electromagnetic models

Transcription

1 Power Conversion & Electromechanical Devices Medical Physics & Science Applications Transportation Power Systems for electromagnetic models Overview Potential types Use of TOTAL and REDUCED potentials Post processing Topological constraints Accuracy Discretization Quadratic and linear elements Coil fields Far-field boundary Skin-effect 2

2 Review of Potential Types (1) TOSCA electrostatics, current flow and SCALA TOTAL electric scalar potential, V, is defined by E = V TOSCA magnetostatics TOTAL magnetic scalar potential, ϕ, is defined by H = ϕ Should be used in magnetic material regions, including permanent magnets Will be set if potential type in a cell is Default 3 Review of Potential Types (2) TOSCA magnetostatics (continued) REDUCED magnetic scalar potential, φ, is defined by H = φ + H S Must be used in regions where source currents (coils) exist ELEKTRA, CARMEN, DEMAG and SOPRANO Magnetic VECTOR potential, A, is defined by A E = t V 4

3 Review of Potential Types (3) ELEKTRA, DEMAG and CARMEN in reduced potential volumes, reduced magnetic vector potential, A m, is given by B = B S + A m Applying Default choice for potential in the Modeller will enforce the correct choice of potential type. In magnetic problems with Biot-Savart coils (TOSCA, ELEKTRA, DEMAG and CARMEN), Default gives: Reduced potential (scalar or vector) in AIR Total potential (scalar or vector) in other materials Includes meshed coils for circuits 5 Choice of Potential To use TOSCA magnetostatics, ELEKTRA, DEMAG and CARMEN with Biot-Savart coils most effectively requires the user to make decisions on which potential to use in regions of free space (material AIR) where there are no coils. Choice of potential can affect: Speed Accuracy Images 6

4 Effect of Potential on Analysis Speed Keep size of REDUCED potential small Program must compute Biot-Savart integral at each node in REDUCED region Make shape of REDUCED region only just fit around coils Make all other AIR total potential BUT!!!!!!! 7 Effect of Potential on Analysis Speed Keep shape of REDUCED potential region simple Programs compute the numerical line and surface integrals of the coil fields on the interface surface between REDUCED and TOTAL potentials Connects TOTAL and REDUCED potentials together See also lecture 1-4 Complex surface shapes generate a complex tree to compute integrals at all nodes on the interface Coil field values and integrals close to the coils take longer to compute More adaptive steps in numerical integration 8

5 Two options for a pair of saddle coils Complex interface surface REDUCED region Simple interface surface Complex interface takes nearly 2 times longer to solve in TOSCA 9 Effect of Potential on Accuracy Keep interface of REDUCED potential far away from coils Integrals can be performed more accurately Make REDUCED potential region large Extract from.res file of simple interface saddle coil model Note: Interface is close to coil in places Discretisation sat inadequate to model coil fields at: (0.250, 0.175, ) (cm), error= 2.374% ****** (0.150, 0.175, ) (cm), error= 1.984% ****** (0.300, 7.500E-02, ) (cm), error= 2.688% ******* (0.275, 7.500E-02, ) (cm), error= 4.788% ******** (0.225, 7.500E-02, ) (cm), error= 1.308% ***** Calculating coil fields: 42.1 s cp, 0.8 m elapsed 10

