using torque-ripple-induced vibration to determine the initial rotor position of a permanent magnet...
TRANSCRIPT
![Page 1: Using Torque-Ripple-Induced Vibration to Determine the Initial Rotor Position of a Permanent Magnet Synchronous Machine Phil Beccue, Steve Pekarek Purdue](https://reader037.vdocument.in/reader037/viewer/2022110206/56649f3e5503460f94c5dfbc/html5/thumbnails/1.jpg)
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
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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
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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
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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
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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
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Sensor Placement
Permanent MagnetSynchronous Machine
PVDFWasher
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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)
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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
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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)
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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