Chapter 5: Synchronous Machines
This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy
- Main features of synchronous machines:
- A synchronous machine is an ac machine whose speed under steady-state conditions is proportional to the frequency of the current in its armature.
- The rotor, along with the magnetic field created by the dc field current on the rotor, rotates at the same speed as, or in synchronism with, the rotating magnetic field produced by the armature currents, and a steady torque results.
Figure 5.1 Schematic views of three-phase generators: (a) two-pole, (b) four-pole, and
(c) Y connection of the windings.
§5.1 Introduction to Polyphase Synchronous Machines
- Armature winding: on the stator, alternating current.
- Field winding: on the rotor, dc power supplied by the excitation system.
- Cylindrical rotor: for two- and four-pole turbine generators.
- Salient-pole rotor: for multipolar, slow-speed, hydroelectric generators and for most synchronous motors.
- Acting as a voltage source:
- Frequency determined by the speed of its mechanical drive (or prime mover).
- The amplitude of the generated voltage is proportional to the frequency and the field current.
λa=kwNphΦpcos((poles2)ωmt)=kwNphΦpcosωmetλa=kwNphΦpcos((poles2)ωmt)=kwNphΦpcosωmetalignl { stack {
size 12{λ rSub { size 8{a} } =k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } "cos" \( \( { { ital "poles"} over {2} } \) ω rSub { size 8{m} } t \) } {} #
" "=k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } "cos"ω rSub { size 8{ ital "me"} } t {}
} } {} (5.1)
ωme=(poles2)ωmωme=(poles2)ωm size 12{ω rSub { size 8{ ital "me"} } = \( { { ital "poles"} over {2} } \) ω rSub { size 8{m} } } {} (5.2)
ea=dλadt=kwNphdΦpdtcosωmet−ωmekwNphΦpsinωmetea=dλadt=kwNphdΦpdtcosωmet−ωmekwNphΦpsinωmet size 12{e rSub { size 8{a} } = { {dλ rSub { size 8{a} } } over { ital "dt"} } =k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } { {dΦ rSub { size 8{p} } } over { ital "dt"} } "cos"ω rSub { size 8{ ital "me"} } t - ω rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } "sin"ω rSub { size 8{ ital "me"} } t} {} (5.3)
ea=−ωmekwNphΦpsinωmetea=−ωmekwNphΦpsinωmet size 12{e rSub { size 8{a} } = - ω rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } "sin"ω rSub { size 8{ ital "me"} } t} {} (5.4)
Emax=ωmekwNphΦp=2πfmekwNphΦpEmax=ωmekwNphΦp=2πfmekwNphΦp size 12{E rSub { size 8{"max"} } =ω rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } =2πf rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } } {} (5.5)
Erms=2π2fmekwNphΦp=2πfmekwNphΦpErms=2π2fmekwNphΦp=2πfmekwNphΦp size 12{E rSub { size 8{ ital "rms"} } = { {2π} over { sqrt {2} } } f rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } = sqrt {2} πf rSub { size 8{ ital "me"} } k rSub { size 8{w} } N rSub { size 8{ ital "ph"} } Φ rSub { size 8{p} } } {} (5.6)
- Synchronous generators can be readily operated in parallel: interconnected power systems.
- When a synchronous generator is connected to a large interconnected system containing many other synchronous generators, the voltage and frequency at its armature terminals are substantially fixed by the system.
- It is often useful, when studying the behavior of an individual generator or group of generators, to represent the remainder of the system as a constant-frequency, constant-voltage source, commonly referred to as an infinite bus.
- Analysis of a synchronous machine connected to an infinite bus.
- Torque equation:
T==π2(poles2)2ΦRFfsinδRFT==π2(poles2)2ΦRFfsinδRF size 12{T"==" { {π} over {2} } \( { { ital "poles"} over {2} } \) rSup { size 8{2} } Φ rSub { size 8{R} } F rSub { size 8{f} } "sin"δ rSub { size 8{ ital "RF"} } } {} (5.7)
where
ΦR=ΦR= size 12{Φ rSub { size 8{R} } ={}} {}resultant air-gap flux per pole
Ff=Ff= size 12{F rSub { size 8{f} } ={}} {}mmf of the dc field winding
δRF=δRF= size 12{δ rSub { size 8{ ital "RF"} } ={}} {}electric phase angle between magnetic axes of
ΦRΦR size 12{Φ rSub { size 8{R} } } {}and
FfFf size 12{F rSub { size 8{f} } } {}
- The minus sign indicates that the electromechanical torque acts in the direction to bring the interacting fields into alignment.
- In a generator, the prime-mover torque acts in the direction of rotation of the rotor, and the electromechanical torque opposes rotation. The rotor mmf wave leads the resultant air-gap flux.
- In a motor, the electromechanical torque is in the direction of rotation, in opposition to the retarding torque of the mechanical load on the shaft.
- Torque-angle curve: Fig. 5.2.
Figure 5.2 Torque-angle characteristics.
