Transmission Parameters of an Infinitely Long Co-Axial Cable

Transmission Parameters of an Infinitely Long Co-Axial Cable
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 an Infinitely Long Co-Axial Cable Prepared By: Dr. Vladimir A Labay, Department of Electrical and Computer Engineering Gonzaga University, Spokane, Washington Telephone: Copyright 2006 Reference Text: Elements of Engineering Electromagnetics, Sixth Edition by N.N. Rao Software: Maxwell 2D Student Version Estimated time to complete This tutorial: 90 minutes

Tutorial and Educational Objectives
The educational goal is to provide undergraduate engineering students with understanding of a specific engineering topic and FE theory, along with an ability to apply commercial FE software to typical engineering problems. The educational goal will be accomplished through four educational objectives based on Bloom’s Taxonomy and ABET Criterion 3 as follows: Engineering Topics (Comprehension; 3a, 3k). Understand the fundamental basis of engineering topics through the use of finite element computer models. FE Theory (Comprehension; 3a). Understand the fundamental basis of FE theory. FE Modeling Practice (Application; 3a, 3e, 3k). Be able to implement a suitable finite element model and construct a correct computer model using commercial FE software. -- integrates 3 and 4 above. FE Solution Interpretation and Verification (Comprehension and Evaluation; 3a, 3e). Be able to interpret and evaluate finite element solution quality, including the importance of verification.

Table of Contents Overview of Computational Electromagnetics
Finite Element Method in EM Problem Definition and Background Overview of Maxwell 2D Creating the RG59 Project Defining Materials and Boundaries Generating a Solution Analyzing the Solution Concluding the Session Further Reading and References

1. Overview of Computational Electromagnetics
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) r: the Volume Charge Density (C/m3) m: is the Permeability of the medium (H/m) e: the Permittivity of the medium (F/m)

1. Overview of CEM (con’t)
Maxwell’s Equations Faraday’s Law: Ampère’s Circuital Law: Gauss’ Laws: 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)

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

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 of Maxwell’s equations. The FE method applied in electromagnetics is a full-wave technique. Full-wave techniques have the potential to be the most accurate of all numerical 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!

Central to all CEM techniques is the idea of discretizing some unknown electromagnetic 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 a number 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 causes limitations that one must keep in mind

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 the electromagnetic 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?

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 did not 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’s Equations 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

2. Finite Element Method in EM (con’t)
The basic concept of the FE method is that although the behavior of a function may be complex when viewed over a large region, a simple approximation may be sufficient 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 Artificial absorbing region (box surrounding the antenna) Antenna Patch Infinite Ground Plane Substrate Material

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 of unknowns) 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

The FE method solution consists of essentially four steps: Discretizing the region of interest into a mesh, that is, finite elements Deriving the governing equations for the individual finite elements Relating the individual finite elements to the assembly of the elements Obtaining and solving the system of equations for the unknown quantity Commercial FE software packages for electromagnetics When using a commercial FE software package, these four steps are done internally with little intervention 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

MAXWELL 2D Maxwell is a high-performance full-wave electromagnetic field simulator for arbitrary 2D electromagnetic problems modeling that takes advantage of a Windows graphical user interface. It integrates simulation, visualization, 2D modeling, and automation in an easy-to-learn environment. The student version of this software is a available at 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 finite elements 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 special quantities of interest, including force, torque, inductance, capacitance, and power loss. You can select any of the following solution types: Electrostatic, Magnetostatic, Electrostatic, Eddy Current, DC Conduction, AC Conduction, Eddy Axial. The student version does not contain thermal, transient, or parametric capabilities.

