4. Fundamentals of AC Machinery - University of Florida · AC Machines: We begin this study by...

Fundamentals ofFundamentals of AC Machinery

Revised October 6, 2008

4. Fundamentals of AC Machinery 1

AC Machines:

We begin this study by first looking at some commonalities that exist for all machines, then look at specific machines such as

synchronous machines

induction machines

DC machinesC ac es

EEL 3211 © 2008, H. Zmuda


AC Machines:

Motors: AC Electrical Energy Mechanical Energy

We apply a current to a wire in the presence of an applied magneticWe apply a current to a wire in the presence of an applied magnetic field which produces a torque – this is motor action. Just like when two permanent magnets are near each other, a force is exerted which makes their magnetic fields want to “line up.”

Generators: Mechanical Energy AC Electrical Energy Ge e ato s: ec a ca e gy C ect ca e gy

We move a wire in the presence of an applied magnetic field which induces a voltage generator actioninduces a voltage – generator action.

In each case, the applied magnetic field for AC machines is

EEL 3211 © 2008, H. Zmuda


produced by a field winding.

AC Machines:

Two major classes of AC machines:

Synchronous Machines are motors and generators whose fieldSynchronous Machines are motors and generators whose field current (or current in the rotor windings) is supplied by a direct connection via a rotating contact to a separate stationary power source.

Induction Machines are motors and generators whose field nduction achines a e o o s a d ge e a o s w ose e dcurrent (current in the rotor windings) is induced in the rotor by magnetic induction (transformer action) and by the relative motion of the rotor and statormotion of the rotor and stator.

We will consider each machine in detail. First we look at the

EEL 3211 © 2008, H. Zmuda


basic commonalities that govern both types of machines.

In rotating machines, voltages are generated in windings or groups of coilscoils

•by rotating these windings mechanically through a magnetic field,

•by mechanically rotating a magnetic field past the winding,

•or by designing the magnetic circuit so that the reluctance varies with rotation of the rotor.

By any of these methods, the flux linking a specific coil is changed cyclically, and a time-varying voltage is generated.

A set of such coils connected together is typically referred to as an armature winding.

EEL 3211 © 2008, H. Zmuda


g

In general, the term armature winding is used to refer to a winding or a set of windings on a rotating machine which carry ac currentsa set of windings on a rotating machine which carry ac currents.

In ac machines such as synchronous or induction machines, the armature winding is typically on the stationary portion of the motor referred to as the stator, and these windings may also be referred to as stator windings.stator windings.

Terminology: The stationary part of a machine is called the stator, while the part that rotates is called to rotorwhile the part that rotates is called to rotor.

The field circuits of most synchronous and induction machines are located on their rotors.

EEL 3211 © 2008, H. Zmuda


With rare exceptions, the armature winding of a synchronous hi i th t t d th fi ld i di i th tmachine is on the stator, and the field winding is on the rotor.

The field winding is excited by direct current conducted to it by means of stationary carbon brushes which contact rotating slip rings or collector rings.

EEL 3211 © 2008, H. Zmuda


A Simple G tGenerator

A highly idealized analysis of this machine would assume a

EEL 3211 © 2008, H. Zmuda


sinusoidal distribution of magnetic flux in the air gap.

AC Machine Basics – Instead of rotating a conductor in a fixed magnetic field as we did in an earlier example (see Set 1 Slidesmagnetic field as we did in an earlier example (see Set 1, Slides 66 and onward), suppose we now (somehow) rotate the magnetic field inside a fixed conductor.

B

i i

Xe

EEL 3211 © 2008, H. Zmuda


AC Machine Basics – For a uniform magnetic flux density B, the total flux linking the circuit isthe total flux linking the circuit is,

cosB da BA

for a constant rotational speed,

andt

BA t

the induced voltage is

cosBA t

sinde BA tdt

EEL 3211 © 2008, H. Zmuda


dt

AC Machine Basics –

It is difficult to have a uniform rotating magnetic field like the one considered so as to achieve a sinusoidal voltage. Instead, practical machines attempt to create a sinusoidally distributed rotating magnetic field in the air gap between the rotor and stator. As we will later see, this produces the desired sinusoidalstator. As we will later see, this produces the desired sinusoidal induced voltage.

