Electrical Machines 2 AC Machines

ELECTRICAL

MACHINES – II

(AC MACHINES)

Presented by

C.GOKUL

AP/EEE

Velalar College of Engg & Tech

,

Syllabus

BOOKS Reference

LOCAL AUTHORS: {For THEORY use this books}

1.Electrical Machines-II by “Gnanavadivel” – Anuradha Publication 2. Electrical Machines-II by “Godse” – Technical Publication

For Problems:

 Electric Machines by Nagrath & Kothari {Refer Solved Problems}

Important Website Reference



Electrical Machines-II by S. B.

Sivasubramaniyan -MSEC, Chennai



http://yourelectrichome.blogspot.in/



http://www.electricaleasy.com/p/electri

NPTEL Reference

• Electrical Machines II by Dr. Krishna

Vasudevan & Prof. G. Sridhara Rao

Department of Electrical Engineering , IIT

Madras.

• Basic Electrical Technology by Prof. L.

Umanand - IISc Bangalore {video}

BASICS OF

ELECTRICAL

Electrical Machine?

Electrical machine is a device which

can convert



Mechanical

energy into

electrical

energy (

Generators/alternators

)



Electrical

energy into

mechanical

energy (

Motors

)



AC current from one voltage level to

other voltage level without changing its

frequency (

Transformers

)

Fundamental Principle..



Electrical Machines (irrespective of

AC or DC) work on the fundamental

principle of Faraday’s law of

Faraday’s Law



Faraday’s Law of Electromagnetic

Induction states that an EMF is

induced in a coil when the magnetic

flux linking this coil changes with time

or



The EMF generated is proportional to

the rate at which flux is changed.

d

d

e

N

dt

dt

ψ

ϕ

Two forms of Induced EMF !



The effect is same if the magnet is

moved and the coil is made stationery



We call it as

statically

induced EMF



The previous case is referred to as

Governing Rules



It becomes evident that there exists a

relationship between mechanical energy,

electrical energy and magnetic field.



These three can be combined and precisely

put as governing rules each for generator

and for motor

Fleming’s Right hand rule

Fleming’s Left hand rule

Fleming's left hand rule (for motors)



First finger

-

direction of magnetic field (N-S)



Second finger

- direction of current

(positive to negative)

Maxwell’s Corkscrew rule



If the electric current is moving away from the

observer, the direction of lines of force of the

magnetic field surrounding the conductor is

clockwise and that if the electric current is

moving towards an observer, the direction of

lines of force is anti-clockwise

Corkscrew (Screw driver) rule

Coiling of Conductor



To augment the effect of flux, we coil the conductor

as the flux lines aid each other when they are in the

same direction and cancel each other when they are

in the opposite direction



Many a times, conductor is coiled around a magnetic

material as surrounding air weakens the flux



We refer the magnetic material

as armature core

Electromagnet



The magnetic property of current carrying

conductor can be exploited to make the

conductor act as a magnet – Electromagnet



This is useful because it is very difficult to

find permanent magnets with such high field



Also permanent magnets are prone to ageing

problems

Whenever current passes through

a conductor…



Opposition to flow of current



_{Opposition to sudden change in current}

Opposition to sudden change in voltage

Inductive Effect



Reactance EMF



_{Lenz Law}

An induced current is always in such a

direction as to oppose the motion or

change causing it

Capacitive effect

( )

1

( )

q t

( )

V t

i t dt

C

C

=

=

∫

( )

( )

( )

dq t

dv t

i t

C

dt

dt

⇒

=

=

Q

C

V

=

Inductive & Capacitive effects

Pure L & C networks – not at all

possible!

Pure L & C networks – not at all

possible! – contd.

Current & Flux



As already mentioned,

Star connection

3

ph

L

ph

V

V

I

I

=

=

Delta Connection

3

ph

L

ph

V

V

I

I

=

=

Maxwell's Right Hand Grip

Rule

Right Handed Cork Screw

Generators



The Generator converts

mechanical

power into

electrical

power.



Synchronous generators (Alternator) are

constant speed

generators.



The conversion of mechanical power into

electrical power is done through a coupling field

(magnetic field).

Magnetic

Mechanical Electrical

Electric Generator

G

Motor



The Motor converts

electrical

power into

mechanical

power.

