Homework # 5 EE517 Fall 2000 Due September 26, 2000

In Woodson & Melcher finish reading chapter 3 and chapter 5 section 5.1.

Problem 1

The iron's permeability may be assumed to be infinite.

a) Write Ampere's law for the above magnetic device.

b) What is the B field in the iron where the winding is?

c) What is the flux linked by the coil and the inductance?

  1. The air gaps g1 and g2 are not independent of each other. What is the relationship between them that is imposed by the geometry? (hint: consider g1 = 0 and g2 = 0)

Problem 2

The mean path length around the iron core is lm. The winding factor is k and the area of the window is Aw.

  1. Assume you know the B field in the iron where the winding is. Use Faraday's law to find the terminal voltage.
  2. How can you measure the B field?
  3. Use ampere's law to find the H field.
  4. How can you measure the H field?
  5. If a stepped voltage of value Vo for t > 0 is applied to the winding and the initial field is zero, what is the B field in the iron as a function of time and how long does it take for the B field to reach Bsat?5 points
  6. Theoretically (from your result in e)) how does the B change for times greater than the time for the iron to saturate. Do you think this happens? If not why not?
  7. If you want the maximum current density in the copper to be to be Jmax, how does the maximum possible winding current depend on the number of turns?

Problem 3

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 x must be less than a.

a) Write down the expression for conservation of energy for the electromagnet. Repeat the same for co-energy. What is the definition of co-energy in terms of the energy, flux and current?

b) What is the co-energy for the electromagnet?

c) What is the magnetic force on the iron ball? Is the force to up or down?

d) At what current will the net force of gravity and magnetic force be zero?