 Applied Electromagnetics Using QuickField™ and MATLAB® is intended
as an introductory level textbook for teaching computer-based electricity,
magnetism and multiphysics. The text is easily accessible to advanced
undergraduates and beginning graduate students in physics and engineering.
Many exercises and demonstrations may be implemented using QuickField and
MATLAB in a traditional introductory level physics course. This second audience
will benefit from the visualization of electric and magnetic field distributions
and force calculations without a working knowledge of the finite element
method or potential theory.
QuickField is a window-based, Finite Element Method (FEM) software package that supports Electrostatics, DC and AC conduction, Magnetostatics, AC and Transient Magnetics, Steady State and Transient Heat Transfer and Stress Analysis problem types. Models are created in a ‘point-and-click’ CAD environment, where material properties and boundary conditions are assigned. Automatic mesh generation and post processing are fast and user-friendly. Solutions to most problems in the textbook can be displayed in a matter of seconds after the model has been created. The textbook is packaged with a companion CD with a student version of the software capable of solving all the problems in the text. Additional examples are included with the software. The student version of QuickField may also be downloaded from the Tera Analysis website at www.quickfield.com. The user’s guide and demonstration videos are also available on the website. Application-based examples in the text and on the website include the calculation of currents in biological tissue under electrical stimulation, superconducting magnetic shielding, magnetic levitation, electromagnetic nondestructive testing as well as the motion of charged particles in electric fields. Multiphysics applications include coupled stress, electromagnetic and thermal analysis. Students taking a course in electromagnetic theory usually concentrate mostly on analytical techniques, e.g., solving differential equations and boundary value problems. Unfortunately, students often come away with a limited understanding of how electromagnetic fields behave. Computer modeling serves to bridge this understanding gap in that it enables visualization of electric and magnetic fields and electrical currents and therefore builds an intuitive and qualitative understanding that is not readily gained in manipulating complex analytical expressions. Analytical methods developed in this text concentrate on separation of variables, conformal mapping, and Laplace transform techniques. Numerical finite difference and Monte Carlo methods are also introduced with examples in MATLAB. Comparison of numerical solutions with theory helps establish confidence in numerical methods and builds experience in establishing the reliability of computational results and the applicability of theoretical approximations. The book includes extensive problem sets that facilitate computer-based learning of electromagnetics and the application of QuickField and MATLAB illustrating some of the basic concepts in electromagnetic theory such as Gauss’ Law and Ampere’s Law. The exercises are designed to allow user selection of different parameters, dimensions, material properties, and initial conditions. Tables of physical properties and characteristic dimensions of engineering materials and biological materials in living cells and the human body are included in Appendices 4 and 5 for the reader’s convenience. The reader is encouraged to conform, modify, and extend these exercises according to his or her own interests. Chapter 1 introduces mathematical preliminaries and MATLAB concurrently with additional MATLAB examples in Appendix 1. The vector analysis component of Chapter 1 provides simple MATLAB examples calculating vector dot and cross products. The divergence, curl, gradient, and Laplacian are also calculated in different coordinate systems. The Laplace Transform introduced in Chapter 1 is used in chapters on transient magnetics, thermal analysis, stress analysis, and electrical circuit modeling. Analytical and computational methods of solving Laplace and Poisson’s equations are developed in Chapter 2. Readers wishing to jump directly into QuickField may begin with Chapter 3 “A Walk Through QuickField.” This chapter will get the reader started simulating simple electrostatic and magnetostatics problems in QuickField with step-by-step visual instructions for plotting electric and magnetic fields, creating contour graphs, and calculating integral values. Chapters 4 through 10 cover electrostatics, magnetostatics, time-harmonic magnetics, transient magnetics, superconductivity, alternating and direct current flow. Chapters 11 and 12 cover thermal and stress analysis and multiphysics examples with coupled heat transfer, stress and electromagnetic coupling. Applications include space capsule atmospheric reentry simulations that couple thermal and stress analysis as well as modeling the temperature distribution resulting from current flow in a fuel cell. The text concludes with Chapter 13 on passive electrical circuits. QuickField includes a CAD-based electrical circuit simulator that simulates circuits with AC or transient time dependence. Applications include filter circuits and equivalent circuit models of neurons and cells under electrical stimulations. |
