An Introduction to MAGNET for Static 2D Modeling Infolytica Corporation PDF

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An Introduction to MAGNET for Static 2D Modeling Infolytica Corporation J D Edwards We welcome your comments regarding Infolytica Corporation documents. You may send comments or corrections to the following address: Post: Documentation Manager Infolytica Corporation 300 Leo Pariseau, Suite 2222 Mont...

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An Introduction to

MAGNET for Static 2D Modeling

Infolytica Corporation

J D Edwards

We welcome your comments regarding Infolytica Corporation documents. You may send comments or corrections to the following address:

Post:

Documentation Manager Infolytica Corporation 300 Leo Pariseau, Suite 2222 Montréal, Québec H2X 4B3 Canada

e-mail: [email protected]

© 2007 Infolytica Corporation All rights reserved. This document may not be reproduced, translated into another language, stored in a retrieval system, or transmitted in any form or by any means, electronic, photocopying, recording, or otherwise, without written permission from Infolytica Corporation. The information in this document is subject to change without notice.

Contents Chapter 1 Introduction

1

Overview .......................................................................................................1 Modeling in 2D and 3D .................................................................................3 Magnetic concepts .......................................................................................4 Using MagNet effectively .............................................................................8 Getting help ..................................................................................................9

Chapter 2 Tutorial: C-core Electromagnet

11

Introduction ................................................................................................11 Device Model ..............................................................................................12 Building the model .....................................................................................14 Solving the model.......................................................................................21 Post-processing .........................................................................................24 Modifying the model...................................................................................28 Postscript....................................................................................................39

Chapter 3 Case Studies: Translational Geometry

41

Introduction ................................................................................................41 E-core electromagnet.................................................................................41 Magnetic latch with a permanent magnet .................................................45 Busbar forces .............................................................................................48 Field in a cylindrical conductor .................................................................54 Cylindrical screen in a uniform field .........................................................56 Transformer equivalent circuit ..................................................................62 Variable Reluctance Stepper Motor ..........................................................67 Linear synchronous motor ........................................................................72

Chapter 4 Case Studies: Rotational Geometry

79

Introduction ................................................................................................79 Mutual inductance of coaxial coils............................................................83 LVDT displacement transducer .................................................................86 Magnetic pull-off force ...............................................................................89 Moving-coil transducer ..............................................................................93

Chapter 5 Scripting

97

Introduction ................................................................................................97 Example model ...........................................................................................97 Script for the model....................................................................................99 Creating a new script ...............................................................................101 Automation with Excel .............................................................................102

Appendix A

Field Equations and Solution

111

Field equations .........................................................................................111 Boundary conditions and symmetry.......................................................113 Numerical solution ...................................................................................115

Appendix B

Energy, Force and Inductance

119

Stored energy and co-energy ..................................................................119 Force calculation......................................................................................121 Inductance calculation.............................................................................122

Appendix C References

Open Boundary Implementation

125 129

Chapter 1

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Introduction

Chapter 1

Introduction Overview The principal aim of this document is to introduce new users to the power of MagNet for solving 2D static magnetic field problems. A tutorial with detailed instructions takes the first-time user through the most important features of MagNet. This is followed by a series of case studies illustrating modeling techniques and introducing further features of the package. The document concludes with an introduction to advanced features that make MagNet a uniquely powerful tool.

What is MagNet? MagNet is the most advanced package currently available for modeling electromagnetic devices on a personal computer. It provides a “virtual laboratory” in which the user can create models from magnetic materials and coils, view displays in the form of field plots and graphs, and get numerical values for quantities such as flux linkage and force. A MagNet user needs only an elementary knowledge of magnetic concepts to model existing devices, modify designs, and test new ideas. MagNet is designed as a full 3D-modeling tool for solving static magnetic field and eddy-current problems. Many devices can be represented very well by 2D models, so MagNet offers the option of 2D modeling, with a substantial saving in computing resources and solution time. With 2D models, MagNet can also handle problems where currents are induced by the motion of part of the system. A feature of MagNet is its use of the latest methods of solving the field equations and calculating quantities such as force and torque. To get reliable results, the user does not need to be an expert in electromagnetic theory or numerical analysis. Nevertheless the user does need to be aware of the factors that govern the accuracy of the solution. One of the aims of this document is to show how the user can obtain accurate results. In 2D, problems can be solved very rapidly, so it is usually not necessary to consider the trade-off between speed and accuracy. In 3D modeling, on the other hand, this is an important consideration. For the advanced user, MagNet offers powerful facilities for user-defined adjustment of the model parameters, calculation of further results from the field solution, and control of the operation of the package with scripts and scripting forms.

