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Synchronous Machines

PART 3

(reference: C.Hubert, pg 305-387)

Figure 1 shows a magnet mounted on a rotor, and a coil on a stator. When the shaft rotates the rotor

flux will induce a voltage in the coil. The frequency of the voltage produced is: )..( sprnf ss = or

)..(60

mprN

f ss = . If one distributes the coil on the winding as shown in

figure 2, one can obtain a quasi sinusoidal induced voltage. Instead of a

stacked coil one can use slotted coils which are distributed in a better way

under a magnetic pole (figure 2). These windings are connecte in series in such

a way that the terminal voltage is near sinusoidal. Winding layouts is a special

topic and quite complicated. In this introductory course let us simply use an

effective number of turns per phase ( eN ). This produces an induced voltage

which can be approximated as:

peo fNE = 44.4

with poleperfluxp =

andfof course is the rated

synchronous frequency.

Finally figure 3 shows that one can

place 3 windings with a PHYSICAL or

GEOMETRIC placement 120o w.r.t. each other. When the magnet (NS) rotates, the voltage induced in

each coil will have the same frequency, but out of phase (time delay) by 120o

One could write the equations of the 3 phases as:

N

S

e1

figure 1

N

S

coils in

series

figure 2

N

S

e1

e2

e3 figure 3

( )

+=

=

=

3

2sin)(

3

2sin)(

sin)(

3

2

1

tEte

tEte

tEte

so

so

so

or draw a FRENEL vector diagram which is a vector

representation of each voltage induced on a plane which

rotates in reverse direct with the angular frequency s .Note the convention of (+) annotation

e1

e3 e2

Ns

Look at phasesgoing by

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SIMPLE EQUIVALENT CIRCUIT

The simplest equivalent circuit is

derived here. The rotor produces a fluxf

and when the shaft rotes at synchronous

speed onr can draw the equivalent circuit on

a PER PHASE BASIS (assumes that the 3phases are balanced). Ra is called the

effective resistance of the armature

winding. This is about 1.6 times the DC resistance because it takes into account the AC resistance due to

skin effect caused by the AC current at 60Hz. Xa is the leakage reactance of the armature winding,

caused by the flux linking winding. This flux does not link with the field winding, hence does not produce any

voltage. (This is a combination of several effects: end connection leakage reactance; slot leakage reactance;

tooth top and zig zag (or differentiel) leakage reactance; belt leakage reactance).

The equation of the synchronous generator, with the output voltageaV taken as the origin of the

phasors: aaaaaf XjIRIVE ++=

But contrary to the DC machine, here we have VECTORS, hence the output voltage Va will depend

upon the load power factor. The figures below show, for a fixed output voltage, the phasors depending upon

the power factor. Note that for a needed output voltage, the internal emf varies a lot, and contrary to the

DC machine, the output voltage can be higher or equal to the internal emf produced.

ARMATURE REACTION

The flux produced by the armature winding reacts with the flux set up by the poles on the rotor. The

total flux will therefore be reduced. This is called the armature reaction. With refeence to the figure next

page, let us examine a sequence of events when the generator deliveres a load at unity power factor:

a) If p is the flux under a pole at no load, the generator voltage aE must lag p by 90o

b) Since the p.f. is unity, the phase current Ia is in phase with the terminal voltage Va.

c) As phase current Ia passes through the armature winding, its magnetomotive force (mmf) produces a flux

ar which is in phase with Ia. The effective flux e per pole in the generator is therefore

arpe +=

Rf

Xf

fVf

exciter

Ef

Ra jX a

ZaVa

Ia

VaIa

p.f. lag

RaIa

jX aIaEf

Va

p.f. unity

jX aIa

RaIa

Ef

Ia Va

p.f. lead

IaEf

RaIa

jX aIa

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d) The flux ar , in turn induces an emf arE in the armature winding. It is called the armature reaction

emf. It lags the flux ar by 90o . hence the effective voltage per phase eE is: arae EEE +=e) The equivalent circuit can be shown and the equation derived as:

