project report (2) 3rd sem n.docx

8/14/2019 Project report (2) 3rd sem n.docx

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7. Poly phase synchronous machine:The synchronous machines are usually built

with field winding on the rotor and armaturewinding on the stator. However in order to

represent the two axis model of the synchronous

machine, the armature is placed on rotor and

field is mount on stator.

7.1. Basic synchronous machineparameter:

A three phase synchronous machine has four

basic s winding; namely three identical and

symmetrical armature windings and one fieldwinding.


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Fig.7.1. Basic two pole

synchronous machine

Fig.(7.1) shows the elementary two pole

synchronous machine. With salient pole

construction. The field winding axis is taken asdirect axis and the inter polar axis as quadrature

axis. The d- and q- axes remain fixed with the

field winding on stator , but three magnetic axes

of three phase armature phases rotate as the

rotor revolves. In order to concentrate more onthe synchronous machine basic features and to

further analysis, the following assumption are

made.


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1.Influences of damper are neglected.

2.Hysteresis and magnetic saturation are

neglected.3.Space distribution of armature m.m.f wave

and field flux wave are assumed sinusoidal.

4.The armature slots dont have any effect on

synchronous machine inductances.

7.1.1. Synchronous machine resistances:The armature circuit resistances for A,B,C

phases are designated by ra,rb, rc respectively.

Since the phase windings are identical, the

magnitudes of all the three resistances are equal,

i.e. . For convenience is used toidentify the winding resistance of phase A aswell as armature circuit resistance for any of the

three phases. The symbol

is used to represent

the field-circuit resistance.

7.1.2. Synchronous machine inductances:


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a) Field self-inductances. The air gaplength seen by the field m.m.f remains

constant, whatever the rotor position w.r.tfield poles may be. As a result of it, the

reluctance seen by the field m.m.f is always

constant and consequently field self-

inductances

is constant.

b) Armature to field mutual inductances.The mutual inductances between armatureand field windingsvary periodically with

space angle. Here the mutual inductances

between phase A and field winding

is

cosine in nature as shown in fig.(7.2) below.Therefore


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c) Armature self-inductances.

is a constant term, whereas is the

amplitude of second-harmonic component.

There is some flux, produced by phase A, which

does not cross the air gap to link the stator. This

flux is called leakage flux and can be accounted

for phase leakage inductance

The variation

of self inductance

with space angle

is

depicted in figure.


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Fig.7.2. Variation of self

inductance of phase A with space angle

d) Armature mutual inductance

(7.3)For round rotor machine, Where are the mutual inductancesbetween phases A, B; are the mutual


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inductances between phases B, C and

are the mutual inductances between phases

C, A. can be found by determining the fluxlinkages with phase B when only phase A isexcited or vice versa. The variation of orwith space angle is shown in figure 7.3

Fig.7.3. Variation of armature mutual

inductances with space angle7.2 Three-phase Synchronous Machine (with

no Amortisseurs)

The synchronous machine voltage equations ind-q can be obtained by a simple graphical

method developed in Art. () and same is done in


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this article. The primitive machine model or

generalized model of a polyphase machine

without amortisseurs is illustrated in figure (7.3).

Fig. 7.3 generalized model of a polyphase

synchronous machine

The stator coil DS represents the field winding

and the rotor coils DR, QR represent the

polyphase armature winding. In the analysis to

follow, the effect of ammortisseur circuits (or

damper bars) is neglected, so that basicimportant results are obtained more clearly. The


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voltage equations in matrix form for the model

of fig (7.3) are written as follows:

=

(7.4)

Now the field winding is designated by F in

place of more general DS. Similarly the

armature winding DR and QR are designated by

D and Q respectively.

Since the armature winding is uniformly

distributed and balanced,

=

=

(say).

With these constraints, the voltage Eq. ()

becomes,


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(7.5)The voltage-current relations given by Eq. (7.5)

are applicable both for steady state and transient

analysis of synchronous macine.

7.2.1 Balanced Steady-State Analysis

With d.c. field winding on the outer fixed

member, the stator field does not revolve and is,

therefore, stationary in space. Polyphase

currents in the armature winding produce

synchronously revolving field and for the

relative speed between the two fields to be zero,

the armature must revolve synchronous speed

opposite to the direction of rotating field. Thus

the speed of resultant air-gap flux is stationary

in space and in view of this, the operator p must


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be replaced by zero. On the steady state the

armature can run at synchronous speed only,

therefore, rotor angular velocity electrical radians per second.Thus the steady-state voltage equations, from

Eq. (7.5) are

(7.6)

The subscript zero indicates constant or d.c.

values. In place of etc. should have beenwritten, but it is not done here merely to use

them at a later stage.

