Homework # 7 EE517 Fall 2000 Due October 17, 2000

In Woodson & Melcher read chapter 5 section 5.1.

Do problem 5.1 in Woodson and Melcher.

Problem 1

In the above figure an electromagnet is being used to levitate an iron ball. The force of gravity Fg is down as shown and is equal to m*g where g is the constant gravitational acceleration. The inductance of the electromagnet depends on the position x of the iron ball. Do not worry about any other direction of movement other than the x direction. The inductance of the electromagnet has been measured for different ball positions and a mathematical function as been fitted to the data. For values of x that are of interest the inductance can be approximated as

Note that x must be greater than zero.

a) Plot the inductance as a function of position.

b) What is the electromagnet's co-energy in terms of x, Lo, a and the current?

c) What is the magnetic force on the iron ball?

1. Write down the systems equations of motion.
2. What is the systems static equilibrium? For the static equilibrium, what is the ball's acceleration?
3. Linearize the iron ball's equations of motion about their static equilibrium position. Is the static equilibrium stable?

Problem 2

In the figure below a magnetic elastic beam is free to vibrate in the +/- x direction as shown. The total air gap in the x direction is l. The permeability of the iron and beam may be approximated as infinite. You may assume the elastic beam has an effective mass M so the motion of the beam can be described by a differential equation that contains this mass and the spatial coordinate x. The elastic force due to the bending of the beam can be modeled as a spring force .

1. Show, that neglecting fringing fields, the inductance of this device is
2. Show that the co-energy and force are given by
3. Show the units for the force are correct. Remember that a (Henry Amp2) is a Joule, a Joule is a (Newton m), and a Newton is a unit of force.
4. Write the equations of motion assuming the current is a constant.
5. What is the static equilibrium of your equations?
6. Linearize your equations of motion about their static equilibrium position. For what values of current is the static equilibrium stable or unstable?
7. How could you measure current with this device?