The University of Manchester
Third Year Project
Modelling Using Finite Element Methods
Author:
Jahangir Mammadov
Supervisor:
Thomas Thomson
Advisor:
Dave Lester
April 29, 1014
1 Dedication
I dedicate this report to you, grandma. It is already a month that you are not with our
family. I hope, you are in the Heaven waiting for us. I love you so much. Rest in peace,
mother!
2 Acknowledgements
I would like to specially thank Thomas Thomson for his continuous support and help. I
also want to thank my advisor Dave Lester for his advice and participation in the
project seminars. Special thanks, to PhD. student August Johansson for his help during
measurements in the laboratory.
3 Abstract
This report mainly discusses the implementation and results of a project proposal,
“Modelling using Finite Element Methods”. It covers a brief introduction to modelling
and simulation as well as Finite Element Method/Analysis (FEM/A). The main part of
the report is devoted to implementation, which is a model of an electromagnet. The
software tool that is used to model the electromagnet is COMSOL Multiphysics, a
commercial FEA package provided by the University of Manchester, Computer
Science School. Additionally, the report includes other electromagnet models and their
comparison with the original model. The additional models and their comparison are
out of the project scope.
4 Table of Contents 1. INTRODUCTION .......................................................................................................................... 6 1.1 INCEPTION ................................................................................................................................................ 6 1.2 AIMS AND OBJECTIVES ........................................................................................................................ 6 1.3 APPROACH ............................................................................................................................................... 7 2. FINITE ELEMENT METHOD MODELLING ...................................................................... 7 2.1 MODELLING AND SIMULATION ............................................................................................................. 7 2.2 FINITE ELEMENT METHOD ................................................................................................................... 10 2.3 FINITE ELEMENT ANALYSIS ................................................................................................................ 12 2.4 INTRODUCTION TO COMSOL ............................................................................................................. 13 3. MODELLING IN COMSOL .................................................................................................... 15 3.1 LABORATORY ELECTROMAGNET ...................................................................................................... 16 3.2 MODELLING AN IRON CORE ELECTROMAGNET ............................................................................. 17 3.2.1 Geometry ............................................................................................................................................... 17 3.2.2 Materials ............................................................................................................................................... 20 3.2.3 Physics Interface – Magnetic Field ............................................................................................ 21 3.2.4 Meshing .................................................................................................................................................. 25 3.2.5 Study ........................................................................................................................................................ 26 4. SIMULATION RESULTS ......................................................................................................... 27 4.1 RESULTS ..................................................................................................................................................... 27 4.2 COMPARING THE LABORATORY AND MODEL ELECTROMAGNETS .......................................... 29 5. NEW DESIGNS OF ELECTROMAGNETS ........................................................................ 30 5.1 ELECTROMAGNET WITH TWO COILS ................................................................................................. 30 5.2 ELECTROMAGNET WITH A ROUNDED GAP ...................................................................................... 32 5.3 ELECTROMAGNET WITH A CHAMFERED GAP ................................................................................. 33 5.4 COMPARING THE MODELS .................................................................................................................... 34 5 Chapter 1
1. Introduction
1.1 Inception
Modelling is a multi-billion per year industry that has become commonplace in
recent years [1]. It is mainly used in manufacturing and science, but has strong
emphasize in medicine, military and other fields as well [1]. Modelling and simulation
are mainly used to improve the productivity in manufacturing [1].
Since the invention of modern day computers modelling cannot be remembered
without computers. Introduction of computers to numerical analysis completely
changed what modelling has been before. Great computational power of computers is
involved in solving numerical problems that made modelling process easy and fast.
Finite Element Method (FEM) is one of the popular numerical analysis methods,
which modelling software packages are built on. Most of the modelling software
packages use FEM to offer multi-physics modelling that is widely used to model
complex engineering problems. FEM’s introduction in industry helped engineers to
avoid cumbersome and expensive prototyping phase in production cycle. Models
replaced prototypes. The result of the project in this report is also a good example for
models and modelling with computer technology.
The proposal of the project was designed to model an object using software
technology, so-called COMSOL Multiphysics. The software tool was provided by the
University of Manchester, Computer Science School. It is a commercial and generalpurpose modelling software that provides functionalities to model multi-physics
systems.
1.2 Aims and Objectives
The key aim of the project was to model an iron cored electromagnet using the
provided tool. The project involved learning how a typical FEM software package
works and apply that knowledge to model the electromagnet. The main aim of the
project identified additional requirements that included research and training.
6 1.3 Approach
The early iterations of the project were mainly devoted to research about FEM and
its implementation in a software package. Additionally, general working principles of
finite element software packages were studied in order to gain the basic understanding
of COMSOL’s work mechanism.
COMSOL is a quite complex modelling tool that requires good training and practice. I
attended to a workshop organized by COMSOL Inc. to learn modelling process. I
modelled a busbar and a wrench in order to gain some experience with modelling in
COMSOL. After gaining some practice, early versions of an electromagnet were
modelled. In fact, they were not a proper electromagnet as the one in the laboratory.
They were solenoid models. Solenoid is a current carrying wire coupled with an iron
core.
As a next step, the laboratory electromagnet was studied and all its parameters were
recorded. After some measurements and experiments on the electromagnet, modelling
process started.
Chapter 2
2. Finite Element Method Modelling
This chapter begins with explaining modelling and simulation as well as its
advantages and disadvantages. The first section compares modelling and prototyping
as two different steps in production cycle and concludes the comparison in the favour
of modelling. The second section examines Finite Element Method which is an
appreciated numerical technique used in industry. The chapter continues with
discussing general work principles of software tools that are built on top of the FEM
method. The last section is an introduction to COMSOL Multiphysics. It gives brief
information about the main features of the software. The section also builds an
understanding of how COMSOL’s GUI is handled by providing screenshot images.
2.1 Modelling and Simulation
A computer model is an abstract representation of any system in a computer
environment intended to build an understanding of the system in real life [7].
Engineers and scientists create models to learn the possible behaviour of a real object
by conducting experiments on the models [8]. Modelling is the process of building a
model [8]. So, modelling allows engineers and scientists to get information about an
7 object without testing it in a real environment. Modelling and simulation terms are
interchangeably used in science [5]. The main reason is that most of simulation
software supports modelling process and modelling software is able to do simulations
by themselves [1].
This definition of modelling is against the notion of prototyping, which is an
expensive and time-consuming phase in production cycle. Advanced technologies,
algorithms and computer power offer wide variety of functionalities that allow people
to model every possible system. Modelling and simulation is as diverse as the
engineering fields and its emergence brought engineering, science and art together that
helped the experts from different areas to make decisions together on different aspects
of a system [1]. Experts are able to test every condition or situation related to any
targeted object by means of computer modelling (simulation) [1]. The useful features
of computer modelling and simulation lead to remarkable decrease in the usage of
prototyping in industry [1]. It requires quite long time to create a prototype of an
object, because all parts of a prototype are created by human power in real life.
