1. Maxwell 2D Boundaries/Sources, Meshing, and Post Processing

2. T3_2D, pg. 2 6/28/02 Boundaries & Sources Toolbar  The tool bar is located on the top of the Boundary Manager window.  Begin “picking” by left clicking  Finish “picking” by right clicking

3. T3_2D, pg. 3 6/28/02 Boundaries & Sources Typical Boundary Conditions  Balloon (for problems with fringing fields)  Value A = 0, V = 0 (no fringing)  Master/Slave (motor problems)  Symmetry (periodic geometries - if possible, model entire model first to see how fields behave) NOTE: boundary condition on the outer edge of problem region must always be specified unless background is excluded

4. T3_2D, pg. 4 6/28/02 Boundaries & Sources Electrostatic Boundaries Boundary Type Electric Field BehaviorUsed to Model… NewmannE and D are tangential to the boundary.Default outer boundary. NaturalE is continuous across the boundary.Default boundary between objects. Value Sets the electric potential, , on the boundary. The behavior of E depends on whether  is constant or functional. Boundaries at known voltages. BalloonTwo options are available: w Charge — The charge at “infinity” balances the charge in the drawing region. The net charge is zero. (Use for capacitance calculations) w Voltage — The voltage at “infinity” is zero. Electrically insulated structures (Charge option) or electrically grounded structures (Voltage option). Even Symmetry E is tangential to the boundary.Planes of symmetry where the signs (plus or minus) of all voltages and charges are the same on both sides of the boundary. Odd Symmetry E is perpendicular to the boundary.Planes of symmetry where the signs (plus or minus) of all voltages and charges on one side of the boundary are opposite those on the other side. “Master” Matching Boundary E has the same magnitude and direction (or the same magnitude and opposite direction) on the master boundary and all slave boundaries that are assigned to it. Planes of symmetry in periodic structures where E is neither tangential to nor perpendicular to the boundary. Slave Matching Boundary The E-field on the boundary is forced to match the magnitude and direction (or opposite direction) of the E- field on the master boundary to which it is assigned. Planes of symmetry in periodic structures where E is neither tangential to nor perpendicular to the boundary.

5. T3_2D, pg. 5 6/28/02 Boundaries & Sources Magnetostatic Boundaries Boundary Type Magnetic Field BehaviorUsed to Model… NewmannB and H are perpendicular to the outer edges of the problem space. Default outer boundary condition. NaturalNormal components of B and tangential components of H are continuous across the edge. Default boundary between objects. Value (Dirichlet) Sets the magnetic vector potential, A Z or rA , on the boundary. The behavior of H depends on whether A Z or rA  is constant or functional. Outer boundaries at specific vector potentials; externally applied magnetic fields. BalloonModels the case where the structure is “infinitely” far away from other magnetic fields or current sources. Magnetically isolated structures. Even Symmetry H is perpendicular to the boundary.Planes of symmetry where the signs (plus or minus) of all currents are the same on both sides of the boundary. Odd Symmetry H is tangential to the boundary.Planes of symmetry where the signs (plus or minus) of all currents are opposite to those on the other side of the boundary. “Master” Matching Boundary H has the same magnitude and direction (or the same magnitude and opposite direction) on the master boundary and all slave boundaries that are assigned to it. Planes of symmetry in periodic structures where H is neither tangential to nor perpendicular to the boundary. Slave Matching Boundary The H-field on the boundary is forced to match the magnitude and direction (or opposite direction) of the H- field on the master boundary to which it is assigned. Planes of symmetry in periodic structures where H is neither tangential to, nor perpendicular to, the boundary.

6. T3_2D, pg. 6 6/28/02 Boundaries & Sources Eddy Current Boundaries Boundary Type Eddy Current Field BehaviorUsed to Model… NewmannB(t) and H(t) are perpendicular to the outer edges of the problem space. Default outer boundary condition. NaturalNormal components of B(t) and tangential components of H(t) are continuous across the edge. Default boundary between objects. Value (Dirichlet) Sets the magnetic vector potential, A Z (t) or rA  (t), on the boundary. The behavior of H(t) depends on whether A Z (t) or rA  (t) is constant or functional. Outer boundaries at specific vector potentials; externally applied magnetic fields. BalloonModels the case where the structure is “infinitely” far away from other magnetic fields or current sources. Magnetically isolated structures. Even Symmetry H(t) is perpendicular to the boundary.Planes of symmetry where the signs (plus or minus) of all currents are the same on both sides of the boundary. Odd Symmetry H(t) is tangential to the boundary.Planes of symmetry where the signs (plus or minus) of all currents are opposite to those on the other side of the boundary. “Master” Matching Boundary H(t) has the same magnitude, phase, and direction (or the same magnitude and opposite direction and phase) on the master all assigned slave boundaries. Planes of symmetry in periodic structures where H(t) is neither tangential to nor perpendicular to the boundary. Slave Matching Boundary The H-field on the boundary is forced to match the magnitude and direction (or opposite direction) of the H-field on the master boundary to which it is assigned. Planes of symmetry in periodic structures where H(t) is neither tangential to nor perpendicular to the boundary. ImpedanceIncludes the effect of induced currents beyond the boundary surface. Conductors whose skin depths are very tiny compared to the dimensions of the rest of the structure.

