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ECE 320
Energy Conversion and Power Electronics
Spring 2009
Instructor: Tim Hogan
Chapter 2: Magnetic Circuits and Materials
Chapter Objectives
In this chapter you will learn the following:
•
How Maxwell’s equations can be simplified to solve simple practical magnetic problems
•
The concepts of saturation and hysteresis of magnetic materials
•
The characteristics of permanent magnets and how they can be used to solve simple problems
•
How Faraday’s law can be used in simple windings and magnetic circuits
•
Power loss mechanisms in magnetic materials
•
How force and torque is developed in magnetic fields
2.1
Ampere’s Law and Magnetic Quantities
Ampere’s experiment is illustrated in Figure 1 where there is a force on a small current element
I
2
l
when it is placed a distance,
r
, from a very long conductor carrying current
I
1
and that force is
quantified as:
l
I
r
I
F
2
1
2
π
μ
=
(N)
(2.1)
I
1
r
I
2
F
Conductor
1
Current Element
of Length
l
Figure 1.
Ampere’s experiment of forces between current carrying wires.
The magnetic flux density,
B
, is defined as the first portion of equation (2.1) such that:
l
BI
F
2
=
(N)
(2.2)
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From (2.1) and (2.2) we see the magnetic flux density around conductor 1 is proportional to the
current through conductor 1,
I
1
, and inversely proportional to the distance from conductor 1.
Looking
at the units and constants handout given in class, or from (2.2) the units of,
B
, are seen as
⎟
⎠
⎞
⎜
⎝
⎛
⋅
m
A
N
,
thus
μ
, called permeability, has units of
⎟
⎠
⎞
⎜
⎝
⎛
2
A
N
.
More commonly, the relative permeability of a given
material is given where
0
μ
r
=
and
⎟
⎠
⎞
⎜
⎝
⎛
⋅
×
=
−
m
A
N
10
400
9
0
π
.
Since a Newtonmeter is a Joule,
and a Joule is a Wattsecond:
⎟
⎠
⎞
⎜
⎝
⎛
⋅
=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅
⋅
=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
⋅
=
⎟
⎠
⎞
⎜
⎝
⎛
⋅
2
2
2
2
m
s
V
m
A
s
A
V
m
A
J
m
A
m
N
m
A
N
.
This shows
B
is
a per meter squared quantity, and the (V·s) units represents the magnetic flux and is given units of
Webers (Wb).
This flux can be found by integrating the normal component of
B
over the area of a
given surface:
∫
⋅
=
S
ds
n
B
ˆ
φ
(2.3)
The magnetic field intensity is related to the magnetic flux density by the permeability of the
media in which the magnetic flux exists.
B
H
≡
(2.4)
For the system in Figure 1,
r
I
B
H
2
1
=
=
and have units of
⎟
⎠
⎞
⎜
⎝
⎛
m
A
.
If there were multiple
conductors in place of conductor 1, for example in a coil, then the units would be ampereturns per
meter.
A line integration of
H
over a closed circular path gives the current enclosed by that path, or
for the system in Figure 1:
1
2
0
1
2
I
dl
r
I
dl
H
H
r
C
=
=
⋅
=
∫
∫
(2.5)
again, if the system contained multiple conductors within the enclosed path, the result would give
ampereturns.
Equation (2.5) is Ampere’s circuital law.
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This note was uploaded on 05/30/2009 for the course ECE ECE 320 taught by Professor Timhogan during the Spring '09 term at Michigan State University.
 Spring '09
 TimHogan

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