electric machine
Chapter 5 Synchronous Machines 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 4.12 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. tNk tNk mepphw mpphwa ω ωλ cos 2 poles cos Φ= ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Φ= (4.45) 1 mme 2 poles ωω ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = (4.46) tNkt dt d Nk dt d e mepphwmeme p phw a a ωωω λ sincos Φ− Φ == (4.47) tNke mepphwmea ω ω sin Φ − = (4.48) pphwmepphwmemax 2 Φ = Φ = NkfNkE π ω (4.49) pphwmepphwmerms 2 2 2 Φ=Φ= NkfNkfE π π (4.50) 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: RFfR 2 sin 2 poles 2 δ π FT Φ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= (5.1) where = resultant air-gap flux per pole R Φ f F = mmf of the dc field winding RF δ = electric phase angle between magnetic axes of R Φ and f 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. 5.1. Figure 5.1 Torque-angle characteristics. 2 An increase in prime-mover torque will result in a corresponding increase in the torque angle. : pull-out torque at . 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. loss of synchronism, pulling out of step. max TT = o 90 RF = δ ⇒ §5.2 Synchronous-Machine Inductances; Equivalent Circuits Figure 5.2 Schematic diagram of a two-pole, three-phase cylindrical-rotor synchronous machine. §5.2.1 Rotor Self-Inductance §5.2.2 Stator-to-Rotor Mutual Inductances §5.2.3 Stator Inductances; Synchronous Inductance §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. s L = effective inductance seen by phase a under steady-state, balanced three-phase machine operating conditions. ses LX ω = : synchronous reactance a R = armature winding resistance af e = voltage induced by the field winding flux (generated voltage, internal voltage) a I = armature current a v = terminal voltage Motor reference direction: faasaaa ˆˆˆˆ EIjXIRV ++= (5.23) Generator reference direction: faasaaa ˆˆˆˆ EIjXIRV +−−= (5.24) 3 Figure 5.3 Synchronous-machine equivalent circuits: (a) motor reference direction and (b) generator reference direction. ϕ XXX + = als (5.25) al X = armature leakage reactance ϕ X = magnetizing reactance of the armature winding R ˆ E = air-gap voltage or the voltage behind leakage reactance Figure 5.4 Synchronous-machine equivalent circuit showing air-gap and leakage components of synchronous reactance and air-gap voltage. 4 §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 5.11 (a) Impedance interconnecting two voltages; (b) phasor diagram. 5 φ cos 22 IEP = (5.34) Z EE I 21 ˆˆ ˆ − = (5.35) δ j eEE 11 ˆ = (5.36) 22 ˆ EE = (5.37) Z j eZXjRZ φ =+= (5.38) () ZZ Z jj j j j e Z E e Z E eZ EeE IeI φφδ φ δ φ −− −= − == 2121 ˆ (5.39) () ( ZZ Z E Z E I φφδφ −−−= coscoscos 21 ) (5.40) () 2 2 221 2 cos Z RE Z EE P Z −−= φδ (5.41) () 2 2 221 2 sin Z RE Z EE P Z −+= αδ (5.42) where ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =−= − X R ZZ 1 tan90 φα o (5.43) () 2 2 121 1 sin Z RE Z EE P Z −−= φδ (5.44) Frequently, 0and, ≈≈<< Z XZZR α , δ sin 21 21 X EE PP == (5.45) Equation (5.45) is commonly referred to as the power-angle characteristic for a synchronous machine. The angle δ is known as the power angle. Note that and are the line-to-neutral voltages. 1 E 2 E For three-phase systems, a factor “3” shall be placed in front of the equation. The maximum power transfer is X EE PP 21 max,2max,1 == (5.46) occurring when . o 90±= δ If 0> δ , leads and power flows from source to . 