Nelson 8.0Induction Heating – AC/RF

Magnetic Fields with Eddy Current

PO Box 13595, Albuquerque, NM 87192 U.S.A.Telephone: +1-505-220-3975E mail: techinfo@fieldp.com

Internet: http://www.fieldp.com

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Contents

1 Introduction 3

1.1 Program functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Learning Nelson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Finite-element solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Scripts and data files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Creating and analyzing a solution with Nelson 10

2.1 Mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Creating the Nelson script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Finite-element solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Solution analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Nelson solution reference 17

3.1 Nelson script format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Program control commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Commands for material properties . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Running Nelson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Format of the Nelson output file . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Eddy current and energy loss in drive coils 25

4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 TriComp analysis functions 31

5.1 File menu commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Plot menu commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.3 User specified contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4 Saving and loading views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.5 Analysis menu commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 Scan plot menu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.7 Vector tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.8 Analysis script commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6 Standard Nelson calculated quantities 48

6.1 Nelson variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.2 Standard analysis configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.3 Additional configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7 Building custom analysis configurations 54

7.1 Configuration file structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.2 Introduction to RPN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567.3 Operators, parameters and variables . . . . . . . . . . . . . . . . . . . . . . . . . 577.4 Building expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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1 Introduction

1.1 Program functions

Nelson is a finite-element tool to calculate AC or RF magnetic fields in conductive media.Applications include simulation of induction heating for material processing and the frequencyresponse of transformers or audio devices. The program handles any cylindrical or rectangulargeometry. You can define up to 250 regions to represent coils, conductors or ferrites. Theamplitude and phase of harmonic currents may be set individually. Materials are characterizedby values of the relative magnetic permeability and electrical conductivity. The graphicalanalysis functions of Nelson provide full information on the amplitude and phase of magneticflux density, electric field and current density throughout the solution regions. The programautomatically calculates and records power density and total current organized by regions. Theinformation can be ported directly to the TDiff program for thermal calculations. Nelson

makes screen and hardcopy plots of potential, field amplitude current density and resistiveenergy deposition.

The intuitive graphical-user-interface makes it easy to learn Nelson and to perform quickapplication setups. In contrast to other field solution software, Nelson features completedata transparency. Input operations are automatically recorded in text scripts that providedocumentation of your work. Scripts make it easy to reconstruct solutions and to share setupswith colleagues.

1.2 Learning Nelson

The size of this manual reflects the extensive capabilities of the Nelson package. It is notnecessary to read the entire document to address most applications. We have organized thechapters to help you get started quickly. You can investigate advanced topics as you gain moreexperience. The first step is to follow the walkthrough example in Chapter 2. This will giveyou an overview of the package organization and the solution process. Then, run some of theprepared examples supplied with the package. The examples give insights into solution optionsand may serve as templates for your own work. Here are suggestions for additional activities:

• Browse the following sections in this chapter to review some basics of RF/AC magneticfields with eddy currents. You will probably return to the material to resolve issues (suchas boundary conditions) as you get more involved in the programs.

• Scan the section on FPController (the component program launcher) in the Mesh

manual.

• Be sure to read Chapter 2 of the Mesh manual which reviews essential concepts forconformal meshes in finite-element solutions.

• Chapter 3 of the Mesh manual follows a walkthrough example to introduce constructionof a solution geometry with the interactive Mesh drawing editor. The exercise reviews

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fundamental tools you will need for you own simulations. Chapter 4 describes how toconvert a script into a conformal mesh.

• As you gain experience you will want to to take advantage of the full range of Mesh/Nelson

capabilities. Chapter 4 of the Mesh manual is a comprehensive reference on the Mesh

drawing editor. Chapter 5 covers processing, plotting and repairs of meshes. Chapter 3in this manual is a complete reference on Nelson control scripts. Chapter 5 gives infor-mation on the analysis functions common to all the TriComp programs. Capabilitiesinclude a variety of plots, interpolations of field values and integrals of field energy andother quantities. Chapter 6 reviews the calculated quantities defined by the standardNelson configuration files. Finally, Chap. 7 describes how to build a customized post-processor that includes your own calculated quantities.

• Additional chapters cover advanced techniques in Mesh. Chapters 6 and 7 of the Mesh

manual show how to make direct entries in scripts to invoke advanced control features.Chapter 8 describes how to create meshes from photographic and data images. Thisfeature is useful to model complex or irregular systems that are difficult to describe withsimple geometric specifications.

1.3 Finite-element solution procedure

An in-depth understanding of finite-element numerical methods is not necessary to use Nelson.Nonetheless, it is important to have a clear idea of fundamental concepts to create effectivesolutions. This section describes background material to understand the steps in an Nelson

solution.The term field indicates a quantity (scalar or vector) defined over a region of space. Ex-

amples of fields include the vector electric field E in an electrostatic solution, electric andmagnetic fields in an electromagnetic solution and the scalar temperature T in a thermal so-lution. Variations of field quantities are usually described by continuous partial differentialequations, such as the complex-value Poisson equation (Sect. 1.5). These equations can besolved directly by analytic methods if the system geometry and material properties are simple(for example, ferromagnetic layers with uniform µr between concentric cylinders). Analyticsolutions are extremely difficult in systems with asymmetric structures or non-linear materials.Furthermore, closed-form results are often expressed in terms of series expansions that mustbe evaluated numerically. For all but the simplest problems, it is usually quicker and moreaccurate to employ a direct numerical approach.

The fundamental issue in numerical field solutions is that digital computers cannot directlysolve continuous equations. On the other hand, computers are well suited to solving large setsof coupled linear equations. The goal of all numerical field methods (finite-difference, finite-element or boundary-element) is to convert the governing differential equations into a set ofcoupled linear equations. The solution of the linear equations approaches the results of thedifferential equation when the set becomes large.

The basis of the finite-element approach is to divide the full solution volume into a numberof small volumes, or elements . Here, the term small indicates that element dimensions aremuch less than the scale length for variations of field quantities. The division of the volume iscalled the computational mesh. Figure 1 shows the type of mesh used for the two dimensionalsolutions of Nelson. The figure defines three terms that will be used throughout this manual:

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Figure 1: Conformal triangular mesh – definition of terms.

• Element. Volume divisions of the system.

• Nodes. Points where elements intersect.

• Facets. Surfaces between two elements.

The elements in Fig. 1 have triangular cross-sections. In a planar solution, the cross-section liesin the x-y plane and the elements extend an infinite distance in the z direction. In a cylindricalsolution, an element is a figure of revolution about the axis with a triangular cross-section inthe z-r plane.

The mesh in Fig. 1 has the important property of conformality. The term means thatthe triangles have been specially shaped to conform to the boundaries between materials (re-gions). As a result, each element has an unambiguous material identity. The finite-elementmethod is based on two approximations: 1) material properties in an element are uniformand 2) elements are small enough so that the field quantities can be approximated by simpleinterpolation functions. With these assumptions, the governing differential equation can beintegrated over elements surrounding a node to yield a linear equation. This equation relatesthe field quantity at the node to those at the surrounding nodes. The coupled set has onelinear equation for each node in the mesh. In Nelson, solution of the set representing thecomplex-value Poisson equation gives the quasi-electrostatic potential at each node. We canthen perform two-dimensional interpolations to find harmonic potential at intervening points,or take numerical spatial derivatives to find the components of the electric field.

With this background, we can understand the steps in an Nelson solution:

1. You define the boundaries of material objects in a solution volume for your application.The task is usually accomplished with the drawing editor of Mesh. You can also specifycontrol information such as the target element sizes that may affect the accuracy and run

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Table 1: Nelson files

Name Function

MName.MIN Mesh input script (definition of foundation mesh andregion outlines)

MName.MLS Mesh diagnostic listingMName.MOU Mesh output (node locations and element identifies)EName.NIN Nelson input script (run control and material proper-

ties)EName.NLS Nelson diagnostic listingEName.NOU Nelson output file (node locations and electrostatic

potential)AName.SCR Nelson script for automatic data analysis

time of the solution. The result is a text record (script) with sets of line and arc vectorsthat outline electrodes and dielectrics.

2. Mesh analyzes the boundary specifications and automatically generates a set of conformaltriangles such as those of Fig. 1. The program creates an output file that lists the locationsof nodes and the identities of elements.

3. You define the material properties of regions in the solution volume. Usually, this task isperformed through an interactive dialog in Nelson.

4. Nelson reads the mesh geometry and applies the material parameters to generate thecomplex-value linear equation set.

5. Nelson solves the equations using iterative methods and records the node potentials andcoordinates in an output file. This file serves as a permanent record of the solution thatcan be re-loaded for latter analysis.

6. You can use the interactive graphical environment of Nelson to explore the solution.The program creates a wide variety of plots and performs quantitative calculations offield quantities. You can also write scripts for automatic control of complex analyses.

1.4 Scripts and data files

Mesh and Nelson read and generate several types of files. For a typical solution it is notnecessary for your to deal directly with the files – the user interface takes care of data orga-nization. On the other hand, if you generate a large amount of data it’s a good practice toarchive application results in individual folders. Furthermore, there are advanced features ofMesh and Nelson that require direct entries to the input scripts. For convenience, all inputscripts and output data files are in text format. Both Mesh and Nelson feature integratedtext editors. Table 1 lists the file types and functions in the Nelson package. Note that thesuffixes indicate the file function.

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1.5 Theoretical background

The reference S. Humphries, Field Solutions on Computers (CRC Press, Boca Raton,1997), Chap. 11 gives detailed information on the theory of non-radiative magnetic fields inconductive materials and the application of finite-element methods in Nelson. This sectionsummarizes basic concepts that may be helpful in preparing solutions.

Static magnetic fields are created by applied currents in coils and atomic currents that resultfrom domain alignment. Time-varying magnetic fields can generate a third type of current.The electric field associated with changing magnetic flux drives currents through conductivematerials. Eddy currents are of particular concern in magnetic materials with high conductivitylike iron. They can significantly reduce the effectiveness of transformers and other devices.