However, in a computer environment a model is easily designed by using built-in tools
and functionalities. The ease of job and accessibility of the required tools in a
computer environment speeds the modelling process remarkably. Additionally,
prototyping is more costly than modelling. Prototypes are real life objects made of
resources, which may have thousands of pounds value in the market. On the other
hand, a single computer with simulation software can be enough to create a model in
most cases. The 2006 National Science Foundation (NSF) Report on “Simulationbased Engineering Science” concludes that computer simulation is well accepted by
the engineering society [6]. According to the report simulation is superior to
prototyping for many reasons:
• Simulations are cheaper and safer than prototypes in conducting experiments
[6]. It is even possible to simulate detonation of nuclear devices and their
effects on surrounding environment using computer power [6]. However, it is
not possible in real world. Effects of hurricanes and other natural disasters can
also be simulated in a computer environment [6].
• Computer simulations are even more realistic than prototyping in some studies
[6]. It is possible to imitate the surface of planets to experiment NASA
missions or simulate deep ocean environment for navy operations [6].
• Simulation is faster than prototyping, when different versions of an object are
experimented [6]. Each small change in a prototype requires additional effort
and time [6]. However it can even take seconds to make that small change in a
computer model [6].
• Computer allows integrating different simulated systems in a coherent
environment, which is quite complex and effort-taking process in real world
[6].
Modelling and simulation is not only time-efficient and financially effective, but it
also offers more capabilities than prototyping [1]. It changed the notion of design and
experimentation in engineering and science [1].
The application domain of modelling and simulation is very large that it is used in
different disciplines of engineering, science, education, medicine and business.
Simulation is used in medicine to gain an understanding of the interaction among the
biological systems such as molecules, cells and tissues [5]. It plays a crucial role in
finding treatments to dangerous disease like cancer, heart and respiratory disease [5].
8 Using simulation and modelling technologies oil and gas exploitation became safer,
cost-efficient and environmentally friendly [5]. Engineers can benefit from the full
potential of the reservoirs, because they are able to learn oil and gas fields by using
advanced simulation technologies such as multi-physics and chemistry modelling [5].
Computer technology is also used in materials engineering to develop new materials
[5]. The technology is mainly used to investigate the molecular structure of the
materials with their physical properties [5]. Scientists interfere the structure and the
properties of materials to invent new materials [5].
It is not possible to stay competitive in manufacturing and production without
applying modelling and simulation technologies to the design cycle of products.
Modelling reduce efforts to design and test a product. It speeds design cycle and lets
the manufacturers to introduce their products to the market earlier [5]. Modelling and
simulation technology is used to model manufacturing process, to analyse the structure
of any object, to model new products using multi-physics technology and to verify the
reliability and the quality of the product among the others [5].
Although, many advantages of modelling and simulation in engineering make it
crucial in design cycle, it can cause dangerous effects, if it is not held carefully. It
requires good knowledge of the application domain as well as computational
mathematics and mathematical modelling techniques [8]. Only the experts in these
fields are involved in modelling processes. Modelling projects can fail because of
many reasons:
• Modelling projects cannot be successful if their requirements are not stated
properly [8]. Modelling process remarkably depends on the preciseness of the
project requirements. A small requirement mistake in any stage may result in
an unwanted model.
• Another problem is the granularity of the final model. It is very important to
model an object in a correct level of abstraction [8]. Excessive details increase
the complexity of the model that results in the usage of vast amount of
resources [8]. The resources are computation power, time and human effort in
this case [8]. Abstract designs, on the other hand, mask the details that prevent
the users to examine the behaviour of the model, which is of critical interest
[8].
• Model is validated when the model shows the required behaviour [8]. That’s
why designers usually ignore the unexpected behaviour that the model
performs [8]. Designers are interested in the correctness of the required
behaviour and do not test whether the additional behaviour is erroneous or not.
Ignorance may result in a faulty model afterwards, when the model is used for
other purposes [8].
Modelling and simulation have some disadvantages, too. It is very difficult to learn
how to model. Simulation tools require special training and knowledge of application
domain. Sometimes it is difficult to interpret simulation results, since they are mostly
random variables [7].
The section discussed the advantages and disadvantages of computer modelling
(simulation) as well as its usage in different areas. It was a general definition of
modelling. However, the next chapter will discuss mathematical models and
mathematical techniques that are used to model systems.
9 2.2 Finite Element Method
Modelling is quite a broad concept, but in the context of this project a model is an
abstract symbolic device built to simulate the behaviour of a real system or an object
[12]. Symbolic means that a model is the interpretation of a real system (object) with
the symbols and language of other disciplines. The project is an example of modelling
in engineering. A model is the translation of an object to the symbols of mathematics
in engineering [12]. Mathematical modelling is a process of building a mathematical
model of an object to certain extent of abstraction [12]. Engineers are not interested in
the complete behaviour of a system [12]. They are trying to simulate some aspects of
the system behaviour. So, mathematical models reinterpret the one or two aspects of
the physical behaviour of the system [12]. There can be more than one model of a
system each interpreting different aspects of it. Dismissing the other behaviour of the
system results in a simpler model [12].
So, mathematical model is a simplifying technique. However, models are not as simple
as it is stated here. They often coupled with complex differential equations in time and
space that results in a model with infinite number of degree of freedom [12]. Such
models can be solved using either analytical or numerical techniques [12]. Solution is
called analytical or numerical model then. Analytical models are restricted to simple
geometries and boundary conditions that make it unpractical for engineering models
[12]. Numerical solutions are mostly used to model systems in engineering [12]. In
order to make the technique practical, degree of freedom is reduced to finite numbers
and this process is called discretization [12]. Discretization is the process of
transferring a continuous system into discrete parts. [12] The aim is to convert those
parts to computer graphics and adhere them there. [12] The product of numerical
model is a discrete model.
FEM is the most popular discretization technique in structural mechanics [12]. Other
techniques such as boundary element methods and finite difference methods can be
applied to small number of problems [12]. FEM is better than other methods, because
it offers greater flexibility to model complex geometries [12]. It can handle boundary
conditions and variable material properties [12]. FEM has a clear structure that
software engineers can design general-purpose software for various applications [12].
It has a solid foundation, which makes it more reliable than other methods [12]. FEM
allows analysing and estimating errors in solutions by using approximation theory [12].
FEM is a technique that divides mathematical models into simple components, which
are called finite elements [12]. Finite element is a disjoint simple geometric figure that
is expressed in degree of freedom (Dof) [12]. Dof is used as a value for mathematical
functions that constitutes the discrete model [12]. Finite elements, together, form a
discrete model, which is a translation of the mathematical model [12]. The assembly of
the elements is called a mesh [12]. Figure 1 is a mesh representing a discrete model of
a plane.
10 Figure 1: A mesh sequence of a discrete plane model [12]
Finite elements are individual entities that one can hold each at a time [12]. The
properties of each element can be developed individually. Elements have attributes
such as element dimensionality, nodes, geometry and element degree of freedom that
forms their property [12]. FEM offers one, two and three-dimensional elements as
well as zero dimensional elements such as lumped string and point masses. [12] The
dimension of one-dimensional elements can be converted by applying kinematic
transformations in order to use them in multi-dimensional models [12]. Nodes are
points located in the corners and endpoints of the elements that form their geometry
[1]. These points are representing Dof of the elements as well. FEM models are built
from fairly simple elements as Figure 2 shows.