M

Basic Construction

Parts

AC MACHINES



Two categories:

1.Synchronous Machines:



Synchronous Generators(Alternator)



Primary Source of Electrical Energy



Synchronous Motor

UNIT-1

Synchronous

Generator

Synchronous Generators

Generator

Exciter

View of a two-pole round rotor generator and exciter.

Synchronous Machines

• Synchronous generators or alternators are used to convert

mechanical power derived from steam, gas, or hydraulic-turbine to ac electric power

• Synchronous generators are the primary source of electrical energy we consume today

• Large ac power networks rely almost exclusively on synchronous generators

• Synchronous motors are built in large units compare to induction motors (Induction motors are cheaper for smaller ratings) and

Construction

 Basic parts of a synchronous generator:

• Rotor - dc excited winding

• Stator - 3-phase winding in which the ac emf is generated

 The manner in which the active parts of a synchronous machine are cooled determines its overall physical size and structure

Armature Windings (On Stator)

• Armature windings connected are

3-phase

and are

either star or delta connected

• It is the stationary part of the machine and is built up of

sheet-steel laminations having slots on its inner

periphery.

• The windings are 120 degrees apart and normally use

distributed windings

Field Windings (on Rotor)

• The field winding of a synchronous machine is always

energized with direct current

• Under steady state condition, the field or exciting

current is given

I

= V

/R

V

= Direct voltage applied to the field winding

Rotor

• Rotor is the rotating part of the machine

• Can be classified as: (a) Cylindrical Rotor and (b) Salient

Pole rotor

• Large salient-pole rotors are made of laminated poles

retaining the winding under the pole head.

Various Types of ROTOR

 Salient-pole Rotor

1. Most hydraulic turbines have to turn at low speeds (between 50 and 300 r/min)

2. A large number of poles are required on the rotor

Hydrogenerator Turbine Hydro (water) D ≈ 10 m Non-uniform air-gap N S S N d-axis q-axis

a. Salient-Pole Rotor

• Salient pole type rotor is used in low and medium speed

alternators

• This type of rotor consists of large number of projected

poles (called salient poles)

• Poles are also laminated to minimize the eddy current

losses.

•

This type of rotor are large in diameters and short in

axial length.

Salient-Pole Synchronous Generator

L ≈ 10 m D≈ 1 m Turbine Steam Stato r Uniform air-gap Stator winding Roto r Rotor winding N S  High speed  3600 r/min⇒ 2-pole  1800 r/min⇒ 4-pole

 Direct-conductor cooling (using hydrogen or water as coolant)

 Rating up to 2000 MVA

Turbogenerator

d-axis

q-axis

• Cylindrical type rotors are used in high

speed alternators (turbo alternators)

• This type of rotor consists of a smooth and

solid steel cylinder having slots along its

outer periphery.

Cylindrical-Rotor Synchronous Generator

Stator

Working of Alternator &

frequency of Induced EMF

Working Principle

• In the synchronous generator field system is rotating and armature winding is steady.

• Its works on principle opposite to the DC generator

Working Principle

• Armature Stator

• Field Rotor

• No commutator is

required {No need for

commutator because

we need AC only}

Every time a complete pair of poles crosses the conductor, the induced voltage goes through one complete cycle. Therefore, the generator frequency is given by

120

60

.

2

pn

n

p

f

=

=

Frequency of Induced EMF

N=Rotor speed in r.p.m P=number of rotor poles

f=frequency of induced EMF in Hz No of cycles/revolution = No of pairs of poles = P/2

No of revolutions/second = N/60

Advantages of stationary

armature

• At high voltages, it easier to insulate

stationary armature winding(30 kV or more)

• The high voltage output can be directly

taken out from the stationary armature.

• Rotor is Field winding. So low dc voltage

can be transferred safely

• Due to simple construction High speed of

Rotating DC field is possible.