To derive approximate equations we assume that real currents in the solution volume aremuch larger than displacement currents. The condition is equivalent to

f ≪ 1

2πǫρ. (1)

In ferrites with high volume resistivity, the limit of Eq. 1 holds for frequencies below about1.0 MHZ. The approximation holds for most frequencies of interest in iron. Dropping thedisplacement current, we can write Ampere’s law as:

∇×B

µ= J0 + Jc. (2)

In Eq. 2, the quantity J0 is the applied current density in coils and Jc is conductive current inmetals and semiconductors. The magnetic flux density is related to the vector potential by:

B = ∇×A. (3)

Using Eq. 3, we can express Faraday’s law as

∇× E = − ∂

∂t(∇×A) . (4)

If there are no free charges to create electrostatic-type fields, then Eq. 4 implies that the totalelectric field is related to the vector potential by

E = −∂A

∂t. (5)

The conductive current is therefore given by

Jc = −σ∂A

∂t. (6)

Combining Eqs. 2, 5 and 6, we find that the vector potential is given by:

∇ ·(

1

µ∇A

)

= −J0 − σ∂A

∂t= −J0 − jσωA. (7)

The final form applies to a system with steady-state harmonic drive currents at angular fre-quency. In two-dimensional solutions, there is only one non-zero component of vector potential.In this case, Eq. 7 has the form of the complex-number scalar Poisson equation. This equation

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is easily solved with finite-element techniques. Given values of A, the magnetic flux density B

and electric field E can be determined from Eqs. 3 and 5.In rectangular problems (variation in x and y with no variation in z) Nelson finds the

vector potential Az created by currents that move in the z direction. The magnetic fields aregiven by:

Bx =∂Az

∂y, (8)

By = −∂Az

∂x. (9)

For problems in cylindrical coordinates, Nelson calculates the quantity rAθ, where Aθ is thevector potential created by azimuthal currents. In this case the magnetic fields are given by

Br = −1

r

∂

∂z(rAθ), (10)

Bz =1

r

∂

∂r(rAθ). (11)

The equations show that changes in values of the vector potential are related to the magneticflux (the surface integral of the normal components of magnetic flux density). For example, ina rectangular problem with boundaries at y1 and y2, the difference in the boundary values ofthe vector potential at position x0 is given by

Az(x0, y2)− Az(x0, y1) =∫ y2

y1

Bx(y′, x0) dy

′. (12)

A similar equation holds for integrals in the x direction. Therefore, if we set Az equal to aconstant value (usually zero) around the complete solution boundary, all magnetic flux mustbe trapped inside the region because the integral of Bx or By over any plane is zero. This isequivalent to generating the magnetic field inside a perfectly conducting metal box. Anotherway to see this is to note that Eq. 3 implies that the magnetic flux density must be parallel toa surface of constant A.

In cylindrical geometry, the symmetry of magnetic field lines dictates that the axis (r = 0.0)must always be a line with rAθ = 0.0. Nelson automatically sets the condition for cylindricalproblems that include r = 0.0 by adding a constant potential region on the axis. The value ofthe stream function at the outer solution radius r0 is related to the flux of axial field by

rAθ|r0 =(

1

2π

)∫ r0

02πr′dr′Bz(r, z) =

Flux

2π. (13)

For a uniform magnetic flux density B0, Eq. 13 implies that r0Aθ0 = B0r20/2, As an example,

Figure 2 shows the magnetic field lines around an iron cylinder immersed in a uniform solenoidalfield of 0.1 tesla. A solenoidal field B0

∼= 0.1 tesla was created by setting rAθ = 5.0 × 10−4

tesla-m2 on the outer boundary at r0 = 0.10 m.With this background, we can now discuss the implications of the two types of boundary

conditions used in Nelson:

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Figure 2: Setting boundary values of the vector potential to create a region of uniform flux.

• Dirichlet. Dirichlet boundary points have a fixed value of vector potential that doesnot change as the Nelson relaxation proceeds. A line region of uniform vector potentialrepresents a perfectly-conducting metal surface with magnetic field lines parallel to theboundary. An internal volume region at fixed vector potential represents a metal bodythat excludes the magnetic field.

• Neumann. A Neumann boundary is one where the normal derivative of the vectorpotential is specified. The boundaries inNelson are limited to the case where ∂A/∂n = 0.The special Neumann condition implies that the magnetic field is normal to the boundary.One advantage of the finite-element method is that all boundaries that are not fixedautomatically satisfy the special Neumann condition, even if they are slanted or curved.Neumann boundaries are often to reduce computation time by representing only half ofa symmetric system.

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Figure 3: Region divisions for the WALKTHROUGH example, dimensions in cm.

2 Creating and analyzing a solution with Nelson

2.1 Mesh generation

As a quick introduction to Nelson techniques and capabilities, we shall work through theexample shown in Fig. 3. The simulation addresses induction heating of the surface of a smallmetal object by a shaped coil. The system has cylindrical symmetry about the z axis (bottom).The five coils create a solenoid magnetic field at 2.0 kHz frequency. All coils are in phaseand each coil caries a peak current of 1000 A. The hollow metal piece has relative magneticpermeability µr = 3.0 and conductivity σ = 8.33× 105 S/m (volume resistivity ρ = 120.0 µΩ-cm). In a finite-element solution, a Neumann or Dirichlet condition must be specified alongthe boundaries of the solution volume. In this case, we use the condition rAθ = 0.0, equivalentto perfectly-conducting walls. The extra space around the assembly ensures that the boundarycondition has a small effect on fields near the work piece. The goal of the calculation is todetermine the distribution of electric and magnetic fields and the total volumetric resistiveheating.

The Mesh script WALKTHROUGH.MIN (supplied in the Nelson example library) is requiredfor the solution. Copy the file to a working directory and be sure that the Data folder in theTC program launcher is set correctly. Run Mesh and load WALKTHROUGH.MIN. Pick the Editscript/graphics command to enter the drawing editor. Here, you can use the display capabilitiesto confirm the following region assignments:

1. Air (Filled).

2. Ferromagnetic object (Filled).

3. Coil 1 (Filled).

4. Coil 2 (Filled).

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Figure 4: Completed mesh for the electron diode example.

5. Coil 3 (Filled).

6. Coil 4 (Filled).

7. Coil 5 (Filled).

8. Dirichlet boundary.

Abandon the drawing and return to the main menu. Click the Process command. A detailof the completed mesh is shown in Fig. 4. Note how the size of the elements varies for goodresolution of skin-depth effects on the surface of the work piece. Choose the Save mesh (MOU)command. You can now close or minimize Mesh.

2.2 Creating the Nelson script

Run Nelson from TC. The screen is initially blank and the status bar indicates that theprogram is waiting for input. Note the prominent tools marked 1, 2 and 3. The notation is areminder of the three steps in a magnetodynamic solution:

1. Set up program controls and material properties.

2. Generate and solve the finite-element equations.

3. Analyze the solution.

To start the first operation, we must identify the Mesh output file that defines the sys-tem geometry. Click the 1 tool or the Setup menu command. In the dialog, choose the file

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Figure 5: Dialog to set Nelson control parameters and region properties.

WALKTHROUGH.MOU. Nelson loads the information and displays the dialog of Fig. 5. Note thatthe grid contains a row for each mesh region with a notation of the Fill status. There are fivecolumns in the grid where you can enter values: MuR (relative magnetic permeability), Sigma(conductivity in S/m), Current (total current through the region in amperes), Potential (vectorpotential Az in tesla-m or stream function rAθ in tesla-m2 for fixed regions) and Phase (phaseof the potential or current in degrees).

The entries in the Control parameter section of the dialog have the following functions:

• GEOMETRY. The symmetry of the solution: planar or cylindrical. A planar solutionvaries in x-y and has infinite length in z, while a cylindrical solution has symmetry in θ.

• FREQUENCY. AC/RF frequency in Hz.

• RESTARGET. Accuracy tolerance for the iterative matrix solution of the finite-elementequations.

• MAXCYCLE. Maximum number of cycles in the iterative solution.

• DUNIT. Choose an entry to set a factor to convert the units used for coordinates in theMesh file to meters. The factor is the number of mesh units per meter: 39.37 for inches,100.0 for cm.

• OMEGA. A parameter in the range 0.0 to 2.0 to control the iterative matrix solution.

The values entered in the region grid box (shown in Fig. 5) define the following character-istics:

• Region 1. Air region with µr = 1.0.

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• Region 2. Ferromagnetic object with µr = 3.0 and σ = 8.333× 105 S/m.

• Regions 3-7. Coils with drive currents of 2000 A.

• Region 8. Fixed boundary with rAθ = 0.0.

Note that regions with a non-zero entry in the MuR column and no entry in the Current columnare assigned zero drive current by default. Similarly, regions with non-zero current are assignedµr = 1.0 by default. Set up the dialog with the values shown in Fig. 5 and click OK. Nelson

uses the information in the dialog to create the script WALKTHROUGH.NIN shown in Table 2.Chapter 3 reviews the script format and advanced program capabilities.

2.3 Finite-element solution

The next step is to generate and to solve the finite-element equations. Click the tool marked2 or the menu command Solve. Accept the default entry of WALKTHROUGH.NIN in the dialogby clicking OK. The screen turns blue to indicate that the program is in solution mode and thestatus bar reports the progress of the operation. When complete, Nelson creates the output fileWALKTHROUGH.NOU which contains full information on the mesh as well as values of the complexvector potential at each node. Section 3.5 describes the file format.

2.4 Solution analysis

We can now use the information in WALKTHROUGH.NOU to create plots and to perform quantitativeanalyses. Choose either the tool marked 3 or the Analyze menu command. In the dialog, acceptthe default choice of data file. Nelson loads the file, sets up the analysis menu and toolbar andcreates the default contour plot shown in Fig. 6. Take a moment to inspect the organizationof the working environment. The status bar at bottom lists the following information: solutionfile name, data file name, current analysis configuration, interpolation method and mouse snapstatus. The data file gives you the option to record the results of analysis operations in textformat. The plot window is divided into four areas. The main plot is at the upper left andthe plot legend is at the upper right. The orientation area at the lower right is a miniaturerepresentation of the solution volume with an outline showing the current view. Values appearin the information area at the bottom left in response to analysis commands such as Pointcalculation.

The contours in the plot of Fig. 6 lies along lines of constant rAθ at a reference phase of 0.0o.

In a two-dimensional solution, the contours correspond to lines of B (magnetic flux density).Because the lines are separated by equal intervals of magnetic flux, the spacing looks larger nearthe axis of a cylindrical solution. To plot another quantity, click on the command Plots/Plotsettings/Type and choose the option Filled contour. Click on Plots/Plot settings/Quantity andchoose |E|, the amplitude of the toroidal electric field. If you zoom the view, you can reconstructthe plot of Fig. 7 which shows electric field in and near the work piece.

The following chapter gives detailed reference material on analysis functions of Nelson. Youcan experiment with the program capabilities to create a wide variety of plots. To complete thesession, we shall run a script that controls automatic calculations. To inspect the file, click onEdit script in the File menu and pick the file WALKTHROUGH.SCR. Table 3 shows the file content.The first command ensures that the file WALKTHROUGH.NOU is loaded while the second command

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Table 2: Nelson solution script for the WALKTHROUGH example

Mesh = walkthrough

Geometry = Cylin

DUnit = 3.2808E+00

ResTarget = 5.0000E-08

Omega = 1.2500E+00

MaxCycle = 5000

Freq = 6.0000E+01

* Region 1: VACUUM

Material(1) = 1.0000E+00 0.0000E+00

* Region 2: OBJECT

Material(2) = 3.0000E+00 8.3300E+05

* Region 3: COIL01

Material(3) = 1.0000E+00 0.0000E+00

Current(3) = 1.0000E+03 0.0000E+00

* Region 4: COIL02

Material(4) = 1.0000E+00 0.0000E+00

Current(4) = 1.0000E+03 0.0000E+00

* Region 5: COIL03

Material(5) = 1.0000E+00 0.0000E+00

Current(5) = 1.0000E+03 0.0000E+00

* Region 6: COIL04

Material(6) = 1.0000E+00 0.0000E+00

Current(6) = 1.0000E+03 0.0000E+00

* Region 7: COIL05

Material(7) = 1.0000E+00 0.0000E+00

Current(7) = 1.0000E+03 0.0000E+00

* Region 8: BOUNDARY

Potential(8) = 0.0000E+00 0.0000E+00

EndFile

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Figure 6: Working environment of Nelson in the analysis mode.