- An increase in prime-mover torque will result in a corresponding increase in the torque angle.
- T=TmaxT=Tmax size 12{T=T rSub { size 8{"max"} } } {}: pull-out torque at
δ=90δ=90 size 12{δ="90"} {}.Any further increase in prime-mover torque cannot be balanced by a corresponding increase in synchronous electromechanical torque, with the result that synchronism will no longer be maintained and the rotor will speed up.
⇒⇒ size 12{ drarrow } {} loss of synchronism, pulling out of step.
§5.2 Synchronous-Machine Inductances; Equivalent Circuits
Figure 5.3 Schematic diagram of a two-pole,
three-phase cylindrical-rotor synchronous machine.
- A cross-sectional sketch of a three-phase cylindrical-rotor synchronous machine is shown schematically in Fig.5.3. The figure shows a two-pole machine; alternatively, this can be considered as two poles of a multipole machine. The three-phase armature winding on the stator is of the same type used in the discussion of rotating magnetic fields in Section 4.5. Coils
aa'aa' size 12{a { {a}} sup { ' }} {},
bb'bb' size 12{b { {b}} sup { ' }} {} and
cc'cc' size 12{c { {c}} sup { ' }} {} I represent distributed windings producing sinusoidal mmf and flux-density waves in the air gap. The reference directions for the currents are shown by dots and crosses. The field winding
ff'ff' size 12{f { {f}} sup { ' }} {}on the rotor also represents a distributed winding which produces a sinusoidal mmf and flux-density wave centered on its magnetic axis and rotating with the rotor.
- When the flux linkages with armature phases a, b, c and field winding f are expressed in terms of the inductances and currents as follows,
λa=Laaia+Labib+Lacic+Lafifλa=Laaia+Labib+Lacic+Lafif size 12{λ rSub { size 8{a} } =L rSub { size 8{ ital "aa"} } i rSub { size 8{a} } +L rSub { size 8{ ital "ab"} } i rSub { size 8{b} } +L rSub { size 8{ ital "ac"} } i rSub { size 8{c} } +L rSub { size 8{ ital "af"} } i rSub { size 8{f} } } {} (5.8)
λb=Lbaia+Lbbib+Lbcic+Lbfifλb=Lbaia+Lbbib+Lbcic+Lbfif size 12{λ rSub { size 8{b} } =L rSub { size 8{ ital "ba"} } i rSub { size 8{a} } +L rSub { size 8{ ital "bb"} } i rSub { size 8{b} } +L rSub { size 8{ ital "bc"} } i rSub { size 8{c} } +L rSub { size 8{ ital "bf"} } i rSub { size 8{f} } } {} (5.9)
λc=Lcaia+Lcbib+Lccic+Lcfifλc=Lcaia+Lcbib+Lccic+Lcfif size 12{λ rSub { size 8{c} } =L rSub { size 8{ ital "ca"} } i rSub { size 8{a} } +L rSub { size 8{ ital "cb"} } i rSub { size 8{b} } +L rSub { size 8{ ital "cc"} } i rSub { size 8{c} } +L rSub { size 8{ ital "cf"} } i rSub { size 8{f} } } {} (5.10)
λf=Lfaia+Lfbib+Lfcic+Lffifλf=Lfaia+Lfbib+Lfcic+Lffif size 12{λ rSub { size 8{f} } =L rSub { size 8{ ital "fa"} } i rSub { size 8{a} } +L rSub { size 8{ ital "fb"} } i rSub { size 8{b} } +L rSub { size 8{ ital "fc"} } i rSub { size 8{c} } +L rSub { size 8{ ital "ff"} } i rSub { size 8{f} } } {} (5.11)
the induced voltages can be found from Faraday's law. Here, two like subscripts denote a self-inductance, and two unlike subscripts denote a mutual inductance between the two windings. The script is used to indicate that, in general, both the self- and mutual inductances of a three-phase machine may vary with rotor angle.
§5.2.1 Rotor Self-Inductance
- With a cylindrical stator, the self-inductance of the field winding is independent of the rotor position 0m when the harmonic effects of stator slot openings are neglected.