Winding

Factors(

K

p

, K

d

)

cos

2

sin

2

sin

2

K

m

K

m

α

β

β

=













=













Pitch factor (

K

p

)



Consider 4 pole, 3 phase machine having 24

conductors



Pole pitch = 24 / 4 = 6 slots



_{If Coil Pitch or}

_{Coil Span = pole}

_{pitch, then it}

is referred to as

full-pitched

winding



If

Coil Pitch < pole pitch

, it is referred to as



Coil Span = 5 / 6 of pole pitch



_{If falls short by 1 / 6 of pole pitch}

or

This is done primarily to



_{Save copper of end connections}



Improve the wave-form of the generated emf

(sine wave)



Eliminate the high frequency harmonics

There is a disadvantage attached to it



_{Total voltage around the coil gets reduced}

because, the emf induced in the two sides of

the coil is slightly out of phase



Due to that, their resultant vectorial sum is less

than the arithmetic sum

Pitch factor – K

p

Vectorsum

K

Arithmaticsum

=

Pitch factor – contd.

Pitch factor – contd.

Pitch factor – contd.

_

_

2

cos

2

2

cos

2

Vector

sum

K

Arithmatic

sum

E

E

α

α

=

=

=

Distribution factor (K

d

)



As we know, each phase consists of

conductors distributed in number of slots to

form polar groups under each pole



The result is that the emf induced in the

conductors constituting the polar group are

not in phase rather differ by an angle equal

to angular displacement of the slots



For a 3 phase machine with 36 conductors, 4 pole,

no. of slots (conductors) / pole / phase is equal to 3



Each phase consists of 3 slots



Angular displacement between any two adjacent

slots = 180 / 9 = 20 degrees



If the 3 coils are bunched in 1 slot, emf induced is

equal to the arithmetic sum (3Es)



Practically, in distributed winding, vector sum has to

be calculated



Kd = Vector sum / Arithmetic sum

_

_

_

_

_

_

d

emf

with distributed winding

K

emf

with concentrated winding

0

0

180

180

.

_

_

_

no of

slots

per

pole

n

2

sin

2

2

sin

2

sin

2

sin

2

m

r

K

m

r

m

K

m

β

β

β

β













=

























=













Problem:

EMF Equation

of Alternator

Equation of Induced EMF



Average emf induced per conductor = dφ / dt

Here,

dφ = φP



If

P

is number of poles and flux / pole is

φ

dt = time for N revolution = 60 / N second

Therefore,

Average emf = dφ / dt = φP / (60 / N)

60

NP

ϕ

=

Equation of Induced EMF – contd.

We know,



_{N = 120 f / P}

Substituting, N we get



Avg. emf per conductor = 2 f φ Volt

If there are Z conductors / ph, then

Avg. emf induced / ph = 2 f φ Z Volt

Equation of Induced EMF – contd.



We know, RMS value / Avg. Value = 1.11



Therefore,



RMS value of emf induced / ph = 1.11 (4 f φ T) V

= 4.44 f φ T Volt



This is the actual value, but we have two other

factors coming in the picture, Kc and Kd



These two reduces the emf induced

Armature

Reaction of

Armature Reaction



Main Flux Field Winding



_{Secondary Flux Armature Winding}



Effect of Armature Flux on the Main Flux is

called

Armature Reaction

Armature Reaction in alternator

I.) When load p.f. is unity

II.) When load p.f. is zero lagging

III.) When load p.f. is zero leading

Armature Reaction in alternator

I.) When load p.f. is unity



distorted but not weakened.- the average flux in the

air-gap practically remains unaltered.

II.) When load p.f. is zero lagging



the flux in the air-gap is weakened- the field

excitation will have to be increased to compensate

III.) When load p.f. is zero leading

the effect of armature reaction is wholly

magnetizing- the field excitation will have to be

reduced

1. Unity Power Factor Load



Consider a purely resistive load connected to the

alternator, having unity power factor. As induced

e.m.f.

E

drives a current of

I

and load power

factor is unity,

E

and

I

are in phase with each

other.



If

Φ

is the main flux produced by the field

winding responsible for producing

E

then

E

lags

Φ

by 90

.



Now current through armature

I

, produces the

armature flux say

Φ

. So flux

Φ

and

I

are always in

the same direction.

• Phase difference of 90o _{between the armature flux and the main flux}

• the two fluxes oppose each other on the left half of each pole while assist each other on the right half of each pole.

• Average flux in the air gap remains constant but its distribution gets distorted.

2. Zero Lagging Power Factor Load



Consider a purely inductive load connected to the

alternator, having zero lagging power factor.



I

driven by E

lags E

by 90

_{which is the power}

factor angle Φ.



Induced e.m.f. E

lags main flux Φ

by 90

_while

Φ

is in the same direction as that of I

.



the armature flux and the main flux are exactly in

opposite direction to each other.

