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 F_{g} 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, L_{o}, a and the current?

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

- Write down the systems equations of motion.
- What is the systems static equilibrium? For the static equilibrium, what is the ball's acceleration?
- 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 .

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