Note that the term is associated with and with . These quantities andare known as


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, direct axis synchronous reactanceand

= quadrature axis synchronousreactance.In order to include these terms, Eq. (7.6) is re-

written as,

(7.7)7.2.3 Phasor equation and Phasor diagrams

In order to determine the performance character

tics of the machine in actual machine in actual

machine axes, i.e. a-b-c co-ordinates, the

transformation from d, q, o to abc variables asgiven by Eq. (7.4) can be used.

Considering phase a only,


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(7.8)Now ] And ] =

.

(7.9)

For phase a, let the voltage be given by Where

is the angle that

makes at

.

Also .


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; Where

(7.10)Where,

and

Similarly (7.11)Where,

and

7.2.4 Open-circuit conditions

On no load, , i.e. the actualarmature currents are zero. These zero values

when transformed to d-q axes currents are . In view of this, Eq. (7.7)becomes


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and

.

The no-load r.m.s. voltage of phase a, from Eq.

(), is (7.12)

Where is called the excitation e.m.f. or excitation

voltage. This voltage can also be expressed in

terms of maximum mutual inductances (between any armature phase and the fieldwinding).

(7.13)

7.2.5 Load conditions


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In this case, the voltage of phase a is obtained

by the substitution of

from Eq. (7.12)

for . (7.14)

[ ( )] (7.15)

In Eq. (7.15); and are only magnitudeand not in complex notation. In order to expressthem as a phasors, refer to fig. 7.4 , where d-axis

is taken as the reference or real-axis.


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Fig. 7.4 (a) Phase-a axis at an angle

from d-axis and

(b) Phasor component of Ia

The phasor notation for armature current , interms of phasors

and

can be written as

From Eq. (7.14); (7.16) and or

With these changes, Eq. (7.15) becomes


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(7.17)

is the terminal voltage of any one phase for asalient-pole type synchronous motor.For a generator, the voltages are generated and

the currents are output currents. In view of this,

the generator voltage equation can be obtainedfrom motor voltage equations by writing in place of . Therefore, the generator voltageequation is

(7.18)

Eq. (7.18) gives phasor diagram for a salient-

pole synchronous generator as shown in figure

7.5.


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Fig. 7.5 Phasor diagram for synchronous

generator

7.2.6 Internal Power Factor Angle

For calculating and , the internal powerfactor angle

for a generator must be

known. For this purpose draw ab normal to asshown in figure 7.6 ; where for simplicitywith is dropped. Since ab is 90 away from , it


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must be a reactance drop, say . Draw acperpendicular to ob as shown in figure 7.6.

Fig.7.6 Resolving of Iainto its d- and q-axes

components Id and Iq for synchronousgenerator

rom generator phasor diagram of figure 76

it is seen that oa and oc are right angled

triangles and, therefore,

.

Thus,


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Or or (7.19)This shows that in fig. (7.6), . If ob isdesignated as

and, therefore the angular

position of excitation voltage with respect to, i.e. the power angle is is known. From this,angle , therefore, and can becomputed. Fig (7.6) also reveals that

(7.20)Eq. (7.18) may now be written as

(7.21)


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Eq. (7.21) include a term which isdue to saliency, i.e. no-uniform air gap. This

quantity reduces to zero for a cylindrical-rotorsynchronous machine in which . Thissaliency factor in salient-pole synchronous

machine is quite appreciable, since isapproximately 60 to 70% larger then

. Even

in turbo alternators, small salient-pole effect is

present due to the effect of field-winding slots

on q-axis reluctance consequently differsslightly (about 10%) from

in cylindrical-

rotor synchronous machines.From the generator phasor diagram of Fig (7.6),

it is seen that

(7.22)

7.2.7 Cylindrical-rotor synchronous

machines


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These machines are characterized by uniform air

gap so that reluctances along d and q axes are

equal. In view of this, Whereis called the synchronous reactance.Thus the voltage equations for a cylindrical-

rotor synchronous generator, from Eq. (), is (7.23)Here

is called the synchronous

impedance. With the help of Eq. (7.23), the

voltage phasor diagram of a cylindrical-rotor

alternator is as shown in fig.7.7 (a). This Eq.

(7.23) also help in representing this type of

machine by the equivalent circuit of fig. 7.7 (b)


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Fig. 7.7 Cylindrical rotor alternator

(a) Phasor diagram and (b) Its equivalent

circuit

7.2.8 Steady-state Power-angle

Characteristics:

The expressions for power are derived from the

torque matrix of synchronous machine.

