Study Guide for Exam - Field Waves I | ECSE 2100, Study notes of Electrical and Electronics Engineering

Download Study Guide for Exam - Field Waves I | ECSE 2100 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! Fields and Waves I ECSE-2100 Spring 2000 K. A. Connor Revised: 4/10/00 Rensselaer Polytechnic Institute Troy, New York, USA 1 Lesson Summaries Version 1.4 For each of the in-class lessons, a summary of what should have been learned is presented. The particular problems and experiments done in each lesson contain examples of each item listed. Other examples are included in the text and class notes. Lesson 1 – Intro to Fields and Vector Mathematics Review In these lessons, the basic vector mathematics necessary to apply Maxwell’s Equations and the related equations (e.g. Coulomb’s Law and the Biot-Savart Law) is investigated. As a general rule, one need only look at Maxwell’s Equations in both integral and differential form to see the mathematical tools we require in this course. 1.1 Intro to Fields and Waves • In a circuit application, capacitive effects can occur due to proximity of wiring, the box enclosing the circuit, etc. • In a circuit application, inductive effects can occur when wire loops surround the same region of space. A transformer, even if it is a poor one, can be created by wrapping a few turns of wire around a conventional solenoidal inductor. • Wires, whether open or closed loops can pick up a large variety of noise signals from standard electrical devices such as power cords, computer screens etc. • Standard cabling, such as a coaxial cable, behaves very differently at low and high frequencies. Low frequencies are those for which the physical dimensions of the system of interest are very small compared to a wavelength. • Transmission lines must be terminated properly to work correctly. 1.2 Basic Math and Coordinate Systems • There are three coordinate systems we work with in an introductory level Electromagnetics course – rectangular, cylindrical and spherical • In these coordinate systems, you should be able to: o Add and subtract vectors and do dot and cross products o Sketch surfaces on which one coordinate is a constant. o Find the area of surfaces on which one coordinate is a constant o Identify the vector surface element dS r for each surface in item 2. o Sketch volumes defined by coordinate surfaces (surfaces on which a coordinate is a constant). o Find the volumes of item 5. o Integrate functions over volumes of the form ( )f r dv∫ r , where ( )f rr is some function expressed in one of the three standard coordinate systems. • You should know the surface areas and volumes of rectangles, cylinders and spheres. Fields and Waves I ECSE-2100 Spring 2000 K. A. Connor Revised: 4/10/00 Rensselaer Polytechnic Institute Troy, New York, USA 2 • You should know the line elements, surface elements and volume elements for each system. • You should be able to convert a vector from one coordinate system to another. 1.3 Gradient, Line Integrals and Curl You should be able to: • Do closed line integrals of the form r r B dl⋅∫ , particularly for paths on which a coordinate is a constant. • Take the curl ∇ × r B of vector functions we will find as electric and magnetic fields. These usually depend on a single coordinate and have only one component, but not in every case. • Take the gradient of a scalar ∇ f to obtain a vector function In addition, you should understand: • Stokes Theorem and how to apply it • That fields with no curl satisfy r r B dl⋅ =∫ 0 • That fields with no curl can be represented by the gradient of a scalar. 1.4 Surface Integrals and Divergence You should be able to: • Evaluate vector area integrals of the form r r A dS⋅∫ for surfaces on which a coordinate is a constant. • Evaluate the divergence ∇ ⋅ r A of a vector. In addition, you should understand: • The divergence theorem and how to apply it. 1.5 Wave Properties and Phasors Will be added in a future version of this list. Lesson 2 – Electric Fields In these lessons we learn how to apply Coulomb’s Law, Gauss’ Law and Poisson’s Equation to find the electric field ( )E rr and the electric potential ( )V rr from a given charge distribution with and without conductors and dielectrics. The methods learned can mostly only be applied to highly symmetric problems (describable with a single coordinate). The exception is direct application of Coulomb’s Law for a very limited set of problems and the application of a simple finite difference numerical technique. We also address the boundary conditions for electric fields and electric flux density ( ) r r D r , Fields and Waves I ECSE-2100 Spring 2000 K. A. Connor Revised: 4/10/00 Rensselaer Polytechnic Institute Troy, New York, USA 5 • Find the analytical solution to one-dimensional electric field problems using Laplace’s or Poisson’s equations. (These are the same problems we can solve using Gauss’ Law.) • Write the finite difference versions of Laplace’s and Poisson’s equations. • Solve the finite difference Laplace’s equation using a spreadsheet . Cases are limited to two-dimensional problems. However, both open and closed boundaries and problems with and without dielectric materials that partially fill the field regions must be considered. 