1 ˆ E 2 ˆ E 1 ˆ E 2 ˆ E When 0 < δ , lags and power flows from source to . 1 ˆ E 2 ˆ E 2 ˆ E 1 ˆ E Consider Fig. 5.12 in which a synchronous machine with generated voltage and synchronous is connected to a system whose Thevenin equivalent is a voltage source in series with a reactive impedance . The power-angle characteristic can be written af ˆ E s X EQ ˆ V EQ jX δ sin EQs EQfa XX VE P + = (5.47) 6 Figure 5.12 Equivalent-circuit representation of a synchronous machine connected to an external system. Note that , 21 EEP ∝ 1− ∝ X P , 21max EEP ∝ , and . 1 max − ∝ XP In general, stability considerations dictate that a synchronous machine achieve steady-state operation for a power angle considerably less than . o 90 7 Figure 5.14 Equivalent circuits and phasor diagrams for Example 5.7. 8 9 10 [...]... Quadrature-axis air-gap fluxes in a salient-pole synchronous machine Figure 5. 22 Phasor diagram of a salient-pole synchronous generator 5. 3.2 Phasor Diagrams for Salient-Pole Machines 19 Figure 5. 23 Phasor diagram for a synchronous generator showing the relationship between the voltages and the currents X d = X al + X ϕ d (5. 55) X q = X al + X ϕ q (5. 56) Figure 5. 24 Relationships between component voltages... (5. 48) Figure 5. 17 Construction used for the derivation of a synchronous generator capability curve ˆ ˆ P − j Q = Va + jX s I a ˆ ˆ ˆ E = V + jX I af ⎛ V2 P + ⎜Q + a ⎜ Xs ⎝ 2 a 2 (5. 49) (5. 50) s a ⎞ ⎛V E ⎟ = ⎜ a af ⎟ ⎜ X s ⎠ ⎝ ⎞ ⎟ ⎟ ⎠ 2 Figure 5. 18 Typical form of synchronous- generator V curves 16 (5. 51) Figure 5. 19 Losses in a three-phase, 45- kVA, Y-connected, 220-V, 60-Hz, six-pole synchronous machine. .. curves Ra ,eff = RT 234 .5 + T = Rt 234 .5 + t short − circuit load loss (short − circuitarmaturecurrent )2 14 (5. 32) (5. 33) 5. 5 Steady-State Operating Characteristics Figure 5. 15 Characteristic form of synchronous- generator compounding curves 15 Figure 5. 16 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 Apparent... machine 17 5. 6 Effects of Salient Poles; Introduction to Direct-And Quadrature-Axis Theory 5. 6.1 Flux and MMF Waves 18 Figure 5. 20 Direct-axis air-gap fluxes in a salient-pole synchronous machine 2 V 3 cos (3 ω e t + φ 3 E 3 ,a = (( ) ) cos(3(ω t − 120 ) + φ ) = ) (5. 52) E3,b = 2V3 cos 3 ω e t − 120 o + φ3 = 2V3 cos(3ω e t + φ3 ) (5. 53) 2V3 cos(3ω e t + φ3 ) (5. 54) E3,c = 2V3 o e 3 Figure 5. 21 Quadrature-axis... 5. 3 Open- and Short-Circuit Characteristics 5. 3.1 Open-Circuit Saturation Characteristic and No-Load Rotational Losses Figure 5. 5 Open-circuit characteristic of a synchronous machine 5. 3.2 Short-Circuit Characteristic and Load Loss 11 Figure 5. 6 Typical form of an open-circuit core-loss curve ˆ ˆ E a f = I a (Ra + jX s ) (5. 26) Figure 5. 7 Open- and short-circuit characteristics of a synchronous machine. .. voltages in a phasor diagram o′a ′ b ′a ′ = oa ba ˆ Iq X q b′a ′ ⎞ ˆ ˆ ′a ′ = ⎛ o Ia = X q Ia ⎜ ⎟ oa = ˆ ⎝ ba ⎠ Iq ˆ ˆ ˆ ˆ ˆ E a f = Va + Ra I a + jX d I d + jX q I q 20 (5. 57) (5. 58) (5. 59) Figure 5. 25 Generator phasor diagram for Example 5. 9 21 22 ... characteristics of a synchronous machine Figure 5. 8 Phasor diagram for short-circuit conditions ˆ ˆ E R = I a (Ra + jX a l ) X s ,u = Xs = 12 Va ,ag I a ,ac Va ,rated ′ Ia (5. 27) (5. 28) (5. 29) Figure 5. 9 Open- and short-circuit characteristics showing equivalent magnetization line for saturated operating conditions Of ′ SCR = Of ′′ AFNL SCR = AFSC 13 (5. 30) (5. 31) Figure 5. 10 Typical form of short-circuit load