The torque matrix G is given by


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Now,

Total synchronous power,

Under steady state, Since the power is invariant in the two systems

of variables, power per phase is

( ) ( ) 724)


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If d.c. excitation is reduced to zero, andEq. (7.24) becomes

( ) 725)

This power given by Eq. 7.25 is called the

reluctance power and is the basis of operation of

a wide variety of reluctance motors.

Many times, it is convenient to include load

angle in the synchronous power expression of

Eq. 7.24. This can be easily done with the helpof phasor diagram


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Fig. 7.8 Phasor diagram

From the above phasor diagram,

and i.e.

726)

and 727)Substitute the values ofId andIq in Eq. (7.24)

728)For a round rotor machine, ,therefore, synchronous power from Eq. (7.24) is 729)


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SubstituteIqfrom Eq. (7.27) in Eq. (7.29) gives

................................................ (7.30)

With the help of Eq. 7.30 the variation of power

P with is plotted in ig1a to give the power-

load angle characteristics of a cylindrical rotor

synchronous machine. Eq. 7.28 gives the power

angle characteristics of a salient pole

synchronous machine, which is illustrated in

Fig. 7.9 (b)

Fig. 7.9 Power-angle characteristics for


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(a) Cylindrical rotor synchronous machine.

(b) Salient-pole machine.

Fig. 7.9(a) reveals that maximum power in a

cylindrical-rotor machine occurs when = 90

For a salient pole machine, the maximum power

as seen from Fig. 7.9(b) is at load angle less

than 90 This value of load angle can be

obtained from Eq. 7.28 as follows:

731)Its solution gives the value of and sustitution

of angle as calculated from Eq 731 in Eq

(7.28) gives the maximum power which is

sometimes called the pull-out power.

7.2.9 Reactive Power:


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The reactive power Q is determined from phasor

diagram of Fig. 1(a) by noting that reactive

power is equal to the product of voltage andquadrature lagging component of armature

current.

Substitution of Id and Iq , from Eq. (3) and (4)respectively, gives

............................................ (7.32)For a cylindrical-rotor machine,

, and reactive power for this machine, From

Eq. (7.32) is given by


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7.33)

Eq. (7.33) shows that

When

i.e. under normal

excitation,

and motor operates at

unity p.f.

When i.e. Motor is under-excited, Q is positive and, therefore, the

motor draws reactive power from supply

lines.When i.e. Motor is over-

excited, Q is negative and, therefore, the

motor delivers reactive power to the supply

lines.

For a salient-pole alternator, it can be shown

that


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.

/

And for a cylindricalrotor alternator

For both the generator and motor operations, it

can be stated in general that an over-excited

( machine delivers or exportsreactive power to the supply system; an under-

excited (

machine absorbs,

imports or draws reactive power from thesupply system.

7.3 Short Circuit Ratio (SCR):

SCR of a synchronous machine helps in

obtaining an estimate of its operating

characteristics.


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Fig. 7.10 Short Circuit Ratio

SCR is defined as the ratio of field current

required to generate rated voltage on open

circuit, to the field current required to circulate

rated armature current on three phase shortcircuit.

As triangles oab and ode are similar, therefore


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734)

From Eq. 7.34 735)It is thus seen that SCR is equal to the reciprocalof per unit value of direct axis synchronous

reactance Xd.SCR effects both the physical size

and operating characteristics of the synchronous

machine. A significance SCR can be gained

from the following considerations.

a) Low SCR: Low value of SCR meansgreater value of Xd. An examination of

Fig.[] with

, shows that low SCR

results in large voltage variations with loadand thus a poor voltage regulation.

Therefore, in order to maintain constant


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terminal voltage, the field current will have

to be varied over a wide range.

Eq. (7.28) shows that low value of SCRmeans less synchronous power P and

consequently a lower stability limit.

A low value of SCR means more Xd (or Xs)

and therefore a low value of synchronizing

power Ps( The synchronizing power is transient in

nature and come into play only when there is

a disturbance in the system. In other words,

the function of Ps is to keep the machine in

synchronism. Therefore, a machine with low

value of SCR has low synchronizing power

and therefore a low tendency for keeping the

machine in parallel with the infinite bus.A low value of SCR results in low value of

armature short-circuits current.


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b) High SCR: The synchronous machinewith high SCR has better voltage regulation

and improved steady-state stability limit.The disadvantage of high SCR is the high

values of armature short-circuit current.

A synchronous generator, feeding a long

transmission, or cable, line, may be

capacitive loaded in which ease of armaturecurrent leads the terminal voltage. The mmf

set up by a leading armature current , aids

the field mmf and consequently the

generated voltage increases. In order to

reduce the effect of armature mmf on thefield mmf and to retain the voltage under

control, air gap length should be increased,

which means a synchronous machine with

high SCR should be used.