6 Effect of Potential on Accuracy Extract from.res file of complex interface saddle coil model Discretisation inadequate to model coil fields at: (0.280, 2.500E-02, ) (cm), error= 4.046% ******** (0.310, 2.500E-02, ) (cm), error= 7.995% ********* (0.310, 2.500E-02, ) (cm), error= 7.984% ********* (0.310, 2.500E-02, ) (cm), error= 8.008% ********* (0.310, 2.500E-02, ) (cm), error= % *********** (0.310, 2.500E-02, ) (cm), error= % ************ (0.259, 2.500E-02, ) (cm), error= 1.297% ***** (0.280, 2.500E-02, ) (cm), error= 3.013% ******* (0.182, 0.172, ) (cm), error= 3.191% ******* (0.286, 4.623E-02, ) (cm), error= 2.488% ****** (0.243, 0.203, )(cm), error= 3.132% ******* (0.280, 2.500E-02,-7.4E-02) (cm), error= 5.257% ******** (0.280, 2.500E-02, ) (cm), error= 4.444% ******** (0.314, 2.500E-02,-5.9E-02) (cm), error= % *********** (0.318, 2.500E-02,-3.6E-02) (cm), error= % ************ (0.310, 2.500E-02, ) (cm), error= 7.954% ********* (0.280, 2.500E-02, ) (cm), error= 3.493% ******* (0.280, 2.500E-02, ) (cm), error= 6.097% ******** (0.311, 2.500E-02, ) (cm), error= % ********** (0.315, 2.500E-02, 4.8E-02) (cm), error= 8.893% ********* (0.313, 2.500E-02, 2.3E-02) (cm), error= % ************ (0.259, 2.500E-02, ) (cm), error= 1.037% ***** (0.206, 0.142, ) (cm), error= 1.316% ***** (0.310, 2.500E-02, ) (cm), error= % ********** (0.310, 2.500E-02, ) (cm), error= % ************ (0.280, 2.500E-02, ) (cm), error= 5.186% ******** (0.310, 2.500E-02, ) (cm), error= % ********** (0.310, 2.500E-02, ) (cm), error= % ************ (0.280, 2.500E-02, ) (cm), error= 3.460% ******* (0.310, 2.500E-02, ) (cm), error= % ********** (0.310, 2.500E-02, ) (cm), error= % ********** (0.428, 4.006E-02, ) (cm), error= 2.981% ******* (0.268, 0.210, ) (cm), error= 1.209% ***** Calculating coil fields : 96.5 s cp, 1.8 m elapsed 11 Effect of Potential on Accuracy Field on line close to TOTAL/REDUCED interface 12

7 Summary Two opposing requirements Small REDUCED potential region close to coils Biot-Savart fields at nodes are fast Biot-Savart field integrals at interface are slow Harder to get accurate integrals Large REDUCED potential region with interface distant from coils Lots of nodes in REDUCED region => Biot-Savart fields at nodes are slow Biot-Savart field integrals at interface are fast Easier to get accurate integrals Default choice of potential usually gives large REDUCED potential region 13 Choice of potential and post processing Potential type also affects possibilities for field evaluation in the Post-processor Default TOTAL potential Total field values are recovered by nodal averaging REDUCED potential Total field values are recovered by summing nodal coil and nodal magnetization (+ eddy current) fields See also lecture

8 Choice of potential and post processing Other possibilities in the Post processor REDUCED potential Coil fields may be re-computed at the evaluation point from Biot-Savart expression Any potential Total fields may also be recovered by integration Biot-Savart expression for fields from currents (source and eddy currents) Integration of magnetization See lecture 2-1 for choosing field recovery method 15 Automatic cuts in TOSCA magnetostatics Iron ring Total magnetic scalar potential Rest of space Reduced magnetic scalar potential Current carrying coil 16

9 Automatic cuts in TOSCA magnetostatics Iron ring completely surrounds current carrying coil a problem! Why? Ampere s law states: H dl = I But in total magnetic scalar potential regions: H = ϕ Contour integral of a gradient around a closed contour is zero Therefore, ϕ.dl = 0 = I 17 Automatic cuts in TOSCA magnetostatics The iron ring is a multiply connected region of TOTAL potential TOSCA introduces a cutting region Extract from RES file Checking connectivity of potential regions: 1 automatic potential cut has been added. 18

10 Multiply connected regions in TOSCA Opera automatically inserts a cut This makes the total potential discontinuous Field is continuous 19 Multiply connected regions Not an issue that needs to be considered in TOSCA current flow TOSCA electrostatics SCALA SOPRANO None of these programs use TOTAL and REDUCED scalar potentials In TOSCA magnetostatics, cuts are inserted automatically. ELEKTRA, CARMEN and DEMAG use reduced/total vector potential Also introduce automatic cuts for coil field integral evaluation 20

