using torque-ripple-induced vibration to determine the initial rotor position of a permanent magnet...

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