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Finite Element Tutorial in Electromagnetics #3

DRAFT Sponsored by NSF Grant #05-559: Finite Element Method Exercises for use in Undergraduate

Engineering Programs

Transmission Parameters of anInfinitely Long Co-Axial Cable

Prepared By: Dr. Vladimir A Labay, Department of Electricaland Computer Engineering

Gonzaga University, Spokane, WashingtonTelephone : 509.323.3553 Email: [email protected]

Copyright 2006Reference Text : Elements of Engineering Electromagnetics,

Sixth Edition by N.N. RaoSoftware : Maxwell 2D Student Version

www.ansoft.com

Estimated time to completeThis tutorial: 90 minutes

x

y

,

a

b

mailto:[email protected]

http://www.ansoft.com/

http://www.ansoft.com/

mailto:[email protected]

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Tutorial and Educational Objectives

The educational goal is to provide undergraduate engineering studentswith understanding of a specific engineering topic and FE theory, alongwith an ability to apply commercial FE software to typical engineeringproblems. The educational goal will be accomplished through foureducational objectives based on Bloom’s Taxonomy and ABET Criterion

3 as follows:

1. Engineering Topics (Comprehension; 3a, 3k). Understand the fundamental basisof engineering topics through the use of finite element computer models.

2. FE Theory (Comprehension; 3a). Understand the fundamental basis of FE theory.

3. FE Modeling Practice (Application; 3a, 3e, 3k). Be able to implement a suitablefinite element model and construct a correct computer model using commercial FE

software. -- integrates 3 and 4 above. 4. FE Solution Interpretation and Verification (Comprehension and Evaluation; 3a,

3e). Be able to interpret and evaluate finite element solution quality, including theimportance of verification.

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Table of Contents

1. Overview of Computational Electromagnetics

2. Finite Element Method in EM

3. Problem Definition and Background

4. Overview of Maxwell 2D

5. Creating the RG59 Project 6. Defining Materials and Boundaries

7. Generating a Solution

8. Analyzing the Solution

9. Concluding the Session

10.Further Reading and References

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1. Overview of ComputationalElectromagnetics

• Engineering Electromagnetics

– The study of electrical and magnetic fields and their interaction

– Governed by Maxwell’s Equations

(Faraday’s Law, Ampère’s Circuital Law, and Gauss’ Laws)

• Maxwell’s Equations relate the following Vector and Scalar Fields E: the Electric Field Intensity Vector (V/M)

H: the Magnetic Field Intensity Vector (A/m)

D: the Displacement Flux Density Vector (C/m2)

B: the Magnetic Flux Density Vector (T)

J: the Current Density Vector (A/m2

) : the Volume Charge Density (C/m3)

: is the Permeability of the medium (H/m)

: the Permittivity of the medium (F/m)

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1. Overview of CEM (con’t)

D

DJHt

0 B

BEt

Faraday’s Law: Ampère’s Circuital Law:

Gauss’ Laws:

HB ED

Constitutive Equations:

Actual solution complex and for realistic problems require approximations

Solutions to Maxwell’s equations using numerical approximations of is known as the

study of computational electromagnetics (CEM)

Maxwell’s Equations

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Applications of CEM• Over the past five decades CEM has been successfully applied to several

engineering areas, including: – Antennas

– Biological electromagnetic (EM) effects

– Medical diagnosis and treatment

– Electronic packaging and high speed circuits – Superconductivity

– Microwave devices and circuits

– Law enforcement

– Environmental issues

– Avionics

– Communications

– Energy generation and conservation – Surveillance and intelligence gathering

– Homeland Security

– Signal Integrity

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• Approximation of Maxwell’s equations may be classified into several categories, for example, low-frequency, quasi-static, full-wave, lumped element equivalent, etc.

• This tutorial deals with the finite element (FE) method to approximate the solution ofMaxwell’s equations. The FE method applied in electromagnetics is a full-wavetechnique. Full-wave techniques have the potential to be the most accurate of allnumerical approximations because they incorporate all higher order interactions and

do not make any initial physical approximations

• Examples of full-wave computational electromagnetic (CEM) techniques include: – Finite difference time domain (FDTD) Method

– Method of Moments (MoM) Method

– Finite Element (FE) Method

– Transmission Line Matrix (TLM) Method

– The Method of Lines (MoL)

– The Generalized Multipole Technique (GMT)

The FDTD, MoM and FE are the most popular today!

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• Central to all CEM techniques is the idea of discretizing some unknownelectromagnetic property, for example:

– In MoM the Surface Current is typically used

– In FE, the Electric Field

– In FDTD, the Electric and Magnetic Field

• As part of discretization, meshing is used to subdivide a large geometry into anumber of nonoverlapping subregions or elements, for example:

– In two dimensional regions triangles maybe used

– In three dimensional geometries a tetrahedral shape may be used

• Within each element, a simple functional dependence (basis functions) is assumed

for the spatial variation of the unknown

• CEM is a modeling process and therefore a study in acceptable approximation

In other words, CEM replaces a real field problem with an approximate one which causeslimitations that one must keep in mind

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• During the creation of the approximate problem, assumptions and simplifications are

generally introduced. These will place limitations on the solution, for example: – Assuming an infinite ground plane in an antenna structure. Is this assumption valid?

– Have you made simplifications on the design that are not valid? For example, simplifying a

thin wire by a current filament.

