chapter 4 introduction to rotating machines · discuss some of the principles ... jinlin gong-...

Jinlin GONG

2013-2014-2 Electric Machinery

Chapter 4 Introduction to

Rotating Machines

山东大学校长办公会议

Jinlin GONG- School of Electrical Engineering

Keynotes

The objective of this chapter is to introduce and

discuss some of the principles underlying the

performance of electric machinery.

2

As will be seen, these principles are common to both

ac and dc machines. Various techniques and

approximations involved in reducing a physical

machine to simple mathematical models, sufficient to

illustrate the basic principles, will be developed.



目录

4.1 Elementary concept

4.2 Introduction to AC and DC Machines

4.3 MMF of Distributed Windings

4.4 Magnetic Fields in Rotating Machinery

4.5 Rotating MMF Waves in AC Machines

4.6 Generated Voltage

4.7 Torque in Nonsalient–pole Machines

4.8 Linear Machines

4.9 Magnetic Saturation

4.10 Leakage Fluxes

4.11 Summary



4.1 Elementary Concept

In rotating machines, voltages are generated in windings or

groups of coils by rotating these windings mechnically through

a magnetic field, by mechanically rotating a magnetic field past

the winding.

Armature winding:

A winding or a set of windings on

a rotating machine which carry ac

currents.

In ac machines Stator

In dc machines Rotor




Field winding:

A winding or a set of windings

which carry dc current and

which are used to produce

main operating flux in the

machine.

In ac machines

Stator In dc machines

Rotor

In order to minimize the effects of eddy currents, the armature

structure is typically built from thin laminations of electrical steel

which are insulated from each other.




S

F S

F S

N

=0°

O

Torque producing characteristic:

The torque is generated in order to align the flux distribution

for both stator and rotor.




Analytically based models:

V

jXs R

I

R’2/g v1

I1 r1 l1. l’2.

X R

I

I’2

Induction machine Synchronous machine

Analytically-based models are essential to the analysis and

design of electrical machines.

One objective is to recognize that physical insight into the

performance ot these devices.



4.2 Introduction to AC and DC machines

Synchronous Machines:

4.2.1 AC machines

Traditional ac machines fall into one of two categories:

synchronous and induction.

Schematic view of a simple synchronous generator

Armature winding--stator

A single coil of N turns, two coil side a

et –a placed in diametrically opposite

narrow slots on the inner periphery.

Field winding--Rotor

It is excited by direct current.

• brushes + collector rings

• brushless excitation system



Synchronous Machines

A highly idealized analysis of this machine would assume a

sinusoidal distribution of magnetic flux in the air gap.

(a) Sinusoidal distribution of flux density (b) Corresponding waveform of the voltage

Sinusoidal flux

distribution

Constant rotor

speed

Sinusoidal

voltage

The electric frequency of the generated voltage is synchronized

with the mechanical speed.




(a) 4-pole, single-phase generator

The generated voltage now goes through two complete cycles

per revolution of the rotor.

(b) Space distribution of the air-

gap flux density

2




The electrical frequency fe of the voltage generated in a

synchronous machine:

Hz

𝑝𝑜𝑙𝑒𝑠

2




Elementary two-pole cylindrical-rotor field winding

Salient-pole Hydroelectric generators

Cylindrical rotor Steam turbines and gas turbines




Three voltages phase-displaced by

120 electrical degrees in time

Three coils phase-displaced by 120

electrical degrees in space

(a) 2-pole 3-phase generator (b) 4-pole (b) Y connection of the winding

The minimum number of coil sets is given by one half the number of poles.




S

F S

F S

N

=0°

O

The armature current creates a magnetic flux wave in the air gap.

This flux reacts with the flux created by the field current and

electromechanical torque results from the tendency of these

two magnetic fields to align.




S

F

In a generator, this torque opposes

rotation, and mechanical torque must

applied from the prime mover to sustain

rotation.

In a synchronous motor:

Alternating current

Stator Rotating magnetic field

Rotor dc excitation current

fixed magnetic field

A steady electromechanical torque is produced when the

rotor rotates in synchronism with the magnetic filed of stator.



Induction Machines

The stator windings are essentially the same as those of a

synchronous machine, the rotor windings are electrically

short-circuited and frequently have no external connections.

(a) Stator iron core (b) Stator winding (c) Rotor squirrel-cage

The rotor windings are actually solid aluminum bars which are

cast into the slots in the rotor and which are shorted together by

cast aluminum rings at each end of the rotor.



Induction Machines

The induction machines are asynchronous machines and

produce torque only when the rotor speed differs from

synchronous speed.

Interestingly, although the rotor operates asynchronously,

the flux wave produced by the induced rotor currents

rotates in synchronism with the stator flux wave.