• As this effect causes reduction in the main flux, the terminal voltage drops. This drop in the terminal voltage is more than the drop

3. Zero Leading Power Factor Load



Consider a purely capacitive load connected to the

alternator having zero leading power factor.



This means that armature current I

driven by E

,

leads E

by 90

, which is the power factor angle Φ.



Induced e.m.f. E

lags Φ

by 90

_{while I}

aph

and

Φ

are always in the same direction.



the armature flux and the main field flux are in the

same direction

• As this effect adds the flux to the main flux, greater

e.m.f. gets induced in the armature. Hence there is

increase in the terminal voltage for leading power factor

loads.

Phasor Diagram

for Synchronous

Phasor Diagram of loaded

Alternator

E_f which denotes excitation voltage

V_t which denotes terminal voltage

I_a which denotes the armature current

θ which denotes the phase angle between V_t and I_a

ᴪ which denotes the angle between the E_f and I_a

δ which denotes the angle between the E_f and V_t

r_a which denotes the armature per phase resistance Two important points:

(1) If a machine is working as a synchronous generator then direction of Ia will be in phase to that of the E_f.

a. Alternator at Lagging PF



E

by first taking the component of the V

in the

direction of

I



Component of

V

in the direction of Ia is V

cos

θ ,

Total voltage drop is

(V

cos

θ+I

r

)

along the

I

.



we can calculate the voltage drop along the direction

perpendicular to

I

.



The total voltage drop perpendicular to

I

is

(V

sinθ+I

X

)

.



With the help of triangle BOD in the first phasor

diagram we can write the expression for

E

as

b. Alternator at Unity PF



E

by first taking the component of the V

in

the direction of I

.



θ = 0

hence we have

ᴪ=δ

.



_{With the help of triangle BOD in the second}

phasor diagram we can directly write the

expression for

E

as

c. Alternator at Leading PF



Component in the direction of

Ia

is

V

cosθ

.



As the direction of Ia is same to that of the

V

thus

the total voltage drop is

(V

cosθ+I

r

)

.



Similarly we can write expression for the voltage

drop along the direction perpendicular to

I

.



The total voltage drop comes out to be (

V

sinθ-I

X

).



With the help of triangle BOD in the first phasor

diagram we can write the expression for

E

as

Determination of the parameters of

the equivalent circuit from test data

 The equivalent circuit of a synchronous generator

that has been derived contains three quantities that

must be determined in order to completely

describe the behaviour of a real synchronous

generator:

The saturation characteristic: relationship between

I

and φ (and therefore between I

and E

)

The synchronous reactance, X

VOLTAGE

REGULATION

Voltage regulation of an alternator is

defined as the rise in terminal voltage of the

machine expressed as a fraction of

percentage of the initial voltage when

specified load at a particular power factor is

reduced to zero, the speed and excitation

Voltage

Regulation

A convenient way to compare the voltage

behaviour of two generators is by their

voltage regulation (VR). The VR of a

synchronous generator at a given load,

power factor, and at rated speed is defined

as

%

V

V

E

VR

fl

fl

nl

100

×

−

=

Voltage

Regulation

Case 1: Lagging power factor

:

A generator operating at a lagging power factor has a

positive voltage regulation.

Case 2: Unity power factor

:

A generator operating at a unity power factor has a small

positive voltage regulation.

Case 3: Leading power factor

:

A generator operating at a leading power factor has a

negative voltage regulation.

Voltage

Regulation

This value may be readily determined from

the phasor diagram for full load operation.

If the regulation is excessive, automatic

control of field current may be employed to

maintain a nearly constant terminal voltage

as load varies

Methods of

Determination of

voltage regulation

Methods of Determination of

voltage regulation

Synchronous Impedance Method / E.M.F.

Method

Ampere-turns method / M.M.F. method

ZPF(Zero Power Factor) Method / Potier

ASA Method

1. Synchronous Impedance

Method / E.M.F. Method

The method is also called E.M.F. method of determining

the regulation. The method requires following data to

calculate the regulation.

1. The armature resistance per phase (R

).

2. Open circuit characteristics which is the graph of open

circuit voltage against the field current. This is possible by

conducting open circuit test on the alternator.