EEL 3211 © 2008, H. Zmuda


Rotating Magnetic Field

Fundamental principle of AC machine operation: If a three-phase set of currents flows in a three-phase winding, it will produce a rotating magnetic field.

EEL 3211 © 2008, H. Zmuda


Three-Phasea Three PhaseWindingsa

bc

b ca

b c

EMPTY STATOR

EEL 3211 © 2008, H. Zmuda


aConvention: Current is positive

Magnetic Field

aCurrent is positive if it flows into the unprimed and out

f h i dMagnetic Field

produced by winding aa’

bc

of the primed winding.

b c

aaH t

ab c

x

EMPTY STATOR

EEL 3211 © 2008, H. Zmuda


a

Magnetic Field

a

x Magnetic Field produced by winding bb’

bc bbH t

b ca

b c

EMPTY STATOR

EEL 3211 © 2008, H. Zmuda


a

Magnetic Field

a

x Magnetic Field produced by winding cc’

bc ccH t

b ca

b c

EMPTY STATOR

EEL 3211 © 2008, H. Zmuda


Recall Set 1, Slide 14

enclosed ccC

NiH d I H Ni H

cC

NiB B H H

c

A

NiAB da

i

c

Ac

iiiiiiii

xx

xxxxxx ix

B H i

EEL 3211 © 2008, H. Zmuda


aAdd all three fields

a

x xybc

x

y

b ca

b cx

H tt tH HH

EMPTY STATOR

EEL 3211 © 2008, H. Zmuda


a ccbba H tt tH HH

bbaa ccB B tB tB t

120

1 3

2400 bbaa ccB tB tB t

y 1 32

1 32bb bbaa B t BB t t x x y

x

y

1 32 2cc ccB t B t x y x

cos 120

0 5 sin 120

0 866

1 12 2aa bb ccB t B t B t

x

cos 120

180

0.5 sin 120180

0.866

cos 240

180

0.5 sin 240

180

0.866

3 322 bb ccB t B t

y

EEL 3211 © 2008, H. Zmuda


22

1 1B B t B t B t x

2 2

3 3

aa bb ccB B t B t B t

x

3 3

2 2bb ccB t B t

y

1 1sin sin 120 sin 240

B H i

B B t t t

x sin sin 120 sin 240

2 2

3 3

MB B t t t

x

3 3sin 120 sin 2402 2MB t t

y

EEL 3211 © 2008, H. Zmuda


Let Mathcad do the algebra...

sin t 12

sin t 120

180

12

sin t 240

180

32

sin t 2

3sin t 120

3sin t 240

2sin t 120

180

2

sin t 240180

332

cos t

EEL 3211 © 2008, H. Zmuda


1 1 1 1sin sin 120 sin 2402 2MB B t t t

x

3 3sin 120 sin 2402 2MB t t

y

3 3sin cos2 2M MB t B t

x y 2 2

EEL 3211 © 2008, H. Zmuda


3 sin cosMB B t t x y

y s cos2 M t t y

t sint cost

yt

MBt sint cost0 0 1/2 1 0

2t

tMB

/2 1 0 0 -1

3/2 1 0

x32

t t

MBMB

3/2 -1 00t

MB

The magnetic flux has constant magnitude (1.5 BM) but rotates in time with angular velocity .

EEL 3211 © 2008, H. Zmuda


The winding has produced an effective (rotating) magnet with a north and a south pole, and is hence termed a two pole winding.

a

north and a south pole, and is hence termed a two pole winding.

bcx x

1.5S MB BSB

N S

ab c

xx

EEL 3211 © 2008, H. Zmuda


The magnetic poles complete one complete mechanical rotation for each electrical cycle of applied current.for each electrical cycle of applied current.