• When analyzing solutions generated from a CEM techniques keep in mind limitations

of the solution introduced by tolerances and manufacturing deviations, for example: – Tolerances are a part of all manufactured devices. How do small changes in dimensions or

material properties affect the performance?

– Do other manufacturing considerations, other than tolerances, affect the performance?

• Limitations are also introduced by the finite discretization

– Is the mesh fine enough so that the basis functions can adequately represent theelectromagnetic fields?

• Finally, numerical approximations and finite machine precision will limit the analysis

– Does double precision provide enough accuracy for your problem, especially if it is ill

conditioned?

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2. Finite Element Method in EM

• Initially used in structural mechanics and thermodynamics dating back to the 1950’s

• First application in electromagnetics appeared in literature in the late 1960’s but didnot see widespread adoption until the 1980’s – A problem of ―spurious modes‖ was not solved until the 1980’s through a theoretical

breakthrough with edge elements

– Widespread availability of powerful main-frame and personal computers also aided the

expansion

• The FE method starts with the partial differential equation form of Maxwell’sEquations

• Solution by the FE method can be viewed from two main perspectives – Variational analysis

• Finds a variational functional whose minimum corresponds to the solution of the PDE

– Weighted residuals• Introduces a ―weighted‖ residual or error and using Green’s function, shift one of the differentials in the

PDE to the weighting functions

– In most applications these two viewpoints result in identical equations

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2. Finite Element Method in EM (con’t)

• The basic concept of the FE method is that although the behavior of a function maybe complex when viewed over a large region, a simple approximation may besufficient for a small subregion

• FEM can handle essentially two different types of EM problems

– Eigenanalysis (source-free)

– Deterministic (driven)

• FEM does not include a radiation condition

– Open regions, such as antennas (see below), requires special treatment

• Introduction of a artificial absorbing region within the mesh

• Example Microstrip Patch Antenna

Antenna Patch

Infinite Ground Plane

Substrate Material

Artificial absorbingregion(box surrounding theantenna)

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Strengths of the FE method• Handles complex geometries and material inhomogeneities easily

• Handles dispersive or frequency-dependent materials easily

• Handles eigenproblems easily

• Has better frequency scaling characteristics that MoM (but usually requires a larger set ofunknowns)

• Easily applicable to ―multi-physics‖ problems by coupling solutions in thermal or mechanical to the

EM solution

Weaknesses of the FE method

• Inefficient treatment of highly conducting radiators when compared to the MoM

• FEM meshes become very complex for large 3-D structures

• More difficult to implement than the FDTD thus limiting their use in commercial software. Little

code development is done by engineers• Efficient preconditioned iterative solvers are required when higher-order elements are used.

Again, restricting the code development by individual engineers

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The FE method solution consists of essentially four steps:1. Discretizing the region of interest into a mesh, that is, finite elements

2. Deriving the governing equations for the individual finite elements

3. Relating the individual finite elements to the assembly of the elements

4. Obtaining and solving the system of equations for the unknown quantity

Commercial FE software packages for electromagneticsWhen using a commercial FE software package, these four steps are done internally with littleintervention by the user. The main tasks of the design engineer is to properly develop the model,assign the material properties, and specify the sources of the electromagnetic fields.

Some Companies that market commercial FEM EM software

• Maxwell, High frequency structure simulator (HFSS) and Designer by Ansoft Corporation

• Emag by Ansys

• Multiphysics with Electromagnetics Module by Comsol

• COSMOSEMS by SolidWorks Corporation

Maxwell by Ansoft will be used solely in this tutorial

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MAXWELL 2DMaxwell is a high-performance full-wave electromagnetic field simulator for arbitrary 2Delectromagnetic problems modeling that takes advantage of a Windows graphical user interface.It integrates simulation, visualization, 2D modeling, and automation in an easy-to-learnenvironment.

The student version of this software is a available at www.ansoft.com/downloads.cfm

Maxwell includes: – A graphical interface to simplify design entry

– A field solving engine with accuracy-driven adaptive solutions

– Powerful post processor for displaying currents, fields and other parameters

– Automatic and adaptive mesh generation and refinement and tangential vector finiteelements

– A comprehensive materials database that contains permittivity-, permeability, electric-,

magnetic-loss tangents for common materials.

– Iteratively calculates the desired electrostatic or magnetostatic field solution and specialquantities of interest, including force, torque, inductance, capacitance, and power loss. Youcan select any of the following solution types: Electrostatic, Magnetostatic, Electrostatic,Eddy Current, DC Conduction, AC Conduction, Eddy Axial. The student version does notcontain thermal, transient, or parametric capabilities.

http://www.ansoft.com/downloads.cfm

http://www.ansoft.com/downloads.cfm

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3. Problem Definition and Background

The Co-axial Transmission Line

• The following tutorial is intended to show how to create, simulate, and analyze a co-axial transmission line using the Ansoft’s Maxwell 2D Design Environment

• From the analysis, you will be able to verify the transmission line’s circuit parametersand field plot

• This tutorial leads you step-by-step through the calculation of a co-axial transmissionline . By following the steps in this tutorial you will be able to: – Draw a geometric model – Modify a model’s design parameters – Assign variables to a model’s design parameters – Specify solution settings for a design – Validate a design’s setup

– Run a Maxwell simulation – Create a 2-D plot of the field pattern – Calculate the capacitance of the line from the fields – Create a field overlay plot of the results – Study the mesh created by Maxwell for the solution

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3. Problem Definition and Background (con’t)

• The lumped-element circuit model for aTransmission Line – As shown to the right, a transmission line is

schematically represented as a two-wire line.This due to the fact that transverseelectromagnetic wave propagation or TEMwaves on a transmission line requires at leasttwo conductors

– A infinitesimal length, see (b), can be modeled

as a lumped-element circuit where R, L, G, Care per unit length quantities

• The Lossless Line – In many line, such as the co-axial

transmission line, the loss of the line is verysmall and can be neglected for the analysis.Thus, setting R and G equal to zero, the TEM

wave propagation can be completelydescribed in terms of L and C.

