DOC PREVIEW
MIT 6 013 - Problem Set #9

This preview shows page 1-2 out of 5 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Problem 9.1 Problem 9.4MIT OpenCourseWare http://ocw.mit.edu 6.013/ESD.013J Electromagnetics and Applications, Fall 2005 Please use the following citation format: Markus Zahn, Erich Ippen, and David Staelin, 6.013/ESD.013J Electromagnetics and Applications, Fall 2005. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/termsMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.013 Electromagnetics and Applications Problem Set #9 Issued: 11/9/05 Fall Term 2005 Due: 11/23/05 Suggested Reading Assignment: Chapters 4 and 8. Quiz 2 will be on Thursday, November 17 at 10-11 a.m. It will cover material through P. S. #8,with a focus on sinusoidal steady state and transient waves on transmission lines; parallel plate,rectangular, and dielectric waveguides. Quiz 2 is a closed book exam. Quiz 2 Formula Sheets(as attached to this problem set) will be provided. No individually prepared formula sheets will beallowed. Problem 9.1 A coaxial cylinder inductor is dipped into a magnetizable fluid with permeability μ and mass density mρ. A current I flows down the center conductor and returns up the outer cylinder. The total height of the cylindrical inductor is s+l where l is the inductor height above the outside fluid level and is the inductor length below the outside fluid level. s a) Calculate Band H in the annulus between cylinders in the free space and magnetic fluid regions. Neglect fringing field effects. b) Calculate the self-inductance of the coaxial inductor as a function of the magnetic fluid height h above the outside fluid level. c) What is the magnetic force on the magnetic fluid? d) How high does the fluid rise within the cylinder against the gravitational acceleration downwards of ? hg 1Adapted from Problem 6.37 in Electromagnetic Field Theory: A Problem Solving Approach, by Markus Zahn, 1987. Used with permission.Problem 9.2 The figure below shows a diagrammatic cross section of a two-phase, salient-pole machine. The windings in an actual machine are distributed in many slots along the periphery of the stator, rather than as shown. The rotor is made of magnetically soft iron which has no permanent magnetism. The electrical terminal relations are given by 210 1 2(cos2)sin2LM iM iθθ=+ +210 2sin 2 ( cos2 ), MiLM iλλθθ+− = (a) Determine the torque of electrical origin 12(, ,)eTiiθ. (b) Assume that the machine is excited by sources such that 12cos , sinssiI tiI tωω== , and the rotor has the constant angular velocity mω such that mtθωγ=+ . Evaluate the instantaneous torque . Under what conditions is it constant? eT(c) The rotor is subjected to a mechanical torque (acting on it in the θ+-direction): , where is a constant. The time-varying part of the torque perturbs the steady rotation of (b) so that 0()TTTt′=+0T0()mttθωγγ′=++ . Assume that the rotor has a moment of inertia J but there is no damping. Find the possible equilibrium angles 0γ between the rotor and the stator magnetic field. Then write a differential equation for ()tγ′, with as a driving function. ()Tt′(d) Consider small perturbations of the rotation ( )tγ′, so that the equation of motion found in (c) can be linearized. Find the response to an impulse of torque 00() ()Tt Iut′= , assuming that before the impulse in torque the rotation velocity is constant. (e) Which of the equilibrium phase angles 0γ found in (c) is stable? Problem 5.11 from Electromechanical Dynamics, by Herbert H. Woodson and James R. Melcher, 1968. Used with permission.Problem 9.3 Electrostatic voltmeters are often constructed as shown in the figure above. N pairs of pie-shaped plates form the stator and rotor of a variable capacitor (the figure shows six pairs of rotor plates and six pairs of stator plates). The rotor plates are attached to a conducting shaft that is free to rotate through an angle θ. In the electrostatic voltmeter a pointer is attached to this shaft so that the deflection θ is indicated on a calibrated scale (not shown). a) Determine (,qv)θ, where q is the charge on the stator and v is the voltage applied between the rotor and the stator. The device is constructed so that fringing fields can be ignored and the area of the plates is large compared with the cross section of the shaft. In addition, it is operated in a region of θ in which the plates overlap but not completely. b) Find the torque of electrical origin on the rotor. c) The shaft is attached to a torsional spring with torque-angle relationship TKθ=− What is the static angular deflection θ as a function of voltage? 3Adapted from Problem 3.13 in Electromechanical Dynamics, by Herbert H. Woodson and James R. Melcher, 1968.Problem 9.4 A magnetic circuit, including a movable plunger of mass , is shown in the above figure. The circuit is excited by an N-turn coil with current and consists of a perfectly permeable yoke and plunger miμ→∞ with a variable air gap ()xt and a fixed non-magnetic gap d . The system, with the cross section shown, has a width w into the paper. a) Find the terminal relation for the flux (),ixλlinked by the electrical terminal pair. Ignore fringing in the nonmagnetic gaps. Note that the coil links the flux through the magnetic material times. N b) Find the energy stored in the electromechanical coupling. This should be done by making use of part (a). (,mWλ)x) c) Use the energy function to compute the force of electrical origin (,mWxλef acting on the plunger. d) Write an electrical (circuit) equation of motion involving andxλas the only dependent variables and ()It as a driving function. e) Write the mechanical equation of motion for the mass. This differential equation should have andxλas the only dependent variables, hence taken with the result of (d) should constitute a mathematical formulation appropriate for analyzing the system dynamics. 4Adapted from Problem 3.4 in Electromechanical Dynamics, by Herbert H. Woodson and James R. Melcher,


View Full Document

MIT 6 013 - Problem Set #9

Documents in this Course
LASERS

LASERS

9 pages

Quiz 2

Quiz 2

7 pages

Quiz 1

Quiz 1

6 pages

Load more
Download Problem Set #9
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Problem Set #9 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Problem Set #9 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?