Homework # 6 EE517 Fall 2000 Due October 10, 2000

In Woodson & Melcher read chapter 5 section 5.1.

Do problem 3.3 in Woodson and Melcher.

Problem 1

Two parallel copper plates are held a fixed distance, d, apart. A shorting bar shorts the two plates and the shorting bar (slider) is free to move in the x direction. You may assume the plates extend to infinity in the x direction and that their width W is much greater than d so that the magnetic field is zero everywhere except between the plates (d = 1cm and W = 8cm). You may neglect resistance.

1. What is the magnetic field between the plates for X < x and for X > x ( x is the value of X where the sliding conductor is located? Draw your Ampere contour and surface for finding the field. Label the direction for the contour integral and the normal to the surface
2. Compute the flux linked by this one turn coil. Show your Faraday contour and surface. What three things can be done to minimize the inductance?
3. Compute the systems co-energy in terms of the geometry and the current.
4. What is the force on the slider in terms of the geometry and the current? Which direction will the slider move in?
5. Write the generalized circuit equations and the equations of motion for the slider. It is easiest to write the equations of motion in state variable form. What order is this system of equations?
6. If the current is produced by a DC current source and at t = 0 the slider's velocity is 0, what is its velocity as a function of time for t > 0 in terms of the current and geometry? How does using a current source effect the order of the system?
7. What current is required to generate 1lb of force on the slider when x = 1m (a number is required)?
8. What is the mechanical power out of the system versus position moving in the x direction for the conditions in part f)?
9. What is the terminal voltage versus position moving in the x direction for the conditions in part f)?
10. What is the electrical power into the system versus position moving in the x direction for the conditions in part f)?
11. How would things change if the current source were a sinusoidal current source instead of a constant (DC) current source?

Problem 2

Consider the line cord to your TV made up of two #14 solid Cu wires 6 ft long running parallel to each with the conductor centers 0.15 inches apart. The inductance of the line cord is

where D is the spacing between the centers of the wire, and l is the length of the two parallel wires .

1. Make a drawing of the path in D-i space that you use to find the co-energy. Label all important quantities along the path.
2. Find an expression for the co-energy of the wires assuming that the distance D between the wires is the mechanical variable.
3. Write an analytic expression for the force between the conductors in the line cord.
4. Do the conductors repel each other or attract each other?
5. Look up the OR (OD) of #14 wire and compute the force for a DC current equal to 15A. Give your answer in both Newton and in lbs (numbers are required).
6. For the case of a sinusoidal current, derive an expression for the average force between the conductors in terms of the rms current. Note the force is proportional to I2 just as the power dissipated in a resistor is (think how you calculate average power or just do the average integral).