– However, L and C are not independent fromeach other. They are related by:

LC

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Basic TEM Wave Propagation Equations

– The phase constant (rad/m)

– The characteristic Impedance (ohms)

– The wavelength on the line (m)

– The phase velocity (m/s)

– The voltage (Volts) and current (amps)

where superscripts (+) and (-) correspond to forward and reverse traveling waves,respectively

LC

C

L Z

o

LC

22

LC v p

1

z jV z jV zV oo expexp)( z j Z

V z j

Z

V z I

o

o

o

o expexp)(

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Calculation of the Inductance or Capacitance

– Given a one meter section of a uniform transmission line with fields E and H and cross-sectional surface area, as shown below, from field theory, the time-average stored magneticand electric energy is given by

– From circuit theory the magnetic and electric energy is also given by

– Thus the self-inductance and capacitance per unit length is given by

– The 2D field problem must be solved to

determine the propagation parameters of a TEM transmission line

S

m ds H H W *

4

Se ds E E W

*

4

4

2

om

I LW 4

2

oe

V C W

S

o

ds H H I

L*

2

S

o

ds E E V

C *

2

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General Procedure for analyzing a TEM line

1. Solve Laplace’s equation for the cross-sectional area of the transmission line. Thesolution will contain unknown constants.

2. Find the constants by applying the boundary conditions for the known voltages on theconductors.

3. Compute the electric field from the following equations

4. Compute the magnetic field from the following equation

5. The capacitance or inductance may now be calculated. Recall that only one needs tobe determined as they are related by

0),(2 y xt

)exp(),(),,( z y xe z y x E

),(ˆ

1),( y xea y xh z

),(),( y x y xe t

)exp(),(),,( z y xh z y x H

LC

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Calculation of the Transmission Line Parameters for a Co-Axial Cable

– Given the geometry shown below, a the outer conductor at V = 0 and the inner conductor atVo, the solution of Laplace’s equation yields the following fields between the conductors.

The material between the conductors is assumed to be a perfect dielectric.

– Thus the capacitance and inductance is given by

– Finally, the characteristic impedance is

)exp( / ln

ˆ

),,( zab

aV z E

o

)exp(

2

ˆ

),,( z Z

aV z H

o

o

H/m) / ln(2

1

)2(

2

0 22abd d L o

b

a

o

F/m) / ln(

21

) / (ln

2

0 22 abd d

abC

b

a

a

b

C

L Z o ln

2

x

y

,

a

b

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Calculation the Inductance:

For this tutorial we will verify the above equations for a popular co-axial

transmission line used for cable television distribution in North America. Thespecifications for the transmission line are as follows:

Type: RG59A/UCenter conductor: copper with diameter 0.58mmDielectric material: polyethylene with a relative dielectric constant of 2.25Outer conductor: copper braid with diameter 4.5 mm

Jacket: PCVII with diameter 6.1 mm

pF/m15.68

)29.0 / 82.1ln(

)25.2)(10854.8(2

) / ln(

212

ab

C

nH/m3.367)29.0 / 82.1ln(2

104) / ln(

2

7

ab L o

4.731015.68

103.36712

9

C

L Z

o

Calculation the Capacitance:

Calculation the Impedance:

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Conventions used in this Tutorial

• Main Procedures are presented in Bold. Detailed procedures and indicated by a numbered listafter the main procedure. Notes are in italics .

• Bold type is used for the following:

– Keyboard entries that should be typed in their entirety exactly as shown. For example,―Inf_GND‖ means to type the Inf followed by a underscore then type GND

– On screen prompts and messages, names of options and text boxes, and menu commands.

For example, click Edit>Select>By name – Labeled keys on the computer keyboard. For example, ―Press Enter‖

• Italic type is used for the following:

– Emphasis

– Keyboard entries when a name or variable muse be typed in place of words in italics. Forexample, ―copy file name ‖ means to type the word copy, to type a space, and then to type a

file name.

• The plus (+) sign is used between keyboard keys to indicate that you should press the keys at thesame time. For example, ―Press ctrl+u‖ means to press the ctrl key and the u key at the sametime.

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4. Overview of Maxwell 2D

Follows this general procedure when using the simulator to solve 2D problems:1. Use the Solver command to specify which of the following electric or magnetic field quantities tocompute:

– Electrostatic , Magnetostatic , Eddy Current, DC Conduction, AC Conduction , Eddy Axial

2. Use the Drawing command to select one of the following model types:

– XY Plane Visualizes cartesian models as sweeping perpendicularly to the cross-section.

– RZ Plane Visualizes axisymmetric models as revolving around an axis of symmetry in the cross-

section. 3. Use the Define Model command to access the following options:

– Draw Model Allows you to access the 2D Modeler and draw the objects that make up the geometricmodel.

– Group ObjectsAllows you to group discrete objects that are actually one electrical object. For instance, twoterminations of a conductor that are drawn as separate objects in the cross-section can be grouped torepresent one conductor.