TABLE OF CONTENTS Section 8.7 - Superconductivity: Hollow Superconducting Shells
Zero Field Cooling If a multiply connected superconductor (MCS), such as a tube or ring, is cooled in the presence of a magnetic field, flux is expelled from the bulk of the superconductor but still penetrates the opening. After turning off the external magnetic field, flux will remain trapped in the opening. If the superconductor is cooled in zero field (ZFC) and the field is then switched on, screening currents will flow that cancel the applied magnetic field in such a way that the total flux through the MCS remains zero. The appropriate MCS boundary condition depends on the field present at the time that the superconducting state is established. In two dimensions, constraining the vector potential to be zero on the surface of the superconductor satisfies the zero flux condition. The field-cooled case is modeled with zero normal field, or permeability. Flux Trapping A multi-connected superconducting body such as a ring or a tube will trap any flux present in the hole of the superconductor upon transition to the superconducting state. If the field is then changed after the superconducting state is established, screening currents will flow so as to keep the amount of trapped flux a constant. In this way a ZFC superconductor will maintain the zero flux condition upon the application of an external magnetic field. Superconducting Ring A field-cooled superconducting washer is simulated in an axial background field
in Figure 8.5 (a). The cross-section of the washer appears as two strips in A ZFC superconducting ring in a uniform axial magnetic field is modeled in r-z coordinates with the boundary condition
is constrained to be zero. In general, flux lines will still penetrate the ring although the total flux will remain zero. In r-z coordinates, for Normalized plots of Superconducting Tube Figure 8.7 (a) illustrates the flux line distribution of a superconducting tube in an axial uniform field. The tube is modeled as two parallel strips in
the axial field. For a superconducting tube with rotational symmetry, the field attenuates exponentially along the axis of the tube according to
and
for axial and transverse components of the incident field, respectively. The z-component magnetic field along the tube axis is plotted in Figure 8.8 showing agreement with the theoretical attenuation factor (8.9). The transverse attenuation factor for a highly permeable tube (see Chapter 5) is the same as the axial attenuation factor for a superconducting tube of the same radius. This suggests combining superconducting and highly permeable materials for shielding applications. |
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coordinates. Here the background field is seen to be diverted by the sides of the
washer but still penetrates the hole of the superconductor. The boundary condition
tion applied here is 
on the superconductor. This boundary condition is
also applicable modeling a superconducting washer with a thin slit so that the
superconducting path is not complete. In this case the flux density is enhanced
in the opening of the washer as is evident by the increased density of flux lines in
that region. The washer then acts as a superconducting flux-focuser. In Figure
8.5 (b) the boundary condition Az = 0 is applied to the strip washer simulating the
ZFC condition, where flux lines are now expelled from the interior.-83.jpg)
applied to the superconductor so that, by Stokes' theorem, the total flux-84.jpg)
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along a radial contour of a superconducting ring with a 5 cm O.R. and a 1 cm I.R. modeled in r-z coordinates (a) zero field-cooled ring (b) field-cooled ring and (c) ring in normal state.-88.jpg)
the sign of-89.jpg)
must change along a radial contour from the center to the edge of the ring. Notably, a locus or ring of points exists, where the axial component of magnetic field vanishes completely inside the ring.-90.jpg)
along a radial contour of a field cooled, superconducting ring (or flux focuser), a ZFC ring, and a normal state ring are shown in Figure 8.6. The ring has an O.D. of 10 cm and an I.D. of 1 cm. The magnetic field is unattenuated by the normal state ring and is increased by a factor of 2.3 at the center of the field-cooled ring. At the center of the ZFC ring, the field is reduced by a factor of 10 and is zero at the nodal point (where -92.jpg)
coordinates with boundary condition Az = 0. As a comparison Figure 8.7 (b) shows the calculated flux density with the same tube in a transverse x-directed field. Note the greater field penetration into the tube by the transverse field compared to
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