Introduction to COMSOL Multiphysics. Numeric and interpolation data formats

Successful engineering calculations are usually based on experimentally validated models, which can replace both physics experiments and prototyping to some extent, and provide a better understanding of the design being developed or the process being studied. Compared to conducting physical experiments and testing prototypes, simulation allows faster, more efficient and more accurate optimization of processes and devices.

Users of COMSOL Multiphysics ® are free from the rigid restrictions that are typically associated with simulation packages and can control every aspect of the model. You can get creative with modeling and solve problems that are complex or impossible with a conventional approach by combining an arbitrary number of physical phenomena and specifying custom descriptions of physical phenomena, equations and expressions through a graphical user interface (GUI).

Accurate multiphysics models take into account a wide range of operating conditions and a large set of physical phenomena. Thus, simulation helps to understand, design and optimize processes and devices, taking into account the real conditions of their work.

Sequential Modeling Workflow

Simulation in COMSOL Multiphysics ® allows you to explore electromagnetic phenomena, structural mechanics, acoustics, fluid dynamics, heat transfer and chemical reactions, as well as any other physical phenomena that can be described by systems of partial differential equations in one software environment. You can combine all these physical phenomena in one model. The COMSOL Desktop ® graphical user interface provides access to a complete integrated simulation software environment. Whatever devices and processes you study, the modeling process will be logical and consistent.

Geometric modeling and interaction with third-party CAD packages

Operations, Sequences, and Selections

The COMSOL Multiphysics ® core package contains geometric modeling tools for creating geometry from solids, surfaces, curves, and Boolean operations. The final geometry is determined by a sequence of operations, each of which can receive input parameters, which facilitates editing and parametric studies of multiphysics models. The relationship between geometry definition and physics settings is two-way - any change in geometry automatically leads to corresponding changes in the associated model settings.

Any geometric objects can be combined into selections for further use in determining physics and boundary conditions, building grids and graphs. In addition, a workflow can be used to create a parameterized geometry part, which can then be stored in the Parts Library and reused in many models.

Import, Processing, Defeaturing and Virtual Operations

The import of all standard CAD and ECAD files into COMSOL Multiphysics ® is supported by the Import CAD Data and Import ECAD Data modules, respectively. The Design module extends the set of geometric operations available in COMSOL Multiphysics ® . The modules Import data from CAD and Design provide the ability to correct geometries and remove some unnecessary details (operations Defeaturing and Repair). Surface mesh models, such as the STL format, can be imported and converted into geometric objects using the core COMSOL Multiphysics ® platform. Import operations work in the same way as all other geometric operations - they can use selections and also associativity in parametric and optimization studies.

As an alternative to the Defeaturing and Repair operations, the COMSOL ® software package also includes so-called virtual operations that allow you to eliminate the influence of a number of geometric artifacts on the finite element mesh, in particular, elongated and narrow boundaries, which reduce the accuracy of the simulation. Unlike defeaturing detail removal, virtual operations do not change the curvature or precision of the geometry, but produce a cleaner mesh.

List of geometric modeling functions

• Primitives
  • Block, sphere, cone, torus, ellipsoid, cylinder, spiral, pyramid, hexagon
  • Parametric curve, parametric surface, polygon, Bezier polygons, interpolation curve, point
• Operations Extrude (Extraction), Revolve (Reversal), Sweep and Loft (create a body along a path or along sections 1
• Boolean operations: union, intersection, difference and division
• Transforms: array creation, copying, mirroring, moving, rotating and scaling
• Transformations:
  • Convert to closed solid, surface, curve
  • Midsurface 1 , Thicken 1 , Split
• Chamfer (Bevel) and Fillet (Rounding) 2
• Virtual Geometric Operations
  • Remove details (Automatic application of virtual operations)
  • Ignore: vertices, edges and borders
  • Form an aggregate object: from edges, boundaries or regions
  • Collapse an edge or border
  • Merge vertices or edges
  • Mesh control: vertices, edges, borders, regions
• Hybrid modeling: solids, surfaces, curves and points
• Work Planes with 2D Geometric Modeling
• Import from CAD and two-way integration with plug-ins Import data from CAD, Engineering and LiveLink™ products
• Repairing and deleting parts from CAD models using plug-ins Import data from CAD, Design and LiveLink™ products
  • Cap faces (Close face), Delete (Delete)
  • Rounding, Getting rid of short edges, narrow edges, borders and ledges
  • Detach faces (Selecting a domain from the boundaries), Knit to solid, Repair (Getting rid of gaps, Processing and correcting geometry)

1 Requires the Design module

2 These 3D operations require the Design module

This bike frame was designed in the SOLIDWORKS ® software package and can be imported into COMSOL Multiphysics ® with a few clicks. You can also import geometry from other third-party CAD packages or create them using COMSOL Multiphysics ® 's built-in geometry tools.

COMSOL Multiphysics ® tools allow you to modify and correct third-party CAD geometries (to match the FE calculation), as in this case in a bicycle frame model. If you wish, you could create this geometry from scratch in COMSOL Multiphysics ® .

finite element mesh for a bicycle frame project. It is now ready for calculation in COMSOL Multiphysics ® .

A mechanical calculation of a bicycle frame model was performed in COMSOL Multiphysics ®. Analysis of the results can suggest what changes to make to the frame design in a third-party CAD package for further work.

Ready preset interfaces and functions for physical modeling

The COMSOL ® software package provides out-of-the-box physics interfaces for modeling a wide variety of physics phenomena, including common interdisciplinary multiphysics interactions. Physical interfaces are specialized user interfaces for a particular engineering or research area that allow you to thoroughly control the simulation of the studied physical phenomenon or phenomena - from setting the initial parameters of the model and discretization to analyzing the results.

After selecting the physical interface, the software package prompts you to select one of the types of studies, for example, using a non-stationary or stationary solver. The program also automatically selects for the mathematical model the appropriate numerical discretization, solver configuration, and visualization and post-processing settings suitable for the physical phenomenon under study. Physical interfaces can be freely combined to describe processes involving multiple phenomena.

The COMSOL Multiphysics ® platform includes a large set of basic physics interfaces, such as interfaces for describing solid mechanics, acoustics, fluid dynamics, heat transfer, chemical transport, and electromagnetism. By extending the base package with additional COMSOL ® modules, you get a set of specialized interfaces for modeling specific engineering problems.

List of available physics interfaces and material property representations

Physical interfaces

• Electric currents (Electric currents)
• Electrostatics (Electrostatics)
• Heat transfer in solids and fluids (Heat transfer in solids and fluids)
• Joule heating
• Laminar flow
• Pressure acoustics (Scalar acoustics)
• Solid mechanics (Solid mechanics)
• Transport of diluted species
• Magnetic Fields, 2D (Magnetic fields, in 2D)
• Additional specialized physical interfaces are contained in expansion modules

materials

• Isotropic and anisotropic materials
• Inhomogeneous materials
• Materials with spatially inhomogeneous properties
• Materials with properties that change over time
• Materials with non-linear properties that depend on some physical quantity

Thermal actuator model in COMSOL Multiphysics ® . The Heat Transfer branch is expanded and shows all relevant physical interfaces. For this example, all plug-ins are enabled, so there are many physical interfaces to choose from.

Transparent and flexible modeling based on user equations

A software package for scientific and engineering research and innovation should not be just a simulation environment with a predefined and limited set of features. It should provide interfaces for users to create and customize descriptions of their own models based on mathematical equations. The COMSOL Multiphysics ® package has this flexibility - it contains an equation interpreter that processes expressions, equations, and other mathematical descriptions before creating a numerical model. You can add and customize expressions in physics interfaces, easily linking them together to model multiphysics phenomena.

More advanced customization is also available. The customization capabilities of the Physics Builder allow you to use your own equations to create new physics interfaces that can then be easily incorporated into future models or shared with colleagues.

List of Functions Available When Using Equation-Based Modeling

• Partial differential equations (PDE) in weak form
• Arbitrary Lagrange - Euler Methods (ALE) for Problems with Deformed Geometry and Moving Meshes
• Algebraic equations
• Ordinary Differential Equations (ODE)
• Differential Algebraic Equations (DAE)
• Sensitivity analysis (requires the optional Optimization module for optimization)
• Calculation of curvilinear coordinates

Model of the wave process in an optical fiber based on the Korteweg - de Vries equation. Partial differential equations and ordinary differential equations can be defined in the COMSOL Multiphysics ® software package in coefficient or mathematical matrix form.

Automated and manual meshing

The COMSOL Multiphysics ® software uses various numerical methods and techniques to discretize and mesh the model, depending on the type of physics or combination of physical phenomena being investigated in the model. The most commonly used discretization methods are based on the finite element method (for a complete list of methods, see the Solvers section of this page). Accordingly, a general purpose meshing algorithm creates a mesh with elements of the type that is suitable for this numerical method. For example, the default algorithm can use an arbitrary tetrahedral mesh or combine it with a boundary layer meshing method to combine elements of different types and provide faster and more accurate calculations.

Mesh refinement, re-meshing, or adaptive meshing operations can be performed during the solution process or a special study step for any type of mesh.

List of available options when building a mesh

• An arbitrary mesh based on tetrahedra
• Swept mesh based on prismatic and hexahedral elements
• Boundary layer mesh
• Tetrahedral, prismatic, pyramidal and hexahedral solid elements
• Custom triangular mesh for 3D surfaces and 2D models

• Free quad mesh and structural 2d mesh (Mapped type) for 3D surfaces and 2D models
• Grid copy operation
• Virtual Geometric Operations
• Dividing meshes into regions, boundaries, and edges
• Import meshes created in other software

Automated unstructured tetrahedral mesh for wheel rim geometry.

Semi-automatically constructed unstructured mesh with boundary layers for micromixer geometry.

A manual mesh for an electronic component model on a printed circuit board. A finite element mesh combines a tetrahedral mesh, a triangular mesh on the surface, and a mesh built by pulling into the volume.

The surface mesh of the vertebral model was saved in STL format, imported into COMSOL Multiphysics ® and converted into a geometric object. An automated unstructured grid was superimposed on it. STL geometry provided by Mark Yeoman of Continuum Blue, UK.

Studies and their sequences, parametric calculations and optimization

Research types

After choosing a physics interface, COMSOL Multiphysics ® offers several different types of studies (or analysis). For example, in the study of solid body mechanics, the software package offers non-stationary studies, stationary studies and studies on natural frequencies. For problems of computational fluid dynamics, only non-stationary and stationary studies will be proposed. You can freely choose other types of studies for your calculation. The study step sequences define the solution process and allow you to select the model variables to be calculated at each step. Solutions from any previous stages of the study can be used as input to the next stages.

Parametric Analysis, Optimization and Estimation

For any stage of the study, you can run a parametric sweep (sweep), which can include one or more model parameters, including geometric dimensions or settings in boundary conditions. You can perform parametric sweeps on various materials and their properties, as well as on the list of specified functions.

The helical static mixer model was created using the COMSOL Multiphysics ® Modeler.