( )aaaae jXRIVE ++=

NOTE: both magnetizing and leakage

reatances are present at the same

time, but it is rather difficult to

separate one from the other. It is

simpler to combine them

amsXXX

+=and call it the

SYNCHRONOUS REACTANCE

We can also define the SYNCHRONOUS IMPEDANCEsas jXRZ +=

Note: Ra

can be measured with DC measurement techinques, and has to be corrected to AC

values which is approximately a factor of 1.5. However, thi sAC value is still much smaller than the

value of the synchronous reactance of the machine.

ar

e

E

EE

I RVIa

a a

jI Xa a

are

a

Rf

Xf

Vf

exciter Ra jXa

ZaVa

Ia

Ea

Ear

-

++

-

Ee

armature

aI

aV

aaIR

aaIXj

arEeE

aE

ar

ea

PF= 0.86 lag

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aV

aaIXj

arE

eE

aE

aI

ar

ea

aaIR

PF= 0.96 lead

Synchronous Reactance Determination:

One usually plots the open circuit characteristic of the generator, and the short circuit characteristic of

the generator.

The unsaturated value

can be calculated from the air

gap line on the figure as:

cd

adXs =

However, a realistic

value shows some saturation

of the open circuit curve.

Hence one takes a corrected

value for synchronous

reactance as:

SC

OCs

I

V

cd

bdX ==

Voltage Regulation:

This is defined for full load: 100% =a

aa

VVEVR

Power Relationships:

The prime mover (turbine, other motor etc..) must supply a mechanical power on the shaft

sshaftinMP =

open circuit

characteristic

short circuit

characteristic

(Ef)

(Ia)

air gap

line

a

b

b

bIf

Ef

Ia

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However, we also have to add to this mechanical input the power needed to create the excitation i the

machine:

ffsshaftin IVP +=

The losses in the machine are rotational losses, magnetic losses, copper losses and stray losses

The DEVELOPED POWER is obtained by subtracting the rotational losses, the field winding lossesand the stray load losses from the imput power.

Furthermore, by subtracting the copper losses in the armature, we obtain the OUTPUT POWER.

The power output of a synchronous generator is: cos3 aao IVP = (Va and Ia are per phase)

Approximate Power Relation in a Cylindrical Rototr Generator:

If we can neglect the resistance in a synchronous generator, the approximate circuit diagram is shown

below:

From the circuit, we can establish:

s

aaa

jX

VEI

= projecting the vector on the (Va) and (jVa) axis gives

s

aa

s

aa

X

VEj

X

EI

=

cossin

but also projecting directly Ia on the Va axis, one getss

aa

X

EI

sincos =

Hence the approximate power output is given by: s

aa

aaout X

EVIVP

sin3cos3

==

When current and voltage is kept constant, the power generated depends upon sin.This angle is called the POWER ANGLE.

It follows that the TORQUE DEVELOPED isss

aad

X

EV

sin3=

Ia

jXs

Ea Za Va

Ea

Va

jIaXs

Ia

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Effects of Loading:

Taking the approximate equivalent circuit (Ra=0), we can see the effect of loading the generator. Since

E=cst, as the power increases with load

current, the terminal voltage decreases.

This is for a unity power factor.

The same occurs with lagging power

factor, but it can be seen that Va

decreases much faster

Conversely, with a leading power

factor, the output voltage will increase

with the power angle!

jIX

Va

E

E

jIX

Va

jIX

E

E

Ia

Va

jIX

Va

Ia

E

Ia

Va

Ia

Va

jIXE

jIX

Va

rated

Pf1 leading

Pf2 leading

Pf3 =1

Pf4 laggingg

Pf5 laggingg

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From the equivalent circuit:

(aaqdaa IREEEV ++= where ( qd EE + is the armature reaction vector

we can state:

ddd XjIE = and qqq XjIE =

and qda III +=If the armature resistance is negligible w.r.t. the reactances, we can simplify to:

qqddaa IjXIjXEV =

The power output is:

cos3 aaIVP =

( )cosdI is the projection of Ia on the Va axis. Since Ia=Id+Iq, let us project Id and Iq on the Vaa

axis also:

( )cosdI = ( ) ( ) ( ) ( ) cossincos90cos qdqd IIII +=+

hence the power cossin3 qda IIVP +=

replace by:q

aq

X

VI

sin= and

d

aad

X

VEI

cos=

The power becomes:

2sin11

2

3sin

32

+=

dq

a

d

aa

XX

V

X

EVP

The 1st element is the same as the power in the cylindrical machine with the synchronous reactancebeing the DIRECT component, and the 2nd term is due to the RELUCTANCE TORQUE ot the machine.