The mechanical speed of rotation of the magnetic field in revolutions per second is equal to the electrical frequency inrevolutions per second is equal to the electrical frequency in Hertz,

f f l,electrical mechanicalf f two poles

EEL 3211 © 2008, H. Zmuda


Note the order of the windings,

a c b a c b

a

x x

bc

b c

SB

ab c

x

EEL 3211 © 2008, H. Zmuda


Now suppose that we repeat the pattern:|a c b a c b a c b a c b

1b1c 2a|a c b a c b a c b a c b

x xx 1

1c2b

1a2a

1b2cx

xx

1a

1

2b 2c

EEL 3211 © 2008, H. Zmuda


1

Here’s how a phase is wound:

x xx 1b1c 2a

b

1a2a x

a

1c

a

2b

2a

1a

1b

2a

2c1aa

x 2ax

xx

1a

1b

2b

2

2c

a1a2a

EEL 3211 © 2008, H. Zmuda

4. Fundamentals of AC Machinery 281ax

The text illustrates windings as follows: a

x x

ab bc cacb

b

c

c

Back endof coil

ab c

x

of coil

1a 1a 2c 2c1c 1c 1b 1b 2a 2a 2b 2b 1 12 21 11 1 2 22 2

b bcounterclockwise

EEL 3211 © 2008, H. Zmuda


a ac cb b

Recall: The winding has produced an effective (rotating) magnet with a north and a south pole, and is hence termed a two pole

a

with a north and a south pole, and is hence termed a two pole winding.

bcx x

a c b a c b

SBN S S N

a c b a c b

ab c

xx

EEL 3211 © 2008, H. Zmuda


|S N S N

a c b a c b a c b a c b

x 1b1c 2aS N S N

x xx 1

1c2bS N

1a2a

N

B

B

1b2cSN

xx

x

1a

1

2b 2c

EEL 3211 © 2008, H. Zmuda


1

The text illustrates windings as follows:

ab bc cacB

Back endof coil

B

B

B

S

of coil

1a 1a 2c 2c1c 1c 1b 1b 2a 2a 2b 2b

N S N1 12 21 11 1 2 22 2

b bcounterclockwise

EEL 3211 © 2008, H. Zmuda


a ac cb b

180o

mechanical90o

mechanical

S

mechanical360o

electricalS SN N N

mechanical180o

electrical

N N NS S S

Though a full 360o of electrical rotation has occurred a l h h i ll d b l 180 hpole has mechanically progressed by only 180o, hence

2electrical mechanicalf f

EEL 3211 © 2008, H. Zmuda


The magnetic poles complete one complete mechanical rotation for two electrical cycles of applied current.for two electrical cycles of applied current.

For the four-pole winding, the electrical frequency of the current in Hertz is t i the mechanical frequency of rotationin Hertz is twice the mechanical frequency of rotation.

2 ,electrical mechanicalf f four poles

Generalization: ,2electrical mechanicalPf f P poles

1 l lFor nm mechanical revolutions per minute (rpm),

n P

1 cycle cycle=60 second minute

,120

melectrical

n Pf P poles

EEL 3211 © 2008, H. Zmuda


The windings have produced two effective (rotating) magnets with two north and two south poles, and is hence termed a four

x 1b1c 2awith two north and two south poles, and is hence termed a four pole winding.

x xx 1

1c2bS Nmech

1a2a

N

Bmech

mech mech

1b2cSN

mechx

xx

1a

1

2b 2c

EEL 3211 © 2008, H. Zmuda


1

S

N NSN

S

2pole 4pole2pole 4pole

EEL 3211 © 2008, H. Zmuda


Interesting: If the current in any two of the three coils is swapped, the direction of the magnetic field’s rotation will be reversed.the direction of the magnetic field s rotation will be reversed.

This means that the direction of an AC motor can be revered by switching the connection on any two of the three coilsswitching the connection on any two of the three coils.

The verification of this fact is left as an exercise (it’s done for you in the book).