Copyright OJSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Bryansk State Technical University L.А. Potapov, I.Yu. Butarev COMSOL MULTIPHYSICS: SIMULATION OF ELECTROMECHANICAL DEVICES Approved by the editorial and publishing board as a textbook Bryansk 2011 Copyright JSC Central Design Bureau BIBCOM & LLC Agency Kniga-Service LBC 31.21 Potapov, L. A. Comsol multiphysics: Modeling of electromechanical devices [Text] + [Electronic resource]: textbook. allowance / L.A. Potapov, I.Yu. Butarev. - Bryansk: BSTU, 2011. - 112 p. ISBN-978–5-89838-520-0 Brief information about the Comsol Multiphysics software package is given. Examples of constructing 2D and 3D models of electromechanical devices are considered. The textbook is intended for full-time students of the specialty 140604 "Electric drive and automation of industrial installations and technological complexes", and can also be useful to graduate students and undergraduates in electrical specialties of higher educational institutions and engineering and technical workers developing electrical devices. Il.116. Bibliography - 3 names. Scientific editor S.Yu. Babak Reviewers: Department of Energy and Automation of Production Processes, Bryansk State Academy of Engineering and Technology; Candidate of Technical Sciences A. A. Ulyanov Editor of the publishing house L.N. Mazhugina Computer typesetting by N.A. Sinitsyna Templan 2011, p 45 Signed for printing 09/30/11 Format 60x84 1/16. Offset paper. Offset printing. Conv. pech.l. 6.51 Uch.-ed.l. 6.51 Circulation 60 copies. Order Bryansk State Technical University 241035, Bryansk, Boulevard im. 50th Anniversary of October, 7, tel. 58-82-49 Operational printing laboratory of BSTU, st. Institutskaya, 16 ISBN 978–5-89838-520-0 Bryansk State Technical University, 2011 Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Kniga-Service 3 FOREWORD Modern personal computers and related software have made 2D accessible to a wide range of specialists - and 3D modeling of various technical devices. This makes it possible to study processes occurring in places inaccessible for physical experiments: inside a massive rotor, in various sections of magnetic circuits, etc., which speeds up and simplifies the development of new devices. At the same time, it is possible to abandon numerous prototype samples that were previously necessary for optimizing and fine-tuning the design being developed. The Comsol Multiphysics software package, developed by the Swedish company Comsol, makes it possible to obtain models of complex technical devices with all the various processes occurring in these devices. However, there are no manuals in Russian for this software package. In the proposed tutorial, the basics of work in one of the sections of this complex (AC / DC) are given and, using the example of several electromechanical devices, the features of obtaining 2D and 3D models are considered in detail. The simulation results obtained in this way, which characterize the processes of distribution of currents and magnetic fluxes in the depth of the rotors, are of interest to specialists involved in the development of similar equipment. The tutorial consists of three chapters. The first chapter covers the basics of working in the Comsol Multiphysics software package. The second chapter provides examples of building 2D models of electromagnetic brakes with massive and hollow rotors. The third chapter provides examples of building 3D models of an electromagnet and an electromagnetic damper with a disk rotor. Copyright OJSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 4 The work in preparing the training manual was distributed as follows: I.Yu. Butarev - development and description of models of electromechanical devices, translation from English of available materials on the Comsol Multiphysics complex; L.A. Potapov - general management of the work, preparation of the manuscript for publication. The textbook is intended for students, graduate students and undergraduates of electrical engineering specialties of higher educational institutions. It can be used in the study of the disciplines "Theory of the electromagnetic field", "Electrical machines", "Electrical devices", etc., as well as in course and diploma design. The manual is also of interest to engineering and technical workers associated with the development of electrical equipment. Copyright OJSC "Central Design Bureau" BIBCOM " & LLC "Agency Kniga-Service" 5 INTRODUCTION There is a large group of electromechanical devices in which electromagnetic processes occur inside massive, hollow or disk rotors. In this case, it is not possible to single out currents or magnetic fluxes. Therefore, it is also impossible to measure them. It is necessary to use the concepts of current density and magnetic fluxes (induction), to consider their distribution over the thickness or depth of the rotor. The interaction of current density with magnetic fields determines the mechanical forces and moments that can be measured and that are of most interest to users. When the rotor speed changes, the pattern of the electromagnetic field changes: the current density increases and becomes more uneven, the magnetic field is carried away by the rotating rotor in the direction of rotation. All these phenomena can be observed and investigated using 2D and 3D modeling of electromagnetic processes using special programs. Some of these programs have been in use for a long time and are geared towards the corresponding hardware, for example, the ANSYS program has been known for about 20 years. Others have appeared recently, such as the Comsol Multiphysics software package, developed by the Swedish firm Comsol. It allows you to obtain models of complex electromechanical devices, taking into account the electromagnetic processes occurring in them. A great advantage of the Comsol Multiphysics software package is its very user-friendly interface. To use it, it is not required to write partial differential equations (you may not know them at all), although it is they that he uses, it is not necessary to build a finite element mesh - he forms it himself, etc. It is enough to draw an object, set the properties of materials, boundary conditions and indicate in what form to display the simulation results. Naturally, it is possible to improve the mesh, change the solver, derive the result from a given equation, and so on. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 6 1. SOME INFORMATION ABOUT COMSOL MULTIPHYSICS The Comsol Multiphysics software package was developed by the Swedish company Comsol. It allows you to simulate several physical processes occurring simultaneously in complex technical devices. 1.1. General Description Comsol Multiphysics (formerly Femlab) is a software suite of technology tools for modeling physical fields in scientific and engineering applications. Its main feature is the ease of modeling and unlimited multiphysics capabilities that allow you to simultaneously study thermal, electromagnetic and other processes on the same model. In this case, it is possible to model one-dimensional, two-dimensional and three-dimensional physical fields, as well as the construction of axisymmetric models. Comsol Multiphysics consists of sections (electromagnetism, acoustics, chemical reactions, diffusion, hydrodynamics, filtration, heat and mass transfer, optics, quantum mechanics, semiconductor devices, strength of materials and many others), which contain partial differential equations and the constants of those or other physical processes (thermal, electromagnetic, nuclear, etc.). Each section consists of subsections focused on a narrower class of fields under study (direct and alternating currents, etc.). ). For each of the subsections, you can select the type of analysis (static, dynamic, spectral). Comsol Multiphysics uses numerical methods of mathematical analysis in simulations based on partial differential equations (PDE) and the finite element method (FEM). The PDE coefficients are given in the form of understandable physical parameters, such as magnetic induction, current density, magnetic permeability, intensity, etc. (depending on the selected physical partition). The PDE conversion is carried out by the program itself. User interaction with Multiphysics is done using a graphical user interface (GUI) either in Comsol Script or MATLAB, in the tutorial using the GUI only. To solve differential equations, Comsol Multiphysics software automatically overlays a given geometric model of the problem with a mesh (mesh) taking into account the geometric configuration. In Comsol Multiphysics, you can choose one of the presented methods for solving algebraic equations, such as UMFPACK, SPOOLES, PARDISO, Cholesky expansion, and others. Since many physical laws are expressed in the form of partial differential equations, it is possible to model scientific and engineering phenomena from many fields of physics or engineering by connecting models in different geometries and linking models of different dimensions using coupling variables. The tutorial covers the basics of modeling in the AC/DC Module section, which uses Maxwell's equation system. The section contains subsections Statics Electric (electrostatics), Statics Magnetic (magnetostatics), Quasi-Statics Electric (electrical quasi-statics), QuasiStatics Magnetic (magnetic quasi-statics), Quasi-Statics Electromagnetic (electromagnetic quasi-statics), Rotating Machinery (rotating machines), Virtual Work ( virtual work), Electro-Thermal Interaction (electrothermal interaction). Each subsection has several models. So, in the subsection Quasi-Statics Magnetic there are models Perpendicular Induction Currents, Vector Potential (perpendicular induction currents, vector potential); In-plane Induction Currents, Vector Potential (plane induction currents, vector potential) and In-plane Induction Currents, Magnetic field (plane induction currents, magnetic field). Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Book-Service" 8 1.2. Basics of modeling When modeling in Comsol Multiphysics, the following sequence of actions is necessary: ​​1. Set up the Model Navigator: select the model dimension in Space Dimension (space dimension); define a section in it (each section corresponds to a certain differential equation) and a subsection, as well as the type of model and the type of its analysis. 2. Determine the working area and set the geometry of the device under study. 3. Set constants (initial data), dependencies of variables on coordinates and time. 4. Indicate electromagnetic properties and initial conditions. 5. Set boundary conditions. 6. Build a grid that takes into account the configuration of the model. 7. Determine the parameters of the solver and start the calculation. 8. Set display mode and get results. Let's consider in more detail the specified sequence of actions. Model Navigator After turning on Comsol Multiphysics, the Model Navigator (Fig. 1.1) appears on the computer screen, in which the dimension of the model is selected - on the first New tab in Space Dimension (space dimension). Then a partition is selected (by clicking the cross in front of the name), for example, the physical partition of the AC / DC Module, and similarly the subsection. When choosing the dimension of the model, it must be remembered that even setting a grid in a three-dimensional model can take tens of minutes (even on a very powerful computer). For most 3D problems, it makes sense to define and calculate the 2D model first, and then calculate the 3D model if necessary. In addition, if you do not import geometry from an external CAD system, but specify it directly in Comsol Multiphysics, then it is more convenient to obtain a three-dimensional model by converting a two-dimensional one. Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Book-Service 9 Fig 1.1. Model Navigator Since we are going to model a DC electromagnetic brake, we select the AC/DC Module physics section, which uses Maxwell's equation system. The section contains subsections Statics, Electric (electrostatics); Statics, Magnetic (magnetostatics), etc. (Fig. 1.1). To create multiphysics models, for example, to take into account heating during the operation of an electromagnetic brake, you must press the Multiphisics button and the Add geometry button (add geometry), in the window that opens, select the dimension and names of the axes. Then click the Add… button and first select one physical section (AC/DC Module → Quasi-Statics, Magnetic → Perpendicular Induction Currents, Vector Potential), and then add the second section to the model (AC/DC Module → Electro-Thermal Interaction → Perpendicular Induction Heating) For each of the subsections, you can select an analysis type by clicking the Application Mode Properties button, such as Steady-state analysis (stationary analysis) or Transient analysis (transient analysis). Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Kniga-Service" 10 Also on the New tab in the Model Navigator, you can select the type of finite elements, the default is Lagrange-Quadratic (Lagrange-quadratic). In this case, Lagrangian elements are proposed, up to the fifth degree. Hermitian elements, Euler elements and many other applied elements are available in some sections. In addition to the New tab, the Model Navigator contains three more tabs. The Model Library tab contains sample models for all physical subsections. The User Models tab stores the created models. Using the Settings tab, you can set the desired language and change the workspace background from white to black. Since COMSOL 3.2, the system of units is also set there. Also in the models navigator there is an Open tab, which, like the User Models tab, allows you to work with files. Workspace and object image After pressing the OK button in the Model Navigator, the Comsol Multiphysics main interface window opens with the workspace (Fig. 1.2), toolbars and the main menu. The buttons on the toolbars repeat the main menu items, so we will consider the main menu items in order: File - contains commands for creating, opening and saving files, printing, as well as importing geometry from external CAD systems and exporting the resulting data to a text file. Edit - contains commands for undoing and redoing operations, working with the clipboard and selection commands. Options - contains commands for setting the workspace Axes / Grid settings (sizes and settings for the axes and construction grid (Grid, not to be confused with the finite element mesh Mesh!), Constants, Expressions, Functions, Coupling Variables and various display settings geometric elements and scale. Draw - contains commands for constructing and transforming geometric objects, as well as commands for turning two-dimensional objects into three-dimensional ones. Copyright OJSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 11 Physics - contains commands for setting the physical properties of Subdomain subdomains, Boundary boundary conditions, including Periodic Conditions, Point Settings and changing the system of differential equations Equation system. Mesh - contains commands for managing a finite element mesh. Solve - contains commands to control the solver. These commands allow you to select time dependence, linearity or non-linearity, solution method, simulation step, relative error, and many other solver parameters. Postprocessing - contains commands for displaying the results of calculations in all possible forms from vectors and over 1.2. The main programming interface for pre-Comsol Multiphysics level plots and boundary integrals. Multiphysics - Opens the Model Navigator and allows you to switch between physics modes in multiphysics models. Help - contains an extensive help system. Copyright JSC "Central Design Bureau" BIBCOM " & LLC "Agency Book-Service" 12 In fig. 1.3 shows a window with a workspace. In the upper part of the window there are buttons (1) for working with the file and clipboard and the main buttons for modeling, which allow you not to use the Mesh, Solve and Postprocessing commands. Most of the window is occupied by the graphics area (2). To the left of it are the drawing buttons (3). In one-dimensional mode, these are the buttons point (point), line (line), mirror (displays the object in a mirror), move (moves the object) and scale (changes the size of the object). Rice. 1.3. Workspace window In 2D mode, buttons for creating Bezier curves, rectangles and ovals are added, as well as an Array button that creates a matrix of objects of any size from one object. Button Rotate (rotation) allows you to rotate the created object to any angle. In 3D mode, using the buttons, you can create parallelepipeds, ellipsoids, cones, cylinders and balls, as well as control the location of the coordinate axes and the lighting of the figure. To set the boundaries of the displayed workspace, you must use the Options command (Fig. 1.2), and then the Axes / Grid settings command (options> axis / grid settings) (Fig. 1.4). As an example, let's limit the work area to 6 cm along the X axis and 4 cm along the Y axis. In this case, the center of the coordinate system will be placed in the center of the graphics area. In the window that opens, select the Axis (axes) tab (the Axis equal checkbox means that the axes will be equal, i.e. one meter along the X axis) the same size as the Y-axis). For extended objects, this checkbox can be unchecked, and then the axes in the window may not be equal. This is useful when the object is disproportionately large in one of the given dimensions. a) b) Fig. 1.4. The window for setting the boundaries of the work area: a - Axis tab, b - Grid tab In the x-y limits section, you need to set the limits for displaying the axes, for us it is -0.03 and 0.03 for the minimum and maximum of the corresponding axes. On the Grid tab (lattice), you can uncheck Auto and set the grid spacing yourself. Why is it necessary? When building a model, you can only specify the coordinates of the corresponding shapes (for example, the coordinates of the center of the circle and its radius), but it is often more convenient to define the shape by marking these coordinates with the mouse, and then it is necessary that the lattice nodes coincide with the key points of the shape. Therefore, if the thickness of the minimum element is one millimeter, then it is advisable to set exactly this grid spacing. The Visible checkbox allows you to turn off the grid display mode. At the bottom of the workspace, you can also turn off the mouse binding to the SNAP lattice, but then when entering an object with the mouse, key points can only be set approximately. In the x–y grid area, you can set the spacing of the grid along the corresponding axes in the x and y spacing fields. The Extra x and Extra y fields allow you to add any number of extra grid lines. The next step after setting the lattice is to determine the geometry of the object of study. If it is not created in advance in an external CAD program (Autodesk, AutoCAD, Compass, etc.) or is not set in the MATLAB program (then it is imported using File>Import), then you will have to set it internally - Copyright JSC Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" with 14 partnerships. Let's say we want to draw a rectangle. You can use the corresponding buttons Rectangle / Square [rectangle / square] and Rectangle / Square (Centered) [rectangle / square (centered)], the first click marks the location of the corner or center, and then the rectangle is stretched to the required size and fixed with the second click. Pressing the Ctrl key creates a square. If you press the Shift key and click on the button, a window will open with all the parameters of the figure (Fig. 1.5). If the figure is built, then it can be similarly edited by double-clicking on it. The same window can be opened via the main menu Draw>Specify objects. The Size command sets the size of an object using the Width (width) and Height (height) fields. The Rotation angle command sets the angle of rotation straight ahead. 1.5 An example of the parameter window for constructing a rectangle in degrees. The Position area determines the location of the object. The Base drop-down list lets you define what the x and y coordinates refer to. Corner means that the location of the corner of the rectangle is specified (if an ellipse is drawn, then the coordinates of the described rectangle must be specified). Center means that the coordinates of the center of the object are set. The Style drop-down list offers options: Solid - a whole shape will be created, Curve - a curve-contour of the shape will be created. A curve is needed to create a complex figure: first, curves and object boundaries are set, and then the selected curves are made into a solid figure using the Coerce to solid command. In 3-D mode, instead of Curve, there is the concept of Face - a shell. In the Name field, you can enter the name of the object. While the Specify objects windows allow you to set precise coordinates and sizes for objects, they are often easier to set with the mouse, and Bezier curves can only be set with the mouse. That is why it is necessary to determine the grating period in advance. Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Kniga-Service" 15 When defining complex shapes, you have to specify dozens of elementary objects (ovals, rectangles, Bezier curves, lines, points), then they need to be combined or divided. This is usually done on physical grounds using the buttons Union (combine), Difference (difference) and Intersection (intersection) or the Draw>Create Composite Object command ... This command opens a window where you can specify from which elements the figure is created. After creating a figure, using the Fillet / Chamfer button or the Draw menu item of the same name, you can set chamfers or rounding corners. You can also replicate the shape with the Array button, flip with Mirror, and resize with Scale. The Rotate and Move buttons rotate and move the selected shape, respectively. All these buttons are repeated as Draw>Modify menu items. When creating three-dimensional models, it is convenient to set elementary figures in 3D mode, while more complex ones are first set in 2D mode, and then transferred to a three-dimensional area. So a rectangle of 1x0.5 meters was created. If you select it and press the Draw> Extrude button, the Extrude window will open (Fig. 1.6), where you can set the object undergoing the operation and the name of the workspace (for one model, you can set several workspaces, usually several 2D- geometries and one composition (Fig. 1.6. Extrude window 3D). The Distance field determines how many times the section will be stretched. If a circle was drawn, then after extrusion there will be a cylinder, if the section is rails, then there will be a rail model. Scale x and y set how many times the section will change along the length of the object. If you set two twos in these fields, then after extrusion (if the section was round) a truncated cone will appear. Displacement defines the shift of the upper plane of the figure relative to the base. Twist twists the figure around its axis. Copyright OJSC «TsKB «BIBCOM» & OOO «Agency Kniga-Service» 16 Draw>Embed will copy a two-dimensional rectangle into a three-dimensional workspace (by default, to the z=0 plane). Another plane is set via Draw>work plane settings. The Draw>Revolve operation will create a rotation figure, i.e. from a rectangle, you can create a ring with a rectangular cross section. In the window that opens, you can specify the angle of rotation along the two axes (in degrees) and the coordinates of the points around which the rotation figure will be created. For clarity, using the Scene Light command, you can set the “object lighting”, the Zoom extents button will place the figure on the entire screen. If during further modeling it is necessary to change any geometry element, then you can return to the geometry input mode using the Draw>Draw Mode command or the Draw Mode button at the top of the screen. Constants, Expressions, Functions Comsol Multiphysics has commands for working with constants and functions. Most of these commands are found in the Options menu. Let's consider some of them. 1. Constants (constants). It is recommended to put the constants used in the model in a table and then to set only the letter designation. So, set the current in the winding Ip=500, and then set Ip instead of a number in all areas of the object. Then, if necessary, it will be possible to change one digit in the Constants menu and not change the numbers for all areas of the object. Also, a list of frequently used constants can be saved in a separate file and transferred from model to model. 2. Expression (expressions) Contains Scalar expression (scalar mathematical expressions), Subdomain, Boundary, Edge (only in 3D mode) and Point expression. You can set the dependence of the electromagnetic parameter on time t; from coordinates x, y, z; from the dimensionless coordinate s (varies from 0 to 1 along the length of each boundary) or from any other calculated values. . For various elements of the system, very often the same parameters are determined according to different laws. It is possible to assign one name to a variable, for example alfa. Having opened the Boundary expression (boundary expressions), set different formulas for calculating alfa for different boundaries. Then for all boundaries it will be possible to set the coefficient alfa, and the program itself will substitute the corresponding expression for each boundary. Similarly for Subdomain, Edge Expressions. 3. Coupling Variables (coupling variables). You can specify complex dependencies between parts of the system, for example, link boundary conditions with a volume integral. 4. Functions (function). You can set your own function, and using not only mathematical expressions. If you choose Interpolation function, then you can set an array of parameters and an array of function values, and build an interpolation function based on them. You can set the interpolation method from the proposed ones (for example, splines), it is possible to import data from an external file. 5. Coordinate systems (coordinate systems). You can create an arbitrary coordinate system with any location of the axes relative to each other. 6. Material / Coefficients Library (library of materials). You can set any physical properties of substances, and even their dependence on electromagnetic parameters (magnetic permeability, electrical conductivity, etc.). 7. Visualization/Selection settings (visualization settings). You can control the display of objects, lighting and selection. 8. Suppress (hiding). You can make any element of the system invisible (for clarity in complex objects). Defining electromagnetic properties of materials and initial conditions Once the geometry is set and all constants are defined, we can begin to define electromagnetic properties. First, open the Physics>Subdomain Settings menu - the settings window for the physical coefficients of the domains will open (Fig. 1.7). Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Kniga-Service" 18 For each of the physical modes, this window has its own view, and all fields will be discussed in the relevant chapters. Here we consider only fields common to all regimes. The figure shows the window for the Perpendicular Induction Currents mode in 2D mode. At the top, the Equation field shows the current equation. In the Subdomain selection field, select the area for which the physical properties are to be determined. Rice. 1.7. The settings window for the physical coefficients of the regions If there are many regions, then it is necessary to select all created from the same material. If identical constants are assigned to areas, then they automatically form a group in the Groups tab, which in the future allows you not to select all areas one by one again, especially if the model is very complex. To select all areas, press Ctrl+A. For the selected areas (Subdomains), the physical properties are set one by one. So, for area 1 (fig.1.7) it is necessary to set 7 values. The Velocity parameter shows how fast (m/s) this or that area is moving. This parameter is divided into two parts, which Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 19 correspond to the speeds along the axes. There will be three parts in three-dimensional mode. The Potential difference Δ V parameter is the potential difference (V) for a given area. The Length parameter specifies the length of the area (m). The External Current density Jez parameter sets the external current density for the region. The Electric Conductivity σ parameter sets the relative electrical conductivity of the area material (S/m). The drop-down list Constitutive Relation allows you to select the relationship between the magnetic induction and the magnetic field strength in the material. In our case, the simplest relationship B= μ0μrH is chosen. The Relative permeability parameter specifies the relative magnetic permeability (a dimensionless number or a certain function). f(B). You can use the built-in approximator in Options>Functions. The syntax in this mode is the same as in MATLAB, but it is more convenient to enter not expressions into the fields, but variable names and define them using Options>Expression. There are 6 tabs in the upper part of the settings window (fig.1.7). In the Physics tab, you must set the universal physical constants, in this case, electromagnetic (μ0,ε0). For common standard materials, you can use the built-in library using the Load button and select the required material there. In the Infinite Element tab, you can select the element type from the list. The Forces tab allows you to set the Maxwellian surface tension tensor for the total electromagnetic force or moment. So, let's enter the name_forcex_q variable in the Name field. The program will define this as a force in the X direction. Similarly, for the moment, the name_torquex_q variable is used, which sets the electromagnetic moment around the X axis. The Init tab is designed to set the initial conditions, in this case it is the magnetic potential along the z-component - Az. Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Kniga-Service" 20 The Element tab allows you to select the type of finite elements and their coefficients. The Color tab allows you to change the color of a given area or group of areas, which greatly simplifies orientation in a complex task with a large number of materials. Specifying boundary conditions and changing differential equations Specifying the physical properties of materials in areas, boundary conditions, and conditions on edges or points occurs in the appropriate modes, which are automatically enabled when opening the windows for entering the properties of these elements. The modes are manually enabled using the Point Mode, Edge Mode, Boundary Mode and Subdomain Mode buttons located in the upper part of the workspace at the right end before the help button or commands from the menu section Physics>Selection Mode>… Boundary conditions are set using the Physics> Boundary Settings command or F7 buttons. In the window that opens (Fig. 1.8), you must select the boundaries in the Boundary selection field. To set Dirichlet boundary conditions on the boundary of two bodies, you must first enable the Interior boundaries checkbox, otherwise the interior boundaries will be unavailable. In the Conditions tab, you must select the type of boundary conditions. The Boundary Conditions list prompts you to select the type of boundary conditions, such as Magnetic Field (magnetic field strength), and set the value of the coefficient on the boundary. Here everything is similar to the Subdomain Settings mode, only instead of the border areas between them. Often, when modeling complex devices, such as multi-pole electric motors, an elementary volume is isolated and a calculation is carried out for this elementary volume. For a correct calculation, it is necessary to set a special type of boundary conditions - periodic boundary conditions. To do this, select Periodic Condition in the Boundary Condition list, specify the coefficients and the type of periodicity. The Color/Style tab gives borders with different boundary conditions different colors and display styles. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 21 Pic. 1.8. Boundary conditions window In addition to the conditions on the Boundary boundary, it is required to set periodic properties for Point points in 2D mode (for example, the current value at the point) and in 3D mode for Edge edges. For some multiphysics problems, where it is necessary to associate two objects with a different mesh type (for example, a rectangular mesh in one part of the system with a triangular one in the other) and continuity boundary conditions, you can apply the Physics>Identity Conditions identity conditions. Comsol Multiphysics has many options for flexible customization of the program for each specific task. You can change the system of partial differential equations (PDE). To do this, use the Physics>Equation system commands. These commands allow you to widely change the initial PDE equations, the methods of specifying the initial and boundary conditions, as well as the parameters of the finite elements. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 22 Building a grid After setting all the properties and boundary conditions, start building a grid. For the simplest models, at the first stage of the estimated calculation, you can set the default mesh Mesh>Initialize Mesh (or the button with the image of a triangle). For a finer mesh, you need to press Mesh>Refine mesh several times and, having obtained a sufficiently fine mesh, proceed to solve the problem. When you click these buttons, the work area switches to Mesh Mode, and the mesh is displayed in the work area. Manually this mode can be called by the corresponding button or menu command Mesh>Mesh Mode. For simple models, you can limit yourself to this (for smaller mesh elements, the system will automatically thicken the mesh), and if you need to thicken the mesh even more in any part of the system, you can click the Refine selection button and specify the required area. In one-dimensional and two-dimensional stationary mode, it is possible to build the finest grid - the calculation speed on modern computers will still be acceptable. In this case, it must be remembered that the size of the finite element must be several times smaller than the thickness of the boundary layer, otherwise the solution may be unstable. Therefore, it is recommended to build a grid of such density that there are at least ten finite elements between any two boundaries. By default, Comsol Multiphysics generates a triangular mesh in 2D and a tetrahedral mesh in 3D. To set the mesh parameters, select Mesh> Free Mesh parameters or press the F9 button. The settings window will open, in the Global tab (Fig. 1.9) you can select one of the preset modes. There are nine modes in the Predefined mash sizes list - from Extremely fine (extremely accurate) to Extremely coarse (very coarse), the rest are located between these extreme modes. In the fields, you can set your own values ​​for the mesh parameters after selecting the Custom mesh size list. Maximum element size specifies the maximum element size. By default, it is equal to 1/15 of the maximum side, it is optional to set it. If nothing is set in the previous field, then the value of the Maximum element size scaling factor field will determine the size of the element (if you set 0.5, then the element size will be equal to 1/30 of the maximum side, if 0.1 then 1/150). Element growth Copyright JSC "TsKB "BIBCOM" & LLC "Agency Kniga-Service" 23 rate (element growth rate) is responsible for the degree of condensation, takes values ​​from one to infinity, the closer the value is to one, the more uniform the grid. The smaller the values ​​of Mesh curvature factor and Mesh curvature cut off, the more precisely the curvature of the border is specified: with large values ​​of these parameters, a broken line will be considered instead of a curve. Resolution of narrow regions sets the minimum number of elements along the shortest border; for accurate calculations, it is recommended to set the value of this parameter to at least ten. Rice. 1.9. The Refinement method mesh settings window is responsible for the mode of operation of the Refine mesh command and takes two values: Regular and Longest. If set to Regular, this command divides each element into four parts in 2D mode and eight parts in 3D mode. The Longest value divides each element into two parts along the longest side. The Subdomain, Boundary, Edge, and Point tabs allow you to set the element size for the corresponding model elements. The Advanced tab allows you to set the mesh anisotropy. The Remesh button rebuilds the mesh with the new parameters. Copyright JSC "TsKB "BIBCOM" & LLC "Agency Book-Service" 24 In 2D mode, for objects that are close to rectangular, you can set a quadrilateral mesh using the Mesh>Mapped mesh Parameters menu item or the Ctrl+F9 keys. Previously, we mentioned ways to convert 2D models to 3D using the Draw>Extrude and Draw>Revolve commands. In this case, after setting the 3D geometry, you will have to re-create the mesh from tetrahedra, which can take a significant amount of time. Sometimes it is advisable to first build a mesh in 2D mode (triangular or quadrilateral), and then use the Mesh>Extrude Mesh commands to stretch the meshed shape or unwind the meshed shape using the Mesh>Revolve Mesh command. Then the elements will not be tetrahedral, but in the form of parallelepipeds or prisms. The time of constructing such a grid is less than constructing a tetrahedral grid from scratch, but the type of grid does not drastically affect the speed of calculating the problem. Decider The choice of a decider and its parameters is very important, since, in general, the reliability of calculations depends on it. Incorrect tuning can lead to gross solution errors or calculation inconsistencies that are very difficult to detect. It is also necessary to optimize the solution correctly, since, for example, even a not very complex three-dimensional model of an electric brake is calculated for about 10 minutes on a computer with an AMD Phenom II X2 processor and 3Gb of RAM, and some non-linear non-stationary models can be calculated for many hours even on a very powerful computer . The Solve button or the Solve>Solve problem menu item launches the solver with the current settings. The Restart button or the menu item Solve>Restart restarts the solver using the current values ​​(distribution of the magnetic field and current in the winding) as initial values. If we are considering a stationary problem, then pressing this button should not change the solution. Fluctuations in the values ​​in this case indicate the instability of the solution. It is expedient to use this command for complex calculations, when it is possible to obtain an approximate solution on a coarse grid and for a linear or stationary solver, and then making a finer grid and at If necessary, changing the solver to a non-linear or transient one, recalculate the problem. Often this allows you to get a solution faster than the direct calculation of a complex problem. To change the parameters, press Solve>Solver parameters… or the corresponding F11 button. A window will open (Fig. 1.10). If the Auto select solver checkbox is checked, then the program, depending on the application mode, has selected the most suitable solver, which most often does not need to be changed for simple calculations. Rice. 1.10. Solver Parameters window (non-stationary analysis) When choosing a solver, you must first determine the stationary or transient process being studied. If the process is non-stationary, then in the vast majority of cases, the Time Dependent solver is suitable (Fig. 1.10). If the process is stationary, then it is necessary to determine the linearity or non-linearity of the model. If there are doubts about the linearity of the model, then it is recommended to immediately install a nonlinear solver: if you install a nonlinear solver for a linear model, the answer will be correct, but it will take more time to calculations; and if a linear solver is installed for a nonlinear problem, then there will certainly be gross errors. If among the given parameters there are variables (for example, magnetic or dielectric permittivity), for which the dependence on the desired field (current) or other variables associated with the desired field was specified, then the problem is nonlinear. Rice. 1.11. Solver Parameters window (parametric analysis) For linear and non-linear stationary problems, you can select a parametric solver (Parametric), in which you must specify the parameters for which several values ​​are set (Fig. 1.11). So, set a number of different speeds of rotation of the rotor (in Fig. 1.11 range (0,1200,6000)), and then build the mechanical characteristic of this electric machine according to the results obtained. . After selecting the solver in the Solver field, set the main properties. For the Time Dependent tab, this is Time stepping. In the Times field in the range (a:x:b) format, time layers are specified, where a is the start time of analysis, b is the end time of analysis, x is the time interval (step). For example, the time interval is set from 0 to 1 s with an intermediate step of 0.1 s. The time unit in this case is the second, but other units can be set in Physics>Subdomain Settings in the Time scaling coefficient field. If you set 1/60 instead of 1, then the unit of time will be equal to 1 minute. You can set the time parameters of the analysis directly by entering them in this line, or use the Edit button. There we set First and Last Value (initial and final values), respectively, select Step Size (step size) or Number of Values ​​(number of intermediate values), and according to the selected type of interval partitioning, we get what we need. You can also use the function of splitting values ​​according to some law in the drop-down list Function to apply to all values ​​(a function applied to the distribution of values). You can choose, for example, exponential or sinusoidal partitioning. Buttons Add (add) and Replace (replace) allow you to add a new or replace an existing temporary layer. The fields Relative and Absolute Tolerance (relative and absolute error) determine the error at each iteration step. The Allow complex number checkbox allows you to use complex numbers in calculations - this is necessary if you set the PDE coefficients in a complex form. The Times to store in output item determines which time steps will be output for post-processing calculations. The default is Specified Times, i.e. times defined on the General tab. To get the values ​​of all solver steps, select Time steps from solver. In general, the solver selects steps arbitrarily, depending on the dynamics of the system, i.e. ignores the times specified in the General tab. In order for the solver to take this list into account (for example, if external influences are impulsive and the solver can “slip past them”), you must set Time steps taken by solver to Strict (then the solver will be used) only these steps) or Intermediate (the solver uses both free steps and those listed on the General tab) instead of the default Free. If it is necessary to force a time step, then this is done in the Manual Tuning of step size field. The Advanced tab is designed for advanced users and allows you to fine-tune the applied numerical method. For parametric solvers (Fig. 1.11), it is necessary to set the name of the parameter that will be changed in the Name of parameter field and the values ​​that it will take in the List of parameter values ​​field. Values ​​can be given as 0:10:100 or given as the range(0:10:100) function. On a specific picture (Fig. 1.11), the rotation parameter of the electromagnetic brake rotor (rpm) is set. The selected values ​​are from 0 to 6000 every 1200. The Stationary tab allows you to select the type of system for linearity / non-linearity in the Linearity drop-down list. The default is Automatic, and the system itself determines the linearity of the task. For a non-linear problem, you can enter, if required, Relative Tolerance (relative error), Number of Iterations (number of iterations), and also check the boxes next to Damped Newton (Newton's damped method) and Higly Nonlnear Problem (significantly non-linear problem). For significantly non-linear processes, it is recommended to check the Highly nonlinear problem checkbox and increase the number of iterations. For all modes, except for Time Dependent, you can check the Adaptive Mesh Refinement checkbox, then the mesh will be refined according to a complex algorithm when solving. If the physics and geometry are quite complex and it is not very clear how to set the mesh parameters, it is recommended to check this box. However, this will increase the calculation time. You can also set Matrix symmetry to Symmetric if the matrix is ​​symmetrical. Most of the calculation time is occupied by solving systems of linear equations, the Linear system solver is responsible for their solution. The default is Direct (UMFPACK). This solver consumes a lot of computer resources and for models that require a long calculation, you can choose a more suitable one. If the previous resolver doesn't work or runs unacceptably long, you can try SPOOLES - it requires less memory, but is unstable. In an extreme case, an iterative solver GMRES is selected. For positive-definite systems with symmetric matrices, Direct Cholesky (TAUCS) or iterative Conjugate Gradients are chosen. Iterative solvers consume less memory, but you need to watch how they converge and increase the number of iterations if necessary. After setting the properties, press the Solve button or the Solve>Solve Problem command. Often, after obtaining a solution, the model and its parameters (physical properties and boundary conditions) need to be slightly modified. And if these changes are not very large, then you can use the Solve>Update model command. Then the task will not be recalculated, and new values ​​will be obtained by interpolation. You can also press the Restart button, then the task will be recalculated, but the initial Init values ​​will be set to those that were obtained at the previous stage. This can slightly reduce the computation time. Also, using this command, you can identify the instability of the solution, if by pressing this button without changing the model parameters, we get different solutions (oscillations of the numerical solution), then this indicates instability. Then you need to reduce the grid. Visualizing the Results After the solution is complete, the Postprocessing mode is automatically turned on, in which you can observe the results of the calculation. This mode can be manually enabled by the corresponding button on the top panel or by the command Postprocessing> Postprocessing mode. By default, in calculations with perpendicular induction currents, the magnetic induction distribution (Tesla) is displayed over the surface, and the equipotentials show the magnetic potential distributions (Weber/meter). The visualization settings are enabled by the Postprocessing>Plot parameters command or the F12 key. The Plot parameters window opens with several tabs (Fig. 1.12). On the General tab, you can tick off all types of visualization that will be displayed on the screen. You can select Surface (surface), Countour (contour, isoline), Boundary (border), Max/min marker (maximum and minimum mark)), Geometry edges geometry edges). In the Surface mode, the distribution of the investigated quantity on the surface is specified by color. The Contour mode outputs the solution as isolines (equipotentials). Arrow plot displays the vector field (magnetic induction flux) in the form of arrows. Streamline plot plots a vector field as streamlines. Animate in the transition mode creates an animation of the solution if you select Surface, then the window will open (Fig. 1.13), where in the listfined quantitites list (predetermined values) it is possible to set almost any possible parameter: Electric Conductivity, Total Current Density (total density currents), etc. (the default value is Magnetic Flux Density, y component). In this case, the designation of the selected variable will be displayed in the Expression (expression) field (for example, By_q). If you select Contour, the value will be displayed in the message line under the workspace along with the coordinates of the point. In Fig. 1. In the Predefined list, you can also set any parameter that is equal to the defined value (isoline). It is possible to combine in one figure (Fig. 2.55) the output of one parameter by color (filling intensity), and the other parameter in the form of isolines (for example, lines of equal magnetic potential). In the field Solution to use (using the solution) (Fig. 1.12) in the transitional analysis mode, you can select the time layer (by default, the last one is displayed) in the drop-down list Solution at time (solution for time). If you select the Interpolated item there, then in the Time field you can specify an intermediate time value and get an interpolated calculation. In parametric solver mode, the list will not be temporary layers, but parameter values, and you will need to select a parameter in the Parameter Value drop-down list (parameter value). With burning Fig. 1.13. Window Plot Parameters >Surface SNAP switch can only view the values ​​in the grid nodes. If you press the Draw Point for Cross-Section Point Plot button, and then place it on the figure, a window will open with a graph of parameter changes over time. The Draw Line for Cross-Section Line Plot button allows you to draw a straight line through the figure and get a graph of the parameter change along this line. These buttons duplicate the Postprocessing>Cross-Section Plot Parameters menu item, which opens a window with three tabs. On the General tab, you can select the time layers or (in the case of a parametric solver) the parameter values ​​for which the graph will be built. The Point tab allows you to set the coordinates of the points for which the graph will be built and the variable from which it is built. The Line tab also sets the variable and coordinates of the line, it is possible to set the number of equally spaced parallel lines. The transient analysis will build a graph for each selected time layer. If you select the Domain Plot parameters menu item in Postprocessing, then you can get a solution in the form of a graph of the distribution of the parameter under study (current density, magnetic induction, etc.) along the previously specified line. In 3D mode, the main visualization is Slice Plot. In this mode, a certain number of sections of the computational domain with the distribution of the given variable is shown. Isosurface Plot shows isosurfaces. Subdomain Plot shows a picture of the distribution of the scalar field of the parameter under study over the entire volume. Boundary Plot shows the distribution of the investigated parameter on all the boundaries of the figure. Other modes are similar to 2D mode. All parameters of the corresponding visualization modes are configured in the Postprocessing>Plot Parameters (F12) window. In addition, in the three-dimensional mode, you can see the buttons responsible for "lighting" and the angle of the object. Often there is a need to integrate some parameter over a volume, surface, or edge. The Postprocessing>Subdomain/ Boundary/Edge Parameters commands allow you to do this: you can select the required element, set a variable or an expression. So, in order to find out the area or volume (for example, to calculate the volumetric power) of an object, it is necessary to set 1 instead of the integrand. LLC "Agency Book-Service" 33 engraving according to this expression. This is convenient for determining the mechanical characteristics of an electrical machine. After the solution, the computer will immediately display this graph. Each of the resulting graphs can be saved both as a picture and as a text file. You can fully export all received data using the File>Export>Postprocessing Data menu item. Self-test questions 1. How is the model navigator set up? 2. What operations can be performed in the Draw menu? 3. How to draw a rectangle in the workspace? 4. In which menu and in which menu item are constants written? 5. How to set the material properties of the model? 6. How to set up the mesh of the 2D model? 7. What decision device should be chosen to set a number of rotation speeds for building a mechanical characteristic? 8. How to set the construction of lines of equal vector potential on the model? 9. How to get a graph of the distribution of magnetic induction over a given section? Copyright OJSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 34 2. SIMULATION OF ELECTROMECHANICAL DEVICES IN 2D MODE Mastering the technique of modeling various electrical devices in Comsol Multiphysics is most effective on specific examples. In this case, it is necessary not only to build a model of an electrical device, but also to explore it most fully. 2.1. DC electromagnet Job. Build a model of a C-shaped electromagnet with the following data: the number of turns in the excitation winding w = 5000, current I = 10 A, working gap δ = 25 mm, magnetic circuit cross section 50x50 mm2, height and width of the magnetic circuit, respectively 400 and 350 mm. Determine the value of the scattering fluxes and the scattering coefficient. Construct graphs of the distribution of magnetic induction: a) along the width of the pole in the middle of the gap and on the surface of the poles; b) in the longitudinal direction at the edge of the pole and away from the pole. Model building. After double-clicking on the icon of the Comsol Multiphysics program, we get to the model navigator window. For our model, we need to select a two-dimensional coordinate space, for which we make sure that the Space dimension pop-up list is set to 2D mode. Then we select the section of the AC / DC Module program that is responsible for modeling electricity. Left-click on the plus sign opposite this section, after which the subsections contained in this section will open. Our simulation requires the mode Statics, Magnetic. Select it - click the cross opposite this mode. There are various modes of operation that allow you to select the type of task. We need the very first one - Perpendicular Induction Currents, Vector Potential. This time we click on the name of the mode with the left mouse button, it should be highlighted in blue. Now click OK. The main working area of ​​the program has appeared. We are currently in drawing mode. This is evidenced by the depressed icon. First, you need to specify the area in which the designed electromagnet will be located. The dimensions of this region should be several times larger than the dimensions of the electromagnet. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 35 The farther the boundaries of this zone from the surfaces of the electromagnet, the less distortion they will introduce into the picture of the electromagnetic field created by the electromagnet. For definiteness, we will create this zone in the form of a rectangle with dimensions of 11m2. There are two ways to create a rectangle. The first is from one of the vertices, and the second is from the center. For convenience, let's take the second one. To do this, on the drawing panel (to the left of the workspace), press the button, move the mouse to the point (0; 0) and press the left button and then move the mouse to one of the vertices of the future rectangle. Let this be the top (0.5; 0.5). After that, click the left button again and the rectangle is ready. Rice. 2.1. Setting up the model navigator Now let's draw the core of the future magnetic circuit. The easiest way to do this is by segments of straight lines, drawing them from point to point, observing the specified dimensions. To do this, press the button, thereby choosing the option of constructing a drawing of the magnetic circuit with a broken line. Let's increase the drawing area using the button on the main panel and take, for example, a point with coordinates x = -0.2; y = -0.05, press the left mouse button. Next, you need to go up 20 cm, then right 35 cm, then down 40 cm, then left 35 cm, then up 15 cm, then right 5 cm, down 10 cm, etc. To do this, move the cursor up from the starting point to the point (-0.2; 0.15) and note that the cursor is followed by a straight line. At the second point, press the left mouse button again and move the cursor to the point (0.15; 0.15) and again notice that the mouse is followed by a line from the previous point. Press the left mouse button again. Now our task is to close the lines into a figure by drawing a core. To do this, go in turn to the following points: (0.15; -0.25); (-0.2; -0.25); (-0.2; -0.1); (-0.15; -0.1); (-0.15; -0.2); (0.1; -0.2); (0.1;–0.2); (0.1;0.1); (0.1;0.1); (–0.15; 0.1); (-0.15; -0.05) - perform the previously described operations and close at the first point (-0.2; -0.05). Press the right mouse button to finish drawing. You should get a shape like in Fig. 2.2. Building by points resulted in the air gap being too large. Of course, it was possible to pre-increase the number of points on the axes using the Options>Axis/Grid Settings window, but we will do it in a different way. To do this, on the resulting figure of the magnetic circuit, double-click the mouse. The Object Properties window should appear, and the shape should break into numbered lines. Rice. 2.2. The first option Let's do it in such a way as to raise the horizontal line at the bottom of the magnetic circuit at number 3. To do this, select it in the list and note that it is highlighted in red. Our task is to shift it upwards, i.e. for two points, set new coordinates along the Y axis. In both cases, enter the coordinates –0.075 and press the Preview button. It can be seen that the red line has moved. But the figure is now not closed. To close it, you need to raise the vertical lines 1 and 7. Define the line with number 1 in the list and for the point (–0.2; –0.1), change the coordinate value –0.1 to –0.075 and press Preview again. Now line 1 is connected to line 3. The line 7 remains. Similarly, we replace the coordinate -0.1 at the point (-0.15; -0.1) with -0.075 and click Preview. The shape is now closed. You can click OK. Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Book-Service 37 6 10 4 9 5 3 7 1 8 2 Pic. 2.3. Making a drawing of the magnetic circuit After that, we will draw two current windings using rectangles. To do this, press the button and select the point (0,1;0). Let's make a left mouse click and drag the cursor to the point (0.05; -0.1). Similarly, create another rectangle using the points (0.15; 0) and (0.2; - R3 R2 0.1). The result should be the following figure, as in (Fig. 2.4). When the geometry is built, you can move on to setting constants and CO1 variables. To do this, go to the menu Fig. 2.4. Final Options>Constants and set in the fields the drawing of the electromagnet of the expression according to the table below. Table 1 Name Imax Sob Expression 10 0.005 Wob 5000 Description Current in the conductor Winding area Number of conductors in the winding After all the constants have been written, you can click OK. Now we go to the Options>Expressions>Global Expressions menu, in which we enter the expression for the current density according to Table. 2. Table 2 Name J Expression (Imax*Wob)/Sob Description Winding current density Press OK. The next step is to set the physical properties for the regions. To do this, open the Physics>Subdomain Settings menu (Fig. 2.5) and see that the program has divided our drawing into 4 areas. Now we need to set the physical properties offered in this menu for these areas. Let's start with area 1, which is air (Fig. 2.6, a). Set the parameter σ (Electric conductivity) to 0.001, and leave the rest of the parameters unchanged. Rice. 2.5. Setting the physical properties of the regions Let's move on to region 2 (Fig. 2.6, b). This area is the core. Let us set the following parameters: σ (Electric conduction) 0.1 and μr (Relative Permeability) – 1000. We leave the other parameters unchanged. a) b) Fig. 2.6. Highlighted areas: a – area of ​​space 1 outside the electromagnet; b-magnetic circuit The next area numbered 3 (Fig. 2.7, a) corresponds to the winding. Let's set the following parameters: σ (Electric conductivity) - 1 and Jez (External Current Density) - J. The remaining parameters are not changed. For the remaining area 4 (Fig. 2.7,b), we will set similar parameters, except that in the Jez (External Current Density) parameter we will set the value to -J. a) b) Fig. 2.7. Selected areas: left side (a) and right side (b) of the excitation winding This completes the setting of the area parameter. You can close the Subdomain Settings window by clicking OK. Usually the program itself exposes them correctly, but it's always worth checking. Let's go to the Groups tab and make sure that two groups are created, the first one is for the outer rectangle. The Boundary Condition line is set to Magnetic Insulation. The second group, which represents the boundaries of the core and windings, is set to Continuity in the Boundary Condition line. Rice. 2.8. Boundary Conditions Setting Window The next step in setting up the model is setting the grid. Since the model is quite simple, we will set the smallest grid. To do this, go to Mesh> Free Mesh Parameters or press F9. A window similar to the one in Fig. 2.9 Set Predefined Mesh sizes to Extemely Fine. Then press Remesh and wait until the mesh is built. After its creation, you can proceed to the configuration of the resolver. Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Book-Service" 41 Pic. 2.9. Mesh Setup Window Let's go to the Solve>Solver Parameters menu or press the F11 key (Fig. 2.10). Let's check which resolver is installed. Stationary must be set in the Solver list and Linear System Solver must be set to Direct (UMFPACK). If so, then you can click OK and proceed to the solution. To do this, click the button on the toolbar and wait a few minutes until this task is completed. Rice. 2.10. Solver settings window Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 42 Model study. At the end of the solution, a picture of the distribution of the field should appear. By default, the distribution of the normal component of magnetic induction appears. Let's go to the Postprocessing>Plot Parameters menu (Fig. 2.11). Rice. 2.11. Result Output Window Next, click on the Surface tab and select Total Current Density, z component from the Predefined Quantities list. Now let's move on to the Contour tab. Put a checkmark next to the inscription Contour Plot. This checkbox will enable the display of lines in the figure. In the Predefined Quantities list, select Magnetic Potential, z component. In Number of Levels we will write the value 30 (Fig. 2.11). Let's put an end to the Uniform Color. Press the button Color.. In the palette that appears, select the blue color and press OK. Now click OK on the Plot Parameters menu. A picture should appear similar to the picture in Fig. 2.12. Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Book-Service" 43 Pic. 2.12. The picture of the distribution of the magnetic field of the electromagnet Let us define the leakage flux, understanding by it that part of the flux that does not reach the working gap. Built in fig. 2.12 lines of equal vector magnetic potential form tubes of equal magnetic flux, therefore, by calculating the number of flux tubes passing inside the excitation winding and in the working gap, one can estimate their difference, which will characterize the leakage flux. The ratio of the stray flux to the total flux will determine the scatter factor. In this example, the number of equal flow tubes in the field winding area is 20, and in the working gap area 8. Thus, the leakage flux is determined by 12 equal flow tubes, and the scattering coefficient for this 2D model is kp = 0.6. To obtain graphs of the distribution of magnetic induction in the gap, it is necessary to draw additional lines along which we will consider the distribution of induction. First, let's set up the drawing grid. To do this, go to the Options>Axes/Grid Settings menu (Fig. 2.13) and select the Grid tab. Let's uncheck Auto and in the line y spacing we will write the value 0.0125. Now it will be convenient to build the necessary lines. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 44 Let's go back to the drawing mode and draw a few straight lines with the button. The first straight line with coordinates (-0.2; -0.075) and (-0.2; -0.05), the second - (-0.15; -0.075) and (-0.15; -0.05), the third - (-0.35; -0.075) and (0; -0.075), the fourth - (-0.35; -0.0625) and (0; -0.0625), the fifth - (-0.35; –0.05) and (0; –0.05), the sixth – (–0.25; –0.075) and Fig. 2.13. Menu Options>Axes/Grid Settings (-0.25; -0.05), seventh - (-0.1; -0.075) and (-0.1; -0.05). The result should be a picture similar to Fig. 2.14. Now let's go back to Pole B5 Physics>Subdomain Settings B7 B1 B2 B4 B6 and set up new subdomains B3 according to the task. For Clearance of this for sub-areas with Fig. 2.14. Additional lines in the gap, numbers 2, 3, 5, 6, 8 and 9 (they are highlighted in color on the lines necessary to obtain the graphs in Fig. 2.15), you must specify characteristics similar to sub-area 1, i.e. set the parameter σ (Electric conductivity) to 0.001, and leave the other values ​​unchanged. Check Physics > Boundary Settings Pole and make sure the outer rectangle Gap 3 5 is set to Magnetic Insulation and the rest of the lines are set to 2 6 8 9 Continuity. Now we need to recalculate the grid. You can use the button. Rice. 2.15. Selected sub-areas with numbers Then you can restart the decisive 2, 3, 5, 6, 8, 9 device with the button. The resulting solution will not differ from the previous one. Now we can investigate the distribution of induction along lines. Let's call them conditionally B1 ... B7 as in fig. 2.14. Go to Postprocessing>Domain Plot Parameters. Go to the Line/Extursion tab. The drawing area will switch to line mode. Now let's allocate Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 45 line B1. It is divided into two arrows. To select it, hold down Ctrl and click on both arrows. This will highlight them (Figure 2.16). Now let's write normB__emqa in Predefined Quantities. This variable shows normal. 2.16. Vymal component of induction modulo. line division You can click OK. A graph will appear, similar to the one shown in Fig. 2.17, a. Let's repeat the data of the manipulation graphs for the remaining six straight lines. B, T B, T 0.2 0.3 0.2 0.1 0.1 0 0 0.01 0.02 a) x, m B, T 0.28 0 0 0.02 x, m b) B, T 0.039 0.26 0.0388 0.22 0.0386 0.18 0 0.01 0.02 y, m 0.01 c) 0.0382 0 0.02 y, m 0.01 d) Fig . 2.17. Distribution of magnetic induction: along the x axis a - in the middle of the gap; b - on the surface of the pole; along the y axis in - at the edge of the pole; d – far from the pole 2.17 shows the distribution of magnetic induction along the x axis in the middle of the gap (line B4) and on the surface of the poles (lines B3 and B5). The distribution of magnetic induction in the middle of the gap (Fig. 2.17, a) is a smooth curve, reaching a maximum under the center of the pole. The curve is slightly asymmetrical. The decline in magnetic induction at the right edge of the pole (located closer to the excitation winding) is slower than at the left edge of the pole. Copyright JSC "Central Design Bureau" BIBCOM " & LLC "Agency Book-Service" 46 In fig. 2.17, c, d are graphs of the distribution of magnetic induction in the longitudinal direction (along the y axis) at the edge of the pole and away from the pole (at a distance equal to the width of the pole). From fig. 2.17,c it can be seen that the magnetic induction at the edge of the pole varies from 0.3 T to 0.2 T (in the middle of the gap). At the same time, on the right and left edges of the pole (lines B1 and B2), the law of change is the same. Away from the pole (lines B6 and B7), the magnetic induction is 5 times less than under the pole and changes insignificantly. 2.2. Electromagnetic brake with a massive rotor based on the stator of an asynchronous motor Task. It is required to obtain a 2D model of a brake with a massive ferromagnetic rotor, made on the basis of the stator of a two-phase asynchronous motor ADP 532, and to study various modes of brake operation, taking into account the stator gearing. Electrical conductivity of the rotor material γ=6106 Sm/m. The magnetization curve of the rotor material is given in the table, the working gap between the stator and the rotor is 0.3 mm. Model building. When building a model using Comsol Multiphysics, we first configure the navigator (Model Navigator). To do this, run the program and select a 2D space in the Space Dimension in the Model Navigator. Next, select the AC / DC Module folder. In it, select Statics, Magnetic, and then Perpendicular Induction Currents, Vector Potential. Next, click the Multiphysics button. Since the rotor rotates in the electromagnetic brake, it is necessary to create a condition for the rotation of the grid. To do this, click Add. Now we go to the Comsol Multiphysics folder, and in it we find the Deformed Mesh folder. In it, select Moving Mesh (ALE). Now both modes have appeared on the right side and it is necessary to set their connection. First select Induction Currents, Vector Potential. Click the Application Mode Properties button. We leave all the settings in place, except for Constraint Type and Frame. Set them to Non-ideal and Frame (ale), respectively. We press OK. Now select Moving Mesh (ALE). It turns out that Perpendicular Induction Currents, Vector Potential and Moving Mesh (ALE)(ale). lie in the same folder, as in Fig. 2.19. Perpendicular Induction Currents, Vector Potential must be the first mode. If the Moving Mesh (ALE)(ale) is ahead of it, select the Moving Mesh (ALE)(ale) and click Remove. And then add Moving Mesh (ALE)(ale) again from the folder. If everything is similar to Fig. 2.19, then click OK. Rice. 2.19. Customizing the Model Navigator Building the model in this example is different from the previous example. Since the graphical capabilities of the Comsol Multiphysics program are limited, and the presence of a powerful internal graphic editor is inappropriate in a fairly complex and powerful complex, it is necessary to use import from external CAD systems: Autodesk AutoCAD, Compass and others to study complex models. Rice. 2.20. Brake drawing Copyright OJSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 48 In the above example, the graphics were imported from one of the CAD systems. On fig. Figure 2.20 is a snapshot of this model in drawing mode in Comsol Multiphysics. After the geometry has been exported, you must enter constants and expressions for the model. To do this, go to the Options>Constants menu. We introduce the following constants according to the table. 3. Table 3 Name Expression Description d 0.38*10^(-3) Excitation wire diameter s ((3.14*(d^2))/4) w 164 Im 0.6[A] Sa w*sa rpm –1909.96 Conductor area excitation windings Number of conductors in the groove of the excitation winding Maximum amplitude of the excitation winding current Total area of ​​the excitation winding conductors Rotor speed, (rpm) omegarot 2*pi*frot TIME frot gammarot c 2.5*pi/omega[s] (rpm/60) 6e6 a/delta radius (19.7e-3) S1 33.370698e-6 Area of ​​the outer part of the groove S2 length delta 31. 177344e-6 (65e-3)[m] (0.3e-3)[m] Groove inside area Machine active length Air gap gamma 5.998e7 Rotor RPM, (rad/s) Time (static mode only) Rotational speed of the rotor Conductivity of the material of the rotor Ratio of the thickness of the rotor to the size of the air gap Radius of the outer surface of the rotor Conductivity of the material of the stator winding Now the constants are written and you can click OK. Let's move on to filling in the global variables of expressions. To do this, go to Options>Expressions>Global Expressions menu. We enter expressions according to the table. 4. Table 4 Name Jv Expression 0.5*Im*w/S1 Jn 0.5*Im*w/S2 dvx dvy Bn omegarot*y -omegarot*x (x*Bx_emqay+y*By_emqa)/sqrt(x^2+ +y ^2) Btn Hn Htn Description Field winding current density in the upper slots Field winding current density in the lower slots magnetic induction (-x*Hx_emqa-y*Hy_emqa)/sqrt(x^2+y^2) Normal component of magnetic field strength (-x*Hy_emqa+y*Hx_emqa)/sqrt(x^2+y^2) Tangential component of the magnetic field After filling in the table, press OK and proceed to the next step. Now let's write the expression H=f(B) for our rotor. To do this, go to Options>Functions. Let's press the New button. The New Function window will appear (Figure 2.21). In it, we write the value func in the Function Name and select the Interpolation value. Leave Table in the list. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 50 Fig 2.21. H=f(B) function setting window In the table that appears, leave the Piecewise Cubic and Interpolation Function values ​​for the Interpolation Method and Extrapolation Method lines, respectively. Fill in the data in the table in the window according to the table. 5. X denotes the induction of the magnetic field B, and f(x) is the strength of the magnetic field H. x -2.09 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0, 6 –0.5 –0.4 0 0.4 0.5 0.6 0.8 1 1.2 1.4 1.6 Table 5 f(x) –44000 –127800 –4100 –2090 –1290 –924 –682 –488 –400 –320 0 320 400 488 682 924 1290 2090 4100 H, A/m 104 0.5 0 –0.5 –1 –2 –1 0 1 V, T 2.22. Curve of magnetization of the rotor material Copyright OJSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 51 1.8 2.09 127800 44000 Let's check the entered data by pressing the Plot button. A graph should appear, as in Figure 2.22. Now it is necessary to describe the properties of subdomains and boundary conditions. Since the embedded CAD model contains the geometry of a two-phase rotor, only the windings of one phase will be energized. Make sure Perpendicular Induction Currents, Vector Potential is selected in the Multiphysics menu at the top. Now go to Physics>Subdomain Settings or press F8. So, in this model there will be nine different groups of subdomains with their own unique properties. First, we select subdomains according to Fig. 2.23, a. To select the specified subdomains, do not close the Subdomain Settings window, but only move it away. Next, select the subdomains with the left mouse click while holding down the Ctrl key. After the subdomains are selected, we set properties for them. Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Kniga-Service" 52 a) c) b) d) Pic. 2.23. Setting the current density positive (a) and negative (b) in the lower layers of the excitation winding; positive (c) and negative (d) in the upper layers of the excitation winding Let's edit the parameters in these subdomains in the Subdomain Settings window (Fig. 2.24). In the constant L we write length, in the constant J ze - Jv, and in the constant σ - gamma. Click the Apply button. Now again, without closing the Subdomain Settings window, select subdomains according to Fig. 2.23b. Copyright OJSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 53 Similarly, we will edit the constants in these subdomains in the Subdomain Settings. In the constant L we write length, in the constant J ze - Jv, and in the constant σ - gamma. Click the Apply button. Now again, without closing the Subdomain Settings, let's select the subdomains according to Fig. 2.23, in Fig. 2.24. Parameters setting window The data of the sub-region (Fig. 2.23, c) correspond to the excitation winding in the lower slots. Let's edit the parameters in the subdomain data in the Subdomain Settings in the same way. In the constant L we write length, in the constant J ze - Jn, and in the constant σ - gamma. Click Apply. Now again, without closing the Subdomain Settings, let's select the subdomains according to Fig. 2.23, g. In the constant L we write length, in the constant J ze - Jn, and in the constant σ - gamma. Click the Apply button. Now again, without closing the Subdomain Settings, let's select the subdomains according to Fig. 2.25 a. These subdomains (Fig. 2.25, a) correspond to a massive rotor. We set the following constant values ​​for it. The constant v (speed) has two fields to fill in. We prescribe in the first dvx, and in the second dvy. We write length in L, and gammarot in the constant σ. We select the line H=f(B) in the law of dependence H ↔ B, and then in the appeared fields H we write func(Bx_emqa) and Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Book-Service 54 fubc(By_emqa) respectively. 2.25, b, c. Now select the subdomains of Fig. a) b) c) Fig. 2.25. Setting the parameters of the massive rotor (a) stator (b) and free areas (c) In Fig. 2.25, b, the outer sub-region is selected, which corresponds to the stator. It has the following constants: L is equal to length, and μτ is equal to 4000. Now go to the Groups tab of the constant and define the remaining group of unselected subareas that correspond to fig. 2.25, g. For a given group of subregions where there are no currents, we set the constant L equal to length. Now we press OK. Let's set up subregions for the Moving Mesh (ALE) mode. To do this, select the menu Multiphysics>2. Moving Mesh (ALE) (ale). Now let's go to Physics>Subdomain Settings and select all subdomains and set them to No displacement. The setting of the sub-area parameters is completed. Let's move on to creating a mesh for the model. To create and configure a mesh, go to the Mesh>Free Mesh Parameters menu or press the F9 button. A menu should appear as shown in Fig. 2.26, a. Select Extremely Fine from the Predefined Mesh Size drop-down list. This will allow you to solve the problem very accurately. Since the problem is two-dimensional and linear, the solution will not be difficult for a sufficiently powerful computer. The program itself will create the most convenient grid for calculation after pressing the Remesh button. Ultimately, you should get something similar to Figure 2.26b. If you are not satisfied with the mesh size, then you can configure it yourself by selecting the checkbox next to Custom mesh size. Also, if you need a higher accuracy of the grid at some point in the task, then you can use the tabs Subdomain (subdomain), Boundary (border), Point (point). b) a) Fig. 2.26. Mesh creation: a - Free Mesh Parameters window, b - model mesh Now let's move on to setting up the solver. Let's go to the Solve>Solver Parameters menu or press the F11 button. A window will appear as shown in Fig. 2.27. Stationary static mode resolver is currently selected. Let's choose Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 56 Parametric in the list. In the Parameter names line, write the rpm parameter. This is the rotational speed of the brake rotor (rpm). In Parameter Values ​​we will write range (0.50, 200), i.e. we will vary the rpm parameter from 0 to 200 rpm every 50 rpm. Let us leave the remaining parameters of the solver as standard, since they are optimally selected for this task. Let's press OK. Let's also try to separately derive graphs of the mechanical characteristics through the following formulas when solving separately: /m is the tangential component of the magnetic field strength, J , A/m2 is the current density, L is the rotor length along the Z axis, R is the rotor radius. Rice. 2.27. Solver Parameters window Copyright OJSC "Central Design Bureau "BIBCOM" & OOO "Agency Kniga-Service" 57 To do this, call the window Postprocessing > Probe Plot Parameters (Fig. 2.28) Fig. 2.28. 2.28. Probe Plot Parameters window Click the New button. In the Plot type pop-up window, select Integration. Leave Domain Type - Subdomain. In Plot Name we will write the name of our chart, for example "Moment". Now we choose the subdomains of the rotor similarly to Fig. 2.25 a. In the Expression field, we write the integrand formula - Jz_emqa*Bn*length *radius. Now, to test, let's create another function Fig. 2.29. Selection of the outer surface of the rotor for determining the integral. Similarly, press the New button. In the Plot type pop-up window, select Integration. Let's select in Domain Type - Boundary. In Plot Name, write the name of the chart - "Moment 2". Let's press OK. Now it is necessary to select the surface of the rotor (Figure 2.29), since integration over the surface is assumed (moment through the tension tensor Maxwell) In the Expression field, write the integrand formula Bn * Htn *length*radius. Now we can start solving. To do this, click Solve>Solve Problem or the = icon on the panel. The solver will start and you will have to wait a few minutes. Conclusion and analysis of the results of calculation. After the 0.3 calculation, Comsol will automatically display the torque graphs (Fig. 2.30), since the 0.2 calculation was registered. To obtain a more visual and 0.1 smooth picture of the dependence of torque on speed in Solver 0 120 160 ω 0 40 80 Parameters in the values ​​of Fig. 2.30. Dependence of the moment Parameter Values ​​it is desirable to prescribe range (0.10, 200) on the rotation speed. However, a large number of points will interfere with obtaining other graphs, so obtaining graphs of induction, current, etc. along the surface and along the depth were carried out in the calculation with five parametric points. Now let's configure the display options for the solution. To do this, go to Postprocessing>Plot Parameters. Select the Surface tab and select Total current density, z component from the Predefined Quantities list. Then let's move on to the Contour tab. In Predefined Quantities, select Magnetic Potential, z component. In Levels we will write 40, and in Contour Fig. 2.31. In the Plot Parameters Color window, select Uniform Col on the Contour or tab, for example, blue color (Fig. Copyright OJSC "Central Design Bureau" BIBCOM " & LLC "Agency Book Service" 59 2.31). Do not forget to check the box in the upper left corner opposite Contour Plot. Now we press OK. Rice. 2.32. The picture of the electromagnetic field in the brake On the graph (Fig. 2.32) you can see the distribution of current density and magnetic potential in the electromagnetic brake. Lines are limited to tubes of equal magnetic flux. Where the lines are drawn thicker, the magnetic induction is greater. The graph shows how the magnetic field is carried away by a rotating rotor. The color shows the current density distribution in the rotor. Let us consider how the brake parameters change along the surface of a massive 2.33. Window Domain Plot Parameters Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Kniga-Service To do this, go to the Postprocessing>Domain Plot Parameters menu and select the Line/Extrusion tab (Fig. 2.33). Now select a line representing the surface of the rotor. To do this, let's alternately enter the values ​​Bn, Btn, Hn, Jz_emqa in the Expression field and, after each new value, press the Apply button, we will get graphs of the distribution of this variable over the selected length. You should get graphs similar to the graphs in Fig. 2.34, a, b and fig. 2.35, a, b. Bn, T Btn, T 1 3 2 5 0.4 0.4 2 0 0 54 3 4 1 –0.2 –0.4 –0.6 –0.8 0 0.04 a) l, m 0, 08 –1 0 0.04 b) 0.08 l, m 2.34, a. Distribution of normal (a) and tangential components of induction along the length of the rotor at different speeds of rotation of the rotor: 1– n = 0 rpm; 2– n = 50 rpm; 3– n = 100 rpm; 4– n = 150 rpm; 5– n = 200 rpm Hn, A/m 106 1 3 2 0 5 4 –2 –4 2 J, A/m2 106 5 2 4 3 0 2 1 –2 –4 0.04 l, m 0, 08 0.08 l, m 0 b) a) Fig. 2.35. Distribution of the normal component of tension (a) and current density (b) along the length of the rotor at different speeds of rotation of the rotor: 1– n = 0 rpm; 2– n = 50 rpm; 3– n = 100 rpm; 4– n = 150 rpm; 5– n = 200 rpm 0 0.04 Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Book-Service 61 2.36. Selection of a line to determine the parameters for the depth of the rotor Now we will get the graphs of the distribution of the same parameters for the thickness of the rotor. To do this, we select the line according to Fig. 2.36 and repeat the manipulations with the introduction of variables. As a result, we get graphs (Fig. 2.37, 2.38). Bn, T Btn, T 0 4 –0.2 3 –0.4 2 –0.6 5 0.3 0.1 0 –0.1 1 –0.3 4 1 2 3 5 –0.5 –0 .8 0 0.004 0.008 0.012 l, m 0 0.004 0.008 0.012 l, m b) a) 2.37. Distribution of normal (a) and tangential (b) components of induction over the thickness of the rotor at different speeds of rotation of the rotor: 1– n = 0 rpm; 2– n = 50 rpm; 3– n = 100 rpm; 4– n = 150 rpm; 5– n = 200 rpm Copyright JSC Central Design Bureau BIBCOM & OOO Agency Kniga-Service 62 Hn, A/m 106 0 –1 –3 –5 J, A/m2 106 3 1 2 3 4 5 2 1 0 2 4 3 1 5 0.004 l, m 0 0.004 0.008 0.012 l, m 0 b) a) Fig. 2.38. Distribution of the normal component of tension (a) and current density (b) over the thickness of the rotor at different speeds of rotation of the rotor: 1– n = 0 rpm; 2– n = 50 rpm; 3– n = 100 rpm; 4– n = 150 rpm; 5– n = 200 rpm –7 –1 Similarly, other parameters can be considered depending on the purpose of the study. 2.3. Electromagnetic brake with a hollow ferromagnetic rotor Task. Run a simulation of an electromagnetic brake with a hollow ferromagnetic rotor, using the model of a brake with a massive rotor as a base. The thickness of the hollow rotor is 1.7 mm. Maximum rotation speed 3000 rpm. Model development. Open a model with a massive rotor and select Draw Mode from the toolbar. Our task is to draw the inner surface of the rotor. Let's leave a gap equal to 0.3 mm, and make the rotor 1.7 mm thick. Therefore, we need to draw a circle with a radius of 18 mm. To do this, in the Draw Mode, select the Ellipse/Circle (centered) button and with the Ctrl key pressed, holding the left mouse button, draw a circle, the center of which is a point with coordinates (0,0). If Grid is set too large, then draw a slightly smaller circle, and then double-click on the resulting circle to open the properties and set the following values ​​for the axes: A-semiaxes: 0.018; B- Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 63 semiaxes: 0.018 (Fig. 2.39). The result should be a model of a hollow rotor. Now let's move on to editing the subdomains of the model in the Subdomain Settings. The hollow rotor is part of the previously existing massive one, so its parameters can not be changed, and for the circle remaining inside, it is necessary to set the parameter. 2.39. Air Ellipse figure settings window. Because of the line drawn in the circle, there were two areas in it. To edit the constants of these subdomains, we select them and in the constant v (speed) in the two available fields to fill in, erase dvx and dvy, and write 0 instead. a) b) Fig. 2.40. Editing subdomains located inside the circle: a – air; b – hollow rotor The subdomains we have identified are now air. It remains to edit the moment definition in Postprocessing>Probe Plot Parameters. From the old model, the definitions of integrals over the circumference and area remained (more precisely, Copyright JSC "Central Design Bureau" BIBCOM " & LLC "Agency Kniga-Service" 64 in terms of volume and surface, since the formula contains multiplication by the length of the rotor), but since the rotor is now hollow, its volume has changed and one more surface (internal) has been added. Therefore, the first formula can be kept unchanged, the second formula must be clarified and a formula for determining the moment along the lower boundary must be added. Together with the moment along the upper limit, it will have to give the same mechanical characteristic as when integrating over volume. Let us edit the moment by volume and select the subdomains for integration shown in Fig. 2.40, b (i.e. Fig. 2.41. Selection of the inner sub-region of the hollow rotor). Let's create a new hollow rotation surface function by clicking the New button in the Probe Plot Parameters window. In the Plot type pop-up window, select Integration. Let's select in Domain Type - Boundary. In Plot Name we will write the name of our chart - "Moment 3". Let's press OK. Now we need to select the inner surface of the rotor (Fig. 2.41). In the Expression field, write the integrand Bn*Htn*length*radius. The last step before calculating the model is to change the parameters of the solver. So, the rotation speed of a hollow rotor is higher than the rotation speed of a massive one, so go to Solver Parameters and edit the Parameter Values ​​field by changing the step and final speed. Let's write the following - range (0,600, 3000). You can click OK. Conclusion and study of simulation results. Run the model by clicking the button on the toolbar. As a result of calculations, we obtain the dependences of the electromagnetic torque on the rotor speed (Fig. 2.42) - the mechanical characteristics of the brake. The first characteristic is obtained by integrating over the volume the product of the rotor current density and the primary magnetic induction, the second and third characteristics - by integrating over the upper and, respectively, the lower surface of the rotor the product of the normal component of the magnetic induction and tangential component of the magnetic field strength (using the Maxwell stress tensor). From the graphs (Fig. 2.42) it can be seen that the sum of the moments on the upper and lower surfaces of the rotor is equal to the moments determined by integration over the volume of the rotor. In this case, the value of the moment on the lower surface of the rotor is much less than on the upper one. Bn, T 0.08 1 2 0.06 0.04 0.02 3 0 0 1000 2000 2.42. Mechanical characteristics of the brake obtained by integration: 1 - by volume; 2 - along the upper surface; 3 – along the lower surface of the hollow rotor By going to the Postprocessing> menu and setting the output of the current density over the rotor cross section, as well as the distribution of lines of equal vector potential, you can get a picture of the electromagnetic field in the brake rotor at a given rotation speed (Fig. 2.43). Tubes of equal magnetic flux, formed by lines of equal magnetic potential, show that the magnetic flux is almost completely closed along the rotor. The current density varies over a wide range both along the circumference of the rotor and along its thickness. Let us consider in more detail how the magnetic induction and current density change along the circumference and along the thickness of the rotor. To do this, go to the Postprocessing>Domain Plot Parameters menu and select the Line/Extrusion tab. Rice. 2.43. Picture of the electromagnetic field in the brake rotor Now let's choose a line representing the upper surface of the rotor (Fig. 2.43). Similarly to the previous example, we will alternately enter the values ​​Bn, Jz_emqa into the Expression field, pressing the Apply button after each new value, and we will get graphs of the distribution of this variable over the selected length. You should get graphs like in Fig. 2.44. Copyright OAO Central Design Bureau BIBCOM & OOO Agency Kniga-Service 67 J, A/m 106 Bn, T 0.2 2 0.1 0 1 4 6 5 –0.1 0 –0.2 –0.3 0 3 1 2 0.02 0.04 x, m –1 0 0.02 0.04 x, m b) a) Fig. 2.44. Distribution of the normal component of induction (a) and current density (b) in the upper layer of the rotor along its circumference at different rotation speeds: 1 - n = 0 rpm; 2 - n=600 rpm; 3 - n=1200 rpm; 4 - n=1800 rpm; 5 – n=2400 rpm; 6 - n=3000 rpm An analysis of the graphs (Fig. 2.44) shows that with an increase in the rotor speed, the magnetic induction decreases in value and shifts in phase in the direction of rotor rotation, and the current density increases with an increase in rotor rotation speed. To determine the laws of distribution of these parameters over the thickness of the rotor, we select whether 2.45. Selection of a line to determine the distribution of parameters over the thickness of the rotor, leaving the center and passing along the rotor (Fig. 2.45). Then we repeat the operations with the definitions of graphs for Bn, Btn, Htn, Jz_emqa and get the graphs (Fig. 2.46, a, b and Fig. 2.47, a, b). Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Kniga-Service 68 Bn, T Btn, T 6 1 0 5 2 4 –0.1 3 3 4 2 –1 1 –0.2 6 5 0.004 y, m b ) a) Fig. 2.46. Distribution of normal (a) and tangential (b) components of induction over the thickness of the rotor at different rotation speeds: 1 – n = 0 rpm; 2 - n=600 rpm; 3 - n=1200 rpm; 4 - n=1800 rpm; 5 – n=2400 rpm; 6 - n \u003d 3000 rpm 0 0.002 0.004 y, m 0 0.002 06 T in the surface layer. In addition, it almost linearly changes along the thickness of the rotor, approaching a value close to zero in the inner layer of the hollow rotor. In this case, the normal component of the magnetic induction on the inner surface of the rotor and in the inner air space inside the hollow rotor changes when the rotation speed changes from 0.02 T to zero. The tangential component of the magnetic induction changes differently: with an increase in the rotation speed, it increases, increases when approaching the inner surface of the hollow rotor, i.e. changes along the thickness of the rotor in the opposite direction. In contrast to the normal component of the magnetic induction (in the surface layer of the rotor 69 the tangential component of the magnetic induction is practically equal to zero). It is characteristic that in the internal space inside the hollow rotor the tangential component of the magnetic induction is also practically equal to zero. The distribution of the tangential component of the magnetic field strength over the thickness of the rotor is similar to the distribution of the tangential component of the magnetic induction. The difference lies in the fact that in the internal space (air) inside the hollow rotor, the tangential component of the magnetic field strength is not equal to zero. Htn, A/m 103 0 J, A/m2 107 –1 1 6 5 4 3 1 –1 2 2 4 5 –9 0 6 1 0.002 0.004 у, m 0 0.004 у, m 2.47. Distribution of the tangential component of the magnetic field strength (a) and current density (b) over the thickness of the rotor at different rotation speeds: 1 - n = 0 rpm; 2 - n=600 rpm; 3 - n=1200 rpm; 4 - n=1800 rpm; 5 – n=2400 rpm; 6 – n=3000 rpm 0.002 Current density distribution over the rotor thickness differs from those considered. The current density increases with increasing speed of rotation and increases, approaching the upper surface of the rotor, while remaining equal to zero on the inner surface of the rotor. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 70 2.4. A simplified model of a salient-pole brake with a hollow non-magnetic rotor Task. Obtain a simplified model of a salient-pole brake with a hollow non-magnetic rotor and study the distribution of magnetic inductions and current density along the surface and along the depth of the rotor at various rotation speeds. Rotor radius 0.024 m, rotor thickness 0.002 m, total clearance 0.003 m, electrical conductivity of the rotor material γ = 6 106 S/m. The current in the excitation winding is 5 A, the number of turns w = 100. Preparation and tuning of the model. On fig. 2.48 shows a structural diagram of the brake (for clarity, one electromagnet out of four is shown). An attempt to build a model close to the given structural diagram leads to the need to build a 3D model and very high computer requirements, which in most cases is unattainable. To simplify the model, you can turn the rotor on a plane, as is done when obtaining analytical dependencies of the moment on the design parameters . We use this approach to build a simplified brake model. To do this, imagine a 2D brake model as an infinite strip moving between the poles of an electromagnet. For greater clarity and Fig. 2.48. Structural scheme to simplify research can electromagnetic brake with take part of the rotor, equal to half of the hollow non-magnetic rotor of pole division and one pole. a - structural diagram; b - computer Using the equality of the boundary conditions from above and the model from below, as well as the right and left of the model (with a change of sign), as it were, they closed the rotor and the magnetic circuit into a ring. By placing a concentrated excitation winding on the magnetic circuit and setting a certain current density in it, we obtain a given value of magnetic induction in the working gap (for example, 0.4 T and 1.2 T) with a stationary rotor. For simulation of the rotor rotation, we set the linear speed of the rotor as a function of the angular velocity or the number of revolutions per minute: 2 nr v  r  . 60 Let's perform the necessary operations to obtain an electromagnetic brake model using Comsol Multiphysics. Let's go to the Model Navigator. For our model, we need to select a two-dimensional coordinate space, for which we make sure that the Space dimension pop-up list is set to 2D mode. After we select the section of the program AC / DC Module, R6 R5 responsible for modeling electricity. Next, select the Statics, Magnetic mode, then Perpendicular Induction Currents, Vector Potential, i.e. The steps are the same as in the first example. We press OK. In drawing mode, go to Options>Axes/Grid Settings and select the Grid tab. Let's uncheck Auto and in the lines x spacing and y spacing we will write the value 5e4. Next, create a rectangle centered at R8 R7 (0;0) using the button and move the mouse to Fig. 2.49. Drawing a point (0.019; 0.03), which will be the coordinate of the simplifications of the model of that corner of the rectangle. Now let's create a brake rectangle with center (0;0) and corner (0.0065; 0.03), a rectangle from center (0; 0) to corner (0.019; 0.0015), and the last rectangle with center (0; 0 ) to the angle (0.019; 0.001). Next, create rectangles using the Draw the first rectangle through the points (-0.0065; 0.03) and (-0.0135; 0.023), the second through the points (0.0065; 0.03) and (0.0135; 0.023) , the third through the points (-0.0065; -0.03) and (-0.0135; -0.023) and the fourth through the points (0.0065; -0.03) and (0.0135; -0.023). Now let's draw straight lines using the button. The first from the point (0; -0.0015) to the point (0; 0.0015), the second from the point (-0.0125; -0.0015) to the point (-0.0125; 0.0015), the third from point (-0.019; 0) to point (0.019; 0). The result should be a picture similar to the picture in Fig. 2.49. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 72 Let's move on to setting constants and variables. To do this, go to the Options>Constants menu and set the expressions in the fields according to the table. 5 Table 5 Name Imax S Expression 5 [A] 4.9*10^-5 Wob 100 L p R 0.06 [m] 4 0.024 Description Conductor current Winding area Number of conductors in winding Rotor length Number of pole pairs Rotor radius After all constants recorded, click OK. Now go to the Options>Expressions>Global Expressions menu. In this menu, enter the formula for the current density according to Table. 6 Table 6 Name J V Expression (Imax*Wob)/S 2*pi*n/60*R Description Winding current density Rotor speed in rad/s Press OK. The next step is to set the physical properties for the regions. To do this, open the Physics>Subdomain Settings menu and get a picture consisting of 30 subdomains. Now you need to set the physical properties offered in this menu for these areas. Let's start with areas 13 and 18, which are a steel stator (Fig. 2.50, a). Set the constant L (Length) to L, the constant σ (Electric conductivity) to 0.001, the constant μr (Relative Permeability) to 1000000, and leave the rest of the constants as they are. For sub-areas 3, 4, 10, 11, 15, 16, 20, 21, 26 and 27, which are the rotor (highlighted in Fig. 2.50,b), set the following parameters: v (Velocity) - in the first field, enter the variable V , and the second is left with 0, the constant L (Length) value L and the constant σ (Electric conductivity) value 6106. For subregions 1, 2, 5, 6, 9, 12, 14, 17, 19, 22, 24, 25, 28 and 29, which are air (highlighted in Fig. 2.50, c), we set the following parameters: σ (Electric conductivity) value of 0.001, and leave the rest of the parameters as they are. Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Kniga-Service 73 For subdomains 7 and 8 (Fig. 2.50, d), we set the following parameters: σ (Electric conductivity) –107 and J ze (External Current Density) +J. For subdomains 23 and 30 (Fig. 2.50, e), we set the following parameters: σ (Electric conductivity) - 107 and J ze (External Current Density) -J. This completes the subregion setup. You can click OK. 13 18 a) b) c) 7 23 8 30 d) e) Fig. 2.50. Setting the properties of various areas: a - stator magnetic circuit; b – hollow rotor (highlighted); c – air (highlighted); d - left side; e – the right part of the excitation winding Let's go to the Physics> Boundary Settings window (Fig. 2.51) and set the boundary conditions for the model. For the boundaries on the left and right sides of the model, marked with a thick line in Fig. 2.51, a, set the value of Periodic Condition to Boundary Condition. In Type of Periodity Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 74 select Antiperiodity. In the Periodic Pair Index, let's set the numbers in order. First, we choose the boundaries 1 and 74, set everything for them and denote them by the number 1. a) b) Fig. 2.51. Setting the properties of various areas: a - stator magnetic circuit; b – hollow rotor (highlighted); Similarly, for the boundaries in pairs 3 and 75, 5 and 76, 7 and 77, 9 and 78, 11 and 79, set the Periodic Condition value to Boundary Condition and select Antiperiodity in the Type of Periodity. Set the Periodic Pair Index to 2, 3, 4, 5, 6, respectively. For the top and bottom borders of the model (highlighted in Fig. 2.51b), set the value of Periodic Condition to Boundary Condition. In Type of Periodity select Continuity. Let's set the numbering in the Periodic Pair Index. First, select the boundaries 2 and 13, set everything for them and denote them by the number 7. Similarly, for the boundaries in pairs 15 and 19, 30 and 43, 54 and 69, 71 and 73, set the Periodic Condition value to Boundary Condition and select Continuity in the Type of Periodity. Set the Periodic Pair Index to 8, 9, 10, 11, respectively. Let's check that the remaining boundaries (Fig. 2.51) have the Continuity value selected in the Boundary Condition. This completes the border setting. You can click OK. Now let's set up the mesh of the model. To do this, go to Mesh> Free Mesh Parameters or press F9. A window similar to the one in Fig. 2.52. Let's set Predefined Mesh sizes to Ex-Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 75 tremely Fine. Then press Remesh and wait for the mesh to be created. Let's set up the resolver. To do this, go to the Solve>Solver Parameters menu or press the F11 key. Install the parametric solver. Set Parametric in the Solver list, and in Linear System Solver - Direct (UMFPACK). In Parameter Names we will introduce the variable n, and in Parameter Values ​​- range (0.2000, 12000), i.e. the parameter n will change from 0 to 12000 in steps of 2000. Before turning on the resolver, go to Multiprocessing>Probe Plot Parameters (Fig. 2.52). Let us set the equations for deriving the dependence of the torque on the rotation velocity M  r  B y H x LRdS ; M  r  JB y LRdV , where M, N m is the electromagnetic moment, By, Tl is the normal component of the magnetic induction, Hx, A/m is the tangential component of the magnetic field strength, J , A/m2 is the current density, L– the length of the rotor along the Z axis, r is the radius of the rotor. Rice. 2.52. Setting the brake torque equation Press the New button. In the Plot type pop-up window, select Integration. Leave Domain Type - Subdomain. In Plot Name we will write the name of our chart, for example, "Moment". Let's select the section of the rotor, similarly to Fig. 2.53, a. In the Expression field, we write the integrand formula Jz_emqa*By_emqa*L*R. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 76 For verification, we will create two more functions for determining the integral. Similarly, press the New button. In the Plot type pop-up window, select Integration. Let's select in Domain Type - Boundary. In Plot Name, write the name of the chart "Moment 2". Let's press OK. We select the upper surface of the rotor (Fig. 2 53, b). In the Expression field, enter - By_emqa*Hx_emqa*L*R. Similarly, we create another odb) well, a function. Let's select in Domain Type - Boundary. In Plot Name, write the name of the chart "Moment 2". Let's select the lower surface of the rotor (Fig. 2.53, c). In the field c) Expression we write down the formula Fig. 2.53. Isolation: a - volume; By_emqa*Hx_emqa*L*R. b, c - integration surfaces If everything is so, then you can press OK and proceed to the solution. To do this, press the button on the toolbar M(n) and wait until this task is solved. Conclusion and analysis of calculation results. Based on the results of solution 2, the program will generate 0.2 three graphs M(n) and, if two graphs, which are determined by 0 1000 2000 n, rpm 0, are along the surfaces of the rotor, Fig. 2.54. The output of the graphs M (n) when added, it will be seen that in the integration: they will give the sum of the third gra1- by volume; 2 - along the upper boundary; fic. On fig. 2.54 graphs 3 - along the lower boundary M(n) are combined on one coordinate field, i.e. moment formulas gave equal results. When you start a previously recorded program, graphics are not automatically displayed, but when you restart the program, they are displayed. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 77 Next, we will configure the display parameters of the problem solution. To do this, go to Postprocessing>Plot Parameters. Select the Surface tab and in the Predefind Quantites list, select Total current density z component (display current density distribution) or Magnetic flux y component (distribution of the y component of magnetic induction). Then let's move on to the Contour tab. In Predefined Quantites, select the Magnetic Potential z component. In the Levels line below, we will write 40 (that is, we will set the number of lines of equal vector magnetic potential). In the Contour Color line, select Uniform Color and set the color, for example, blue (the color of the lines of the vector magnetic potential). Do not forget to check the box in the upper left corner opposite Contour Plot. Now click OK. The graph (Fig. 2.55) shows the distribution of the current density in the rotor and the magnetic field in the magnetic circuit and in the air. Lines of equal vector magnetic potential formed tubes of equal magnetic fields. 