Lff=Lff=Lff0+Lf1Lff=Lff=Lff0+Lf1 size 12{L rSub { size 8{ ital "ff"} } =L rSub { size 8{ ital "ff"} } =L rSub { size 8{ ital "ff"0} } +L rSub { size 8{f1} } } {} (5.12)
where the italic L is used for an inductance which is independent of
θmθm size 12{θ rSub { size 8{m} } } {}. The component
Lff0Lff0 size 12{L rSub { size 8{ ital "ff"0} } } {}corresponds to that portion of
LffLff size 12{L rSub { size 8{ ital "ff"} } } {} due to the space-fundamental component of air-gap flux
§5.2.2 Stator-to-Rotor Mutual Inductances
- The stator-to-rotor mutual inductances vary periodically with
θmeθme size 12{θ rSub { size 8{ ital "me"} } } {}, the electrical angle between the magnetic axes of the field winding and the armature phase a as shown in Fig.5.2 and as defined by Eq.4.54. With the space-mmf and air-gap flux distribution assumed sinusoidal, the mutual inductance between the field winding f and phase a varies as
cosθmecosθme size 12{"cos"θ rSub { size 8{ ital "me"} } } {}; thus
Laf=Lfa=LafcosθmeLaf=Lfa=Lafcosθme size 12{L rSub { size 8{ ital "af"} } =L rSub { size 8{ ital "fa"} } =L rSub { size 8{ ital "af"} } "cos"θ rSub { size 8{ ital "me"} } } {} (5.13)
θme=poles2θm=ωet+δe0θme=poles2θm=ωet+δe0 size 12{θ rSub { size 8{ ital "me"} } = left [ { { ital "poles"} over {2} } right ]θ rSub { size 8{m} } =ω rSub { size 8{e} } t+δ rSub { size 8{e0} } } {} (5.14)
Laf=Lfa=Lafcos(ωet+δe0)Laf=Lfa=Lafcos(ωet+δe0) size 12{L rSub { size 8{ ital "af"} } =L rSub { size 8{ ital "fa"} } =L rSub { size 8{ ital "af"} } "cos" \( ω rSub { size 8{e} } t+δ rSub { size 8{e0} } \) } {} (5.15)
§5.2.3 Stator Inductances; Synchronous Inductance
- With a cylindrical rotor, the air-gap geometry is independent of
θmθm size 12{θ rSub { size 8{m} } } {} if the effects of rotor slots are neglected. The stator self-inductances then are constant; thus
Laa=Lbb=Lcc=Laa=Laa0+La1Laa=Lbb=Lcc=Laa=Laa0+La1 size 12{L rSub { size 8{ ital "aa"} } =L rSub { size 8{ ital "bb"} } =L rSub { size 8{ ital "cc"} } =L rSub { size 8{ ital "aa"} } =L rSub { size 8{ ital "aa"0} } +L rSub { size 8{a1} } } {} (5.16)
§5.2.4 Equivalent Circuit
- Equivalent circuit for the synchronous machine:
- Single-phase, line-to-neutral equivalent circuits for a three-phase machine operating under balanced, three-phase conditions.
Ls=Ls= size 12{L rSub { size 8{s} } ={}} {}effective inductance seen by phase a under steady-state, balanced three-phase
machine operating conditions.
Xs=ωeLsXs=ωeLs size 12{X rSub { size 8{s} } =ω rSub { size 8{e} } L rSub { size 8{s} } } {}: synchronous reactance
Ra=Ra= size 12{R rSub { size 8{a} } ={}} {}armature winding resistance
eaf=eaf= size 12{e rSub { size 8{ ital "af"} } ={}} {}voltage induced by the field winding flux (generated voltage, internal voltage)
Ia=Ia= size 12{I rSub { size 8{a} } ={}} {} armature current
va=va= size 12{v rSub { size 8{a} } ={}} {} terminal voltage
Motor reference direction:
Vˆa=RaIˆa+jXsIˆa+EˆafVˆa=RaIˆa+jXsIˆa+Eˆaf size 12{ { hat {V}} rSub { size 8{a} } =R rSub { size 8{a} } { hat {I}} rSub { size 8{a} } + ital "jX" rSub { size 8{s} } { hat {I}} rSub { size 8{a} } + { hat {E}} rSub { size 8{ ital "af"} } } {} (5.17)
Generator reference direction:
Vˆa=−RaIˆa−jXsIˆa+EˆafVˆa=−RaIˆa−jXsIˆa+Eˆaf size 12{ { hat {V}} rSub { size 8{a} } = - R rSub { size 8{a} } { hat {I}} rSub { size 8{a} } - ital "jX" rSub { size 8{s} } { hat {I}} rSub { size 8{a} } + { hat {E}} rSub { size 8{ ital "af"} } } {} (5.18)
Figure 5.4 Synchronous-machine equivalent circuits:
(a) motor reference direction and (b) generator reference direction.
X
s
=
X
al
+
X
ϕ
X
s
=
X
al
+
X
ϕ
size 12{X rSub { size 8{s} } =X rSub { size 8{ ital "al"} } +X rSub { size 8{ϕ} } } {}
Xal=Xal= size 12{X rSub { size 8{ ital "al"} } ={}} {}armature leakage reactance
XϕXϕ size 12{X rSub { size 8{ϕ} } } {}=magnetizing reactance of the armature winding
EˆREˆR size 12{ { hat {E}} rSub { size 8{R} } } {}= air-gap voltage or the voltage behind leakage reactance
Figure 5.5 Synchronous-machine equivalent circuit showing air-gap and
leakage components of synchronous reactance and air-gap voltage.
§5.4 Steady-State Power-Angle Characteristics
- The maximum power a synchronous machine can deliver is determined by the maximum torque that can be applied without loss of synchronism with the external system to which it is connected.
- Both the external system and the machine itself can be represented as an impedance in series with a voltage source.
