Excitation 研發(fā)培訓(xùn),設(shè)計(jì)培訓(xùn)

1、Synchronous Machines - Reactance and Excitation CalculationMark FanslowTeco Westinghouse Engineering Training Nov. 2005Stator Revolving Magnetic FieldMagnetic Pole Pairs rotate at Synchronous SpeedThree Phase AC VoltageSynchronous RotorDC input to the rotor creates electromagnetic poles pairsPoles o

2、f different polarity are created by winding around the pole in different directionsSynchronous MotorRotor “Locked” into power supply, rovloves at synchronous speedPhasors and Phasor DiagramsnThe concept of Phasors is related to sinusoidal waveforms that are distributed in space and vary with time.nP

3、hasors are complex (as in complex numbers) quantities used for simplified calculation of time varying and traveling waveforms. The magnitude and phase relationship between the vectors and can be shown simply in phasor diagrams.Phasor Diagram ExamplenConsider the simple RL (resistive inductive) circu

4、itV=Vmsin(t)22)sin(rLtVimVoltage function of time. Current function of time and displaced by angle VILMagnetic Flux in Inductor 90 out of phase with voltage waveformi90Phasor Diagram Cont.VIFLLRecall: HBroB: Magnetic Flux Density denote H: Magnetic Flux Intensity or mmf denote Fo: Constant of permea

5、bility of free spacer: Constant of permeability of electrical steel Cylindrical Rotor Synchronous Machine: Due to the even air gap, the formulations for basic machine quantities is simplified. We will consider the cylindrical machine theory first and then extend the analysis to the salient pole case

6、.Axis of FieldAxis of Phase A90lagArmature current mmf FarResultant mmf FrField mmf FfRotor RotationStatorRotorNSStator I.D.Rotor O.D.IaVtIaraEintEaar-arffRjIaxAjIax1r- ararErCylindrical Rotor Phasor Diagram For Synchronous GeneratorVt:Terminal VoltageIa:Stator Current Lags terminal Voltage by Angle

7、 Ea:Air Gap Voltage, the voltage induced into the armature by the field voltage acting alone, the “open circuit” voltageEr:Reactive voltage produced by armature fluxEint:Resultant Voltage in the Air Gapar:Flux from Armature current (Armature Reaction)f:Flux due to field currentr:Resultant flux ar+ f

8、Ff:mmf of Field Aar:mmf of Armature Current (Armature Reaction)FR:Resultant mmf, the sum of F+AIara:Voltage drop of armature resistanceIaxl:Reactive voltage drop of Armature leakage reactanceIaxA:Reactive voltage drop of reactance of Armature reactionFrom the diagram it is evident that Er, the volta

9、ge induced in the stator by the effect of the armature reaction flux, is in phase with IaXl. The summation of Er and IaXl gives the total reactive voltage produced in the armature circuit by the armature flux. The ratio of this total voltage to armature current is defined as Xs or the synchronous re

10、actance. lAalarsXXIXIEXSalient Pole Synchronous MachinesSalient Pole Synchronous MachineTwo Reaction TheorynOne of the concepts used in cylindrical rotor theory was the summation of both flux and mmf wave forms. nr= f+ ar and Fr = Ff+FarnThis is allowed since B = o rH and the magnetic permeability o

11、f the air gap is constant around the rotornHowever for salient pole machines , the permeability of the flux path varies significantly as the ratio of air gap to steel changes. Therefore r f+ ar Salient Pole Synchronous MachinenWith a salient pole machine, a sinusoidal distribution of flux cannot be

12、assumed in the air gap due to the variation in magnetic permeance along the air gap.nHowever, owing to the symmetry of a salient pole machine along the direct and quadrature axis, a sinusoidal distribution of mmf can be assumed along each axis.nBreak the mmf of the armature current Aar into two comp

13、onents, Ad and Aq. Ad is the component of Aar that works along the direct axis and Aq is the component of Aar centered on the quadrature axis.Two Reaction Theory: IntroductionIf you accept that Aar can be broken into two components Ad and Aq, it follows that for each mmf waveform, a emf waveform exi