2.7 Method of Images (Note: We did not cover this material this term.) You should be able to: • Solve for the electric potential V and electric field E of either a point charge or a line charge located near a conducting plane. In the case of the line charge the line should be parallel to the plane. • Use the results of this analysis to find the capacitance of wires over ground planes or charged spheres over ground planes. Lesson 3 – Magnetic Fields In these lessons we learn how to apply Ampere’s Law and the Biot/Savart Law to find the magnetic field ( ) r r B r and the magnetic vector potential ( ) r r A r from a given current distribution with and without conductors and magnetic materials. The methods learned can mostly only be applied to highly symmetric problems (describable with a single coordinate). The Biot/Savart Law will be used primarily to address the symmetries of the problems. We will also learn to apply Ohm’s Law in point form to find the resistance of conductors. We also address the boundary conditions for the magnetic flux density ( ) r r B r and the magnetic field intensity ( ) r r H r , particularly at boundaries between two magnetic materials or between a magnetic material and a conductor. We address the application of Faraday’s Law to find the voltage induced in a circuit due to a changing magnetic flux. We address how to determine the mutual and self inductance, the stored energy of and forces exerted on a conductor/magnetic material configuration. Finally, we learn a little bit about permanent magnets and the magnetic circuit analysis technique. 3.1 Currents and Resistance You should be able to: • Find the total current carried by a conductor from the current density. • Determine the resistance of a conductor with a uniform current distribution • Determine the resistance of a conductor with a non-uniform current distribution • Measure the resistance of a cable and compare with calculations Fields and Waves I ECSE-2100 Spring 2000 K. A. Connor Revised: 4/10/00 Rensselaer Polytechnic Institute Troy, New York, USA 6 3.2 Magnetic Fields and Ampere’s Law You should be able to: • Determine from a field line plot or from a field expression whether a given field could or could not be a magnetic field. • Determine the symmetries of standard geometries such as coaxial cables, solenoids and toroids. • Determine the magnetic field (either r B or r H ) from a given current density for any of the simple one-dimensional standard geometries. 3.3 Flux and Magnetic Vector Potential You should be able to: • Interpret a flux diagram (a collection of typical magnetic field lines) to determine, for example, the flux at one location in terms of the flux at another. • Determine the internal or external flux produced by a given current distribution using either the magnetic field ( ) r r B r or the magnetic vector potential ( ) r r A r . That is, by apply either ψ m B dS= ⋅∫ r r or ψ m A dl= ⋅∫ r r . 3.4 Faraday’s Law You should be able to: • Find the magnetic field (either r B or r H ) from a given standard current distribution • Find the flux in a typical coil cross-section using either ψ m B dS= ⋅∫ r r or ψ m A dl= ⋅∫ r r . • Find the total flux linked by a coil Λ = N mψ • Determine the emf ( )V t d dt = − Λ induced in a coil by a changed flux linkage • Perform a simple experiment that shows the flux from one circuit linking another • Find the emf produced in a loop moving through a static magnetic field. 3.5 Inductance You should be able to: • Find the self-inductance of a simple configuration such as a coaxial cable, a solenoid or a torus using the expression L I = Λ after first finding Λ following the procedures of lesson 3.4. • Find the mutual-inductance of two simple configurations located in reasonably close proximity to one another. Fields and Waves I ECSE-2100 Spring 2000 K. A. Connor Revised: 4/10/00 Rensselaer Polytechnic Institute Troy, New York, USA 7 • Find the approximate self or mutual inductance of a simple circuit using information provided on the magnetic field at some typical points. This information could either have been determined analytically or using numerical methods. • Perform an experiment to show that the calculation of the mutual inductance is in reasonable agreement with actual application 3.6 Magnetic Materials You should be able to: • Calculate the magnetic field (either r B or r H ) from a given standard current distribution including the effects of magnetic materials. • Determine the change in direction of a magnetic field at the boundary between two magnetic materials • Perform an experiment that shows the ability of a magnetic material to conduct magnetic flux from one coil to another and thus improve coupling between the coils 3.7 Magnetic Circuits You should be able to: • Apply the magnetic circuit method to determine the magnetic field and flux in a configuration of coils and magnetic materials. • Use the magnetic field information obtained using the magnetic circuit method to determine inductance and energy storage of a simple configuration 3.8 Magnetic Energy and Force You should be able to: • Determine the energy stored in a magnetic field configuration • Find the inductance from the energy stored in a magnetic field configuration • Qualitatively describe the force between a magnetic field and a magnetic material (for example between a permanent magnet and a piece of iron). That is, estimate changes in energy as the magnetic field and the magnetic material move with respect to one another. Lesson 4 – Transmission Lines In these lessons, we address what happens to voltages and currents on transmission lines (cables at sufficiently high frequency that wave phenomena become important). More class time is spent doing experiments than in any other part of the course. We study quite thoroughly the properties of 100 meter lengths of RG58A/U coaxial cable, since that is the cable we use to connect the instruments in the studio classroom. Also, this is very much a standard cable with many, many uses in the electrical and electronics industry. For