Chapter 1
Introduction to Electromechanical
Energy Conversion
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1.1 Magnetic Circuits
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Magnetic Field Concept
Magnetic Fields:
• Magnetic fields are the fundamental mechanism
by which energy is converted (transferred)
from one form to another in electrical machines.
Magnetic Material
• Definition : A material that has potential to attract other
materials toward it, materials such as iron, cobalt, nickel
• Function: Act as a medium to shape and direct the
magnetic field in the energy conversion process
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Magnetic Field Concept
• Magnetic field around a bar
magnet
• Two “poles” dictated by
direction of the field
• Opposite poles attract
(aligned magnetic field)
• Same poles repel (opposing
magnetic field)
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Magnetic Field Concept
Magnetic Flux/ Flux Line Characteristic
1. Outside - Leaves the north pole (N) and enters
the south pole (S) of a magnet. Inside - Leaves
the south pole (S) and enters the north pole (N)
of a magnet.
2. Like (NN, SS) magnetic poles repel each other.
3. Unlike (NS) magnetic poles attracts each other.
4. Magnetic lines of force (flux) are always
continuous (closed) loops, and try to make as
shortest distance loop.
5. Flux line never cross each others
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Magnetic Field Concept
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Machines Basic Requirements
• Presence of a “magnetic fields” can be
produced by:
– Use of permanent magnets
– Use of electromagnets
• Then one of the following method is needed:
– Motion to produce electric current
(generator)
– Electric current to produce motion (motor)
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Ampere’s Law
• Any current carrying wire will produce
magnetic field around itself.
Magnetic field around a wire:
• Thumb indicates direction of current flow
• Finger curl indicates the direction of field
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Ampere’s Law
Ampere’s law: the line integral of magnetic
field intensity around a closed path is equal to
the sum of the currents flowing through the
surface bounded by the path
H

H  dl 
i
I1 I
2

dl
Recall that the vector dot product is given by

H  dl  Hdl cos(  ) in which  is the angle
between H and dl.
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Ampere’s Law
If the magnetic intensity has constant
magnitude and points in the same direction as
the incremental length dl everywhere along
the path, Ampere’s law reduces to
Hl 
i
in which l is the length of the path.
Examples
of such cases: (i) Magnetic field

around a long straight wire, (ii) Solenoid
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Example 1:
( a long straight
Wire)
Example 2:
(Solenoid)
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Flux Density
• Number of lines of magnetic force (flux)
passing through unit area
or Wb/m2
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Field Intensity
• The effort made by the current in the wire
to setup a magnetic field.
• Magnetomotive force (mmf) per unit length
is known as the “magnetizing force” H
• Magnetizing force and flux density related
by:
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Permeability
• Permeability  is a measure of the ease by
which a magnetic flux can pass through a
material (Wb/Am). The higher the better flux
can flow in the magnetic materials.
• Permeability of free space o = 4 x 10-7
(Wb/Am)
• Relative permeability, r :
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Reluctance
• Reluctance, which is similar to resistance, is the
opposition to the establishment of a magnetic
field, i.e." resistance” to flow of magnetic flux.
Depends on length of magnetic path , crosssection area A and permeability of material .
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Magnetomotive Force
• The product of the number of turns
and the current in the wire wrapped
around the core’s arm. (The ability of a
coil to produce flux)
N
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Magnetomotive Force
• The MMF is generated by the coil
• Strength related to number of turns and current,
Symbol F, measured in Ampere turns (At)
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Magnetization Curve
Behavior of flux density compared with
magnetic field strength, if magnetic intensity H
increases by increase of current I, the flux
density B in the core changes as shown.
B(T) 
flux ()
Near
saturation
Saturation
B   0 r H
linear
H(A/m) 
current (I)
Magnetization curve (B-H characteristic)
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Magnetic Equivalent Circuit

i
lc
N
+
F
-

i
Analogy between magnetic
circuit and electric circuit
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Magnetic Circuit with Air Gap
l
c

c
i
N
lg
c 
 
lc
 cA c
Ni
C  g
;

+
F
-
g 
g
lg
 0A g
Ni  H c l c  H g l g
Flux density
Bc 
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c
Ac
;
Bg 
g
Ag
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Parallel Magnetic Circuit
l2
l1
I
l3
3
1
S1
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II 
2
S3
I
N
S2
+ NI
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Loop I
NI = S33 + S11
= H3l3 + H1l1
Loop II
NI = S33 + S22
= H3l3 + H2l2
Loop III
0 = S 1  1 + S2  2
= H1l1 + H2l2
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Electric vs Magnetic Circuit
Magnetic circuit
Electric circuit
Term
Symbol
Term
Symbol
Magnetic flux

Electric current
I
Flux density
B
Current density
J
Magnetomotive
force
F
Electromotive
force
E
Permeability

