IJPEDS)
Vol. 8, No. 1, March 2017, pp. 156~166
ISSN: 2088-8694, DOI: 10.11591/ijpeds.v8i1.pp156-166  156
Journal homepage: http://iaesjournal.com/online/index.php/IJPEDS
A Novel Sensorless Initial Rotor Position Estimation Method for
Permanent Magnet Synchronous Motors
Chien Feng Wu1
, Shir Kuan Lin2
1-2
Department of Electrical Engineering, National Chiao Tung University, Taiwan, Province of China
Article Info ABSTRACT
Article history:
Received Oct 03, 2016
Revised Dec 09, 2016
Accepted Dec 19, 2016
This paper presents a high efficiency initial rotor position estimation method
for Permanent Magnet Synchronous Motors (PMSM). The approach uses the
viable inductance model to analyze the optimal motor injection sine wave
frequency as the motor’s test signal. Unlike other high-frequency injection
methods, this approach does not require trial and error experiments. The
injection frequency is identified by programmable simulations using Matlab.
Experimental evaluation of 3-phase PMSM showed that this injection
frequency optimization method works successfully. The proposed method
can find the optimal injection frequency without experimentation, and
outperforms other test signals in terms of accuracy, vibration quantity and
noise. This microprocessor-developed 3-phase PMSM control driver can be
applied to electrical appliances, machine tools and automation.
Keyword:
High frequency signal injection
Initial rotor position estimation
Optimal injection frequency
PMSM
Copyright © 2017 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Shir-Kuan Lin,
Department of Electrical Engineering,
National Chiao Tung University,
Taiwan, Province of China.
Email: sklin@mail.nctu.edu.tw
1. INTRODUCTION
Sensorless Permanent-Magnet (PM) motors are commonly used in industrial machinery and home
appliances. Their advantages include a simple structure, good reliability, high power density, good efficiency
and low maintenance costs. In addition, such motors will not be damaged by arcing or friction at high-speed
rotation. PM motors use non-contact phase commutation instead of mechanical phase-changing devices such
as brushes. To activate and rotate the PM motor smoothly, position sensors are needed to detect the rotor
position. Position sensors can be photo- or magnet-based, but rotor position detection may be costly and
unreliable.
Implementing motor control without position sensors can enhance the reliability and reduce the cost
of motor sensors. In a motor control system, rotor position must be known to obtain better control
performance. In recent years, several methods have been developed to estimate rotor position, including the
counter electromotive method [1]-[4], the extended Kalman filter [5],[6], the artificial neural networks [7],
Finite Element method [8],[9] and sliding mode current observer [10]-[12]. These methods are based on
modern control theory and realized through the use of microprocessors.
However, these methods can only work when the rotor is rotating. As noted by Haque [13], when
the rotor is still, measuring the initial position of the rotor is difficult in several respects: 1) A still motor
produces no electrical-field or magnetic-field change which can be measured to evaluate rotor-position, 2)
position estimation is load dependent [14], and 3) it entails high calculation complexity and sophisticated
hardware requirements [15].
In 1996, Matsui [16] reviewed five methods of estimating the initial position of sensorless PM
brushless DC motor, including: 1) Auxiliary sensors, 2) open loop controls, 3) specific gate patterns, 4)
arbitrary starting and 5) salient-pole motors. Compared with Methods 2) - 4), Method 5) starts the motor