EEL 3211 © 2008, H. Zmuda


Magnetomotive Force and Flux Distribution on AC Machines

So far we’ve considered only an empty stator.

The direction of the magnetic flux density produced by the coilsThe direction of the magnetic flux density produced by the coils was assumed perpendicular to the plane of the coil, determined via the right-hand-rule.

In a real machine (i.e., one with a rotor) things are a bit more complicated.complicated.

In a real machine there is a ferromagnetic rotor with a (small) gap between the rotor and the statorbetween the rotor and the stator.

The rotor can be a simple cylinder or something more complicated.

EEL 3211 © 2008, H. Zmuda


Stator Stator

Gap

Gap

Rotor

Gap

RotorNS

The cylindrical rotor is called a non-salient pole machine

The non-cylindrical rotor is called a salient pole machine

EEL 3211 © 2008, H. Zmuda


non-salient pole machine. called a salient pole machine.

variablespacing airspacing air gap

For the salient pole machine, note how the gap width varies withFor the salient pole machine, note how the gap width varies with angle . Since the total reluctance is dominated by the air gap, is possible to design a shape that produces a flux density which varies appro imatel as cos

EEL 3211 © 2008, H. Zmuda


approximately as max cos


Two observations, as per our earlier discussion of magnetic circuits:

•only the reluctance of the gap will be non-negligible

•the magnetic flux density vector B will be the shortest path possible or perpendicular to both the rotor and stator (path of least reluctance).least reluctance).

EEL 3211 © 2008, H. Zmuda



Consider a non-salient pole machine:

We wish to produce a sinusoidal voltage in the stator windings.

EEL 3211 © 2008, H. Zmuda


How do we do this?


If the induced potential (emf) at the terminals is to be sinusoidal, then the (magnitude) of the flux density vector B must vary sinusoidally along the surface of the gap Recall again the resultsinusoidally along the surface of the gap. Recall again the result from Note Set 1 (Slide 14) that the flux is:

N AY Yoo

gap gap gap

Ni ABA HA A

Y Ye

This sinusoidal variation will happen if H and in turn, the mmf Yvary sinusoidally along the surface as shown on the next slidevary sinusoidally along the surface as shown on the next slide...

EEL 3211 © 2008, H. Zmuda



This is the ideal case that we would like to achieve:

sinB B max sinB B

EEL 3211 © 2008, H. Zmuda


Magnetomotive Force and Flux Distribution on AC MachinesA simple way to accomplish this is to allow the number of turns of h i di i i i id l f hithe stator winding to vary in an approximate sinusoidal fashion as

shown, but this is rarely, even never done in practice.cosn N

7 7

3 3cosc cn N

cos induced c cd de n Ndt dt

1010

Number of conductors nc in each slot

Y

1010

Y

360

7 7

0 180Approximate sinusiodal

EEL 3211 © 2008, H. Zmuda

4. Fundamentals of AC Machinery 453 3

Approximate sinusiodalmmf distribution

Distributed Windings – How it’s really done.

Most armatures have a distributed winding where the coils are spread over a number of slots around the air-gap periphery.

The individual coils are connected so that the result is a magnetic field having the same number of poles as the field winding.

EEL 3211 © 2008, H. Zmuda


g p g

MMF of Distributed Windings – The study of the magnetic fields of distributed windings can be approached by examining theof distributed windings can be approached by examining the magnetic field produced by a winding consisting of a single N-turn coil (called a concentrated winding) which spans 180 electrical d hdegrees, as shown

180

x

EEL 3211 © 2008, H. Zmuda


MMF of Distributed Windings – A coil which spans 180 electrical degrees is known as a full-pitch winding (as opposed to a fractionaldegrees is known as a full pitch winding (as opposed to a fractional pitch winding).

180

x

EEL 3211 © 2008, H. Zmuda


MMF of Distributed Windings – The magnetic field produced by the current in the coil is shown by the dashed lines Since thethe current in the coil is shown by the dashed lines. Since the permeability of the armature and field iron is much greater than that of air, assume that all the reluctance of the magnetic circuit is in the iair gap.