2.55. Pattern of distribution of magnetic fluxes. This makes it possible to see in the brake that the magnetic field is unevenly distributed under the pole, that part of the magnetic flux is closed outside the poles. The magnetic flux is carried away by the rotating rotor, while the induction increases under the edge of the pole. Consider the distribution of magnetic induction and current density over the thickness of the rotor. Let's go to Multiprocessing > Domain Plot Parameters. Go to the Line/Extursion tab. The drawing area will switch to line mode. Now we select the line under the center of the pole (Fig. 2.56, a). It is divided into four arrows. To select it, hold down Ctrl and click on all the arrows. This will make them stand out. Now in Predefined Quantities we will write Bu_emqa. This variable shows the Y-component of the modulo induction, which in this model will be the normal component of the induction. Copyright JSC "TsKB "BIBCOM" & LLC "Agency Book-Service" 78 You can click OK. A graph similar to the one in Fig. 2.56b. Note that under the middle of the pole, the normal component of the magnetic induction is practically unchanged in value at a given rotation speed. With an increase in the rotation speed, it decreases, remaining the same throughout the entire thickness of the rotor. a) B, T 0.4 1 2 3 3 4 5 0.3 2 6 7 y, mm 0 7 5 4 3 1 6 0.2 0.15 J, A/m2 107 2 1 2 2 1 y, mm 0 c) b) Fig. 2.56. Selection of a line under the center of the pole to determine the change in magnetic induction and current density across the thickness of the rotor (a); distribution of magnetic induction (b) and current density (c) at different rotor speeds: 1 – n = 0 rpm; 2 - n = 2000 rpm; 3 - n = 4000 rpm; 4 - n = 6000 rpm; 5 - n = 8000 rpm; 6 - n = 10000 rpm; 7 – n = 12000 rpm 0 1 Let us also consider the current density distribution over the thickness of the rotor on the selected line. To do this, we will write Jz_emqa in Predefined Quantities. The graph in fig. 2.56, c. The current density, as well as the normal component of the magnetic induction, remains the same throughout the entire thickness of the rotor at a given rotation speed, but increases with an increase in the rotation speed, remaining unchanged throughout the thickness of the rotor. Let us study the distribution of the normal component of the magnetic induction of the current density at other points of the rotor. Let's select a line on the right edge of the pole (Fig. 2.57, a) and consider for it the distribution of magnetic induction (Fig. 2.57, b) and current density (Fig. 2.57, c). Copyright OJSC Central Design Bureau BIBCOM & OOO Agency Kniga-Service 79 Note that under the edge of the pole there is a completely different nature of the distribution of these quantities. They vary along the thickness of the rotor, increase significantly with increasing rotation speed. Compared to the previous graph, the current density has almost doubled. The magnetic induction over the thickness of the rotor did not increase very significantly, but in the air gap it increased almost 2 times near the surfaces of the pole. a) J, A/m2 107 3 Bn, T 7 0.6 5.6.7 0.5 2 4 0.4 2 0.3 5 4 3 6 3 1 1 0 2 1 2 y, mm 1 0 c ) b) Fig. 2.57. Selection of a line on the right edge of the pole to determine the change in magnetic induction and current density across the thickness of the rotor (a); distribution of magnetic induction (b) and current density (c) at different rotor speeds: 1 – n = 0 rpm; 2 - n = 2000 rpm; 3 - n = 4000 rpm; 4 - n = 6000 rpm; 5 - n = 8000 rpm; 6 - n = 10000 rpm; 7 - n \u003d 12000 rpm 0 1 2 y, mm Similarly, we repeat the manipulations for the selected line on the left edge of the pole (Fig. 2.58, a) and outside the pole at a distance equal to half the width of the pole (Fig. 2.58, d), and consider for them the distribution of magnetic induction (Fig. 2.58, b, e) and current density (Fig. 2.58, c, e). In the first case, the distribution of the normal component of magnetic induction over the thickness of the rotor is uneven, its value is much less than under the center of the pole, and decreases with increasing rotor speed. The current density with an increase in the speed of rotation of the rotor first increases, and then begins to decrease. In the second case, outside the pole, the picture changed again. The normal component of the magnetic induction has become an order of magnitude smaller, almost does not change over the thickness of the rotor, decreases with increasing rotation speed and changes sign at high rotation speeds. The current density first increases with an increase in the rotor rotation speed, and then begins to decrease and changes sign at high rotation speeds. a) Bn, T 0.4 2 0.3 1 4 7 0.8 4 5 5 3 6 1.2 3 0.2 J, A/m2 107 6 2 0.4 0.1 0 2 7 y, mm 1 1 0 0 1 b) Bn, T 1 0.04 5 4 6 7 1 2 J, A/m2 106 1 0 2 3 0.02 –0.02 0 y, mm c) d) 0 2 y, mm 3 2 1 5 –1 6 –2 7 0 1 4 2 y, mm f) e) 2.58. Selection of a line on the left edge of the pole (a) and outside the pole (d) to determine the change in magnetic induction and current density across the thickness of the rotor; distribution of magnetic induction (b, e) and current density (c, f) at different rotor speeds: 1 – n = 0 rpm; 2 - n = 2000 rpm; 3 - n = 4000 rpm; 4 - n = 6000 rpm; 5 - n = 8000 rpm; 6 - n = 10000 rpm; 7 – n = 12000 rpm Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Kniga-Service 81 Let us further consider the distributions of magnetic induction and current density along the rotor. To do this, we select a line along the surface of the rotor (Fig. 2.59) at the level of the middle of the thickness of the rotor. With an increase in the speed of rotation of the rotor, the normal component of the magnetic induction under the left edge of the pole decreases, and under the right edge it only slightly increases. At some distance to the left of the pole, it changes sign. The current density of the rotor with an increase in the speed of rotation increases significantly under the right edge of the rotor, and under the left edge of the rotor it increases slightly. At the highest speeds of rotation, the current density of the rotor under the right edge of the pole is 4 times greater than under the left. At some distance to the left of the edge of the pole, the rotor current density changes sign. a) By, T 0.4 3 4 5 0.3 0.2 1 2 4 5 4 3 2 0 0 7 6 6 6 7 0.1 J, A/m2 107 1 2 0.02 0.03 x, m 0.01 0.02 0.03 x, m 0 c) b) Fig. 2.59. Selection of a line along the rotor (a); distribution of magnetic induction (b) and current density (c) at different rotor speeds: 1 – n = 0 rpm; 2 - n = 2000 rpm; 3 - n = 4000 rpm; 4 - n = 6000 rpm; 5 - n = 8000 rpm; 6 - n = 10000 rpm; 7 – n = 12000 rpm 0 0.01 Analyzing the obtained graphs of the distribution of magnetic induction and current density, the following features can be noted. 1. Magnetic induction and current density in the rotor under the center of the pole do not change along the thickness of the rotor at a given rotation speed. With an increase in the rotor speed, the magnetic induction decreases from 0.42 to 0.2 T, and the rotor current density increases from 0 to 3.5 107 A /m2. 2. Under the edges of the pole, the magnetic induction and the current density in the rotor differ significantly in value. With an increase in the rotation speed, this difference increases, while the distribution of these values ​​over the thickness of the rotor becomes uneven. 3. Outside the pole piece at a distance equal to half the pole, the magnetic induction has significantly decreased and, with an increase in the rotation speed, changes from 0.05 to -0.02 T with a change of sign. The current density of the rotor also varies from 1.3·106 A/m2 to -2.4·106 A/m2 Questions for self-test 1. Do the graphs of the distribution of the magnetic induction of the electromagnet in the middle of the gap and on the surface of the poles differ? 2. How does the distribution of the normal and tangential components of the magnetic induction change over the thickness of the massive rotor at different rotation speeds? 3. If you draw a line along the radius of a massive rotor, does the current density always retain its sign on it, if not, then when and why? 4. Does the current density on the inner surface of the hollow rotor change at different rotation speeds? 5. According to what law are the magnetic induction and current density distributed under the center of the pole of a salient-pole brake across the thickness of the rotor at various rotor speeds? Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 83 3. MODELING OF ELECTROMECHANICAL DEVICES IN 3D MODE When modeling electromechanical devices in 3D mode, high demands are placed on the computer. This, in turn, significantly limits the number of devices for which such simulation is possible. Below we consider the main methods of modeling in 3D mode using the examples of an electromagnet and a damper with a disk rotor. 3.1. 3D model of an electromagnet Task. Obtain a 3D model of the electromagnet using the 2D model obtained earlier (Section 2.1). Determine the law of change of magnetic induction in the middle of the working gap and on the surface of the electromagnet pole. Model building. One easy way to define a 3D model is to stretch the 2D model. To create a three-dimensional version of the electromagnet, let's return to the finished model from paragraph 2.1. Having opened the model, we will switch to the drawing mode with the button and delete the areas of the coils (Fig. 3.1, a) by selecting them and pressing the Delete keys. Using the electromagnet prepared earlier as a blank, draw its upper half along the old lines. To do this, select the line drawing on the toolbar and draw a half of the electromagnet (Fig. 3.1, b, highlighted in bold). Since the drawing grid is too coarse, we will draw a figure a little more than half of the electromagnet, and then, using a double mouse click, go to the properties of the figure and select line 7. In the y coordinate for each point, write the value -0.0625 (Fig. 3.1, in ). Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 84 7 b) a) c) Fig. 3.1. Preparation of a 3D model of an electromagnet according to the existing 2D model: a - removal of the excitation winding; b - drawing of the upper half of the model; c – change of line coordinates 7 Press OK. Select the lower half of the electromagnet with the left mouse button and press the Delete button. The result is half an electromagnet. Select it with the left mouse button. Let's use the Mirror button. In the menu that appears, replace the value from the Normal Vector field in the x coordinate 1 with 0, and in the y coordinate, replace 0 with 1 (Fig. 3.2). Rice. 3.2. The Mirror program window Since the Mirror tool reflects shapes relative to the line of the coordinate axis, we will get a reflected shape superimposed on the original fig. 3.3, a. Due to the discrepancy between the upper part of the electromagnet and the X axis, the reflected figure is partially superimposed on the upper half of the electromagnet and it will need to be shifted down. To do this, we select the lower half of the electromagnet. To obtain the correct arrangement of the halves of the electromagnet, move the selected figure down with the left mouse button pressed. As a result, we get Fig. 3.3b. b) a) Fig. 3.3. Obtaining a flat model of an electromagnet: a - imposition of a reflected figure on an existing one; b – figure of the model after displacement of the lower half Let's select both halves of the electromagnet. To do this, hold down the Ctrl key and press alternately on the upper and lower halves of the electromagnet. Next, go to the Draw>Extrude menu (Fig. 3.4). Rice. 3.4. Draw > Extrude command window Make sure CO1 and CO2 are selected. In the Distance field, write the value 0.05. This means that the electromagnet will be stretched by 0.05 m along the z-axis. Press OK and get a three-dimensional model, similar to Fig. 3.5. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 86 Pic. 3.5. Electromagnet 3D model Now let's use the menu File>Export>Geometry Objects to File. In the window that appears, click OK. And then we save our geometry in a separate file in any folder (Fig. 3.6) under the name electromagnit. Comsol will save the geometry in a special mphbin format. This will be necessary in order to later import this geometry into a new 3D model. Rice. 3.6. Saving the 3D model in a separate folder Copyright OJSC "Central Design Bureau "BIBCOM" & LLC "Agency Kniga-Service" 87 Now start Comsol and create a new model in the model navigator (Fig. 3.7). In the Space Dimension list, select the 3D mode. Click the cross next to the AC/DC Module folder. Next, open the Statics, Magnetic folder and select Magnetostatics, Vector Potential. Click OK Fig. 3.7. Launching a 3-D model for modeling Import geometry using the File>Import>CAD Data from File menu. Select the previously saved electromagnit.mphbin file and click Open. Given the peculiarities of the location of the electromagnet in the previous task, we will try to move it symmetrically relative to the center. To do this, use the Move button on the drawing panel and set the offset coordinates (0.025; 0.0625; -0.025). Now the magnet is symmetrical about the center. Let's create an outer sphere that defines the boundary conditions. To do this, use the button on the drawing panel. In the menu that opens (Fig. 3.8), set the Radius value to 1, and leave the rest of the parameters by default and click OK. Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Book-Service" 88 Pic. 3.8. Creating an Outer Sphere that Insulates The geometry is created. Let's move on to setting constants. To do this, go to the Options>Constants menu. In the menu that appears, fill in the data according to the table: Name Ip mu Expression 0.5 * 10^5 100 Description Winding current Relative magnetic permeability of the electromagnet Go to Options> Expressions> Subdomain Expressions and select subdomains 2 and 3 (Fig. 3.9) and write them in Name the core variable, and in Expression - the value 1. Let's move on to setting the physical parameters. To do this, open the Physics>Subdomain Settings menu. There are three subregions of space for which it will be necessary to set their own physical parameters. For subdomain 1, which is the outer sphere, we leave all the settings as standard. For subdomains 2 and 3 (Fig. 3.9), we leave all the parameters as they are, and set the value mu in the parameter μr. Let's move on to setting the boundary values. 3.9. Specifying condition areas. Let's go to the Phys-magnetic core menu ics>Boundary Settings and go to the Groups tab. Verify that Comsol automatically splits the model into two groups. For the first group, which is the outer sphere, make sure that the Magnetic Insulation value is set. For the second group, which is the surface of the electromagnet, the Continuity condition must be set. Let's set the current in the coil. Open the Physics>Edge Settings menu. Let's select the edges numbered 44 and 48 (Fig. 3.10, a) and set the Value/Expression to Ip. Similarly, we select edges 46 and 53 (Fig. 3.10, b) and set the Value / Expression to the value minus Ip. 46 44 53 48 b) a) Fig. 3.10. Setting the current in the excitation winding (coil): a - faces 44 and 48; b – faces 46 and 53 To create a mesh and save estimated time, you can compose it in parts with different partitioning parameters. To begin with, we select an electromagnet (Fig. 3.11). b) a) Fig. 3.11. Setting the grid: a – program window; b – area of ​​the magnetic core Let's go to the Subdomain tab and select the upper and lower subdomains of the electromagnet 2 and 3 (Fig. 3.11, b). Let's write the value 0.02 in Maximum Element size. Press the Remesh button. Then we select subdomain 1 and in Maximum Element size we write the value 0.2. Press the Remesh button again. Let's move on to the solver in the Solve>Solver Parameters menu (Fig. 3.12). Make sure the mode is set to Static and the Solver analysis mode is set to Stationary. The Linear System Solver must be set to FMGRES mode and the Preconditioner must be set to Geometric Multigrid. After making sure of this, you can click OK. Rice. 3.12. Solve Solve window Now let's launch the solution using the button on the control panel. After the solution, a rather uninformative Slice graph will appear, showing the distribution of induction in some sections. Since we have an outer sphere, the choice of other graphical representations will be inconvenient. Therefore, it is necessary to get rid of the mapping of the outer sphere. To do this, go to the Options>Supress>Supress Edges menu (Fig. 3.13). Select lines 1-4 and 33-40 and press OK. Now let's go to the menu Options>Supress>Supress Copyright JSC "Central Design Bureau" BIBCOM" & LLC "Agency Book-Service" 91 Boundaries (Fig. 3.14). Select surfaces 1–4 and 19–22 corresponding to the sphere and press OK. Now the sphere will not interfere when viewing the results. Rice. 3.13. Options>Supress>Supress Edges menu window 3.14. Menu window Options>Supress>Supress Boundaries Let's go to the menu Postprocessing>Cross-Section Plot Parameters (Fig. 3.15). Let's go to the tab Extrusion / LineExtrusion and Preference FluxxMagnitude Density norms. In the Crosssection line data section, write the value –0.3 in x0. This straight line is shown in Fig. 3.16, a. It is directed in the longitudinal direction from the excitation winding to the working gap. Then press Apply and get the distribution of magnetic induction along this straight line (Fig. 3.16, b). Analyzing the graph, it can be noted that the magnetic induction distribution curve is not symmetrical. At the right Fig. 3.15. The Postprocessing> menu window of the edge of the pole facing the Cross-Section Plot Parameters inside the electromagnet, the magnetic induction decays more slowly than at the left edge. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 92 Bn, T 0.3 0.2 0.1 0 0 0.1 0.2 x, m b) a) Fig. 3.16. Obtaining a graph of the distribution of magnetic induction in the middle of the gap under the center of the pole in the direction of the x axis: a - setting the line; b – graph of the distribution of magnetic induction Now let's leave x0 as it is, and in y0 and y1 we will introduce the values ​​-0.015. The straight line passes in Fig. 3.17 a. Click Apply. We get the distribution of magnetic induction at the pole (Fig. 3.17, b). The graph of the distribution of magnetic induction near the surface of the pole differs significantly from the graph (Fig. 3.16, b) obtained in the middle of the air gap. On the corner faces of the electromagnet, a significant increase in magnetic induction is obtained. Bn, T 0.6 0.4 0.2 0 0 0.1 0.2 x, m a) b) Fig. 3.17. Obtaining a graph of the distribution of magnetic induction in the middle of the gap on the surface of the pole in the direction of the x axis: a - setting the line; b – graph of distribution of magnetic induction Let's return zero values ​​in y0 and y1. Let us write the values ​​–0.15 in x0 and x1. We write –0.15 in z0, and 0.15 in z1. Let's get a straight line, presented in fig. 3.18, a. This line is perpendicular to the line drawn in Fig. 3.16, a. The distribution of induction along this straight line is shown in fig. 3.18b. We can note the symmetry of the graph of the distribution of magnetic induction in this direction. Bn, T 0.3 0.2 0.1 0 0 0.1 0.2 x, m a) b) Fig. 3.18. . Obtaining a graph of the distribution of magnetic induction in the middle of the gap under the center of the pole in the direction of the z axis: a - setting the line; b - graph of the distribution of magnetic induction Now we will write in y0 and y1 the values ​​​​-0.015. We get the straight line shown in Fig. 3.19, a. The distribution of magnetic induction is given in fig. 3.19b. This graph, which characterizes the distribution of magnetic induction on the surface of the pole in the transverse direction, shows a significant increase in magnetic induction at the edges of the pole, similar to Fig. 3.17b. Bn, T 0.6 0.4 0.2 0 a) 0 0.1 b) 0.2 x, m 3.19. Obtaining a graph of the distribution of magnetic induction in the middle of the gap on the surface of the pole in the direction of the z axis: a - setting the line; b – graph of distribution of magnetic induction then gradually decreases towards its edges. Outside the pole, the magnetic induction decreases sharply. A completely different distribution of magnetic induction on the surface of the pole (Fig. 3.17 and 3.19). At the edges of the poles in the direction of the x and z axes, the magnetic induction is significantly (almost 2 times) increased. 3.2. 3D model of a damper with a disk rotor Task. Obtain a 3D model of a damper with a disc rotor. The rotor is a copper disk with a thickness of 1 cm and a radius of 10 cm, which rotates with an initial angular velocity of 1000 rpm in a magnetic field (B=1T) created by a permanent magnet. The working gap is 1.5 cm. Determine the law of change in the braking torque and the speed of rotation of the rotor in time. Model building. Figure 3.20 shows the structural diagram of the damper. The damper consists of a disk made of conductive material and a permanent magnet. The magnet creates a constant magnetic field in which the disk rotates. When a conductor moves in a magnetic field, a current is induced in it and the Lorentz force slows down. 3.20. Constructive em rotation of the disk. damper circuit For a disk rotating with an angular velocity ω perpendicular to the Z-axis, the velocity V at the point (x, y) has the form v  ( y, x, 0) . Maxwell's equation is written using the vector magnetic potential A and the scalar electric potential U: 0 n  A  0; n J  0. Consider now how the system changes over time. The induced moment slows down the rotation of the disk and is described by an ordinary differential equation (ODE) for the angular velocity ω d Tz  , dt I where the moment Tz is described as the Z-component of the vector. T  r  J  B dV . disk The moment of inertia I for a disk with radius R of unit thickness is r 2 r 4 . I m  2 2 Here m is the mass of the disk, and  is the density of the disk. Modeling. To build the model, launch Comsol Multiphysics and select 3D mode from the Space Dimension list. Click the cross next to the AC/DC Module folder. Next, open the following folders in sequence: Statics, Magnetic>Magnetostatics, Vector Potential>Reduced Potential>Ungauged potentials. This mode allows you to well simulate permanent magnets by setting the initial magnetization. Now you can click OK and wait for the simulation window to start. Let's create a cylinder by clicking on the button on the drawing panel. In the window that appears (Fig.3.21) select the following settings for the cylinder: Radius 0.1, Height 0.01 and Axis base point z: 0.005. Leave all other parameters as default and click OK. Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Book-Service" 96 Pic. 3.21. Creating a Cylinder Let's create a sphere (Figure 3.22) using the pop button. on the panel ri-Fig. 3.22. Creating a sphere In the settings window (Fig. 3.23) set the Radius to 0.3, and leave the rest of the parameters unchanged and click OK. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 97 Pic. 3.23. Sphere Settings Window Let's go to the Draw>Work-Plane Settings menu to make it easier to draw a magnet in a plane. In the dialog box (Fig. 3.24) select the value y-z in Plane and leave x = 0. Click OK. The Geom2 plane will appear, in which we can easily build a magnet, just like in 2D models. Rice. 3.24. Draw>Work-Plane Settings Window Let's go to Draw> Specify Objects>Rectangle to create a rectangle. Its settings are Width 0.02, Height -0.0075+0.06, Base Corner, x 0.06, y -0.06 (Figure 3.25). Let's repeat Draw> Specify Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 98 Objects>Rectangle to create the second rectangle. settings Width 0.06, Height 0.02, Base Corner, x 0.08, y -0.06. His Fig. 3.25. Creating a Rectangle From the Draw menu, choose Create Composite Object. In the dialog box (Fig. 3.26) uncheck Keep interior boundaries and select both rectangles R1 and R2. Then click OK. This will create one object from these rectangles. Rice. 3.26. Create Composite Object window On the toolbar, select the button to reflect our shape. In the window that appears (Fig. 3.27), set the following parameters: Point on line x 0 y 0, Normal Vector x 0 y 1. Using Draw> Specify Objects>Rectangle, create another rectangle with the following characteristics: Width 0.02 Height 0, 08 Base Corner x 0.12 y -0.04. Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 99 Pic. 3.27. Mirror window 3.28. Selecting three objects Let's select all three objects (Fig. 3.28). Let's go to the Draw>Exturude menu. The dialog box (Fig. 3.29) will allow you to get this three-dimensional figure. For Distance, select 0.02 and click OK. The resulting figure must be shifted from the Ox axis using the button on the toolbar. Set x to -0.01 and click OK. Rice. 3.29. Obtaining a three-dimensional figure The creation of geometry is completed. You can go to the settings for constants, variables and scopes. To do this, go to Options > Constants and set the constants there according to the table. 3.1. Table 3.1 Name Expression description rpm 1000 Initial disk rotation speed, rpm W0 2*pi*rpm Initial angular velocity, rad/s I0 0 External moment of inertia menu Options>Expressions>Scalar Expressions and write the variables according to the table. 3.2. Table 3.2 Name Fx Fy Fz Expression Jy_emqav*Bz_emqavJz_emqav*By_emqav Jz_emqav*Bx_emqavJx_emqav*Bz_emqav Jx_emqav*By_emqavJy_emqav*Bx_emqav In the Options>Expressions>Subdomain Expressions menu, select the subdomains from the third to the fifth, and write the variable for them in the core (Fig. 3.30) and in Expression the value is 1. 3.30. View of a three-dimensional figure Next, go to the Draw > Integration Coupling Variables > Subdomain Variables menu, in which we write the data for subdomain 2 according to the table (Fig. 3.31). Name Iz Tqz Expression 8700*(x^2+y^2) x*Fy-y*Fx Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Book-Service 101 Pic. 3.31. Subdomain Variables window Let's move on to setting the physical properties of subdomains. Let's call the menu Physics > Subdomain Settings. Let's set the properties using the table. 3.3. Settings Table 3.3 Subdomain Subdomain 2 Subdomains 3,4 Subdomain 5 1 (Air) (Disk) (Magnetic Core) (Permanent Magnet) 0 -y*W 0 0 0 x*W 0 0 0 0 0 0 Velocity x Velocity y Velocity z Electric conductivity 1 5.998e7 Constitutive relation B = μ0μrH B = μ0μrH Rel. permeability 1 1 Rem. flux density x – – Rem. flux density y – – Rem. flux density z – – 1 1 B = μ0μrH B = μ0μrH + Br 4000 1 – 0 – 0 – 1 Now let's move on to setting the boundary conditions by calling the Physics>Boundary Settings menu (Fig. 3.32). Let's go to the tab Groups and Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 102 select group one, which is the outer sphere. For it, select the Electric Insulation value in the Boundary Condition. a) b) Fig. 3.32. Boundary conditions setting: a – menu; b - view of the outer sphere Let's set up a function to determine the rotation speed in time. To do this, open Physics>Global Equations. In the window that appears, fill in the data according to the table, and also make sure that SI Name Equation Init(u) W WtW0 Tqz/(Iz+I0) Init(ut) description is selected in the Base Unit System. to the subdomain display mode with the button on the taskbar. Let's choose a copper disk. To select it in 3D mode, it is necessary to click on this subdomain with the left mouse button similarly to the 2D mode, but the program will prompt you to select the closest subdomain for the observer. Then you need to press the left button again and the program will select the next area. For this task, you need to make two clicks to select the disk (Fig. 3.33). Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 103 Having selected the disk, go to the Mesh>Free Mesh Parameters menu (Fig. 3.34). In Predefined mesh sizes, select Extremely fine. Next, go to the Advanced tab. In zdirection scale factor we will enter the value 1.1. Then press the Mesh Selected button to create a mesh for the disk. Then go back to the Global tab and set Predefined mesh sizes to Coarser. Let's press OK. Now select the button to switch to the grid display mode. Then Fig. 3.33. Selecting a disk area, press the button - Mesh Remaining (Free). The mesh has been created. Rice. 3.34. Defining a grid for a disk Copyright JSC Central Design Bureau BIBCOM & OOO Agency Kniga-Service 104 Let's set up the resolver. To do this, go to the Solve > Solver Parameters menu (Fig. 3.35). Let's choose the Time dependent mode. Set Times to range (0.25), Relative Tolerance to 0.001, Absolute Tolerance to W 0.1 V 1e-5 tA* 1e-7, those absolute errors for different variables are set to different values. Let's move on to the Time Stepping tab. Here we select the Intermediate value in Time steps taken by solver and check the box next to Manual tuning of nonlinear solver. Click on the Nonlinear Settings button and write 0.2 in the Tolerance factor, as well as 7 in the Maximum number of iterations. Uncheck the box next to Use limit on convergence rate and select Once per time step in the Jacobian update list. Click OK. Let's go to the Advanced tab. In it, select Manual in the Type of scaling list, and in Manual scaling, write W 0.01 V 1e-5 tA* 1e-7. Click OK. Rice. 3.35. Configuring the resolver Copyright JSC "Central Design Bureau" BIBCOM " & LLC "Agency Kniga-Service" 105 Let's move on to the output of graphs during the solution. To do this, go to the Postprocessing > Probe Plot Parameters menu. Press the New button and in the menu that appears, select Global in the Plot Type list. Let's write Omega in Plot Name. The value W should appear in Expressions. If it does not appear, then we will write it down. Let's create another chart in the same way. Let's write Torque in Plot Name. In the Expression field, write - Tqz. Let's choose to create another graph. This time choose Integration as Plot Type and Subdomain as Domain Type. Let's write Power in Plot Type. Let's select subdomain 2 and write Q_emqav in Expression. Click OK. Now you can start solving the problem. To do this, press the button. This problem is solved for quite a long time on modern computers due to the complexity of the model, so you have to wait about 10 ... 20 minutes. After the ω, s-1 th decision is made, the program will display three graphs that were set earlier. The first (Fig. 3.36) 60 graph shows the change in rotation speed in rad/s during braking. Note that 20 the speed of disk rotation during 10 s rapidly 5 20 0 10 15 t, s decreases, then 3.36. The change in speed decreases more slowly than the slowness of the rotor during braking, and by 20 s the rotation of the filament field of the disk stops. The second graph (Fig. 3.37, a) shows the change in torque. First, for 5 s, the torque increases rapidly, and then slowly decreases and approaches zero by 20 s. Graph fig. 3.38b describes the change in time of power dissipation in the disk. Over time, the dissipated power decreases rapidly and approaches zero by 13 s. Copyright OAO Central Design Bureau BIBCOM & OOO Agency Kniga-Service 106 Q, W 12 M, Nm 0.12 8 0.08 4 0.04 0 0 0 t, s 0 10 15 20 10 5 15 20 t, c a) b) Fig. 3.37. Change of braking torque (a) and power dissipation (b) in the rotor during braking 5 In fig. 3.38 shows a picture of the distribution of currents in the rotor (a larger value of the arrow corresponds to a higher current density) 3.38. Picture of the current density distribution in the damper rotor 3. 39 (the electromagnet is made invisible - it is indicated by lines). Analyzing this figure, it is possible to establish an uneven distribution of current density under the pole - under one edge of the pole, the current density reaches 5104 A/m2, and under the other - less than 104 A/m2. At the edge of the rotor (above the pole), the current density remains quite high (about 2104 A/m2. J,A/m2 106 Fig. 3.39. Current density distribution on the disk surface at t=1 s. Let's go back to the Postprocessing>Plot Parameters menu "Uncheck the Subdomain and Edge sections. Click OK. This will allow you to better see the straight lines, along which we will look at the distribution of magnetic induction and current density. To do this, go to the Postprocessing> Cross-Section Plot Parameters menu (Fig. 3.40, a). Select time values ​​0, 5, 10, 15, 20 and 25 s by clicking on these values ​​while holding down the Ctrl key. In the Line/Extrusion tab, click on the Line/Extrusion button and in the left corner of the Settings window, check the box next to Legend. we will leave the other values ​​at zero. This straight line is shown in Fig. 3.40 b. Then press Apply and get the distribution of magnetic induction along this straight line (Fig. 3.41, a). b) a) Fig. 3.40. Postprocessing>Cross-Section Plot Parameters menu window (a), line selection for determining the change in magnetic induction (b) B, T 4 3 0.08 4 3 0.06 0.04 0.02 J, A/m2 106 2 1 1 2 2 1 3 4 0 0.02 0.04 0.06 0.08 r, m 0 0.02 0.04 0.06 0.08 r, m 0 b) a) Fig. Fig. 3. 41. Distribution of magnetic induction (a) and current density (b) along the disk radius at different times after switching on: 1– t = 0 s; 2– t = 5 s; 3– t = 10 s; 4– t = 25 s 0 Copyright OJSC Central Design Bureau BIBCOM & LLC Agency Kniga-Service 109 Let's return to the Line/Extrusion tab. In Predefined totals, current quantities, densities norms, and click on OK. Let's get the current density distribution along this straight line (Fig. 3.41, b). In the Cross-section line data, we write the values ​​-0.07 and 0.07 in x0 and x1, respectively, in y0 and y1 - the value 0.07, and leave the remaining fields with zero values. Let's get the straight line of distribution fig. 3.42. Let's go back to Predefined Flux Density magnetics, norms, and on the left we choose OK. We obtain the distribution of the magnetic 3.42. Construction of the line for induction in fig. 3.43 a. determination of changes in magnetic induction and current density B, T 3 0.08 0.6 0.06 0.04 4 J, A/m2 107 0.8 2 1 0.4 2 0.2 1 3 4 0 0 0 0, 02 0.04 0.06 0.08 x, m 0 0.02 0.04 0.06 0.08 x, m Fig. 3. 43. Distribution of magnetic induction (a) and current density under the center of the pole in the direction perpendicular to the radius, at different times after switching on: 1– t = 0 s; 2– t = 5 s; 3– t = 10 s; 4– t = 25 s 0.02 Let's go back to the Line/Extrusion tab. In Predefined, we take the total current quantities of densities norms and press the key. Let us obtain the current density distribution along this straight line in Fig. 3.43b. In the Cross-section line data, we will write the value 0 in x0 and x1, leave the value 0.07 in y0 and y1, and -0.01 and 0.01 in z0 and z1, respectively. Agency Book-Service» 110 We get a line under the center of the pole in the direction of the y axis, on which we consider the distribution of magnetic induction and current density over the thickness of the rotor (Fig. 3.44). Back in Predefined, select Flux Density norms and click OK. We obtain the distribution of magnetic induction along the y-axis (Fig. 3.45, a). Rice. 3. 44. Definition of the line under Analyzing fig. 3.45, a, with the center of the pole in the direction of the y-axis, we note that the magnetic induction in the gap and in the rotor in the direction of the y-axis remains almost unchanged at a given disk rotation speed. With a decrease in the rotation speed after 5, 10. and 25 s. magnetic induction increases from 0.025 to 0.1 T. Let's go back to the Line/Extrusion tab. In Predefined totals, current quantities, densities norms, and click on OK. We get the current density distribution over the thickness of the rotor (Fig. 3.45, b). B, T J, A/m2 106 0.08 3 2 4 0.06 0.04 0.02 2 3 1 2 1 3 1 4 0 0 0.02 0.04 0.06 0.08 y, m 0 .02 0.04 0.06 0.08 y, m b) a) Fig. 3. 45. Distribution of magnetic induction (a) and current density (b) under the center of the pole in the direction of the y axis at different times after switching on: 1– t = 0 s; 2– t = 5 s; 3– t = 10 s; 4– t = 25 s 0 Analyzing the graph of the distribution of current density over the thickness of the disk rotor, we note that in the first time after starting at a high speed of rotation of the rotor, the current density is distributed unevenly over the thickness of the rotor. With a decrease in the rotation speed, the current density tends to a uniform distribution over the thickness of the rotor. Copyright JSC "Central Design Bureau "BIBCOM" & OOO "Agency Kniga-Service" 111 Questions for self-examination 1. How to use the Extrude command to get 3D models from 2D models? 2. How to get a graph of the distribution of any physical quantity along any known straight line? 3. What can be achieved using the Supress menu? 4. How to get a different finite element mesh using the settings in Free Mesh Parameters? Copyright JSC "Central Design Bureau "BIBCOM" & LLC "Agency Book-Service" 112 LIST OF USED AND RECOMMENDED LITERATURE 1. Roger, W. Pryor. Multiphysics modeling using COMSOL: First Principles Approach. Jones and Bartlett Publishers, 2010. 2. Bul, O.B. Methods for calculating the magnetic systems of electrical apparatus. ANSYS program: textbook. allowance for students. higher textbook institutions / O.B. Bul.–M.: Academy, 2006. 3. Egorov, V.I. The use of computers for solving problems of heat conduction: textbook. allowance / V.I. Egorov.–SPb: SPb GU ITMO, 2006. Copyright OJSC Central Design Bureau BIBCOM & OOO Agency Kniga-Service 113 CONTENTS FOREWORD …………………………..……………………. ……..3 INTRODUCTION…………………………..…………………………………..5 1.1. General characteristics ………………………………………………………………6 1.2. Basics of Modeling……………………………………….8 Model Navigator……………………………………………….8 Workspace and Image of the Object of Study…..10 Constants , expressions, functions …………………………………………………………………………………..16 Setting the electromagnetic properties of materials and initial conditions……………………………………. …………………....20 Building a grid …………………………………………. ..22 Solver ………………………………………...24 Visualization of results ………………………………....29 Self-test questions…………… …………………...33 2. SIMULATION OF ELECTROMECHANICAL DEVICES IN 2D MODE …………………………..…….34 2.1. DC solenoid…………………………...34 2.2. Electromagnetic brake with a massive rotor based on the stator of an asynchronous motor……………….46 2.3. Electromagnetic brake with hollow ferromagnetic rotor………………………..62 2.4. Simplified model of a salient-pole brake with a hollow non-magnetic rotor……………………………….69 Self-test questions…………………………………………………………………………… ....81 3. MODELING OF ELECTROMECHANICAL DEVICES IN 3D MODE …………………………..……..82 3.1. 3D model of an electromagnet……………………………………..82 3.2. 3D model of damper with disc rotor………………....93 Questions for self-examination……………………………………..110 LIST OF USED AND RECOMMENDED LITERATURE………………… …………………………………111