Figure 2: The examples of one, two and three-dimensional finite elements [12].
One-dimensional elements are straight and curved lines. Bi-dimensional elements
usually have triangular and rectangular shapes. Elements in 3D are tetrahedral,
pentahedral and hexahedral figures that are also called prisms and bricks [12].
Elements are connecting to each other at their Dofs [12]. Degree of freedom of an
element shows its state and position in space [12].
11 2.3 Finite Element Analysis
FEM has a broad range of application domain. Using FEM one can model systems
in solid mechanics, heat conduction, electrostatics, magnetostatics and others [13].
Wide range of functionality in FEM modelling allows engineers to model complex
systems.
Computational data can be written manually if a model is simple enough. However,
when the amount of elements gets bigger in the models of complex systems (almost all
the models in engineering), it becomes very hard and time-consuming to process the
model data [10].
The majority of the models are designed by using computer technologies nowadays.
There are some projects that require, even, supercomputers and very expensive
software tools to model and simulate systems. The process of FEM modelling using
software packages is called Finite Element Analysis (FEA) [10]. The design of FEA
packages can be simpler than any word processing software that a special package can
be designed for each model [10]. However, the packages are complex enough that the
majority of users refer to general-purpose FEA packages. A typical software package
requires the following data about the mode to start modelling process [14]:
• Nodal points which are spatial locations (of the elements) that forms model
geometry
• Elements connecting nodal points
• Mass properties and boundary conditions
• Loading or forcing functions
• Analysis options
A typical FEA procedure in commercial software is completed in three steps as shown
in Figure 3.
Figure 3: FEA processes [14]
12 Users are usually allowed to interfere with the procedure in “preprocess” stage [14].
The first action in the stage is analysis type selection [14]. Users can select one or
more analysis out of structural, thermal, modal and other analysis types depending on
model domain. Secondly, the geometry of the model is created using the tools provided
by the package [11]. In the next step material properties are assigned to the model
components [14]. Mesh creation is the final step in preprocessing stage, which can be
implemented manually or automatically by a computer depending on the software
package [14]. During meshing process firstly, nodal points are defined and connected
to build the elements [11]. Then boundary conditions and loads are applied in order to
create the mesh sequence [14].
Computer power shows itself during “proprocess” stage. This is the main reason
behind the emergence of FEA packages. All equations are solved during this stage by a
computer [14]. Computer displays the results in “postprocess” stage, which is also
called simulation [14]. The user sees the results of the simulation and analyses the
model.
COMSOL Multiphysics is also a typical FEA software package that works with
the general principles stated here.
2.4 Introduction to COMSOL
One of the most popular FEA packages currently used in industry is COMSOL
Multiphysics. COMSOL provides a broad range of functionalities in terms of physics
interfaces that allow the users to model any physics-based system. COMSOL’s
multiphysics environment makes it to stand alone among FEA software packages. Its
multiphysics environment is capable of modelling and studying multiple physics
modules simultaneously. For example, it allows the users to model induction heating
along with magnetism. The physics-based modules augment the core physics interfaces
of COMSOL Multiphysics and provide additional interfaces for electrical, mechanical,
fluid flow, and chemical applications [9]. Our model will use Magnetic Field (mf)
interface of the AC/DC module under electrical applications libraries.
COMSOL has a quite complex Graphical User Interface and APIs. Even to use a
simple built-in geometric operation, some reading and practice are required.
Otherwise, the user can get unwanted results in a small mistake. Users should read the
reference guides and physics interface manuals to model proper applications. Now the
readers do not have a chance to study those materials. That’s why, a quick manual will
be provided below before starting the implementation part. The manual will include
detailed explanation of GUI elements, built-in geometry operations and physics
interfaces.
COMSOL’s graphical user interface is composed of three main windows as it is
shown in Figure 4. The most important components of the initial window are Ribbon
and Model Builder.
13 Figure 4: The print-screen of COMSOL desktop that shows Model Builder, Properties
and Graphics windows.
Ribbon contains the first two rows with tabs locating at the top of the window. The
ribbon tabs have buttons and drop-down lists for controlling all steps of the modelling
process [2]. Steps in the modelling process can be easily controlled either via Ribbon
or via Model Builder. The ribbon tabs reflect the modelling steps by providing easy
access to the parameters and functions of each step. Steps during the modelling process
are geometry design, material properties and physics definition, mesh creation, study
selection and the visualization of the simulation results [2].
Model Builder is in the left of COMSOL desktop that gives an access to all the steps of
modelling process. Model Builder is a main component for controlling the modelling
steps, analysing the results and generating reports. Modelling steps are defined and
controlled by building a Model Tree under Model Builder. Model is created starting
with the default model tree, continuing with node addition and settings change [2].
As it is stated before, modelling process is completed in six steps: Geometry,
Materials, Physics, Mesh, Study and Results. The steps will elaborately be discussed
throughout the implementation part, but now a quick review of the steps will be given
below.
The steps under Model Tree are also called nodes. The small window at the right
of Model Builder displays the properties of the selected node. So the alternate
processes will be node addition and parameter definition. Compute and Build buttons
are called to implement the properties defined in the property window.
The hardest and time-consuming steps in the modelling process are geometry creation
and physics definition. COMSOL has wide variety of tools and built-in operations to
14 design the geometry of a model. By right clicking the mouse on Geometry node users
can access all the tools and operations. While designing the geometry the user can
either choose to build the object among primitive objects like cube, cylinder, cone,
sphere, etc. or can choose to add a 2D plane (called Work Plane in COMSOL) in order
to draw the geometry manually. Geometries of the objects can be quite complex that
primitive objects may not satisfy the user. So, I used two Work Planes to design a coil
and a core, separately. It is not too difficult to draw on a Work Plane, because
COMSOL provides a separate window for that. But, COMSOL eases the user’s effort
by providing simple figures such as rectangles, circles, lines, etc. COMSOL has
auxiliary built-in operations, which are called Boolean operations. These operations are
capable to subtract, unite, intersect and compose two or more objects. Property
window near Model Builder shows the selected objects.
You will often meet with “Build Selected”, “Build All”, and “Compute” expressions
in the following chapters and sections. These are the buttons in the property window.
The user clicks build buttons to implement either the selected operation or all the
operations from the beginning. Compute button is in Study step. It is clicked to
compute all equations.
Physics nodes are also added and controlled through Model Tree. When an
interface is added, its built-in nodes come with that. The user accesses desired nodes
by right clicking on the parent interface node as it is in other steps. Any physical
property can be assigned to domains, boundaries, edges or points in the object figure.
Modelled object usually have different physical properties in its different parts and
COMSOL lets the user to assign various physical properties to different regions in the
same object.
Mesh, Study and Results steps are software-controlled steps unlike Geometry and
Physics steps, which are mainly built by the user. Those three steps are the main reason
that engineers refer to software package like COMSOL. Otherwise, engineers could
model the applications manually without using computer power.