Magnetic axis ofstator coil

x

EEL 3211 © 2008, H. Zmuda


MMF of Distributed Windings – From symmetry it is evident that the magnetic field intensity H in the air gap at angle under onethe magnetic field intensity Hag in the air gap at angle a under one pole is the same in magnitude as that at angle a + under the opposite pole, but the fields are in the opposite direction.

Remember that H isalways perpendicularto the stator/rotor


surface though thediagram doesn’treally show it.

stator coil

a

x ˆH H ˆag agH H a

EEL 3211 © 2008, H. Zmuda


x ˆag agH H a

MMF of Distributed Windings – Around any of the closed paths shown by the flux lines the mmf is Ni Since all the reluctance ofshown by the flux lines the mmf is Ni. Since all the reluctance of this magnetic circuit is in the air gap, the mmf drops associated with the magnetic circuit inside the iron can be neglected.


a BA Ye

x

agap

gap

e

Y e

EEL 3211 © 2008, H. Zmuda


g p

MMF of Distributed Windings – Since the air-gap fields Hag on opposite sides of the rotor are equal in magnitude but opposite inopposite sides of the rotor are equal in magnitude but opposite in direction, so will the air-gap mmf. Since each flux line crosses the air gap twice, the mmf drop across the air gap equals half of the

l / h i f di ib i i lik di ib itotal or Ni/2. The air-gap mmf distribution is a step-like distribution of amplitude Ni/2. On the assumption of narrow slot openings, the mmf jumps abruptly by Ni in crossing from one side to the other of f j p p y y ga coil.

This is shown on the next slideThis is shown on the next slide...

EEL 3211 © 2008, H. Zmuda


MMF of Distributed Windings –

Note how themmf ‘flips’ here

and here

x

EEL 3211 © 2008, H. Zmuda


MMF of Distributed Windings – “Unwound ”

actual mmf in gapNi

a2

Ni0 2

3a

2

X X

Rotor Surface

Stator Surface

Gap

EEL 3211 © 2008, H. Zmuda


Gap

MMF of Distributed Windings – “Unwound ”4f d Ni

actual mmf in gapNi

sinusoidal fundamental4 cos

2fund

gap aNi

Y

a2

Ni0 2

3a

2

X X

Rotor Surface

Stator Surface

Gap

EEL 3211 © 2008, H. Zmuda


Gap

MMF of Distributed Windings –

Rotor Surface aConcentrated Winding (one phase)

XStator Surface

a

Rotor Surface aDistributed Winding (one phase)

Stator Surface

X X X XX

EEL 3211 © 2008, H. Zmuda


Distributed Windings -

A distributed winding consists of coils distributed in several slots. This helps reduce the harmonics which, in general, are quite undesirable, leaving only the fundamental.u des ab e, eav g o y t e u da e ta .

Ph f th tPhase a of the armature winding of a simplified two-pole, three-phase ac machine. Phases b and c occupy the empty slots. The windings of the threewindings of the three phases are identical and are located with their magnetic

120 d tEEL 3211 © 2008, H. Zmuda


axes 120 degrees apart.

Distributed Windings -

EEL 3211 © 2008, H. Zmuda


Distributed Windings – Note that the mmf will be less for a distributed winding than would be for a concentrated winding.distributed winding than would be for a concentrated winding.

EEL 3211 © 2008, H. Zmuda


Distributed Windings – Fractional Pitch Windings further serve to reduce the harmonic content, but these will not be considered in thisreduce the harmonic content, but these will not be considered in this course.

FULL PITCH FRACTIONAL PITCH

EEL 3211 © 2008, H. Zmuda


FULL PITCH FRACTIONAL PITCH

Non-salient pole machines – Distributed Rotor Windings – In a similar manner the rotor windings are also distributed to reduce higher h iharmonics.