An induction machine may be regarded as a generalized

transformer in which electric power is transformed between

rotor and stator together with a change of frequency and a

flow of mechanical power.



4.2.2 DC machines

Nφ

φ S

e

a

b

c

de

+

A

B

-

电刷

换向片

n

发电机模型

n原动机

(a) DC generator (b) Schematic of DC generator

Amature winding Rotor Carbon brush

Field winding Stator



4.2.2 DC machines

(a) Space distribution

of air-gap flux density

in an elementary dc

machine

(b) Waveform of Voltage

between brushes

commutator

Flat topped



4.2.2 DC machines

The direct current in the field winding of a dc machine

creates a magnetic flux distribution which is stationary with

respect to the stator.

The direct current flows through the brushes, the armature

creates a magnetic flux distribution which is also fixed in

space and perpendicular to the axis of the field flux.

It is the interaction of these two flux distributions that

creates the torque of the dc machine.



4.3 MMF of distributed winding

(a) Flux produced by a concentrated, full-pitch winding

A coil which spans 180

electrical degrees is known

as a full-pitch coil.

The individual coils are interconnected so that

Most armatures have distributed windings, i.e. windings which

are spread over a number of slots around the air-gap periphery.

Npole-mag=Npole-field.

𝐻𝑎𝑔 𝜃𝑎 = −𝐻𝑎𝑔 𝜃𝑎 + 𝜋



4.3 MMF of distributed winding

(b) The air-gap mmf produced by current in the winding

Each flux line

crosses the air gap

twice, the mmf drop

is equal to Ni/2

Around any closed paths shown by the

flux lines, the mmf is Ni, the mmf drop in

the iron can be neglected, and all of mmf

drop will appear across the air gap.

𝐻𝑎𝑔 𝜃𝑎 = −𝐻𝑎𝑔 𝜃𝑎 + 𝜋 ℱ𝑎𝑔 similary



4.3.1 AC machines

Air-gap mmf fundamental High-order

harmonic

With its peak aligned with the magnetic axis of the coil.



4.3.1 AC machines

Distributed winding:

Distributed two-pole three-phase winding with

full pitch coil

Consisting of coils

distributed in several slots

The winding is arranged in

two layers, each full-pitch

coil of Nc turns.

The windings of the three

phases are identical and

located with their magnetic

axes 120 degrees apart.



4.3.1 AC machines

mmf of one phase of a distributed two-pole

three-phase winding with full pitch coil

The mmf wave is a series of steps each of height 2Ncia.

The distributed winding produces an mmf wave which is closer

approximation to a sinusoidal mmf wave than that of the

concentrated winding.

The fundamental

component Fag1

𝑁𝑝𝑕 series turns per phase



Example 4.1

Each slot is separated by 360° 24 = 15° .

Four slots containing the coil sides labeled a are at

𝜃𝑎 = 67.5°, 82.5°, 97.5°, 112.5° and the opposite sides of each coil are

thus at −112.5°, −97.5°, −82.5°, −67.5°

(a) Write an expression for the space-fundamental mmf produced by the

two coils whose sides are in the slots at 𝜃𝑎 = 112.5°𝑎𝑛𝑑 −67.5°.

(b) Write an expression for the space-fundamental mmf produced by the


(c) Write an expression for the space-fundamental mmf of the complete

armature winding.

(d) Determine the winding factor 𝑘𝑤for this distributed winding.



Example 4.1

(a) Write an expression for the space-fundamental mmf produced by the


(b) Write an expression for the space-fundamental mmf produced by the


The magnetic axis of this pair of coils is at𝜃𝑎 =112.5°−67.5°

2= 22.5°

The total ampere-turns in each slot is equal to 2𝑁𝑐𝑖𝑎

ℱ𝑎𝑔1 22.5° =4

𝜋

2𝑁𝑐𝑖𝑎2

cos 𝜃𝑎 − 22.5°

ℱ𝑎𝑔1 −22.5° =4

𝜋

2𝑁𝑐𝑖𝑎2

cos 𝜃𝑎 + 22.5°



Example 4.1

ℱ𝑎𝑔1 𝑡𝑜𝑡𝑎𝑙= ℱ𝑎𝑔1 −22.5° + ℱ𝑎𝑔1 −7.5° + ℱ𝑎𝑔1 7.5° + ℱ𝑎𝑔1 22.5°

(c) Write an expression for the space-fundamental mmf of the complete

armature winding.

(d) Determine the winding factor 𝑘𝑤for this distributed winding.

