ELE 361
Electric Machines I
http://www.ee.hacettepe.edu.tr/~usezen/ele361/
1
Textbooks
• A.E. Fitzgerald, C. Kingsley, S.D. Umans, Electric
Machinery, McGraw-Hill, 6th Ed., 2003, (5th Ed. 1991)
• S.J. Chapman, Electric Machinery Fundamentals, McGrawHill, 2nd Ed., 1991 (3rd Ed., 1993)
• G.R. Slemon, A. Straughen, Electric Machines, Addison
Wesley, 1980.
• P.C. Sen, Principles of Electrical Machinery and Power
Electronics, J. Wiley, 1989
• S.A. Nasar, L.E. Unnewehr, Electromechanics and Electric
Machines, J. Wiley, 2nd Ed., 1983.
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Contents
• Basic concepts of magnetic circuits (Ch.1, Text 1)
– magnetization, energy storage, hysteresis and eddy-current losses
• Single-phase transformers (Ch.2, Text 1)
– equivalent circuit, open-and short circuit tests, regulation, efficiency
• Electromechanical energy conversion (Ch.3, Text 1)
– field energy, co-energy, force, torque, singly and doubly-excited systems
• Principles of rotating machines (Ch.4, Text 1)
– Construction and types of rotating machines, induced emf, armature mmf,
torque production
• Direct-current machines (Ch.7, Text 1)
– emf and torque production, magnetization characteristic, methods of
excitation, DC generator and motor analysis, ratings and efficiency
• Single-phase induction motors (Ch.9, Text 1)
– equivalent-circuit, s/s operation, starting. linear induction motor, splitphase, capacitor type, shaded pole motors
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I. Basic concepts of
Magnetic Circuits (M.C.)
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1. Basic principles
• Electromechanical energy conversion device (E.M.D)
– links electrical & mechanical systems
• or Electromechanical transducer (E.M.T)
– converts electrical energy to mechanical energy and vice versa
Electric Motors
Electrical energy
Mechanical energy
Generators
• The energy conversion is reversible
5
• Most energy forms are converted to electrical energy, since
it can be
– transmitted & distributed easily
– controlled efficiently and reliably in a simple manner
Primary sources
of energy
Electrical Energy
hydropower, fossile fuel, natural gas
wind, nuclear power etc.
mechanical
energy
Turbine
primary
source
mechanical , heat,
chemical, light etc.
electrical
energy
G
Ultimately desired
Output
electrical
energy
process
EMT
mechanical
energy
M
Pumps,
fans etc
output
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• Coupling between electrical systems and mechanical
systems is through the medium of fields of electric
currents or charges.
– MAGNETIC FIELDS
• Electromagnetic machine
– ELECTROSTATIC FIELDS
• Electrostatic machine (not used in practice due to low power
densities, resulting in large m/c sizes)
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Principle phenomena in
Electromechanical Energy Conversion (E.M.C)
1. Force on a conductor
2. Force on ferromagnetic materials
(e.g. iron)
3. Generation of voltage
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Force on a conductor
•
A mechanical force is exerted on
current carrying conductor in a
magnetic field (MF) and also between
current carrying conductors by means
of their MF
–
Reversibly voltage is induced in a circuit
undergoing motion in a MF
 

F l i B
NB. In left-hand rule, B and i exchange fingers
Right-hand rule
9
Ex1.
Ex3.
Ex4.
Ex2.
Induced voltage
e
d
dt
 (t )  flux linkage
10
Force on a ferromagnetic materials
•
A mechanical force is exerted on a ferromagnetic
material tending to align it with the position of the
densest part of MF.
11
Generation of voltage
•
A voltage is induced in a coil when there is a change
in the flux linking the coil
eN
d d

dt dt
e2  N 2
d
dt
'iron  'air
12
•
The change in flux linkage is either due to changing
flux linking the coil (i.e. transformer voltage) or by
relative motion of coil and MF with respect each other
Single-coil rotor
Flux linkage of the coil
(i.e. flux captured by the coil)
    NB0 A cos ,
  t
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Classification of E.M.D.
E.M.D.
Continuous energy
conversion devices
electric motors,
generators
Force producing
devices
Devices used for
measurement and
control
Relays, solenoids,
electromagnets
Electromechanical
transducers
14
An E.M.D involves energy in 4 forms:
•
Motoring action
Energy input from
electrical sources
•
=
Mechanical
energy output
+
Energy converted
into heat due to losses
+
Increase in energy
stored in magnetic field
–
Energy converted
into heat due to losses
–
Increase in energy
stored in magnetic field
Generating action
Electrical energy
output
=
Mechanical
energy input
Irreversible conversion to heat
occurs due to
– heat in i2R losses (copper losses)
– magnetic losses (core losses)
– mechanical losses (friction & windage losses)
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Rewriting the energy balance equation (motoring convention):
Electrical energy input
–
Copper losses
=
Net electrical energy input
Mechanical energy output
+
+
Friction & windage losses
Increasing energy stored in M.F.
+
Core losses
Gross mechanical energy output
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2. Analysis of Magnetic Circuits (M.C.)
g   c
c  air  0
0  4  10  7 H/m
c   r 0
2000   r  80000
Magnetomotive force (F):
F  Ni
Core flux density (Bc):
Bc 
Airgap flux density (Bg):
Bg 