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 parameters and field plot This tutorial leads you step-by-step through the calculation of a co-axial transmission line . 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

3. Problem Definition and Background (con’t)
The lumped-element circuit model for a Transmission Line As shown to the right, a transmission line is schematically represented as a two-wire line. This due to the fact that transverse electromagnetic wave propagation or TEM waves on a transmission line requires at least two conductors A infinitesimal length, see (b), can be modeled as a lumped-element circuit where R, L, G, C are per unit length quantities The Lossless Line In many line, such as the co-axial transmission line, the loss of the line is very small and can be neglected for the analysis. Thus, setting R and G equal to zero, the TEM wave propagation can be completely described in terms of L and C. However, L and C are not independent from each other. They are related by:

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

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 magnetic and 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

General Procedure for analyzing a TEM line Solve Laplace’s equation for the cross-sectional area of the transmission line. The solution will contain unknown constants. Find the constants by applying the boundary conditions for the known voltages on the conductors. Compute the electric field from the following equations Compute the magnetic field from the following equation The capacitance or inductance may now be calculated. Recall that only one needs to be determined as they are related by

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 at Vo, 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

For this tutorial we will verify the above equations for a popular co-axial transmission line used for cable television distribution in North America. The specifications for the transmission line are as follows: Type: RG59A/U Center conductor: copper with diameter 0.58mm Dielectric material: polyethylene with a relative dielectric constant of 2.25 Outer conductor: copper braid with diameter 4.5 mm Jacket: PCVII with diameter 6.1 mm Calculation the Inductance: Calculation the Capacitance: Calculation the Impedance:

Conventions used in this Tutorial
Main Procedures are presented in Bold. Detailed procedures and indicated by a numbered list after 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. For example, “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 the same time. For example, “Press ctrl+u” means to press the ctrl key and the u key at the same time.

4. Overview of Maxwell 2D Follows this general procedure when using the simulator to solve 2D problems: Use the Solver command to specify which of the following electric or magnetic field quantities to compute: Electrostatic , Magnetostatic , Eddy Current, DC Conduction, AC Conduction , Eddy Axial 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. 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 geometric model. Group Objects Allows you to group discrete objects that are actually one electrical object. For instance, two terminations of a conductor that are drawn as separate objects in the cross-section can be grouped to represent 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 which they appear. For example, you must first create a geometric model with the Define Model command before you specify material characteristics for objects with the Setup Materials command. A check mark appears on the menu next to the completed steps.

4. Overview of Maxwell 2D (con’t)
Use the Setup Materials command to assign materials to all objects in the geometric model Use the Setup Boundaries/Sources command to define the boundaries and sources for the problem Use the Setup Executive Parameters command to instruct the simulator to compute the following special quantities: Matrix (capacitance, inductance, admittance, impedance, or conductance matrix, depending on the selected solver) Force Torque Flux Linkage Current Flow Use the Setup Solution Options command to specify how the solution is computed. 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. 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 and can be plotted a number of ways. For instance, you can display a plot of equipotential contours or you can graph potential as a function of distance. Use the calculators. The post processor allows you to take curls, divergences, integrals, and cross and dot products to derive special quantities of interest.

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 Manager The Project Manager enables you to create and manage Ansoft products. You can add new project directories, create projects in existing directories, and rename and copy projects. To access the Project Manager, click PROJECTS from the Maxwell Control Panel. The Project Manager appears.

5. Creating the RG59 Project (con’t)
Create a Project Directory Click Add from the Project Directories list. The Add a new project directory window appears, listing the directories and subdirectories. Type the following in the Alias field: my_projects Select Make New Directory near the bottom of the window. By default, my_projects appears in this field. Click OK The my_projects directory is now created under the current default project directory. You are now returned to the Project Manager, and my_projects now appears in the Project Directories list.

Create a Project In the Project Directory box click my-projects to access the newly created directory In the Project Manager, click New in the Projects list. The Enter project name and select project type window appears. Type RG59 in the name field. Make sure that Maxwell SV Version 9 appears in the Type field. If not select it. You may enter your name in the Created By field. Clear Open project upon creation. You do not wish to automatically open the project at this time, so you may enter project notes first. 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: Leave Notes selected by default. Click in the area under the Notes option. This places an I-beam cursor in the upper-left corner of the Notes area. Enter your notes: This is an analysis of a RG59 75 ohm co-axial cable. Click Save Notes. You are now ready to open the new Maxwell SV project and run the software.