IJPEDS ISSN: 2088-8694 
A Novel Sensorless Initial Rotor Position Estimation Method for Permanent Magnet .... (Chien Feng Wu)
157
more smoothly and prevents reverse rotation. Method 5) is mainly based on the magnetic saturation of the
magnetic conductive material. The test signal, voltage or current are applied to the motor, and then the
difference of the corresponding current peak value is measured to calculate the current position and rotation
speed of the rotor [17]. In practice, to maintain the rotor position at standstill, the energy within the motor
must be offset. Once the test signals are input, an identical counter-directional signal must be input to zero
out the net torque. This signal injection method is referred to as the Salient Pole method.
The Salient Pole method can avoid undesirable events such as rotor back-kicking or rotor oscillation
prior to smooth running, making this method superior to the other four methods. Based on the Salient Pole
method, several approaches are provided to further enhance the precision of initial-position estimation.
In research on the Salient Pole method, Schmidt et al. [18] established a table of input-voltages and
corresponding peak currents of a motor. When the motor starts, a three-phase voltage is applied to it. The
induced peak current is then measured and the initial position of the rotor can be obtained based on the table.
Tursini et al. [19] found that evaluations of rotor position based only on peak current are subject to several
uncertainties. They proposed an improved method which applies the same voltage waveform 16 to measure
the feedback current, using fuzzy theory to determine rotor position, resulting in estimations with errors of
less than 20 degrees.
Hatakeyama [20] evaluated the rotor position by applying two sine waves. The amplitude of the first
high-frequency sine wave is small, so the rotor remains still. However, the amplitude of the second sine wave
is high, providing sufficiently strong magnetic saturation. Since the first sine wave cannot induce a strong
current, the detected rotor position cannot be determined to be in the one-to-two quadrant or the three-to-four
quadrant. Hence the second sine wave is applied to determine the quadrant of the rotor position. However,
the strong second sine wave will result in some rotor rotation.
2. RESEARCH METHOD
The line inductance of a PMSM, denoted by t
L , can be described as t eq end
L L L
  where eq
L is the 2-
D equivalent inductance and end
L is the end-turn inductance of windings. end
L is caused by three-dimensional
(3-D) magnetic flux in a PMSM, and is usually much less than end
L . Thus, this paper is concerned only with
eq
L . Some assumptions are listed here to simplify the analysis.
The line inductance of a PMSM, denoted by t
L , can be described as t eq end
L L L
  where eq
L is the 2-
D equivalent inductance and end
L is the end-turn inductance of windings. end
L is caused by three-dimensional
(3-D) magnetic flux in a PMSM, and is usually much less than end
L . Thus, this paper is concerned only with
eq
L . Some assumptions are listed here to simplify the analysis.
A1. We ignore the eddy current effect on inductance.
A2. The shape of the shoes, the teeth and the slots of the stator is simplified as rectangular shapes.
A3. For the calculation of the air-gap permeance, the field pattern assumes that the flux lines are always
orthogonal to the intersection surface of the air-gap and the steel.
A4. The B-H curve of the PM is linear with the constants of the relative permeability ( r
 ) and the
remanence ( r
B ).
A5. The relative permeability of the rotor and the stator back iron are infinite ( Fe
   ), and the nonlinear
interaction of these two fields is considered only to calculate the air-gap inductance.
A6. The input coil current is constant.
As shown in Figure 1, in this paper, all three phases are excited by different winding currents, as
opposed to the approach used in [21]. It is known that eq
L is the sum of the air-gap ( g
L ) and the slot
inductances ( slot
L ).
( )
3
s
eq g slot
N
L L L
  (1)
B where s
N is the total number of slots, and the denominator 3 is the number of the phases of a
PMSM. The inductances g
L and slot
L are respectively functions of the flux linkage g
 and s
 .

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IJPEDS Vol. 8, No. 1, March 2017 : 156 – 166
158
Figure 1. Simplified shoes, teeth, and slots structure
Figure 2 shows the flux linkages of the windings divided into five parts: g
 , 1
s

, 2
s

, 3
s

and 4
s

,
where g
 is the flux crossing the air-gap, while s

consists of 1
s

, 2
s

, 3
s

and 4
s

.
Figure 2. Flux pattern of winding current
The slot
L can also be divided into four parts as:
1 2 3 4
2 2
slot s s s s
L L L L L
   
(2)
Figure 3. (a) equivalent magnetic circuit of 1
s
 . (b) equivalent magnetic circuit of 2
s
 . (c)
Equivalent magnetic circuit of air-gap flux
Figure 3(a) and (b) respectively show the equivalent magnet circuits of 1
s
 and 2
s
 . Note that N is
the number of turns of each stator tooth, 1
s
R is the magnetic reluctance of 1
s
 , and 2
s
R is the magnetic

159
reluctance of 2
s
 . It follows from Figure 3 that 1 1
3 / 2
s s
Ni R
  . According to the definition of the inductance,
we obtain
2
1
1
1
3 9
2 4
s
s
s
d N
L N
di R

