4 Finite Element Analysis of a three-phase PM … Element Analysis of a three-phase PM synchronous machine . ... to link to the entire available excitation field. For the practical

Assignment 4 1-10

4 Finite Element Analysis of a three-phase PM synchronous machine

The goal of the third assignment is to extend your understanding on electromagnetic analysis in FEM. This assignment is a continuation from the previous design task, where the initial design is selected for a detailed study in FEMM. Your task is to analyse single machine geometry from EMK_task_3 in order to analyse various aspects by the help of FEMM. From this assignment you should be able to calculate torque, obtain flux linkage and to establish experience on

― Winding distribution for a three-phase machine and derivation of back emf waveform from flux linkage waveforms

― Flux density waveforms in the core and initial data for core loss estimation

― Vector diagram of the machine and parameter estimation for field oriented control

― Magnetic forces in the air-gap and torque computation of loaded and unloaded machine

4.1 Getting started, assignment flowchart Quick progressing guide

― Specify the machines that you studied previously with EMK_task_3.m. Continue the analysis in FEMM by using EMK_task_4.m. in Matlab.

― Assignment tasks are highlighted after the tutorial text

― Study instantaneous values of torque, phase currents and fluxes at the predefined rotor position. Construct vector diagram where you can show, fluxes, currents and voltages.

― Take a closer look to flux density distributions along the air-gap and draw some conclusions about stress components

― Hands on learning instructions on tmp_magn_ini.fem for deeper study.

4.2 Analysis object As an outcome from previous home assignment you are asked to continue the analysis of your specified surface mounted permanent magnet synchronous machine (SPMSM). This continuation brings you closer to detailed analysis and different electromagnet quantities such as flux density variation in different parts of the machine and the use of this information in machine analysis, magnetic flux linkage from the rotor field to the stator coils, three-phase winding distribution, current supply and resulting torque waveforms as a function of rotor position.

3-phase winding and coil distribution

The angular span of the coils is generally as wide as the excitation pole pitch of the rotor in order to link to the entire available excitation field. For the practical arrangements of the coils to the three phase windings the coils can be overlapped or non-overlapped. According to that the three-phase winding can be either distributed or concentrated Figure 4.1. For conventional alternating sinusoidal 3-phase supply is that the currents and the voltages are symmetric at fundamental waveform frequency without high harmonic content Figure 4.2. Consequently the distribution of the excitation field, which is perpendicular to and along the rotor surface, supposes to be sinusoidal as well as the distribution of current carrying coils around this surface.

EIEN20 Design of Electrical Machines, IEA, 2016

Assignment 4 2-10

Figure 4.1 Two types of windings: distributed winding (left) and concentrated winding (right). Phases are marked with different colours that are al in accordance with the following figure. The positive phase current supposes to produce phase field along the phase ‘projection’ axis. The permanent magnets are shown as a number of green filled regions and the magnetization direction aligns (in the figure) with phase a-axis.

phase aphase bphase c

0 1/6 1/3 1/2 2/3 5/6 1-1

-0.866

-0.5

0

0.5

0.8661

Figure 4.2 Three phase projection space and the corresponding excitation waveforms. The initial rotor position and current values are selected according to time instant ‘0’.

Practical arrangement of rotor and stator “magnets”

The coils are distributed in (machine) space so that the time dependent phase currents create a travelling/rotating field that counteracts with excitation field and produces electromagnetic torque. This torque is ideally continuous and constant at each position of the rotor. In prior, to complete three phase system a single phase system with a single turn coil is observed as a temporal magnet that magnetization gap ge is larger than the one (g) for permanent magnets.

θ2

θ1

θ

P

Q R

S

g

ge

θ

0 π 2πrotor

stator

MMF(θ), B(θ)

0.5NIB

Figure 4.3 Circular and opened machine with surface mounted magnets with excitation B-wave and single coil with armature MMF-wave.


