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. 20051Stator Revolving Magnetic FieldMagnetic Pole Pairs rotate at Synchronous SpeedThree Phase AC Voltage2Synchronous RotorDC input to the rotor creates electromagnetic poles pairsPoles

2、 of different polarity are created by winding around the pole in different directions3Synchronous MotorRotor “Locked into power supply, rovloves at synchronous speed4Phasors and Phasor DiagramsThe concept of Phasors is related to sinusoidal waveforms that are distributed in space and vary with time.

3、Phasors 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.5Phasor Diagram ExampleConsider the simple RL (resistive inductive) circ

4、uitV=Vmsin(t)Voltage function of time. Current function of time and displaced by angle VILMagnetic Flux in Inductor 90 out of phase with voltage waveformi906Phasor Diagram Cont.VIFLLRecall: B: Magnetic Flux Density denote H: Magnetic Flux Intensity or mmf denote Fo: Constant of permeability of free

5、spacer: Constant of permeability of electrical steel 7Cylindrical 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.8Axis of Fiel

6、dAxis of Phase A90lagArmature current mmf FarResultant mmf FrField mmf FfRotor RotationStatorRotorNSStator I.D.Rotor O.D.9IaVtIaraEintEaar-arffRjIaxAjIax1r- ararEr10Cylindrical Rotor Phasor Diagram For Synchronous GeneratorVt:Terminal VoltageIa:Stator Current Lags terminal Voltage by Angle Ea:Air Ga

7、p 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+ fFf:mmf of F

8、ield 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 reaction11From the diagram it is evident that Er, the voltage induce

9、d 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 reactance.

10、12Salient Pole Synchronous Machines13Salient Pole Synchronous Machine14Two Reaction TheoryOne of the concepts used in cylindrical rotor theory was the summation of both flux and mmf wave forms. r= f+ ar and Fr = Ff+FarThis is allowed since B = o rH and the magnetic permeability of the air gap is con

11、stant around the rotorHowever 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 15Salient Pole Synchronous MachineWith a salient pole machine, a sinusoidal distribution of flux cannot be assumed in the air g

12、ap due to the variation in magnetic permeance along the air gap.However, 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.Break the mmf of the armature current Aar into two components, Ad and Aq. Ad

13、is the component of Aar that works along the direct axis and Aq is the component of Aar centered on the quadrature axis.16Two 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 exists 90 degrees out o

14、f 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 currentXad: Direct axi

15、s armature reaction reactanceXaq: Quadrature axis armature reaction reactance17IaVtIaraEintEaarjIaxAAjIax1IdxldIqxlqIdxAdIqxAqFor Cylindrical Rotor:Ia2=Id2+Iq2Xld=XlqXAd=XAq18Two Reaction DiagramSince for a Cylindrical Rotor:Ia2=Id2+Iq2 and (IaXs)2 = (IdXd)2 + (IqXq)2Xld=XlqXAd=XAqXs=Xd=XqIt is not

16、necessary to use two reaction theory to describe the quantities of a cylindrical rotor machine.19EaIqIaIdfraqadarPhasor Diagram of a Salient Pole Synchronous Generator20IaVtIdEintIqK1EintIdxadIqxaqEaIaraIaxlIa: Stator currentVt: Stator terminal VoltageIra: Voltage of stator resistanceIxl: Voltage of

17、 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 voltageId =

18、Iasin(+)Iq=Iasin(+)21Pole Face Design - Magnetic FieldsThe 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 side mmf to

19、and equivalent rotor side mmf.90lagArmature current mmf FarResultant mmf FrField mmf FfRotor RotationStatorRotorNS22Pole 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 field wind

20、ing only23Pole 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 center24Pole Face Design Magnetic FieldCq1: If peak of the fundamental is unity, then Cq1

21、 is peak of acutal waveform. Note that Wieseman calls this Aq1Armature mmf Sine wave whose axis coincides with the gap between poles25List of Pole Constants Cd1 Ratio of the fundamental of the airgap flux produced by the direct axis armature current to that which would be produced with a uniform gap

22、 equal to the effective gap at the pole centerCq1 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 centerC1 The ratio of the fundamental to the actual maximum value o

23、f the field form when excited by the field only (no-load)Cm 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/C1K Flux distribution coefficie

24、nt; the ratio of the area of the actual no load flux wave to the area of its fundamental26Pole ConstantsWhat 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 note a

25、re identical to graphs that first appeared in a 1927 AIEE paper titled Graphical Determination of Magnetic Fields by Robert Wieseman. 27Pole 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 intensive nat

26、ure 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 outsi

27、de 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.28Pole Face Design Magnetic FieldsDetermination of K29Pole Face Design Magnetic FieldsDetermination of C130Pole Face Design Magnetic FieldsDetermin

28、ation of Cq131Pole Face Design Magnetic FieldsDetermination of Cd132Pole 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 f

29、ield flux waveform.33Reactance 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 Constant34T = Common Reactance Factorm = #

30、 phasesL = Stator Core Lengthf= frequencyZ = Series Conductors per PhaseKw = Winding factor StatorP = # PolesReactance Calculations35a = Permeance FactorD = Diameter of Stator BoreP = # PolesKg = Carters Gap Coefficeintgmin = Minimum air gap at center of poleReactance Calculations36Armature Leakage

31、Reactance is determined using a methodology identical to the induction machine.Synchronous ReactancesXd = Xl + XadXq = Xl + Xaq37Excitation Calculations1. Calculate total Magnetic Flux2. Convert armature magnetic Flux to Field Equivalent3. Calculate Eint by adding armature resistance and leakage rea

32、ctance 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 the results of 4 and 5 calcula

33、te 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 saturation mmf11. Total direct axis e

34、xcitation is the sum of 8 and 938Excitation CalculationsStep One: Total fundamental flux per poleEph:Phase voltage at stator terminalsKp:Pitch FactorKd:Distribution FactorFreq:Frequency NSPC:Armature series turns per phase per circuit39Excitation Calculation Step 2: Calculate total flux per pole on

35、open circuitK relates fundamental flux per pole to total flux per pole using factors. 40Excitation 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 phasor diagram as Eint.Step 3a: Calculate the stator

36、leakage reactance using same formulas derived for the Induction motor stator.41Field Excitation Cont.Portion of the previous phasor diagram is redrawnFrom diagram it is evident that:IaxlIaraIaVtIdEintIq42IaVtEintIqK1EintIdxadIqxaqIaxaqIaraIaxlIdK1Eintsin()43Field Excitation Cont.Step 4: Calculate th

37、e ampere turns needed to magnetize the air gap at rated voltage: Fg. Same gapfactor used from induction motor theory (i.e. Carters coefficient)44Field Excitation Cont.Step 5: Calculate the stator core + stator teeth ampere turns at the value of flux corresponding to Eint.Ac: Area of stator coreAt: A

38、rea of stator teethBCmax: maximum value of flux density in stator coreBTmax: maximum value of flux density in stator teeth45Field Excitation Cont.Using 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.46Field Excitation Cont.Step 6: Calculate the component mmf in the direct axis corresponding to K1eintStep 6a: Calculate saturation factor K1 from B-H curve for electrical stee