Investigation on Torque in PM Machines - OperaFEA · Investigation on Torque in PM Machines ... Radial airgap flux density ... Q-axis flux linkage Flux density distribution Flux linkage

©Prof. Z. Q. Zhu

EUGM 2014

1

Investigation on Torque in PM Machines- with particular reference to frozen permeability

The University of Sheffield, UK

[email protected]

Prof. Z. Q. Zhu and Dr. W. Q. Chu

04-06-2014

EUGM 2014

©Prof. Z. Q. Zhu

One of the largest research groups in the world, specialising on permanent magnet machines and control systems (>100 personnel, including 6 full professors, 12 Academic staff, >60 PhD students, >20 RAs)

Host• Rolls-Royce University Technology Centre in ‘Advanced Electrical Machines & Drives’• Sheffield Siemens Wind Power Research Centre (S2WP)• Romax Technical Centre

Birthplace 3 university spin-off companies (MagTec, Magnomatics, Red Deer Technologies Group)

Strong industrial collaboration, particularly in automotive, wind power, aerospace, and domestic appliance sectors

Electrical Machines and Drives Research Group

Headed by Prof. Z.Q. Zhu, Fellow IEEE, Fellow IET

2

EUGM 2014

©Prof. Z. Q. Zhu

• Domestic• Industrial• Transportation• Aerospace • Renewable energy• …..

Applications of PM Machines

Air conditioner compressor(Panasonic)

Lawn Mower (Bosch)

Aerospace (Airbus)EV/HEV (Toyota)

PM machines have high torque density and efficiencyUsed to be considered only suitable for low power application (Prof.TJEMiller’s book), and expensive & restricted to special applications, e.g. aerospace

Now, applied to numerous market sectors and at MW power level

Elevator (Otis)Servo (Yaskawa)

Power Steering (TRW/Brose)Wind Power (Siemens)

3

©Prof. Z. Q. Zhu

EUGM 2014Contents

4

1. Introduction

2. Torque Calculation Methods

3. Torque Components Separation

Frozen permeability

Average torque: PM and reluctance torques

Torque ripple: On-load cogging torque etc

4. Influence of Skewing

5. Summary

©Prof. Z. Q. Zhu

EUGM 20141. Introduction

5

Torque

Average torque

Torque ripple

PM torque

Reluctance torque

Cogging torque

Maxwell stress tensor

Virtual work……

……

Finite element

Lump circuit model

Analytical

……

Calculation Analysis Improvement

EUGM 2014

©Prof. Z. Q. Zhu 6

• Maxwell stress tensor

• Virtual work method

Torque is calculated from an integral directly from the flux density components

ΔWin

Lossless machine

Δ Wm Δ Wout=Δ(Win-Wm )=Tout Δ θm

2. Torque Calculation Theories

µ0 : permeability of free space Lef : effective axial length r : radius of integration path Bn , Bt : normal and tangential flux density components

Torque is obtained from the energy conservation law.

Widely used in FE software, since flux density results are available

Not widely used, since differential operation is required/sometimes needs to be approximated

W'm : magnetic co-energy Wm : stored magnetic energy in the machine Win , Tin : input energy and corresponding torqueθm : rotor position in mechanical angle

EUGM 2014

©Prof. Z. Q. Zhu 7

• Calculation of magnetic energy


PM (demagnetization curve)

wiron=∫HdB

wiron

Air

• Calculation of input torque (or power)

Iron (B-H curve)

Br : remanence flux densities. Bp : parallel flux densities.Bt : tangential flux densities.µrp, µrt: relative recoil permeability in parallel and tangential directions

Only valid for average torque calculation, not for torque ripple

PM

Iron

EUGM 2014

©Prof. Z. Q. Zhu 8

• Torque equations


Based on abcia, ib, and ic : 3 phase currentsua, ub, and uc : 3 phase voltagesΩ : mechanical angular speed

Based on dq0

General form (mathematical mapping from abc torque equation by dq0 transformation)

Classical form (or simplified form) is based on assumptions: The windings are sinusoidally distributed around the stator periphery, giving rise to

a sinusoidal induced voltage and sinusoidal variation of self- and mutual inductance with rotor position.

The magnetic circuit is linear.

Hence, ψd and ψq are constant. ψd/dθ=0 and ψq/dθ =0.