7. T3_2D, pg. 7 6/28/02 Boundaries & Sources Thermal Boundaries Boundary Type Thermal Field BehaviorUsed to Model… TemperatureAssigns a fixed temperature to the model.Temperatures. Convection and Radiation Defines the convective and radiative properties of the model. Temperature distributions and thermal radiation Heat Flux Density Defines the heat flux density of an object in water per meter squared. Heat flux on planar objects.

8. T3_2D, pg. 8 6/28/02 Boundaries & Sources Transient Boundaries Boundary Type Transient Field BehaviorUsed to Model… NewmannB and H are perpendicular to the outer edges of the problem space. Default outer boundary condition. NaturalNormal components of B and tangential components of H are continuous across the edge. Default boundary between objects. Value (Dirichlet) Sets the magnetic vector potential, A Z, on the boundary. The behavior of H depends on whether A Z is constant or functional. Outer boundaries at specific vector potentials; externally applied magnetic fields. BalloonModels the case where the structure is “infinitely” far away from other magnetic fields or current sources. Magnetically isolated structures. Even Symmetry H is perpendicular to the boundary.Planes of symmetry where the signs (plus or minus) of all currents are the same on both sides of the boundary. Odd Symmetry H is tangential to the boundary.Planes of symmetry where the signs (plus or minus) of all currents are opposite to those on the other side of the boundary. “Master” Matching Boundary H has the same magnitude and direction (or the same magnitude and opposite direction) on the master boundary and all slave boundaries that are assigned to it. Planes of symmetry in periodic structures where H is neither tangential to nor perpendicular to the boundary. Slave Matching Boundary The H-field on the boundary is forced to match the magnitude and direction (or opposite direction) of the H- field on the master boundary to which it is assigned. Planes of symmetry in periodic structures where H is neither tangential to, nor perpendicular to, the boundary.

9. T3_2D, pg. 9 6/28/02 Boundaries & Sources Typical Source Conditions  Voltage or Charge (for electrostatic problems)  Current (for magnetostatic and eddy problems)  Permanent Magnet (for magnetostatic problems)  Heat Source (for thermal problems) NOTE: Every problem needs at least ONE source of excitation

10. T3_2D, pg. 10 6/28/02 Boundaries & Sources Electrostatic Sources Electrostatic Source Type of Excitation VoltageTotal DC voltage on a closed geometric object. Edge voltageTotal DC voltage on an edge. ChargeCharge on an object (total charge or charge density). The object’s potential is computed during the solution. Charge sheetCharge on an edge (total charge or charge density). The edge’s potential is computed during the solution. Permanently polarized materials also act as sources of electric field

11. T3_2D, pg. 11 6/28/02 Boundaries & Sources Magnetostatic Sources Magnetostatic Source Type of Excitation Current or perfect current DC current flowing in an object (either the total current or the current density). Current sheet or perfect current sheet DC surface current on an edge or edges (either the total surface current or the surface current density). Permanent magnets also act as sources of magnetic fields.

12. T3_2D, pg. 12 6/28/02 Boundaries & Sources Eddy Current Sources Eddy Current Source Type of Excitation Current sheet or perfect current sheet Magnitude and phase of AC surface current on an edge or edges (either the total current or the current density). Current or perfect current Magnitude and phase of AC current flowing in an object. Can be one of the following: Solid* — Models eddy currents in a conductor. Parallel — Connects two or more conductors in parallel to an outside source. The total current flowing through all selected conductors (including eddy currents) is specified. Stranded — Models current as being carried on strands within the conductor, with no eddy or displacement currents. Either the total current or the current density may be specified. Current density is uniform, unless a functional current density is defined. Eddy effects are not modeled in a perfect conductor, but current is distributed on its surface so that no fields penetrate. *Specify the total current in the conductor: I Total = I Source + I Eddy + I Displacement