Figure 7: Amplitude of Eθ

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Table 3: Nelson analysis script WAKLTHROUGH.SCR.

INPUT WalkThrough.NOU

OUTPUT WalkThrough.DAT

NSCAN 100

SCAN 0.0 0.0 0.0 1.5

VOLUMEINT 2

ENDFILE

Figure 8: Amplitude of Eθ

opens the data file WALKTHOUGH.DAT. The program sets the number of scan points to 100 andcomputes a set of values at z = 0.0 ft from r = 0.0 ft to r = 1.5 ft. Finally, Nelson analyzesRegion 2 (the work piece) and closes the output file. It is easy to capture the values in theresulting text file and transfer them to other software. The plot of |E| in Fig. 8 was createdwith PsiPlot. Nelson also has internal plot capabilities with digital oscilloscope features. TheVolumeInt command initiates a volume integral and lists the resistive power dissipation as 167.5W.

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3 Nelson solution reference

3.1 Nelson script format

You can create a control script for Nelson interactively using the interactive dialog described inSect. 2.2 or by writing the commands directly with a text editor. You can easily alter any scriptwith an editor. This chapter gives a detailed description of the syntax and functions of scriptcommands. Section 3.2 covers commands that control program operation. These commandsare created by the entries in the Control parameters section of the dialog (Fig. 5). Section 3.3reviews commands to set simple material properties. These commands are created by entries inthe Region properties grid of the dialog. The remaining sections describe advanced capabilitiesof Nelson as well as program operation and output file format.

The Nelson script is a text file composed of data lines that contain commands and pa-rameters. You can construct a script using the Setup dialog in Nelson. You can also writeand modify scripts with an editor. Direct script editing is required for some advanced Nelson

capabilities.The script must end with the EndFile command. The entries on a line may be separated

by the standard delimiters introduced in the Mesh manual:

• Space [’ ’]

• Comma [’,’]

• Tab

• Colon [’:’]

• Left parenthesis [’(’]

• Right parenthesis [’)’]

• Equal sign [’=’]

Any number of delimiters may be used in a line. Blank lines and comment lines are ignored.Comment lines begin with an asterisk (*). Nelson accepts commands in any order. Thefollowing example illustrates a complete script:

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* File: REACTOR.WIN

* 1000.0 A per coil

* Skin depth: 0.15 m

* Omega = 8.52E7

* Rho = 1.208

Freq = 13.56E6

MaxCycle = 3000

Omega = 1.90

Geometry = Cylin

DUnit = 1000.0

* Solution volume

Material(1) = 1.0

* Dielectric chamber wall

Material(2) = 7.3, 0.0

* Plasma volume

Material(3) = 1.000, 1.208

* Right cap

Material(4) = 1.0, 0.0

* Mid Coil

Material(5) = 1.0, 0.0

Current(5) = 500.0, 0.0

* Right coil

Material(6) = 1.0, 0.0

Current(6) = 1000.0, 0.0

* Axis

Potential(7) = 0.0, 0.0

ENDFILE

There are two classes of commands: program control and region properties. A control com-mand contains a keyword and a value. Region commands set the physical properties associatedwith elements and nodes. They have the format:

Keyword RegNo Value

Here, the integer RegNo is the region number defined in the Mesh input file. The stringKeyword specifies the physical property. The value may be one or more numbers. As anexample, the command

MATERIAL 2 50.0 3400.0

sets the properties of elements with region number 2 to µr = 50.0 and σ = 3400.0 S/m. (Note:For back-compatibility, Nelson recognizes command formats from Version 1.0 through 5.0where control commands start with the keyword Set and region commands start with Region.)

3.2 Program control commands

In the following sections, commands are written symbolically and as they might appear in theNelson script:

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MESH MPrefixMESH PlasmaReactorSpecify the Mesh output file that defines the geometry of the simulation. The file must havea name of the form MPrefix.MOU and be available in the working directory. If this commanddoes not appear in a script with the name FPrefix.NIN, then Nelson will seek the defaultmesh file FPrefix.MOU.

DUNIT DUnitDUNIT = 1.0E4You can use any convenient distance units in Mesh. This command defines a factor to convertcoordinates supplied by Mesh to the standard distance units of meters used in Nelson. Thequantity DUnit equals the number of Mesh units per meter. For example, if the Mesh

dimensions are entered in microns, set DUnit = 1.0 × 106. The default value is 1.00. (Note.Spatial quantities recorded in the output file FPrefex.NOU are always in meters. In an analysissession with Nelson, spatial quantities in graphs and listing files are scaled to the Mesh units.For example, if the Mesh dimensions are in cm and DUnit = 100.0, the spatial quantities inplots will be in cm.)

GEOMETRY [Rect, Cylin]GEOMETRY = CylinNelson handles problems in rectangular or cylindrical geometries. Rectangular systems havevariations in x and y with infinite length in z. Cylindrical systems have variations in r and zwith azimuthal symmetry. The parameter options are Rect and Cylin. In cylindrical solutionsthe program takes the z axis along the Mesh x direction and the r axis along y. In this casethe program issues an error message if any node has a y coordinate less than 0.0.

FREQ FreqFREQ = 3.45E6Specify the frequency (in Hz) for current and specific values of vector potential This commandwith a non-zero value is required for a solution.

OMEGA OmegaOMEGA = 1.96This command controls the over-relaxation factor for the iterative matrix solution, a numberin the range 0.0 to 2.0. Generally, higher values give faster convergence. Reduce the over-relaxation factor if the solution does not converge. If this command does not appear, theprogram uses the Chebyshev acceleration method to pick optimal values of the over-relaxationfactor.

MAXCYCLE MaxCycleMAXCYCLE = 500The maximum number cycles for the iterative matrix solution. The default value is 2500.

19

RESTARGET ResTargetRESTARGET = 5.0E-6Nelson calculates the relative error in the magnitude of the vector potential during the iterativematrix solution. The program stops when the error falls below ResTarget. Solutions generallyhave sufficient accuracy for ResTarget less than about 1.0×10−6. The default value is 5.0×10−8.

3.3 Commands for material properties

The following three commands define the properties of regions. The commands may be gener-ated with the Setup dialog in Nelson or directly with a text editor.

POTENTIAL RegNo VecPot [Phase]POTENTIAL(3) = 0.0POTENTIAL(5) = 5.0E-3 45.0The keyword Potential designates that the vector potential or stream function of the regionnodes has fixed amplitude and phase and will not change during the matrix relaxation. Thecommand sets the amplitude of the potential in tela-m or tesla-m2 and the phase in degrees.The program uses a value 0.0o if the phase is omitted.

MATERIAL RegNo MuR [Sigma]MATERIAL(5) = 1.0MATERIAL(3) = 200.0 5000.0Set the properties of a ferromagnetic or air material. Enter the relative magnetic permeability(µr = µ/µ0) and the conductivity σ in S/m. If the conductivity value is omitted, the default isσ = 0.0

3.4 Running Nelson

Nelson can run as an interactive program in a window or as a background task. To run theprogram from the Command Prompt, use a command of the form:

[ProgPath\]Nelson [DataPath\]FPrefix <ENTER>

where the file FPrefix.NIN and the Mesh output file are available in the data directory. Withthis capability, you set up extended autonomous Nelson runs using a DOS batch file or aPython script.

The remainder of this section discusses commands in the main menu when Nelson runs inthe interactive mode. The program enters this mode when launched from TC or run with noinput file prefix. The following commands appear in the File popup menu:

EDIT SCRIPT (NIN)EDIT LISTING (NLS)EDIT FILEThese commands call the internal editor to inspect or to modify Nelson input and output files.

20

Choosing a file from an alternate directory does not change the working directory. The Editscript (NIN) command shows a list of all files with names of the form FPREFIX.NIN while Editlisting (NLS) displays files with names FPREFIX.NLS.

RUN ANALYSIS SCRIPTAn analysis script allows you to perform complex or repetitive analyses on a set of similarsolutions. This command displays a dialog listing files with the suffix SCR. Pick a file andclick OK. The script can load data files, open and close data records, and perform any ofthe quantitative analysis functions described in this chapter. The script command language isdescribed in Sect. 5.8. Note that the analysis script must be in the same directory as the datafiles.

SETUPThe function of this command is to create a Nelson script to control an magnetodynamicsolution. The program first prompts for a Mesh output file to define the system geometry.The prefix of the file will be used as the argument of the Mesh script command. The programthen displays the dialog shown in Fig. 5. The number of regions in the dialog is determinedby the Mesh file. The functions of the control parameters in the upper box are described inSect. 3.2. You can enter basic physical properties of regions in the grid box (see Sect. 3.3).

The Solve popup menu includes the following two commands.

RUNPick an input file (such as FPrefix.NIN) to start a solution. The working directory is changedif you pick a file from an alternate directory. The run begins if the file FPrefix.MOU or the filespecified in the Mesh command is present. During the solution, the screen color is blue andthe progress is shown in the status bar.

STOPThis command terminates Nelson and saves the current state of the relaxation solution. Forexample, you may want to stop a relaxation solution at a moderate value of convergence tocheck whether the problem has been correctly defined.

SOLUTION DISPLAYWhen solution display is active, Nelson plots |Az| or |rAθ| during the matrix inversion to showsolution convergence. The number of relaxation steps per plot is initially small and increasesas the solution nears its final state. This feature is useful to verify convergence or to identifyproblem areas. (Note that the plot operations increase the run time.)

ANALYZEPick a file of the type FPrefix.NOU and call up the analysis menu for plotting and numericalanalysis.

21

NELSON MANUALDisplay this manual in your default PDF viewer. The file nelson.pdf must be available in thesame directory as nelson.exe.

SKIN DEPTH CALCULATORA handy utility to estimate the magnetic skin depth described in Sect. 4.2.

3.5 Format of the Nelson output file

The Nelson output file FPrefix.NOU is in text format. The file has three sections:

• Header with general information on the run

• Node and element information

• Region information

The header section consists of a title line and 10 data lines:

--- Run parameters ---

XMin: -1.06681297E+00

XMax: 1.06681297E+00

KMax: 111

YMin: 0.00000000E+00

YMax: 1.21921483E+00

LMax: 61

DUnit: 3.28080000E+00

NReg: 8

ICylin: 1

Frequency: 6.00000000E+01

Lines 2 and 3 list xmin and xmax, the limits along horizontal axis (x or z) of the solution volume.Dimensions are given in meters. The quantity kmax in Line 4 is the number of nodes along thehorizontal direction. Lines 5-7 describe the vertical axis (y or r). Line 8 contains the quantityDUnit, the conversion factor from dimensions used inMesh to meters. Line 9 gives the numberof regions in the solution, while Line 10 specifies the symmetry (0: planar, 1: cylindrical). Line11 gives the RF frequency in Hz.

The node section consists of 4 title lines and kmax × lmax data lines, one for each node ofthe solution space.