Introduction to COMSOL and its features gave us a good insight how it works and
how the features are controlled. Now we can model a simple iron core electromagnet
using COMSOL’s features.
Chapter 3
3. Modelling in COMSOL
This chapter mainly discusses how an electromagnet is modelled using COMSOL
MultiPhysics interfaces and libraries. A model of an electromagnet is created after
examining a real life example, which is a simple iron core electromagnet provided by
the university.
15 The first section of this chapter describes the tools and technologies used during the
experimentation as well as the parameters of the electromagnet. The section also
provides some information about how electromagnets work. The second section goes
through the design phases and interfaces used to build a model of the electromagnet.
Each step of the modelling process in COMSOL will be discussed in the following
subsections of the second section. The first subsection describes how the geometry of
the model is created using the built-in geometry tools provided by the software. The
second subsection discusses material assignment to the domains of the model, while
the third one is about physics definition of the object and its individual parts. The
fourth subsection is about mesh creation. The last subsection discusses solver
configurations set for the equations and data sets automatically generated by the
software.
3.1 Laboratory Electromagnet
Electric currents flowing through a wire generates magnetic field. A solenoid is
a cylindrical wire that generates magnetic field B when it carries electric currents. A
ferromagnetic material iron core multiplies magnetic field ten and even thousand times
when it is added to a solenoid. All electromagnets work with the same principle of iron
core solenoid. The laboratory electromagnet has also the same working principle [15].
There were many electromagnet designs in the laboratory differing in their geometry,
power and materials. In this project the requirement was to model a simple iron core
electromagnet, shown in Figure 5. The electromagnet has a simple geometry, which is
made of two elements. It is composed of an iron core and a multi-turn coil. The
electromagnet is used as part of a larger experimental apparatus, where there is a need
to create a magnetic field, which can be controlled by changing the current in coils. A
typical use would be to measure the magnet-optical response of novel data storagemedia such as bit pattern media (BPM).
Figure 5: The laboratory electromagnet. The electromagnet in the image is 15 times
smaller than the real one.
16 The electromagnet is experimented by measuring the dimensions of the coil and the
iron core using a vernier calliper. Looking from top, the iron core is 165mm in length
and 94mm in width. Its height and thickness are measured to be 51.5mm and 25mm.
The gap in the core is 32.2mm wide.
The multi-turn coil is 107mm in length and 80mm in width and its height is
124mm.
The core is made of laminated iron, a special kind of iron used in electromagnets
production, which has relative permeability of Mur = 200 [15]. The coil is made of
insulated copper wire. The wire has 1.18mm cross-section area and it is turned 1614
times.
Having measured the physical dimension the next step would be the measurement
of magnetic flux density the electromagnet produces in the gap. I am very thankful to
PhD. student August Johansson, who helped by providing the tools and assisting with
the measurements. In order to undertake the experiment we used two devices and
software that has an interface to enter current value and display magnetic flux density
influencing the probe. Firstly, the probing device was adjusted in the gap of the iron
core and plugged in a measurement device. Then, the coil was excited by a current
controlled power supply. The software was used for controlling current value and
reading the values from the probe. The electromagnet generated nearly 0.056 T of
magnetic flux density in the middle of the gap applying 2A current to the coil.
All required experimentations and measurements were taken already to build a model,
which is explained in the following section.
3.2 Modelling an iron core electromagnet
COMSOL has well-defined interfaces and well-structured libraries to easily build
a model and simulate the results. The software provides libraries for geometry design,
materials and physics interface, mesh creation and solver generation, which will be
explained in the following sub-sections. The Results node presented in the last
subsection, handles simulation by providing detailed graph, images and numbers. This
is a key output for verifying the model against the real life example. Bold words in the
remaining sections will represent modelling steps, functionalities and interfaces of
COMSOL.
3.2.1 Geometry
The Geometry definition function (node) was used to create the electromagnet
geometry using tools in COMSOL. The geometry of the model is composed of two
main elements, a multi-turn coil and an iron core. For building each part of the model a
Work Plane is added to Geometry node to convert a 2D geometry drawn in the plane
to a 3D object in the space. The position of the newly added plane is defined
automatically as (0,0,0) in XYZ coordinates or it can be defined manually by entering
appropriate coordinates. 2D geometry objects and features are added to the plane to
create a 2D object sequence. All the length units used in Geometry are in mm, while
angular units are in deg.
17 Two rectangles and one Bezier polygon are used for building the geometry of the
multi-turn coil. Firstly, a rectangle (Rectangle 1) with 80mm width and 124mm height
is drawn and it is centred about the position (0,0).
A smaller rectangle (Rectangle 2) with 34mm width and 58mm height is added and
centred about the same position with the first rectangle. So, the two rectangles
coincide. Next, by using Difference Boolean operation smaller rectangle is subtracted
from the bigger to draw a final 2D object sequence. Inner and outer corners of the
object are rounded in the next step by adding two Fillet (Fillet 1 and Fillet 2)
operations. The inner and outer corners are filleted with 4mm and 25mm circular fillet
arches, respectively. Then, a line between the triangles is drawn using Bezier Polygon
(Bezier Polygon 1), as it is shown in Figure 6.
This line is an internal boundary, which will represent an input for coil excitation
during physics definition.
Figure 6: Final geometry sequence of the multi-turn coil in 2D Work Plane (1).
After two-dimensional object sequence created in Work Plane 1, it is converted to a
three-dimensional object using Extrude (Extrude 1) operation. Object is extruded
107mm. It means its distance from the plane is 107mm.
The second Work Plane (Work Plane 2) is added to Geometry node for drawing an
iron core object sequence using three different-sized rectangles. Two Difference
operations are used to subtract smaller two rectangles (Rectangle 4 and Rectangle 5)
from the biggest one (Rectangle 3). Rectangle 4 is 114.2mm in length and 44mm in
width. Rectangle 5 is 32.2mm in length and 40mm in width while Rectangle 3, the
biggest rectangle is 164.2mm in length and 94mm in width.
After subtraction operations, inner and outer corners of the object except the corners
surrounding the gap are rounded. The inner corners are filleted 10mm, while the outer
corners filleted 25mm each. Figure 7 shows the final version of the iron core in 2D
format.
18 Figure 7: Iron Core geometry sequence in 2D Work Plane (2).
Finally, 2D core is extruded 51.5mm. After creating the coil and the core on different
planes, they are repositioned in space to form a proper electromagnet figure.
Otherwise, those two objects stay as unrelated objects. The only way to reconfigure the
object sequence is to relocate the objects on different planes and move them to
different directions using XY coordinates. The Work Plane of the multi-turn coil
(Work Plane 1) is located on YZ plane and has X and Y displacements of 132mm,
25.5mm, respectively. The plane (Work Plane 2), where the core was drawn is
relocated on XY plane and has -47mm and -1mm of X and Y displacements.
It is recommended to add a sphere (Sphere 1) to the geometry and put the
electromagnet inside. The added sphere will be filled with air during material
allocation in order to simulate room environment. This would be important when
considering thermal effects such as thermal heating of the electromagnet. However, in
the work these effects are not taken into account and it remains as a future goal. The
model is in vacuum currently, which is completely different to the environment the real
electromagnet experimented.