Assignment 4 3-10

The circular machine is opened and rolled to planar machine in order to visualise MMF(θ) and B(θ) waves along the gap. By assuming infinitely permeable magnetic core in the stator and in the rotor, the flux path can be assumed and the resulting magnetic field intensity H drops across the gap d MMF and it changes abruptly in the middle current conducting wire where the direction of H field is changed.

(4.1)

ge. All flux lines that enclose Nt I ampere turns buil

12 rere

S

R

Q

P

HgHgHdlHdlNi

0 HgHgNiF ee (4.2)

The rectangular waveform of full pi inding, where the angular width of coil is the same as the angular width of the pole (π electrical radians) determines the fundamental component and the harmonic content of the MMF(θ) wave according to

tch w

,5,3,1

2sin

2

4

h

tsmh h

hIN

MMF

(4.3)

Similar to the MMF waveform the rectangular shape of the gap flux density waveform is mainly determined by the angular width of permanent magnets. The fundamental and the harmonic components for the peak value of the gap magnetic flux density are given by

2

sin4 m

gm

khB

,5,3,1h h

Sinusoidally fed PM machine

In a rotary motor of interest (in this assignment), the rotating part (rotor) includes permanent magnets that establish the magnetic field and the stationary part (stator

gmhB

. (4.4)

) includes the current carrying coils. The excitation wave of the B(θ) field and the armature sheet current wave K(θ) rotate in synchronism. The target for the machine sinusoidally distributed and the magnetic forces are frequency ωe. The geometric arrangements of permanent magnets and their magnetisation direction cause a radial flux density in the g

construction is that these waves are both built on the fundamental electric angular

ap that peak value for fundamental wave is Bgm1,

tN

BtB eP

gmg 2

sin, 1 , (4.5)

The set of balanced sinusoidal currents I(θ) in the axially oriented stator winding of Nt turns gives a MMF(θ) distribution around the stator periphery, where MMFsm1 is peak value for the fundamental stator MMF wave.

tMMFtMMF esms 2sin, 1 , (4.6)

Only the fundamental space component Bgm1 of the air-gap flux density, which interacts with the fundamental of the MMF F , produces the desired torque. The waves MMF(

N P

sm1 θ) and MMF(θ) are not necessarily orthogonal and this is counted by electrical angle β. The total force over the entire surface of the rotor and therefore the torque can be found by:


Assignment 4 4-10

dlrtMMFtBT rreP

smegm

0

11 2sin

2sin (4.7)

which after integration and simplifications becomes:

N P 2 N

sin2 11 ersmgmP lrMMFB

NT (4.8)

Multi-phase winding and field oriented control

Single coil is able to produce only biintroduce new field orientation so that t

on the phase-A builds its field along a-axis, the applied Lorenz forces on the stator A and B coils forces the rotor opposite to the rotation direction, which means that the machine is in generator operationcurrent vector is orthogonal to field vector (along x-axis) and opposite to back emf vector (along

directional alternating field. Additional displaced coils he coil displacement in combination with current supply

can create a desired direction of the field and also a rotating magnetic field. A two pole 3-phase PM machine is shown (Figure 4.4) where the wiring directions of the phase coils are shown out from the plane direction (+) and in to the plane direction (-). According to the definition of coil orientation and the current excitati

. According to the field oriented control the

y-axis).

Figure 4.4 Principle drawing of a two pole-pair permanent magnet synchronous motor. The stationary three phase system together with the armature phase coils are shown, where each coil produces field in the respective axis. Apart from that the two phase systems: stationary (αβ) and rotary (xy) are drawn. The control theory in accordance with Lorenz force law is established on the rotor (xy) coordinate system.

Observation of calculation results

EMK_task_4 is configured to execute FEMM directly in Matlab and carry out electromagnetic analysis. The outcome of the initial task shows flux density distribution in FEMM, various data in Matlab workspace and figures with waveforms.

1. Study the flux density distribution in the middle of air-gap. Try to find values of peak value for fundamental gap flux density Bgm1 estimated in A3 and evaluated in A4.