Transient Analysis:

Transient behavior analysis is important because

Determine the shaft stress.


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Winding and bus bar stress. Protective relay setting. Circuit breaker rupturing duties etc.

For transient analysis two assumption, usually

made during transient analysis, are as follows.

The machine is running initially on noload and under steady-state condition.

Speed before and after the short circuitremains unchanged from its synchronous

speed Voltage equations for a synchronous generator

are ( )

( )736)For transient analysis above equation havelengthy manipulations. In order to avoid this


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some term in equation may be omitted to obtain

an easier solution of the transient problem.

A. All resistances neglected.With the neglect the resistances of both the

field ( ) and armature windings and with alternator speed

737)Before short circuit i.e. at

The armature currents 738)The field current before short circuit is


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Now from Eq. (7.37) and Eq. (7.38)

Before the short circuit After the short circuit

Thus the effect of short circuit is to

reduceto zero suddenly and this can betaken as equivalent to the sudden applicationof a unit step-function voltage tothe q-axis terminal.

Both before and after the short circuit,the voltage applied to the field winding is therefore superimposed voltage for fieldwinding is zero.


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Now from above discussion the voltage

equation after balanced

short circuit, in

terms of superimposed quantities, are 739

740

741From eq. (7.39)

742

Now put in eq. (7.40) then 743Where,

744 axis transient inductance.


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Now put in eq. (7.41) and then

745From eq. (7.43)

7.46)

Now put the value of from eq. (7.43) in eq.(7.45) and in complex frequency domain is

747After taking partial fraction eq. (7.46) becomes

Take Laplace inverse of above equation we get

a result


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748

Fig.7.11 Variation of Id (t) for a sudden 3-phase short circuit

Note:- transient d-axis current consist of A D.C. component

.

Fundamental frequencycomponent .

Put in Eq. (7.46)and we get 749


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Fig.7.12 Variation of Iq (t) for a sudden 3-

phase short circuit

Note:-transient q-axis current contain only

fundamental component of current.

The actual phase current when angle between d-

axis and axis of phase a is

after

short circuit, at any time t second is

Fig. 7.13 Phase-a axis displaced from d-axis

by an angle


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(a) at the instant of short circuit and(b)

at any time t

[ ]750From equation (7.48),(7.49) and (7.50) we get

751Where

=An angle made by phase-a with d-axis when

short circuit occurs.

Before the short circuit, i.e. at , and with this equation Eq.(7.51) becomes


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. /

752Similarly current forPhase-b, Phase-c,

From Eq. 7.52 we seen that phase current( ) consist of First term as fundamental frequency

component Second term as the D.C. component. Third term as the second harmonic

component.


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Note:-

The neglect of field and armatureresistance imply that none of the term in Eq.(7.52) decay with time. Magnitude of, fundamental, D.C. and

second harmonic component depends upon

the Pre-fault excitation

.

Magnitude of fundamental componentdepends upon the d-axis transient reactance.

Magnitude of D.C. component dependsupon

.

The magnitude of second harmoniccomponent depends on the term ,which is called the transient saliency. If short circuit occurs at then

D.C. component of current

is zero.

D.C. component of becomesfundamental frequency component of phasecurrent


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Fundamental frequency component of

and

get transformed to D.C. and

second harmonic component of Transient field current can be obtain bysubstituting from eq.(7.48) in eq.(7.42)

Total field current is

..(7.53)Now put in Eq. (7.53) thenwe finally

754


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Fig.7.14 Variation of If (t) for a sudden 3-

phase short circuit

B. In this, the term containing are the transformervoltages of armature voltage and neglect this

term from armature voltage equation alsoneglect the resistances of both the field( )and armature windings .So in this case after balance 3-short circuit Eq.

(7.39), (7.40) and (7.41) becomes

755 756


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757

From Eq. (7.55)

From Eq. (7.56)

From Eq. (7.57)

Field current,

Total field current


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758Current in phase-a is

[ ]

Put the value of and in above equation

759Note:- If transformer voltages are neglected

from the armature voltage equation, the

phase currents consist only fundamental

terms(D.C. and second harmonic terms are

absent).


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Current and purely in D.C. innature i.e. it is constant.

Dynamic analysis of synchronous machine is

usually carried out by neglecting the armature

transformer voltages.