EEL 3211 © 2008, H. Zmuda


Induced Voltage in AC Machines

Just as a three-phase set of currents can produce a rotating magnetic field, a rotating magnetic field can produce a three-phase set of voltages in the coils of a stator.

EEL 3211 © 2008, H. Zmuda



Consider first a single coil.cStator

dGap

i d de

Rotor Stator

b

inducede

Rotor StatorWinding

a

EEL 3211 © 2008, H. Zmuda



Consider first a single coil, and consider a rotating rotor with a sinusoidally distributed magnetic field in the center of a stationary coil. How this magnetic field is generated will be discussed later.

Specifically assume that the magnitude of the flux density in the airSpecifically assume that the magnitude of the flux density in the air gap varies sinusoidally with mechanical angle while the direction of B is radial– this is the ideal the ideal situation.

This angle will be measured with respect to the peak rotor flux density.

All this is shown as...

EEL 3211 © 2008, H. Zmuda


Induced Voltage in AC MachinesThe magnetic flux density measuredA snapshot in time:

B

density measured around the rotor is:

cosMB B MBB

Rotor magnetic flux axisaxis.

Here is measured with respect to the rotor.

EEL 3211 © 2008, H. Zmuda



But since the rotor is rotating, the magnetic flux density vector in the gap at any angle measured with respect to the stator is

cosMB B t

B

MBB

EEL 3211 © 2008, H. Zmuda



Recall that the induced voltage in the wire is given by

B

where is the velocity of the wire relative to the magnetic field

inducede v B

vwhere is the velocity of the wire relative to the magnetic fieldand is the length of the conductor in the magnetic field. Recall that points to the “positive” end.

v

v B

When we derived this equation (in Note Set 1), the wire was moving and the magnetic field was stationary. Here the magnetic g g y gfield is moving and the wire is stationary, so some modification is needed.

EEL 3211 © 2008, H. Zmuda


Lorentz Force Law inducede B v

x x x x x x x x x x x x x xx wire

+++ F q v B

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

v

+++

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

vinducede

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

__

F B

x x x x x x x x x x x x x xx

B into page

___ F q v B

Recall Note Set 1EEL 3211 © 2008, H. Zmuda


B into page Recall Note Set 1


We establish a moving reference frame in that we move (rotate) with the magnetic field. Then to us the magnetic field is stationary and the coil will appear to go by us with an apparent velocity ofpp g y pp y

relativev

and the original equation applies.

hi i d b h f ll i iThis is captured by the following picture...

EEL 3211 © 2008, H. Zmuda



B

relativevB Note here that is shown as

being directed radially outward. This is because its actual

B

This is because its actual direction will be obtained with the from the factor

cosMB B t

B i ll i th th di ti

l tiv B

B is really in the other direction here since = at this point and

cos cost t EEL 3211 © 2008, H. Zmuda


relativev B cos cost t


B 0

crelativevBv B 0

d

MB

inducede

M

relativev

a

b relative

B t

a

lv bae v B t

B t

180

EEL 3211 © 2008, H. Zmuda


relativevB vB t


,v B v B

bae v B t

This is why we want a radial B

cos

vB t

vB t

cos

cos 180M

M

vB t

vB t

cosMvB t

EEL 3211 © 2008, H. Zmuda



B

crelativevB

B

d

MB relativev

B

relativev

inducede

M

B

ab

B

a

lv 180

EEL 3211 © 2008, H. Zmuda


relativevB


B dce v B

B

B

crelativevB vB

v B B

d

MB

relativev

inducede

M

a

b

a

lv 180

EEL 3211 © 2008, H. Zmuda


relativevB

cose vB t Induced Voltage in AC Machines

c

d

cosba Me vB t

d

b

inducede

a

b

cosdc Me vB t

2 cosinduced ba dc Me e e vB t

EEL 3211 © 2008, H. Zmuda



2 cosinduced Me vB t

v r2 cos

cos , 2induced M

M

e r B tAB t A r

loop area

for an NC - turn coil,cos ,

M

Mt AB o a NC tu co ,

cosinduced Se N t

EEL 3211 © 2008, H. Zmuda



For three coils, each with NC-turns, and place around the rotor 120o

apart, the magnitude of each voltage will the same as for the case of a single coil but will differ in phase by 120o, ora single coil but will differ in phase by 120 , or