M.: NRNU MEPhI, 2012. - 184 p. Description:
Designed to study the mathematical modeling environment Comsol Multiphysics. The manual discusses in detail the key methods of working with this system and understands specific typical tasks. The book also contains a guide to mathematical programming in Comsol Script and features of the interaction of the Comsol Multiphysics package with the Matlab system.
This manual is the first Comsol Multiphysics manual in Russian.
Useful for 3rd and 4th year students studying mathematical modeling. Contents:
Finite element method.
Theoretical introduction.
Types of finite elements. Getting started with FEMLAB.
Installation.
General principles of work.
Application modes.
The process of setting and solving a problem.
Comsol Multiphysics 3.5a environment.
Model navigator.
Working environment of the program.
Setting areas.
Drawing basic geometric objects.
Object transformations.
Logical operations with objects.
Analytical assignment of objects.
Formulation of the problem.
Specifying equation coefficients.
Setting of boundary conditions.
Mesh generation.
triangular grid.
Quadrangular elements.
Choice of basic functions.
The solution of the problem.
Stationary solvers.
Visualization of results.
Construction of the main graph.
Export graph to file.
Construction of graphs on sections and points.
Construction of graphs on the borders and at the key points of the area.
Expressions and functions in FEMLAB.
Introduction.
Setting constants and regular expressions.
Using constants and regular expressions.
Functions.
Axes and grid properties. Practical simulation on FEMLAB.
Solution of non-stationary problems.
Formulation of the problem.
The solution of the problem.
Solution visualization.
Accounting for the initial conditions of the problem.
Solution of differential-algebraic systems of equations.
Solving problems for eigenvalues.
Solving problems with a parameter.
Solution of acoustics equations.
General information.
Mathematical statement of the problem.
Applied mode of acoustics equations.
Border conditions.
An example of a sound propagation problem. Reactive silencer acoustics.
Solving problems of structural mechanics.
Theoretical introduction.
Applied mode of structural mechanics equations.
Fixings.
Loads.
An example of the problem of stress distribution in a trapezoidal membrane.
Solution of the problem of finding ice flow velocities by the FEMLAB system.
Theoretical information.
Statement and solution of the problem.
Implementation of the multiphysics mode.
Solving problems with changing geometry.
Solution of the problem of heating a liquid drop.
Forms of equations.
General information.
Application modes.
The coefficient form of the equation.
General form.
Weak form.
Solution of one-dimensional problems.
Solution of three-dimensional problems.
Specifying 3D geometry.
Defining Equations and Mesh Generation.
Visualization of results.
Transition from two-dimensional geometry to three-dimensional. Communication with matlab. Comsol Script.
Introduction.
Launching joint work with Matlab and Comsol Script.
Getting started with Comsol Script.
Basic information.
Working with memory Comsol Script.
Vectors, matrices and arrays in Comsol Script.
Elements of programming in Comsol Script.
The if branch operator.
Conditional loop.
Cycle with a counter.
Choice operator.
Task modeling in Maltab and Comsol Script.
FEMLAB object model.
Solution of the Poisson equation.
Import and export of the model.
Creation of geometric objects.
Creation of basic geometric objects.
Creation of complex objects.
Object transformations and logical operations.
Interpolation of geometric objects.
Model assignment.
Basic provisions.
Formulation of the problem.
Setting equations.
Mesh generation.
test functions.
Constants and expressions.
Solver choice.
Visualization and data processing.