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 in the project area. The Maxwell SC Executive Commands menu appears.

Executive Commands Window The Executive Commands window is divided into three sections: the Executive Commands menu, display area, and the solution monitoring area. The Executive Commands menu acts as a entry point to each step of creating and solving the model problem. You select each module through the Executive Commands menu, and the software returns you to this menu when you are finished. You also view the solution process through 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 window allow 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 possible without 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 the problem is solving.

Specify the Solver Type Before you start drawing your model, you need to specify which field quantities to compute. By default, Electrostatic appears as the Solver type. Because you will be solving an electrostatic problem, 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 extends into the z-direction. This is known as a cartesian or XY plane model. By default, XY Plane appears as the Drawing plane. Because the model you will be creating is in the XY plane, leave this 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 you draw 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 the project’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.

Check the Drawing Units Make sure the drawing units are specified to millimeters. If not, click Model>Drawing Units and select 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 the geometry is neither to large or small for the window. Select Model>Drawing Size For Minima enter X: -5 Y: -5 and for Maxima enter X: 5 Y: 5. You may Tab between boxes on when entering the coordinates. Click OK Keyboard Entry In the following sections, the radius of the co-axial cable lie between grid points. You can position these 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.

Create the center conductor To create the circle to represent the center conductor. Click Object>Circle>2 point. After you do, the pointer changes to crosshairs. 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. To select a point on the circumference of the circle, use keyboard entry because, 0.29 mm falls between 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 window opens. In the Number of Segments field, enter The Angular increment automatically changes to 2. Click OK. The New Object window appears. 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.

Create the dielectric Click Object>Circle>2 point. After you do, the pointer changes to crosshairs. 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. To select a point on the circumference of the circle, use keyboard entry because, 1.82 mm falls between grid points 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 window opens. In the Number of Segments field, enter The Angular increment automatically changes to 2. Click OK. The New Object window appears. Assign a Name and Color to the newly created object.. Type dielectric 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 green boxes. Click OK. The object now appears in the drawing region. It is green and has the name dielectric.

Create the outer conductor Click Object>Circle>2 point. After you do, the pointer changes to crosshairs. 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. To select a point on the circumference of the circle, use keyboard entry because, 2.25 mm falls between grid points 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 window opens. In the Number of Segments field, enter The Angular increment automatically changes to 2. Click OK. The New Object window appears. Assign a Name and Color to the newly created object.. Type outer in the name field. Do not press return. Click the solid red square next to Color. A palette of 16 colors appears. Click the black boxes. Click OK. The object now appears in the drawing region. It is black and has the name outer.

Save the Geometry Maxwell SV does not automatically save your work. Therefore, periodically save the geometry while 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 the pointer reappears, the geometry has been saved in a disk file in the RG59.pjt directory.

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: Click File>Exit. Ig you have made further modifications to the model, a window appears, prompting you to save the changes before exiting. 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, indicating that this step has been completed. Note: Because none of the objects are electrically connected at any point in a 3D rendering of the model, you do not need to use the Define>Model>Group Objects command.

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: If necessary, add materials with the properties of the objects in your model to the material database. 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 you will 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 Click dielectric in the Object list, or click on the substrate object in the geometric model. Click polyethylene in the Material list and Click Assign. polyethylene now appears next to dielectric in the Object list.

6. Defining Materials and Boundaries (con’t)
Assign a Material to the center and outer conductor Click Multiple Select at the top of the window, if it is not already enabled. 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. Click copper in the Material list and click Assign. The conductors have now been assigned the properties of copper. Copper appears next to those objects’ names. Assigning Materials to the Background The background object is the only object that is assigned a material by default. Include it as part of the problem region in which to generate the solution. When a material name — such as vacuum — appears next to background in the Objects list, the background object is included as 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 Click Exit at the bottom-left of the Material Setup window. A window with the following prompt appears: Save changes before closing? Click Yes. You are returned to the Executive Commands window.