 
(3)
It is well known that 1 0 0 0
/
s
R b h z


, where 0
b
and 0
h
are defined in Figure 1 The current direction of
faces b and c are the same, therefore 2
s

and 2
s
L
are almost equal to zero:
2 2
0 0
s s
L
  
(4)
In Figure 2, i.e., the slot between the teeth of phases b and c. The magnetic motive force (MMF) at
the top of the slot is equal to 2 i
N , while it is zero at the bottom. This is because the turns are uniformly
distributed in the slot, and the magnetic field 3( )
s
H x crossing the slot increases linearly from the bottom of
the slot to the top. Since 3(0) 0
s
H  and 3 0
( ) 2 / ( )
s
H h Ni b t

   , we have 3 0
( ) 3 / 2[( ) ]
s
H x Ni b t h

   , where x is
the distance from the bottom of the slot (see Figure 2).
This allows us to express 3
s
L from the energy viewpoint as
2 2
0 3 0
3 2
0
3
4 ( )
s
VOL
s
H dV N hz
L
b t
i
 

 
 

(5)
It is well known that 4 2
s s
L L

, therefore 4
s

and 4
s
L
are almost equal to zero:
4 0
s
  4 0
s
L 
(6)
The equivalent magnetic circuit in Figure 3(c) describes the air-gap flux g
 that goes from the shoe
of phase a to the reluctance of phase a Fa
R and the PM reluctance g
R , then 1/2 g
 separately pass through
reluctance of phase a and phase b. Fa
R , Fb
R , Fc
R and g
R can be approximated as:
0 0
0
1 1 0 2 2
0
1 1 0 2 2
0
1 1 0 2 2
( ) ( )
( ) ( )
( ) ( )
m
g
r
Fa
r a a a a
Fb
r b b b b
Fc
r c c c c
l g
R
z z
h h
R
B z B tz
h h
R
B z B tz
h h
R
B z B tz
    
    
    
    
 
 
 
 
(7)
1
a

, 1
b

, 1
c

are the relative degrees of permeability in the tooth, and 2
a

, 2
b

, 2
c

are those in the
shoe. It then follows from Figure 3(c) that 3 / 2[ ]
g g Fa ss
Ni R R R
   
( )( )/( ) ( )
ss g Fb g Fc g Fb g Fc
R R R R R R R R R
      . Thus
2
3 9
2 4[ ]
g
g
g Fa ss
d N
L N
di R R R

 
 
(8)
However, by assumption A5, the relative permeability of the stator back iron is a nonlinear function
of the flux densities in the shoes and the teeth of the stator. These flux densities are defined as follows:

 ISSN: 2088-8694
160
1 2
1 2
1 2
/ 2 / 2
/ 2 / 2
ma g sa ma g sa
a a
mb g sb mb g sb
b b
mc g sc mc g sc
c c
B B
z tz
B B
z tz
B B
z tz



     
 
     
 
     
 
(9)
where m
 are the fluxes of the PM flowing through the three phase. s
 are the equivalent slot linkage fluxes
of the three phases.
It is easy to calculate ma

, mb

, mc

, sa
 sb
 sc

with the following equations:
2 4 4
ma mga mb mgb mc mgc
l l l
s s s
sa sb sc
z B dl z B dl z B dl
L i L i L i
N N N
     
       
  
(10)
It is apparent that flux densities are functions of g
 , so the flux g
 should make the following
error function ( )
g
E  zero from the equivalent circuit shown in Figure 3 (c):
3
( ) [ ] 0
2
g g g Fa s
E Ni R R R
     
(11)
( )
g
E  is a nonlinear equation with a single variable g
 . Applying the iteration method to the above
equation, we can easily obtain g
 that makes ( ) 0
g
E   , provided that m
 is known.
We substitute the flux g
 into (8) to calculate the air-gap equivalent inductance g
L . The slot
inductance slot
L and the air gap inductance g
L can be substituted into Eq. 1 to calculate the magnetic
saturation and the motor’s effective equivalent inductance eq
L . Finally, substitute into (1) to obtain eq
L
Figure 1 shows the d-q winding model of a PMSM. In this figure, 1d and 1q represent the armature
d-q windings, 2d the constant field winding corresponding to the permanent magnet mounted on the rotor and
3d and 3q the equivalent windings for eddy current. In this paper, the iron loss arises from the resistance of
the equivalent winding for eddy current [1].
3. ROTOR POSITION ESTIMATION METHOD
As noted in [2], the motor inductance changes with the rotor angle, thus the relationship between the
inductance value and the current can be written as:
di
v L
dt

