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8/8/2019 Mquina Sncrona_texto
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xformer part1 pg 1
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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xformer part1 pg 2
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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xformer part1 pg 3
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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xformer part1 pg 4
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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xformer part1 pg 5
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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xformer part1 pg 6
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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xformer part1 pg 8
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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xformer part1 pg 9
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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xformer part1 pg 10
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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xformer part1 pg 11
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.