22

--- Vertices ---

k l RgNo RgUp RgDn x y

==============================================================

1 1 9 1 0 -1.06681297E+00 0.00000000E+00

2 1 9 1 0 -1.03540759E+00 0.00000000E+00

3 1 9 1 0 -1.00406601E+00 0.00000000E+00

4 1 9 1 0 -9.72817581E-01 0.00000000E+00

5 1 9 1 0 -9.41696476E-01 0.00000000E+00

6 1 9 1 0 -9.10751862E-01 0.00000000E+00

7 1 9 1 0 -8.80057925E-01 0.00000000E+00

...

A(Real) A(Imag) PowerUp PowerDn

================================================================

0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00

0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00

0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00

0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00

0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00

0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00

0.00000000E+00 0.00000000E+00 0.00000000E+00 0.00000000E+00

Each data line contains the following quantities:

• The indices of the node (K,L)

• The region number of the node (RgNo) and region numbers for two associated elements(RgUp and RgDn). The upper element lies above the line between nodes (K,L) and(K+1,L) and the lower element lies below the line.

• The coordinates of the node in meters, (x,y) or (z,r).

• The real and imaginary parts of the vector potential Az at the node in tesla-m or thestream function rAθ in tesla-m2.

• The time-averaged resistive power density in the upper and lower elements at the nodein W/m3.

The region section consists of four title lines following by NReg data lines, one for eachregion.

--- Regions ---

RegNo Fix VecPotAmp VecPotPhase

====================================================================

1 0 0 0 0 0 0.00000000E+00 0.00000000E+00

2 0 0 0 0 0 0.00000000E+00 0.00000000E+00

3 0 0 0 0 0 0.00000000E+00 0.00000000E+00

...

Mu Sigma CurrentReal CurrentImag Area

===============================================================================

1.25663700E-06 0.00000000E+00 0.00000000E+00 0.00000000E+00 2.37766263E+00

3.76991100E-06 8.33000000E+05 0.00000000E+00 0.00000000E+00 1.07554126E-01

1.25663700E-06 0.00000000E+00 4.30545946E+04 0.00000000E+00 2.32263249E-02

...

23

A one in the second column designates a fixed-potential region (electrode). Finally, Nelson

records the region names passed from Mesh:

--- Region names ---

VACUUM

OBJECT

COIL01

COIL02

COIL03

COIL04

COIL05

BOUNDARY

24

4 Eddy current and energy loss in drive coils

4.1 Theory

In calculations of previous sections, drive coils were represented as regions with a specifiedcurrent and zero conductivity. The solutions determined magnetic field penetration and energylosses in surrounding conductive structures. The underlying assumption was that regions carriedeither applied current density J0 or induced conductive current density Jc (see Eq. 2). In thislimit, Nelson simply carries out a direct solution of Eq. 7.

In many applications, we want to find power loss in windings when the skin depth is compa-rable to or less than the width of the conductor. These solutions involve real coils with non-zeroconductivity. In this case, regions carry both a virtual drive current I0 and a conductive currentIc. The quantities I0 and Ic are complex numbers that represent amplitude and phase. Thetotal current that flows in a region is It = I0 + Ic.

A self-consistent solution presents the following problem: although we must specify I0,to initiate the calculation, the known quantity is It (the current supplied by the generator).The resolution is simple when the calculation includes a single conductor. We make an initialsolution assigning a normalized virtual drive current I0 = [1.0, 0.0]. The calculated vectorpotential gives the induced electric field and the conductive current density over the region.We can then perform an integral to find the total current for the normalized drive, it1. Thetotal current generally has reduced amplitude and a different phase from the vitual drive current.Let the desired total current be represented by the complex number It. We could achieve thevalue by setting the virtual drive current to

I0 =Itit0

. (14)

Calculations are more challenging with multiple drive coils. Here, the fields of one coilmay induce conductive current in others. Because of the interactions, we cannot apply Eq. 14individually to each region. Fortunately, solutions generated by Nelson are linear so we canuse the principle of superposition. Suppose there are N coil regions with σ > 0.0 and let I0nrepresent the virtual drive current for coil n. The set of virtual drive currents can be representedas a column vector I0 of dimension N . The drive currents gives rise to a set of total currentsIt. Invoking the linearity of Nelson solutions, we can write the relationship as:

It = Ai × I0, (15)

where Ai is a matrix with dimensions N ×N . We can invert Eq. 15 to find a relationship thatgives the drive current necessary to achieve a given set of total currents:

I0 = A−1i × It. (16)

If the vector It represents the set of known total currents supplied by the generator, then Eq. 16determines the virtual drive currents necessary to achieve the solution.

25

The main effort is to find the matrix Ai. To evaluate the components, Nelson performsN preliminary calculations with unity drive currents. For example, suppose that N = 3. Thevirtual drive current vector

I02 =

(0.0, 0.0)(1.0, 0.0)(0.0, 0.0)

. (17)

gives a set of normalized total currents that may be associated with the components in thesecond column of the matrix:

a12 = it1, (18)

a22 = it2, (19)

a32 = it3. (20)

The program must perform N preliminary calculations to fill all columns of the matrix.In summary, Nelson performs the following operations when it detects that the solution

contains regions with drive current and non-zero conductivity:

1. Count and order the coil regions.

2. Perform N preliminary calculations with excitation by unit drive vectors to evaluate theN2 complex number matrix coefficients.

3. Invert the matrix and determine the virtual drive currents to achieve the target totalcurrents.

4. Carry out the main solution with the calculated virtual drive currents and save the results.

All operations are performed automatically so that no additional information is required fromthe user. The calculation does requires more time. For example, a run with 10 conductive driveregions takes 11 times longer than a simple solution.

4.2 Examples

Two following two examples illustrate code features to find self-consistent current distributionsin conductive coils. The first involves a single conductor, a copper bus bar that carries an ACcurrent with peak amplitude 1000.0 A. The input file BBAR.MIN describes a long cylindrcal withradius 1.0 cm in a large solution volume to approximate free space. Region 1 represents air,the bus bar is Region 2 and Region is the solution volume boundary with Az = 0.0

The Skin depth calculator (Fig. 9) in the Nelson Tools menu is useful to estimate regimeswhere we can expect strong contributions from eddy currents. It solves the equation:

δ =

√

1

πfσµrµo

, (21)

where δ is the skin depth, σ is the electrical conductivity, f is the frequency and µr is therelative magnetic permeability. To use the tool, fill in three known values and leave one field

26

Figure 9: Skin depth calculator.

Table 4: Contents of the file BBAR.NIN

Mesh = BBar

Geometry = Rect

DUnit = 1.0000E+02

ResTarget = 2.0000E-07

MaxCycle = 5000

Freq = 175.0

Omega = 1.50

* Region 1: AIR

Material(1) = 1.0000E+00 0.0000E+00

* Region 2: COIL

Material(2) = 1.0000E+00 5.814E7

Current(2) = 1.0000E+03 0.0000E+00

* Region 3: BOUNDARY

Potential(3) = 0.0000E+00 0.0000E+00

EndFile

blank. Nelson fills the missing field when you press the Calculate button. In this case, weuse the values σ = 5.814 × 107 S/m (copper), µr = 1.0 and δ = 0.5 × 10−2 m. The resultingtransition frequency is 174.3 Hz. Accordingly, we shall investigate the frequency range 5.0 Hzto 250 Hz. Note that there is a practical upper limit on the frequency. If the skin depth iscomparable to or less than the element width in Region 2 (0.05 cm), the calculation may notconverge.

Table 4 shows the complete Nelson input script. Region 2 has the properties µr = 1.0and σ = 1/ρ = 5.814 × 107. The target total current is 1000.0 A at 0.0o phase. The codewill automatically adjust the drive current to achieve this value. The calculation involves twomatrix solutions: one to find the normalization factor and one to generate the normalized fielddistribution. We can check the validity of the code by comparing values at low frequency wherethe current density is approximately uniform with radius. At a current amplitude of 1000.0A, we predict the following values: current density jz = 3.183 × 106 A/m2, electrical fieldEz = 0.0548 V/m and power density p = 8.713 × 104 W/m3. The power loss integrated overthe cross section is 27.37 W/m. An inspection of the listing at the end of the file BBAR.NLSshows the Nelson result at 2.0 Hz is 27.39 W/m.

At high frequency, the current density is forced toward the outer surface of the rod over a

27

Figure 10: Resistivite power density in the copper rod at different frequencies for example BBAR.Top: 75.0 Hz. Bottom: 175.0 Hz.

28

Figure 11: Power loss per length as a function of frequenct in the copper rod of example BBAR.The dashed line is the predicted zero-frequency value.

width comparable to the skin depth. Figure 10 shows the resulting distribution of resistive powerloss. The non-uniform current density also leads to an increase in the total power dissipation.Power loss per length in the bus bar as a function of frequency is plotted in Fig. 11.

The MULTICOIL example illustrates Nelson capabilities for in a more complex system. A4-turn copper solenoid creates an axial magnetic field that acts on an enclosed stainless steelpipe. The target total current in each coil is 2000.0 A at 0.0o phase. Figure 12 shows the highlynon-uniform power profile. As a result of eddy-current effects, the current density is highlyenhanced at ends of the outer turns. At the end of the run, Nelson creates the following listingof information in the NLS file:

Global solution properties

Volume (m3): 1.41371669E-02

Power (W): 7.18692965E+02

Solution analysis

NReg Drive current (A) Total current (A) Power (W)

Amplitude Phase Amplitude Phase

=================================================================

2 0.0000E+00 ( 0.00) 2.2830E+03 (-106.40) 2.6125E+02

3 1.7134E+04 ( 78.31) 2.0000E+03 ( 0.00) 1.1917E+02

4 2.0217E+04 ( 78.80) 2.0000E+03 ( 0.00) 1.0957E+02

5 2.0217E+04 ( 78.80) 2.0000E+03 ( 0.00) 1.0956E+02

6 1.7134E+04 ( 78.31) 2.0000E+03 ( 0.00) 1.1914E+02

The coils drive a current with amplitude 2.83 kA and -106.4o phase in the pipe. To achieve thedesired total currents at the high frequency, the drive currents require strong adjustment. Thedrive current value in the outer coils is 17.13 kA at phase 78.31o.

29

Figure 12: Geometry of the MULTICOIL example showing the power deposition distribution inthe coil and workpiece at f = 300.0 Hz.

Note: The following conventions are applied in the analysis routines described in the fol-lowing chapter for regions that represent drives with non-zero conductivity. Nelson calculatesinduced electric field and current density from the vector potential using Eq. 5. To find thetotal electric field and current density, we must add the resistive components created by thedrive current. The total values of electric field and current density are displayed for conductivecoil regions in element, surface and scan plots. In these plots, the electric field appears tobe discontinuous passing from a conducting coil to an insulator. The code does not computeresistive fields in insulators because they play no role in inductive coupling to other structures.Only the inductive components of E and j are displayed in contour plots.