Ac

Ag
[ Ampere-turns (AT) ]
[ Wb/m2 or Tesla (T) ]
[ Wb/m2 or Tesla (T) ]
where  represents the magnetic flux
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Fringing effects:
Due to fringing effects
Ag > Ac
Normally, we ignore fringing effects, so
Ag  Ac
Since Ag  Ac 
Bg  Bc
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Magnetomotive Force
 
F  C H  d
For the M.C. on the right
F   H  H c  c  H g g  Ni
where H represents the magnetic field intensity
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Relationship between Bc and Hc
In the linear region
Bc  c H c
linear region
Bc   , H c  F
linear region
20
F  H c c  H g g
Assuming operating in the linear region, we can rewrite the above equation as :
F 
Bc
c
c 
Bg
0
g
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Magnetomotive Force - 2
F 
Bc
c
c 
Bg
0
g
Noting that B = /A, we can rewrite the above equation as
F 

 c Ac
c 

 0 Ag
g
 c
g 

F 

 c Ac  0 Ag 


We can further simplify the notation

F   Rc  Rg

where R represents the magnetic resistance
of the medium against flux, called reluctance
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Reluctance

F   Rc  Rg

where:
Rc 
c
c Ac
and
Rg 
g
0 Ag
[ AT/Wb ]
Magnetic resistance of a medium against magnetic flux is called RELUCTANCE
Note the analogy between the electrical circuits

F   Rc  Rg


V  i  R1  R2 
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Analogy between electric and magnetic circuits
Correspondence of conductance in magnetic circuits is called permeance:
P 
1
R
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Simplifications:

F   Rc  Rg
Rc 
c
c Ac
Rg 

g
 0 Ag
Noting that c = r 0 and 2000 < r < 80000
Rc << Rg
in the linear region of Bc-Hc curve, i.e. in linear M.C.s
so
F   Rg
 
Ni Ni 0 Ac
F


R g Rg
g
Nearly all magnetomotive force (F ) is used to overcome the airgap portion of the MC
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3. Flux Linkage and Inductance
Flux linkage
  N
For linear magnetic circuits
and induced voltage e is given by e 
d
dt
  Li
where L indicates the self-inductance of coil
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Self inductance of the N-turn coil:
  N  Li

L
N NBc Ac

i
i
For non-linear magnetic circuits
or
N N F N 2
L


i
i Rc Rc
L
d
d
N
di
di
L  N 2 Pc
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Ex1.
Find:
a) the inductance of the winding
b) flux density in gap g1 (B1)
Equivalent magnetic circuit:
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Ex2.
plastic = 0
N = 200 turns
i = 50A
Consider the plastic ring above and assuming rectangular cross section area
a) Find B at the mean diameter of coil
b) Find inductance of coil, assuming flux density inside ring is uniform
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Self and Mutual Inductances
1  N 1
F1  N 1 i1
2  N 2
F2  N 2 i2

F1  F2
F F
 1 2
Rg  Rc
Rg
Assumption:  c   air   0
1 
1 
N 1 N 1i1
Rg

N 12 0 Ag
g
L11
Self-inductance of coil
N 1 N 2 i2
Rg
i1 
N 1 N 2  0 Ag
g
i2
L12
Mutual-inductance between
coils 1 & 2
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Self and Mutual Inductances - 2
1  L11i1  L12 i2
2  L21i1  L22 i2
L11  N 12 Pg
L22  N 22 Pg
L12  L21  N 1 N 2 Pg
where
Pg 
1
Rg

0 Ag
g
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Leakage Flux
core
flux
     m
Leakage flux:
Magnetizing flux:
(core flux)
l
m
leakage
flux
Not all the flux closes its path from the magnetic core, but
some portion closes its path through air.
This is called the leakage flux, l .
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4. Magnetic Stored Energy
• Stored energy in a magnetic circuit in a time
interval between t1 and t2 :
W  tt12 p dt

t
t 12 e
t
 t 12
i dt
pei
eN
d d

dt dt
d
i dt
dt
W  12 i d
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
W  12 i d
For a linear magnetic circuit:
  Li

i

L
W 
1 2
  d
L 1
W 
1 2
2  12
2L


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Similarly

  L i  d  L di
W  12 i d
i
 Li12 i di
1
 Li22  i12 
2
With i1= 0, i2= i or 1= 0, 2= 
1
W  Li 2
2
or
W 
1 2

2L
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
W  12 i d
F Ni  i
F
N
  N   d  N d

