Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-0

PowerPoint Slidesto accompany

Electric MachinerySixth Edition

A.E. Fitzgerald

Charles Kingsley, Jr.

Stephen D. Umans

Chapter 4

Introduction to Rotating

Machines

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-1

4.1 ELEMENTARY CONCEPT

Horizontal axis

Magnetic Field

e(t)

Electromechanical energy conversion occurs when changes in the flux linkages λ resulting from mechanical motion.

dt

dte

λ=)(

•Rotating the winding in magnetic field

•Rotating magnetic field through the winding

•Stationary winding and time changing magnetic field (Transformer action)

Producing voltage in the coil

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-2

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.

VRM (Variable Reluctance Machines) No windings on the rotor

Stepper Motors (non-uniform air-gaps)

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-3

4.2 INTRODUCTION TO AC AND DC MACHINES

AC Machines: Synchronous Machines and Induction Machines

(a) Space distribution of flux density and (b) corresponding waveform of the generated voltage for the single-phase generator.

Synchronous Machines:•Two-pole, single phase machine

•Rotor rotates with a constant speed

•Constraction is made such that air-gap flux density is sinusoidal

•Sinusoidal flux distribution results with sinusoidal induced voltage

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-4

•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.

aae

pθθ2

=602

fnp

e =n: rpm

fe: Hz

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-5

Hydroelectric power plant (D. Yıldırım, İTÜ Lecture Notes)

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-6generator sets giant shaft connecting turbine to generator

generator

turbine

hydropower-plant-generator.swf

Hydroelectric power plant

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-7

Diameter of rotor:

16 meters

Rotating mass:

2650 ton

715 MW generator

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-8

Elementary two-pole cylindrical-rotor field winding.

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.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-9

Schematic views of three-phase generators: (a) two-pole, (b) four-pole, and (c) Y connection of the windings.Figure 4.12

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-10

Typical induction-motor speed-torque characteristic.

Figure 4.15

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

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-11

Typical Induction Motor

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-12

Windings placed in stator slots

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-13

Inside View of An Induction Motor

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-14

Elementary dc machine with commutator.

Figure 4.17

DC Machines:

Armature winding on the rotor with current conducted from it by means of carbon brushes

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-15

(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

4.3 MMF OF DISTRIBUTED WINDINGS

aag

iNF θ

πcos

2

41

=

Fourier Analysis

=

2

4)( 1

iNF peakag π

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-16

The mmf of one phase of a distributed two-pole, three-phase winding with full-pitch coils.

Figure 4.20

AC Machines:

= a

aph

ag

p

p

iNkF θ

π

ω

2cos

41

ωk Winding factor (usually between 0.85 and 0.95)

phN Series turns per phase

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-17

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.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-18

The air-gap mmf of a distributed winding on the rotor of a round-rotor generator.

= r

rrrag

p

p

INkF θ

π 2cos

41

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-19

Cross section of a two-pole dc machine.

DC Machines:

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-20

(a) Developed sketch of the dc machine(b) mmf wave; (c) equivalent sawtooth mmf wave, itsfundamental component, and equivalent rectangular current sheet.

Sawtooth waveform because of restrictions imposed by the commutator.

Peak value of fundamental component

81.082=

π

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-21

(a) Cross section of a four-pole dc machine; (b) development of current sheet and mmf wave.

aa

peakag ip

NF

=)( a

apeakag i

p

NF

=

21

8)(

π

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-22

Four Pole Stator of a DC Motor:

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-23

Rotor of a DC Motor:

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-24

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-25

Parts of a small DC motor

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-26

The air-gap mmf and radial component of Hag for a concentrated full-pitch winding.

4.4 MAGNETIC FIELDS IN ROTATING MACHINERY

aagg

NiH θ

πcos

2

41

=

= a

aphw

ag

p

pg

iNkH θ

π 2cos

41

Distributed winding:

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-27

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.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-28

Structure of typical salient-pole machines: (a) dc machine and (b) salient-pole synchronous machine.

Machines with non-uniform air gaps.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-29

Finite-element solution of the magnetic field distribution in a salient-pole dc generator. Field coils excited; no current in armature coils. (General Electric Company.)

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-30

Flux distribution in a 4-pole salient-pole generator

Colors represent the strength of B.

Blue to Red : The flux density increases

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-31

Single-phase-winding space-fundamental air-gap mmf: (a) mmf distribution of a single-phase winding at various times; (b) total mmf FFFFag1

decomposed into two traveling wavesF F F F – and F F F F +; (c) phasor decomposition of FFFFag1.

4.5 ROTATING MMF WAVES IN AC MACHINES

= a

aphw

ag

p

p

iNkF θ

π 2cos

41

tIi eaa ωcos=

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-32

Simplified two-pole three-phase stator winding.

MMF Wave of a Polyphase Winding

tIi ema ωcos=

)120cos( 0−= tIi emb ω

)120cos(0+= tIi emc ω

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-33

Instantaneous phase currents under balanced three-phase conditions.Figure 4.30

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-34

The production of a rotating magnetic field by means of three-phase currents.Figure 4.31

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-35

Cross-sectional view of an elementary three-phase ac machine.Figure 4.32

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-36

Voltage between the brushes in the elementary dc machine of Fig. 4.17.Figure 4.33

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-37

Elementary two-pole machine with smooth air gap: (a) winding distribution and (b) schematic representation.Figure 4.34

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-38

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 δδδδsr positive, i.e., with the rotor mmf wave Fr leading that of the stator Fs.Figure 4.35

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-39

The mmf and H field of a concentrated full-pitch linear winding.Figure 4.36

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-40

Typical open-circuit characteristic and air-gap line.Figure 4.37

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-41

Finite-element solution for the flux distribution around a salient pole. (General Electric Company.)Figure 4.38

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-42

Flux-density wave corresponding to Fig. 4.38 with its fundamental and third-harmonic components.Figure 4.39

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-43

Three-coil system showing components of mutual and leakage flux produced by current in coil 1.Figure 4.40

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-44

Flux created by a single coil side in a slot.Figure 4.41

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-45

Problem 4.8: (a) full-pitch coil and (b) fractional-pitch coil.Figure 4.43

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-46

Elementary generator for Problem 4.13.Figure 4.44

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-47

Elementary cylindrical-rotor, two-phase synchronous machine for Problem 4.22.Figure 4.45

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4-48

Schematic two-phase, salient-pole synchronous machine for Problem 4.24.Figure 4.46