3. Short circuit characteristics which is the graph of short

circuit current against field current. This is possible by

conducting short circuit test on the alternator.

The alternator is coupled to a prime mover capable

of driving the alternator at its synchronous speed.

The armature is connected to the terminals of a

switch. The other terminals of the switch are short

circuited through an ammeter. The voltmeter is

connected across the lines to measure the open

circuit voltage of the alternator.

 The field winding is connected to a suitable d.c.

supply with rheostat connected in series. The field

excitation i.e. field current can be varied with the

help of this rheostat. The circuit diagram is shown

in the Fig.

a.

Open Circuit Test

Procedure to conduct this test is as follows :

i) Start the prime mover and adjust the speed to the synchronous

speed of the alternator.

ii) Keeping rheostat in the field circuit maximum, switch on the d.c.

supply.

iii) The T.P.S.T switch in the armature circuit is kept open.

iv) With the help of rheostat, field current is varied from its

minimum value to the rated value. Due to this, flux increasing

the induced e.m.f.

Hence voltmeter reading, which is measuring line value of open

circuit voltage increases. For various values of field current,

Open-circuit test Characteristics

The generator is turned at the rated speed

The terminals are disconnected from all loads, and

the field current is set to zero.

Then the field current is gradually increased in

steps, and the terminal voltage is measured at each

step along the way.

It is thus possible to obtain an open-circuit

characteristic of a generator (E

or V

versus I

)

from this information

Open-Circuit

Characteristic

Short-circuit

test

Adjust the field current to zero and

short-circuit the terminals of the generator

through a set of ammeters.

Record the armature current I

as the field

current is increased.

Such a plot is called short-circuit

characteristic.

Short-circuit test

 After completing the open circuit test observation, the field

rheostat is brought to maximum position, reducing field

current to a minimum value.

 The T.P.S.T switch is closed. As ammeter has negligible

resistance, the armature gets short circuited. Then the field

excitation is gradually increased till full load current is

obtained through armature winding.

 This can be observed on the ammeter connected in the

armature circuit. The graph of short circuit armature

current against field current is plotted from the observation

table of short circuit test. This graph is called short circuit

characteristics, S.C.C.

Short-circuit

test

Adjust the field current to zero and short-circuit

the terminals of the generator through a set of

ammeters.

Record the armature current I

as the field current

is increased.

Connection for Short

Circuit Test

Open and short circuit

characteristic

Curve feature

The OCC will be nonlinear due to the

saturation of the magnetic core at higher

levels of field current. The SCC will be

linear since the magnetic core does not

saturate under short-circuit conditions

.

Determination of X

_s

 For a particular field current I_fA, the internal voltage E_f (=V_A) could be found from the occ and the short-circuit current flow I_sc,A could be found from the scc.

 Then the synchronous reactance X_scould be obtained using

I_fA E_f or V_t (V) Air-gap line OCC I_sc (A) SCC I_f (A) V_rated V_A I_sc,B I_{sc, A} I_fB

(

)

: R_a is known from the DC test.

X

_s

under saturated condition

(

)

Z

R

X

=

−

I_fA E_for V_t(V) Air-gap line OCC _I sc(A) SCC I_f(A) V_rated V_A I_sc,B Isc, A I_fB

Advantages and Limitations of

Synchronous Impedance Method

 The value of synchronous impedance Z

for any load

condition can be calculated. Hence regulation of the

alternator at any load condition and load power factor can

be determined. Actual load need not be connected to the

alternator and hence method can be used for very high

capacity alternators.

 The main limitation of this method is that the method

gives large values of synchronous reactance. This leads to

high values of percentage regulation than the actual results.

Hence this method is called pessimistic method

Equivalent circuit & phasor diagram under

condition

Short-circuit Ratio

 Another parameter used to describe synchronous generators is the

short-circuit ratio (SCR). The SCR of a generator defined as the ratio of the field current required for the rated voltage at open circuit to the

field current required for the rated armature current at short circuit. SCR is just the reciprocal of the per unit value of the saturated

synchronous reactance calculated by

[

]

Synchronous Generator Capability

Curves

 Synchronous generator capability curves are used to

determine the stability of the generator at various points of

operation. A particular capability curve generated in Lab

VIEW for an apparent power of 50,000W is shown in Fig.

The maximum prime-mover power is also reflected in it.

