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

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Oswin Ira Nicholson
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1 Assignment 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 threephase 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.

Assignment 4 2-1 Figure 4.1 Two types of windings: distributed winding (left) and concentrated winding (right).

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 a phase b phase c.866 1.5 -.

2 Assignment 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 a phase b phase c /6 1/3 1/2 2/3 5/6 1 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. 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. MMF(θ), B(θ) Q R θ stator P θ 2 θ 1 S ge.5ni B θ g rotor π 2π Figure 4.3 Circular and opened machine with surface mounted magnets with excitation B-wave and single coil with armature MMF-wave.

3 Assignment 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 ge. All flux lines that enclose N t I ampere turns build MMF and it changes abruptly in the middle current conducting wire where the direction of H field is changed. F Ni Q P Hdl S R Hdl g e H Ni g H g H e e r g H 2 e r 1 (4.1) (4.2) The rectangular waveform of full pitch winding, 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 MMF smh 4 Nt I 2 h1,3,5, sinh 2 h 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 B gmh 4 B gm h1,3,5, sinh k 2 h m 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) 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 construction is that these waves are both sinusoidally distributed and the magnetic forces are built on the fundamental electric angular frequency ω e. The geometric arrangements of permanent magnets and their magnetisation direction cause a radial flux density in the gap that peak value for fundamental wave is B gm1, B g P t B sin t (4.3). (4.4) N, gm1 e, (4.5) 2 The set of balanced sinusoidal currents I(θ) in the axially oriented stator winding of N t turns gives a MMF(θ) distribution around the stator periphery, where MMF sm1 is peak value for the fundamental stator MMF wave. MMF s N, P 1, (4.6) t MMFsm sin et 2 Only the fundamental space component B gm1 of the air-gap flux density, which interacts with the fundamental of the MMF F sm1, produces the desired torque. The waves MMF( 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:

4 Assignment T B gm1 N P sin P et N MMFsm1 sin et rr lrd (4.7) 2 2 which after integration and simplifications becomes: N P T Bgm 1MMFsm1rr le sin (4.8) 2 Multi-phase winding and field oriented control Single coil is able to produce only bidirectional alternating field. Additional displaced coils introduce new field orientation so that the 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 excitation 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 operation. According to the field oriented control the current vector is orthogonal to field vector (along x-axis) and opposite to back emf vector (along 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 5-1 Figure 4.5 flux density distribution over the cross-section of the loaded and un-loaded machine Magnetic flux density inthe airgap B g, [T] 1.5.8 B gnl (), [T] B gtl (), [T] 1 B gn (), [T] B gt (), [T].

5 Assignment Figure 4.5 flux density distribution over the cross-section of the loaded and un-loaded machine Magnetic flux density inthe airgap B g, [T] B gnl (), [T] B gtl (), [T] 1 B gn (), [T] B gt (), [T] M agnetic flux density in the airgap B g, [T] B gnl (), [T] B gtl (), [T] B gn (), [T] B gt (), [T] harmonic order, [-] Figure 4.6 flux density waveform and spectrum in the air-gap 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 dominating compared to the tangential component. Alternatively, 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 t n and the shear stress t t relate to the gap flux density normal component B n and tangential component B t in respect of the pole surfaces. 1 tn B 2 2 n B 2 t (4.9) t t B B (4.1) n t Magnetic shear stress in the airgap () or t(), [N/m 2 ] x (), [N/m 2 ] 2 (), [N/m 2 ] 3 (), [N/m 2 ] 4 (), [N/m 2 ] B n t n α α B t angle, [deg] B t t t Figure 4.7 magnetic stress waveforms in the gap and vector presentation on stress and flux density vectors

Assignment 4 6-1 There is one more plot that is calculated from the flux density distribution by using the force tensor expressions. 2.

torque from the loaded machine share stress waveform.

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 computed on a number of contours in the air-gap stator Torque integration contours Torque integration line rotor Figure 4.

Notice that a tenser FE-mesh is generated in the airgap region in order to obtain more accurate field distribution in the gap region. 3.

Vector diagrams Study the three phase circuit parameters: currents, voltage drops and flux linkages.

6 Assignment 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 in the middle of air-gap 2. Torque is integrated from the electromagnetic stress, which is computed on a number of contours in the air-gap stator Torque integration contours Torque integration line rotor Figure 4.8 torque computation around a selected body from a number of contours and weighted results or a single circular line in the middle of air-gap. Notice that a tenser FE-mesh is generated in the airgap 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 with 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 j j j k a be ce (4.11) When the current and flux vectors are on their place, then construct also voltage vector by defining rotation direction and speed ω. With the initial values the obtained voltage is across a single turn and this per turn quantity can be easily scaled to the actual supply voltage. Current in wire total current 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 magnetics does?

7 Assignment Construct space vector diagram of currents, fluxes and voltages per single winding turn 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. 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 2. Turn the rotor -3 degrees: From and -3 degrees 3. Save the file with different tmp_magn_study.fem menu Edit Move, select Rotation around, name: From menu File Save As, e.g. 4. Solve and open analysis results: From menu Analysis Analyze and View Results. If flux density plot or flux lines are not visible, then they need to be activated from the (post-processor) menu Density Plot, Contour Plot, Vector Plot 5. Define circular contour in the middle of gap (A:1-2-3): From menu Operation Contours Right mouse click on the right side and in the middle of gap, the second click on the opposite (left side) and in the middle of gap and press on Shiftkey. In this alert define curvature angle of 18 degrees. In order to close the circle you should click close by starting point and with Shift-Key define the curvature angle of 18 degrees.

8 Assignment A B 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

9 Assignment Solve and open analysis results: From (pre-processor) menu Analysis Analyze and View Results. 1. Calculate torque and show that 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. 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 718 or something higher from to zero.

Assignment 4 1-1 13. Solve and open analysis results: From (pre-processor) menu Analysis Analyze and View Results. Take a copy of flux density distribution, B-field 14.

10 Assignment Solve and open analysis results: From (pre-processor) menu Analysis Analyze 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 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. 16. Take a copy of field intensity distribution, H-field, are you able to explain what do you see?

Unified Torque Expressions of AC Machines. Qian Wu

Unified Torque Expressions of AC Machines Qian Wu Outline 1. Review of torque calculation methods. 2. Interaction between two magnetic fields. 3. Unified torque expression for AC machines. Permanent Magnet