1、Reactance of Induction MachinesFundamental ConceptsL = Instantaneous Inductance of Coil (Henries)e = Instantaneous value of voltage induced in the coil by the change of current (Volts)di/dt = Instantaneous value of the rate of change of the current which flows through the coilFaradays Lawepath = ind

2、uced voltage around an enclosed pathE = Vector of Electric Field around the pathdl = Vector of enclosed pathd/dt = Rate of Change of Flux enclosed by the pathA more useful form of Faradays law:whereIs the sum of all fluxes x linking corresponding turns nx of the coilCombining EquationsReactance of a

3、n inductive circuitX=2fL (Ohms)f=freqency of voltage across inductance (Hz)orAmperes Circuit LawH = Vector of Magnetic Field Intensity (Amp/meter)dl = Vector along the length of the pathJ = Vector field of current density enclosed by path dA = area enclosed by the pathTo apply Amperes Circuit law, c

4、onsider the simple magnetic circuit of a torus with a non-ferromagnetic coreHr(2r) = Ni B=uruoHB = Vector Quantity of Magnetic Flux Density (Tesla)uo = constant relating Flux Density to Flux Intensity in free space or non-ferrous materials = 4x10-7 (Tm/A)ur = proportionality constant for ferromagnet

5、ic materials (unit-less)Rearranging:Also:Amp/meterThe integration of flux density over a unit area gives total flux . For a torroid =BA WebersCombining Equations for L and , the inductance of the coil is shown to be determined entirely by the dimensions of the coil and independent of the current in

6、the wire. The reactance of the torroid is thenFor Ferromagnetic Materials, a value of r o is read from a Saturation CurveEffect of Saturation on InductanceEquivalent Circuit of Induction Motor R1 Primary resistanceX1 Leakage reactance of primary windingXM Magnetizing Reactancegh = Resistance path to

7、 account for core lossX2 Secondary leakage reactanceR2/s Resistance of squirrel cage windingThere is no standard method for reactance calculations as simplifying assumptions are made at many steps in their calculation. The formulas to follow have many simplifying assumptions (such as no saturation)

8、that I will try to point out as we go along. However, the dependency of reactance to physical parameters is the same for all calculation methods.Magnetizing Reactance XMT=Reactance FactorD= Stator Bore Diameter (I.D.)L=Core Length in InchesKW = Stator Winding Factor Kd x Kp g = single air gap inches

9、Kg = Carters AirGap CoefficientP = # PolesT = Reactance Factor = Wheref= frequencym=# of phasesZ = # stator conductors in series per phaseThe resultant voltage in the air gap available to create the rotating magnetic field is less than the peak voltage of the sine wave at the motor terminals Because

10、 of 2 factors: Kp: Accounts for the pitch of the winding Kd: Accounts for the distribution of the windingTo account for the slot openings of the stator and rotor surface, a gap factor is used as a multiplier to create an “effective air gap. This factor is called the “Carters Coefficient Kg KW=KpKdTo

11、tal Leakage Reactance X1 +X2The Primary slot reactanceThe Secondary slot reactanceThe zig-zag reactanceThe belt-leakage reactanceThe Coil End Leakage ReactanceThe peripheral leakageBecause relatively few stators or rotors are skewed, the leakage reactance caused by either a skewed stator or skewed r

12、otor will not be considered. It is sufficient to know that an additional leakage reactance exists for skewed motors.Reactance 3 and 4 are sometimes combined and called Differential LeakageStator Slot Leakage ReactanceT = Reactance FactorL = Length of stator coreS = # stator SlotsKs = Factor to accou

13、nt for coils of different phases being in the same slot1 = Permeance Factor of Stator Slot Based on slot geometry For Stator:For Rotor:Rotor Slot Leakage ReactanceXSR = Rotor Slot Leakage Reactance Referred to the StatorT = Reactance FactorL = Length of rotor coreR = # rotor SlotsKw = stator winding

14、 factor2 = Permeance Factor of rotor slot Based on slot geometry Rotor Leakage Reactance Cont.The value of XSR is effected by the operating condition of the motor. The value calculated is for “running conditions when rotor frequency is 1-2 Hz and rotor current is relatively low. During running condi

15、tions, the rotor bar current can be though of as distributing evenly throughout the bar. During locked rotor, the rotor frequency is 60 Hz and due to the “Deep Bar Effect the current crowds toward the top of the bar. The current crowding along with the saturation effects of the high inrush current c

16、ause the starting value of XSR to be significantly reduced. Differential LeakageBesides the fundamental air gap flux wave form there are harmonic wave forms that are multiples of the number of poles and rotate at sub-multiples of synchronous speed. These harmonics induce voltages in the windings tha

17、t produced them and therefore add to the reactance of the winding.The differential leakage is sometimes broke into two components: The zig-zag and phase belt leakage. The following notation combines the belt leakage with the zig-zag.XM = Magnetizing ReactanceS= # Stator SlotsP = # PolesR + # Rotor S

18、lotsKgS Carters airgap coefficient for the statorKgR Carters airgap coefficient for the rotorKS Short Pitch Correction FactorKW Stator Winding Correction FactorCoil end Leakage ReactanceStatorRotorAS = Mean stator overhangAR = Mean Rotor OverhangAS = Mean Stator overhangL.M.H.T. = Length of mean hal

19、f turn stator coilL = Stator core length = per unit coil pitch = Pole PitchKS = Short pitch correction factorPeripheral Air Gap LeakageLines of flux that enter the air gap but do not link the rotor but instead flow peripherally back into the stator. If we imagine a induction motor with the rotor rem

20、oved, all the flux in the stator bore will be of this air leakage character.D = Stator I.D.g = Single air gapPeripheral reactance is negligible for induction machines with small air gaps but can be relevant on high speed synchronous motors with very large air gaps.Consider a 4 pole induction motor,

21、800 frame with D=32.75 and g=.175: XP/XM = .00091Consider a 4 pole synchronous motor, 800 frame with D=33.5 and g=1.0: XP/XM = .028 Summary of Induction Motor Leakage ReactanceXSS = Stator slot leakage reactanceXRS = Rotor slot leakage reactanceXZZ = differential leakage reactance (Zig-Zag + Belt Le

22、akage)XES = Stator End Winding Leakage ReactanceXES = Rotor End Winding Leakage ReactanceX2 = XSR +XZZ + XERX1 = XSS +XZZ + XESLeakage Reactance Summary Continued The Leakage reactance varies as the square of effective conductors or turns per phase. If the number of stator slots and the number of parallel circuits are not changed, it varies as the square of the number of turns per coilIn Order to decrease the leakage reactance do one or more of the following: Decrease the stator bo