Although, these 3 objects are positioned properly to model the laboratory
electromagnet, COMSOL considers them as three unrelated objects. We need to
specifically define those objects as a single object by calling Form a union built-in
operation under Geometry node. The software then forms a union from all geometry
objects. The union is divided into domains, separated by boundaries according to the
participating geometry objects [2]. It is also possible but often not necessary to specify
boundary conditions on interior boundaries among domains in the geometry [2].
COMSOL ensures continuity in the physics fields across interior boundaries by
default. Uniting the objects is the last step in forming the geometry, which results in
Figure 8.
19 Figure 8: Complete geometry of the model, an electromagnet in a sphere.
According to COMSOL’s geometry statistics shown in Table 1, the model contains 3
domains, which are built from 51 boundaries. Domain 1 is sphere, while Domain 2 and
Domain 3 are the iron core and the multi-turn coil, respectively.
Property
Value
Space dimension
3
Number of domains
3
Number of boundaries
51
Number of edges
128
Number of vertices
82
Table 1: The table shows the amount of each geometric property in the model.
3.2.2 Materials
After creating the geometry of the model, the second step in the process is to
assign materials to each object. The sub-nodes under Materials are used to add
predefined or user-defined materials, to specify specific material properties using
model inputs, functions, values, and expressions or to create a custom material library
[2]. COMSOL lets the user either to choose an object as a domain, which automatically
assigns a material to all the boundaries of the object or to select more granular
elements like boundaries, edges and points. In the model, domains are used to assign
20 materials to the objects, as the coil has the same material property in all their
boundaries like the iron core.
COMSOL will return an error if an object left undefined, because it will not be able to
calculate the functions required for finite elements in Physics interface (the third step
in the process). So, if an object is remained unassigned an automatic error will remind
the user.
Materials are grouped according to physics interfaces in COMSOL. For assigning
materials to the multi-turn coil, the iron core and the sphere, three materials are chosen
from the built-in materials group. COMSOL even provides the functionality to change,
remove or add properties to materials and this functionality is used to make the
materials more similar to the material properties of the laboratory electromagnet.
Copper with relative permeability of Mur = 1 and relative permittivity of εr = 1
is assigned to the multi-turn coil, while soft iron (with losses) is assigned to the core.
Soft iron is used for different purposes in industry, that’s why some properties,
especially, relative permeability of the material is not given initially by COMSOL and
requires the user to define them. Relative permeability of soft iron can vary depending
on the application for which it was produced. Soft iron with relative permeability of
Mur = 200 is the material mainly used in electromagnets [15]. The user must define it
manually in COMSOL. Finally, the sphere object is filled with air to imitate the
laboratory environment during simulation.
3.2.3 Physics Interface – Magnetic Field
The AC/DC module in COMSOL is widely used by engineers and scientists to
understand, predict and design electric and magnetic fields in statics and lowfrequency applications [3]. The AC/DC module includes stationary and dynamic
electric and magnetic fields in two-dimensional and three-dimensional spaces along
with traditional circuit-based modelling of passive and active devices [3]. All
modelling formulations are based on Maxwell’s equations [3]. The AC/DC module
supports modelling with its various physics interfaces [3]. The AC/DC interfaces cover
electrostatics, DC current flow, magnetostatics, AC and transient current flow, AC and
transient magnetodynamics, and AC electromagnetic formulations [3].
Magnetic field is one of the physics interfaces under the AC/DC module, which
allows users to compute magnetic field and induced current distributions in and around
coils, conductors and magnets [3]. The interfaces of the module support stationary,
frequency-domain, small-signal analysis and time-domain modeling, which provide
features to design time-dependent and stationary models [3]. This feature of Magnetic
Field (mf) physics lets the user easily model an electromagnet in stationary mode [3].
According to default settings of the interface, all domains in a model are selected to
define the magnetic vector potential parameters and solve the equations to compute
magnetic field [3]. However, if a certain domain is in the center of interest, it can be
selected manually to disregard other domains during simulation.
When Magnetic Field (mf) physics interface is added to the model, three nodes,
Ampère’s Law, Magnetic Insulation and Initial Values nodes are automatically added
under the interface to define the basic principles and equations to compute the
magnetic field.
21 The Ampère’s Law node adds Ampère’s law for the magnetic field and provides an
interface for defining the constitutive relation and its associated properties as well as
electric properties [3].
Domain selection for Ampère’s law node is predefined by the parent node
(Magnetic Field interface) and cannot be changed. As all the domains were chosen in
the physics interface for the electromagnet model, equations will be calculated for each
domain. Users are required to choose and define some fields in the node to customize
Ampère’s law properties to their needs. These fields are “Model Inputs”, “Material
Types”, “Coordinate System Selection”, “Conduction Current”, “Electric Field” and
“Magnetic Field” with their subfields. In the current model (a simple iron core
electromagnet) “Temperature”, “Absolute pressure” and “Magnetic Flux Density (B)”
are variables, which were set as model inputs for these simulations. “Temperature” and
“Absolute pressure” are included as 293.15K and 1atm respectively. “Magnetic Flux
Density” subfield is entered as an initial guess of simulation results or a good start
point for solvers. No value is entered to this subfield. Coordinate system is selected as
“Global Coordinate System”, while “Material Type” selected as ‘From material’. In
this context, “Material Type” decides how materials behave and how material
properties are interpreted when the mesh is deformed. “From material” is chosen to get
the corresponding properties from the domain materials. “Conduction Current” defines
“Electrical Conductivity σ (SI unit: S/m)” for the model, and chosen to be picked up
from the material properties. “Electric Field” gets “Relative Permittivity” from
material properties as well. “Magnetic Field” specifies constitutive relation that
describes the macroscopic properties of the medium (relating the magnetic flux density
B and the magnetic field H) and the applicable material properties, such as relative
permeability [3]. “Constitutive relation” is specified as “Relative permeability”, which
is obtained from the material properties. “Magnetic Insulation” is another component
under the physics interface, which is added automatically according to the default
settings. It sets magnetic vector potential to zero at the selected boundaries. As other
default nodes it inherits its selection from “Magnetic Field (mf)” parent node. Thus, all
boundaries are selected, but insulation is not applicable to the boundaries constituting
the coil and the core. Only the sphere is insulated and magnetic potential at its
boundaries vanishes. Magnetic vector potential of the electromagnet (coil and core)
cannot be zeroed, because by default the interface calculates magnetic field for those
domains when the user assigns proper materials (iron and copper). “Initial Values”
node is provided by the “Magnetic Field” interface to add initial values for “Magnetic
Vector Potential A (Wb/m)” that can serve as an initial value for the simulation results
or a good guess for the non-linear solver [3]. Default XYZ components of the vector
are 0 Wb/m and they are unchanged for this model, too.