EUGM 2014

©Prof. Z. Q. Zhu 9

• Numerical results


T(dq0_g) and T(abc) are identical for torque calculation T(dq0_c) is simple but only able to predict the average torque To predict the torque ripple, it must use T(abc) or T(dq0_g)

• Examples

• Summary

If

Then

6-pole/18-slot inset SPM

0

1

2

3

4

5

0 30 60 90 120

Torq

ue (N

m)

Rotor position (degree)

T(abc) T(dq0_g) T(dq0_c)-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0 30 60 90 120

Torq

ue (N

m)


T(abc) T(dq0_g) T(dq0_c)

Inset SPM with Ia=4A and β=-30˚ Inset SPM with Ia=4A and β=-30˚ (no PM, current only)

EUGM 2014

©Prof. Z. Q. Zhu

-2

-1

0

1

2

3

4

5

0 30 60 90 120

Torq

ue (N

m)


TmwTvir=Tin-d(Wm)/dθmTind(-Wm)/dθm-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 30 60 90 120

Torq

ue (N

m)


TmwTvir=Tin-d(Wm)/dθmd(-Wm)/dθmTin

10

• Comparison of Maxwell stress tensor and virtual work


Open circuit Inset SPM with Ia=4A and β=-30˚

Maxwell stress tensor and virtual work principle are identical for torque calculation innormal FE simulations.

The variation of stored magnetic energy (∂Wm/∂θm) contributes significantly to the torqueripple.

• Summary

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©Prof. Z. Q. Zhu 11

• Purposes

• Principle

To separate the on-load PM and armature field when considering the saturation and cross-coupling

To separate the on-load torque components, both average and ripples

3. Frozen Permeability

Normal conditions (without using FP)

Ball = B(FP, PM) + Bi(FP, i)

µall < µPM

HPM PM only (B) BPM

Current only (C) Hi Bi

Current and PM (A) Hall = HPM + Hi Ball

Due to nonlinearity µall < µi Ball < BPM + Bi

Cannot decompose the on-load PM and armature field components

Using FPCurrent and PM (A) Hall = HPM + Hi Ball and µall

µall and PM only (D) HPM B(FP, PM)µall and current only (E) Hi B(FP, i)

All based on µall

On-load PM and armature field components can be decomposedH (A/m)

B (T)

HallHPMHi

Ball

B(FP, PM)

B(FP,i)

µallµPMµi

BPMBi

AB

C

E

D

EUGM 2014


• Procedures

Non-linear FE with all excitations(PM + current)

Save and freeze the permeability (every element and step)

Linear FE with single excitation(either PM or current)

3. Frozen Permeability

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Field distribution

• All field quantities agree well.• Field distribution on load is asymmetric about d-axis, which is not a problem for field

calculation, but, as will be shown later, is a problem for torque calculation based on FP

Full load permeability

On-load PM field

Radial airgap flux density

Tangential airgap flux density

D-axis flux linkage

Q-axis flux linkage

Flux density distribution Flux linkage

3. Frozen Permeability• FE results

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-180 -120 -60 0 60 120 180R

adia

l flu

x de

nsity

(T)

Angle (degree)

Whole(FP,PM)+(FP,i)(FP,PM)(FP,i)

-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5

-180 -120 -60 0 60 120 180

Tang

entia

l flu

x de

nsity

(T)

Angle (degree)


-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 30 60 90 120

D-a

xis f

lux

linka

ge (W

b)



-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 30 60 90 120

Q-a

xis f

lux

linka

ge (W

b)



EUGM 2014


• Two different ways of torque separation

Maxwell stress tensor: FP based flux density results and direct torque output of FE

Virtual work principle: indirect results based on FP obtained flux linkage, energy and currents

3.0 Torque Separation

Whole model(PM+current)

Frozen permeability (PM only)

Tmw

ψd , ψqTvir

Frozen permeability (current only)

Tmw(FP, PM)

ψd (PM), ψq(PM)Tvir(FP, PM)

Tmw(FP, i)

ψd (i),ψq(i)Tvir(FP, i)


Virtual work


Virtual work


Virtual work

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• Two different ways of average torque separation

3.1 Average Torque Separation

Maxwell stress tensor Virtual workTotal torque Tmw Tvir

Reluctance torque Tmw(FP,i) Tvir(FP,i)PM torque Tmw-Tmw(FP,i) Tvir(PM) or Tvir-Tvir(FP,i)On-load cogging Tmw(FP,PM) Tvir(FP,PM)