C. Armature transformer voltages andits resistances neglected ( )Under this condition,

760

761

762From Eq. (7.60)

763From Eq. (7.62)


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0

1

Ratio field time constant.d-axis open circuit transient time constantThus

* + 764

is called the d-axis short circuit

transient time constant

From Eq. (7.64)

Now convert above equation in s-domain and

finally in time domain so


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765Now

( ) 766So total field current is

767The phase current is given by


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768Note:- The effect of including the field

resistance

is to make the amplitude of

short circuit current equal to after thetransient have decay and its R.M.S value is . The equation of envelope of the short

circuit current, from eq.(32) is . / The R.M.S value of short circuit current

as function of time is


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769Now from above equation

Just after the short circuit i.e. at thearmature phase current is

, which is

depends on transient reactance and itsfinal steady state current is , whichdepend on d-axis synchronous reactance.

The decay from

to

is an

exponential and is determine by time

constant.


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Fig. 7.15 Variation of rms short-circuitarmature current during transient

D. Armature transformer voltages andits resistances neglected ( ) butamortisseur circuits included.

The synchronous machines fitted with damperbars, or amortisseurs, are studied with one coil

KD in d-axis and another coil KQ in q-axis

Due to damper winding effect

770


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Where

=Sub transientcomponent of current due to damper winding.This component of current decay with a time

constant

,which is called the d-axis short

circuit sub-transient time constant. In case the effect of armature

transformer voltage consider

Then examine the Eq. (7.48) reveals that

,with damper bar present must include a

fundamental frequency component 771 In case armature resistance also

consideredThen fundamental frequency component of would decay with the armature timeconstant .Thus complete expression of current


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772Similarly

773Now

0

.


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/

.

/ 1774In above equation second harmonic component

is usually small. In modern synchronous

generator

therefore eq. (39) becomes

775Note:-

In case the fault occurs at

and at time


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776From eq. (7.76) we note that first term is A.C.

component of and second term is D.C.component of. . Maximum possible value of armature

phase current is after one half ofcycle i.e. and i.e. if axis ofphase-a along d-axis at the instant of fault

and attenuation does not take place. R.M.S. value of symmetrical armature

short-circuit current is given by


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777

Fig. 7.16 Variation of rms short-circuitarmature current during sub-transient

In the above graph at Just after the short circuit, current is

equal to .


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Current change exponentially to with a time constant

and to with time constant. The asymmetrical or D.C. component of

total short circuit armature current

The R.M.S or effective value of the total

short circuit armature current

778)

The effective value of greatest armaturecurrent including A.C. and D.C. component

is


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7.5 Transient Torque:

The object of this article is to calculate the

maximum value of torque at the instant of short

circuit. In order to do this the following

assumptions are made:

Field current remains constant at Maximum value of ; reachedafter sudden 3-phase short circuit dont

decay.

The torque in a cylindrical-rotor synchronousmachine is given by 7.79)Therefore the short-circuit torque is

( ) where


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( after shortcircuit)

And Therefore,

= .(7.79)

If the synchronous machine has

as its

rated phase values of voltage and currentrespectively, then rated torque is given by .(7.80)From Eqs. (7.79) and (7.80)

.(7.81)If With this,


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(7.82)In Eq. (5), ratio of short-circuit torque to rated

torque is maximum, when sint=1,

i.et=90o

Therefore

If ,then

It is seen from above that maximum value of

torque, after sudden 3-phase short circuit, is

several times greater than its rated torque.

7.6 Sudden reactive loading and unloading:

In this case, variation of currents and voltages is

investigated when sudden reactive loading or


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unloading of synchronous generators carried

out.

7.6.1 Sudden reactive loading:

It is assumed that the synchronous generator is

initially unloaded and is developing a nominal

terminal voltage at synchronous speed. The

nature of balanced reactive loading may be

inductive or capacitive.

(a)Sudden inductive loading:

For a sudden inductive loading, the load

reactance XL is taken to be lumped with the

alternator internal reactances and then sudden

short circuit is assumed to occur at its

terminals. Thus the r.m.s value of armature

phase current at any time t is


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=

.(7.83)

Where load time constant TL , is ..(7.84)The voltage across the load reactance XLat anytime t is given by Therefore, 0 1..(7.85)


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The variation of field current after sudden

inductive loading XL,is given by

0

1..(7.86)The variation of excitation voltage with time is

given by

=

0

1


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=

0 1(7.87)Starting of large induction motors is a fairly

good example of sudden inductive loading of

synchronous generators.

Examination of Eq. (8) shows that just after

sudden inductive loading , the terminal voltage

at once drops from Efoto OA=. This value

decreases finally to

with a load time

constant of , as depicted in fig. (7.17)


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Fig. 7.17 Variation of alternator terminal

voltage with sudden inductive loading

(b)Sudden capacitive loading:

The behavior of an alternator for sudden

capacitive loading can be obtained simply by

replacing XLwith (-XC).