Permitivity
e
Resistance
R
Reluctance
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Leakage Flux
• Part of the flux generated by a current-carrying
coil wrapped around a leg of a magnetic core
stays outside the core. This flux is called
leakage flux.
Useful
flux
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Fringing Effect
• The effective area provided for the flow of
lines of magnetic force (flux) in an air gap is
larger than the cross-sectional area of the core.
This is due to a phenomenon known as fringing
effect.
Air gap
– to avoid flux
saturation when
too much current
flows
- To increase
reluctance
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Example 1
Refer to Figure below, calculate:1) Flux 2) Flux density 3) Magnetic intensity
Given r = 1,000; no of turn, N = 500; current, i = 0.1 A.
cross sectional area, A = 0.0001m2 , and means length
core lC = 0.36 m.
lc
i
N
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1.
2.
3.
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1.75x10-5 Wb
0.175 Wb/m2
139 AT/Wb
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Example 2
lg

i
Movable
part
Data- 1T –
700 at/m
N
• The Figure represents the magnetic circuit of a relay. The coil has
500 turns and the mean core path is lc = 400 mm. When the airgap lengths are 2 mm each, a flux density of 1.0 Tesla is required
to actuate the relay. The core is cast steel.
a. Find the current in the coil.
(6.93 A)
b. Compute the values of permeability and relative permeability
of the core.
(1.14 x 103 , 1.27)
c. If the air-gap is zero, find the current in the coil for the same
flux density (1 T) in the core.
( 0.6 A)
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Electromagnetic Induction
• An emf can be induced in a coil if the magnetic flux
through the coil is changed. This phenomenon is known
as electromagnetic induction.
d
• The induced emf is given by
e  N
dt
• Faraday’s law: The induced emf is proportional to the
rate of change of the magnetic flux.
• This law is a basic law of electromagnetism relating to
the operating principles of transformers, inductors, and
many types of electrical motors and generators.
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Electromagnetic Induction
• Faraday's law is a single equation describing two
different phenomena: The motional emf generated by a
magnetic force on a moving wire, and the transformer
emf generated by an electric force due to a changing
magnetic field.
• The negative sign in Faraday's law comes from the fact
that the emf induced in the coil acts to oppose any
change in the magnetic flux.
• Lenz's law: The induced emf generates a current that
sets up a magnetic field which acts to oppose the change
in magnetic flux.
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Lenz’s Law
An induced current has a direction such that the magnetic
field due to the induced current opposes the change in
the magnetic flux that induces the current.
As the magnet is moved toward the
loop, the B through the loop
increases, therefore a counterclockwise current is induced in the
loop. The current produces its own
magnetic field to oppose the motion of
the magnet
If we pull the magnet away from the
loop, the B through the loop
decreases, inducing a current in the
loop. In this case, the loop will have a
south pole facing the retreating north
pole of the magnet as to oppose the
retreat. Therefore, the induced current
will be clockwise.
Self-Inductance
• From Faraday’s law
e N
d

dt
dl
dt
• Where l is the flux linkage of the winding is defined as

l  N
• For a magnetic circuit composed of constant magnetic
permeability,
the relationship between  and i will be

linear and we can define the inductance L as
L
l
i
• It can be shown later that
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
L 
N
2
 eq
Self-Inductance
• For a magnetic circuit having
constant magnetic permeability

F

Ni

l
 o r A
• So,
L
N
i

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
N
i
.
Ni
l
 o r A

 o  r AN
l
2

N
2

Henry
Mutual Inductance
i1
+
l1
-

N1
turns
g
Notice the current i1 and i2
have been chosen to produce
the flux in the same direction.
It is also assumed that the flux
is confined solely to the core
and its air gap.
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N2
turns
i2
+
l2
-
Magnetic core
Permeability ,
Mean core length lc,
Cross-sectional area Ac
Mutual Inductance
The total mmf is therefore F  N 1i1  N 2 i 2
with the reluctance of the core neglected and assuming that Ac
= Ag the core flux
is

  (N 1i1  N 2 i 2 )
o Ac
g
If the equation is broken up into terms attributable to the
individual current, the flux linkages of coil 1 can be
expressed as

 A 
 A 
o c
o c
l1  N 1  N 
i1  N 1 N 2 
i 2
 g 
 g 
2
1
 l1  L11 i1  L12 i 2
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Mutual Inductance
where L11
 A 
 N  o c  is the self-inductance of coil 1
 g 
2
1
and L11 i1 is the flux linkage of coil 1 due to its own current i1.
 The mutual inductance between coils 1 and 2 is

L12
 A 
 N 1 N 2  o c 
 g 
and L12 i 2 is the flux linkage of coil 1 due to current i2.

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
Mutual Inductance
Similarly, the flux linkage of coil 2 is
 A 


2 o Ac
o c
l 2  N 2   N 1 N 2 
i1  N 2 
i 2
 g 
 g 
 l 2  L 21 i1  L 22 i 2
where L 21  L12 is the mutual inductance and

L 22
 A 
 N  o c 
 g 
2
2
 is the self-inductance of coil 2.
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Mutual Inductance: Example
i1
+
l1
-

N1
turns
g
N2
turns
i2
+
l2
-
Magnetic core
Permeability >>o,
Cross-sectional area Ac = Ag = 1 cm X 1.5916 cm
Air gap length, g = 2 mm
N1 = 100 turns, N2 =200 turns
Find L11, L22, and L12 = L21 = M
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Magnetic Stored Energy
We know that for a magnetic circuit with a single winding
l  N   Li
and
e
dl
dt

d
dt
(N ) 
d
( Li )
dt

For a static magnetic circuit the inductance L is fixed

eL
di
dt
For a electromechanical energy device, L is time varying

e L
di
dt

i
dL
dt
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Magnetic Stored Energy
The power p is
p  ie  i
dl
dt
Thus the change
in magnetic stored energy

l2
t2
W 

t1

pdt 

l2
id l 
l1

l1
l
dl 
L
1
2L
l 2  l1 
2
2
The total stored energy at any l is given by setting l1 = 0:
W 
1
2L

l 
2
L
i
2
2
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