W  12 F d
B 
H F
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4. Magnetic Materials
• Magnetic
– Ferrimagnetic (2000 < r < 10000)
• e.g. Mn-Zn alloy
– Ferromagnetic (r around 80000)
• Hard (permanent magnet)
– e.g. Alnico, Neodimium-Iron-Boron, etc.
(rare-earth magnets)
• Soft (electrical steel)
– e.g. FeSi, FeNi and FeCo alloys
• Non-magnetic
– Paramagnetic (r slightly > 1)
• e.g. aluminum, platinum and magnesium
– Diamagnetic (r slightly < 1)
• e.g. copper and zinc
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Properties of Magnetic Materials:
• Become magnetized in the same
direction of the applied magnetic field
• B varies nonlinearly with H (doublevalued relationship between B and H )
• Exhibit saturation and hysteresis
• Dissipate power under time-varying
magnetic fields
38
Terminology:
• Magnetization curve
• Magnetic hysteresis
• Residual flux density, Br and coercive
field intensity, Hc
• Cyclic state
39
S
S
N
N
Magnetic Hysteresis
40
Normal (DC) magnetization curve (n.m.c)
for a ferromagnetic core:
The curve used to describe a magnetic material is called
the B-H curve, or the hysteresis loop:
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Hysteresis Loop:
Br : residual flux density
Hc : coercive field intensity
42
Hysteresis Loop
Magnetic performance of magnetic
material depends on their previous
history
During measurements, the material should be
put to a definite magnetic cycle:
H is varied in a cyclic manner
{ +Hm  0  –Hm  0  +Hm}
2.
3.
4.
From a demagnetized state (B = 0) while mmf F or field intensity H is
gradually increased, B moves on n.m.c. from a  b :
[ H = 0  Hm  B = 0  Bm]
B moves from b  c : [ H = Hm  0
 B = Bm  Br ]
B moves from c  d : [ H = 0  –Hc
 B = Br  0 ]
B moves from d  e : [ H = –Hc  –Hm  B = 0  –Bm ]
5.
B moves from e  f : [ H = –Hm  0

B = –Bm  –Br ]
6.
7.
B moves from f  g : [ H = 0  Hc
B moves from g  b : [ H = Hc  Hm


B = –Br  0]
B = 0  Bm]
1.
43
Ex:
 r  70000
Find:
a.
b.
c.
d.
e.
f.
g.
The exciting current ie for Bc = 1.0 T.
The flux  and flux linkage  (ignore leakage fluxes).
The reluctance of the airgap Rg and magnetic core Rc.
The induced emf e for a 60 Hz core flux of Bc = 1.0 sin 377 t, Tesla
The inductance L of the winding (neglect fringing fluxes)
The magnetic stored energy W at Bc = 1.0T
Assuming that core material has a DC magnetization curve, find the
exciting current i for Bc = 1.0 T
44
Ex:
 r  70000
Magnetization curve of the core
45
6. AC Excitation and Losses
a)
Relation between periodic exciting current ie and
flux  in a magnetic circuit
v t   e t   N
d
dt
v t   Vm cos t
where ω  2πf
 ( t )   m sin t
e t   N
d
 N m cos t  E m cos t
dt
E m  2fN m
E rms 
2
fN m
2
E rms  4.44 fN m
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Due to non-linear B-H characteristic (or  - F ch.) of a magnetic
material, the exciting current ie (or i ) is a distorted sine wave
although flux  is sinusoidal.
47
Distorted sine wave exciting current waveform
48
Expanding ie using Fourier series
ie ( t )  I c1 cos t  I m 1 sin t  I c 3 cos 3t  I m 3 sin 3t  I c 5 cos 5t  I m 5 sin 5t  
Neglecting high order harmonics:
ie ( t )  I c1 cos t  I m1 sin t
ie ( t )  I c cos t  I m sin t
rc: core loss resistance
xm: magnetizing reactance
Steady-state equivalent circuit model of the
exciting branch
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b) Energy (power) losses in magnetic circuits
Power loss in M.C. is due to:
• Hysteresis loss
• Eddy current losses
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Hysteresis Loss

W  12 F d
F  H c lc
  Bc Ac
B
W  Ac lc B12 H dB
B
W  Vc B12 H dB
Vc: Volume of the magnetic core
51
For one cycle of ac excitation:
Wab  Vc BBm HdB
0
W bc  Vc BBc HdB
0
a
m
Wcd  Vc B Bm HdB  0
c
Wda  Vc BBa HdB  0
m
Hysteresis loss per cycle of ac excitation:
Wh  Wab  Wbc  Wcd  Wda
Ph  Wh f
Empirical eqn: Ph  Vc Bm f
: constant depending on material type
1.5  x  2.5
x
52
Eddy Current Loss
In general, M.C. have
- very high magnetic permeability,
c  air  0
- high electrical conductivity
(low resistivity), which causes
extra I2R losses (Pe) within the
magnetic materials when they are
subject to time-varying MF.
Eddy current loss:
Pe  K eVc d 2 Bm2 f 2
53
Eddy Current Loss
Pe  K eVc d 2 Bm2 f 2
d: thickness of lamination
Ke: constant depending on material
resistivity 
Core loss:
Pcore  Ph  Pe
Stacking factor Fs in a laminated material
Ac (effective)  Fs Ac (actual)
0.95 < Fs < 1
54
Core Loss
Pcore  Ph  Pe
Core Loss is given in manufacturer’s data sheets for each specific core material as
Pcore vs Bm curves in log. scale, with operating frequency as a parameter:
Core loss increases with increasing frequency
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