11 Discretization Opera-3d allows the user to choose which parts of the model use LINEAR (1st order) and which parts use QUADRATIC (2nd order) elements ELEKTRA, CARMEN, DEMAG & SOPRANO only use first order edge elements irrespective of settings Generally, it is too expensive to make all of a problem QUADRATIC e.g tetrahedral elements make 6004 nodes - all linear elements nodes - mixed linear and quadratic elements (typically) nodes - all quadratic elements Comments on discretization also apply to TEMPO, STRESS and QUENCH 21 Discretization magnetic field models Confine the QUADRATIC element regions to: Parts of the model where a high accuracy is required e.g. image region of an MRI magnet Parts of the model where the field gradients are very high e.g. end region of an electrical machine Concentrate other elements in areas of interest Use larger elements in the far-field fi Use enough elements to model variation of permeability in non-linear analyses 22

12 Curved surface elements Forces quadratic elements on curved surfaces of model 23 Curved surface elements Linear elements throughout Curved surface elements Note also the difference in the range of values 24

13 Discretization for skin-effect in ELEKTRA, CARMEN, DEMAG and SOPRANO In eddy current models, fields and currents decay exponentially from the surface of conducting materials. Skin-depth δ = π 1 f 0 μ μ σ r J = J 0 e x δ where J 0 is the surface value and x is the normal direction to the surface into the conductor 25 Discretization for skin-effect in ELEKTRA, CARMEN, DEMAG and SOPRANO The finite element discretization of the model must be adequate to represent the exponential decay. Minimum discretization: 2 elements in the first skin-depth 1 element in the second skin-depth Larger than skin-depth below second skin-depth Better discretization leads to more accurate loss calculations 26

14 Discretization for skin-effect in ELEKTRA, CARMEN, DEMAG and SOPRANO Cell element size = 5 mm Top face element size = 1 mm elements in block Eddy currents induced in copper plate at 500 Hz δ 3 mm Use layered mesh 1720 elements in block 27 Layering types Two types of layering are available Geometric Constructs new cells and copies surface mesh from top of cell to bottom of cell -> top of next cell Planar, cylindrical and spherical surfaces only Mesh Maps mesh non-uniformly into original cell Available in regions that can be meshed with hexahedral or prism elements Any shape surface 28

15 Geometric and mesh layering 29 Layering in the Modeller Layering is also useful to aid the modelling of thin structures in such applications as: Machine air gaps Magnetic wall shielding Ship hulls Layering is specified as a face property: Type of layering Number of layers Depth of layers May be functional Direction of layering Forward/Backward 2 layers with geometric layering 30

16 Surface Impedance Boundary Condition In some models, layering cannot produce elements that can support accurate calculation of eddy currents Elements are so thin that mesh generation fails Unsaturated steel at 10 khz, δ 100 μm ELEKTRA-SS can use a surface impedance boundary condition (SIBC) Assumes 1-d skin-effect theory Interior of conducting medium is not included in the model SIBC switched on in material properties 31 Plate at 500 khz using SIBC Distribution of current t/fi field inside plate is not available Integral values such as force, power dissipation etc can be calculated 32

17 Far-field boundary The position of the far-field boundary affects the solution Real space extends to infinity but the model is artificially truncated The boundary condition defines the types of image of the model produced by truncation NORMAL MAGNETIC says that field will be normal to far-field boundary TANGENTIAL MAGNETIC says that t field will be tangential to far-field boundary If REDUCED potential is at the boundary, these conditions only apply to the field component represented by the reduced potential 33 Far-field boundary Permanent magnet with close boundary Tangential magnetic Normal magnetic 34

18 Far-field boundary The boundary needs to be far-enough away that it does not affect the results in the area of interest. Run 2 models with different types of boundary condition. Does the solution in important regions change beyond acceptable limit? If it does, move the boundary 2 times as far away Look at B on boundary (only needs 1 solution) Half field from model Half field from images Important in shielding calculations 35 Far-field boundary Application dependent MRI magnet Open system Boundary at least 100 times magnet size Motor and generator Closed system Tangential magnetic boundary at back of stator if not saturated t Suggestion - start at 10x object size if you are unsure 36