Maxwell stress tensor:

Direct results based on flux density results

Virtual work principle:

Indirect results based on flux linkage, energy and currents

** Tvir(rel), Tvir(FP,i), Tvir(PM),Tvir(FP,PM) are based on classic dq0torque equation for average torqueestimation

All field components and flux linkages are FP results

EUGM 2014

©Prof. Z. Q. Zhu

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

0 30 60 90 120

Torq

ue (N

m)


Tmw(FP,PM)

Tvir(FP,PM))

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0 30 60 90 120

Torq

ue (N

m)


Tmw(FP,i)

Tvir(FP,i)

16

• Problems

• Numerical results of inset SPM machine when Ia=4.0A, β=-30˚

Average Tmw(FP,PM) (cogging torque) is not zero

Average torque components from virtual work and Maxwell stress tensor are different but the resultant average torque is the same

Torque components cannot be obtained by FE directly based on Maxwell stress tensor.

FP + PM (cogging torque)


FP + i

-2

-1

0

1

2

3

4

5

6

-90 -60 -30 0 30 60 90

Torq

ue (N

m)

Current phase advance angle (degree)

TvirTmwTvir(PM)Tmw(PM)Tvir(FP,i)Tmw(FP,i)

EUGM 2014


• Why the average of Tmw(FP, PM) is not zero?

Rotor position 0˚ Rotor position 30˚

Non-zero average torque of Tmw(FP, PM) is due to the interaction between PM & equivalent rotational magnetic saliency in the stator (permeability distribution is asymmetric about d-axis!)


• Based on permeability distributions

Rotor position 0˚ Rotor position 30˚

Open circuit

On-load(Ia=4.0A, β= - 30˚)

Zero average cogging torque is due to the symmetrical permeability distribution about d-axis

EUGM 2014

©Prof. Z. Q. Zhu

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-90 -60 -30 0 30 60 90

Torq

ue (N

m)


Tmw(FP,i)Tvir(FP,i)

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 30 60 90 120

T qou

t(FP,

PM)

(Nm

)


Open circuit

Loaded

18

• Why the average of Tmw(FP, PM) is not zero?• Based on on-load PM flux linkage results when Ia=4.0A, β = -30˚

As a motor,Tmw(FP, PM) < 0Tmw(FP, i) > Tvir(FP, i)

As a generator,Tmw(FP, PM) > 0Tmw(FP, i) < Tvir(FP, i)

Negative average q-axis flux linkage & hence negative average torque of Tmw(FP, PM)

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0 30 60 90 120

ψq(

PM) (

Wb)


Open circuit

Loaded


-0.8

-0.6

-0.4

-0.2

0.0

0.2

-90 -60 -30 0 30 60 90

Torq

ue (N

m)


Tmw(FP,PM)Tvir(FP,PM)

• Variation of average torque components with β

q-axis flux linkages by PM Tmw(FP, PM)

Ia=4.0A

EUGM 2014


• Conclusions on average torque separation

• Numerical results

Average torque components should be separated based on virtual work principle

PM torque:

Reluctance torque:

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

-40 -30 -20 -10 0 10 20 30 40

Torq

ue (N

m)

Advance angle (degree)

TmwTvirTvir(FP,PM)+Tvir(Fp,i)Tvir(FP,PM)=1.5p[ψd(PM)*Iq-ψq(PM)*Id]Tvir(FP,i)=1.5p[ψd(i)*Iq-ψq(i)*Id]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-40 -30 -20 -10 0 10 20 30 40

Torq

ue (N

m)


Tr=Tmw-1.5p[ψd(PM)*Iq-ψq(PM)*Id]

Tvir(FP,i)=1.5p[ψd(i)*Iq-ψq(i)*Id]Tmw(FP,i)


Resultant total torques from Maxwell stress and virtual work are the same

Torque components obtained from Maxwell stress is incorrect

EUGM 2014

©Prof. Z. Q. Zhu

3.2 Torque Ripple Separation

20

• Problem

• Incorrect magnetic energy calculation in conventional FP method

Average Tmw(FP, PM) ≠0. Hence, it cannot be the on-load cogging torque

Using the magnetic energy variation instead of Tmw(FP, PM) :