Figure 5.6 (a) Impedance interconnecting two voltages; (b) phasor diagram.
P2=E2IcosφP2=E2Icosφ size 12{P rSub { size 8{2} } =E rSub { size 8{2} } I"cos"φ} {} (5.19)
Iˆ=Eˆ1−Eˆ2ZIˆ=Eˆ1−Eˆ2Z size 12{ { hat {I}}= { { { hat {E}} rSub { size 8{1} } - { hat {E}} rSub { size 8{2} } } over {Z} } } {} (5.20)
Eˆ1=E1ejδEˆ1=E1ejδ size 12{ { hat {E}} rSub { size 8{1} } =E rSub { size 8{1} } e rSup { size 8{jδ} } } {} (5.21)
Eˆ2=E2Eˆ2=E2 size 12{ { hat {E}} rSub { size 8{2} } =E rSub { size 8{2} } } {} (5.22)
Z=R+jX=∣Z∣ejφzZ=R+jX=∣Z∣ejφz size 12{Z=R+ ital "jX"= \lline Z \lline e rSup { size 8{jφ rSub { size 6{z} } } } } {}(5.23)
Iˆ=Iejφ=E1ejδ−E2∣Z∣ejφz=E1∣Z∣ej(δ−φz)−E2∣Z∣e−jφzIˆ=Iejφ=E1ejδ−E2∣Z∣ejφz=E1∣Z∣ej(δ−φz)−E2∣Z∣e−jφz size 12{ { hat {I}}= ital "Ie" rSup { size 8{jφ} } = { {E rSub { size 8{1} } e rSup { size 8{jδ} } - E rSub { size 8{2} } } over { \lline Z \lline e rSup { size 8{jφ rSub { size 6{z} } } } } } = { {E rSub {1} } over { size 12{ \lline Z \lline } } } size 12{e rSup {j \( δ - φ rSub { size 6{z} } \) } } size 12{ - { {E rSub {2} } over { size 12{ \lline Z \lline } } } } size 12{e rSup { - jφ rSub { size 6{z} } } }} {}(5.24)
Icosφ=E1∣Z∣cos(δ−φz)−E2∣Z∣cos(−φz)Icosφ=E1∣Z∣cos(δ−φz)−E2∣Z∣cos(−φz) size 12{I"cos"φ= { {E rSub { size 8{1} } } over { \lline Z \lline } } "cos" \( δ - φ rSub { size 8{z} } \) - { {E rSub { size 8{2} } } over { \lline Z \lline } } "cos" \( - φ rSub { size 8{z} } \) } {}(5.25)
P2=E1E2∣Z∣cos(δ−φz)−E22R∣Z∣2P2=E1E2∣Z∣cos(δ−φz)−E22R∣Z∣2 size 12{P rSub { size 8{2} } = { {E rSub { size 8{1} } E rSub { size 8{2} } } over { \lline Z \lline } } "cos" \( δ - φ rSub { size 8{z} } \) - { {E rSub { size 8{2} } rSup { size 8{2} } R} over { \lline Z \lline rSup { size 8{2} } } } } {} (5.26)
P2=E1E2∣Z∣sin(δ+αz)−E22R∣Z∣2P2=E1E2∣Z∣sin(δ+αz)−E22R∣Z∣2 size 12{P rSub { size 8{2} } = { {E rSub { size 8{1} } E rSub { size 8{2} } } over { \lline Z \lline } } "sin" \( δ+α rSub { size 8{z} } \) - { {E rSub { size 8{2} } rSup { size 8{2} } R} over { \lline Z \lline rSup { size 8{2} } } } } {} (5.27)
Where
αz=90o−φz=tan−1(RX)αz=90o−φz=tan−1(RX) size 12{α rSub { size 8{z} } ="90" rSup { size 8{o} } - φ rSub { size 8{z} } ="tan" rSup { size 8{ - 1} } \( { {R} over {X} } \) } {} (5.28)
P1=E1E2∣Z∣sin(δ−φz)−E12R∣Z∣2P1=E1E2∣Z∣sin(δ−φz)−E12R∣Z∣2 size 12{P rSub { size 8{1} } = { {E rSub { size 8{1} } E rSub { size 8{2} } } over { \lline Z \lline } } "sin" \( δ - φ rSub { size 8{z} } \) - { {E rSub { size 8{1} } rSup { size 8{2} } R} over { \lline Z \lline rSup { size 8{2} } } } } {} (5.29)
Frequently,
R<<∣Z∣R<<∣Z∣ size 12{R"<<" \lline Z \lline } {},
∣Z∣≈X and αz≈0∣Z∣≈X and αz≈0 size 12{ \lline Z \lline approx X" and "α rSub { size 8{z} } approx 0} {},
P1=P2=E1E2XsinδP1=P2=E1E2Xsinδ size 12{P rSub { size 8{1} } =P rSub { size 8{2} } = { {E rSub { size 8{1} } E rSub { size 8{2} } } over {X} } "sin"δ} {} (5.30)
Equation (5.30) is commonly referred to as the power-angle characteristic for a synchronous machine.