© 2007 Infolytica Corporation

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Introduction to MagNet

Limitations The information given in this document has been prepared specifically for the entry-level version of MagNet. This version is restricted to static magnetic fields and 2D models, without facilities for parameterization: the automatic solution of sequences of problems with modified model parameters. Because parameterization is such a useful feature, the document includes examples of its use, but alternative methods are also provided for users who have only the entry-level version of MagNet.

A guide to the document The next sections in chapter 1 give some background information for first-time users of software for electromagnetics, particularly for users whose knowledge of elementary magnetism is insecure. It is helpful but not essential to read some of this before proceeding to the next chapter. Chapter 2 is a practical introduction to MagNet in the form of a tutorial. It takes the user through all the steps of modeling a simple magnetic device, with full explanations of the operations and the interpretation of the results. This chapter is an essential prerequisite for chapters 3 and 4. Chapters 3 and 4 contain case studies in which MagNet is applied to a variety of magnetic problems. These can be used in two ways: as reference material, and as a series of graded exercises for developing skills after completion of the tutorial. Chapter 5 introduces scripting in MagNet, including the use of Microsoft Excel to control MagNet. Scripting is now available in all versions of MagNet, and chapter 5 indicates some of the ways of using this feature. Chapter 6 gives an introduction to the powerful Calculator facility that is built in to MagNet, which enables the user to carry out further processing of the solution. The latest release of MagNet includes some new facilities for displaying field quantities that formerly required the Calculator, so most users will not need to refer to this chapter. Appendix A contains further information about the magnetic field equations and the solution methods used in MagNet for 2D problems. Novice users do not require most of this material, but advanced users may find the additional insight helpful. The discussion of boundary conditions is relevant to all users, and includes the basis of the Kelvin transformation technique for openboundary problems. Appendix B covers energy, force and inductance calculation. This includes the derivation of some of the equations used in the case studies, and further information about the methods used in MagNet. Appendix C explains how to implement the Kelvin transformation for open-boundary problems.

Second edition This edition has been revised to take account of the latest features of MagNet (version 6.21.1 at the time of writing). The Tools and Extensions menus now include field-plotting facilities, which in most cases make it unnecessary to use the Calculator for such purposes, so the chapter on the Calculator has been moved to the end. An important addition to the document is the use of the Kelvin transformation technique for handling some open-boundary problems. This technique is used in the case study on busbar forces in chapter 3, and for the problem of forces between permanent-magnet blocks in chapter 5. Finally, the opportunity has been taken to make the dimensions more realistic in some of the case studies.

© 2007 Infolytica Corporation

Chapter 1

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Introduction

Modeling in 2D and 3D Some practical problems are essentially three-dimensional – examples include the rotor of a clawpole alternator and the end-winding regions of rotating AC machines. Problems of this kind require the full 3D modeling capability of MagNet. In many cases, however, a 2D model will give useful results. There are two common types of device geometry that allow 3D objects to be modeled in two dimensions: translational geometry and rotational geometry.

Translational geometry Translational geometry means that the object has a constant cross-sectional shape generated by translation – moving the shape in a fixed direction. The diagram below shows a C-core formed in this way.

With translational geometry, any slice perpendicular to the axis has the same shape. Rotating electrical machines can often be represented in this way, and so can many other devices such as transformers and actuators. Inevitably this 2D approximation neglects fringing and leakage fields in the third dimension, so the model must be used with caution. The shape is usually drawn in the x-y plane, with the z-axis as the axis of translation.

Rotational geometry Rotational geometry means that the object has a shape formed by rotation about an axis, like turning on a lathe. The diagram below shows an object formed in this way from the same basic C shape used in the diagram above.

Objects with rotational geometry are usually described in cylindrical polar coordinates, with the z-axis as the axis of rotation. The rotated shape is then defined in an r-z plane, which makes an angle θ with the 3D x-axis. This geometry differs from translational geometry in two important respects. First, it is a true representation of a real 3D object, so highly accurate solutions are possible. Secondly, there are different equations to be solved, and different methods required for calculating quantities such as force and inductance. For all built-in calculations MagNet handles these differences automatically, but the user needs to be aware of the difference when using the Calculator facilities. In MagNet, the 2D cross-section of a rotationally symmetric model must be drawn in the x-y plane, with the y-axis as the axis of rotation. The x-y coordinates then correspond to the r-z coordinates of the conventional cylindrical polar coordinate representation. © 2007 Infolytica Corporation

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Introduction to MagNet

Magnetic concepts MagNet can be used to model practical devices without knowing anything about the differential equations of electromagnetism or the numerical methods used to solve them. This section reviews some basic magnetic concepts that are required for making effective use of MagNet; more advanced topics are covered in appendix A. The system of units used is the SI or MKSA system, although other systems will be mentioned in the context of magnetic materials.