Magnetic energy calculation in the iron with frozen permeability

It is due to that it is the linear FP FE solution

Procedure




Incorrect energy area in the iron

EUGM 2014


• Solution:Using nonlinear FP FE with iteration step =1 for energy calculation, instead of linear FP FE

(available in OPERA)

Resultant full load (PM+i)

Non-linear FE with , iteration step=1 (either PM or current)






Linear FP+PM Nonlinear FP+PM(iteration step=1)

Same permeability Same field

Different permeability

Different field


See next page

EUGM 2014

©Prof. Z. Q. Zhu

0.00

0.05

0.10

0.15

0.20

0.25

0 30 60 90 120

Ene

rgy

(W)


Wiron(FP,PM) ∫HdB

Wiron(FP,PM) HB/2

0.00

0.05

0.10

0.15

0.20

0.25

0 30 60 90 120

Ene

rgy

(W)


Wiron'(FP,PM) ∫HdB

Wiron'(FP,PM) HB/2

22

• Magnetic energy calculation in the ironNonlinear FP (iteration step=1)

Using nonlinear FP method (available using OPERA) can obtain the magnetic energy in the iron correctly


∫HdB=HB/2 ∫HdB<HB/2

Linear FP

EUGM 2014


• On-load cogging torque calculation-∂Wm(FP,PM)/∂θm

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 30 60 90 120

Torq

ue (N

m)


β=-40° β=-30°β=-20° β=-10°β=0° β=10°β=20° β=30°β=40° Tc(open)

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

0 30 60 90 120

Torq

ue (N

m)



-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

0 30 60 90 120

Torq

ue (N

m)



On-load cogging torque should be calculated by correct energy method from -∂Wm(FP,PM)/∂θm

PM flux linkage -∂Wm(FP,PM)/∂θm

Tmw(FP, PM) Ripple of Tmw(FP, PM) (average is removed)

Tmw(FP, PM)

Zero average torque Dominated by fundamental

as open circuit one Magnitude changes

accordingly with flux linkage results

Non-zero average torque Dominate component is

changed to second harmonic when β is lower

Magnitude changes much more than expected


-100

102030405060708090

100110

-40 -30 -20 -10 0 10 20 30 40

Flux

link

age

ratio

(%)


ψd(PM)/ψd(open)

ψq(PM)/ψd(open)

©Prof. Z. Q. Zhu

EUGM 20143. Summary

24

• The results from Maxwell stress tensor and virtual work principle areidentical for torque calculation in normal FE simulations.

• To predict the torque ripple, it must use T(abc) or T(dq0_g), whilstT(dq0_c) is simple but only able to predict the average torque.

• Non-zero average torque of Tmw(FP, PM) (non-zero on-load coggingtorque obtained by Maxwell stress) is due to the interaction between PMand equivalent rotational magnetic saliency in the stator.

• On-load cogging torque can be predicted by -∂Wm(FP,PM)/∂θm combinedwith nonlinear FP method.

• PM and reluctance torques should be separated based on the combinationof virtual work principle and frozen permeability, using correct energycalculation.

EUGM 2014


• Skewing is one of most widely used method to reduce the torque ripple


• Skewing angle (θsk) is usually chosen to be equal to one torque ripple period

• For 3-phase PM machines, both the on-load torque ripple and cogging torque areexpected to be eliminated when θsk is 60° electrical

• When the end effect is negligible, instead of complicated and time consuming 3Dmodelling, multi-slice model can be used

• OPERA can create all the slices automatically and all the slices are solvedsimultaneously (Their torque waveforms are different. If the result for different slice isobtained simply by shifting, the resultant torque with skewing will be incorrect no matterwhether the machine is saturated or unsaturated, as will be shown later).

30º elec.

60º elec.

Axial

S

S

60º elec.

Axial

……

S

S

60º/N elec.N-slice

or N-steps

2-slice or

2-steps

EUGM 2014


• 20-slice skewing with θsk = 60° using OPERA.


• Two machines (with/without shaping), three operations (open circuit, 60A and 120A).

M1

N-slice or

N-steps

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

0 30 60 90 120 150 180

Torq

ue (N

m)


Without skewingWith skewing

Open circuit

Half load

Full load

0.00

0.01

0.02

0.03

1 2 3 4

Torq

ue r

ippl

e (p

.u.)