– Couple Model Allows you to define thermal coupling for a project.

Note: The commands shown on the Executive Commands menu must be chosen in the sequence in whichthey appear. For example, you must first create a geometric model with the Define Model command beforeyou specify material characteristics for objects with the Setup Materials command. A check mark appearson the menu next to the completed steps.

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4. Overview of Maxwell 2D (con’t)

4. Use the Setup Materials command to assign materials to all objects in the geometric model5. Use the Setup Boundaries/Sources command to define the boundaries and sources for the

problem

6. Use the Setup Executive Parameters command to instruct the simulator to compute thefollowing special quantities:

– Matrix (capacitance, inductance, admittance, impedance, or conductance matrix, depending on the selectedsolver)

– Force

– Torque

– Flux Linkage

– Current Flow

7. Use the Setup Solution Options command to specify how the solution is computed.

8. Use the Solve command to solve for the appropriate field quantities. For electrostatic problems,the simulator computes, the electric potential, from which it derives E and D.

9. Use the Post Process command to analyze the solution, as follows:

– Plot the field solution. Common quantities (such as , E, and D) are directly accessible from menus andcan be plotted a number of ways. For instance, you can display a plot of equipotential contours or you cangraph potential as a function of distance.

– Use the calculators. The post processor allows you to take curls, divergences, integrals, and cross and dotproducts to derive special quantities of interest.

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5. Creating the RG59 Project

Access the Maxwell Control Panel To access Maxwell SV, you must first open the Maxwell Control Panel, which allows you to create and

open projects for all Ansoft projects.To start the Maxwell Control Panel, do one of the following:• Use the Start menu to select Programs>Ansoft>Maxwell SV.• Double-click the Maxwell SV icon.The Maxwell Control Panel appears. If not, refer to the Ansoft installation guides for assistance.

Access the Project ManagerThe Project Manager enables you to create and manage Ansoft

products. You can add new project directories, createprojects in existing directories, and rename and copyprojects.

To access the Project Manager, click PROJECTS from theMaxwell Control Panel. The Project Manager appears.

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Create a Project Directory

1. Click Add from the Project Directories list.The Add a new project directory windowappears, listing the directories andsubdirectories.

2. Type the following in the Alias field:my_projects

3. Select Make New Directory near the bottomof the window. By default, my_projects appears in this field.

4. Click OK

The my_projects directory is now createdunder the current default project directory.You are now returned to the ProjectManager, and my_projects now appears inthe Project Directories list.

5. Creating the RG59 Project (con’t)

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Create a Project

1. In the Project Directory box click my-projects to access the newly created directory2. In the Project Manager, click New in the Projects list. The Enter project name and selectproject type window appears.

3. Type RG59 in the name field.

4. Make sure that Maxwell SV Version 9 appears in the Type field. If not select it.

5. You may enter your name in the Created By field.

6. Clear Open project upon creation. You do not wish to automatically open the project at this

time, so you may enter project notes first.7. Click OK.

This information should now be displayed in the corresponding field on the Projects list.

Save Project Notes

It is a good idea to save notes about your new project for future reference. To enter notes:

1. Leave Notes selected by default.

2. Click in the area under the Notes option. This places an I-beam cursor in the upper-left corner ofthe Notes area.

3. Enter your notes: This is an analysis of a RG59 75 ohm co-axial cable .

4. Click Save Notes.

You are now ready to open the new Maxwell SV project and run the software.


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Open the RG59 Project and Run the Simulator

Make sure that the RG59 project is highlighted in the Projects list. To run Maxwell SV, click Open inthe project area. The Maxwell SC Executive Commands menu appears.


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Executive Commands Window

• The Executive Commands window is divided into three sections: the Executive Commandsmenu, display area, and the solution monitoring area.

• The Executive Commands menu acts as a entry point to each step of creating and solving themodel problem. You select each module through the Executive Commands menu, and thesoftware returns you to this menu when you are finished. You also view the solution processthrough this menu.

• The display area shows either the project’s geometry in a model window or the solution to the

problem once a solution has been generated. The commands along the bottom of the windowallow you to change your view of the model:

– Zoom In Zooms in on an area of the window, magnifying the view.

– Zoom Out Zooms out of an area, shrinking the view.

– Fit All Changes the view to display all items in the window. Items will appear as large as possiblewithout extending beyond the window.

– Fit Drawing Displays the entire drawing space.

– Fill Solids Displays objects as solids rather than outlined objects. Toggles with Wire Frame.

– Wire Frame Displays objects as wire-frame outlines. Toggles with Fill Solids.• The buttons along the top of the window are used when you are generating and analyzing a

solution.

• Solution Monitoring Area displays solution profile and convergence information while theproblem is solving.

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Specify the Solver Type

Before you start drawing your model, you need to specify which field quantities to compute. Bydefault, Electrostatic appears as the Solver type. Because you will be solving an electrostaticproblem, leave this type as it appears.

Specify the Drawing Plane

The RG59 model you will be drawing is actually the XY cross-section of a the coaxial that extendsinto the z-direction. This is known as a cartesian or XY plane model. By default, XY Planeappears as the Drawing plane. Because the model you will be creating is in the XY plane, leavethis type as it appears.

Now, you are ready to draw the model.

Access the 2D Modeler

To draw the geometric model, use the 2D Modeler, which allows you to create 2D structures.

To access the 2D Modeler, click Define Model>Draw Model. The 2D Modeler appears.

