Homework # 3 EE517 Fall 2000 Due September 14, 2000

In Woodson & Melcher finish reading chapter 2 and start reading chapter 3.

In Woodson and Melcher do 2.5, and 3.1.

Problem 1

a) What is the B field in the core in terms of the current density in the wire and the area of the window? Assume a uniform magnetic core cross sectional area Ac. The area available to the winding is called the window area A_{w}. The winding factor K is defined as where a_{w} is the area of the wire.

b) What is the current density in the wire in terms of the current in the wire? You do not want the area of the wire in your expression, just A_{w}.

c) What is the inductance of this device assuming N turns?

d) What is the energy stored in the inductor in terms of circuit quantities?

e) Write the energy in part d) in terms of field variables only. Replace the inductance in the energy equation of part d) with the inductance from part c) and group terms so you can identify the field quantities. Obtain an expression for the area product (A_{c} A_{w}) of the core in terms of the B field, the current density, and the energy to be stored by the inductor. Note the maximum B field occurs when the current is a maximum and thus the stored energy is a maximum.

Problem 2

You are to design an inductor with a value of 0.5 mH using a gapped core similar to the one in problem 1. You may assume the iron is infinitely permeable. The inductor must keep this value up to a maximum current of 3.0 A (it can not saturate). You may assume the current is sinusoidal. The iron you are using saturates at a magnetic field of 0.3 Tesla. The inductor is air cooled so a reasonable maximum rms wire current density is 500 A / cm2. The winding factor is expected to equal to 0.5.

a) What is the maximum energy this inductor must be able to store?

b) What is the minimum area product required for this inductor? See problem 1.

Assume the area of the inductor's window is 2 times the area of the core.

c) What is the area of the window and what is the area of the core?

d) How many turns are required and what is the air gap length?

e) Assuming all areas are square, draw the core and dimension it.

f) Verify your results are correct.

Problem 3

- Write down Faraday's law in integral form for this problem making all suitable approximations.
- Sketch the ideal electric field lines between the parallel plates. Write the form this field will take.
- Write down Gauss's law in integral form. Using the electric field with the form in b) compute the electric field in terms of the charge on the upper plate? Sketch the surface and volume you use indicating the normals on the surface.
- Integrate the electric field from one plate to the other using the field in c) and using the potential. What is the electric field equal to in terms of the applied voltage? What is the charge equal to in terms of the applied voltage?
- Recall that the divergence of Ampere's law gives conservation of charge. Write down the integral form of conservation of charge valid for this problem. Sketch the surface and volume to use to find the current going to the upper plated in terms of the charge on the upper plate.