You are here: Home » Content » Chapter 5: Synchronous Machines
Content Actions

Chapter 5: Synchronous Machines

Module by: NGUYEN Phuc

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.
Hình 1
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
  • 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.
Hình 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
Hình 3
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)
Hình 4
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
Hình 5
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.
Hình 6
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=E1E2Xsinδ