Assignment 4 5-10

Figure 4.5 flux density distribution over the cross-section of the loaded and un-loaded machine

0 20 40

1.5 0.8

60 80 100 120 140 160 180-1.5

-1

-0.5

0

0.5

1

Mag

netic

flu

x de

nsity

inth

e ai

rgap

Bg,

[T]

BgnL

(), [T]

BgtL

(), [T]

Bgn0

(), [T]

Bgt0

(), [T]

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

BgnL

(), [T]

BgtL

(), [T]

Bgn0

(), [T]

Bgt0

(), [T]

0.7

agne

tic f

lux

dens

ity in

the

airg

ap B

g, [T

]

harmonic order, [-]

M

Figure 4.6 rm and spectrum in the air-gap

ating compared to the tangential component.

d the shear stress tt relate to the gap flux density normal component Bn and tangential component Bt in respect of the pole surfaces.

flux density wavefo

Gap forces

Magnetic forces in an air-gap of conventional electrical machine cause magnetic attraction and tensile forces between the bodies i.e. between permanent magnets and iron parts. Hereby, it is assumed that the normal field component is dominAlternatively, the compressive stresses would take place. The magneto-mechanic load to the bodies becomes the biggest when the poles are aligned, the gap flux is the largest and the magnetic potential surfaces are perpendicular to the pole surfaces. The normal stress tn an

22

02

1tnn BBt

(4.9)

0tn

t

BBt

(4.10)

0 30 60 90 120 150 180 210 240 270 300 330 360

-3

-2

-1

0

1

2

3

x 105

B 4

angle , [deg]

Mag

netic

she

ar s

tres

s in

the

airg

ap

( )

or t

( ),

[N

/m2 ]

t

1(), [N/m2]

2(), [N/m2]

3(), [N/m2]

4(), [N/m2]

Bn

tn

Bt tt

α α

Figure 4.7 magnetic stress waveforms in the gap and vector presentation on stress and flux density vectors


Assignment 4 6-10


There is one more plot that is calculated from the flux density distribution by using the force tensor expressions.

2. This plot shows stress tensors and your task is to distinguish which belongs to unloaded or loaded machine and which of them is shear stress and whish is tensile stress and calculate torque from the loaded machine share stress waveform.

Torque computation with FEMM

The accuracy of the field representation is vital when calculating electromagnetc stress tensor in the air-gap. There are two approaches used in the post-processing file and in order to obtain a good result a tenser FE-mesh has been used in the air-gap region:

1. Torque is integrated from the electromagnetic stress, which is calculated on a circle line

e is integrated from the electromagnetic stress, which is computed on a number of

in the middle of air-gap

2. Torqucontours in the air-gap

rotor

stator Torque integ-

ration contours Torque integ-

ration line

Figure 4.8 torque compusingle circular line in the middle of air-gap. Notice that a tenser FE-mesh is generated in the air-gap region in order to obtain more accurate field distribution in the gap region.

3. Your task is to compare line integral, weighted and previously calculated torque h expected analytic outcome in A3.

Vector diagrams

Study the three phase circuit parameters: currents, voltage drops and flux linkages. Use this information and derive ψm and ψs space vectors, which correspond to the resultant flux linkages for the unloaded and loaded machine, respectively. Visualize also current vector and observe the angle between the magnetization flux linkage and current vector.

tation around a selected body from a number of contours and weighted results or a

wit

3

4

3

2

j

c

j

ba eekj (4.11)

When the current and flux vectors a e on their place, ining rotation

the obtained voltage is across a single turn and this per

ply

t in “insulated” slot → current distribution → torque as a result of current and magnetisation flux T~ψ I, Are you able to formulate force and torque any other way than FE for ma

rthen construct also voltage vector by defdirection and speed ω. With the initial values

turn quantity can be easily scaled to the actual supvoltage.

Current in wire → total curren

gnetics does?