- The angle
δδ size 12{δ} {} is known as the power angle.
- Note that
E1E1 size 12{E rSub { size 8{1} } } {} and
E2E2 size 12{E rSub { size 8{2} } } {} are the line-to-neutral voltages.
- For three-phase systems, a factor “3” shall be placed in front of the equation.
- The maximum power transfer is
P1,max=P2,max=E1E2XP1,max=P2,max=E1E2X size 12{P rSub { size 8{1",max"} } =P rSub { size 8{2",max"} } = { {E rSub { size 8{1} } E rSub { size 8{2} } } over {X} } } {} (5.31)
occurring when
δ=±90oδ=±90o size 12{δ= +- "90" rSup { size 8{o} } } {}.
- If
δ>0δ>0 size 12{δ>0} {},
Eˆ1Eˆ1 size 12{ { hat {E}} rSub { size 8{1} } } {} leads and power flows from source to
Eˆ2Eˆ2 size 12{ { hat {E}} rSub { size 8{2} } } {}
- When
δ<0δ<0 size 12{δ<0} {},
Eˆ1Eˆ1 size 12{ { hat {E}} rSub { size 8{1} } } {} lags
Eˆ2Eˆ2 size 12{ { hat {E}} rSub { size 8{2} } } {} and power flows from source
Eˆ1Eˆ1 size 12{ { hat {E}} rSub { size 8{1} } } {} to
Eˆ2Eˆ2 size 12{ { hat {E}} rSub { size 8{2} } } {}
- Consider Fig. 5.7 in which a synchronous machine with generated voltage Êaf and synchronous
XsXs size 12{X rSub { size 8{s} } } {} is connected to a system whose Thevenin equivalent is a voltage source
VEQVEQ size 12{V rSub { size 8{ ital "EQ"} } } {} in series with a reactive impedance
jXEQjXEQ size 12{ ital "jX" rSub { size 8{ ital "EQ"} } } {}. The power-angle characteristic can be written
P=EafVEQXs+XEQsinδP=EafVEQXs+XEQsinδ size 12{P= { {E rSub { size 8{ ital "af"} } V rSub { size 8{ ital "EQ"} } } over {X rSub { size 8{s} } +X rSub { size 8{ ital "EQ"} } } } "sin"δ} {} (5.32)
Figure 5.7 Equivalent-circuit representation of
a synchronous machine connected to an external system.
- Note that
P∝E1E2,P∝X,Pmax∝E1E2P∝E1E2,P∝X,Pmax∝E1E2 size 12{P prop E rSub { size 8{1} } E rSub { size 8{2} } ,P prop X,P rSub { size 8{"max"} } prop E rSub { size 8{1} } E rSub { size 8{2} } } {} , and
Pmax∝XPmax∝X size 12{P rSub { size 8{"max"} } prop X} {}.
- In general, stability considerations dictate that a synchronous machine achieve steady-state operation for a power angle considerably less than
90o90o size 12{"90" rSup { size 8{o} } } {}.
§5.3 Open- and Short-Circuit Characteristics
§5.3.1 Open-Circuit Saturation Characteristic and No-Load Rotational Losses
Figure 5.8 Open-circuit characteristic of a synchronous machine.
§5.3.2 Short-Circuit Characteristic and Load Loss
Figure 5.9 Typical form of an open-circuit core-loss curve.
Eˆaf=Iˆa(Ra+jXs)Eˆaf=Iˆa(Ra+jXs) size 12{ { hat {E}} rSub { size 8{ ital "af"} } = { hat {I}} rSub { size 8{a} } \( R rSub { size 8{a} } + ital "jX" rSub { size 8{s} } \) } {} (5.33)
Figure 5.10 Open- and short-circuit characteristics of a synchronous machine.
Figure 5.11 Phasor diagram for short-circuit conditions.
EˆR=Iˆa(Ra+jXal)EˆR=Iˆa(Ra+jXal) size 12{ { hat {E}} rSub { size 8{R} } = { hat {I}} rSub { size 8{a} } \( R rSub { size 8{a} } + ital "jX" rSub { size 8{ ital "al"} } \) } {} (5.34)
Xs,u=Va,agIa,acXs,u=Va,agIa,ac size 12{X rSub { size 8{s,u} } = { {V rSub { size 8{a, ital "ag"} } } over {I rSub { size 8{a, ital "ac"} } } } } {} (5.35)
Xs=Va,ratedIa'Xs=Va,ratedIa' size 12{X rSub { size 8{s} } = { {V rSub { size 8{a, ital "rated"} } } over {I rSub { size 8{a} } rSup { size 8{'} } } } } {} (5.36)
Figure 5.12 Open- and short-circuit characteristics showing
equivalent magnetization line for saturated operating conditions.