Electrical cables are characterized by parameters such as impedance and attenuation coefficient. This topic will consider an example of modeling a coaxial cable, for which there is an analytical solution. We'll show you how to calculate cable parameters from electromagnetic field simulations in COMSOL Multiphysics. Having dealt with the principles of building a model of a coaxial cable, in the future we will be able to apply the knowledge gained to calculate the parameters of transmission lines or cables of an arbitrary type.

Electrical cable design issues

Electrical cables, also called transmission lines, are now widely used for the transmission of data and electricity. Even if you are reading this text from a screen on a cell phone or tablet computer using a “wireless” connection, there are still “wired” power lines inside your device connecting various electrical components into a single whole. And when you return home in the evening, you will most likely connect the power cable to the device for charging.

A wide variety of power lines are used, from small, made in the form of coplanar waveguides on printed circuit boards, to very large high-voltage power lines. They must also function in various and often extreme modes and operating conditions, from transatlantic telegraph cables to electrical wiring on spacecraft, the appearance of which is shown in the figure below. Transmission lines must be designed with all the necessary requirements in mind to ensure their reliable operation under given conditions. In addition, they can be the subject of research in order to further optimize the design, including meeting the requirements for mechanical strength and low weight.

Connecting wires in the cargo hold of the OV-095 shuttle mock-up at the Shuttle Avionics Integration Laboratory (SAIL).

When designing and using cables, engineers often work with distributed (or specific, i.e. per unit length) parameters for series resistance (R), series inductance (L), shunt capacitance (C), and shunt conductance (G, sometimes called insulation conductivity). These parameters can be used to calculate the quality of the cable, its characteristic impedance and losses in it during signal propagation. However, it is important to keep in mind that these parameters are found from the solution of Maxwell's equations for the electromagnetic field. To solve Maxwell's equations numerically to calculate electromagnetic fields, as well as to take into account the influence of multiphysics effects, you can use the COMSOL Multiphysics environment, which will allow you to determine how the parameters of the cable and its efficiency change under various operating modes and operating conditions. The developed model can then be converted into an intuitive application like this one, which calculates parameters for standard and commonly used transmission lines.

In this topic, we will consider the case of coaxial cable - a fundamental problem that is usually contained in any standard curriculum on microwave technology or power lines. The coaxial cable is such a fundamental physical entity that Oliver Heaviside patented it in 1880, just a few years after Maxwell formulated his famous equations. For students of the history of science, this is the same Oliver Heaviside, who first formulated Maxwell's equations in the vector form that is now generally accepted; the one who first used the term "impedance"; and who made a significant contribution to the development of the theory of power lines.

