25471_ENERGY_CONVERSION_11456789785.pptx

ENERGY CONVERSION ONE
(Course 25741)
CHAPTER FIVE
SYNCHRONOUS GENERATORS
….continued

Phasor Diagram of Equivalent Circuit
• Voltages in Syn. Gen. are ac & expressed in
phasor which have both magnitude and angle,
therefore need to be presented in 2
dimensional plot.
• KVL relation of one phase can be plotted in
terms of phasor voltages through this 2
dimensional plot named phasor diagram
• which show the relationship between different
voltages and currents

• Figure below show the phasor diagram when
supplying a load at unity power factor
• EA differs from Vφ by resistive & inductive
voltage drops, all referenced to Vφ which
arbitrarily assumed at an angle of 0◦

• Phasor diagram when generators operating at
lagging & leading power factors

• Note: at specific phase voltage & armature current, a
larger internal voltage EA required for lagging loads
than for leading loads
• Therefore a larger field current required with lagging
loads to get same terminal voltage
EA=Kφω
• Alternatively for a given IF and load current , Vφ is
lower for lagging loads & higher for leading loads
• In real synchronous machines, XS normally much
larger than RA & often neglected in qualitative study

Power and Torque
• A synchronous generator is a synchronous
machine used as generator
• converts mechanical energy to 3 phase
electrical energy
• Source of mechanical energy is prime mover (a
diesel engine , a steam turbine, …)
• Any source employed should have an almost
constant speed regardless of load
• If it were not constant, the power system
frequency would wander

Power and Torque
• Not all mechanical power going to a
synchronous generator becomes electrical
power the difference between input & output
power represent losses, Power flow Diagram

Power and Torque
• input mechanical power is shaft power in
generator
• Pin=Тprime-mover ωm
(Тprime-mover ≡ Tapp in text book)
• Pconv=Tgenerated ωm = 3EA IA cos γ
(γ angle between EA and IA)
(Tgenerated ≡Tind in text book)
• Difference between Pin and Pconv in generator
represents mechanical, core, and stray losses
of machine

Power and Torque
• Real elect. output power of syn. Gen. in line quantities:
Pout=√3 VT IL cosθ
in phase quantities:
Pout=3 Vφ IA cosθ
reactive power output:
Qout=√3 VT IL sinθ
Qout=√3 Vφ IA sinθ
ignoring armature resistance RA (XS>>RA), a useful
relation can be derive to approximate output power of
Gen.

Power and Torque
• Generator Equivalent circuit for Δ connection

Power and Torque
• To derive that useful equation, when stator
resistance ignored, phasor diagram employed

Power and Torque
• The vertical segment bc is EA sinδ or XS IA cosθ
 IA cosθ = EA sinδ / XS
• Substituting this in equation of Pout 
P = 3Vφ EA sinδ / XS
• since resistances assumed zero, losses not included
in this equation (& it is both Pconv ,Pout)
• Above equation shows power produced by a Syn.
Gen. depends on angle δ (between Vφ,EA), the torque
angle
• Maximum power that Gen. can supply occurs when
δ=90◦. At this angle sinδ=1 
Pmax=3Vφ EA / XS (1)

Power and Torque
• Maximum power in last equation called “static
stability limit” of Gen.
• Real or practical Gen. never get close to this
limit
• Full load torque angles of 15 to 20◦ are typical
angle of real machines
• If Vφ assumed constant, real power output
directly proportional to IA cosθ and EA sinδ
• These are useful for plotting phasor diagram of
Syn. Gen. as load changes

Power and Torque
• From chapter 4, torque developed in a Gen.
can be expressed as:
• Тind= k BR X BS
• Tind=k BR X Bnet
• Magnitude of torque in this equation is:
• Тind=kBRBnet Sinδ
• δ: angle between rotor and net magnetic fields
• Since BR produces voltage EA, and Bnet
produces voltage Vφ, angle δ between EA and Vφ is
same as angle between BR and Bnet

Power and Torque
• Alternative expression for this torque in syn.
Gen. is derived (employing Pconv= Tind ωm) and
Equation (1):
Tind=3Vφ EA sinδ/(ωm XS) (2)
• This equation gives torque in terms of circuit
parameters, while the equation in last chapter
expressed it in terms of magnetic quantities

MEASUREMENT of MODEL PARAMETERS
• Equivalent circuit of synchronous generator contains 3
quantities that must be determined to describe
behavior of a real Syn. Gen.
1- relation between field current & flux (IF & EA)
2- Synchronous reactance
3- Armature resistance
• To determine these parameters, the first step is to do
open-circuit test
1- Generator is turned at rated speed, and terminals
disconnected from loads, field current set to zero
2- Field current gradually increased in steps, and
terminal voltage measured at each step along the way

• Leave terminals open IA=0,
Vφ= EA and thus construct
a plot of EA or VT versus IF
• This plot is open circuit
characteristic (OCC) of Gen.
• With this characteristic,
internal generated voltage of
Gen. for any given IF can be
determined
• typical characteristic shown


• Note : in this characteristic, as If increases first curve
is almost perfectly linear until some saturation
observed at high field current
• Unsaturated iron in frame of synchronous machine
has a reluctance several thousand times lower than
air-gap reluctance
 at first all mmf is across air gap and resulting flux
increase is linear
• When iron saturate reluctance of iron increase
profoundly and flux increases more slowly with a
similar increase in mmf
• Linear section of OCC called air-gap line

• Second step is to perform a
short circuit test
1- adjust field current to zero
2- short circuit terminals of
generator through a set of
ammeter
3- armature current IA or line
current IL measured as IF
increased
4- such a plot named short
circuit characteristic (SCC)
• It is a straight line

• It is a straight line, since employing the
equivalent circuit developed in this section:
IA= EA/(RA+jXS)
• Phasor diagram shown:
• Flux density vectors shown:

• Bnet is very small since almost BR and BS cancels
• The machine is unsaturated and SCC is linear
• The magnitude of current:
IA= EA / √[RA ^2+jXS ^2]
• internal Impedance of machine
ZS=√RA^2+XS^2=EA/IA
• Since XS>>RA above relation reduces:
XS≈ EA/IA =Vφ,OC/ IA (3)
• if EA and IA known for given situation, synchronous
reactance XS can be found

• Approximate method for determining
synchronous reactance XS :
1- measure internal voltage EA open circuit test
at the specific field current (OCC)
2- measure short circuit current flow IA,SC at the
same field current (SCC)
3- Find XS using the approximate equation (3)

• Concern:
• the EA is determined from OCC where machine is
partially saturated for large field currents
• While IA is determined form SCC , where machine is
unsaturated at all field currents
• Therefore at high field currents (that core is saturated
in OC) EA measured from OCC at a specific If ≠ EA
measured at same If under short-circuit conditions
• This makes resulting value of XS only approximate

• However, answer found by
this method is accurate up
to the saturation point
• So, unsaturated
synchronous reactance XS,u
can be found at any If in
linear portion of OCC
• Approximate Xs varies with
degree of saturation of
OCC, so it should be
calculated at approximate
load on machine
• Plot of approximate XS as
function of If shown 

• It is important to know winding resistance as well as
synchronous reactance
• Resistance can be approximated by applying dc
voltage to winding while machine is stationary &
measuring the resultant current flow
• use of dc voltage means reactance of windings will be
zero in measurement
• However, this is not perfect since ac resistance is
larger than dc resistance due to skin effect
• Measured value of resistance can be inserted in
impedance equation to improve determination of
synchronous reactance, however this doesn’t help
since the error due to saturation has much larger
effect in Xs

25471_ENERGY_CONVERSION_11456789785.pptx

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