SCR=Of'Of''SCR=Of'Of'' size 12{ ital "SCR"= { {O { {f}} sup { ' }} over {O { {f}} sup { '' }} } } {} (5.37)
SCR=AFNLAFSCSCR=AFNLAFSC size 12{ ital "SCR"= { { ital "AFNL"} over { ital "AFSC"} } } {} (5.38)
Figure 5.13 Typical form of short-circuit load loss and stray load-loss curves.
RTRT=234.5+T234.5+tRTRT=234.5+T234.5+t size 12{ { {R rSub { size 8{T} } } over {R rSub { size 8{T} } } } = { {"234" "." 5+T} over {"234" "." 5+t} } } {} (5.39)
Ra,eff=short−circuit load loss(short−circuit armature current)2Ra,eff=short−circuit load loss(short−circuit armature current)2 size 12{R rSub { size 8{a, ital "eff"} } = { {"short" - "circuit load loss"} over { \( "short" - "circuit armature current" \) rSup { size 8{2} } } } } {} (5.40)
§5.5 Steady-State Operating Characteristics
Figure 5.14 Characteristic form of synchronous-generator compounding curves.
Figure 5.15 Capability curves of an 0.85 power factor, 0.80 short-circuit ratio,
hydrogen-cooled turbine generator. Base MVA is rated MVA at 0.5 psig hydrogen.
Apprent power
=
P
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Q
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Apprent power
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size 12{"Apprent power"= sqrt {P rSup { size 8{2} } +Q rSup { size 8{2} } } =V rSub { size 8{a} } I rSub { size 8{a} } } {}
Figure 5.16 Construction used for the derivation of a synchronous generator capability curve.
P−jQ=Vˆa+jXsIˆaP−jQ=Vˆa+jXsIˆa size 12{P - ital "jQ"= { hat {V}} rSub { size 8{a} } + ital "jX" rSub { size 8{s} } { hat {I}} rSub { size 8{a} } } {} (5.41)
Eˆaf=Vˆa+jXsIˆaEˆaf=Vˆa+jXsIˆa size 12{ { hat {E}} rSub { size 8{ ital "af"} } = { hat {V}} rSub { size 8{a} } + ital "jX" rSub { size 8{s} } { hat {I}} rSub { size 8{a} } } {} (5.42)
P2+(Q+Va2Xs)2=(VaEafXs)2P2+(Q+Va2Xs)2=(VaEafXs)2 size 12{P rSup { size 8{2} } + \( Q+ { {V rSub { size 8{a} } rSup { size 8{2} } } over {X rSub { size 8{s} } } } \) rSup { size 8{2} } = \( { {V rSub { size 8{a} } E rSub { size 8{ ital "af"} } } over {X rSub { size 8{s} } } } \) rSup { size 8{2} } } {} (5.43)
Figure 5.17 Typical form of synchronous-generator V curves.
§5.6 Effects of Salient Poles; Introduction to Direct-And
Quadrature-Axis Theory
§5.6.1 Flux and MMF Waves
Figure 5.18 Direct-axis air-gap fluxes in a salient-pole synchronous machine.
E3,a=2V3cos(3ωet+φ3)E3,a=2V3cos(3ωet+φ3) size 12{E rSub { size 8{3,a} } = sqrt {2} V rSub { size 8{3} } "cos" \( 3ω rSub { size 8{e} } t+φ rSub { size 8{3} } \) } {} (5.44)
E3,b=2V3cos(3(ωe−120o)+φ3)=2V3cos(3ωet+φ3)E3,b=2V3cos(3(ωe−120o)+φ3)=2V3cos(3ωet+φ3) size 12{E rSub { size 8{3,b} } = sqrt {2} V rSub { size 8{3} } "cos" \( 3 \( ω rSub { size 8{e} } - "120" rSup { size 8{o} } \) +φ rSub { size 8{3} } \) = sqrt {2} V rSub { size 8{3} } "cos" \( 3ω rSub { size 8{e} } t+φ rSub { size 8{3} } \) } {} (5.45)
E3,c=2V3cos(3(ωet−120o)+φ3)=2V3cos(3ωet+φ3)E3,c=2V3cos(3(ωet−120o)+φ3)=2V3cos(3ωet+φ3) size 12{E rSub { size 8{3,c} } = sqrt {2} V rSub { size 8{3} } "cos" \( 3 \( ω rSub { size 8{e} } t - "120" rSup { size 8{o} } \) +φ rSub { size 8{3} } \) = sqrt {2} V rSub { size 8{3} } "cos" \( 3ω rSub { size 8{e} } t+φ rSub { size 8{3} } \) } {} (5.45)
Figure 5.19 Quadrature-axis air-gap fluxes in a salient-pole synchronous machine.
Figure 5.20 Phasor diagram of a salient-pole synchronous generator.
§5.6.2 Phasor Diagrams for Salient-Pole Machines
Figure 5.21 Phasor diagram for a synchronous generator showing
the relationship between the voltages and the currents.