• The main 2D Modeler window contains the Drawing Region, the grid-covered area where youdraw the objects that make up your model. This main window in the 2D Modeler is called the

project window. A project window contains the geometry for a specific project and displays theproject’s name in its title bar. Subwindows are allowed within the project window so you can have

several view of the model available at one time.

• Note the Status Bar on the bottom of the screen. U and V displays the coordinates of the mouse’s

position and allows you to enter the coordinates using the keyboard.

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Check the Drawing Units

Make sure the drawing units are specified to millimeters. If not, click Model>Drawing Units andselect mm and click OK.

You are now ready to draw the objects that make up the geometric model

Set the Drawing Size

Given that the largest dimension of the cable is 6.1 mm, modify the drawing size so that thegeometry is neither to large or small for the window.

1. Select Model>Drawing Size

2. For Minima enter X: -5 Y: -5 and for Maxima enter X: 5 Y: 5. You may Tab between boxes onwhen entering the coordinates.

3. Click OK

Keyboard Entry

In the following sections, the radius of the co-axial cable lie between grid points. You can positionthese points in one of two ways:

– Change the grid spacing so that the object’s dimensions lie on grid points.

– Use ―keyboard entry‖ — that is, enter the coordinates directly into the U and V fields in the status bar.

We will use keyboard entry to enter some of the dimensions of the geometry.

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Create the center conductor

To create the circle to represent the center conductor.

1. Click Object>Circle>2 point. After you do, the pointer changes to crosshairs.

2. Select the center of the circle as follows:• Move the crosshairs to the point on the grid where the U- and V- coordinates are (0,0). Remember that the

coordinates of the cursor’s current location are displayed in the status bar.

• Click the point to select it.

3. To select a point on the circumference of the circle, use keyboard entry because, 0.29 mm fallsbetween grid points

– Double-click in the Rad field in the status bar. – Type 0.29 and press Return or click Enter in the status bar. After you do, the New Arc/Circle windowopens.

4. In the Number of Segments field, enter 180. The Angular increment automatically changes to2.

5. Click OK. The New Object window appears.

6. Assign a Name and Color to the newly created object. By default, the object is assigned object1 and the color red. Be sure to change the name of the object to indicate its function and to assign

a different color to object. This is important in identifying the object later.• Type center in the name field. Do not press return.

• Click the solid red square next to Color. A palette of 16 colors appears.

• Click one of the blue boxes.

• Click OK.

The object now appears in the drawing region. It is blue and has the name center.

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Create the dielectric


2. Select the center of the circle as follows:

• Move the crosshairs to the point on the grid where the U- and V- coordinates are (0,0). Remember that thecoordinates of the cursor’s current location are displayed in the status bar.



– Double-click in the Rad field in the status bar.

– Type 1.82 and press Return or click Enter in the status bar. After you do, the New Arc/Circle windowopens.



6. Assign a Name and Color to the newly created object..

• Type dielectric in the name field. Do not press return.


• Click one of the green boxes.

• Click OK.

The object now appears in the drawing region. It is green and has the name dielectric.

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Create the outer conductor


2. Select the center of the circle as follows:

• Move the crosshairs to the point on the grid where the U- and V- coordinates are (0,0). Remember that thecoordinates of the cursor’s current location are displayed in the status bar.



– Double-click in the Rad field in the status bar.

– Type 2.25 and press Return or click Enter in the status bar. After you do, the New Arc/Circle windowopens.



6. Assign a Name and Color to the newly created object..

• Type outer in the name field. Do not press return.


• Click the black boxes.

• Click OK.

The object now appears in the drawing region. It is black and has the name outer.

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Save the Geometry

Maxwell SV does not automatically save your work. Therefore, periodically save the geometrywhile you are working on it. To save the RG59 model now:

• Click File>Save. The pointer changes to a watch while the geometry is written to files. When thepointer reappears, the geometry has been saved in a disk file in the RG59.pjt directory.

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Exploring the 2D Modeler

Try using the Window>Change View>Zoom In command to zoom in on certain object.

To return to a pre-zoomed view of the drawing region Click Window>Change View>Fit Drawing.After you do, all objects are displayed in the drawing region.

Exit the 2D Modeler

To exit the 2D Modeler:

1. Click File>Exit. Ig you have made further modifications to the model, a window appears,

prompting you to save the changes before exiting.2. Click Yes. The geometry is saved to a disk file in the RG59.pjt project directory, and the

Executive Commands window appears. A check mark appears next to Define Model, indicatingthat this step has been completed.

Note: Because none of the objects are electrically connected at any point in a 3D rendering of themodel, you do not need to use the Define>Model>Group Objects command.

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6. Defining Materials and Boundaries

Set up Materials

To define the material properties for the objects in the geometric model, you must:

– Assign the properties of a copper to both conductors.

– Assign polyethylene to the dielectric material to the co-axial cable.

In general, to assign materials to objects:

1. If necessary, add materials with the properties of the objects in your model to the materialdatabase.

2. Assign a material to each object in the geometric model as follows:

– Select the object(s) for which a specific material applies. – Select the appropriate material.

– Click Assign to assign the selected material to the selected object(s).

In this tutorial, you do not have to add materials to the material database — all materials that youwill need are already included in the global material database.

Note: You must assign a material to each object in the model.