=4

𝜋

7.66𝑁𝑐

2𝑖𝑎 cos 𝜃𝑎 = 4.88𝑁𝑐𝑖𝑎 cos 𝜃𝑎

Recognizing that, for this winding 𝑁𝑝𝑕 = 8𝑁𝑐, th total mmf of part (c)

can be rewritten as:

ℱ𝑎𝑔1 𝑡𝑜𝑡𝑎𝑙=

4

𝜋

0.958𝑁𝑝𝑕

2𝑖𝑎 cos 𝜃𝑎 𝑘𝑤 = 0.958



4.3.1 AC machines

distributed winding on the rotor of a round-rotor

The winding is symmetric

with respect to the rotor

axis

The number of turns per

slot can be varied to

control the various

harmonics

ℱ𝑎𝑔1 =4

𝜋

𝑘𝑤𝑁𝑝𝑕

𝑝𝑜𝑙𝑒𝑠𝑖𝑎 cos


2𝜃𝑎 𝑖𝑎 = 𝐼𝑚𝑎𝑥 cos𝜔𝑡 Mmf wave which

is stationary in space and varies

sinusoidal both with respect to 𝜃𝑎

and in time.



4.3.1 AC machines

distributed winding on the rotor of a round-rotor

There are fewer turns

in the slots nearest the

pole face.

The fundamental

mmf ℱ𝑎𝑔1:

=4

𝜋

𝑘𝑟𝑁𝑟

𝑝𝑜𝑙𝑒𝑠𝐼𝑟 cos


2𝜃𝑟

𝑘𝑟: winding factor

𝑁𝑟: series turns

𝐼𝑟: winding current



4.3.2 DC machines

Cross section of a two-pole dc machine

The armature winding

produces a magnetic field

whose axis is vertical and

perpendicular to the axis of

the field winding.

The armature flux is always

perpendicular to that

produced by the field

winding and a continuous

unidirectional torque results.



4.3.2 DC machines

(a) Developped sketch of the dc machine

(b) mmf wave of sawtooth form

The height of

each step:

2𝑁𝑐𝑖𝑐

The number of ampere-

turns in a slot

The peak value

of the mmf wave:

6𝑁𝑐𝑖𝑐

Along the magnetic axis

of the armature, midway

between the field poles.



4.3.2 DC machines

(c) Equivalent sawtooth

mmf wave, its

fundamental component,

and equivalent

rectangular current sheet

For a more realistic winding with a large number of armature

slots per pole, the triangular distribution becomes a close

approximation.

This mmf wave would be produced by a rectangular

distribution of current density at the armature surface.



4.3.2 DC machines

(a) Cross section of a four-pole

dc machine (b) Development of current sheet and mmf wave

The peak value of the sawtooth armature mmf wave:

𝐹𝑎𝑔 𝑝𝑒𝑎𝑘=

𝐶𝑎

2𝑚 × 𝑝𝑜𝑙𝑒𝑠𝑖𝑎

𝐶𝑎-total number of conductors in armature winding

𝑚-number of parallel paths through armature winding



4.4 Magnetic fields in rotating machinery

Investigations of both ac and dc machines on the

assumption of sinusoidal spatial distributions of mmf

It is easiest way to begin by examination of a two-pole

machine, in which the electrical and mechanical

angles and velocities are equal.

The behavior of electric machinery is determined by

the magnetic fields created by currents in the various

windings of the machine.



4.4.1 Machines with uniform air gaps

A single full pitch, 𝜇 → ∞

(a)

(b) Air gap mmf and radial component

With the path C: 𝐻 𝑑𝑙 = 𝑁𝑖 = ℱ

𝐻𝑎𝑔 ∙ 𝑔 = ℱ𝑎𝑔

( 𝐻𝑎𝑖𝑟−𝑔𝑎𝑝 ≫ 𝐻𝑓𝑒𝑟 , 𝐵𝑎𝑖𝑟−𝑔𝑎𝑝

𝜇𝑎𝑖𝑟≫

𝐵𝑓𝑒𝑟

𝜇𝑓𝑒𝑟)

𝐻𝑎𝑔 =ℱ𝑎𝑔

𝑔

ℱ = 𝑁𝑖 ℱ𝑎𝑔 =ℱ

2=

𝑁𝑖

2

𝐻𝑎𝑔1 =ℱ𝑎𝑔1

𝑔=

4

𝜋(𝑁𝑖

2𝑔) cos 𝜃𝑎

The fundamental component:



4.4.1 Machines with uniform air gaps

(c) Air gap mmf and radial component

For a distributed winding with winding factor 𝑘𝑤:

𝐻𝑎𝑔1 =4

𝜋


𝑔 × 𝑝𝑜𝑙𝑒𝑠𝑖𝑎cos 𝜃𝑎𝑒



Example:

A four-pole synchronous ac generator with a smooth air gap has a

distributed rotor winding with 264 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 neglibible, find the rotor winding current required to produce a peak, space-

fundamental magnetic flux density of 1.6T in the machine air gap.