Xd=Xal+XϕdXd=Xal+Xϕd size 12{X rSub { size 8{d} } =X rSub { size 8{ ital "al"} } +X rSub { size 8{ϕd} } } {} (5.46)
Xq=Xal+XϕqXq=Xal+Xϕq size 12{X rSub { size 8{q} } =X rSub { size 8{ ital "al"} } +X rSub { size 8{ϕq} } } {} (5.47)
Figure 5.22 Relationships between component voltages in a phasor diagram.
o'a'oa=b'a'bao'a'oa=b'a'ba size 12{ { { { {o}} sup { ' } { {a}} sup { ' }} over { ital "oa"} } = { { { {b}} sup { ' } { {a}} sup { ' }} over { ital "ba"} } } {} (5.48)
o'a'=(b'a'ba)oa=∣Iˆq∣Xq∣Iˆq∣∣Iˆa∣=Xq∣Iˆa∣o'a'=(b'a'ba)oa=∣Iˆq∣Xq∣Iˆq∣∣Iˆa∣=Xq∣Iˆa∣ size 12{ { {o}} sup { ' } { {a}} sup { ' }= \( { { { {b}} sup { ' } { {a}} sup { ' }} over { ital "ba"} } \) ital "oa"= { { \lline { hat {I}} rSub { size 8{q} } \lline X rSub { size 8{q} } } over { \lline { hat {I}} rSub { size 8{q} } \lline } } \lline { hat {I}} rSub { size 8{a} } \lline =X rSub { size 8{q} } \lline { hat {I}} rSub { size 8{a} } \lline } {} (5.49)
Eˆaf=Vˆa+RaIˆa+jXdIˆd+jXqIˆqEˆaf=Vˆa+RaIˆa+jXdIˆd+jXqIˆq size 12{ { hat {E}} rSub { size 8{ ital "af"} } = { hat {V}} rSub { size 8{a} } +R rSub { size 8{a} } { hat {I}} rSub { size 8{a} } + ital "jX" rSub { size 8{d} } { hat {I}} rSub { size 8{d} } + ital "jX" rSub { size 8{q} } { hat {I}} rSub { size 8{q} } } {} (5.50)
5.7 Power-Angle Characteristics Of Salient-Pole Machines
- For the purposes of this discussion, it is sufficient to limit our discussion to the simple system shown in the schematic diagram of Fig.5.23a, consisting of a salient pole synchronous machine SM connected to an infinite bus of voltage
VˆEQVˆEQ size 12{ { hat {V}} rSub { size 8{ ital "EQ"} } } {}through a series impedance of reactance
XEQXEQ size 12{X rSub { size 8{ ital "EQ"} } } {}. Resistance will be neglected because it is usually small. Consider that the synchronous machine is acting as a generator. The phasor diagram is shown by the solid-line phasors in Fig.5.23b. The dashed phasors show the external reactance drop resolved into components due to
IˆdIˆd size 12{ { hat {I}} rSub { size 8{d} } } {}and
IˆqIˆq size 12{ { hat {I}} rSub { size 8{q} } } {}. The effect of the external impedance is merely to add its reactance to the reactances of the machine; the total values of the reactance between the excitation voltage
EˆafEˆaf size 12{ { hat {E}} rSub { size 8{ ital "af"} } } {}and the bus voltage
VˆEQVˆEQ size 12{ { hat {V}} rSub { size 8{ ital "EQ"} } } {} is therefore
XdT=Xd+XEQXdT=Xd+XEQ size 12{X rSub { size 8{ ital "dT"} } =X rSub { size 8{d} } +X rSub { size 8{ ital "EQ"} } } {} (5.50)
XqT=Xq+XEQXqT=Xq+XEQ size 12{X rSub { size 8{ ital "qT"} } =X rSub { size 8{q} } +X rSub { size 8{ ital "EQ"} } } {} (5.51)
- If the bus voltage
VˆEQVˆEQ size 12{ { hat {V}} rSub { size 8{ ital "EQ"} } } {} is resolved into components its direct-axis component
Vd=VEQsinδVd=VEQsinδ size 12{V rSub { size 8{d} } =V rSub { size 8{ ital "EQ"} } "sin"δ} {} and quadrature-axis component
Vq=VEQcosδVq=VEQcosδ size 12{V rSub { size 8{q} } =V rSub { size 8{ ital "EQ"} } "cos"δ} {} in phase with
IˆdIˆd size 12{ { hat {I}} rSub { size 8{d} } } {} and
IˆqIˆq size 12{ { hat {I}} rSub { size 8{q} } } {}, respectively, the power P delivered to the bus per phase (or in per unit) is
P=IdVd+IqVq=IdVEQsinδ+IqVEQcosδP=IdVd+IqVq=IdVEQsinδ+IqVEQcosδ size 12{P=I rSub { size 8{d} } V rSub { size 8{d} } +I rSub { size 8{q} } V rSub { size 8{q} } =I rSub { size 8{d} } V rSub { size 8{ ital "EQ"} } "sin"δ+I rSub { size 8{q} } V rSub { size 8{ ital "EQ"} } "cos"δ} {}(5.52)
Id=Eaf−VEQcosδXdTId=Eaf−VEQcosδXdT size 12{I rSub { size 8{d} } = { {E rSub { size 8{ ital "af"} } - V rSub { size 8{ ital "EQ"} } "cos"δ} over {X rSub { size 8{ ital "dT"} } } } } {}(5.53)
Iq=VEQsinδXqTIq=VEQsinδXqT size 12{I rSub { size 8{q} } = { {V rSub { size 8{ ital "EQ"} } "sin"δ} over {X rSub { size 8{ ital "qT"} } } } } {}(5.54)
P=EafVEQXdTsinδ+VEQ2(XdT−XqT)2XdTXqTsin2δP=EafVEQXdTsinδ+VEQ2(XdT−XqT)2XdTXqTsin2δ size 12{P= { {E rSub { size 8{ ital "af"} } V rSub { size 8{ ital "EQ"} } } over {X rSub { size 8{ ital "dT"} } } } "sin"δ+ { {V rSub { size 8{ ital "EQ"} } rSup { size 8{2} } \( X rSub { size 8{ ital "dT"} } - X rSub { size 8{ ital "qT"} } \) } over {2X rSub { size 8{ ital "dT"} } X rSub { size 8{ ital "qT"} } } } "sin"2δ} {}(5.55)
Figure 5.23 Salient-pole synchronous machine and series impedance: (a) single-line diagram and (b) phasor diagram.