30

5 TriComp analysis functions

To create plots and to perform numerical calculations, click the Analyze command in the mainmenu and choose a solution file. The Analysis menu contains the following main entries: File,Plots, Analysis, Scans, Export and Return. The commands of the Export menu (which generatehardcopy output and plot files) are similar to those in Mesh. The Return command restoresthe main menu where you can run additional solutions. This chapter gives a general descriptionof post-processing capabilities for all TriComp programs. The following chapter covers specificplot quantities and analysis capabilities of EStat.

5.1 File menu commands

LOAD SOLUTION FILERead a different solution file for analysis without returning to the main menu. Pick a new filein the dialog. Changing the directory in the dialog changes the program working directory.

LOAD CONFIGURATION FILEA configuration file is a text file of information that controls operation in the Analysis menu.The file defines quantities for plots, interpolations, volume integrals and surface integrals. Theprogram loads a default configuration the first time you run it. This file contains a broadspectrum of useful quantities for the solution type. Some programs may automatically switchbetween two default configurations, depending on the solution type (e.g., dielectric or con-ductive solutions in EStat). The default quantities are sufficient for most applications. Usethis command if you want to load a specialized configuration or one that you have prepared.Chapter 7 gives detailed information on the format of configuration files and how to preparethem.

LOAD FROM SERIESInitial-value programs like TDiff may produce several solutions files in a run at different sim-ulation times. Use this command to load other files in a series of solutions. The programdisplays the dialog of Fig. 13 which lists the full set of output files created by the run alongwith the simulation times. Pick the desired file and click OK to load the solution. Note thatthis command does not appear in the menus of boundary-value programs like EStat.

OPEN DATA RECORDCommands such as Point calculation and Line scan generate quantitative information. Youcan automatically record the data generated during an analysis session by opening a data file.Pick a file in the dialog or accept the default. The text-format data file will be stored in theworking directory. You can use an editor to view the file or to extract information to send tomathematical analysis programs or spreadsheets. The suggested suffix for data records is DAT.

31

Figure 13: Dialog to pick a solution in a series generated by an initial-value program.

CLOSE DATA RECORDClose the current data file. Use this command if you want to start a new file. You must closethe data file before opening it with the internal editor.

RUN SCRIPTA analysis script allows you to perform complex or repetitive calculations on a set of similarsolutions. This command displays a dialog listing files with the suffix SCR. Pick a file andclick OK. The script can load data files, open and close data records, and perform any ofthe quantitative analysis functions described in this chapter. The script command language isdescribed in Sect. 5.8. Note that the analysis script must be in the same directory as the datafiles.

CREATE SCRIPTUse this command to create an analysis script with the internal text editor. Supply a file prefixSPrefix in the dialog – the resulting script will be saved with the name SPrefix.SCR. Theprogram opens the file in the editor and writes a reference list of allowed commands. The listfollows the EndFile command and will be ignored by the script parser. Enter commands abovethe EndFile command.

EDIT FILESUse this command to view or to modify an existing file with the internal program editor. Usethe dialog to choose a text file. Changing directories in the dialog does not change the workingdirectory of the program. You must exit the editor to resume normal operation of the program.

EXITExit the program.

32

5.2 Plot menu commands

Spatial plots show variations of quantities over the two-dimensional space of the simulation.The following plot types are available:

• Mesh. Element facets of the computational mesh.

• Region. Computational mesh with elements color-coded by region number.

• Filled contour. Discrete bands of color coded according to values of the current plotquantity.

• Contour lines. Lines that follow constant values of the current plot quantity.

• Element. Elements of the solution space color-coded by values of the current plot quan-tity.

• Surface. A three-dimensional plot representing the current quantity as height over aregion in the x-y or z-r plane. The spatial limits of the plot correspond to the currentview window for Mesh, Region, Filled contour, Contour or Element plots.

The Settings popup menu contains the following commands.

PLOT TYPEChange the plot type.

PLOT QUANTITYA dialog shows a list of the available quantities defined in the INTERPOLATION section of thecurrent configuration file.

PLOT LIMITSIn the default Autoscale mode the program picks limits in Filled contour, Contour lines, Elementand Surface plots that span the full range of the current quantity. With this command you canset specific limits. In the dialog uncheck the Autoscale box and supply minimum and maximumvalues. Note that the program does not check the consistency of the values. Check the box toreturn to Autoscale mode.

TOGGLE GRID DISPLAYUse this command to activate or to suppress the display of grid lines in Mesh, Region, Contourlines, Filled contour and Element plots. Grid lines corresponding to the axes (x = 0.0 ory = 0.0) are plotted as solid lines.

GRID CONTROLThis command displays the dialog of Fig. 14 to set properties of the grid. In the defaultAutoscale mode, the program automatically chooses intervals and positions so that lines occurat convenient values of x or y (for example, 0.01 rather than 0.01153). The grid intervals changeas the view is zoomed. To set the grids manually, uncheck the Autoscale box and enter valuesfor the intervals in x and y.

33

Figure 14: Grid control dialog

MOUSE/KEYBOARDBy default the program uses mouse entry of coordinates for commands like Line scan and Zoom.This command toggles between mouse and keyboard input. Enter keyboard coordinates in thedistance units used in Mesh. In other words, if the solution program has DUnit = 1.0× 106,then enter dimensions in microns. You can temporarily switch to keyboard entry when enteringcoordinates with the mouse by pressing the F1 key.

TOGGLE SNAP MODEWhen snap mode is active, the mouse returns the coordinate values closest to an integer multipleof the quantity DSnap. In other words, if DSnap = 0.5 and the mouse position is (5.4331,-2.6253), the returned coordinates are (5.500, -2.500). Exercise caution when using commandslike Point calculation. If snap mode is active, the calculation is performed at the snap pointrather than the current location of the mouse cursor. You can also toggle the snap mode bypressing the F2 key during coordinate entry.

SNAP DISTANCESet the distance scale DSnap for the mouse snap mode.

TOGGLE ELEMENT OUTLINEThis command determines whether element facets are displayed.

TOGGLE FIXED POINT DISPLAYIn the default mode, the programs generate contour line, filled contour and element plots fromelement information. Therefore, isolated nodes (representing structures like fixed-potentialgrids or sheets) do not appear. In response to this command, the program plots circles atfixed-potential nodes that are surrounded by material elements.

CONTOUR STYLEThis command is active only when the current plot type is Contour lines. There are four

34

Figure 15: Monochrome contour plot with labels

choices: monochrome, monochrome with labels, colored and colored with labels. In the coloredmode, the lines are color-coded according to the value of the plotted quantity. A legend isincluded in the information window to the right of the plot. In the labeled modes, contour linesare numbered according to their values (Fig. 15). Overlapping labels on closely-spaced linesmay look better in a zoomed view.

NUMBER OF CONTOURSChange the number of plotted contour lines. This command is active only when the currentplot type is Filled contour or Contour lines.

TOGGLE NUMBER FORMATThe program automatically chooses real-number notation for axis labels when the range ofdimensions is close to unity. Use this command to switch to scientific notation.

The following commands (described in the Mesh manual), change the view limits in Mesh,Region, Filled contour, Contour line and Element plots. The current view limits of the two-dimensional plots are used when creating three-dimensional Surface plots.

ZOOM WINDOWZOOM INEXPAND VIEWGLOBAL VIEWPAN

The following commands control the appearance of Surface plots. The commands are activeonly when a Surface plot is displayed.

ROTATE 3D IMAGERotate the Surface plot by 90o in the spatial plane.

35

VIEW ANGLE 3DSet the elevation angle for the view point.

SET GRID 3DChange the resolution of the surface plot. To create a Surface plot, a quantity is mapped to arectangular grid with dimensions Nx × Ny. The numbers also determine the total number ofgrid lines in the Surface plot. The default values are Nx = Ny = 75.

5.3 User specified contours

In the default mode of Contour line and Filled contour plots, Nelson chooses a linearly-spacedset of plot values. The values depend on the specified number of lines and the limits. Theminimum and maximum values are set by the program when Autoscale is active or by the user.Although this method is adequate for general use, there are circumstances where more controlis desirable:

• Logarithmic intervals are useful when the plot quantity has a large variation.

• In applications like shielding, it may be necessary to plot a specific value of field magni-tude.

Nelson features a simple but versatile method for user control of contour plots.The first step is to prepare a text file of numerical values for contour lines. You can place

one value per line or put all values on a single line separated by spaces. Nelson recognizes anyvalid real-number format (including scientific notation). Here is an example:

0.0000001

0.0000002

0.0000005

0.000001

0.000002

0.000005

0.00001

0.00002

0.00005

0.0001

The minimum number of entries is 1 and maximum is 100. The values must be arranged inascending order. The plot listing is easiest to understand if the number is 15 or less. Save thefile if you do extended work with standard intervals. Select the intervals you want to displayand copy them to the clipboard.

Next, choose Plot/Plot settings/Plot limits (or click the Plot limits tool) to open the dialogof Fig. 16. Note the checkbox marked From clipboard at the bottom-right. When you checkthis box, Nelson performs the following actions:

36

Figure 16: Plot limits dialog.

• Reads and analyzes the contents of the clipboard, checking that the entries are validnumbers in ascending order.

• Sets the desired contour values.

• Exits the dialog and updates the plot.

Uncheck the box to return to normal operation (linear intervals with automatic or user-specifiedlimits). To enter a new set of values from the clipboard, uncheck and then recheck the box.

Note that the user-specified values affect only Contour line and Filled contour type plots(not plots of type Element and Surface). Figure 17 illustrates how the values are interpreted.In a Contour line plot (upper), each line is displayed with a corresponding listed value on theright if the number of lines is 15 or less. When the number is greater, the value list containsa partial set of selected values. In the Filled contour plot (lower), the list gives the minimumvalue for each colored interval. In other words, an interval covers the range from its listed valueto the listed value of the next interval. For example, the dark violet band in the figure extendsfrom 1.0 × 10−7 to 2.0 × 10−7. Values below the first are plotted in the next lowest color (inthis case, light violet) and values above the top value are plotted in white.

5.4 Saving and loading views

The creation of plots for presentations may involve some effort. With the following two com-mands, you can save all the current view parameters and immediately restore the plot.

SAVE NAMED VIEWSave the view parameters for the current plot. Quantities such as the zoom limits, plottedquantity and plot type (spatial or scan) are saved. The information is stored in a text file inthe current directory with a name of the form FPREFIX.FPV.

LOAD NAMED VIEWLoad a view file and refresh either a spatial or scan plot.

37

Figure 17: Display of user-specified contour intervals. Top: Contour line plot. Bottom: Filledcontour plot.

38

The file contains the complete set of plot parameters. This excerpt illustrates the format:

Program: TriComp

PlotStyle: Spatial

Outline: ON

Grid: ON

Scientific: OFF

FixedPoint: OFF

Vectors: OFF

XGMin: -1.000000E+00

XGMax: 1.000000E+01

YGMin: 0.000000E+00

YGMax: 1.000000E+01

PlotType: Elem

NPQuant: 1

...

If a specific solution file is loaded, the plot will be restored exactly. The saved view feature hastwo useful features if a different mesh is loaded:

• Dynamic adaptation to different solutions.

• Option for user control of the view parameter set.