This depends upon the factor

dq XX

11

called the saliency of the machine. Note that

in well constructed machines Xd is

approximately twice he value of Xq.

The adjacent figure shows the torques

produced by the cyclindrical machine ascompared to the salient pole machine.

Salient-Pole Rotor

TorqueTotal Torque

-180o

180o

Cylindrical-Rotor

Torque

o0

max

Td

MotorGenerator

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Parallel Operation of Synchronous Generators

Assume that Generator A is feeding the full load. If we want to connect a second generator B in

parallel (to share the load), there are a series of steps to be taken:

1) Generator B must have the same phase sequence as Generator A ! (use a phase sequencer)

2) The voltage of the incoming generator must be matched to the bus voltage (adjust to have the samereading)

3) The frequency of the incoming generator must be the same as the bus frequency. (use a

synchroscope).

Consult Lab3 to find out the procedure to synchronize a generator on the bus.

When the primemover of a generator is set to

deliver a certain power on the shaft, and the voltage is

set to deliver that power to an electrical load, a certain

operating point is reached [speed, Voltage, Power]. If

the load increases, the generator speed (governor) willdecrease (not enough power to move the shaft). Hence

we can see the typical primemover/governor

characteristic. The characteristic starts at the no load

speed, and droops. The droop rate is a parameter of

the generator:rated

loadfullloadno

P

ff

P

fGD

=

= and

defines the slope of the governor characteristic.

If 2 generator characteristics are shown, and they are connected in parallel on the same bus, they must

have the same frequency of operation, hence the operating point. In the figure we can see that Generator Adelivers twice the power of generator B.

In order to change the

power in a generator for a given

frequency of opration, one has

to change the primemover

(change the value of the no-load

frequency). Changing the

governor will cause the

characteristic to move with the

same slope.

NOTE: if the governor and

exciter are unchaged, any

change of speed of one

generator will cause a circulating

current between the 2 machines in such a way as to oppose the change, hene it is called a synchronizing

torque. These torques can be enormous and will always make sure that the mchaines are in synchronism

(same frequency).

60Hz

3600

62

500 Power(kW)

Nominal

F(Hz)

Speed(rpm) No load speed (frq)

PBPA

frq

Fixed Frequency

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Synchronous Motors

The power now flows into the machine, hence the equation of the cyclindrical rotor motor becomes

asaaa IjXREV )( ++=

The power output depends upon the mechanical load on the shaft. Since the speed depends upon thefrequency (fixed) when the load is constant, varying the field i

fcannot change the output power. However,

the vector E will be affected, and the vector diagram will change; currents and power factors will change.

The diagrams below illustrate this:

Note: as the magnitude of E varies, the pf will

vary.

The synchronous motor characteristics (Armature Current/ Excitation) is called the V curves

Ia

Ea(a) pf=1

Va

RaIa

jXsIaRaIa

(a) pf lead

Ia

Va

jX sIa

Ea

Va

(a) pf lag

IajXsIa

Ea

RaIa

2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

pf=1

halfL

oad

NoLo

ad

FullL

oad

pf=0.8

leadpf=0.8 lag

Stability

limit

If

Ia

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When one analyses the Vcurves one can see that for a given Power delivered, the excitation will

control the power factor. Hence the synchronous motor can be set to operate at any desired power factor.

Usually one sets it to be at unity power factor since it is the one giving the less current magnitude, hence less

Joules losses.

A special application would be a synchronous motor running at NO LOAD !! By varyng the excitation

one can control a leading/laging power factor, hence this becomes either a Capacitor, a Small Resistor,

or a Reactor. This can be controlled continously with the excitation current. It is called aSYNCHRONOUS COMPENSATOR since it can compensate for REACTIVE POWER.