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to accompany
Sixth Edition
A.E. Fitzgerald
Charles Kingsley, Jr.
Stephen D. Umans

1-1
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Simple magnetic circuit.
Figure 1.1
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Magnetic circuit with air gap.
Figure 1.2
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Analogy between electric and magnetic circuits.
(a) Electric circuit, (b) magnetic circuit.
Figure 1.3
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Air-gap fringing fields.
Figure 1.4
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Simple synchronous machine.
Figure 1.5
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1-7
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1-8
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Faraday’s Law
•
When magnetic field varies in time an electric field is produced in space as determined by Faraday’s Law:

E .
ds
  d 
B .
da dt
C S
• Line integral of the electric field intensity E around a closed contour C is equal to the time rate of the magnetic flux linking that contour.
• Since the winding (and hence the contour C) links the core flux N times then above equation reduces e ( t )
 d
 dt

N d
 dt
• The induced voltage is usually refered as electromotive force to represent the voltage due to a time-varying flux linkage.

1-9
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The direction of emf: If the winding terminals were short-circuited a current would flow in such a direction as to oppose the change of flux linkage.

( t )
  max sin
 t

A c
B max sin
 t e ( t )
 
N
 max cos
 t

E max cos
 t
E max
 
N
 max

2
 f NA c
B max
E rm s

2
 f NA c
B max
+ e(t)
-
N

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(a) Magnetic circuit and (b) equivalent circuit for Example 1.3.
Figure 1.6
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MATLAB plot of inductance vs. relative permeability for
Example 1.5.
Figure 1.7
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Magnetic circuit with two windings.
Figure 1.8
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B-H loops for M-5 grain-oriented electrical steel 0.012 in thick.
Only the top halves of the loops are shown here. (Armco Inc.)
Figure 1.9
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Dc magnetization curve for M-5 grain-oriented electrical steel
0.012 in thick. (Armco Inc.)
Figure 1.10
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1-16
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Excitation phenomena. (a) Voltage, flux, and exciting current;
(b) corresponding hysteresis loop.
Figure 1.11
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Exciting rms voltamperes per kilogram at 60 Hz for M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.)
Figure 1.12
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1-19
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CORE LOSSES
Hysteresis Losses: hysteresis loss is proportional to the loop area
(shaded).
Figure 1.13
W
  i  d
  
H c l c
N
A c
NdB c

A c l c

H c dB c
Eddy Current Losses:
Time-varying magnetic fields give rise to electric fields in the material resulting in induced currents. These induced currents cause Eddy Current
Losses. These losses can be reduced by using thin sheets of laminations of the magnetic material.

1-20
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Core loss at 60 Hz in watts per kilogram for M-5 grain-oriented electrical steel 0.012 in thick. (Armco Inc.)
Figure 1.14
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Example 1.8.
The magnetic core is made from laminations of M-5 grain-oriented electrical steel. The winding is excited with a 60 Hz voltage to produce a flux density in the steel of
B=1.5sin wt T, where w=377 rad/sec. The steel occupies 0.94 of the core crosssectional area. The mass-density of the steel is 7.65 g/cm3. Find a) The applied voltage, (Note that 1 meter is 39.4 inch) b) The peak current, c) The rms exciting current, and d) The core loss.
Solution:
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1-25
(a) Second quadrant of hysteresis loop for Alnico 5;
(b) Second quadrant of hysteresis loop for M-5 electrical steel;
(c) hysteresis loop for M-5 electrical steel expanded for small B. (Armco Inc.)
Figure 1.16

1-26
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Magnetization curves for common permanent-magnet materials.
Figure 1.19
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Portion of a B-H characteristic showing a minor loop and a recoil line.
Figure 1.21
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1-29
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Example (
Final Exam Question
).
Consider the magnetic circuit shown on the right. The permanent magnet material has cross-sectional area A m
=4 cm 2 and length l m
=3.45 mm. The air-gap has the same cross-sectional area as the magnet and its length is g=2 mm. The iron core has infinite permeability. Assume no leakage flux, no fringing.
Answer the following.
a) Express the area and the length in meters correctly (if you get the units
incorrectly, it will be carried out to the remaining of the problem) and then obtain the load line equation.
b) Draw the load line in the graph and give the magnetic field intensity value and magnetic flux density value of the operating point for Samarium-Cobalt .

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Load Line

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Example (
MidTerm 1 Exam Question
).
It is desired to achieve a time-varying magnetic flux density in the air gap of the magnetic circuit in the figure of the form B g
=B o
+B
1 sin wt where B
0
=0.5 Wb/m 2 and B
1
=0.25 T. The dc field B
0 is to be created by a Neodymium-Iron-Boron (characteristic is shown below) magnet, whereas the time-varying field is to be created by a time-varying current. Please, answer the following for A g
=6 cm 2 , g=0.4 cm, and N=200 turns.
a) Write down the flux density of NdFeB as a function of the magnetic field intensity using the characteristic given b) The magnet length and the magnet area A m density and minimize the magnet volume.
that will achieve the desired dc air gap flux
1.4
B
Wb/m
2
1.2
294 kJ/m
3
20
0 kJ/ m
3
10
0 kJ/ m
3
1
0.8
0.6
0.4
0.2
-1000 -800 -600
H, (kA/m)
-400 -200
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Example:
Consider the figure below and answer the following.
a) Draw magnetic equivalent circuit, b) Find the reluctances as a function of x, c) Calculate the flux, d) Calculate the flux linkage of the coil, e) Calculate the inductance of the coil.
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Magnetic circuit for Example 1.10.
Figure 1.18
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Magnetic circuit including both a permanent magnet and an excitation winding.
Figure 1.20
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Magnetic circuit for Example 1.11.
Figure 1.22
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(a) Magnetization curve for Alnico 5 for Example 1.11;
(b) series of load lines for A g
= 2 cm 2 and varying of values of i showing the magnetization procedure for Example 1.11.
Figure 1.23
(a)
(b)
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Magnetic circuit for Problem 1.1.
Figure 1.24
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Magnetic circuit for
Problem 1.9.
Figure 1.26
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Inductor for Problem 1.12.
Figure 1.27
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Pot-core inductor for Problem 1.15.
Figure 1.28
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Inductor for Problem 1.17.
Figure 1.29
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Toroidal winding for Problem 1.19.
Figure 1.30
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Iron-core inductor for Problem 1.20.
Figure 1.31
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Magnetic circuit for Problem 1.22.
Figure 1.32
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Symmetric magnetic circuit for Problem 1.23.
Figure 1.33
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Reciprocating generator for Problem 1.24.
Figure 1.34
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Configuration for measurement of magnetic properties of electrical steel.
Figure 1.35
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Magnetic circuit for Problem 1.28.
Figure 1.36
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Magnetic circuit for the loudspeaker of Problem 1.34
(voice coil not shown).
Figure 1.37
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Magnetic circuit for Problem 1.35.
Figure 1.38
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