lecture 24 induction machines (1)

Copyright © 2017 Hassan Sowidan

ECE 330

POWER CIRCUITS AND ELECTROMECHANICS

LECTURE 24

INDUCTION MACHINES (1)

Acknowledgment-These handouts and lecture notes given in class are based on material from Prof. Peter

Sauer’s ECE 330 lecture notes. Some slides are taken from Ali Bazi’s presentations

Disclaimer- These handouts only provide highlights and should not be used to replace the course textbook.

12/4/2017


INDUCTION MACHINES

• Most widely used machines in the industry

• Both the stator and the rotor carry alternating currents

• Used in pumps, compressors, fans, etc.

• Motor inductance can be controlled to have excellent

torque-speed capabilities.

• Analysis based on equivalent circuits derived from the

energy point of view

,

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INDUCTION MACHINES

,

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PHYSICAL FEATURES AND FREQUENCY

RELATIONSHIPS • It has a three-phase winding on both the stator and the

rotor, wound for P poles

• The three-phase winding on the rotor is short circuited

either through external means or on the rotor itself

Wound rotor

Squirrel cage rotor

• Two basic components: the stator and the rotor

• Analogous to a transformer with rotating secondary.

,

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WOUND ROTOR

The rotor windings that rotate with the rotor are connected

to slip rings.

The fixed brushes provide a short-circuited path through an

external resistance

,

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WOUND ROTOR

External resistance is inserted to achieve a desired torque-

speed characteristic. A higher starting torque can be

achieved together with limited control over speed.

Source: pinterest.com

,

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SQUIRREL CAGE ROTOR

• Rotor windings shorted on the rotor itself

• A fixed-torque speed characteristic

• The conductors through the slots are simply shorted

through a ring at the ends

,

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OPERATION OF AN INDUCTION MACHINE

Three-phase currents are supplied to the stator windings,

resulting in a rotating field.

Each phase may be wound for poles so that the rotating

field revolves at the synchronous speed given by

where is the frequency in Hz and is in revolutions

per minute (rpm)

,

p

=120

spNfsN

f

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sN



• Currents are induced in the rotor windings, which interact

with the rotating field to produce torque.

• The speed increases to a point where the torque

developed between rotor and stator is just enough to

balance the opposing mechanical load torque

• So long as the opposing torque (due to bearing friction,..

etc., under no-load conditions) exists, the rotor speed can

never equal the speed of the stator rotating field speed.

,

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The rotor currents have a frequency .

Mechanical speed in mechanical rads per second satisfies

the relation by

The frequency of induced currents in the rotor is

We define a new dimensionless quantity called slip as

,

r

=2

s r m

p

= /2

r s m

pelectrical rads sec

a= s ct

s

N Ns

N

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: The synchronous speed in rpm,

: the actual speed of the rotor

Multiplying the right side by we get

,

2

120

p

a

22

120 120=

2

120

sct

s

pN pN

spN

a22

2 60 2= =

2

cts m

s

Np pf

sf

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sN

actN



So,

Since

So,

,

r ss

= /2

r s m

pelectrical rads sec

1 2= / 1

2

s

m m s

smechanical rads sec s

p p

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From this

Where is the frequency of current in the rotor

Slip is dimensionless, and it gives an idea of the speed of

the machine with respect to the synchronous speed.

corresponds to synchronous speed

corresponds to standstill or starting conditions

is a very small number between 0.01 to 0.05.

,

=rf s f

rf

= 0s

=1s

s

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EXAMPLE 7.1

Consider a three-phase, 60 Hz, six-pole induction machine

run at 1125 rpm. Find the slip and the rotor current

frequency .

,

rf=

120

spNf

60 120= = 1200

6sN rpm

1200 1125= = 0.0625

1200s

= = (0.0625)60 = 3.75rf sf Hz

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ANALYSIS OF A TWO-POLE MACHINE

Just as in the case of the synchronous machine, we first do

the analysis for a two-pole machine and then generalize to a

P pole machine. The stator has three-phase windings.

Unlike in the synchronous machine, there are three

windings on the rotor,

• Label s for stator windings

• Label r for rotor windings

,

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DERAVATION OF AN EQUIVALENT CIRCUIT

,

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,

= ˆ

ˆ

ˆ

asas

bsbs

cscs

arar

br br

crcr

i

i

iA

i

i

i

i relation is

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where the matrix is

,

A

cos cos( 120 ) cos( 120 )2 2

cos( 120 ) cos cos( 120 )2 2

cos( 120 ) cos( 120 ) cos2 2

2 2

2 2

2 2

ms mss

ms mss

ms mss

mr mrr

T mr mrr r

mr mrr

L LL M M M

L LL M M M

L LL M M M

L LL

L LM L

L LL

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is the self-inductance of the stator coil

is the self-inductance of the rotor coil

The leakage inductance ,

The magnetizing inductance

, ,

Mutual inductance

,

,

=r mr rL L L

2

0=2

sms

N rL

g

0=4

s rN N rM

g

=s ms sL L L

2

0=2

rmr

rNL

g

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sLrL

= ss s

r

NL L M

N = r

r r

s

NL L M

N



Machine parameters:

is the air gap length, is the radius of the air gap, is

the axial length of the core, and are the effective

turns per phase on the stator and rotor, respectively.

Note that for both the stator and rotor windings, the current

directions are into the windings. The rotor currents are

denoted as , , and into the coils

,

g r

sNrN

ˆari ˆ

bri ˆcri

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For two-phase machine

Let the angle between the rotor and stator axes of phase a be

Electric torque requires

Instead of (synchronous)

,

= mt

=r s m

, 0m s r

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0 cos sin

0 sin cos

ˆcos sin 0

ˆsin cos 0

asas s

bsbs s

ar r ar

br rbr

iL M M

iL M M

M M L i

M M L i

lecture 24 induction machines (1)

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