Capability

Curve

2. MMF method (Ampere turns method)

Tests: Conduct tests to find

 OCC (up to 125% of rated voltage)

 refer diagram EMF

3. ZPF method (Potier method)

Tests: Conduct tests to find

 OCC (up to 125% of rated voltage)

 refer diagram EMF

 SCC (for rated current)

 refer diagram EMF

 ZPF (for rated current and rated voltage)

 Armature Resistance (if required)

4. ASA method

Tests: Conduct tests to find

 OCC (up to 125% of rated voltage)

 refer diagram EMF

 SCC (for rated current)

 refer diagram EMF

 ZPF (for rated current and rated voltage)

Losses and

Efficiency

The losses in synchronous generator include:

1. Copper losses in

a) Armature

b) Field winding

c) The contacts between brushes

2. Core losses, Eddy current losses and

Hysteresis losses

Losses

3. Friction and windage losses,the brush

friction at the slip rings.

4. Stray load losses caused by eddy currents in

the armature conductors and by additional

core loss due to the distribution of magnetic

field under load conditions.

synchronous generator power flow

diagram

Synchronization

& Parallel

operation of

Alternator

Parallel operation of synchronous generators

There are several major advantages to operate generators in parallel:

• Several generators can supply a bigger load than one machine by itself.

• Having many generators increases the reliability of the power system.

• It allows one or more generators to be removed for shutdown or preventive maintenance.

Before connecting a generator in parallel with another

generator, it must be synchronized. A generator is said to be

synchronized when it meets all the following conditions:

•

The rms line voltages of the two generators must be

equal.

•

The two generators must have the same phase sequence.

•

The phase angles of the two a phases must be equal.

•

The oncoming generator frequency is equal to the

running system frequency.

Synchronization

Parallel operation of

synchronous generators

Most of synchronous generators are operating in parallel with other synchronous generators to supply power to the same power system. Obvious advantages of this arrangement are:

1. Several generators can supply a bigger load;

2. A failure of a single generator does not result in a total power loss to the load increasing reliability of the power system;

3. Individual generators may be removed from the power system for maintenance without shutting down the load;

4. A single generator not operating at near full load might be quite inefficient. While having several generators in parallel, it is possible to turn off some of them when operating the rest at near full-load condition.

Conditions required for

paralleling

A diagram shows that Generator 2

(oncoming generator) will be connected in parallel when the switch S₁ is closed. However, closing the switch at an

arbitrary moment can severely

damage both generators!

If voltages are not exactly the same in both lines (i.e. in a and a’, b and b’ etc.), a very large current will flow when the switch is closed. Therefore, to avoid this, voltages coming from both generators must be exactly the same. Therefore, the following conditions must be met:

1. The rms line voltages of the two generators must be equal. 2. The two generators must have the same phase sequence. 3. The phase angles of two a phases must be equal.

4. The frequency of the oncoming generator must be slightly higher than the frequency of the running system.

Conditions required for

paralleling

If the phase sequences are different, then even if one pair of voltages

(phases a) are in phase, the other two pairs will be 1200 _{out of phase creating}

huge currents in these phases.

If the frequencies of the generators are different, a large power transient may occur until the generators stabilize at a common frequency. The frequencies of two

machines must be very close to each other but not exactly equal. If frequencies differ by a small amount, the phase angles of the oncoming generator will change slowly with respect to the phase angles of the running system.

If the angles between the voltages can be observed, it is possible to close the switch S₁ when the machines are in phase.

General procedure for

paralleling generators

When connecting the generator G₂ to the running system, the following steps should be taken:

1. Adjust the field current of the oncoming generator to make its terminal voltage equal to the line voltage of the system (use a voltmeter).

2. Compare the phase sequences of the oncoming generator and the running system. This can be done by different ways:

1) Connect a small induction motor to the terminals of the oncoming generator and then to the terminals of the running system. If the motor rotates in the same direction, the phase sequence is the same;

2) Connect three light bulbs across the open terminals of the switch. As the phase changes between the two generators, light bulbs get brighter (large phase difference) or dimmer (small phase difference). If all three bulbs get bright and dark together, both generators have the same phase sequences.

General procedure for

paralleling generators

If phase sequences are different, two of the conductors on the oncoming generator must be reversed.