Set Up Boundaries and Sources After setting material properties, the next step in creating the co-axial model is to define boundary conditions and sources. Initially, all object surfaces are defined as natural boundaries, which simply means that E is continuous across the surface. All outside edges are defined as Neumann boundaries, which means that the tangential components of E and the normal components of D are continuous across 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 scalar potential 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 away from all other electromagnetic sources. Since our outer conductor is set at 0 volts, we will not need 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..

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 ways to assign values to surfaces. 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. Click Edit>Select>Object>By Clicking. The menu bar commands are disabled, and the system expects you to select an item by clicking on it in the model. Click on the center conductor. After you do, it is highlighted. Right-click anywhere in the display area to stop selecting objects. The commands in the menu bar are enabled again, and the center conductor is the only highlighted object on the screen. Now you are ready to assign a voltage to the surface of this conductor. Click Assign>Source>Solid. The name source1 appears in the Boundary list, and NEW appears next to it, indicating that it has not yet been assigned to an object or surface. In the properties section below the model diagram, verify that Voltage is selected. Change the Value field to 1 V. Click Assign. A value of one volt has now been specified for the left microstrip, and voltage replaces NEW next to source1 in the Boundary list.

Set the Voltage on the Outer Conductor Click Window>Change View>Fit All to make all objects appear as large as possible in the subwindow. Click Edit>Select>Object>By Name. A prompt with the following message appears: Enter item name/regular expression. Enter outer, and click OK. The outer conductor appears highlighted in the model. Click Assign>Source>Solid. The name source2 appears in the Boundary list, and the source information appears below the model. Verify that Voltage is selected. 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 source2. Set the boundary on the dielectric Click Edit>Select>Object>By Name. Enter dielectric, and click OK. The outer conductor appears highlighted in the model. Click Assign>Boundary>Value. The name value1 appears in the Boundary list, and the source information appears below the model. 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 Click File>Exit. Save changes when prompted. You are returned to the Executive Commands.

7. Generating a Solution Set Up a Matrix Calculation
After solving the problem, you will use a capacitance matrix to calculate the capacitance. The matrix calculation is defined using the Setup Executive Parameters command. To set up the capacitance matrix calculation: Click Setup Executive Parameters>Matrix from the Executive Commands menu. The Capacitance Matrix Setup window appears. 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. Assign the outer conductor as ground: Click outer in the Object list. Select Ground, and click Assign. Click Exit to close the Capacitance Matrix Setup window. A message appears, asking if you want to save your changes. Click Yes. You return to the Executive Commands window. A check mark now appears next to the Setup Executive Parameters and Setup Executive Parameters>Matrix commands.

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 and sources. As a result, a check mark automatically appears next to the Setup Solution Options button on the Executive Commands menu after you use the Setup Boundaries/Sources command. To access and set up the solution options, click Setup Solution Options. The Solve Setup window appears. Specify the Starting Mesh For this problem, you will use the coarse mesh that is first generated when you begin the solution process. 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 that are used to generate the solution. For this model, the default setting is sufficient. Leave the Solver Residual 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 Auto position, the software makes the choice. The ICCG solver is faster for large matrices, but occasionally fails to converge (usually on magnetic problems with high permeabilities and small air-gaps). The Direct solver will always converge, but is much slower for large matrices. In the Auto 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 fails to converge while the solver choice is in the Auto position, the software will fall back to the Direct solver automatically. Leave the Solver Choice option set to Auto.

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 and parameters in this example. Specify the Adaptive Analysis Settings Set the adaptive refinement settings. 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. Leave Percent refinement per pass to 15. This causes 15 percent of the mesh with the highest error energy to be refined during each adaptive solution (that is, each solve-refine cycle). 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 passes has been completed, or if the percent error and the change in percent error between the last two passes match the requested values. If either of the criteria have been met, the solution process is complete and no more iterations are done. To save your changes and exit the Solve Setup window, click OK. You return to the Executive Commands menu.