Solution:

𝐵𝑎𝑔1 𝑝𝑒𝑎𝑘= 𝜇0 𝐻𝑎𝑔1 𝑝𝑒𝑎𝑘

=𝜇0 𝐹𝑎𝑔1 𝑝𝑒𝑎𝑘

𝑔=

4𝜇0

𝜋𝑔

𝑘𝑟𝑁𝑟

𝑝𝑜𝑙𝑒𝑠𝐼𝑟

Solving for Ir gives:

𝐼𝑟 =𝜋𝑔 × 𝑝𝑜𝑙𝑒𝑠

4𝜇0𝑘𝑟𝑁𝑟𝐵𝑎𝑔1 𝑝𝑒𝑎𝑘

= 11.4𝐴



4.4.2 Machines with nonuniform air gaps

(a) salient-pole machines—dc machine (b) salient-pole synchronous machine

Detailed analysis of the magnetic field distribution in such

machines requires complete solutions of the field problem.



4.5 Rotating MMF waves in AC machines

Polyphase ac

machines

Mmf wave of

polyhase winding

4.5.1 mmf wave of a single phase winding

=4

𝜋


𝑝𝑜𝑙𝑒𝑠𝑖𝑎cos 𝜃𝑎𝑒

ℱ𝑎𝑔1 =

Space fundamental mmf

mmf distribution of a single phase winding at various times




The winding is excited by a current varying sinusoidally in time

at electrical frequency 𝜔𝑒:

𝑖𝑎 = 𝐼𝑎 cos 𝜔𝑒𝑡

The mmf distribution is given by:

ℱ𝑎𝑔1 =4

𝜋


𝑝𝑜𝑙𝑒𝑠𝐼𝑎cos 𝜃𝑎𝑒 cos 𝜔𝑒𝑡 with 𝐹𝑚𝑎𝑥 =

4

𝜋


𝑝𝑜𝑙𝑒𝑠𝐼𝑎

The mmf distribution remains

fixed in space with an amplitude

that varies sinusoidally in time

at frequency 𝜔𝑒




Use of a common trigonometric identity:

ℱ𝑎𝑔1 = 𝐹𝑚𝑎𝑥

1

2cos 𝜃𝑎𝑒 − 𝜔𝑒𝑡 +

1

2cos 𝜃𝑎𝑒 + 𝜔𝑒𝑡

The mmf of a single-phase winding can be resolved into

two rotating mmf waves:

ℱ𝑎𝑔1+ =

1

2𝐹𝑚𝑎𝑥 cos 𝜃𝑎𝑒 − 𝜔𝑒𝑡

In the +𝜃𝑎𝑒direction

ℱ𝑎𝑔1− =

1

2𝐹𝑚𝑎𝑥 cos 𝜃𝑎𝑒 + 𝜔𝑒𝑡

In the −𝜃𝑎𝑒direction




Phasor decomposition of ℱ𝑎𝑔1

Both flux wave rotate in their respect direction with electrical

angular velocity 𝜔𝑒, corresponding to a mechanical angular

velocity 𝜔𝑚:

𝜔𝑚 =2

𝑝𝑜𝑙𝑒𝑠𝜔𝑒 =

𝜋

30𝑛

The positive-traveling flux wave produces useful torque while

the negative traveling flux wave produces both negative and

pulsating torque as well as losses.



4.5.2 mmf wave of a polyphase winding

Simplified two-pole three-phase

stator winding

The windings of individual

phases are displaced from

each other by 120 electrical

degrees in space.

The space-fundamental

sinusoidal mmf waves of the

three phases are displaced

120 electrical degrees in

space.

Each phase is excited by an

alternating current



4.5.2 mmf wave of a polyphase winding

Instantaneous phase currents under

balanced three-phase condition

The alternating instantaneous

currents:

𝑖𝑎 = 𝐼𝑚𝑎𝑥 cos 𝜔𝑒𝑡

𝑖𝑏 = 𝐼𝑚𝑎𝑥 cos 𝜔𝑒𝑡 − 120°

𝑖𝑐 = 𝐼𝑚𝑎𝑥 cos 𝜔𝑒𝑡 + 120°

The mmf of phase a:

ℱ𝑎1 = ℱ𝑎1+ + ℱ𝑎1

−

ℱ𝑎1+ =

1

2𝐹𝑚𝑎𝑥 cos 𝜃𝑎𝑒 − 𝜔𝑒𝑡 ℱ𝑎1

− =1

2𝐹𝑚𝑎𝑥 cos 𝜃𝑎𝑒 + 𝜔𝑒𝑡

chapter 4 introduction to rotating machines · discuss some of the principles ... jinlin gong-...

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