Results of analytical solution for coaxial cable

Let's start our consideration with a coaxial cable, which has the characteristic dimensions indicated on the schematic representation of its cross section, presented below. The dielectric core between the inner and outer conductor has a relative permittivity ( \epsilon_r = \epsilon"-j\epsilon"") equal to 2.25 – j*0.01, relative magnetic permeability (\mu_r ) equal to 1 and zero conductivity, while the inner and outer conductors have a conductivity (\sigma ) equal to 5.98e7 S/m (Siemens/meter).

2D cross-section of a coaxial cable with characteristic dimensions: a = 0.405 mm, b = 1.45 mm, and t = 0.1 mm.

The standard solution for power lines is that the structure of electromagnetic fields in the cable is assumed to be known, namely, it is assumed that they will oscillate and attenuate in the direction of wave propagation, while in the transverse direction the field section profile remains unchanged. If then we find a solution that satisfies the original equations, then by virtue of the uniqueness theorem, the solution found will be correct.

In mathematical language, all of the above is equivalent to the fact that the solution of Maxwell's equations is sought in the form ansatz-forms

for an electromagnetic field , where (\gamma = \alpha + j\beta ) is the complex propagation constant, and \alpha and \beta are the damping and propagation coefficients, respectively. In cylindrical coordinates for coaxial cable, this leads to the well-known field solutions

\begin(align)
\mathbf(E)&= \frac(V_0\hat(r))(rln(b/a))e^(-\gamma z)\\
\mathbf(H)&= \frac(I_0\hat(\phi))(2\pi r)e^(-\gamma z)
\end(align)

from which the distributed parameters per unit length are then obtained

\begin(align)
L& = \frac(\mu_0\mu_r)(2\pi)ln\frac(b)(a) + \frac(\mu_0\mu_r\delta)(4\pi)(\frac(1)(a)+ \frac(1)(b))\\
C& = \frac(2\pi\epsilon_0\epsilon")(ln(b/a))\\
R& = \frac(R_s)(2\pi)(\frac(1)(a)+\frac(1)(b))\\
G& = \frac(2\pi\omega\epsilon_0\epsilon"")(ln(b/a))
\end(align)

where R_s = 1/\sigma\delta is the surface resistance, and \delta = \sqrt(2/\mu_0\mu_r\omega\sigma) is an .

It is extremely important to emphasize that the relationships for capacitance and shunt conductance hold for any frequency, while the expressions for resistance and inductance depend on the skin depth and, therefore, are applicable only at frequencies at which the skin depth is much less than the physical thickness. conductor. That is why the second term in the expression for inductance, also called internal inductance, may be unfamiliar to some readers, as it is usually neglected when metal is considered an ideal conductor. This term is the inductance caused by the penetration of a magnetic field into a metal of finite conductivity and is negligible at sufficiently high frequencies. (It can also be represented as L_(Internal) = R/\omega .)

For subsequent comparison with the numerical results, the ratio for the DC resistance can be calculated from the expression for the conductivity and the cross-sectional area of ​​the metal. The analytical expression for inductance (with respect to direct current) is a little more complicated, and therefore we include it here for reference.

L_(DC) = \frac(\mu)(2\pi)\left\(ln\left(\frac(b+t)(a)\right) + \frac(2\left(\frac(b) (a)\right)^2)(1- \left(\frac(b)(a)\right)^2)ln\left(\frac(b+t)(b)\right) – \frac( 3)(4) + \frac(\frac(\left(b+t\right)^4)(4) – \left(b+t\right)^2a^2+a^4\left(\frac (3)(4) + ln\frac(\left(b+t\right))(a)\right) )(\left(\left(b+t\right)^2-a^2\right) ^2)\right\)

Now that we have the C and G values ​​over the entire frequency range, the DC values ​​for R and L, and their asymptotic values ​​in the high frequency region, we have excellent benchmarks to compare with the numerical results.

Modeling cables in an AC/DC module

When formulating a problem for numerical simulation, it is always important to consider the following point: is it possible to use the symmetry of the problem to reduce the size of the model and increase the speed of calculations. As we saw earlier, the exact solution will be \mathbf(E)\left(x,y,z\right) = \mathbf(\tilde(E))\left(x,y\right)e^(-\gamma z). Since the spatial change of fields of interest to us occurs primarily in xy-plane, then we only want to model the 2D cross section of the cable. However, this raises a problem, which is that for the 2D equations used in the AC/DC module, it is assumed that the fields remain invariant in the direction perpendicular to the simulation plane. This means that we will not be able to obtain information about the spatial variation of the ansatz solution through a single 2D AC/DC simulation. However, with the help of simulation in two different planes, this is possible. Series resistance and inductance depend on the current and energy stored in the magnetic field, while shunt conductance and capacitance depend on the energy in the electric field. Let's consider these aspects in more detail.

Distributed Parameters for Shunt Conductance and Capacitance

Since the shunt conductance and capacitance can be calculated from the distribution of the electric field, we start by applying the interface Electric currents.

Boundary Conditions and Material Properties for the Simulation Interface Electric currents.

Once the model geometry is defined and the material properties are assigned values, the assumption is made that the surface of the conductors is equipotential (which is absolutely justified, since the difference in conductivities between a conductor and a dielectric is typically almost 20 orders of magnitude). We then set the values ​​of the physical parameters by assigning the electrical potential V 0 to the inner conductor and ground to the outer conductor to find the electrical potential in the dielectric. The above analytical expressions for the capacitance are obtained from the following most general relations

\begin(align)
W_e& = \frac(1)(4)\int_(S)()\mathbf(E)\cdot \mathbf(D^\ast)d\mathbf(S)\\
W_e& = \frac(C|V_0|^2)(4)\\
C& = \frac(1)(|V_0|^2)\int_(S)()\mathbf(E)\cdot \mathbf(D^\ast)d\mathbf(S)
\end(align)

where the first relation is the electromagnetic theory equation and the second is the circuit theory equation.

The third relation is a combination of the first and second equations. Substituting the above known expressions for the fields, we get the analytical result given earlier for C in a coaxial cable. As a result, these equations allow us to determine the capacitance through the field values ​​for an arbitrary cable. Based on the simulation results, we can calculate the integral of the electrical energy density, which gives the capacitance a value of 98.142 pF/m, which is consistent with the theory. Since G and C and are related by the expression

G=\frac(\omega\epsilon"" C)(\epsilon")

we now have two of the four parameters.

It is worth repeating that we made the assumption that the conductivity of the dielectric region is zero. This is a standard assumption that is made in all textbooks, and we also follow this convention here, because it does not significantly affect the physics - in contrast to our inclusion of the internal inductance term, which was discussed earlier. Many materials for a dielectric core have non-zero conductivity, but this can easily be taken into account in modeling by simply substituting new values ​​into the material properties. In this case, in order to ensure a proper comparison with the theory, it is also necessary to make appropriate corrections to the theoretical expressions.

Specific parameters for series resistance and inductance

Similarly, series resistance and inductance can be calculated by simulation using the interface Magnetic fields in the AC/DC module. Simulation settings are elementary, which is illustrated in the figure below.

Conductor regions are added to a node Single Turn Coil In chapter Coil group , and, the selected reverse current direction option ensures that the direction of the current in the inner conductor will be opposite to the current in the outer conductor, which is indicated in the figure by dots and crosses. When calculating the frequency dependence, the current distribution in the single-turn coil will be taken into account, and not the arbitrary current distribution shown in the figure.

To calculate the inductance, we turn to the following equations, which are the magnetic analogue of the previous equations.

\begin(align)
W_m& = \frac(1)(4)\int_(S)()\mathbf(B)\cdot \mathbf(H^\ast)d\mathbf(S)\\
W_m& = \frac(L|I_0|^2)(4)\\
L& = \frac(1)(|I_0|^2)\int_(S)()\mathbf(B)\cdot \mathbf(H^\ast)d\mathbf(S)
\end(align)

To calculate the resistance, a slightly different technique is used. First, we integrate the resistive losses to determine the power dissipation per unit length. And then we use the well-known relation P = I_0^2R/2 to calculate the resistance. Since R and L change with frequency, let's look at the calculated values ​​and the analytical solution in the DC limit and in the high frequency region.

“Analytical solution for direct current” and “Analytical solution for high frequencies” graphs correspond to the solutions of analytical equations for direct current and high frequencies, which were discussed earlier in the text of the article. Note that both dependences are given on a logarithmic scale along the frequency axis.

It is clearly seen that the calculated values ​​smoothly pass from the solution for direct current in the low-frequency region to the high-frequency solution, which will be valid at a skin depth much smaller than the conductor thickness. It is reasonable to assume that the transition region is located approximately at the place along the frequency axis where the skin depth and conductor thickness differ by no more than an order of magnitude. This region lies in the range from 4.2e3 Hz to 4.2e7 Hz, which exactly corresponds to the expected result.

Characteristic impedance and propagation constant

Now that we have completed the tedious work of calculating R, L, C, and G, there are two other important parameters for power line analysis that need to be determined. These are the characteristic impedance (Z c) and the complex propagation constant (\gamma = \alpha + j\beta ), where \alpha is the damping factor and \beta is the propagation factor.

\begin(align)
Z_c& = \sqrt(\frac((R+j\omega L))((G+j\omega C)))\\
\gamma& = \sqrt((R+j\omega L)(G+j\omega C))
\end(align)

The figure below shows these values ​​calculated using analytical formulas in DC and RF modes, compared with the values ​​determined from the simulation results. In addition, the fourth relationship in the graph is the impedance calculated in the COMSOL Multiphysics environment using the RF module, which we will briefly discuss a little later. As can be seen, the results of numerical simulation are in good agreement with the analytical solutions for the corresponding limit modes, and also give the correct values ​​in the transition region.

Comparison of the characteristic impedance calculated using analytic expressions and determined from simulation results in the COMSOL Multiphysics environment. Analytic curves were generated using the appropriate DC and RF limit expressions discussed earlier, while the AC/DC and RF modules were used for simulations in COMSOL Multiphysics. For clarity, the thickness of the “RF module” line has been specially increased.

Modeling a cable in the high frequency region

The energy of the electromagnetic field propagates in the form of waves, which means that the operating frequency and wavelength are inversely proportional to each other. As we move into higher and higher frequencies, we have to take into account the relative size of the wavelength and the electrical size of the cable. As discussed in the previous entry, we need to change AC/DC to an RF module at an electrical size of approximately λ/100 (see ibid about the concept of "electrical size"). If we choose the diameter of the cable as the electrical dimension, and instead of the speed of light in vacuum, the speed of light in the dielectric core of the cable, we get a frequency for the transition in the region of 690 MHz.

At such high frequencies, the cable itself is more appropriately considered as a waveguide, and the excitation of the cable can be considered as waveguide modes. Using waveguide terminology, so far we have considered a special type of mode called TEM a mode that can propagate at any frequency. When the cable cross-section and wavelength become comparable, we must also take into account the possibility of the existence of higher order modes. Unlike the TEM mode, most guiding modes can only propagate at an excitation frequency above a certain characteristic cutoff frequency. Due to the cylindrical symmetry in our example, there is an expression for the cutoff frequency of the first higher order mode - TE11. This cutoff frequency is f c = 35.3 GHz, but even with this relatively simple geometry, the cutoff frequency is the solution to a transcendental equation that we will not consider in this article.

So what does this cutoff frequency mean for our results? Above this frequency, the wave energy transported in the TEM mode we are interested in has the potential to interact with the TE11 mode. In an idealized geometry like the one modeled here, there will be no interaction. In a real situation, however, any defects in the cable design can lead to mode interaction at frequencies above the cutoff frequency. This can be the result of a range of uncontrollable factors, from manufacturing errors to gradients in material properties. This situation is most easily avoided at the cable design stage by designing to operate at frequencies known to be lower than the cutoff frequency of the higher order modes, so that only one mode can propagate. If it's of interest, you can also use the COMSOL Multiphysics environment to simulate the interaction between higher order modes, as done in this one (although this is beyond the scope of this article).

Modal Analysis in the Radio Frequency Module and the Wave Optics Module

Modeling of higher order modes is ideally implemented using modal analysis in the RF Module and the Wave Optics Module. The ansatz form of the solution in this case is the expression \mathbf(E)\left(x,y,z\right) = \mathbf(\tilde(E))\left(x,y\right)e^(-\gamma z), which exactly matches the mode structure, which is our goal. As a result, modal analysis immediately provides a solution for the spatial distribution of the field and the complex propagation constant for each of a given number of modes. In this case, we can use the same model geometry as before, except that it is enough for us to use only the dielectric core as the modeling area and .

The results of calculating the damping constant and the effective refractive index of the wave mode from the Mode Analysis. The analytical curve on the left graph, damping factor versus frequency, is calculated using the same expressions as for the RF curves used to compare with simulation results in the AC/DC module. The analytic curve in the right plot, the effective refractive index versus frequency, is simply n = \sqrt(\epsilon_r\mu_r) . For clarity, the size of the "COMSOL - TEM" line has been deliberately increased on both graphs.

It is clearly seen that the results of the TEM Mode Mode Analysis agree with the analytical theory and that the calculated higher order mode appears at a predetermined cutoff frequency. It is convenient that the complex propagation constant is directly calculated during the simulation and does not require intermediate calculations of R, L, C, and G. This becomes possible due to the fact that \gamma is explicitly included in the desired form of the ansatz solution and is found in the solution by substituting it into the main equation. If desired, other parameters can also be calculated for the TEM mode, and more information about this can be found in the Application Gallery. It is also noteworthy that the same method of modal analysis can be used to calculate dielectric waveguides, as implemented in .

Final Notes on Cable Modeling

By now, we have thoroughly analyzed the coaxial cable model. We calculated the distributed parameters from the constant current mode to the high frequency region and considered the first higher order mode. It is important that the results of modal analysis depend only on the geometric dimensions and properties of the cable material. The results for simulation in the AC/DC module require more information about how the cable is driven, but hopefully you are aware of what is connected to your cable! We have used analytical theory solely to compare the results of numerical simulations with well-known results for the reference model. This means that the analysis can be generalized to other cables, as well as adding relationships for multiphysics simulations that include temperature changes and structural deformations.

A few interesting nuances for building a model (in the form of answers to possible questions):

• “Why didn’t you mention and/or give plots of the characteristic impedance and all distributed parameters for the TE11 mode?”
  • Because only TEM modes have uniquely defined voltage, current, and characteristic impedance. In principle, it is possible to assign some of these values ​​to higher-order modes, and this issue will be considered in more detail in future articles, as well as in various works on the theory of transmission lines and microwave technology.
• “When I solve a mod problem using Modal Analysis, they are labeled with their working indexes. Where do the designations TEM and TE11 modes come from?”
  • These notations appear in the theoretical analysis and are used for convenience in discussing the results. Such a name is not always possible with an arbitrary waveguide geometry (or a cable in waveguide mode), but it should be borne in mind that this designation is just a “name”. Whatever the name of fashion, does it still carry electromagnetic energy (excluding, of course, non-tunneling evanescent waves)?
• “Why do some of your formulas have an extra factor of ½?”
  • This happens when solving problems of electrodynamics in the frequency domain, namely, when multiplying two complex quantities. When performing time averaging, there is an additional ½ multiplier, as opposed to time domain (or DC) expressions. For more information, you can refer to the works on classical electrodynamics.

Literature

The following monographs were used in writing this note and will serve as excellent references when looking for additional information:

• Microwave Engineering (microwave technology), by David M. Pozar
• Foundations for Microwave Engineering (Fundamentals of Microwave Engineering), by Robert E. Collin
• Inductance Calculations by Frederick W. Grover
• Classical Electrodynamics (Classical electrodynamics) by John D. Jackson

The latest release of COMSOL Multiphysics® and COMSOL Server™ provides a state-of-the-art integrated engineering analysis environment that enables numerical simulation professionals to create multiphysics models and develop simulation applications that can be easily deployed to employees and customers around the world.

Burlington, Massachusetts June 17, 2016. COMSOL, Inc., a leading provider of multiphysics simulation software, today announces the release of a new version of its COMSOL Multiphysics® and COMSOL Server™ simulation software. Hundreds of new user-requested features and enhancements have been added to COMSOL Multiphysics®, COMSOL Server™, and add-on modules to improve the accuracy, usability, and performance of the product. From new solvers and methods to application development and deployment tools, the new COMSOL® 5.2a software release expands the power of electrical, mechanical, fluid dynamic, and chemical simulation and optimization.

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For chemical modeling, a new multiphysical flow interface with chemical reactions has appeared, as well as the possibility of calculating a surface reaction in a layer of reagent granules. Battery manufacturers and designers can now model complex 3D battery pack assemblies using the new Single Particle Battery interface. The discharge and charge of the battery are modeled using a single-particle model at each point of the geometric construction. This makes it possible to estimate the geometric distribution of the current density and the local state of charge in the battery.

Overview of new features and tools in version 5.2a

• COMSOL Multiphysics®, Application Builder, and COMSOL Server™: The appearance of the user interface of the simulation applications may change while they are running. Centralized unit management to help teams working in different countries. Support for hyperlinks and videos. The new Add Multiphysics window allows users to easily create a multiphysics model step by step by providing a list of available predefined multiphysics links for selected physics interfaces. For many fields, including fields for entering equations, the ability to automatically complete input has been added.
• Geometry and mesh: The improved tetrahedral meshing algorithm in the new version can easily create coarse meshes for complex CAD geometries consisting of many fine details. The new optimization algorithm included in the meshing function improves the quality of the elements; this increases the accuracy of the solution and the rate of convergence. Anchor points and coordinate display are now improved in interactive drawings of 2D geometries.
• Tools for mathematical modeling, analysis and visualization: The new version adds three new solvers: smoothed algebraic multigrid, domain decomposition solver, and discontinuous Galerkin (DG) method. Users can now save data and plots in the Export node of the Results section in VTK format, allowing them to import COMSOL simulation results and meshes into other software.
• electrical engineering: The AC/DC module now includes a built-in Giles-Atherton magnetic hysteresis material model. New interconnections of lumped quadripoles, which appeared in the "Radio Frequencies" module, allow modeling lumped elements to represent parts of a high-frequency circuit in a simplified form, without the need to model details.
• Mechanics: The Structural Mechanics module includes new adhesion and cohesion functions available as a sub-node in the Contact extension. A Magnetostriction physics interface is available that supports both linear and non-linear magnetostriction. The possibility of non-linear modeling of materials has been extended with new models of plasticity, mixed isotropic and kinematic hardening, and viscoelasticity with large deformations.
• Hydrodynamics: The CFD module and the Heat Transfer module now take into account gravity and simultaneously compensate for hydrostatic pressure at boundaries. A new density linearization feature is available in the Non-Isothermal Flow interface. This simplification is often used for free-convective flows.
• Chemistry: Battery manufacturers and designers can now model complex 3D battery pack assemblies using the new Single Particle Battery physics interface available in the Batteries and Fuel Cells module. In addition to this, the new Reacting Flow Multiphysics physics interface is available in the new version.

Using COMSOL Multiphysics®, the Application Builder, and COMSOL Server™, simulation professionals are well positioned to create dynamic, easy-to-use, fast-to-develop, and scalable applications for a given manufacturing area.

Availability

To view an overview video and download COMSOL Multiphysics® and COMSOL Server™ 5.2a software, visit https://www.comsol.com/release/5.2a .

About COMSOL

COMSOL is a global provider of computer simulation software used by technology companies, science labs and universities for product design and research. The COMSOL Multiphysics® software package is an integrated software environment for creating physical models and simulation applications. The special value of the program lies in the possibility of taking into account interdisciplinary or multiphysical phenomena. Additional modules extend the capabilities of the simulation platform for electrical, mechanical, fluid dynamic and chemical application areas. A rich import/export toolkit allows you to integrate COMSOL Multiphysics® with all the major CAD tools available on the engineering software market. Computer simulation professionals use COMSOL Server™ to provide design teams, manufacturing departments, test labs, and company customers with applications anywhere in the world. COMSOL was founded in 1986. Today we have over 400 employees in 22 locations worldwide and we partner with a network of distributors to promote our solutions.

COMSOL, COMSOL Multiphysics, Capture the Concept, and COMSOL Desktop are registered trademarks of COMSOL AB. COMSOL Server, LiveLink, and Simulation for Everyone are trademarks of COMSOL AB. Other product and brand names are trademarks or registered trademarks of their respective holders.