Access the Material Manager

Click Setup Materials.Assign a Material to the Dielectric

1. Click dielectric in the Object list, or click on the substrate object in the geometric model.

2. Click polyethylene in the Material list and Click Assign.

polyethylene now appears next to dielectric in the Object list.

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6. Defining Materials and Boundaries (con’t)

Assign a Material to the center and outer conductor

1. Click Multiple Select at the top of the window, if it is not already enabled.

2. Do one of the following to select center and outer from the Object list:

– Press and hold down Ctrl, and then click each of the object names.

– Press and hold down Shift, and then drag the pointer over the object names.

– To deselect an object, click it.

3. Click copper in the Material list and click Assign.

The conductors have now been assigned the properties of copper. Copper appears next tothose objects’ names.

Assigning Materials to the Background

The background object is the only object that is assigned a material by default. Include it as partof the problem region in which to generate the solution. When a material name — such asvacuum — appears next to background in the Objects list, the background object is includedas part of the solution region.

Because the model is assumed to be surrounded by a vacuum, accept the default material,vacuum, for the background.

Exit the Material Manager

1. Click Exit at the bottom-left of the Material Setup window. A window with the following promptappears: Save changes before closing?

2. Click Yes. You are returned to the Executive Commands window.

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Set Up Boundaries and Sources

After setting material properties, the next step in creating the co-axial model is to define boundaryconditions and sources.

Initially, all object surfaces are defined as natural boundaries, which simply means that E iscontinuous across the surface. All outside edges are defined as Neumann boundaries, whichmeans that the tangential components of E and the normal components of D are continuousacross the surface.

Display the 2D Boundary/Source Manager

Click Setup Boundaries/Sources. The 2D Boundary/Source Manager appears.

Types of Boundary Conditions and Sources

The type of boundary condition and sources that you will use in this tutorial is

a Voltage source. This specifies the voltage on an object in the model. The electric scalarpotential is set to a constant value, forcing the electric field to be perpendicular to the objects’

surfaces

There are outer possible boundary conditions, such as a Balloon boundary. This can only be

applied to the outer boundary, and models the case in which the structure is infinitely far awayfrom all other electromagnetic sources. Since our outer conductor is set at 0 volts, we will notneed to use this boundary condition.

Center conductor: This surface is to be set to 1 volt.

Outer conductor and outer surface of the dielectric: This surface is to be set to zero volts..

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Set the Voltage on the center conductor

There are several ways to select objects’ surfaces, but in this sample problem you will select each

object individually. As a result, the object’s surface will be selected. There are also several waysto assign values to surfaces.

1. Zoom in on the center conductor.

– Click Window>Change View>Zoom In. The cursor changes to crosshairs.

– Select a window that encloses the center conductor.

2. Click Edit>Select>Object>By Clicking. The menu bar commands are disabled, and the systemexpects you to select an item by clicking on it in the model.

3. Click on the center conductor. After you do, it is highlighted.

4. Right-click anywhere in the display area to stop selecting objects. The commands in the menu barare enabled again, and the center conductor is the only highlighted object on the screen. Now youare ready to assign a voltage to the surface of this conductor.

5. Click Assign>Source>Solid. The name source1 appears in the Boundary list, and NEWappears next to it, indicating that it has not yet been assigned to an object or surface.

6. In the properties section below the model diagram, verify that Voltage is selected.7. Change the Value field to 1 V.

8. Click Assign. A value of one volt has now been specified for the left microstrip, and voltagereplaces NEW next to source1 in the Boundary list.

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Set the Voltage on the Outer Conductor

1. Click Window>Change View>Fit All to make all objects appear as large as possible in thesubwindow.

2. Click Edit>Select>Object>By Name. A prompt with the following message appears: Enter itemname/regular expression.

3. Enter outer, and click OK. The outer conductor appears highlighted in the model.

4. Click Assign>Source>Solid. The name source2 appears in the Boundary list, and the sourceinformation appears below the model.

5. Verify that Voltage is selected. Verify that the Value field is set to 0 volts.6. Click Assign. Now the voltage has been specified for the ground plane, and voltage replaces

NEW next to source2.

Set the boundary on the dielectric

1. Click Edit>Select>Object>By Name. Enter dielectric, and click OK. The outer conductorappears highlighted in the model.

2. Click Assign>Boundary>Value . The name value1 appears in the Boundary list, and the source

information appears below the model.3. Verify that the Value field is set to 0 volts. Click Assign. Now the voltage has been specified for

the ground plane, and voltage replaces NEW next to value1.

Exit the Boundary Manager

1. Click File>Exit. Save changes when prompted. You are returned to the Executive Commands.

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7. Generating a Solution

Set Up a Matrix Calculation

After solving the problem, you will use a capacitance matrix to calculate the capacitance. Thematrix calculation is defined using the Setup Executive Parameters command.

To set up the capacitance matrix calculation:

1. Click Setup Executive Parameters>Matrix from the Executive Commands menu. TheCapacitance Matrix Setup window appears.

2. Assign the center conductor as a signal line.

– Click center in the Object list.

– Select the Include in matrix check box.

– Select Signal Line, and click Assign.

3. Assign the outer conductor as ground:

– Click outer in the Object list.

– Select the Include in matrix check box.

– Select Ground, and click Assign.

4. Click Exit to close the Capacitance Matrix Setup window. A message appears, asking if youwant to save your changes. Click Yes.

You return to the Executive Commands window. A check mark now appears next to the SetupExecutive Parameters and Setup Executive Parameters>Matrix commands.