Figure 5.24 Power-angle characteristic of a salient-pole synchronous machine showing the fundamental component due to field excitation and the second-harmonic component due to reluctance torque.
- The general form of this power-angle characteristic is shown in Fig.5.24. The first term is the same as the expression obtained for a cylindrical-rotor machine. The second term includes the effect of salient poles. It represents the fact that the airgap flux wave creates torque, tending to align the field poles in the position of minimum reluctance. This term is the power corresponding to the reluctance torque and is of the same general nature as the reluctance torque. Note that the reluctance torque is independent of field excitation. Also note that, if
XdT=XqTXdT=XqT size 12{X rSub { size 8{ ital "dT"} } =X rSub { size 8{ ital "qT"} } } {}as in a uniform-air-gap machine, there is no preferential direction of magnetization, the reluctance torque is zero and Eq.5.55 reduces to the power-angle equation for a cylindrical-rotor machine.
5.8 Permanent-Magnet Ac Motors
- Permanent-magnet ac motors are polyphase synchronous motors with permanentmagnet rotors. Thus they are similar to the synchronous machines discussed up to this point in this chapter with the exception that the field windings are replaced by permanent magnets.
- Figure 5.25 is a schematic diagram of a three-phase permanent-magnet ac machine. Comparison of this figure with Fig.5.1 emphasizes the similarities between the permanent-magnet ac machine and the conventional synchronous machine. In fact, the permanent-magnet ac machine can be readily analyzed with the techniques of this chapter simply by assuming that the machine is excited by a field current of constant value, making sure to calculate the various machine inductances based on the effective permeability of the permanent-magnet rotor.
Figure 5.25 Schematic diagram of a three-phase permanent-magnet ac machine. The arrow indicates the direction of rotor magnetization.
- Figure 5.26 shows a cutaway view of a typical permanent-magnet ac motor. This figure also shows a speed and position sensor mounted on the rotor shaft. This sensor is used for control of the motor. A number of techniques may be used for shaft-position sensing, including Hall-effect devices, light-emitting diodes and phototransistors in combination with a pulsed wheel, and inductance pickups.
Figure 5.26 Cutaway view of a permanent-magnet ac motor. Also shown is the shaft speed and position sensor used to control the motor. (EG&G Torque Systems.)
- Permanent-magnet ac motors are typically operated from variable-frequency motor drives. Under conditions of constant-frequency, sinusiodal polyphase excitation, a permanent-magnet ac motor behaves similarly to a conventional ac synchronous machine with constant field excitation.
- An alternate viewpoint of a permanent-magnet ac motor is that it is a form of permanent-magnet stepping motor with a nonsalient stator. Under this viewpoint, the only difference between the two is that there will be little, if any, saliency (cogging) torque in the permanent-magnet ac motor. In the simplest operation, the phases can be simply excited with stepped waveforms so as to cause the rotor to step sequentially from one equilibrium position to the next. Alternatively, using rotor-position feedback from a shaft-position sensor, the motor phase windings can be continuously excited in such a fashion as to control the torque and speed of the motor.
- As with the stepping motor, the frequency of the excitation determines the motor speed, and the angular position between the rotor magnetic axis and a given phase and the level of excitation in that phase determines the torque which will be produced.
- Permanent-magnet ac motors are frequently referred to as brushless motors or brushless dc motors. This terminology comes about both because of the similarity, when combined with a variable-frequency, variable-voltage drive system, of their speed-torque characteristics to those of dc motors and because of the fact that one can view these motors as inside-out dc motors, with their field winding on the rotor and with their armature electronically commutated