From Eq. (7.83) the r.m.s value of armature

current is


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(7.88)Where

(7.89)

From Eq. (7.85) , the voltage across the load

reactance Xcis given by

.(7.90)From Eq.(7.86) the field current is given by

0 1..(7.91)


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From Eq. (7.87)

0 1..(7.92)Sudden switching of a long transmission line is

a common example of sudden capacitive

loading of synchronous generators.

Equation (7.90) reveals that just after the

sudden capacitive loading ,i.e at t=0+ the

voltage at once varies from Efoto .Thisvalue changes from

to its final value

with a load time constant of

.

7.6.2 Sudden reactive unloading:


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The synchronous generator ,running at

synchronous speed is initially connected to a

reactive load.Here the behavior of alternator isinvestigated first with sudden removal of

inductive load and then with the sudden

removal of capacitive load.

(a)Sudden inductive unloading:

In this case since the armature current ,after

the load has been disconnected, is zero,its

expression is not of much importance.For the

field current, the voltage expression is given by

( ) .(7.93)We know from generalized theory of dc

machine


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And

Here and , before the inductive load isremoved, are and . After the load isremoved and are and Zerorespectively. Due to this the superimposed field

voltage .Therefore the total field current =

..(7.94)After the load is disconnected, excitation

voltage and terminal voltage are equal.


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Therefore

0 1 .(7

.95)

If

is the load terminal voltage before the

load is disconnected,then Or

* +(7.96)

Substituting in Eq. (7.95) , the value of fromEq. (7.96), we get


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*

+ 0 1

.(7.97)

An examination of Eq. (7.97) shows that just

after the inductive load is disconnected,

terminal voltage at once rises to * +=OB. The increase from * + to its finalvalue

* + with a time constant

as shown in fig 7.18.


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Fig. 7.18 Variation of alternator terminal

voltage with sudden inductive unloading

(b)Sudden capacitive unloading:

The behavior of an alternator for sudden

capacitive unloading can again be obtained

simply by replacing XLwith (-XC) as before.

In view of this ,field current from Eq. (7.94) is


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(7.98)

from Eq. (7.96) is,

The terminal voltage from eq. (7.97), is * + 0

1

.(7.99)

7.7 Transient Analysis: a qualitative

approach


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In this article e, a simple qualitative treatment of

the transient behavior of the alternator, after a

sudden three-phase short circuit is presented.For simplicity, the synchronous machine

resistances are neglected. With this assumption,

the total flux linkages with any close circuit

cant change suddenly at the time of any

disturbance. In other words, the flux linkagesafter sudden disturbance in any close circuit

remain constant at their predisturbance value-

this fact has come to be known as constant flux

linkage theorem.


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Fig. 7.19 The Symmetrical waveforms for

armature short-circuit current for a 3-phase

synchronous generator

With zero armature resistance, the steady-state

armature short circuit current lags the excitation

voltage

by 90 as shown in fig(). This figure

shows that short circuit armature m.m.f.

is in

the d-axis of the machine, consequently d-axis

parameter are mainly involved during the short

circuit analysis. This conclusion is revealed by

Eq. (), which shows that the armature short-

circuit is affected by d-axis parameter

and.The Symmetrical waveforms for armature short-

circuit current for a 3-phase synchronous

generator is shown in fig. 7.19.

7.7.1 Reactances and Time-constants from

Equivalent circuits


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Expressions for the reactances

and time constants

can be derived fromthe transformer-type equivalent circuits of asynchronous machine.

7.7.2 Reactances from equivalent-circuits

d-axis synchronous reactance of a synchronousmachine is given by ; is the d-axismagnetizing reactance.

d-axis transient reactance of a synchronousmachine is given by


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Fig.7.20 d-axis equivalent circuit for a

synchronous machine without damper bars

d-axis sub-transient reactance of a synchronous

machine is given by

Fig 7.21 (a)d-axis equivalent circuit with

damper bars but field winding assumed

absent

(b) d-axis equivalent circuit for a

synchronous machine


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7.7.3 Time constants from equivalent-circuits

(a) Direct-axis open circuit transient time

constant,

(b) Direct-axis short circuit transient time

constant,


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Fig 7.22 Determination of time constant (c) Direct-axis open-circuit sub-transient time

constant, 0 1(d) Direct-axis open-circuit sub-transient

time constant,


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7.7.4 Transient power angle characteristics:

In this article, it is assumed that a sudden 3phase short circuit occur at terminals of

alternator which is initially working under

loaded conditions. After the disturbance,

according to constant flux linkage theorem in a

closed circuit, here the flux remain constant atits pre-disturbance value. Alternatively, it may

be assumed that the voltage regulator action

takes place so as to keep the field flux linkages

constant at their pre-disturbance value.