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7. Generating a Solution (con’t)

Access and Modifying the Setup Solution Menu

Maxwell 2D automatically assigns a set of default solution criteria after you assign boundaries andsources. As a result, a check mark automatically appears next to the Setup Solution Optionsbutton on the Executive Commands menu after you use the Setup Boundaries/Sourcescommand. To access and set up the solution options, click Setup Solution Options. The SolveSetup window appears.

Specify the Starting Mesh

For this problem, you will use the coarse mesh that is first generated when you begin the solutionprocess. This is referred to as the initial mesh. Leave the Starting Mesh option set to Initial.

Specify the Solver Residual

The solver residual specifies how close each solution must come to satisfying the equations thatare used to generate the solution. For this model, the default setting is sufficient. Leave the SolverResidual field set to the default.

Specify the Solver Choice

You can specify which type of matrix solver to use to solve the problem. In the default Autoposition, the software makes the choice. The ICCG solver is faster for large matrices, butoccasionally fails to converge (usually on magnetic problems with high permeabilities and smallair-gaps). The Direct solver will always converge, but is much slower for large matrices. In theAuto position, the software evaluates the matrix before attempting to solve; if it appears to be ill-conditioned, the Direct solver is used, otherwise the ICCG solver is used. If the ICCG solver failsto converge while the solver choice is in the Auto position, the software will fall back to the Directsolver automatically. Leave the Solver Choice option set to Auto.

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Specify the “Solve for” Options

The Solve for options tell the system what types of solutions to generate.

Leave both the Fields and Parameters check boxes selected, to solve for both fields andparameters in this example.

Specify the Adaptive Analysis Settings

Set the adaptive refinement settings.

1. Leave Adaptive Analysis selected.

This allows the simulator to solve the problem iteratively, refining the regions of the mesh in which

the largest error exists. Refining the mesh makes it more dense in the areas of highest error,resulting in a more accurate field solution.

2. Leave Percent refinement per pass to 15.

This causes 15 percent of the mesh with the highest error energy to be refined during eachadaptive solution (that is, each solve-refine cycle).

3. Change Number of requested passes and Percent error set to 25 and 0.1, respectively.

After each iteration, the simulator calculates the total energy of the system and the percent of this

energy that is caused by solution error. It then checks to see if the number of requested passeshas been completed, or if the percent error and the change in percent error between the last twopasses match the requested values. If either of the criteria have been met, the solution process iscomplete and no more iterations are done.

4. To save your changes and exit the Solve Setup window, click OK. You return to the ExecutiveCommands menu.

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Generate the Solution

Now that you have set up the solution parameters, the problem is ready to be solved. To executethe solution, click Solve. The solution process begins, and the following actions occur:

– The system creates the initial finite element mesh for the microstrip structure. A bar labeled Making InitialMesh appears in the Solution Monitoring box at the bottom of the screen. It shows the system’s progress

as it generates the mesh.

– A button labeled Abort appears next to the progress bar and remains there throughout the entire solutionprocess. You can click it to stop the solution process.

– A bar labeled Setting up solution files appears.

– After the system makes the initial mesh, the electrostatic field solution process begins

Monitoring the Solution

The following two monitoring bars alternate in the Solution Monitoring area at the bottom.

Solving Fields: Displayed as the simulator computes the field solution. After computing asolution, identifies the triangles with the highest energy error.

Refining Mesh: Displayed as the simulator refines the regions of the finite element mesh with the

highest error energy. Since you specified 15% as the portion mesh to refine, the simulator refinestriangles with the top 15% error.

To monitor the solution after a few adaptive passes are completed, click the Convergence button.

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Solution Criteria

Information about the solution criteria is displayed on the left side of the convergence display.

Number of passes: Displays how many adaptive passes have been completed and still remain.

Target Error: Displays the percent error value that was entered using the Setup SolutionOptions command — in this case, 0.1 percent.

Energy Error: Displays the percent error from the last completed solution — in this case, 0.0036percent. Allows you to see at a glance whether the solution is close to the desired error energy

Delta Energy: Displays the change in the percent error between the last two solutions — in this

case, 0.0101 percentCompleted Solutions

Information about each completed solution is displayed on the right side of the screen.

Pass: Displays the number of the completed solutions.

Triangles: Displays the number of triangles in the mesh for a solution.

Total Energy (J): Displays the total energy of a solution.

Energy Error (%): Displays the percent error of the completed solutions.

Completing the Solution Process

When the solution is complete, a window with the following message appears: Solution Process is complete .

Click OK to continue.

You are now ready to view the final convergence for the completed solution.

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8. Analyzing the Solution

Access the Post Processor

Click Post Process. The 2D Post Processor appears.

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8. Analyzing the Solution (con’t)

Plot the Voltage in the Dielectric

1. Click Plot>Field. The Create New Plot window appears.

2. Click phi in the Plot Quantity list.

3. Click Surface dielectric in the On Geometry list.

4. Click –all- on the In Area list.

5. Click OK.

The Scalar Surface Plot window appears.

6. Verify that the Show color Key and Filled

check boxes are selected.

7. Change the number of division to 21.

8. Click OK.

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Plot the Mesh in the Dielectric

1. Click Edit>Select and select dielectric from the list that appears. Click OK.

2. Plot>Mesh. The Mesh Plot window appears.

3. Name the plot dielectric_mesh and select Wire Frame radio button..

4. Click OK.

The Mesh Plot appears.

5. Use the zoom function to

enlarge the details of the plot.

