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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
2
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
3
PM Sychronous Machine
cos sin
cos 120 sin 120
cos 120 sin 120
as iqn r idn rn N
bs iqn r idn rn N
cs iqn r idn rn N
i n n
i n n
i n n
cos
cos 120
cos 120
as r mag em rm M
bs r mag em rm M
cs r mag em rm M
e m m
e m m
e m m
The harmonic content of the currents and back-EMF can be expanded as a Fourier series
Back-EMF equations
Current equations
Torque equation
2e as as bs bs cs cs ecog
r
PT i e i e i e T
1,5,7,11,13,...M 1,5,7,11,13,...N
4
Torque Produced by PMSM
Torque is modeled as sum of the average torque and the torque ripple harmonics
cos sin
3
4
3
4
3
4
e e eqy r edy ry Y
mage en iqn
n N
mageqy iqn cqye y n e y n
n N
magedy idn cdye y n e y n
n N
T T T y T y
PT
PT T
PT T
Torque
Average Torque
Harmonics
6,12,18,24,...Y 1,5,7,11,13,...N
5
Sensing Torque Ripple
A polyvinylidene fluoride (PVDF) film produces voltage in response to deformation
sCA
h
s 3V * *n ng Stress h
Vs
Cs
• 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
6
Sensor Placement
Permanent MagnetSynchronous Machine
PVDFWasher
7
Torque Ripple SensorIsolating 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
cos ry1
s
sin ry1
s
sVr
*eqyT
*edyT
* *
* *
cos
sin
eqy sensor r eqy
edy sensor r edy
sensor sensor e e
T v y T dt
T v y T dt
v k T T
8
Closed-Loop Controller
Cost function is defined to be a function of measured quantities (in steady state)
Expression for measured torque ripple is expanded
T Teq eq ed edG T QT T QT
1 1 2
3
( )
( )
eq iq e e qh cq
ed e d cd
T K K i T
T K i T
9
Closed-Loop Controller
The desired current harmonics are then chosen as a function of the measured torque ripple
qh iqh
dG
dt i
dh idh
dG
dt i
22 Tqh e q
d
dti K Qx
32 Td e d
d
dti K Qx
10
Closed-Loop Controller
Diagram of torque ripple mitigation control-loop
Hysteresis Current Controller
PMSMMachine
2
sensork
1
s
1
s
Measured Currents
eqyy Y
T
r
sensorv*eqyx
qh
d
dti
qhi
2TeK Q
GaineT
1iq
*sin r ydelayy
1
s
*edyx
*cos r ydelayy
s
Hall-EffectSensors
Position Observer
11
Initial Position Estimator
cos
cos
0
as s e
bs s e
cs
i I t
i I t
i
Only two stator phases are energized
Produces a torque harmonic, but zero average component
cos2
cos2
asm r bsm re s e ecog r
r r
asm r bsm rsensor s s e s
r r
PT I t T
Pv I k t
12
Initial Position Estimator
Three commanded stator currents
Produces three torque ripple amplitudes at the commanded electrical frequency
cos , 0
cos , 0
cos , 0
as bs s e cs
bs cs s e as
cs as s e bs
i i I t i
i i I t i
i i I t i
13
Initial Position Estimator
The ratio of two vibration waveforms provides position information
Substituting in fundamental component of influence of flux on the stator winding from the permanent magnet
2 cos
2 cos
asm r bsm rs s et s
r rsensorab
sensorbc bsm r csm rs s et s
r r
PI kv
vPI k
cos cos 120
cos 120 cos 120r rsensorab
sensorbc r r
v
v
14
Initial Position Estimator
Using trig identities to simplify
Closed form expression for the tangent of the position observer
3 1cot
2 2sensorab
rsensorbc
v
v
1
1
1
tan 3 2 1
tan 60 3 2 1
tan 60 3 2 1
sensorabr
sensorbc
sensorbcr
sensorca
sensoracr
sensorab
v
v
v
v
v
v
15
Experimental Verification
• 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
The control was tested in hardware using the following setup
16
Initial Position Estimator
Calculated rotor position
Rotor position error
0 50 100 150 200 250 300 3500
100
200
300
Rotor Position (r )R
otor
Pos
itio
n ( r
)
Calculated Rotor Position vs. Actual Rotor Position
ActualCalculated - no-loadedCalculated - loaded
0 50 100 150 200 250 300 350
-2
0
2
Rotor Position (r )
Posi
tion
Err
or (
r )
Estimation Error vs. Rotor Position
17
Measured Start-up Performance
Start-up performance comparison of position observer to an optical encoder
0 0.2 0.4 0.6 0.8 10
500
1000
Rotor Velocity - Measured
RPM
Time (s)
InitialPositionObserver
Position ObserverOptical Encoder
0 0.2 0.4 0.6 0.8 1
-20
-10
0
10
20
Phase-a Stator Current Using Optical Encoder - Measured
Am
ps
Time (s)0 0.2 0.4 0.6 0.8 1
-20
-10
0
10
20
Phase-a Stator Current Using Position Observer - Measured
Am
ps
Time (s)
InitialPositionObserver
18
Torque Ripple Mitigation ImplementationSimulated steady-state results before and after torque ripple mitigation algorithm
0 0.005 0.01 0.0150
2
4
6Torque Before Mitigation - Simulated
N*m
Time (s)
0 0.01 0.02 0.03 0.04-20
-10
0
10
20Phase-a Stator Current After Mitigation - Simulated
Am
ps
Time (s)
0 0.01 0.02 0.03 0.04-20
-10
0
10
20Phase-a Stator Current Before Mitigation - Simulated
Am
ps
Time (s)
0 0.005 0.01 0.0150
2
4
6Torque After Mitigation - Simulated
N*m
Time (s)
19
Torque Ripple Mitigation ImplementationMeasured steady-state results before and after torque ripple mitigation algorithm
0 0.005 0.01 0.015-4
-2
0
2
4Torque Ripple Before Mitigation - Measured
Vol
ts
Time (s)
0 0.01 0.02 0.03 0.04-20
-10
0
10
20Phase-a Stator Current After Mitigation - Measured
Am
ps
Time (s)
0 0.01 0.02 0.03 0.04-20
-10
0
10
20Phase-a Stator Current Before Mitigation - Measured
Am
ps
Time (s)
0 0.005 0.01 0.015-4
-2
0
2
4Torque Ripple After Mitigation - Measured
Vol
ts
Time (s)
20
Torque Ripple Mitigation Implementation
Steady-State FFT of Electromagnetic Torque
0 500 1000 15000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Torque Harmonic Amplitude - Simulated
N*m
6th
harmonic
12th
harmonic
Frequency (Hz)
Before MitigationAfter Mitigation
0 500 1000 15000
0.5
1
1.5Torque Ripple Amplitude - Measured
Vol
ts
6th
harmonic
12th
harmonic
Frequency (Hz)
Before MitigationAfter Mitigation
21
Measured Transient Response
Measured torque ripple and current during step change in commanded torque from 1.25 Nm to 5.0 Nm
0 0.05 0.1 0.15 0.2-20
-10
0
10
20Phase-a Stator Current Transition Response - Measured
Am
ps
time(s)0 0.05 0.1 0.15 0.2
-4
-2
0
2
4Torque Ripple Transition Response - Measured
Vol
tstime(s)
22
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