Using Torque-Ripple-Induced
Vibration to Determine the Initial
Rotor Position of a Permanent
Magnet Synchronous Machine
Phil Beccue, Steve Pekarek
Purdue University
November 6, 2006
Outline
• Background information
– Source of torque ripple in a surface mounted
Permanent Magnet Synchronous Machine
(PMSM)
– Method for measuring torque ripple
– Algorithm used to mitigate torque ripple
• Utilizing Torque Ripple to Determine Rotor
Position
2
PM Sychronous Machine
Back-EMF equations
Torque equation
Te 
P
ias eas  ibs ebs  ics ecs   Tecog
2r

eas  r mag

ebs  r mag

ecs  r mag
The harmonic content of the
currents and back-EMF can be
expanded as a Fourier series
 m
em
 m
em
 m
em
mM
mM
mM
cos  m r 

cos  m 

 120  
cos m  r  120 

r
Current equations
ias    iqn cos  n r    idn sin  n r  
nN

cos  n 

 120    

sin  n 

 120   

ibs    iqn cos n  r  120    idn sin n  r  120  


nN
ics    iqn

nN

r
M 1,5,7,11,13,...
idn

r
N 1,5,7,11,13,...
3
Torque Produced by PMSM
Torque is modeled as sum of the average torque and the
torque ripple harmonics
Torque
Te  Te   Teqy cos  y r   Tedy sin  y r  
Average Torque
Te 
Harmonics
yY
Teqy 
Tedy 

3Pmag
4

3Pmag
4

3Pmag
4

nN
 iqn
en
 
e y n
  e y  n  iqn  Tcqy
 
e y n
  e y  n  idn  Tcdy
nN
nN
Y 6,12,18,24,...


N 1,5,7,11,13,...
4
Sensing Torque Ripple
A polyvinylidene fluoride (PVDF)
film produces voltage in response
to deformation
Cs
Vs
Cs  
A
h
Vs  g3n * Stressn * h
•
•
•
The PVDF film is metallized on both sides
The film acts as a dialectic – forms a capacitance
Modeled by a voltage source with a series capacitor
5
Sensor Placement
Permanent Magnet
Synchronous Machine
PVDF
Washer
6
Torque Ripple Sensor
Isolating Torque Ripple Harmonics
• Values for harmonics of torque are acquired by multiplying the
sensor voltage by cos(yθr) and sin(yθr)
• The result of the multiplication is then passed through a lowpass
filter
*
*
 dt
Teqy
  vsensor cos  y r    Teqy
*
*
 dt
Tedy
  vsensor sin  y r    Tedy
vsensor  k sensor Te  Te 
cos  yr 
r
1
s 
*
Teqy
1
s 
*
Tedy
Vs
sin  y r 
7
Closed-Loop Controller
Cost function is defined to be a function of measured
quantities (in steady state)
G  TeqT QTeq  Ted T QTed
Expression for measured torque ripple is expanded
Teq  ( iq1K e1  K e 2 i qh )  Tcq
Ted  (K e 3i d  Tcd )
8
Closed-Loop Controller
The desired current harmonics are then chosen as a function
of the measured torque ripple
d
i qh  iqhG
dt
d
i qh   2K e 2T Qx q
dt
d
i dh  idhG
dt
d
i d   2K e 3T Qx d
dt
9
Closed-Loop Controller
Te
 2
d
i qh
dt
Gain
 iq1
Measured Currents
i qh
1
s
Hysteresis
Current Controller
r
1
s 
*
xedy

s
PMSM
Machine
Hall-Effect
Sensors
Position Observer
T

yY
*
sin  y r   ydelay

eqy
ksensor
*
cos  y r   ydelay

K Te2Q
*
xeqy
1
s 
vsensor
Diagram of torque ripple mitigation control-loop
10
Initial Position Estimator
Only two stator phases are energized
ias  I s cos et 
ibs   I s cos et 
ics  0
Produces a torque harmonic, but zero average component
P  asm  r  bsm  r  
Te  I s 

 cos et   Tecog  r 
2   r
 r 
vsensor
P  asm  r  bsm  r  
 Is 

 ks cos et  s 
2   r
 r 
11
Initial Position Estimator
Three commanded stator currents
ias  ibs  I s cos et  , ics  0
ibs  ics  I s cos et  , ias  0
ics  ias  I s cos et  , ibs  0
Produces three torque ripple amplitudes at the
commanded electrical frequency
12
Initial Position Estimator
The ratio of two vibration waveforms provides position
information
 asm  r  bsm  r  
2 PI s ks 

 cos et  s 
 r
 r 
vsensorab


vsensorbc
 bsm  r  csm  r  
2 PI s ks 

 cos et  s 
 r 
  r
Substituting in fundamental component of influence of
flux on the stator winding from the permanent magnet
cos  r   cos  r  120 
vsensorab