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Calculate the capacitance of the Co-axial Cable

You have probably noticed that after you have generated the solution and selected

Solutions>Matrix in the Executive Commands window, Maxwell 2D calculated the capacitanceof the structure to be 6.815e-11 F/m. If not, go back and check the result.

In this exercise, we will calculate the capacitance of the co-axial cable from the electric field usingthe integral equation presented earlier in the tutorial. To do so, access the 2D field calculator byclicking Data>Calculator.

The calculator is divided into two parts: the top portion displays the contents of the register stack,and the bottom portion displays the functions of the calculator. Note that the calculator already

contains the results of the previous field plot, which is the voltage plotted on the dielectric.

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Compute the Capacitance

The first step in computing capacitance is to load the E-field and the D-field into the register stack.

To load the E-field and the D-field:

1. If any entries remain on the calculator stack, click Clear to remove them. A message appears,asking you to confirm your command. Click Yes. The calculator stack is cleared of any existingvalues.

2. Click Qty/E from the Input column to load the electric field vector E into the top register of thecalculator first. After E is loaded, the top register appears as follows: Vec: <Ex,Ey,0>

3. Click Qty/D to load the electric flux density vector D into the top register of the calculator. The topregister appears as follows: Vec: <Dx,Dy,0>

4. To calculate the dot product, click Dot from the Vector column of calculator commands. After thedot product has been calculated, the top register of the calculator appears as follows:

Scl: Dot(<Ex,Ey,0>, <Dx,Dy,0>)

5. Click Geom>Surface from the Input column. The Select Surface window appears.

6. Select dielectic from the list, and click OK.

7. Click the integrate button from the Scalar column.Scl: Integrate(ObjectFaces(dielectric),Dot(<Ex,Ey,0>,<Dx,Dy,0>))

8. Click Eval from the Output column. The register now contains the value of the integral.

9. Since the voltage Vo was set to 1 V. This is the capacitance of the Co-axial Cable.

The register should read 6.81161143942141e-011.

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Compute the Inductance and Characteristic Impedance

1. Click the Push button to duplicate the value of the capacitance on the stack.

2. Select Const>Epsi0 from the Input column. The permittivity of free space appears on the stack.

3. Click Num>Scalar from the Input column. The Scalar Constant window appears. Enter 2.25,the relative dielectric constant of polyethylene, in the Scalar Value field. Click OK.

4. Click the multiplication button from the General column.

5. Select Const>Mu0 from the Input column. The permeability of free space appears on the stack.

6. Click the multiplication button from the General column.

7. Click the Exch button to exchange the top two entries of the stack.8. Click the division button from the General column.

This top value is the inductance of the co-axial cable. L = 3.6725910681998e-007

9. Click the Exch button to exchange the top two entries of the stack.

10. Click the division button from the General column.

11. Click the square root button from the Scalar column.

The stack now contains the Characteristic Impedance of the Co-axial cable.Zo = 73.4 ohms.

12. To exit the calculator, Click Done for the bottom of the calculator. You return to the 2D PostProcessor.

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9. Concluding the Session

Exit the Post Processor

1. Click File>Exit from the post processor. A window with the following prompt appears:

Exit Post Processor?

2. Click Yes. The Executive Commands window appears.

Exit Maxwell 2D

1. Click Exit from the bottom of the Executive Commands menu. The following prompt appears:

Exit Maxwell 2D?

2. Click Yes. The Executive Commands window closes, and the Project Manager reappears.

Exit the Maxwell Software

1. To exit the Project Manager, click Exit. The Project Manger closes, and the Control Panelreappears.

2. Click EXIT. A window with the following prompt appears:

Exit Maxwell?

3. Click Yes. You return to Microsoft Windows.

This tutorial is now concluded.

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10. Further Reading and References

Design a 300 ohm Two Wire Transmission Line

Originally, a 300 ohm Two Wire Line was used to carry the television signal from the antenna to

the set inside the house. The geometry is given below.

From field analysis, the capacitance is calculated to be

Using a 20 gauge copper wire (diameter of 32 mils), design a 300 ohm transmission line (in air)and verify its performance with Maxwell 2D.

In Maxwell 2D, you will need to include the background in the analysis. You wish to read aboutthe balloon boundary condition in the help section.

Once complete, calculate the characteristic impedance using the 2D field calculator. Also, plot thevoltage around the structure with one wire at 1 volt and the other at -1 volt.

a a

d

)2 / (cosh1

ad C

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10. Further Reading and References (con’t)

Electromagnetics

– N.N. Rao, Elements of Engineering Electromagnetics , Pearson Prentice Hall, Upper Saddle

River, NJ, 2004

– W.H. Hayt and J.A. Buck, Engineering Electromagnetics, McGraw-Hill, New York, NY, 2006

Computational Electromagnetics

– A. Taflove and S. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method , Artech House, Boston, MA, 2000

– J.Jin, The Finite Element Method in Electromagnetics , 2nd edition, Wiley, New York, NY,2002

– P.P Silvester and R.L. Ferrari, Finite Elements for Electrical Engineers , 3rd edition,Cambridge University Press, Cambridge, 1996

RF/wireless engineering

– D.M. Pozar, Microwave Engineering , 3rd edition, Wiley, New York, NY, 2005

This work is partially supported by the

National Science Foundation grant

Division of Undergraduate Education Course

Curriculum and Laboratory Improvement (CCLI)

Award Number 0536197.

Documents

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