Assignment 4 7-10

4. Construct space vector diagram of currents, fluxes and voltages per single winding turn

4.3 FEMM hands on learning The last group of exercises is given in order to improve your skills on FEM software. These exercises are giving you a chance to try out different aspects on field computation that are related to model definition, result interpretation and not leas active learning.

1. Select rotor: From menu →Operation→Group, press Tab-key, alert opens, write 2 into window and close the alert. Group 2 will be selected and marked read

per phase and slot from the data given by FEMM

Static characteristics

Run the second half of the post-processing code (con.femm=1;) and obtain the current, flux linkage, flux density in core and the torque as a function of rotor position.

5. Copy the calculated waveforms and estimate the back emf. Calculate the torque from flux and current cross-product.

2. Turn the rotor -30 degrees: Fromand -30 degrees

3. Save the file with different tmp_magn_study.fem

4. Solve and open analysis resulResults. If flux density plot or flux lifrom the (post-processor) menu →De

5. Define circular contour in th

grees. In order to close the circle you y starting point and with Shift-Key define the curvature angle of 180

menu →Edit→Move, select Rotation around 0,0

name: From menu →File→Save As…, e.g.

ts: From menu →Analysis→Analyze and View nes are not visible, then they need to be activated nsity Plot, Contour Plot, Vector Plot

e middle of gap (A:1-2-3): From menu e click on the rig

→Operation→Contours Right mous ht side and in the middle of gap, the second click on the opposite (left side) and in the middle of gap and press on Shift-key. In this alert define curvature angle of 180 deshould click close bdegrees.


Assignment 4 8-10

1

3

2

A B3

6. Plot flux density distribution in the gap (B): From menu →Plot X-Y→Plot type: normal flux density

7. Record magnetic flux linkage (from magnets only as current is zero): From menu →View→Circuit Props: use winA, winB and winC values to determine flux linkage space vector from magnets (or from excitation field in general) Ψm

8. Define current vector orthogonal to flux vector so that it gives negative torque: From (pre-processor) menu →Properties→Circuits→Property Definition: select winA, winB and winC →Modify Property where you are asked to specify this instantaneous phase current value that corresponds to the predefined position of the space current vector. Please notice that the peak value for the total current (huge current per single winding turn) is Ae*con.Jm


Assignment 4 9-10

9. Solve and open analysis results: From (pre-processor) menu →Analysis→Analyze

at the value is acceptable: From ((post-processor) menu →Operations→Areas: double-click on rotor so that all regions become selected. Then again from menu→Integrate→Block Integrals→Operations→Torque via weighted stress tensor. Record the value and compare it with previous results.

and View Results.

10. Calculate torque and show th

11. Record magnetic flux linkage (from magnets and stator field as current differs from zero): From menu →View→Circuit Props: use winA, winB and winC values to determine the resultant (stator) flux linkage space vector Ψs

12. Switch off the permanent magnets: From (pre-processor) menu →Properties→Materials→Property Definition: select mag1 →Modify Property where you are asked to reduce the coercivity Hc from 718000 or something higher from 0 to zero.


Assignment 4 10-10

13. Solve and open analysis results: From (pre-processor) menu →Analysis→Analyze

15. Repeat steps 5 and 6 Define circular contour in the gap and plot flux density distribution in the gap (B): From menu →Plot X-Y→Plot type: normal flux density. Estimate approximately “flat” top value of the B-waveform, calculate H-top value and multiply it with the wide gap length (excluding permanent magnets), how this value is related to the total current.

and View Results. Take a copy of flux density distribution, B-field

14. Record magnetic flux linkage (from stator field only as permanent magnet field is zero): From menu →View→Circuit Props: use winA, winB and winC values to determine flux linkage space vector from stator electromagnets only LsIs

16. Take a copy of field intensity distribution, H-field, are you able to explain what do you see?


4 Finite Element Analysis of a three-phase PM … Element Analysis of a three-phase PM synchronous machine . ... to link to the entire available excitation field. For the practical

Documents