Generate the Solution Now that you have set up the solution parameters, the problem is ready to be solved. To execute the 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 Initial Mesh 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 solution process. 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 a solution, 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 refines triangles with the top 15% error. To monitor the solution after a few adaptive passes are completed, click the Convergence button.

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 Solution Options command — in this case, 0.1 percent. Energy Error: Displays the percent error from the last completed solution — in this case, percent. 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, percent Completed 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.

8. Analyzing the Solution
Access the Post Processor Click Post Process. The 2D Post Processor appears.

8. Analyzing the Solution (con’t)
Plot the Voltage in the Dielectric Click Plot>Field. The Create New Plot window appears. Click phi in the Plot Quantity list. Click Surface dielectric in the On Geometry list. Click –all- on the In Area list. Click OK. The Scalar Surface Plot window appears. Verify that the Show color Key and Filled check boxes are selected. Change the number of division to 21.

Plot the Mesh in the Dielectric Click Edit>Select and select dielectric from the list that appears. Click OK. Plot>Mesh. The Mesh Plot window appears. Name the plot dielectric_mesh and select Wire Frame radio button.. Click OK. The Mesh Plot appears. Use the zoom function to enlarge the details of the plot.

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 capacitance of 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 using the integral equation presented earlier in the tutorial. To do so, access the 2D field calculator by clicking 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.

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: 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 existing values. Click Qty/E from the Input column to load the electric field vector E into the top register of the calculator first. After E is loaded, the top register appears as follows: Vec: <Ex,Ey,0> Click Qty/D to load the electric flux density vector D into the top register of the calculator. The top register appears as follows: Vec: <Dx,Dy,0> To calculate the dot product, click Dot from the Vector column of calculator commands. After the dot product has been calculated, the top register of the calculator appears as follows: Scl: Dot(<Ex,Ey,0>, <Dx,Dy,0>) Click Geom>Surface from the Input column. The Select Surface window appears. Select dielectic from the list, and click OK. Click the integrate button from the Scalar column. Scl: Integrate(ObjectFaces(dielectric),Dot(<Ex,Ey,0>,<Dx,Dy,0>)) Click Eval from the Output column. The register now contains the value of the integral. Since the voltage Vo was set to 1 V. This is the capacitance of the Co-axial Cable. The register should read e-011.

Compute the Inductance and Characteristic Impedance Click the Push button to duplicate the value of the capacitance on the stack. Select Const>Epsi0 from the Input column. The permittivity of free space appears on the stack. 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. Click the multiplication button from the General column. Select Const>Mu0 from the Input column. The permeability of free space appears on the stack. Click the Exch button to exchange the top two entries of the stack. Click the division button from the General column. This top value is the inductance of the co-axial cable. L = e-007 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. To exit the calculator, Click Done for the bottom of the calculator. You return to the 2D Post Processor.

9. Concluding the Session
Exit the Post Processor Click File>Exit from the post processor. A window with the following prompt appears: Exit Post Processor? Click Yes. The Executive Commands window appears. Exit Maxwell 2D Click Exit from the bottom of the Executive Commands menu. The following prompt appears: Exit Maxwell 2D? Click Yes. The Executive Commands window closes, and the Project Manager reappears. Exit the Maxwell Software To exit the Project Manager, click Exit. The Project Manger closes, and the Control Panel reappears. Click EXIT. A window with the following prompt appears: Exit Maxwell? Click Yes. You return to Microsoft Windows. This tutorial is now concluded.

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 about the balloon boundary condition in the help section. Once complete, calculate the characteristic impedance using the 2D field calculator. Also, plot the voltage around the structure with one wire at 1 volt and the other at -1 volt.

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

Transmission Parameters of an Infinitely Long Co-Axial Cable

Similar presentations

Presentation on theme: "Transmission Parameters of an Infinitely Long Co-Axial Cable"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Transmission Parameters of an Infinitely Long Co-Axial Cable

Similar presentations

Presentation on theme: "Transmission Parameters of an Infinitely Long Co-Axial Cable"— Presentation transcript:

Similar presentations

About project

Feedback