Magnetic flux density B The fundamental magnetic concept is the magnetic field described by the vector B, which is termed the magnetic flux density. In two dimensions this field is commonly represented by curved lines, known as flux lines, which show both the direction and the magnitude of B. The direction of a line gives the direction of B, and the spacing of the lines indicates the magnitude; the closer the lines, the greater the magnitude. The diagram below shows the flux plot for a simple electromagnet where the C-shaped steel core on the left attracts the steel bar on the right. The two sides of the magnetizing coil are represented by squares.

Although the magnetic field is an abstract concept, the effects of B are concrete and physical. The force in a device such as this electromagnet can be expressed in terms of B. In simplified terms, the flux lines can be treated as elastic bands pulling the bar towards the magnet with a tensile stress (force per unit area) given by B2 / 2μ0. In this expression, B is the magnitude of the vector B, and μ0 = 4π × 10–7 is a fundamental constant. The unit of B is the tesla (T), and the unit of μ0 is the henry per meter (H/m). A direct physical interpretation of B is given by the Lorentz equation for the magnetic force on an electric charge q moving with velocity u: f = qu × B

(1-1)

If the moving charge is an electric current flowing in a conductor, then equation 1-1 leads to the familiar expression f = Bli for the force on a conductor of length l carrying a current i. If the conductor itself is moving with velocity u, then the Lorentz force causes a displacement of charge in the conductor, leading to the expression e = Blu for the induced voltage. Frequently it is not the flux density B that is required, but the magnetic flux φ and the flux linkage λ. Flux is defined as φ = BA when the flux density B is constant and perpendicular to a surface of area A. If the field is not constant or perpendicular to the surface then the flux is given by an integral, but the principle is the same. Flux linkage is the sum of the fluxes for all the turns of a

© 2007 Infolytica Corporation

Chapter 1

5

Introduction coil; this is λ = Nφ for a coil of N turns where each turn links a flux φ. The concept of flux gains its value from Faraday’s law of electromagnetic induction, which states that the voltage induced in a coil is e = dλ / dt. If the flux linkage results from current flow, either in the same coil or in a different coil, this leads to the definition of inductance as flux linkage per ampere. The calculation of inductance is discussed in appendix B.

Magnetic intensity H Electric currents give rise to magnetic fields. The currents may flow in conductors or coils, or they may take the form of electron spin currents in the atoms of a magnetic material. In either case the problem is to define the relationship between the magnetic field described by B, and the currents which are the source of the field. In seeking a mathematical form for this relationship that can be used to solve practical problems, it is useful to introduce a new magnetic quantity H, which is related both to B and to the currents that are the source of B. For a magnetic field in free space, set up by currents flowing in conductors, H is defined through the equation B = μ0 H . The relationship between H and the currents is then given by Ampère’s circuital law:

∫ H.dl = ∑ i

(1-2)

where the integral on the left is taken round a closed path, and the summation on the right is the sum of all the currents enclosed by the path. This equation makes it easy to calculate the field of a simple system such as a long straight conductor or a toroidal coil, and it is the basis of the magnetic circuit concept, which is widely used for approximate calculations in electromagnetic devices. In its differential form it leads to general methods that are applicable to any problem; this point is expanded in appendix A. The quantity H is known as the magnetic intensity; from equation 1-2 it has units of amperes per meter (A/m). For magnetic fields in free space, there would be little advantage in using H; equation 1-2 could be expressed in terms of B and μ0. When magnetic materials are present, however, the situation is completely different.

Magnetic materials When a coil is wound on a core of magnetic material such as iron or steel, instead of a nonmagnetic material such as wood, the behavior changes dramatically. If the core is closed, the coil has a much higher inductance. If the core is open, so that the coil behaves as a magnet, the external magnetic field is greatly increased. The material of the core has itself become the source of a magnetic field that reinforces the effects of the coil. The behavior of magnetic materials can be described by modifying the relationship between B and H. We may put: B = μ 0 (H + M)

(1-3)

where H is the magnetic intensity given by equation 1-2, and M is an induced magnetization in the material which depends on H. Thus H can be regarded as the cause, which is related to currents in conductors; B is the effect, giving rise to forces and induced voltages.

© 2007 Infolytica Corporation

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Introduction to MagNet

Flux density B (T)

From the point of view of the device designer, the magnetization M is unimportant; what matters is the relationship between H and the resulting B. This relationship can be extremely complex; the vectors may not be in the same direction, and the present value of B may depend on the past history as well as the present value of H. For many practical purposes, however, these complexities can be ignored and the properties of the material expressed by a simple curve relating B to H. This is the B-H curve or magnetization characteristic of the material. A typical example is the curve for transformer steel shown below. This curve has three distinct regions: t...