3. The frequency of the oncoming generator is adjusted to be slightly higher than the system’s frequency.

4. Turn on the switch connecting G₂ to the system when phase angles are equal. The simplest way to determine the moment when two generators are in phase is by observing the same three light bulbs. When all three lights go out, the voltage

across them is zero and, therefore, machines are in phase. A more accurate way is to use a synchroscope – a meter measuring the difference in phase angles between two a phases. However, a synchroscope does not check the phase sequence since it only measures the phase difference in one phase.

Synchronization

Concept of the infinite bus

When a synchronous generator is connected to a power system, the power system is often so large that nothing, the operator of the generator does, will have much of an effect on the power system. An example of this situation is the connection of a single generator to the power grid. Our power grid is so large that no reasonable action on the part of one generator can cause an observable change in overall grid frequency. This idea is idealized in the concept of an infinite bus. An infinite bus is a power system so large that its voltage and frequency do not vary regardless of how much real or reactive power is drawn from or supplied to it.

Steady-state

power-angle characteristics

Active and reactive power-angle characteristics

• P>0: generator operation • P<0: motor operation

• Positive Q: delivering inductive vars for a generator action or receiving inductive vars for a motor action

• Negaive Q: delivering capacitive vars for a generator action or receiving capacitive vars for a motor action

P_m

P_e, Q_e

V_t

Active and reactive power-angle characteristics

• The real and reactive power delivered by a synchronous generator or consumed by a synchronous motor can be

expressed in terms of the terminal voltage V_t, generated voltage

E_f, synchronous impedance Z_s, and the power angle or torque angle δ.

• Referring to Fig. 8, it is convenient to adopt a convention that makes positive real power P and positive reactive power Q delivered by an overexcited generator.

• The generator action corresponds to positive value of δ, while the motor action corresponds to negative value of δ.

Pm

Pe, Qe

The complex power output of the generator in volt-amperes per phase is given by

* a t _ I V jQ P S = + = where:

V_t = terminal voltage per phase

I_a* _{= complex conjugate of the armature current per phase}

Taking the terminal voltage as reference

0 j V V t _t _ + =

the excitation( at stator in case of motor) or the generated voltage,

(

)

E f _f

_

Active and reactive power-angle characteristics

Pm

Pe, Qe

Active and reactive power-angle characteristics

Pm

Pe, Qe

Vt

and the armature current,

(

)

where X_s is the synchronous reactance per phase.

(

)

Active and reactive power-angle characteristics

• The above two equations for active and reactive powers hold good for cylindrical-rotor synchronous machines for negligible resistance

• To obtain the total power for a three-phase generator, the above equations should be multiplied by 3 when the voltages are line-to-neutral

• If the line-to-line magnitudes are used for the voltages, however, these equations give the total three-phase power

Steady-state power-angle or torque-angle characteristic of a

cylindrical-rotor synchronous machine (with negligible

armature resistance).

Steady-state stability limit

Total three-phase power: = sinδ X E V P s f t 3

The above equation shows that the power produced by a synchronous generator depends on the angle δ between the V_t and

E_f. The maximum power that the generator can supply occurs when δ=90o_. s f t X E V P = 3

The maximum power indicated by this equation is called steady-state

stability limit of the generator. If we try to exceed this limit (such as by

admitting more steam to the turbine), the rotor will accelerate and lose synchronism with the infinite bus. In practice, this condition is never reached because the circuit breakers trip as soon as synchronism is lost. We have to resynchronize the generator before it can again pick up the load. Normally, real generators never even come close to the limit. Full-load torque angle of 15o _{to 20}o _{are more typical of real}

Pull-out torque

The maximum torque or pull-out torque per phase that a two-pole round-rotor synchronous motor can develop is

      π = ω = 60 2 s max m max max n P P T

where n_s is the synchronous speed of the motor in rpm

P

δ

P or Q

Q

BLONDELS

TWO REACTION

THEORY

BLONDELS TWO REACTION

THEORY

In case of cylindrical pole machines, the direct-axis

and the quadrature axis mmfs act on the same magnetic

circuits, hence they can be summed up as complexors.

However, in a salient-pole machine, the two mmfs do not

act on the same magnetic circuit.

The direct axis component F

operates over a

magnetic circuit identical with that of the field system,

while the q-axis component F

is applied across the

interpole space, producing a flux distribution different

from that of F

or the Field mmf.