14、sts 90 degrees out of phase with it. So Ad has an associated Ed, and Aq has an associated Eq. These voltage drops can be though of has being created by a fictitious reactance drop Ed=XadId and Eq=XaqIq , where Id: Direct axis component of armature current Iq: Quadrature axis component of armature cu

15、rrentXad: Direct axis armature reaction reactanceXaq: Quadrature axis armature reaction reactanceIaVtIaraEintEaarjIaxAAjIax1IdxldIqxlqIdxAdIqxAqFor Cylindrical Rotor:Ia2=Id2+Iq2Xld=XlqXAd=XAqTwo Reaction DiagramSince for a Cylindrical Rotor:Ia2=Id2+Iq2 and (IaXs)2 = (IdXd)2 + (IqXq)2Xld=XlqXAd=XAqXs

16、=Xd=XqIt is not necessary to use two reaction theory to describe the quantities of a cylindrical rotor machine.EaIqIaIdfraqadarPhasor Diagram of a Salient Pole Synchronous GeneratorIaVtIdEintIqK1EintIdxadIqxaqEaIaraIaxlIa: Stator currentVt: Stator terminal VoltageIra: Voltage of stator resistanceIxl

17、: Voltage of stator leakage reactanceEint: Internal air gap voltageK1Eint: Extra mmf required to overcome stator saturation, K1 is saturation factorIdxad: reactive voltage drop direct axisIdxaq: Reactive voltage drop quadrature axisEa: Total sum of direct axis voltage, air gap voltage, open circuit

18、voltageId = Iasin(+)Iq=Iasin(+)Pole Face Design - Magnetic FieldsnThe mmf wave of armature reaction and the mmf wave of the pole are created on two different sides of the air gap but must be combined to determine a resultant mmf. To do this we must determine conversion factors to convert an stator s

19、ide mmf to and equivalent rotor side mmf.90lagArmature current mmf FarResultant mmf FrField mmf FfRotor RotationStatorRotorNSPole Face Design Magnetic FieldsC1: If peak of the fundamental is unity, then C1 is peak of acutal waveform. Note that Wieseman calls this A1No-Load: motor is excited by the f

20、ield winding onlyPole Face Design Magnetic FieldCd1: If peak of the fundamental is unity, then Cd1 is peak of acutal waveform. Note that Wieseman calls this Ad1Armature mmf Sine wave whose axis coincides with the pole centerPole Face Design Magnetic FieldCq1: If peak of the fundamental is unity, the

21、n Cq1 is peak of acutal waveform. Note that Wieseman calls this Aq1Armature mmf Sine wave whose axis coincides with the gap between polesList of Pole Constants nCd1 Ratio of the fundamental of the airgap flux produced by the direct axis armature current to that which would be produced with a uniform

22、 gap equal to the effective gap at the pole centernCq1 Ratio of the fundamental of the airgap flux produced by the quadrature axis armature current to that which would be produced with a uniform gap equal to the effective gap at the pole centernC1 The ratio of the fundamental to the actual maximum v

23、alue of the field form when excited by the field only (no-load)nCm Ratio of fundamental airgap flux produced by the fundamental of armature mmf to that produced by the field for the same maximum mmf. This is the armature reaction conversion factor for the direct axis. Cm=Cd1/C1nK Flux distribution c

24、oefficient; the ratio of the area of the actual no load flux wave to the area of its fundamentalPole ConstantsnWhat follows are graphs that relate the physical geometry of the pole to the pole constants. These graphs can be found in the appendix of Engineering Note 106. The graphs in the engineering

25、 note are identical to graphs that first appeared in a 1927 AIEE paper titled Graphical Determination of Magnetic Fields by Robert Wieseman. Pole ConstantsWieseman used hand plotting techniques to plot the flux fields of several hundreds of pole shapes to come up with the graphs. Due to the intensiv

26、e nature of the work, the graphs are plotted for a limited range of pole geometry: Pole arc/Pole pitch = 0.5 to .75 Gmax/Gmin = 1.0 to 3.0 Minimum gap/pole pitch = .005 to .05 Since these curves are used by SMDS to calculate motor performance, SMDS will not run with any one of these three variables