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Minus sign before is due to the fact thatshort-circuit armature m.m.f opposes the field

flux. The -axis armature flux linkage

From Eq.(7.101) Substitution of this value in Eq. (1) gives

. /


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Here

Is the -axis transient inductance.Multiplication of Eq. (7.102) by gives

or

[

]

or The generalized form of voltage equation of

alternator is ( )


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From the above equation of , if armatureresistance and transformer voltage are

neglected, then ( ) After putting the value of Eq.(70.104) willbecome

( ) Dividing every term of the above equation bywe get its rms value.

(

) Now this equation is suitable to draw the phasor

diagram of synchronous machine which is

shown in below.


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Fig.7.23 Synchronous generator phasor

diagram under transient condition


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Eq.(7.106) shows that when field flux linkage

are considered constant ,the voltage

remains constant during the transient state.

In order to analyse sudden disturbance or

sudden load application a power angle

expression is useful.

The power for synchronous machine is given by But And

( )


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.

/ The power angle expression of Eq.(7.107) isapplicable for a period of time short as

compared with the transient time constant .This period of time less than , is sufficientto cover the critical period in many transient and

dynamic analysis. If the field flux linkagesare maintained constant, then Eq.(7.107) isapplicable till the period of time ismaintained substantially constant by the voltage

regulator action.


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Fig. 7.24 The steady-state and transient

power angle characteristics of synchronous

machine

In Fig.(7.24) the steady-state and transient

power angle characteristics of a typical

synchronous machine, from which it is seen that

the general shape of the two curves is almost

similar. Since is greater than , thefundamental term of power has larger

magnitude during transient state. The physical

reason of the transient amplitude being greaterthan the steady state amplitude is that, just after

the short-circuit, there is a suddenly induced

component of field current which results in


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greater excitation of the machine. Transient

power angle curve of figure.(7.24) shows that a

synchronous machine in a practical system canwithstand a large suddenly applied power

overload, provided its duration is relatively

short as compared with . The transient powerangle expression of Eq.(7.107) is usually

simplified by neglecting the second harmonic

term. The Fig.(7.23) shows isalmost equal to in magnitude. Though

is less than

, yet it is almost

equal to the power given by Eq.(7.107). This is

due to the fact that numerical value of second

harmonic term in Eq.(7.107) is negative and the

total transient power is equal to the difference of

first term and the second harmonic term. Inview of this the transient the transient power is

given by


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is more accurate than the use of Eq.(7.107)withtransient saliency ignored ( ) .Thevoltage equation

leads to the

equivalent circuit of Fig. (7.25),

Fig.7.25. Approximate transient equivalentcircuit of a synchronous machine.

which is commonly employed when transients

and dynamics of a complex multi machine

system are to be investigated even with the help

of computers. The voltage is called thevoltage behind transient reactance.


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Note that the use of the equivalent circuit is

based on two assumptions as under:

(i) The voltage behind transient reactanceduring the transient state is assumed constant.

This means that the field flux constants are

maintained constant during the transient time

much less as compared to

.

(ii) The reactance and under transientconditions equal. This assumption onlypermits the neglect of transient saliency and

enable to model the synchronous machine

under transient condition by a single transient

reactance of Fig.(7.25)

7.7.5 Phasor diagram under transient and

sub transient condition:

Under transient conditions, the direct axis

transient reactance is and q-axis transient


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reactance is . Fig.(7.23) reveals thatunder transient conditions,

Here is the q-axis component of terminalvoltage .If is the d-axis component of,sothat

,then under transient

conditions, In magnitude, , therefore .It is seen from Fig.(7.23) that is almost equalto

, where

Here voltage is called the voltage behindtransient reactance.


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Fig 7.26 Synchronous generator (a) Phasor

diagram (b) Equivalent circuit diagram

under sub-transient condition

Under sub transient conditions, the d-axis and

q-axis sub transient reactances are andrespectively. The phasor sum of


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And

is shown in the Fig.(7.26.a). The resultant of and is equal to .It is seen from thefigure that is almost equal to ,where

The voltage

is called the voltage behind

subtransient reactance

.Eq.(7.109) results in

subtransient equivalent circuit of a synchronous

machine. With the help of Figs.(7.23) and

(7.26.a), one phasor diagram showing

is illustrated in Fig.(7.27).The voltage

, is

called vopltage behind d-axis synchronous

reactance,is given by .


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Note that and are almost are equal inmagnitude. This simplified phasor diagram of

Fig.(7.27) is often for doing the analysis of asynchronous machine connected to a power

system network.

Note that in fig.(7.27)

Fig. 7.27 Phasor diagram for the initial

current and voltages under overloaded

alternator


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for , , i.e. Sub-transientsaliency is ignored,

for , , i.e. transientsaliency is ignored,and for , , i.e. Steady-statesaliency is ignored.

