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PowerPoint Slides
to accompany
Electric Machinery
Sixth Edition
A.E. Fitzgerald
Charles Kingsley, Jr.
Stephen D. Umans
Chapter 4
Introduction to Rotating
Machines
4-0
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4.1 ELEMENTARY CONCEPT
Electromechanical energy conversion occurs when changes in the flux
linkages λ resulting from mechanical motion.
e(t ) 
Magnetic
Field
d
dt
Producing voltage in the coil
Horizontal
axis
•Rotating the winding in magnetic
field
•Rotating magnetic field through the
winding
e(t)
4-1
•Stationary winding and time
changing magnetic field (Transformer
action)
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Armature winding: AC current carrying winding
Synchronous machine
Armature winding is
Induction machine
stator winding (stationary)
DC machine
Armature winding is on
the rotor
Field winding: DC current carrying winding
DC machine
Field winding is on the stator
Synchronous machine
Field winding is on the rotor
Note: Permanent magnets produce DC magnetic flux and are used
in the place of field windings in some machines.
4-2
VRM (Variable Reluctance Machines)
No windings on the rotor
Stepper Motors
(non-uniform air-gaps)
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4.2 INTRODUCTION TO AC AND DC MACHINES
AC Machines: Synchronous Machines and Induction Machines
Synchronous Machines:
•Two-pole, single phase machine
•Rotor rotates with a constant speed
•Constraction is made such that airgap flux density is sinusoidal
•Sinusoidal flux distribution results
with sinusoidal induced voltage
(a) Space distribution of flux density and
(b) corresponding waveform of the generated voltage for the singlephase generator.
4-3
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•Four-pole, single phase machine
•a1,-a1 and a2,-a2 windings connected
in series
•The generator voltage goes through
two complete cycles per revolution of
the rotor. The frequency in hertz will
be twice the speed in revolutions per
second.
p
 ae   a
2
4-4
p n
fe 
2 60
n: rpm
fe: Hz
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Hydroelectric power plant (D. Yıldırım, İTÜ Lecture Notes)
4-5
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Hydroelectric power plant
generato
r
turbine
generator sets
4-6
hydropower-plant-generator.swf
giant shaft
connecting turbine
to generator
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715 MW generator
Diameter of rotor:
16 meters
Rotating mass:
4-7
2650 ton
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Field winding is a two-pole
distributed winding
Winding distributed in multiple
slots and arranged to produce
sinusoidal distributed air-gap
flux.
Why some synchronous
generators have salient-pole
rotor while others have
cylindirical rotors?
Answer: In salient-pole
machines the number of poles
can be large therefore they will
be able to operate in slow speed
to produce 50 Hz voltage.
Elementary two-pole
cylindrical-rotor field winding.
4-8
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Schematic views of three-phase generators: (a) two-pole,
(b) four-pole, and (c) Y connection of the windings.
Figure 4.12
4-9
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Induction Machines:
•The stator winding excited
by ac current. The current
produces a rotating magnetic
field which in turn produces
currents in rotor conductors
due to induction.
•These machines mostly used
as motors.
•Rotor windings are short
circuited (electrically) and
frequently have no external
connections.
•Stator and rotor fluxes rotate
in synchronism with each
other and that torque is
related to the relative
displacement between them.
•Rotor does not rotate
synchronously
4-10
Typical induction-motor speed-torque
characteristic.
Figure 4.15
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Typical Induction Motor
4-11
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Windings placed in stator slots
4-12
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Inside View of An Induction Motor
4-13
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DC Machines:
Armature winding on the rotor
with current conducted from it
by means of carbon brushes
Elementary dc machine
with commutator.
Figure 4.17
4-14
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4.3 MMF OF DISTRIBUTED WINDINGS
(a) Schematic view of flux
produced by a concentrated,
full-pitch winding in a machine
with a uniform air gap. (b) The
air-gap mmf produced
by current in
this winding.
Figure 4.19
Fourier Analysis
Fag1 
4Ni

 cos  a
 2 
( Fag1 ) peak 
4-15
4Ni


 2 
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AC Machines:
The mmf of one phase of a distributed two-pole,
three-phase winding with full-pitch coils.
Figure 4.20
4  k N ph ia 
p 

 cos  a 
Fag1  

p
2 

k
Winding factor (usually
between 0.85 and 0.95)
N ph Series turns per phase
4-16
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Example 4.1: The phase-a two-pole armature winding of figure below can be
considered to consists of 8 Nc-turn full-pitch coils connected in series, with
each slot contaning two coils. There are a total of 24 armature slots, and thus
each slot is separated by 3600 /24=150. Assume angle θa is measures from the
magnetic axis of phase a such that the four slots containing the coil sides
labeled a are at 67.50, 82.50, 97.50, and 112.50. The opposite sides of each coil
are thus found in the slots found at -112.50, -97.50, -82.50, and 67.50,
respectively. Assume this winding to be carrying current ia.
a) Write an expression for the space-fundamental mmf produced by the two coils
whose sides are in the slots at θa=112.50 and -67.50.
b) Write an expression for the space-fundamental mmf produced by the two coils
whose sides are in the slots at θa=67.50 and -112.50.
c) Write an expression for the space-fundamental mmf of the complete armature
winding.
d) Determine the winding factor kw for this distributed winding.
4-17
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The air-gap mmf of a distributed winding on the rotor
of a round-rotor generator.
Fag1 
4-18
4  kr N r I r 
p 