vsensorbc cos  r  120   cos  r  120 
13
Initial Position Estimator
Using trig identities to simplify
vsensorab
3
1

cot  r  
vsensorbc
2
2
Closed form expression for the tangent of the position
1
observer
 vsensorab 
tan  r   3  2
 1
 vsensorbc

 vsensorbc 
tan  r  60   3  2
 1
 vsensorca

1
 vsensorac

tan  r  60   3  2
 1
 vsensorab 
1
14
Experimental Verification
The control was tested in hardware using the following setup
• Test motor is a 2.5 kW, 16 Amp 8-pole surface mount
PMSM with non-sinusoidal back-emf
• A 4096 counts per revolution encoder used to obtain an
accurate rotor position
• Commanded stator current had a frequency of 1000 Hz
and a peak amplitude of 1 A (6.25% of rated)
• The response time was less than 50 ms
15
Initial Position Estimator
Calculated rotor
position
Rotor Position ( r )
Calculated Rotor Position vs. Actual Rotor Position
Actual
Calculated - no-loaded
Calculated - loaded
300
200
100
0
0
50
100
150
200
250
Rotor Position ( r )
300
350
300
350
Rotor position
error
Position Error ( r )
Estimation Error vs. Rotor Position
2
0
-2
0
50
100
150
200
250
Rotor Position ( r )
16
Measured Start-up Performance
Start-up performance comparison of position observer to an
optical encoder
Rotor Velocity - Measured
RPM
1000
Initial
Position
500 Observer
Position Observer
Optical Encoder
0
0
0.2
0.4
0.6
Time (s)
20
20
10
10
0
-10 Initial
Position
-20 Observer
0
1
Phase-a Stator Current Using Optical Encoder - Measured
Amps
Amps
Phase-a Stator Current Using Position Observer - Measured
0.8
0
-10
-20
0.2
0.4
0.6
Time (s)
0.8
1
0
0.2
0.4
0.6
Time (s)
0.8
1
17
Torque Ripple Mitigation Implementation
Simulated steady-state results before and after torque ripple
mitigation algorithm
Torque Before Mitigation - Simulated
Phase-a Stator Current Before Mitigation - Simulated
6
20
4
N*m
Amps
10
0
2
-10
-20
0
0
0.01
0.02
0.03
0.04
Time (s)
Phase-a Stator Current After Mitigation - Simulated
0.01
0.015
6
10
4
N*m
Amps
0.005
Time (s)
Torque After Mitigation - Simulated
20
0
2
-10
-20
0
0
0.01
0.02
Time (s)
0.03
0.04
0
0
0.005
0.01
0.015
Time (s)
18
Torque Ripple Mitigation Implementation
Measured steady-state results before and after torque ripple
mitigation algorithm
Torque Ripple Before Mitigation - Measured
20
4
10
2
Volts
Amps
Phase-a Stator Current Before Mitigation - Measured
0
-10
-20
0
-2
0
0.01
0.02
Time (s)
0.03
-4
0.04
0
4
10
2
Volts
Amps
Phase-a Stator Current After Mitigation - Measured
20
0
-10
-20
0.005
0.01
Time (s)
Torque Ripple After Mitigation - Measured
0.015
0
-2
0
0.01
0.02
Time (s)
0.03
0.04
-4
0
0.005
0.01
0.015
Time (s)
19
Torque Ripple Mitigation Implementation
Steady-State FFT of Electromagnetic Torque
Torque Ripple Amplitude - Measured
Torque Harmonic Amplitude - Simulated
1.5
0.5
Before Mitigation
After Mitigation
Before Mitigation
After Mitigation
0.45
6th harmonic
0.4
0.35
1
th
6 harmonic
Volts
N*m
0.3
0.25
0.2
0.5
0.15
12th harmonic
0.1
12th harmonic
0.05
0
0
500
1000
Frequency (Hz)
1500
0
0
500
1000
Frequency (Hz)
1500
20
Measured Transient Response
Measured torque ripple and current during step change in
commanded torque from 1.25 Nm to 5.0 Nm
Torque Ripple Transition Response - Measured
4
10
2
Volts
Amps
Phase-a Stator Current Transition Response - Measured
20
0
-10
-20
0
-2
0
0.05
0.1
time(s)
0.15
0.2
-4
0
0.05
0.1
time(s)
0.15
0.2
21
Conclusions
• Initial position observer is developed that utilizes
torque ripple measurement to determine position
– Requires no knowledge of machine parameters
– Applicable to surfarce or buried-magnet machines
– Relatively straightforward to implement
• Initial position observer can potentially enable
sensorless operation over the full speed range of
the motor
• Torque ripple mitigation can be achieved without
in-line position encoder
22