(12)
When the voltage is kept constant in the above formula, the inductance value is inversely
proportional to the current differential, so the rotor position can be determined by calculating the difference
of the inductance value from the speed of the current response.
As shown in Figure 4, the drive circuit of the three-phase permanent magnet synchronous motor is a
three half-bridge circuit. We use a three-phase leg with a total of six transistors and flywheel diodes ( 1
S and
2
S , 3
S and 4
S , 5
S and 6
S ) to form a three-phase inverter, where a
v is the input voltage of phase a, b
v is the
input voltage of phase b, c
v is the input voltage of phase c, the neutral point voltage at the intersection of the
three phases is n
v and dc
V is the DC voltage source. The three-phase permanent magnet synchronous motor
coil Y passes phase a as phase current a
i , phase b as phase current b
i , and phase c as phase current c
i .

161
Figure 4. Three-phase motor and drive circuit
The three-phase voltage input method includes six modes, where the a-phase injection definite
voltage signal mode is taken as an example:
( ) ( )
ab an bn ac an cn dc
v v v v v v V
     
( )
a a
i i 

、 ( )
(1/ 2)
b c a
i i i 
 
(13)
where dc
V is the signal amplitude
( ) ( )
3 3
( )
2 2
ab dc s a a
d
v V R i L M i
dt
 
   
(14)
Similarly, assuming the a-phase injection is a negative voltage signal mode
( ) ( )
3 3
( )
2 2
ab dc s a a
d
v V R i L M i
dt
 
    
(15)
The positive value’s square wave test signal dc
V is input to the motor model, the simplified
resistance is 3
2 s
R R
 and the signal is the positive inductance value.
Thus we can rewrite (13) as a voltage signal relationship
( )
( ) ( )
a
ab dc a eqa
di
v V i R L
dt

 
  
(16)
/
( ) ( ) )
( ) (1 ) , /
( )
t
dc
a a eqa(
V
i t e L R
R
a



  
  

(17)
The negative value’s square wave test signal dc
V
 is input into the motor model. The negative signal
input uses the negative inductance value ( )
eq
L  ; similarly, the voltage is
( )
( ) ( )
a
ab dc a eqa
di
v V i R L
dt

 
   
(18)
The current response equation is:
/
( ) ( ) )
( ) (1 ) , /
( )
t
dc
a a eqa(
V
i t e L R
R
a



  

  

(19)

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162
We set a current sampling time p
t , the negative current ( )( )
p
i t
 and positive current ( )( )
p
i t
 .
Subtracting the absolute values from each, the three-phase current difference i
 is:
( ) ( )
( ) ( )
( ) ( )
| ( )| | ( )|
| ( )| | ( )|
| ( )| | ( )|
a a p a p
b b p b p
c c p c p
i i t i t
i i t i t
i i t i t
 
 
 
  


  


  

 (20)
Through this method, we can use the magnitude relationship of the three i
 values to separate the
360 degree electrical angle into six parts, each with an electrical angle resolution of sixty degrees, as shown
in Figure 5 In an electrical angle of 30 degrees b
i
 and c
i
 rise and fall nearly linearly, making this area a
dividing point, and the other five blocks are treated with the same logic.
Figure 5. Electrical angle partioning
This square-wave method can be used to successfully identify the initial rotor position. However, if
the motor has a large stator coil resistance value, the current response will be little changed, resulting in an
excessively large judgment error in determining the initial rotor position. Therefore, this paper seeks to select
an appropriate injection frequency to maximize the current distance and minimize the rotor position
estimation error.
4. OPTIMUM FREQUENCY SEARCH
The proposed optimum frequency search method is described in following flow chart. Figure 6
illustrates the process of determining the optimal injection frequency. The first step is to input the motor’s
geometric parameters. Step 2 sets the coil’s initial current value. Of particular note, the coil current ( )
n
i is not
arbitrary, but is rather determined iteratively. The current iteration calculation seeks to determine the optimal
current ( )
a optimal
i f as we approach the optimal frequency optimal
f , and thus calculate the circuit current ( )
n
i at
the time of analysis to optimize the accuracy of current value calculations.
Step 3 inputs the data produced in Steps 1 and 2 to the model derived in Section 2 to calculate the
motor’s equivalent inductance.
Step 4: The optimal injection frequency search method inputs different frequency sine wave signals
(from 1Hz to 500Hz). When the voltage s
v is positive, ( )( )
eqa e
L 
 is the inductance value; when s
v is
negative, ( )( )
eqa e
L 
 is the inductance value, and this signal is input into the Simulink motor model.
Subtracting the absolute value of the half-cycle coil current ( )
i  from the negative half cycle coil current ( )
i 
obtains i
 , and the frequency at which the maximum current max( )
a
i
 occurs is the optimal current optimal
f
Finally, compare the optimal frequency current and the current at the time. If they are not similar,
then add a current error error
i and continue with further iterations. If they are similar, terminate the operation
with the optimal frequency simulation. Where the current error value error
i is ( )
( )/ 2
opt n
i i
 , if they are almost
equal, the operation terminates to complete the optimal frequency search.