Regarding the first feature, there are situations where you want to create consistent viewsof a set of solutions with different geometries, maintaining a similar appearance. Some plotproperties (like the plot type or quantity) are applicable to any solution, but others (like zoomlimits) depend on the geometry. The program checks each plot parameter for validity. If aparameter is outside the allowed range for the currently-loaded solution, the program computesan alternative. The goal is to preserve as many features of the view as possible.

You can modify view files with an editor. The order of entries is not rigid. On input,the programs use a free-form parser. If a parameter is missing, the program simply makes nochange from the value current in the program. The implication is that you can modify a savedview to include only elements essential to your application.

5.5 Analysis menu commands

The commands in the Analysis popup menu generate numerical data. Most of the functionsrequire coordinate input from the user, usually through the mouse. Therefore, the analysismenu is active only when a Mesh, Region, Contour, Element or Vector plot is displayed.

POINT CALCULATIONClick on the command and move the mouse cursor to a position inside the solution volume.Current coordinates are displayed at the bottom of the screen. Press F2 if snap mode isactive and you want to specify an intermediate point. Press F1 to type in the coordinates.The program writes a subset of quantities to the information area below the plot. If a data

39

Figure 18: Surface integral dialog

file is open, the program also records complete information on all quantities defined in theInterpolation section of the configuration file.

LINE SCANAfter clicking on the command, supply two points with the mouse or keyboard to define ascan line. The snap mode is useful in this application (for example, you may want a scan toextend from 0.000 to 5.000 rather than 0.067 to 4.985.) The program computes a series ofvalues of field quantities at equal intervals along the line. The program makes a screen plot ofthe currently-selected quantity versus distance along the scan and activates the commands inthe Scan plot menu (Sect.5.6). The information is recorded if a data file is open. The defaultis to include all computed quantities. Use the Set recorded quantities command to limit theinformation.

VOLUME INTEGRALSDetermine volume integrals of quantities defined in the VOLUME section of the configuration fileand record the results. The program will prompt if a data file is not currently open. Theprogram first records the global and regional volumes (or areas for planar solutions) and thenintegrals of defined quantities organized by region.

SURFACE INTEGRALSCalculate surface integrals of quantities defined in the SURFACE section of the configurationfile over region boundaries. In response to the command, the program displays the dialog ofFig. 18. Pick one or more regions as internal regions and bounding regions as external. Theprogram computes the integral over the surfaces of the internal region(s) in contact with theexternal regions.

LINE INTEGRALSTake integrals of vector quantities specified in the SURFACE section of the current configuration

40

Figure 19: Dialog to generate a matrix file.

file over a line in the solution volume. Specify the line with the mouse or keyboard as in theLine scan command. The program reports both parallel and normal integrals in the data file(i.e., the vector quantity is taken both parallel and normal to the differential line segments).

MATRIX FILEThe program can write matrix files of field values to help you create your own analysis routines.Although information is available in the output file of the solution program, it may be difficultto deal with the conformal triangular mesh. The Matrix file command uses the interpolationcapabilities of the program to create a text data file of field quantities on a rectangular grid inx-y or z-r. The command displays the dialog of Fig. 19. Here you can set the matrix file name,the dimensions of the box and the number of intervals along x and y (or z and r). The programcreates the file in the current directory. The default is to include all computed quantities. Usethe Set recorded quantities command to limit the information.

The Analysis settings popup menu contains the following entries.

INTERPOLATION METHODThe default interpolation method for plots and analysis commands such as Point calculationand Line scan is a second-order least-squares fit with a selective choice of data points. Forexample, only points on the side of a material boundary that contains the target point areincluded to give the correct field discontinuity. The least-squares fit may fail in very smallregions or enclosed areas if the program cannot identify enough data points. In this case,toggle to the linear mode. Here, field values are determined by a first order fit in the elementthat contains the target point. The status bar reports the current interpolation type.

SCAN PLOT QUANTITYWith this command you can choose the quantity to display in the screen plots of line scans.Pick the quantity from the list box and click OK. All quantities defined in the INTERPOLATION

41

Figure 20: Dialog to set quantities recorded in response to the Line scan and Matrix commands.

section of the current configuration file are available. This setting has no effect on the data filerecords which may include all field quantities.

NUMBER OF SCAN POINTSThis command sets the number of line scan points in plots and data file records. The defaultis 50 and the maximum number is 500.

SET RECORDED QUANTITIESThe default for data records of line scans and matrix files is to include all quantities defined inthe INTERPOLATION section of the current configuration file. This state may result in large fileswith unnecessary information. Use this command to set the quantities that will be included.In the dialog of Fig. 20, activate the check box to include a quantity. There are a few rules:

• Quantity selection is available when the program runs interactively in a window. It doesnot apply when the program is run in the background. In this case, you can limit recordedquantities by editing the INTERPOLATION section of a custom configuration file.

• In the interactive mode, quantity selection applies to direct calls to the Line scan orMatrix file commands and indirect calls from an analysis script.

• All quantities are active when the program starts.

• The state of active quantities is preserved when a new data file is loaded.

5.6 Scan plot menu

The commands of the Scan menu become active following the Line scan command.

OSCILLOSCOPE MODEIn the oscilloscope mode, a scan plot assumes characteristics of a digital oscilloscope (Fig. 21).

42

Figure 21: Scan plot in the oscilloscope mode

The program superimposes a cross-hair pattern on the graph. Plot values at the intersection aredisplayed in the information window. Move the marker along the plot by moving the mouse. Ifyou click the left mouse button at a point, the program displays the plot values along with thenumerical derivative and integral of the curve. The definite integral is taken from the left-handside of the plot to the current point. Values are displayed on the screen and written to the datafile if open. Press the right mouse button to exit the oscilloscope mode.

TOGGLE SCAN SYMBOLSThe setting determines whether plot symbols are added to the scan plot showing calculatedpoints.

TOGGLE GRIDThe setting determines whether grid lines are added to the screen and hardcopy scan plots.

CLOSE SCAN PLOTClose the scan plot plot and return to the Analysis menu.

43

Figure 22: Tools to display vector information in the slice plot menu.

5.7 Vector tools

The analysis menu includes useful tools to display directional quantities in the VECTOR sectionof the current configuration file. Figure 22 shows the corresponding entries on the toolbar.

PROBEThis feature was inspired by the familiar Magnaprobe illustrated in Fig. 22. When you click onthe tool and move the mouse cursor into the plot area, it changes to a semi-transparent probethat rotates about a pivot point to show the local direction of the vector quantity. The statusbar shows the coordinates and the magnitude of the quantity.

FIELD LINES AT POINTSUse this command to add lines of the vector quantity to any two-dimensional plot. The programenters coordinate mode when you click the command. Move the mouse to a point in the solutionvolume and click the left button. The program calculates and plots the path of the vector linethat passes through the point. You may continue to add any number of lines. Click the rightmouse button or press Esc to exit coordinate mode.

VECTOR SCATTER PLOTSuperimpose a uniform distribution of vector arrows pointing in the direction of the vectorquantity. Vectors may be added to any of the two-dimensional plot types, including Region.

REMOVE VECTORSUse this command to turn off the vector display. The program removes vector arrows andcalculated field lines from the current plot.

PICK VECTOR QUANTITYChoose a quantity defined in the VECTOR section of the current configuration file for displaywith the vector tools.

44

5.8 Analysis script commands

Scripts to control analysis sessions have a name of the form FPREFIX.SCR. They should be inthe same directory as the data files. Scripts are text files that follow the TriComp syntaxconventions. The program ignores blank lines and indentations. Data lines use the standarddelimiters and comment lines begin with an asterisk [*]. Processing ends when the EndFilecommand is encountered.

To run a script, choose the Run script command in the File menu. The program shows alist of available scripts. Pick a file and click OK. The script operates on the presently-loadeddata file or you can load other files from within the script. You may also sequentially open oneor more data output files.

The program can perform analyses autonomously under script file control from the commandprompt. If the file GTEST.SCR is in the data directory, then use a command of the form:

[ProgPath]\ProgName GTEST.SCR <Enter>

In the command prompt mode, the analysis functions of the program may be invoked frombatch files, Perl or Python scripts or you own programs. The main application of the commandprompt mode is to generate data files and to perform extended analyses under batch file control.

The following commands may appear in a script:

INPUT FileNameINPUT Switch1.EOUClose the current data file and load a file for analysis. The parameter is the full name of thedata file. You can load several files for sequential analysis.

OUTPUT FileNameOUTPUT SW02.DATClose the current data file and open an output file FileName.

CONFIGURATION [datapath\] FileNameCONFIGURATION = \FieldP\TriComp\ESTAT FORCE.CFGLoad a new configuration file to change the analysis characteristics. If a [datapath] is notspecified, the configuration file must be available in the working directory. If this commanddoes not appear, the program uses the current configuration or the one in force during the lastrun.

SCAN Xs Ys Xe YeSCAN Zs Rs Ze ReSCAN = (0.00, 0.00) (10.00, 0.00)Write the results of a line scan between the specified points to the data file. The four realnumber parameters are the starting and end coordinates in Mesh units.

45

GENSCANPerform any number of calculations along an arbitrary path in space. At each point, writevalues of quantities defined in the INTERPOLATION section of the configuration file to the datarecord file. The command must be part of a structure with the following form:

GENSCAN

xp1 yp1

xp2 yp2

...

xpn ypn

END

Each data line contains two real numbers separated by spaces to define a point in the solutionvolume (x-y or z-r). Enter coordinates in Mesh units.

VOLUMEINT [NReg]VOLUMEINTVOLUMEINT 8Perform a volume integral of quantities defined in the VOLUME section of the configuration fileand write the results to the current data file. If a region number does not appear, integrals aretaken over all regions in the solution volume. Otherwise, the integral extends over elementswith region number NReg. If quantities in the VOLUME section of the configuration file haveunits of C/m3, then the output quantities will have units of C for cylindrical solutions andC/m for planar calculations.

SURFACEINT Reg1 Reg2 ... RegNSURFACEINT 5 7 -9 -12Perform a surface integral of vector quantities defined in the SURFACE section of the configurationfile and write the results to the data file. Positive integer values define the set of Internal regionsand negative values define the External regions. With no specification, all regions that are notInternal are taken as External. The integral is taken over the surface facets between any Internaland External elements. The command may include any number of regions. The regions in theInternal set may or may not be contiguous. Note that the program does not include facets onthe boundary of the solution volume in surface integrals.

MATRIX FName Nx Ny Xs Ys Xe YeMATRIX FPrefix Nz Nr Zs Rs Ze ReMATRIX = Switch1 (10, 20) (0.00, 0.00, 5.00, 10.00)Open a matrix file and record values. The command requires seven parameters: 1) The nameof the matrix file (string), 2) the number of intervals along the x or z direction (integer). 3)the number of intervals along the y or r direction (integer), 4-7) coordinates of the corners ofa box in the solution volume (real). The suggested suffix for matrix files is MTX.

NSCAN NScanNSCAN = 150Set the number of points in a line scan. The default is 50 and the maximum number is 500.

46

INTEPOLATION [LSQ,LINEAR]INTERPOLATION = LinearSet the interpolation method for subsequent Point, Line scan and Matrix commands. Theoptions are LSQ (least-squares fit) and Linear.