3.2.3.1 Multi-­‐turn coil A Multi-Turn Coil represents the current carrying coil (Domain 2) and as the
name suggests it consists of a strand of Copper wire coated with an insulator. Shorting
does not occur between conductors due to insulation [3]. Current flows along the wire
and is negligible in other directions [3]. The interface requires the selected domain to
have magnetic and electric properties in order to be treated as a coil. This node also has
fields and properties that are specified by the user. Multi-Turn Coil node is the most
critical part in defining the physics interface, because the accuracy of the results is
highly dependent on this. As in Ampère’s law, “Temperature” is 293.15K, while
22 “Absolute pressure” is 1atm. Moreover, “Relative Permittivity” and “Relative
Permeability” and “Material Type” are picked up from material properties.
“Coordinate System” is defined as “Global Coordinate System” as it was in previous
nodes. The most important part in the Multi-Turn Coil node under the interface is to
specify the type of the coil. COMSOL provides three coil types: Linear, Circular and
Numeric. Users are allowed to define the direction of the wire as a vector field and the
length of the coil, if they select “User Defined” option under “Coil Type” field. Users
need to choose a proper coil type. Otherwise, COMSOL can fail to solve the equations
for the simulation and may produce erroneous results. Coil current direction is the only
reason that, the node offers three coil types. So, current can flow straightly, circularly
or the direction can be calculated in the Study step.
Linear Coil is formed as multiple straight wires bundled in a sleeve and the geometry
must have a straight longitudinal axis. [4] Direction of the current flow is modeled by
specifying a reference edge. The end surfaces of the coil should touch the external
walls of the air domain surrounding the conductor as in Figure 9 [4].
Current direction in the coil
Figure 9: Linear multi-turn coil. [4]
Circular Coil has a circular cross section and formed using multiple wires
arranged as circular coil and put in a potting material. [4] The geometry of a circular
coil should also have straight longitudinal axis and form a closed loop. Direction of the
current flow is modeled using more than one reference edge that should form a closed
curve [4]. Figure 10 shows an example of a circular coil.
23 Current direction in the coil
Figure 10: Circular multi-turn coil. [4]
In a Numeric coil, current flow is automatically calculated in the Study step. For
this to happen, Automatic Current Calculation sub-node should be added under
Multi-turn coil node. Numeric coil is a general case in COMSOL. Its geometry forms
a closed loop, but unlike linear and circular coil types it can be in any arbitrary shape.
It is preferable to fillet the corners to avoid unhealthy results. It must have an internal
boundary, which is perpendicular to the wire. An internal boundary was created in the
Geometry step to conduct current to the coil. So, the created boundary is used as an
excitation source. Other boundaries in the coil are insulated. Electric Insulation subnode is added to insulate the wire and it also prevents the wire to be parallel with the
coil boundaries. As it is stated before, the wire should be perpendicular to the coil
boundaries. Input sub-node is added to the Model Tree under Multi-turn Coil node to
define the internal boundary as an excitation source. It forces the wire to be orthogonal
to the selected (Input) boundary and also defines the direction of the wire.
It is suggested to select “Numeric coil” type while assigning physics to the
coil, because it is the general form of all coil models in COMSOL. Linear and
Circular coils are the special cases where the coil is straight and circular. After
examining the geometry of the coil, Numeric coil type is selected to define the
multi-turn coil domain in the model.
After coil type selection, values included to “Number of Turns”, “Coil Conductivity”,
“Coil cross-section area” and “Coil Excitation” fields in order to specify the
parameters of the coil that will be used for calculations. “Number of Turns” are 1614
in the model as it is in the real electromagnet. “Coil Conductivity” for wire is entered
as 6 x 107[S/m], which is the conductivity of copper. The cross-section area of the
wires is defined as “User Defined” and entered to be 1.18mm. COMSOL uses “Coil
Conductivity” and “Cross-section area” to compute coil resistance. The current density
flowing in the coil domain is computed from a lumped quantity that constitutes the coil
excitation. [3] The coil can be excited either by current excitation or voltage excitation.
In this case, current excitation is selected and “Coil Current” is entered as 2A for the
very first simulation results.
24 3.2.4 Meshing
After defining the physics interface for the model, the next step in the process is
mesh creation. Meshing a geometry is an essential part of the simulation process, and
can be crucial for obtaining the best results in the fastest manner [9].
The geometric model is divided into thousands of tiny finite elements, which can be in
different shapes. The elements constituting the model mesh are mostly in tetrahedral
shape, pyramid like figure. COMSOL offers two mesh sequence types: “Physicscontrolled mesh”, “User defined” meshed. “Physics-controlled” mesh is preferred for
the model due to the simplicity of the geometry. “User Defined” mesh sequence types
is usually preferred when a model has a complex geometry. By selecting physicscontrolled mesh as the mesh sequence type, the mesh is adapted to the current physics
settings in the model. The user is allowed to choose one of nine element sizes, from
extremely fine to extremely coarse. The predefined element sizes are simply sets of
parameters, five parameters to be exact that are available for modification [9]. The
following five parameters (in bold format) define element sizes according to
COMSOL’s Reference Guide [2]:
• The value in the Maximum element size field specifies the maximum allowed
element size [2].
• The value in the Minimum element size field specifies the minimum allowed
element size. This value can also be used to prevent the generation of many
elements around small curved parts of the geometry. It is not available in 1D
[2].
• The Maximum element growth rate determines the maximum rate at which the
element size can grow. The value must be greater or equal to one. For example,
with a maximum element growth rate of 1.5, the element size can grow nearly
fifty percent from one element to another [2].
• The value in the Resolution of curvature field determines the size of boundary
elements compared to the curvature of the geometric boundary [2].
• In the Resolution of narrow regions field you control the number of layers of
elements that are created in narrow regions (approximately) [2].
The software automatically selected “Normal” element size for the mesh.
Normal size is adequate for model geometries in most cases. However, if a model has a
boundary smaller than the defined element size, then COMSOL returns an error and
advise to select smaller element size. “Normal “ predefined element size allows a
maximum element size of 40mm. The smallest element created in the mesh sequence is
7.2mm. The maximum element growth rate is 1.5. So, it means elements can get 50%
bigger from one to another. The curvature factor for “Normal” size is defined as 0.6
while the resolution of narrow regions is 0.5. Using normal-sized elements the mesh is
built from 233592 elements as it is in Figure 11.
25 Figure 11: Mesh sequence of the model electromagnet
According to Table 2 the majority of the elements are in tetrahedral shape, but other
shapes are also used where the tetrahedral elements did not fit.
Property
Value
Tetrahedral elements
36524
Triangular elements
4869
Edge elements
750
Vertex elements
82
Table 2: The table shows the numbers of different finite element geometries in the
mesh.
3.2.5 Study
Study step is the sixth step in COMSOL after mesh creation. Equations and data
specified in the previous steps are solved in this step to give the simulation results.
Study node holds all the sub-nodes to solve the model. It is the most abstract part of
COMSOL, because users usually are not required to change the settings or enter
parameters. Although, it includes many sub-nodes building a large hierarchy,
COMSOL wants users to define only the study steps they wish. Among the steps the
most popular ones are Stationary and Time-dependent steps. Stationary step is used,
when field variables do not change over time. On the other hand, Time-dependent
study is utilized for simulating the behavior of the model over time. In
26 electromagnetics, Stationary step is used for calculating static electric and magnetic
fields, as well as direct currents. In the model study Stationary step is used for
calculating magnetic field.