27、outside of the given range. There is no reason, besides the limitations of the original curves, why variables outside the ranges listed above couldnt be used.Pole Face Design Magnetic FieldsDetermination of KPole Face Design Magnetic FieldsDetermination of C1Pole Face Design Magnetic FieldsDetermina

28、tion of Cq1Pole Face Design Magnetic FieldsDetermination of Cd1Pole Face Design Magnetic FieldsPole face designs come in two flavors, single radius and double radius. The reason for this is the shape of the pole head relative to the stator bore radius has a large influence of the shape of the field

29、flux waveform.11qaaqdaadCTXCTXReactance CalculationsXad = Reactance of armature reaction directed along the direct axisXaq = Reactance of armature reaction directed along the quadrature axisT = Common Reactance Factora = Permeance FactorCd1 = Pole ConstantCq1 = Pole ConstantPKZfLmTw82210T = Common R

30、eactance Factorm = # phasesL = Stator Core Lengthf= frequencyZ = Series Conductors per PhaseKw = Winding factor StatorP = # PolesReactance Calculationsmin38. 6gKPDgaa = Permeance FactorD = Diameter of Stator BoreP = # PolesKg = Carters Gap Coefficeintgmin = Minimum air gap at center of poleReactance

31、 CalculationsArmature Leakage Reactance is determined using a methodology identical to the induction machine.Synchronous ReactancesXd = Xl + XadXq = Xl + XaqExcitation Calculations1. Calculate total Magnetic Flux2. Convert armature magnetic Flux to Field Equivalent3. Calculate Eint by adding armatur

32、e resistance and leakage reactance drops to terminal voltage.4. Using step 2, calculate the amphere turns required to magnitize the air gap.5. Using step Eint from step 3 and electrical steel magnitization curves calculate the ampere turns required to magnitze the stator core and air gap.6. Using th

33、e results of 4 and 5 calculate the saturation factor. 7. Using the results of 3 and 5 calculate the direct axis component of mmf.8. Calculate the direct axis component of armature reaction 9. Using the results of 6 and 7 calculate the mmf requirements at the pole face10. Calculate the pole saturatio

34、n mmf11. Total direct axis excitation is the sum of 8 and 9Excitation CalculationsStep One: Total fundamental flux per poleSPCdpphfundNfreqKKE44. 4Eph:Phase voltage at stator terminalsKp:Pitch FactorKd:Distribution FactorFreq:Frequency NSPC:Armature series turns per phase per circuitExcitation Calcu

35、lation Step 2: Calculate total flux per pole on open circuitfundfKK relates fundamental flux per pole to total flux per pole using factors. Excitation Calculation Cont.Step 3: Calculate total air gap flux per pole at the specific voltage and load of interest. This voltage was shown on the previous p

36、hasor diagram as Eint.Step 3a: Calculate the stator leakage reactance using same formulas derived for the Induction motor stator.Field Excitation Cont.alIrEIxEcossintanintint1Portion of the previous phasor diagram is redrawnFrom diagram it is evident that:IaxlIaraIaVtIdEintIq22intsincosltatIxVIrVEIa

37、VtEintIqK1EintIdxadIqxaqIaxaqIaraIaxlIdK1Eintsin()cos()sin(tanint1int11EKEKxIaqaField Excitation Cont.Step 4: Calculate the ampere turns needed to magnetize the air gap at rated voltage: Fg. hstacklengtpolepitchBfmaxoggapfactorgBFminmaxSame gapfactor used from induction motor theory (i.e. Carters co

38、efficient)Field Excitation Cont.Step 5: Calculate the stator core + stator teeth ampere turns at the value of flux corresponding to Eint.tfTABmaxcfCABmaxAc: Area of stator coreAt: Area of stator teethBCmax: maximum value of flux density in stator coreBTmax: maximum value of flux density in stator te

39、ethField Excitation Cont.nUsing B-H curves for magnetic steel used for stator laminations, read off a value of mmf in amphere turns for the value of BCmax and BTmax calculated in previous steps. Make sure units match.Field Excitation Cont.nStep 6: Calculate the component mmf in the direct axis corresponding to K1eintnStep 6a: Calculate saturation factor K1 from B-H curve for ele