PLOT FSaveView FOutput Nx NyPLOT (XAxis3D VIEW001 800 600)Write a plot file based on data in the currently loaded solution. The string FSaveView is theprefix of the file FSaveView.FPV, a set of view parameters created with the Save view command(Sect. 5.4). The view file must be available in the working directory. The plot is saved in theworking directory. It is in Windows Bitmap format and has the name FOutput.BMP. Theintegers Nx and Ny give the width and height of the image in pixels. To optimize compatibilitywith graphics format converters, pick values that are multiples of 16 (e.g., 1024× 768).

ENDFILETerminate execution of the script. You may add descriptive text in any format after thiscommand.

The following is an example of a script to compare field values along the axes of four differentsolutions and to write the results to the file COMP.DAT.

NSCAN 200

OUTPUT COMP.DAT

INPUT SWITCH01.EOU

SCAN 0.00 -50.00 0.00 50.00

INPUT SWITCH02.EOU

SCAN 0.00 -50.00 0.00 50.00

INPUT SWITCH03.EOU

SCAN 0.00 -50.00 0.00 50.00

INPUT SWITCH04.EOU

SCAN 0.00 -50.00 0.00 50.00

ENDFILE

47

6 Standard Nelson calculated quantities

Table 5 shows the standard units used for Nelson analyses. The package includes four standardconfiguration files for the analysis of complex-number solutions of AC/RF magnetic fields:

• nelson standard.cfg: amplitudes of quantities and instantaneous values at phases of0.0o and 90.0o.

• nelson reference.cfg: instantaneous quantities at a specified phase.

• nelson phase.cfg: phases and amplitudes of field quantities

• nelson complex.cfg: complex number representation of field quantities.

The default nelson standard.cfg is loaded the first time you run the program. Thereafter,Nelson loads the configuration used in the last analysis session. The next section reviewsNelson variables that may be included in the definition of configuration quantities. Section 6.2covers the content of nelson standard.cfg while Sect. 6.3 summarizes the other configurations.

Table 5: Nelson standard units

Quantity Unit

Spatial dimensions meters or units set by DUnitVector potential tesla-mPhase DegreesElectric field V/m (volts/meter)Current A (amperes)Current density A/m2 (amperes per square meter)Conductivity S/m (Siemens/m)Power density W/m3 (watts per cubic meter)

6.1 Nelson variables

Nelson supplies values of the following field quantities at the present location for use in ex-pressions of the INTERPOLATION, VECTOR, VOLUME and SURFACE sections of the configuration file.Values are calculated using the current interpolation method (least-squares fit or linear).

• &AR: the real part of the quasi-static vector potential Az (tesla-m, rectangular geometry)or rAθ (tesla-m2, cylindrical geometry) V.

• &AI: the imaginary part of the quasi-static vector potential: Az (tesla-m, rectangulargeometry) or rAθ (tesla-m2, cylindrical geometry) V.

48

• &BxzR: the x (planar) or z (cylindrical) component of the real part of the magnetic fluxdensity in tesla.

• &BxzI: the x (planar) or z (cylindrical) component of the imaginary part of the magneticflux density in tesla.

• &ByrR: the y (planar) or r (cylindrical) component of the real part of the magnetic fluxdensity in tesla.

• &ByrI: the y (planar) or r (cylindrical) component of the real part of the magnetic fluxdensity in tesla.

• &ER: the real part of the electric field Ez (planar) or Eθ (cylindrical).

• &EI: the imaginary part of the electric field Ez (planar) or Eθ (cylindrical).

• &PDens: the time-averaged resistive power dissipation in W/m3.

The following region variables are available for use in expressions. The value corresponds tothe region that contains the current location.

• $RegAAmp: the amplitude of Az (planar, tesla-m) or rAθ (cylindrical, tesla-m2) for a region

of fixed vector potential.

• $RegAPhase: the phase of Az (planar) or rAθ (cylindrical) for a region of fixed vectorpotential in radians.

• $RegMu: the relative magnetic permeability of a material region.

• $RegSigma: the electrical conductivity of a material region in S/m.

• $RegCDensR: the real part of the current density assigned to a region in A/m2.

• $RegCDensI: the imaginary part of the current density assigned to a region in A/m2.

• $RegArea: the cross-section area of the region in m2.

Nelson records a single run parameter in the solution files. The parameter $Freq is the RFfrequency in Hz.

6.2 Standard analysis configuration

The configuration file nelson standard.cfg is shown in Table 6. It defines the program pa-rameters µ0 that may be used in expressions. The solution engine passes one run parameter,the AC/RF frequency. To avoid confusion, we must be careful to distinguish between themagnitude of a complex quantity (peak value during the harmonic variation) and the spatialmagnitude of a vector quantity, such as:

|B| = B2x +B2

y . (22)

49

Table 6: Contents of the file nelson standard.cfg

PROGPARAM

$Mu0 = 1.256637E-6

END

RUNPARAM

$Freq = 0.0

END

REALTIMEPARAM

END

INTERPOLATION

* Magnetic field lines at 0 deg

|BLine00| = &AR

* Magnetic field lines at 90 deg

|BLine90| = &AI @CHS

* Amplitude of magnetic field

|B| = &BxzR 2 ^ &BxzI 2 ^ + &ByrR 2 ^ + &ByrI 2 ^ + @SQRT

* Amplitude of electric field

|E| = &ER 2 ^ &EI 2 ^ + @SQRT

* Resistive current density

|JRes| = &ER 2 ^ &EI 2 ^ + @SQRT $RegSigma *

* Drive current density

|JDrive| = $RegCDensR 2 ^ $RegCDensI 2 ^ + @SQRT

* Time-averged power density

PowerDens = &PDens

END

VECTOR

B00 = &BxzR;&ByrR

B90 = &BxzI @CHS;&ByrI @CHS

END

VOLUME

Power = &PDens

END

SURFACE

END

ENDFILE

50

Table 7: Reference-phase quantities in nelson reference.cfg.

Section Quantity Units

Interpolation Az or rAθ (lines of B) tesla-m/tesla-m2

Bx or Bz (magnetic flux density) teslaBy or Br (magnetic flux density) tesla|B| (flux density magnitude) teslaµr (relative magnetic permeability)σ (electrical conductivity) S/mjzc or jθc (eddy current density) A/m2

Vector B (magnetic flux density) tesla

We shall use the following notations: 1) a quantity in bold type enclosed between single verticallines represents a vector magnitude as in Eq. 22, 2) a non-vector quantity enclosed betweenvertical lines represents the amplitude of the harmonic variation and 3) a vector quantityenclosed between double vertical lines is the peak value of the vector magnitude.

The first quantities in the INTERPOLATION section are instantaneous vector potential values(Az or rAθ) at reference phases φref = 0.0o and 90.0o. Contour lines of these quantities followlines of magnetic field at the two phases. Note that lines of |Az| or |rAtheta| do not necessarydefine magnetic field lines.

The amplitude of the magnetic flux density is given by

||B|| =√

B2xR +B2

xI + B2yR + B2

yI . (23)

The amplitude of the eddy current density is related to the amplitude of the electric fieldby

||jc|| = σ ||E||. (24)

The drive current density has non-zero values only in regions that represent coils.The vector tools show the direction of magnetic flux density at phases of 0.0o and 90.0o. To

plot field direction at other phases, load the configuration nelson reference.cfg. The singlevolume-integral quantity is the total resistive power dissipation (in W/m for planar solutionsand W for cylindrical solutions).

6.3 Additional configurations

The remaining configurations provide good illustrations of how to manipulate and to displaycomplex-number quantities. You can inspect the text files with an editor. They can serveas useful templates for building your own expressions. Use nelson reference.cfg to showsnapshots of the harmonic quantities at a specified reference phase. The configuration filedefines the real-time parameter $Phi0, the reference phase in degrees. The default is 0.0o.You can change the value during an analysis with the reference-phase tool or the commandAnalysis/Analysis settings/Set reference phase. Table 7 lists the available quantities.

51

The file nelson phase.cfg illustrates how to display harmonic quantities by amplitude andphase. Use the file to plot or to calculate the amplitude and phase of components of the magneticflux density, Bx/Bz and By/Br. Finally, the file nelson complex.cfg uses complex-numberquantities directly. Table 8 lists the contents.

52

Table 8: Contents of the file nelson complex.cfg.

PROGPARAM

END

RUNPARAM

$Freq = 0.0

END

REALTIMEPARAM

END

INTERPOLATION

AzReal/rAtReal = &AR

AzImag/rAtImag = &AI

BxReal/BzReal = &BxzR

BxImag/BzImag = &BxzI

ByReal/BrReal = &ByrR

ByImag/BrImag = &ByrI

|B|Real = &BxzR 2 ^ &ByrR 2 ^ + @SQRT

|B|Imag = &BxzI 2 ^ &ByrI 2 ^ + @SQRT

EzReal/EThetReal = &ER

EzImag/EThetImag = &EI

JzReal/JThetReal = &ER $RegSigma *

JzImag/JThetImag = &EI $RegSigma *

JDriveReal $RegCDensR

JDriveImag $RegCDensI

END

VECTOR

END

VOLUME

END

SURFACE

END

ENDFILE

53

7 Building custom analysis configurations

You can create configuration files to customize analysis operations for your application. Youcan then switch between configurations for different types of solutions. To build a configuration,copy and rename one of the standard files supplied with the program to act as a template. Youcan put your new configuration file anywhere, but we suggest you store all configurations inthe same directory as the executable programs (c:\fieldp \tricomp). Note that the programremembers the last configuration used and attempts to reload it at the next run. This chapterdescribes the format and function of configuration files and how you can define calculatedquantities.

7.1 Configuration file structure

An analysis configuration is a text file with a name of the form FPREFIX.CFG. Table 7.1 showsthe contents of estat dielectric.cfg. The file includes seven sections with the followingfunctions:

• PROGPARAM: define fixed numerical parameters to use in the expressions of the INTERPOLATION,VECTOR, VOLUME and SURFACE sections.

• RUNPARAM: numerical parameters passed by the program (such as the RF frequency inREF2 or the dump time in TDiff). Do not change this section. Doing so could causeerrors reading solution files.

• REALTIMEPARAM: numerical parameters used in the expressions of the INTERPOLATION,VECTOR, VOLUME and SURFACE. The difference from a PROGPARAM is that these values maybe changed by the user while the program is running. An example is the reference phasein RFE2.

• INTERPOLATION: definition of scalar quantities for plots and scans.

• VECTOR: definition of vector quantities for use with the vector tools.

• VOLUME: definition of scalar quantities used for volume integrals.

• SURFACE: definition of vector quantities for surface integrals.

You may define custom quantities for display and analysis in the last four sections. Inplotting, run speed is a critical issue. The generation of a single element plot may requireover 50,000 interpolations. To ensure fast operation, expressions for calculated quantities inthe configuration file are written in reverse Polish notation (RPN). The program parses thefunction strings once while loading the configuration file and saves them in a coded form.Thereafter, operations are performed at the speed of compiled code. The following sectionreviews the basics of RPN notation. Section 7.3 covers parameters and variables that mayappear in expressions.