It is highly recommended not to forget Coil Current Calculation step while solving
the equations, because it computes the current of a Multi-Turn Coil domain and
produces a current density corresponding to a strand of wire [3]. Coil Current
Calculation study step is only available for 3D models using Magnetic Field
interfaces and Multi-Turn Coil domain nodes. Added Automatic Current Calculation
sub-node to the Multi-Turn Coil domain sets automatic calculation of the current flow
in the coil domain. The boundary conditions of Electric Insulation and Input provide
the needed data for Coil Current Calculation to solve the equations.
After Study steps (Coil Current Calculation, Stationary) added, the computation can
begin.
Chapter 4
4. Simulation Results
The model was completely designed and all the equations and solver
configurations were ready to start the simulation. Compute button was left clicked and
the simulation started.
The result of the simulation is provided in this chapter by means of graphs, tables
and images. The chapter starts by discussing Results node, which is the final node in
the model tree. It also includes the statistics of computation time and mesh elements as
well as occurred errors and warnings in addition to the simulation results. The second
section discusses how the model and real electromagnet are similar by comparing their
behaviour under different conditions. Overall four simulations were held to verify the
model. Finally, the results were compared against the laboratory electromagnet.
4.1 Results
Results is the last branch in COMSOL, containing tools and functionalities for
post processing and result visualization. Tools like 2D/3D plots, Tables, Reports and
Derived Values use data sets located in the branch to generate reports, tables, images
for the simulation results.
Results branch contains two Solution data sets for the model, because two Study
steps, Coil Current Calculation (Eigenvalue Solver) and Stationary studies computed
the model. According to COMSOL’s self-generated statistics 35256 Degrees of
Freedom solved for only Coil Current Direction. Stationary step computed 233592
elements for Magnetic Vector Potential, which results in the data set generating
simulation for Magnetic Flux Density Normal. Overall, Solver solved 268848 elements
in 58 seconds using an Intel i5 dual-core processor.
27 The study steps worked without an error. However, COMSOL generated a warning
report stating that mesh elements are inverted near coordinates (0.009, 0.14, 0.055). It
means COMSOL cannot solve the equations for those elements. That’s why it inverts
their shapes to simpler 2D figures. If the amount of inverted elements is too much,
results may include erroneous answers. However, the warning can be avoided now,
because the amount of inverted elements is not a threat to the accuracy of the results.
Note: 2A of current applied to the coil for the first simulation. Figure 12 shows the
simulation results for the modeled iron core electromagnet. It has the bar on the right
illustrating the values of Magnetic Flux Density (T) in certain parts of the model by
means of coloring.
Figure 12: The simulation result of the model electromagnet. Image shows magnetic
flux (T) density the model generates
Magnetic flux density (T) is low in blue areas, outside the immediate vicinity of the
iron core and coil. A residual of 2mT of magnetic flux density can be measured in
those areas, as expected from a simple interpretation of Maxwell’s equations. The
magnetic flux density is quite high at meeting points of the coil and iron core where it
ranges between 0.6 T and 0.7 T. The magnetic flux density is diminishing in the arms
of the core, because those parts are far from the coil.
The main concern for the study is the amount of magnetic lux density (T) the
electromagnet generates in the middle of the gap. The model electromagnet generates
(0.056 ± 2) Tesla magnetic flux density at the midpoints of the gap.
The statistics of the maximum and minimum amount of magnetic flux density is
provided in Table 3. According to the table, the maximum magnetic flux density is
nearly 0.7 T, which is in the inner right corner of the iron core, where it meets with the
coil. This is also in agreement with Maxwell’s laws where a geometric discontinuity
leads to magnetic flux concentration. Using this property we sharpened the gap field in
other models, discussed in the next chapter, to get more magnetic flux density than the
28 base model. It is not a surprise that, the minimum magnetic flux density is in the far
corner of the sphere as it is consistent with Maxwell’s equations.
X (mm)
Y (mm) Z (mm)
Magnetic flux density norm (T)
-184.77
100
76.53
3.59816e-5
105.27
119.55
28.94
0.71
Table 3: Table shows the maximum and minimum magnetic flux density generated in
the model
Moreover, other properties of the electromagnet and the coil itself can be found
using COMSOL’s useful simulation features. According to the simulation results, the
coil has 5.54Ω of resistance, when 2A of current applied to it. In addition to resistance,
the simulation shows that current density in the coil is 9.14 x 105 A/m^2. Nearly 1300
J/m^3 of energy density is generated in the gap field of the iron core according to the
results.
4.2 Comparing the laboratory and model electromagnets
The model electromagnet generated the same amount of magnetic flux density in
lower currents as the laboratory electromagnet. Although, it behaves like the real
electromagnet in lower currents, the accuracy of the model cannot be verified with
only this. In order to verify that the model behaves as the real electromagnet it is
simulated three times by conducting different amounts of electric current to the coil.
The coil is conducted 3A in the first, 5A and 10A in the second and the third
simulations respectively. Possible magnetic flux density the model and laboratory
electromagnets can generate by applying 3A, 5A and 10A current is given in Table 4.
The table verifies that the model electromagnet is designed properly, because both
electromagnets give the same results under the same conditions.
Current (A)
Magnetic flux density (T)
Laboratory electromagnet Model electromagnet
2A
0.056 ± 0.002
0.057 ± 0.002
3A
0.083 ± 0.002
0.084 ± 0.002
5A
0.139 ± 0.002
0.140 ± 0.002
10A
0.275 ± 0.002
0.276 ± 0.002
Table4: Magnetic Flux Density generated by the laboratory and model electromagnets
while 2A, 3A, 5A and 10A of current applied.
There is a small difference in the geometry of the model and the laboratory
electromagnets. The iron core and coil domains have slightly different geometric
parameters than the real coil and core. The domains initially had the same parameters
with the real life objects. However, COMSOL gave errors during mesh creation. The
reason for the errors was the spatial relationship between the coil and core. In the
initial geometry the core was touching to the coil that prevented COMSOL to generate
mesh elements for each domain separately. The domains were hiding each other’s
mesh elements. It was required by the software to resize the domain geometries. That’s
why the hole in the coil is widened to protect a small space between the coil and core.
The domains did not override each other’s elements after the changes applied. The
29 accuracy of the simulation results was not affected by this change, because it is small
enough to make big differences in the magnetic flux density the model can produce.
This change was expected from the beginning of the design phase, because COMSOL
had given similar errors when simpler models were created before the project.
Overall, the model is quite successful, because it behaves like the laboratory
electromagnet under the same circumstances. It means the methodology and the way
the model created were correct. The requirement of the project, to design a simple iron
cored electromagnet, was accomplished, but there was enough time to make some
experiments and to model other electromagnets. So, we decided to design other
electromagnet models.