163
Figure 6. Flow chart of optimum frequency search
Figure 7 shows the result of the frequency search from 1Hz to 500Hz. The largest i
 occurs at 21Hz,
followed by 20.05mA.
Figure 7. Result of frequency search
5. EXPERIMENTAL RESULTS
Experiments used three-phase, 16-pole, and 9-slot PMSMs, with microcontrollers to produce the
injection signal and measure the coil current to verify the simulation results. The injection signal is a
sinusoidal function with a peak value of 75 V. To find the optimal input sign wave, the frequency of the input
signal was increased from 10 Hz to 50 Hz. The relation between the three-phase current difference ( i
 ) and
the rotation angle is shown in Figure 8 The amplitude of the three-phase current difference ( i
 ) is maximized
when the frequency of input sin wave is 20 Hz. This result is identical to the simulation result.

 ISSN: 2088-8694
164
Figure 8. The relation between feedback current difference ( i
 ) and rotation angle. The input signal is a
sinusoidal function with a frequency of (a) 10 Hz, (b) 20 Hz, (c) 30 Hz, (d) 40 Hz, and (e) 50 Hz
As shown in Figure 5 rotor partition estimate accuracy increases with the three-phase current ( i
 )
error. To improve estimate resolution, this paper uses the arctangent function to reverse the initial electrical
angle. This method uses sum-to-product formula. We already know that , ,
a b c
i i i
   each have 120 degrees
difference of electrical angle. Using Eq. 21, we can obtain , ,
u v w
i i i
   and eliminate the DC offset.
( ) / 3
( ) / 3
( ) / 3
u a b
v b c
w c a
i i i
i i i
i i i
   


   


   


(21)
We again use the plot difference to obtain the ,
x y for each 90 degree difference of electrical angle:
( )/ 3
u
v w
x i
y i i
 



  


(22)
Finally, the electrical arctangent function is used to obtain the initial angle:
arctan( )
e
x
y
 
(23)
Figure 9 shows the relationship between the actual position and the estimated rotor position to
evaluate the estimated accuracy. We use two injection frequencies of the developed method to compare with
the two sine waves method. The result proves that our Sensorless Initial Rotor Position Estimation Method
(Figure 9(a)) get the better result than Two Sine Waves Method [20] (Figure 9(c)) at the error of actual rotor
position and the estimated value. The experimental RMS error is also calculated in Table 1, which tells the
frequency of 20Hz in injection can get the most accurate rotor position.

165
(a) (b) (c)
Figure 9. Rotor position estimates for (a) novel method with 20Hz injection frequence (b) novel method with
50Hz injection frequence (c) two sine waves method [20]
Table 1. Comparison of Different Injection Frequence and Methods
Novel method with 20Hz Novel method with 50Hz Two sine waves method
RMS Error 11.44 18.95 18.37
6. CONCLUSION
In this paper, we analyze the methodology of Magnetic Circuit Analysis to establish an equivalent
inductance model of permanent-magnet synchronous motors. With this model, different rotor positions are
found to correspond to different inductor values. Therefore, The optimum injection frequency is obtained by
the simulation process of the Matlab. The RMS error of the experimental shows the optimum injection
frequency(20Hz), which is much lower than the other injection frequency and injection method. Our method
can help the driver developer in their early motor developmrnt process to avoid using trial and error to find
the optimum injection frequency. Besides, using this injection method, there is only one frequency injection
applied. This can exempt the excessive starting of the motor that would usually generate the noise and
vibration. We conclude that this novel invention can be widely used in the intelligent appliances, that the
permanent magnet synchronous motor is one of the most important components in the device.
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