54

Table 9: Configuration file estat dielectric.cfg.

PROGPARAM

$Epsi0 = 8.854187E-12

END

RUNPARAM

$CondFlag = 0.0

END

REALTIMEPARAM

END

INTERPOLATION

Phi = &Phi

Ex/Ez = &Exz

Ey/Er = &Eyr

|E| = &EMag

Dx/Dz = $Epsi0 &Exz0 *

Dy/Dr = $Epsi0 &Eyr0 *

|D| = $Epsi0 &EMag0 *

EngDens = &Exz &Exz0 * &Eyr &Eyr0 * + 0.5 * $Epsi0 *

END

VECTOR

EVect = &Exz;&Eyr

DVect = &Exz0 $Epsi0 *;&Eyr0 $Epsi0 *

END

VOLUME

Energy = &Exz &Exz0 * &Eyr &Eyr0 * + 0.5 * $Epsi0 *

Charge = &Rho

END

SURFACE

Charge = &Exz0 $Epsi0 *;&Eyr0 $Epsi0 *

END

ENDFILE

55

Figure 23:

7.2 Introduction to RPN

Consider the expression

5.0× 4.53 +[

9.2 + 0.6× 0.9(2+0.67)]

. (25)

Equation 25 could be represented in algebraic notation by the string:

5.0 * 4.5^3 + (9.2 + 0.6 * 0.9 ^ (2 + 0.67))

The rules for parsing such a string are involved, requiring an interpreter with recursive logic.The equation has the following form in RPN:

0.9 0.67 2 + ^ 0.6 * 9.2 + 4.5 3 ^ 5.0 * +

The string is parsed in strict order from left to right for expressions of any degree of complexity.A calculator with RPN logic uses the stack shown in Fig. 23. Numbers are pushed and poppedat the bottom. The first two stack registers have the special names X and Y . Three simplerules govern the evaluation of RPN expressions:

• If the entry is a number, push it on the stack.

• If the entry is a unary operator (exp, ln, sin,...), apply it to the number in the X register.

• If the entry is a binary operator (+,*,ˆ,...) combine the numbers in the X and Y registersand move the stack down.

The final result is given by the number remaining in the X register. By convention, the binaryoperators act in the following way:

• Addition(+): Y +X

56

Table 10: Unary operators

Name Operation Comments

@SIN sin(X) Angle in radians@COS cos(X) Angle in radians@TAN tan(X) Angle in radians@ASIN sin−1(X) Returns angle in radians@ACOS cos−1(X) Returns angle in radians@ATAN tan−1(X) Returns angle in radians@EXP eX

@LN lnX@LOG log10(X)

@SQRT√X

@ABS |X|@EXCH X Y@OVERX 1/X@XSQ X2

@ENTER X ⇒ X, Y Copy X and push on stack@CHS X = −X

• Subtraction (-): Y −X

• Multiplication (*): Y ×X

• Division (/): Y/X

• Exponentiation (ˆ): Y X

Reverse Polish notation eliminates all issues with regard to parsing order. Furthermore, it issimple to compile expressions. Entries are represented either by variables or operator codes.

7.3 Operators, parameters and variables

Expressions to define quantities may contain numbers, unary and binary operations, parameters,standard variables and special variables for the program. Unary operators have names thatbegin with ’@’. Table 10 lists the available set. The following considerations apply with respectto the @CHS operator:

• You may include negative numbers (such as -3.1456 and -8.9E-09) in expressions. Thenumber -22.56 is equivalent to 22.56 @CHS.

• Expressions like -$Epsi0 and -&grady[3] that mix a minus sign with a variable areinvalid. Instead, use forms like $Epsi0 @CHS.

You may define up to 10 constants in the PROGPARAM section that can be used in yourexpressions. For example:

57

PROGPARAM

$Epsi0 = 8.854187E-12

$Epsi0Inv = 1.1294E11

END

A data line contains a parameter name, an equal sign and a value in any valid real-numberformal. A parameter name must begin with a dollar sign ($) and may contain a maximum of 14characters. Depending on the program, additional parameters may be defined in the RUNPARAMsection:

RUNPARAM

$CondFlag = 0.0

END

Here, a data line consists of a name, an equal sign and a default value. The actual valuedepends on the properties of the finite-element solution and is recorded in the output solutionfile. The parameter is set when a data file is loaded. Although you may use run variables inyour expressions, the form of the RUNPARAM section should not be changed. Be sure that theform in a custom configuration is that same as that in the standard files.

The standard variables are

$X, $Y, $Z, $NREG, $DUNIT

Note that the names of variables start with a dollar sign. Calls to defined functions for plots,line scans and other operations are always made at a specific location in the solution space. Theanalysis program sets the current position [$X,$Y,$Z] before any calls are made. To illustrate,the following expression gives the distance from the origin in meters:

RDIST = $X 2 ^ $Y 2 ^ $Z ^ 2 + + @SQRT

The standard variable $DUNIT is a dimension conversion factor passed from the solution pro-gram. For example, suppose dimensions in Mesh were defined in centimeters and convertedto meters for use in the solution program. In this case, an entry in the header of the solutionfile would set $DUNIT = 100.0. The following expression returns the distance from the origin incentimeters:

RDIST = $X 2 ^ $Y 2 ^ $Z ^ 2 + + @SQRT $DUNIT *

The special program variables give interpolated values of field quantities at the current loca-tion. As an example, the following quantities are available for use in expressions in EStat: &Phi(electrostatic potential), &Exz (the x or z component of electric field, &Eyr (the y or r componentof electric field), &EMag (magnitude of the electric field), &Exz0 (the x or z component of thenormalized electric displacement, D/ǫ0). &Dyr (the y or r component of the normalized electricdisplacement), &EMag0 (the magnitude of the normalized electric displacement), &EpsSig1 and&EpsSig2 (the relative dielectric constant or the electrical conductivity), and &Rho (the space-charge density for dielectric solutions). In addition, region variables may be defined. These arequantities that have fixed values over a region. The current value corresponds to the region thatcontains the current location. As an example, EStat supports the following region variables:

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$RegEps1 (the relative dielectric constant or electric conductivity along the primary axis foranisotropic materials), $RegRho (the space-charge density for uniform materials in dielectricsolutions), $RegPot (the voltage of a fixed-potential region), $RegEps2 (the relative dielectricconstant or electric conductivity along the normal axis for anisotropic materials), $RegAng (theangle of the primary axis relative to the x or z axis for anisotropic materials), $RegSinAng and$RegCosAng (the sine and cosine of the angle of primary axis).

7.4 Building expressions

You can add your own expressions to the INTERPOLATION, VECTOR, VOLUME and SURFACE sec-tions of the configuration file, one expression per line. Lines for a scalar quantities in theINTERPOLATION and VOLUME sections have three components:

• A name for display in plots and listings.

• An equal sign (=).

• A valid RPN expression.

For a quantity that has the same name in planar and cylindrical solutions, the name is a stringup to 14 characters in length. Alternatively, the name may have the form:

RectName/CylinName

Here, two strings (each up to 14 characters in length ) are separated by a slash (/). The stringRectName is used for labels when a planar solution has been loaded, and the string CylinNameis used for cylindrical solutions. For example:

FVolX/FVolZ = $RegCurrent &Byr * $RegArea / @CHS

Expression lines in the VECTOR and SURFACE sections have the following form:

NAME = EXPRESSION01;EXPRESSION02

A line consists of a name, an equal sign and two valid RPN expressions separated by a semicolon.The first expression gives the x or z component of the vector, while the second expression givesthe y or r component.

It is best to build a configuration one expression at a time. Modify and save the file, thenuse the Load configuration file command. The routine will report the location of the first syntaxerror encountered. If the configuration file loads successfully, test the new expression using plotand point calculation commands. At shutdown, the program stores the current configurationfile name in the registry and attempts to load it at the next session. If the file is missing orhas a syntax error, the program displays the Load configuration dialog so that you can pick analternative. An error message is displayed if the alternate file is invalid.

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Index

Analysis commands3D angle, 363D grid, 36Close data record, 32Contour style, 34Create analysis script, 32Field lines at points, 44Grid control, 33Interpolation method, 41Line integrals, 40Line scan, 40Load configuration file, 31Load from series, 31Load named view, 37Load solution file, 31Matrix file, 41Number display format, 35Number of contours, 35Number of scan points, 42Open data record, 31Oscilloscope mode, 42Pick vector quantity, 44Plot limits, 33, 36Plot quantity, 33Plot type, 33point calculation, 39Probe, 44Remove vectors, 44Rotate 3D, 35Run analysis script, 32Save named view, 37Scan plot quantity, 41Set recorded quantities, 42Snap distance, 34Surface integrals, 40Toggle fixed point, 34Toggle grid display, 33Toggle keyboard, 34Toggle outline, 34Toggle snap mode, 34Vector scatter plot, 44Volume integrals, 40

Analysis script

Configuration, 45GenScan, 46Input, 45Interpolation, 47Matrix, 46NScan, 46Output, 45Plot, 47Scan, 45SurfaceInt, 46VolumeInt, 46

analysis script commands, 45analysis script example, 47Analysis script format, 45analysis, batch file control, 45analytic solutions, 4autoscale, 33

calculated quantitiesstandard variables, 58

coil losses, 25comment lines, 17complex number

display quantities, 51magnitude and amplitude, 49

computational mesh, 4configuration file, 54

calculated quantities, 54expressions, 59load errors, 59standard, 48structure, 54

conformal mesh, 5contour lines, user-specified, 36

data export, 41delimiters, 17

eddy current, 7, 25element, 4

facet, 5fiducials, 33field

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definition, 4equations, 4

field line plots, 44field probe, 44finite-element method, 4

basis, 5frequency run parameter, 49

grids, 33

interpolation methods, 41

keyboard/mouse control, 34

line integral, 41

matrix file, 41mouse controls, 40

Nelsonanalysis screen, 13analysis script, 13command prompt, 20DUnit, 12, 19file types, 6finte-element solution, 13Geometry, 12, 19instructions, 22MaxCycle, 12Maxcycle, 19Mesh command, 19Mu, 20Omega, 12, 19output file format, 22Potential, 20program function, 3Restarget, 12, 20Run, 21Run analysis script, 21running in a window, 20script, 13, 17script dialog, 11, 21self-consistent drive current, 26skin depth calculator, 22Solution display, 21solution procedure, 5steps in a solution, 11Stop, 21

units, 48Nelson analysis variables, 48node, 5node equations, 4, 5, 13number notation, 35numerical solutions, 4

plot control, 33, 36plot types, 33plot view control, 35Plot views, saving, 37Poisson equation, 5

recorded quantities, 40recorded quantities, rules, 42reference phase, 51reverse Polish notation, 56rpn

binary operators, 56parsing rules, 56

scan plot menu, 42script control of analyses, 32skin depth, 26, 27snap mode, 34standard analysis functions, 31surface integral, 40surface plot control, 35symmetry

cylindrical, 5planar, 5

TDiff, export to, 3text editor

internal, 20, 32

user-specified contour lines, 36

vector display, 44vector tools, 44virtual drive current, 25volume integral, 40

winding losses, 25

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