13. T3_2D, pg. 13 6/28/02 Boundaries & Sources Thermal Sources Eddy Current Source Type of Excitation SolidHeat source on a closed geometric object. Sheet Heat source sheet on an edge or edges. Thermal sources can be linked to a solved eddy current project

14. T3_2D, pg. 14 6/28/02 Setup Executive Parameters Executive Parameters are “common” calculations which can be automatically performed by the solution process if selected  Force  Torque  Flux Linkage  Current Flow (conduction solvers)  Inductance/Capacitance matrices  Post Processing macros (must be created first)  Core loss (eddy current solver)

15. T3_2D, pg. 15 6/28/02 Executive Parameter Core Loss  At a given frequency, the core loss for electrical steel is based on: where: K h is the hysteresis coefficient. K c is the classical eddy coefficient. K e is the excess or anomalous eddy current coefficient due to magnetic domains. B max the maximum amplitude of the flux density. f is the frequency.  The power ferrite core loss is based on: where: C m is constant value determined by experiment. f x is the frequency. B y max is the maximum amplitude of the flux density

16. T3_2D, pg. 16 6/28/02 Executive Parameter Core Loss  Kh, Kc, Ke coefficients derived for Armco M19 silicon steel

17. T3_2D, pg. 17 6/28/02 Setup Solution Options Options  Mesh (initial or current)  Residuals  Percentage refinement  Number of passes  Stopping criteria

18. T3_2D, pg. 18 6/28/02 Meshing Adaptive Refinement Process  Triangles are automatically refined to reduce energy error  Solution continues until one of two stopping criteria is met: 1. the specified number of passes are completed OR 2. percent error energy AND delta energy are less than specified Start Field Solution Generate Initial Mesh Compute Fields Perform Error Analysis Stop Field Solution Has Stopping Criteria been met? Refine Mesh Yes No

19. T3_2D, pg. 19 6/28/02 Meshing Aspect Ratio Limit  Aspect Ratio Limit: X = 10,000Y Y X

20. T3_2D, pg. 20 6/28/02 Meshing Manual Meshing Toolbar  The tool bar is located on the left side of the Manual Mesh window.  Two techniques possible  Seeding  Mesh Refinement

21. T3_2D, pg. 21 6/28/02 Meshing QuadTree Seeding

22. T3_2D, pg. 22 6/28/02 Post Processing  Allows solution results to viewed and exported in either graphical or numerical format  Graphical - Contour plots, Shade plots, Arrow plots, and Line plots  Numerical - Single complex number vector calculator for field quantity manipulation on plane, line, or point  Quantities such as Voltage, Flux, B, H, J can be analyzed in terms of vector components (X or tangential component)  Output to file (grid or user coordinates)  Macros (avg. Bx of an object)

23. T3_2D, pg. 23 6/28/02 Post Processing Standard Plots  Use Plot/Field for standard plots  Need to specify:  Quantity (B, H, J, loss, …)  On Geometry (point, line surface, or area)  In Area  For animated plots in Eddy Current solver, check phase animation

24. T3_2D, pg. 24 6/28/02 Post Processing Special Plots  Use Data/ Calculator for special plots  Manually load Quantity (B, H, J, loss, …)  Select Geometry (point, line surface, or area)  Select Plot (for plot on a surface or plane)  Select 2D Plot (for plot along a line)

25. T3_2D, pg. 25 6/28/02 Post Processing Calculator Stack and Registers Calculator Stack  The calculator is made up of a stack of registers, each of which can hold:  Field quantities such as the B-field or E-field.  Functional or constant scalars and vectors.  Geometries — points, lines, surfaces, or areas — on which a field quantity is to be evaluated.