Harmonics

Open circuitOpen circuit + skewingHalf loadHalf load + skewingFull loadFull load + skewing

-3500

-3000

-2500

-2000

-1500

-1000

-500

0

500

0 30 60 90 120 150 180

Torq

ue (N

m)


Without skewingWith skewing

Open circuit

Half load

Full load0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4

Torq

ue r

ippl

e (p

.u.)

Harmonics

Open circuitOpen circuit + skewingHalf loadHalf load + skewingFull loadFull load + skewing

M2

Cogging torque: eliminated Half load torque ripple: reduced but not eliminated. Full load torque ripple: reduced for M2 but increased for M1.

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4. Influence of Skewing• Effects of skewing

Skewing Equations2-step

3-step

ContinuousT2=T(β= β0 +15º,θ0= -15º)

T1=T(β= β0 -15º,θ0= 15º)

In each slice, the current amplitude is the same but the equivalent phase angle is different,either advanced or retarded, compared to the middle slice (or emf)

This axial variation of equivalent advanced or retarded current phase angle causes (1) differentEM torque ripples in different slices, as well as (2) axial variation of magnetic saturation,which make the skewing less effective or even failed

This is true, no matter whether the machine is saturated [due to (1) and (2)] or unsaturated [dueto (1) only]

It is also true when the stator is skewed instead of rotor

• Overall torque equation considering two effects

• Further explanations are based on 2-slice skewing and θsk=60º.

EUGM 2014


4. Influence of Skewing• Open circuit of M1

-15

-10

-5

0

5

10

15

0 30 60 90 120 150 180

Torq

ue (N

m)


T1 (θo=15) T2 (θo= -15) Tsk

No EM torque. T1 and T2 for Segments 1 and 2 have the same waveform

After skewing, T1 and T2 are anti-phase in terms of the fundamental torque ripple

The fundamental torque ripple are eliminated in Tsk while all the other harmonics remain

Mathematically, it can be concluded that only kN order cogging torque harmonics remain after N-step skewing. By using continuous skewing, the cogging torque can be eliminated.

The mechanism of skewing for reducing the torque ripples is the cancellation of harmonics based on the waveform shifting, which is due to the geometric shifting

In order to eliminate the torque ripple by skewing,

All slices have the same torque waveform

The phase difference between the torque components of different slices is only due to the geometric shifting introduced by skewing

EUGM 2014


4. Influence of Skewing• Half load of M1

Without skewing:

-2000

-1900

-1800

-1700

-1600

-1500

-1400

0 30 60 90 120 150 180

Torq

ue (N

m)


β= -30 β= -20 β= -10 β=0 β=10 β=20

-1900

-1850

-1800

-1750

-1700

-1650

0 30 60 90 120 150 180

Torq

ue (N

m)


T1 (β= -25, θo=15) T2 (β=5, θo= -15)Tall T0 (β= -10, θo=0)

With skewing:

The armature current and hence EM torque ripple are not zero

The on-load torque ripple, especially its magnitude, changes with β

The torque ripples of T1 and T2 can partially cancel each other and the resultant half load torque ripple is still reduced

The torque ripple, even its fundamental, cannot be eliminated, due to the extra variation of on-load torque ripple, both magnitude and phase, with β.

T0 - without skew

EUGM 2014


4. Influence of Skewing• Full load of M1

Without skewing: With skewing:

The magnetic saturation due to armature reaction and EM torque ripple are much higher.

The on-load torque ripple changes significantly with β.

The torque ripples are almost anti-phasewhen β changes by 30º.

The torque ripples of T1 and T2 are not anti-phase but almost in-phase.

The resultant full load torque ripple is not eliminated but increased, since the ripple of T2 is much higher than T0.

-3000

-2900

-2800

-2700

-2600

-2500

-2400

-2300

-2200

0 30 60 90 120 150 180

Torq

ue (N

m)


β= -30 β= -20 β= -10 β=0 β=10 β=20 -2900

-2850

-2800

-2750

-2700

-2650

-2600

0 30 60 90 120 150 180

Torq

ue (N

m)


T1 (β= -25, θo=15) T2 (β=5, θo= -15)Tall T0 (β= -10, θo=0)

T0 - without skew

EUGM 2014


4. Influence of Skewing• Full load of M2

Without skewing: With skewing:

The phase of full load torque ripple without skewing does vary with current phase advance angle but only slightly, due to high cogging torque

The full load torque ripple is reduced but not eliminated, due to the extra torque ripple variation with β.

-3500

-3300

-3100

-2900

-2700

-2500

-2300

-2100

0 30 60 90 120 150 180

Torq

ue (N

m)


β= -30 β= -20 β= -10 β=0 β=10 β=20

-3400

-3200

-3000

-2800

-2600

-2400

0 30 60 90 120 150 180

Torq

ue (N

m)


T1 (β= -25, θo=15) T2 (β=5, θo= -15)Tsk T0 (β= -10, θo=0)

T0 - without skew

EUGM 2014

©Prof. Z. Q. Zhu

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4

Mag

nitu

de (p

.u.)

Harmonics

β= -30β= -20β= -10β=0β=10β=20

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 2 3 4

Mag

nitu

de (p

.u.)

Harmonics

β= -30β= -20β= -10β=0β=10β=20

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4

Torq

ue r

ippl

e (p

.u.)

Harmonics

Without skewingSkewed

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4

Mag

nitu

de (p

.u.)

Harmonics

Without skewingSkewed

32

• Comparison between M1 and M2 at full load


M1 M2

M1 M2

Without skewing:The fundamental torque ripples of M1 and M2 change with β by almost the same amount (in EM torque).e.g. from -30º to 20º, the changes are about 0.067 p.u.

With skewing (β=-10º):The resultant fundamental torque ripples are almost at the same level (full load+skewing = ~0.0249 p.u.)

The on-load torque ripple reduction in M2 is mainly due to reduction of its remarkable cogging torque.

The skewing still fails to eliminate the EM torque ripple at full load.

≈0.067p.u.

≈0.067p.u.

0.015 p.u.

≈0.0249 p.u.

EUGM 2014

©Prof. Z. Q. Zhu

-2900

-2850

-2800

-2750

-2700

-2650

-2600

0 30 60 90 120 150 180

Torq

ue (N

m)


without 2-step 3-step 4-step5-step 6-step continuous 0.00

0.01

0.02

0.03

0.04

0.05

1 2 3 4 5 6 7 8

P-p

torq

ue r

ippl

e (p

.u.)

Number of steps

Open circuitIa=60A, β= -10Ia=120A, β= -10

-15

-10

-5

0

5

10

15

0 30 60 90 120 150 180

Torq

ue (N

m)


without 2-step 3-step 4-step5-step 6-step continuous

-1900

-1850

-1800

-1750

-1700

0 30 60 90 120 150 180

Torq

ue (N

m)


without 2-step 3-step 4-step5-step 6-step continuous

33

• In order to ease the manufacturing and reduce the cost

4. Step Skewing

Open circuit

M2

Full load• 3-step skewing is the most promising alternative in low cost application.• Torque ripple only eliminated on open-circuit, but not on load, depending on load conditions.

• Based on M1 and θsk = 60°• Influence of number of steps

Half load

Torque ripple variation

EUGM 2014


4. Summary

• The effectiveness of skewing on the torque ripple reduction largely depends on the axial variation of torque ripple phase but less on the magnitude.

• Skewing one on-load torque ripple period, the cogging torque is eliminated. However, the EM and hence on-load torque ripple cannot be eliminated.

• After skewing, the EM torque ripple can be reduced if there is no saturation, but may be reduced or even increased if it is saturated depending on the electric loading.

• Skewing is effective in reducing the cogging torque. Hence, when the cogging torque is higher, the skewing is more effective on the on-load torque ripple reduction. In contrast, when the EM torque ripple is high, it is less effective.

• The on-load torque ripple can be even increased by skewing especially when the cogging torque is low and the EM torque ripple is high.

EUGM 2014


5. Conclusions

• Torque components can be separated based on the combination of virtualwork principle and frozen permeability using correct energy calculation.

• On-load cogging torque cannot be calculated by Tmw(FP, PM) but can bepredicted by -∂Wm(FP,PM)/∂θm combined with nonlinear FP method.

• Skewing is effective in reducing the cogging torque.• After skewing, the on-load torque ripple is not necessary reduced.

• The on-load torque ripple may be even increased by skewing.

Documents

Investigation on Torque in PM Machines - OperaFEA · Investigation on Torque in PM Machines ... Radial airgap flux density ... Q-axis flux linkage Flux density distribution Flux linkage