cosaa Se N t

cos 120

cos 240bb Se N t

e N t

hence a three-phase set of currents can generate a uniform rotating

cos 240cc Se N t

e ce a t ee p ase set o cu e ts ca ge e ate a u o otat gmagnetic field in the stator, and a uniform rotating magnetic field can generate a three-phase set of voltages in the stator as claimed,

EEL 3211 © 2008, H. Zmuda



The peak voltage in any phase is

max 2S SE N N f

Note that the voltage at the terminals of the machine will depend on

2rms SE N f Note that the voltage at the terminals of the machine will depend on whether the stator is Y or connected. For a -connected machine,

while for a Y-connected machine,

2phase rms SV E N f

w e o a co ected ac e,

3 6phase rms SV E N f

EEL 3211 © 2008, H. Zmuda


Induced Torque in AC Machines

In AC machines there are normally two magnetic fields – one from the rotor and one from the stator.

It is the interaction between these two fields that produces the torque in the machine.

Consider the following simple single coil machine...

EEL 3211 © 2008, H. Zmuda



Consider a sinusoidal stator flux distribution and a single coil of wire mounted on the rotor.

The stator flux distribution is,

SB

iB B

How much torque is produced?

SsinS SB B

i

x i

EEL 3211 © 2008, H. Zmuda




Conductor 1: Conductor 1:

1indF SB

1indF i B

x

1r

2r

F

sinSi B

2indF1 11

iind indr F

i B

sinSri B

counterclockwise

EEL 3211 © 2008, H. Zmuda




Conductor 2: Conductor 2:

1indF SB

2indF i B

x

1r

2r

1

F

sinSi B

x2indF

2 22

iind indr F

i B

sinSri B

counterclockwise

EEL 3211 © 2008, H. Zmuda



But the current in the rotor produces its own magnetic field, whose direction is determined via the right-hand-rule.

SB

180

S

i

xRBi

R RB H i

EEL 3211 © 2008, H. Zmuda


Induced Torque in AC MachinesNote the net magnetic axis for the stator.

iB B

SB

max

sinS SB B

S

EEL 3211 © 2008, H. Zmuda


Induced Torque in AC Machines 2 sinind Sri B counterclockwise

max

max2 sin

ind S

ind R Sr B B

sin sin sin

sin sinR S R SR S B B B BB B

R RB H i

B

x B

iSB

RB

EEL 3211 © 2008, H. Zmuda



2 sin

2 sinind Sri B counterclockwise

B B t l k i

2 sinind R Sr B B counterclockwise

ind R SB B

This is a general result though qualitative in nature. It applies to any AC machine and will be used to develop a qualitative understanding of torque in AC machines.o to que C ac es.

EEL 3211 © 2008, H. Zmuda



This can be put into a form that will be useful later. The net magnetic field in the machine is the vector sum of the rotor and stator fields,stator fields,

B B B

net R SB B B SB

netB

RB

EEL 3211 © 2008, H. Zmuda



net R S S net RB B B B B B

B B B B B B B

ind R S R net R R netB B B B B B B

B B

ind R netB B

EEL 3211 © 2008, H. Zmuda


Induced Torque in AC Machines – A qualitative Example –Consider the simple (salient pole) machine whose fields are rotating in a counterclockwise direction.

RB

SB

ind R SB B

x

xx

The torque is clockwise, via the right-hand-rule, opposite the direction of rotation so this machine is a generator

EEL 3211 © 2008, H. Zmuda


direction of rotation, so this machine is a generator.

4. Fundamentals of AC Machinery - University of Florida · AC Machines: We begin this study by...

Documents

Transcript of 4. Fundamentals of AC Machinery - University of Florida · AC Machines: We begin this study by...