26. T3_2D, pg. 26 6/28/02 Post Processing Calculator Basic Data Types Each register is labeled with the basic data types as follows: QUANTITIES VecVector quantities, which have both direction and magnitude at each point in space. The components of these quantities are stored in the register SclScalar quantities, which have a magnitude only CvcComplex vector quantities CscComplex scalar quantities GEOMETRIES PntPoints LinLines SrfSurfaces

27. T3_2D, pg. 27 6/28/02 Post Processing Calculator Stack Commands PushReloads the quantity in the top register onto the top of the stack, creating a new register. The contents of the top two registers are then identical. Pop Deletes the top register from the stack. RlDn Rolls the bottom register to the top of the stack, moving the other registers down the stack. RlUp Rolls the top register to the bottom of the stack, moving the other registers up the stack. ExchExchanges the top two registers in the stack. ClearClears the contents of the stack. UndoUse this command to undo the effect of the last operation you performed on the contents of the top register. Successive Undo commands act on any previous operations. Note:Quantities are loaded top-down instead of bottom-up as in an HP calculator

28. T3_2D, pg. 28 6/28/02 Post Processing Calculator Functionality To perform a computation on the field solution, you must first load a basic field quantity into a register on the stack. Once a quantity is loaded into a register, it can be:  Manipulated using mathematical operations such as curls, gradients, cross products, divergences, and dot products.  Integrated over lines, surfaces, or areas of the solution region — either predefined surfaces, volumes, and lists, or lines, surfaces, and areas that were defined using the Geometry/Create commands.  Plotted on a point, line, or area. Plotting derived quantities directly from the calculator lets you bypass the Plot/Field command.  Exported to a file, allowing you to superimpose saved solutions.

29. T3_2D, pg. 29 6/28/02 Post Processing Calculator Evaluate Command Numerically evaluates and displays the results of calculator operations such as integrations, maximum or minimum field computations, field values at points, and so forth. The quantity to be evaluated must be in the top register. The Eval command computes the numerical results of the operation, which replace the contents of the register. For instance, to find the magnetic field energy, U, in a subregion of the problem space, you must numerically evaluate the following integral for that area: Note: To perform a volume integration in 2D, use the integrate button in XY and scale for depth. In RZ problems, use the RZ_Integral button which performs the integration for 360 degrees.

30. T3_2D, pg. 30 6/28/02 Post Processing Calculator Export Command Exports the field quantity in the top register to a file, mapping it to a grid of points. Use this command to save field quantities in a format that can be read by other modeling or post-processing software packages. Two options are available: To FileMaps the field quantity to a customized grid of points. Before using this command, you must create a file containing the points. On GridMaps the field quantity to a three-dimensional cartesian grid. You specify the dimensions and spacing of the grid in the x, y, and z directions.

31. T3_2D, pg. 31 6/28/02 Post Processing Calculator Export to Grid Export to Grid - Real and Imaginary components of B(vector)  Vector data  Min: [2 -10]  Max: [2 10 ]  Grid Spacing: [1 2 ]  Space delimited ASCII file saved in project subdirectory

32. T3_2D, pg. 32 6/28/02 Post Processing Capacitance Calculation  A capacitance matrix represents the charge coupling within a group of conductors  Lumped capacitance is equivalent to 2*Energy with +1 and 0 volts on conductors of interest and all others floating where U e is the energy stored in the electric field C is the capacitance V is the voltage across the dielectrtic  diagonal terms in matrix include line to ground terms Conductor 1Conductor 2 Conductor 3 C 10 C 12 C 13 C 23 C 20 C 30

33. T3_2D, pg. 33 6/28/02 Post Processing Inductance Calculation  The inductance matrix can be expressed as:  The inducatnce coupling conductors i and j is: Note: The inductance calculation is bassed on the apparent permeability in EACH triangle at the specified excitation level  The energy stored in the magnetic field that couples two conductors is: Where  U ij is the energy stored in the magnetic field linking conductor i with conductor j.  I is the current in conductor i.  B i is the magnetic flux density associated with the case in which one amp is allowed to flow through conductor i.  H j is the magnetic field associated with the case in which one amp is allowed to flow through conductor j.

34. T3_2D, pg. 34 6/28/02 Post Processing Inductance Calculation  For multiturn conductors, the net value of inductance is the value given by: where N is the number of turns in the coil.

35. T3_2D, pg. 35 6/28/02 Post Processing Skin Depth  Induced currents allow magnetic fields to penetrate conductors only to a certain depth, which is approximated by the formula: where:   is the angular frequency, which is equal to 2  f. (f is the frequency at which source currents and voltages oscillate during the solution).   is the conductor’s conductivity, (siemens/meter).   r is the conductor’s relative permeability, (amperes/meter).    is the permeability of free space, which is equal to 4  10 -7 A/m.  Currents will be concentrated near the surface of the conductor, decaying rapidly past the skin depth. As the formula above indicates, the skin depth gets smaller as the frequency increases.

36. T3_2D, pg. 36 6/28/02 Post Processing Power Loss  The ohmic loss is:  The ohmic loss is related to the resistance by:  The resistance is therefore: