1 CHAPTER 3 EMT 113: V-2008 School of Computer and Communication Engineering, UniMAP Prepared By: Prepared By: Amir Razif b. Jamil Abdullah Basic AC Motor

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CHAPTER CHAPTER 33

EMT 113: V-2008EMT 113: V-2008

School of Computer and School of Computer and Communication Engineering, UniMAPCommunication Engineering, UniMAP

Prepared By: Prepared By: Amir Razif b. Jamil AbdullahAmir Razif b. Jamil Abdullah

Basic AC Basic AC Motor.Motor.

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3.1 Synchronous Machine.3.1 Synchronous Machine.3.1.1 Introduction to Synchronous Generator.3.1.1 Introduction to Synchronous Generator.3.1.2 Synchronous Generator Construction.3.1.2 Synchronous Generator Construction.3.1.3 Synchronous Machine Characteristics.3.1.3 Synchronous Machine Characteristics.3.1.4 Induce Voltage in Armature Windings.3.1.4 Induce Voltage in Armature Windings.3.1.5 Synchronous Generator and Circuit Model.3.1.5 Synchronous Generator and Circuit Model.3.1.6 Synchronous Motor and Circuit Model.3.1.6 Synchronous Motor and Circuit Model.

3.2 3.2 Induction Motors.Induction Motors.3.2.1 Construction of Induction Motor.3.2.1 Construction of Induction Motor.3.2.2 Basic Induction Motor Concept.3.2.2 Basic Induction Motor Concept.3.2.3 Equivalent Circuit of Three-Phase Induction 3.2.3 Equivalent Circuit of Three-Phase Induction Motor.Motor.3.2.4 Performance Calculation Using Circuit Model.3.2.4 Performance Calculation Using Circuit Model.3.2.5 Power and Torque in Induction Motors. 3.2.5 Power and Torque in Induction Motors. 3.2.6 Induction Motor Torque-Speed Characteristics.3.2.6 Induction Motor Torque-Speed Characteristics.

3.0 Application of 3.0 Application of Alternate Current (AC) Alternate Current (AC) Motor.Motor.

3

3.0 AC Machinery 3.0 AC Machinery Fundamental.Fundamental. AC machines are the generators and motors.

Two major classes of machines; (i)(i) Synchronous machines. Synchronous machines.(ii)(ii) Induction machines Induction machines..

3.1 Synchronous Machine.3.1 Synchronous Machine.

4

Synchronous generatorsSynchronous generators or alternators are synchronous machines used to convert mechanical power to ac electric power, depending on the mode of operation of the machine.

A machine has two modes of operation;(a) motor(a) motorAn electric motor converts electrical energy into mechanical energy. (b) generator. (b) generator. An electric generator is used to convert mechanical energy into electrical energy. For this reason, electric machines are also called electromechanical energy converters.

3.1.1 Introduction to a 3.1.1 Introduction to a Synchronous GeneratorsSynchronous Generators..

5

Below is the condition of motor and generator operations.

The magnetic field in a machine forms the energy link between the electrical and mechanical systems.

Current flowing in coils of wire inside the machine produces the magnetic field. The magnetic field performs two functions:(a) Magnetic attraction and repulsion produces mechanical torque (motor operation),(b) The magnetic field by Faraday’s law induces voltages in the coils of wire (generator operation).

Figure 3.1. Machine Figure 3.1. Machine Modes of Operation: Modes of Operation: (a) Motor, and (a) Motor, and (b) Generation.(b) Generation.

Cont’d…Cont’d…

6

Motor operation in Figure 3.1(a), the field-induced voltage, E, permits the motor to draw power from the line to be converted into mechanical power.

While the magnetic field is inducing this voltage, it is also developing the mechanical output torque, . Note that the induced voltage is in opposition to the current flow. It is often called the counter emf .

Generator operation in Figure 3.1(b), the field induced voltage E is in the same direction as the current and is called the “generated voltage.” In this case the machine torque opposed the input mechanical torque that is trying to drive the generator, and it is called counter torque.


7

In a synchronous generator, a dccurrent is applied to the rotor winding, which produces a rotor magnetic field.

The rotor is turned by a prime mover which producing a rotating magnetic field within the machine.

The outside stationary part, called the stator. The inner rotating part is called the rotor.

The rotor is mounted on a stiff rod, called a shaft. The rotor is solidly fastened to the shaft so that the rotor and the shaft turn at the same speed.

3.1.2 Synchronous 3.1.2 Synchronous Generator Construction.Generator Construction.

Figure 3.2: Cut-Way View of a Synchronous

Machine.

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The terms “rotor speed,” “shaft speed,” and “machine speed” all mean the same thing.

The rotor and the stator have three basic parts;(i) Core.(i) Core.(ii) Windings,(ii) Windings,(iii) Insulation. (iii) Insulation.

The purpose of the rotor and stator cores is to conduct the magnetic field through the coils of the windings.

(i) (i) Cores.Cores. The cores are always made of iron steel, similar

to the transformer core.


9

Windings.Windings. Two common term for windings; “field windings”

and “armature windings”. The “field windings” applies to the winding the

produce the main magnetic field to the machine. The “armature windings” applies to the

windings where the main voltage is induced. The “field windings” = “rotor windings” is on

the rotor which carries the magnetization current. The “armature windings” = “stator windings”

is on the stator which carries the load current.Rotor.Rotor. Rotor can either be salient or non-salient pole. Salient pole is a magnetic pole that protruding

or stick out from the surface of the rotor. It is normally used for the rotor with four or more poles.


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Non-salient pole is a magnetic pole constructed flush with the surface of the rotor. It is normally used for the rotor with two- and four-poles.

There are two common approaches to supplying dc power to the field circuit of the rotor.(1) From the external dc source, by means of slipping and brushes.(2) From a special dc power source mounted directly on the shaft of the synchronous generator.


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Figure 3.3 shows the two types of rotors in a four-pole synchronous machine.

The rotor carries the field winding. The field current or the excitation current is provided by an external dc source. Synchronous machine rotors are simply rotating electromagnets built to have as many poles as are produced by the stator windings.

Figure 3.3: Four Pole Synchronous Machines: (a) with Salient-Pole Rotor, (b) with Cylindrical Rotor

Having three Coils per Pole.


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Synchronous machines are called “synchronous” because their mechanical shaft speed is directly related to the power system’s line frequency.

Stator electric frequency;

where, P is the number of magnetic poles.nm is the mechanical speed of magnetic field in r/min.fs is electrical frequency in Hz.

Typical machines have two-poles, four-poles, and six-poles. There is a set of relationships between angular

quantities (position and velocity) of the mechanical shaft and the electrical system.

3.1.3 Synchronous 3.1.3 Synchronous Machine Characteristics.Machine Characteristics.

)/(.4

120

)(120

sradP

f

orP

fn

rpmPn

f

es

es

se

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The electrical angle between two consecutive magnetic poles (north pole and south pole) is 180°, but the mechanical angle is based on the number of poles. The angle is found as follows;

For example, a two-pole machine has a mechanical angle of 180°, and a four-pole motor has a mechanical angle of 90°. Therefore, the relation between the electrical angle and the mechanical angle is written as:

The angular velocity is the time-derivative of the angular position,


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The relation between the mechanical speed and the electrical speed can be obtained as:

If one machine has two-poles and another has eight-poles, the eight-pole machine will run at precisely one-forth the speed of the two-pole machine.

The synchronous machines, may be operated either as motors or as generators.

Since electrical energy can be transported most economically by three-phase transmission systems, nearly all synchronous generators larger than 10 kVA, and most industrial motors are designed for three-phase operation.


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Distribution FactorDistribution Factor Each phase winding consists of Nf-turns of

conductor, which is made up of coils and groups. There is a group of coils for each pole in the

machine. Within each group, coils are distributed over several slots in the stator. An example is shown in Figure 3.4.

A voltage is induced in each coil with the movement of the rotating air-gap flux. The coil voltages are displaced from one another in phase by the slot angle a. The slot angle is the distance between adjacent slots, and is given (in electrical degrees) by

where, P is the number of poles,Q is the number of slots, m is the number of phases, and q is the number of slots per pole per phase.

3.1.4 Induce Voltage in 3.1.4 Induce Voltage in the Armature Winding.the Armature Winding.

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The resultant voltage for each group of coils is the phasor sum of the coil voltages. It is defined as,

The phasor addition of the voltages is shown in Figure 3.8. From the geometry, the distribution factor can be obtained as;

Where Nf is the total number of series turns for a phase winding.


17

Example 3.1: Example 3.1: Synchronous Synchronous Motor.Motor.A three-phase, wye-A three-phase, wye-connected, 60 Hz connected, 60 Hz synchronous machine synchronous machine has six poles and 36 has six poles and 36 slots in its stator. The slots in its stator. The flux per pole (fmax) is flux per pole (fmax) is 0.024 Wb and each coil 0.024 Wb and each coil in the stator winding in the stator winding has 2 turns. Each coil has 2 turns. Each coil has a span of 5 slots. has a span of 5 slots. Find: Find: (a)(a)the coil pitch the coil pitch and and the pitch factor the pitch factor kkpp, and , and the induced coil the induced coil voltage voltage EEcc and and (b) the induced phase (b) the induced phase voltage voltage EEff on a line-to- on a line-to-line basis.line basis.Solution:

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A synchronous generator functions on the basis of Faraday’s Law. If the flux linking the coil changes in time, a voltage is induced in a coil. The voltage is induced in a conductor if it cuts magnetic flux lines.

Figure 3.4 is a three-phase machine.

The flux density in the air gap is uniform implies that sinusoidally varying voltages will be induced in the three coils aa’, bb’, and cc’ if the rotor carrying dc, rotates at a constant speed, ns.

3.1.5 Synchronous 3.1.5 Synchronous Generator and its Circuit Generator and its Circuit Model.Model.

Figure 3.4: A Three-Phase Machine with a Graph of its Associated Flux Density Distribution Produced by the Rotor Excitation and a Graph of the Three-Phase Voltages Generated.

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The voltage induced in phase a is given by;

Where, is the flux per pole, e is the electrical angular frequency, and N is the number of turns in phase a or the coil aa’.

The induced voltages for the other two phases are equal in magnitude and displaced by ±120°.

The phase voltages may be written as;

and fe is the frequency of the induced voltage.


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E be defined as the open-circuit, induced phase voltage for a certain field current If.

Then E is the internal voltage of the generator. It is assumed that If is such that the machine is operating under unsaturated conditions.

The combination of the leakage reactance, the armature magnetizing reactance and the winding resistance forms the “synchronous impedance”, Zs.

The summation of the two reactance is called the “synchronous reactance”, Xs, but is often designated as Xd for the direct-axis reactance (from the more detailed models of machines).

The winding resistance is also called the armature resistance, Ra.

The equivalent circuit of a synchronous generator and the combined phasor/vector diagram with a lagging power-factor current flow are shown in Figure 3.5.


21

If the generator operates at a terminal voltage VT while supplying a load corresponding to an armature current Ia, then;

In an actual synchronous machine, the reactance is much greater than the armature resistance, in which case;

Among the steady-state characteristics of a synchronous generator, its voltage regulation and power-angle characteristics are the most important ones.

The voltage regulation of a synchronous generator is defined at a given load as;


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where VT is the terminal voltage for a given loading condition, and E is the no-load terminal voltage.

The voltage regulation can be illustrated by the following example.

Figure 3.5: Synchronous Generator Model and Phasor-Figure 3.5: Synchronous Generator Model and Phasor-Vector Diagram.Vector Diagram.


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Example 3.2: Example 3.2: Synchronous Generator.Synchronous Generator.A three-phase, wye-connected 2500 kVA and 6.6 kV A three-phase, wye-connected 2500 kVA and 6.6 kV generator operates at full-load. The per-phase generator operates at full-load. The per-phase armature resistance armature resistance RRaa and the synchronous and the synchronous reactance, reactance, XXdd, are (0.07W+j10.4W)., are (0.07W+j10.4W).Calculate the percent voltage regulation at Calculate the percent voltage regulation at (a) 0.8 power-factor lagging, and (a) 0.8 power-factor lagging, and (b) 0.8 power-factor leading.(b) 0.8 power-factor leading.

Solution.Solution.

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Distribution FactorDistribution Factor Each phase winding consists of Nf- turns of conductor,

which is made up of coils and groups. There is a group of coils for each pole in the machine.

Within each group, coils are distributed over several slots in the stator. An example is shown in Figure 3.4.

A voltage is induced in each coil with the movement of the rotating air-gap flux. The coil voltages are displaced from one another in phase by the slot angle a. The slot angle is the distance between adjacent slots, and is given (in electrical degrees) by

where,P is the number of poles,Q is the number of slots, m is the number of phases, and q is the number of slots per pole per phase.


25

The resultant voltage for each group of coils is the phasor sum of the coil voltages. It is defined as,

The phasor addition of the voltages is shown in Figure 3.8. From the geometry, the distribution factor can be obtained as;

Where Nf is the total number of series turns for a phase winding.


26

Figure 3.6: Synchronous Motor Circuit Model (lossless) and Phasor-Vector Diagram.

The rotor (or field) winding 17 be fed by a dc source that produces the rotor magnetic field with definite polarities.

The rotor will tend to align with the stator field and will tend to rotate with the rotating magnetic field. To recapitulate, it can be said that the stator rotating magnetic field has a tendency to “drag” the rotor along, as if the north pole on the stator “locks in” with a south pole of the rotor.

The armature resistance (Ra) may be neglected as compared to the synchronous reactance.

3.1.6 Synchronous Motor 3.1.6 Synchronous Motor and Circuit Model.and Circuit Model.

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The steady-state per-phase equivalent circuit of a synchronous machine simplifies to the one shown in Figure 3.6.

In the figure, the armature current Ia goes into the machine for motor operation, and out of it for generator operation where VT is the terminal voltage for a given loading condition, and E is the no-load terminal voltage.


28

3.2 Induction Motor.3.2 Induction Motor. In the induction machine the rotor voltage which

produces the rotor current and the rotor magnetic field is induced in the rotor windings rather than physically connected by wires, Figure 3.7.

No dc field current is required to run the machine. Induction machine are cheap to manufacture, rugged and

reliable and find their way in most possible applications. Variable speed drives require inexpensive power

electronics and computer hardware, and allowed induction machines to become more versatile.

In particular, vector or field oriented control allowsinduction motors to replace DC motors in many applications

Figure 3.7: Induction Motor.

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3.1.1 Construction of 3.1.1 Construction of Induction Motor.Induction Motor. Induction motor has the same physical stator as a

synchronous machine, with a different rotor construction.

There are two different types of induction motor rotors ;(a) Cage rotor.(a) Cage rotor.(b) Wound rotor.(b) Wound rotor.

30

(a) Cage rotor(a) Cage rotor. . Cage induction Motor rotor consists of a series of

conducting bars laid into slot carved in the face of rotor and shorted at either end by large shorting ring; Figure 3.8.

Figure 3.9: Rotor Components of the Squirrel Cage Induction Motor.

Figure 3.8: Rotor of the Squirrel Cage Induction Motor.


31

(b) (b) Wound rotor. Wound rotor. A wound rotor has a complete set of three-phase winding that are

mirror images of the winding on the stator. The three phases of the rotor windings are usually Y-connected, the

end of the three rotor wires are tied to slip ring on the rotor shaft. Rotor windings are shorted through brushes riding on the slip rings.

Wound-rotor induction motors are more expansive than the cage induction motors, they required much more maintenance because the wear associated with their brushes and slip rings.

Figure 3.10: Wound-Rotor Induction Motor.Figure 3.10: Wound-Rotor Induction Motor.


32

3.1.2 Basic Induction 3.1.2 Basic Induction Motor Concept.Motor Concept.(a) (a) The Development of Induce Torque in an The Development of Induce Torque in an

Induction Motor.Induction Motor. The three-phase of voltages has been applied to

the stator, and three-phase set of stator current is flowing . These currents produce a magnetic field Bs, rotating counterclockwise direction.

The speed of the magnetic field’s rotation is given by;where,

- fe is the system frequency in hertz.

- P is the number of poles in the machine.

P

fn esync

120

33

The rotating magnetic field Bs passes over the rotor bars and induce a voltage in them.

The voltage induced in a given rotor bar is given by the equation;where,

v = velocity of the bar relative to the magnetic field.B = magnetic flux density vector.I = length of conductor in the magnetic field.

The relative motion of the rotor compared to the stator magnetic field that produces induce voltage in a rotor bar.

The induce torque in the machine is given by;

The rotor induced torque in counterclockwise, the rotor accelerates in that direction.

IBveind )(

SRind BkB


34

Speed.Speed. There is a finite upper limit to the motor speed. If the induction motor rotor were tuning at

synchronous speed, then the rotor bar would be stationary relative to the magnetic field (no induce voltage).

If eind is zero, there would be;- no rotor current,- no rotor magnetic field, - the induce torque is zero,- and the rotor would slow down as a result of friction lost.


35

(b) (b) The Concept of Rotor Slip.The Concept of Rotor Slip. The voltage induced in a rotor bar of an

induction motor depends on the speed of the rotor relative to the magnetic field.

Slip speed is defined as the difference between synchronous speed and rotor speed;Where,

nslip = slip speed of the machine.nsync = speed of the magnetic fields.nm = mechanical shaft speed of motor.

Another term used to describe the relative motion is slip;

Slip in angular velocity, ;

msyncslip nnn

%100

%100

sync

mslip

sync

slip

n

nns

n

ns

%100

sync

msync

ns


36

(c) (c) The Electrical Frequency on the Rotor.The Electrical Frequency on the Rotor. The induction motor works by inducing voltages and

current in the rotor of the machine. If the rotor of a motor is locked so that it cannot move,

the rotor will have the same frequency as the stator. If the rotor turns at synchronous speed, the frequency

on the rotor will be zero. For nm= 0 r/min & the rotor frequency fr = fe slip, s

= 1 nm=nsync & the rotor frequency fr = 0 slip, s = 0- For any speed in between, the rotor frequency is directly proportional to the difference between the speed of the magnetic field nsync and the speed of the rotor nm.

Rotor frequency can be express as;

er sff

)(120 msyncr nnP

f


37

Example 3.3: Example 3.3: Induction Motor.Induction Motor.A 208-V, 10-hp, four-pole, 60-Hz, Y- connected A 208-V, 10-hp, four-pole, 60-Hz, Y- connected induction motor has a full-load slip of 5 percent.induction motor has a full-load slip of 5 percent.

(a)(a) What is the synchronous speed of this motor?What is the synchronous speed of this motor?

(b)(b) What is the rotor speed of this motor at the rated What is the rotor speed of this motor at the rated load?load?

(c)(c) What is the rotor frequency of this motor at the What is the rotor frequency of this motor at the rated load?rated load?

(d)(d) What is the shaft torque of this motor at the rated What is the shaft torque of this motor at the rated load?load?

Solution:Solution:(a)The synchronous speed of this motor is,

(b)The rotor speed of the motor is given by, min/1710min)/1800)(95.01(

)1(

rrn

nsn

m

syncm

min/18004

)60(120

120

rPoles

Hzn

fP

n

sync

esync

38

(c) The rotor frequency of this motor is given by ,

or

(d) The shaft load torque is given by,

.

mNsrradr

hpWhp

P

load

m

outload

.7.41)60min/1)(/2min)(/1710(

)/746)(10(

HzHzsff er 3)60)(05.0(

Hzrrf

nnP

f

r

msyncr

3min)/1710min/1800(120

4

)(120


39

3.1.3 The Equivalent 3.1.3 The Equivalent Circuit of an Induction Circuit of an Induction Motor.Motor. Induction motor operates on the induction of

voltage and current in its rotor circuit from the stator circuit (transformer action).

An induction motor is called a singly excited machine, since power is supply to only the stator circuit.

Figure 3.11 is the equivalent circuit of an induction motor.

Figure 3.11: The Two Forms of Exact Equivalent Circuit of an Induction Motor.

40

R1


41

R1


42

(a) (a) The Transformer Model of an Induction The Transformer Model of an Induction Motor.Motor.

R1 is the stator resistance and X1 is the stator leakage reactance.

The flux in the machine is related to the integral of the applied voltage E1.

The curve of magnetomotive force versus flux (magnetization curve) for this machine is compared to a similar curve for a power transformer.


43

The Final Equivalent Circuit.The Final Equivalent Circuit. The core loss resulting from stator leakage flux is not

negligible, as it is in the transformer.

Figure 3.12: (a) An Equivalent Circuit of an Induction Motor with Rc Neglected.

(b) The Power Flow and Associated Losses of


44

Example 3.4:Example 3.4: Induction Motor Induction Motor Equivalent Circuit.Equivalent Circuit.An induction motor draws 25 A from a 460 V, three-An induction motor draws 25 A from a 460 V, three-phase line at a power of 0.85 lagging. The stator phase line at a power of 0.85 lagging. The stator copper lost is 1000W and the rotor copper loss is copper lost is 1000W and the rotor copper loss is 500W. The rotational losses include 250W of 500W. The rotational losses include 250W of friction and wind-age, 800W of core and 200W of friction and wind-age, 800W of core and 200W of stray load losses. Calculate,stray load losses. Calculate,(a)(a)The air-gap power,The air-gap power,(b)(b)The developed mechanical power, PThe developed mechanical power, Pdd,,(c)(c)The output horse power,The output horse power,(d)(d)The efficiency.The efficiency.Solution:Solution:(a)The air-gap power,

(b)The developed mechanical power, Pd,

KW

WKWP

RCLPP

d

gd

43.15

50093.15

KW

WpfAVP

SCLPP

g

ing

93.15

1000)85.0)(25)(460(3

45

(c) The output horse power,

(d) The efficiency,

.

Figure 4.4: (a) An Equivalent Circuit of an Induction Motor with Rc Neglected. (b) The Power Flow and Associated Losses of

hpKWhp

KWP

KW

WWWKWP

PPPPPPP

out

out

slljwcdrotdout

0.19/746

18.14

18.14

)200250800(43.15

)(

%3.8393.16

18.14

KW

KW

P

P

in

out


46

Example 3.5:Example 3.5: Induction Motor Equivalent Induction Motor Equivalent Circuit.Circuit.If the frequency of the source in the above example is If the frequency of the source in the above example is 60Hz, and the machine has four poles, fine60Hz, and the machine has four poles, fine(a)(a)The slip,The slip,(b)(b)The operating speed,The operating speed,(c)(c)The developed torque,The developed torque,(d)(d)The output torque.The output torque.Solution:Solution:(a)The slip,

(b)The operating speed,

rpmsnn

sradsrads

rpmHz

P

fn

sradHz

P

f

s

s

es

es

1744)0314.01(1800)1(

/6.182)0314.01(/5.188)1(

18004

)60(120120

/5.1884

)60(44

0314.093.15

500

KW

W

P

RCLs

g

47

(c) The developed torque,

(d) The efficiency,

note: that the torque is 8% less than the developed torque.

.

mNKWPd

d .5.846.182

43.15

mNKWPout

d .7.776.182

18.14


48

3.1.4 Performance 3.1.4 Performance Calculation Using Circuit Calculation Using Circuit Model.Model.

The performance of the motor at any speed may be calculated on the basis of relationships derived in previous session. Below is the steps;

(1) Calculate the synchronous speed.(1) Calculate the synchronous speed.

(2) Calculate the slip for the desired speed.(2) Calculate the slip for the desired speed.

(3) Calculate the rotor impedance.(3) Calculate the rotor impedance.

P

for

P

fs

1204 1

s

s

s

s

n

nns

22

2 ''

jXs

Rz

49

Figure 3.13: The Field Impedance Concept in the Induction Machine.

(a) the Field Impedance, Zf. (b) The Model Employing ,Zf.


50

(4) Calculate the field impedance, Z(4) Calculate the field impedance, Zff. . This is the effect of the resultant magnetic field on the impedance of one phase of the stator. It is the parallel combination of the rotor impedance, z2, and the core’s magnetizing reactance, jXm, referring to Figure 4.5(a).

(5) Calculate the impedance looking in from the (5) Calculate the impedance looking in from the phase terminals of the machinephase terminals of the machine,, as shown in Figure 4.5(b).

(6) Calculate the phase current.(6) Calculate the phase current.

)'('

''

22

22

XXjs

R

jXs

RjX

jXRZ

m

f

fff

in

o

Z

VI

011

)()( 111 fffin XxjRrZzZ


51

(7) Calculate the power factor. (7) Calculate the power factor.

(8) Calculate the input power.(8) Calculate the input power.

(9) Calculate the stator copper loss.(9) Calculate the stator copper loss.

(10) Calculate the air gap power.(10) Calculate the air gap power.

(11) Calculate the rotor copper loss.(11) Calculate the rotor copper loss.

(12) Calculate the developed mechanical power.(12) Calculate the developed mechanical power.

)cos( fpf

1213 rISCL

)cos(3 LLLin IVP

fg RIP 213

)cos(3 LLLin IVP

gd PsP )1(

52

(13) Calculate the developed torque.(13) Calculate the developed torque.

(14) Calculate the output power.(14) Calculate the output power.

(15) Calculate the output torque.(15) Calculate the output torque.

(16) Calculate the efficiency.(16) Calculate the efficiency.

in

out

P

P

rotdout PPP

..04.7. lbftn

PormN

P

s

g

s

gd

..04.7. lbftn

PormN

P outoutout


53

3.1.6 Induction Motor 3.1.6 Induction Motor Torque-Speed Torque-Speed Characteristics.Characteristics.

Apply the Thevenin’s theorem to the exact equivalent circuit model of an induction motor.

According to Thevenin’s theorem, the equivalent source voltage VTh is the voltage that would appear across the terminals a and b of Figure 4.6 with the rotor circuit open.

(3) Calculate the rotor impedance.

)(

)(

11 m

minTh Xxjr

jXVV

54

The Thevenin equivalent stator impedance RTh + j XTh is the impedance between the terminals a and b of Figure 3.14, viewed towards the source with the source voltage Vin short circuited.

The impedance is calculated as:

The torque is;

Figure 3.14: Application of Thevenin’s Theorem to the Circuit Model of an Induction Motor.

)()(

))((

)//()(

11

11

11

m

m

m

ThThTh

jXjxr

jXjxr

jXjxr

jXRZ

s

RIP

sd

s

gd

222

'3


55

By combining equation 4.31 and 4.32, results in,

Figure 3.15 is the general shape of the torque-speed (torque-slip) curve. The region of (s>0) and the generator region (s<0) are shown in the Figure.

)'('

22

2

XXjs

RR

VI

ThTh

Th

s

R

XXs

RR

V

ThTh

Th

sd

2

22

2

2 '

)'('

3


56

The internal torque is maximum when the power delivered to R’2/s is at maximum, Figure 3.14(b).

The maximum torque and power happen at the matching of the source impedance and the slip value, smax,

The slip at maximum torque, smax,

22

22

2 )'(''

XXRjXZs

RThThTh

22

2

2max

)'(

'

XXR

Rs

ThTh Figure 3.15: Torque of an Induction Machine

(a) Developed Torque as a Function of Slip. (b) Developed Torque as a Function of Speed.


57

The maximum torque is independent of the rotor resistance while the slip at which it occurs is proportional to the rotor resistance.

222

2

max)'(2

3

XXRR

V

ThThTh

Th

s


58

Example 3.6: Example 3.6: Torque Speed Characteristics on an Torque Speed Characteristics on an Induction Motors.Induction Motors.A 220 V line-to-line, three-phase, wye-connected, 10 hp, 60 Hz A 220 V line-to-line, three-phase, wye-connected, 10 hp, 60 Hz 6-pole induction motor has the following constants in ohms per 6-pole induction motor has the following constants in ohms per phase with reference to the stator:phase with reference to the stator:

rr1 = 0.294 W 1 = 0.294 W R'R'2 = 0.144 W2 = 0.144 Wxx1 = 0.503 W 1 = 0.503 W X'X'2 = 0.209 W 2 = 0.209 W Xm Xm = =

13.25 W13.25 W(a) For a slip (a) For a slip s s = 0.03 determine the load current, = 0.03 determine the load current, II22, the , the developed torque, tdeveloped torque, tdd, and the developed power , and the developed power PPdd. . (b) Find the maximum torque and the corresponding speed. (b) Find the maximum torque and the corresponding speed. (c) Calculate the starting torque and the corresponding current (c) Calculate the starting torque and the corresponding current II22..Solution:Using the circuit in Figure 3.14,

(b) At the maximum torque point, the slip is calculated,-Note how the resistance part dominates this impedance.

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1 CHAPTER 3 EMT 113: V-2008 School of Computer and Communication Engineering, UniMAP Prepared By: Prepared By: Amir Razif b. Jamil Abdullah Basic AC Motor