 cos  r 
 p  2 
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DC Machines:
Cross section of a two-pole dc
machine.
4-19
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(a) Developed sketch of
the dc machine
(b) mmf wave;
(c) equivalent sawtooth
mmf wave, its
fundamental
component,
and equivalent
rectangular current
sheet.
Sawtooth waveform
because of restrictions
imposed by the
commutator.
Peak value of
fundamental component
8

4-20
2
 0.81
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(a) Cross section of a four-pole dc machine;
(b) development of current sheet and mmf wave.
 Na 
 ia
( Fag ) peak  
 p 
4-21
8  Na 
 ia
( Fag1 ) peak  2 
  p 
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Four Pole Stator of a DC Motor:
4-22
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Rotor of a DC Motor:
4-23
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4-24
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Parts of a small DC motor
4-25
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4.4 MAGNETIC FIELDS IN ROTATING
MACHINERY
The air-gap mmf and
radial component
of Hag for a
concentrated
full-pitch winding.
4  Ni 
H ag1    cos  a
  2g 
Distributed winding:
4  k w N ph ia   p 
 cos  a 
H ag1  
 gp  2 
4-26
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Example 4.2: A four-pole synchronous ac generator
with a smooth air gap has a distributed rotor
winding with 263 series turns, a winding factor of
0.935, and an air gap of length 0.7 mm. Assuming
the mmf drop in the electrical steel to be negligible,
find the rotor-winding current required to produce a
peak, space-fundamental magnetic flux density of
1.6 T in the machine air gap.
4-27
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Machines with non-uniform air gaps.
Structure of typical salient-pole machines: (a) dc machine and (b)
salient-pole synchronous machine.
4-28
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Finite-element solution of the magnetic field distribution in a salientpole dc generator. Field coils excited; no current in armature coils.
(General Electric Company.)
4-29
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Flux distribution in a 4-pole salient-pole generator
Colors represent
the strength of B.
Blue to Red : The
flux density
increases
4-30
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4.5 ROTATING MMF WAVES IN AC MACHINES
Single-phase-winding
space-fundamental
air-gap mmf:
(a) mmf distribution of
a single-phase winding at
various times;
(b) total mmf Fag1
decomposed into two
traveling wavesF – and F +;
(c) phasor decomposition
of Fag1.
4  k w N ph ia   p 
 cos  a 
Fag1  

p
 2 
ia  I a cos et
4-31
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MMF Wave of a Polyphase Winding
Simplified twopole three-phase
stator winding.
ia  I m cos et
ib  I m cos(et 1200 )
ic  I m cos(et  1200 )
4-32
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Instantaneous phase currents under balanced
three-phase conditions.
Figure 4.30
4-33
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The production of a rotating magnetic field by means
of three-phase currents.
Figure 4.31
4-34
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Cross-sectional
view of an
elementary
three-phase
ac machine.
Figure 4.32
4-35
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Voltage between the brushes
in the elementary dc machine
of Fig. 4.17.
Figure 4.33
4-36
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Elementary two-pole machine with smooth air gap:
(a) winding distribution and (b) schematic representation.
Figure 4.34
4-37
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Simplified two-pole machine: (a) elementary model and
(b) vector diagram of mmf waves. Torque is produced
by the tendency of the rotor and stator magnetic fields
to align. Note that these figures are drawn with sr positive,
i.e., with the rotor mmf wave Fr leading that of the stator
Fs.
Figure 4.35
4-38
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The mmf and
H field of a
concentrated
full-pitch linear
winding.
Figure 4.36
4-39
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Typical
open-circuit
characteristic
and air-gap
line.
Figure 4.37
4-40
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Finite-element
solution for the
flux distribution
around a
salient pole.
(General
Electric
Company.)
Figure 4.38
4-41
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Flux-density
wave
corresponding
to Fig. 4.38
with its
fundamental
and thirdharmonic
components.
Figure 4.39
4-42
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Three-coil system showing components of mutual
and leakage flux produced by current in coil 1.
Figure 4.40
4-43
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Flux created
by a single
coil side in
a slot.
Figure 4.41
4-44
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Problem 4.8: (a) full-pitch coil and (b) fractional-pitch coil.
Figure 4.43
4-45
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Elementary
generator for
Problem 4.13.
Figure 4.44
4-46
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Elementary
cylindrical-rotor,
two-phase
synchronous
machine for
Problem 4.22.
Figure 4.45
4-47
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Schematic twophase, salient-pole
synchronous
machine for
Problem 4.24.
Figure 4.46
4-48