If the effect of damper bar is included then the

rms value of symmetrical, or a.c. component of

short-circuit armature current is

.

/ .

/


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In in a power system network, asynchronous

machine is represented by Fig.(7.26.b) under

transient conditions.

7.8 Synchronous machine dynamics (electro-

mechanical transients):

Under normal running condition, the relative

velocity between stator and rotor fields is zero.Whenever this relative position is disturbed,

synchronizing restoring torques tending to

maintain this equality come into play. This

tendency of rotor to attain synchronous speed

after a slight disturbance, is the basic reason forthe analysis of synchronousmachine dynamics.

The phenomena involving rotor oscillations

about its final equilibrium position is called

hunting. The nature of synchronous machine

oscillations or hunting can be known only if its

electro-mechanical transients are investigated.


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The cause of disturbances in synchronous

machines may be any one of the following:

(i) A sudden change of load.(ii) A fault in the supply system.(iii) A sudden change in the field current,(iv) Load torque containing harmonic

torques in case of motors, or prime-mover

torque containing harmonic torques in caseof synchronous generators.

The effect of rotor oscillations on the

performance of the synchronous machines is

given below:(i) It increases the mechanical stresses and

fatigue of the shaft.

(ii) When working as a generator, the outputvoltage fluctuates which is an undesirable

phenomenon(iii) It increase the machine losses and,

therefore, increases its operating

temperature.


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(iv) The rotor oscillations may cause themachine to fall out of step.

7.8.1Electro-mechanical equation:

In a synchronous machine the general torque

balance equation is

Where = electromagnetic torque=mechanical torque= torque due to inertia of rotor

=damping torque

Neglect of the mechanical damping torquegives 7.8.2 Generator operation:


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In generator, electric power flows out of it.

Likewise electric torque

, corresponding to

electric power, flows out of the generator, should, therefore, be treated as negative. In viewof this, Eq.(7.111) becomes;

Here , is the mechanical torque input tothe generator. Also Where

power due to inertia of rotor =

In above equation it is given that

has two

component.


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One component is synchronous power, is due to

the presence of load angle

between

and

.

That is given by Where And Where damping constant inwatts per radian per second.

and= Power input to the shaft of alternatorTherefore the required electromechanical

equation is


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Subtraction of Eq.(7.113) from Eq.(7.114) gives

After some mathematical manipulation the

Eq.(7.115) can be reduced to

Though


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Where synchronizing power per electricalradian.

Therefore Eq.(7.116), can be re-written as,

Eq.(7.117) is a second order linear differentialequation and may be written as,

.

/ where .It is usually convenient to write Eq.(7.118) as


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A comparison of Eqs.(7.118) and (7.119)gives,

and

The solution of Eq.(7.119) consists of two parts,

complementary function(C.F) and the particular

integral(P.I).

Complementary function:C.F. Particular integral:


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P.I The complete solution of Eq.(7.118) is C.F+ P.I

The solution of Eq.(7.119) depends upon thevalue of damping factor or damping ratio which may be less than, equal to, or greater thanunity. For different value of , the solution ofEq.(7.119) is as under:

0

1


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For

,

For ,With initial conditions,

and For ,


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With initial conditions,

and

Therefore , - , -Where

natural frequency of oscillation

natural damped angularfrequency and damping ratio


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The plot of Eqs.(7.122),(7.123),(7.124) and

(7.125) are given for different values of

damping ratio and it is seen that the magnitudeof oscillations can be controlled by varying .

The value of (or ); as stated before, can beincreased by using large cross-sectional damperbars. This will decrease the magnitude of rotor

oscillations.

7.9 Cyclic variation of shaft torque (orForced Oscillations)


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An alternator driven by an internal combustion

engine or a synchronous motor driving a

reciprocating compressor is subjected tovariable shaft torques. The variable torque is

equivalent to an average torque plus a large

variety of harmonic torques. If any one of the

harmonic torque component has a frequency

approaching , the amplitude of oscillationsincreases and the machine may fall out of step.Such a state of affairs must be carefully

investigated. If the frequencydiffers by morethan 25% from the frequency of forced

oscillations, it is then assumed that stable

operation remains unaffected.

Let the harmonic-torque component variable

shaft torque be represented by

;

where is the maximum amplitude ofharmonic torque and its angular frequency is


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. Also, (synchronous speed of themachine)

( )

The differential equation of motion governing

the performance of synchronous machine under

forced oscillation is

d

or

d

or

d

where, as before,


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||

For a given value of , the correspondingchange in synchronous power is ||watts

project report (2) 3rd sem n.docx

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