Chapter 5
5. New Designs of Electromagnets
The model electromagnet worked properly and gave the same results as the
laboratory electromagnet. It means all modelling steps were implemented correctly.
Knowing the modelling procedures it is possible to make geometry and physics
changes to the base model or to design new electromagnet models. The gap of the
electromagnet can be reshaped and resized as a good start point. Using a cylinder
shaped core instead of a rectangular shaped core can be a good experiment, too. Using
circular coil type instead of the numeric coil type will require changes to the geometry
of the coil as well as its physics interface. Designing a model with multiple multi-turn
coils is also a good idea.
Possible electromagnet models were evaluated and as a result four electromagnets
with different geometries are proposed to test their suitability as laboratory
electromagnets. Three of them are quite similar to the original model and have some
changes on the dimensions and the geometry surrounding their gap fields. But, one of
them is quite different in the shape. It has two separate coils on the iron core. This
model required making some alterations on the physics interface applied and the
geometry drawn. In the last section newly designed models are compared with the base
model and the laboratory electromagnet. The main purpose is to find a model that
generates more magnetic flux density than the real electromagnet. Of course, the newly
designed models are not expected to have big differences in magnetic flux generation.
However, small differences will certainly occur.
5.1 Electromagnet with two coils
To design the electromagnet with two coils, the coil of the original model is
divided into two pieces. It means all the properties of the original coil are divided into
two, to create two coils out of one. As it is seen in Figure 13, there are two coils on an
iron core. The coils are created in the same way with the coil of the original model, but
different numbers are used for parameters.
30 Figure 13: Proposed geometry of the electromagnet with two coils.
Each coil is created using the same techniques and the same simple objects. However,
the parameters of the rectangles are changed in this model. The big rectangle is 80mm
in length and 124 mm in width. A rectangle with 34mm length and 58mm width is
used as a smaller rectangle. Inner and outer corners of the coils were rounded 4mm and
25mm respectively. After fillet operations created 2D objects were extruded 53.5mm.
The iron core is created on a separate work plane with the parameters that are slightly
different from the parameters of the iron core of the original model. Two coils did not
fit in the original core, that’s why the parameters rectangles used for the new iron core
were changed. The biggest rectangle is changed to be 94mm in width, 184mm in
length. Smaller rectangle is 44mm in width and 134.2mm in length. The smallest
rectangle has the parameters of 70mm width, 32.2mm length.
Its corners are filleted (inner corners – 10mm, outer corners – 25mm) and then
the 2D object sequence is extruded 51.5mm to finish in a 3D iron core. After assembly
of the coils and core, the electromagnet is put in a sphere and the materials are
assigned to each as it was in the original model.
Magnetic Field (mf) interface is added to define physics for the final geometry as
it was in the original model. However, two multi-turn coil domains are used each with
807 turns. In total, 1614 turns. Both coils are excited with 2A current for the first
results. In Study phase, two Coil Current Calculation study steps are added to a
Stationary step, because the current model has two separate coils and each requires
calculating their equations separately.
31 Figure 14 shows the magnetic flux density the electromagnet with two coils can
generate when each coil is excited with 2A current. The model generates about 53 ±
2mT of magnetic flux density at the midpoints of the gap field.
Figure 14: The simulation result of the model with two coils. Image shows magnetic
flux (T) density the model generates
Moreover, the coils have 2.77Ωof resistance each and 9.14 x 105 A/m^2 of current
density together.
Remaining electromagnet models are designed by making alterations on the
geometry of the original model, mainly on the gap field.
5.2 Electromagnet with a rounded gap
Firstly, the gap field of the electromagnet is rounded to examine whether a change
happens in magnetic flux density generated there. The gap is rounded as it is shown in
Figure 15. However, rounding the gap does not make a big difference in the amount of
magnetic flux density generated in the gap. The model produces 55 ± 2mT of magnetic
flux density in the middle of the gap. According to the results it is easily seen that, if
we design an electromagnet with rounded gap, we will not be successful in our aim of
producing more powerful electromagnet than the laboratory electromagnet.
32 Figure 15: The geometry of the electromagnet with its gap rounded/filleted
5.3 Electromagnet with a chamfered gap
The gap is chamfered in the next example shown in Figure 16. The gap is
chamfered 8mm by using Chamfer built-in operation provided by the software.
Chamfered electromagnet produced nearly 0.052 T of magnetic flux density in the gap.
33 Figure 16: The electromagnet with its gap chamfered
Moreover, the gap field without any geometry change reduced 5mm. It is expected to
produce more magnetic flux density than the original model, because the gap is
smaller. 5mm reduction in the distance resulted in 0.019 T of increase in magnetic flux
density produced in the gap. Now the gap is producing approximately 0.075 T of
magnetic flux density.
5.4 Comparing the models
The main purpose in trying different models was to find a model with the same
properties with the laboratory electromagnet, but producing more B field (Magnetic
Flux density) than that. As examined in the previous subsection to produce an
electromagnet with two coils without increasing the number of wire turns in total is not
efficient. It does not produce more B field than the laboratory model. Filleted and
chamfered models are also unsuccessful, because they are slightly weak than the
original electromagnet as Graph 1 shows.
34 Graph1: shows magnetic flux density the electromagnets generate when different
currents applied.
On the other hand, simply reducing the gap can remarkably strengthen the
electromagnet making 0.019T of change when 2A of current is applied. The difference
is larger in higher currents as it is interpreted in Table 5.
Current
Magnetic Flux Density norm (Tesla)
Laboratory
Model
Model with 2
coils
Gap chamfered
Gap reduced
2A
0.056 ± 0.002
0.057 ± 0.002
0.053 ± 0.002
0.052 ± 0.002
0.075 ± 0.002
3A
0.083 ± 0.002
0.084 ± 0.002
0.080 ± 0.002
0.079 ± 0.002
0.111 ± 0.002
5A
0.139 ± 0.002
0,140 ± 0.002
0.134 ± 0.002
0.133 ± 0.002
0.185 ± 0.002
6A
0.275 ± 0.002
0.276 ± 0.002
0.266 ± 0.002
0.260 ± 0.002
0.365 ± 0.002
Table 5: Magnetic flux density norm. produced in the gap fields of the models
However, it is know that geometric discontinuity leads to magnetic flux concentration
according to Maxwell’s laws. It means that the electromagnet with a chamfered gap
should produce the highest magnetic flux density among the models. In order to prove
that idea, the chamfered gap was reduced along with the electromagnet with the
smallest gap.
Chamfered electromagnet began to generate more magnetic flux density than the
simply reduced one when the gap was 18mm and below. For 10A current applied,
35 chamfered electromagnet gives 0.53T of B field, while the simply reduced one gives
0.505T.
We can draw a conclusion from the experiments driven that, chamfered electromagnet
with ≤ 18mm gap generates more magnetic flux density than other electromagnets.
Although, there is no big difference in the amount of B field produced by the
electromagnets in low currents, conducting 10A of current clears all the doubts and
makes it easy to identify the strongest electromagnet.
References:
1.
Andreas Tolk, Engineering Management Challenges for Applying Simulation
as a Green Technology. Old Dominion University
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