The Blondel's two reaction theory hence

considers the results of the cross and

direct-reaction components separately and if saturation

is neglected, accounts for their different effects

by assigning to each an appropriate value for

armature-reaction "reactive" respectively X

and

X

.

Considering the leakage reactance, the combined reactance values becomes

X_ad = X + X _ad and X _sq = X _aq

X_sq < X_sd as a given current component of the q-axis gives rise to a smaller flux due to the higher reluctance of the magnetic path.

•

Let l

and I

be the q and d-axis components

of the current I in the armature reference to the

phasor diagram in Figure. We get the following

relationships

• I

= I cos (σ+θ) I

= I cosφ

• I

= I sin (σ+ φ) I

= I sinφ

I =

√(I

_{+ I}

q2

)= =

√(I

+ I

)

• where I

and I

are the active and reactive

components of current I.

Short Circuit Transients

for Synchronous

Short Circuit Phenomenon

Consider a two pole elementary single phase alternator with concentrated stator winding as shown in Fig. 4. Consider a two pole elementary single phase alternator with concentrated stator winding as shown in Fig. 4.

The corresponding waveforms for stator and rotor currents are shown in the Fig

Let short circuit occurs at position of rotor shown in Fig. 4(a) when there are no stator linkages. After 1/4 Rev as shown Fig. 4(b), it tends to establish full normal linkage in stator winding. The stator opposes this by a current in the shown direction as to force the flux in the leakage path. The rotor current must increase to maintain its flux constant. It reduces to normal at position (c) where stator current is again reduces to zero. The waveform of stator current and field current shown in the Fig. 5. changes totally if the position of rotor at the instant of short circuit is different. Thus the short circuit current is a function of relative position of stator and rotor.

Using the theorem of constant linkages a three phase short circuit can also be studied. After the instant of short circuit the flux linking with the stator will not change. A stationary image of main pole flux is produced in the stator. Thus a d.c. component of current is carried by each phase.

The magnitude of d.c. component of current is different for each phase as the instant on the voltage wave at which short circuit occurs is different for each phase. The rotor tries to maintain its own poles

The rotor current is normal each time when rotor poles occupy the position same as that during short circuit and the current in the stator will be zero if the machine is previously unloaded. After one half cycle from this position the stator and rotor poles are again coincident but the poles are opposite. To maintain the flux linkages constant, the current in rotor reaches to its peak value.

The stationary field produced by poles on the stator induces a normal frequency emf in the rotor. Thus the rotor current is fluctuating whose resultant a.c. component develops fundamental frequency flux which rotates and again produces in the stator winding double frequency or second harmonic currents. Thus the waveform of transient current consists of fundamental, a.c. and second harmonic components of currents.

Thus whenever short circuit occurs in three phase generator then the stator currents are distorted from pure sine wave and are similar to those obtained when an alternating voltage is suddenly applied to series R-L circuit.

Stator Currents during Short Circuit

• If a generator having negligible resistance, excited and running on no load is suddenly undergoing short circuit at its terminals, then the emf induced in the stator winding is used to circulate short circuit current through it. Initially the reactance to be taken into consideration is not the synchronous reactance of the machine. The effect of armature flux (reaction) is to reduce the main field flux.

• But the flux linking with stator and rotor can not change instantaneously because of the induction associated with the windings. Thus at the short circuit instant, the armature reaction is ineffective. It will not reduce the main flux. Thus the synchronous reactance will not come into picture at the moment of short circuit. The only limiting factor for short circuit current at this instant is the leakage reactance.

After some time from the instant of short circuit, the

armature reaction slowly shows its effect and the alternator then

reaches to steady state. Thus the short circuit current reaches to

high value for some time and then settles to steady value.

It can be seen that during the initial instant of short circuit

is dependent on induced emf and leakage reactance which is

similar to the case which we have considered previously of

voltage source suddenly applied to series R-L circuit. The

instant in the cycle at which short occurs also affects the short

circuit current. Near zero e.m.f. (or voltage) it has doubling

effect. The expressions that we have derived are applicable only

during initial conditions of short circuit as the induced emf also

reduces after some tome because of increased armature

reaction.

The short circuit currents in the three phases during short

circuit are as shown in the Fig(next slide)

Capability Curves of

Synchronous