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Reactance of Induction Machines,Fundamental Concepts,L = 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 coil,Faradays Law,epath = induced voltage around an enclosed path E = Vector of Electric Field around the path dl = Vector of enclosed path d/dt = Rate of Change of Flux enclosed by the path,A more useful form of Faradays law:,where,Is the sum of all fluxes x linking corresponding turns nx of the coil,Combining Equations,Reactance of an inductive circuit X=2fL (Ohms) f=freqency of voltage across inductance (Hz),or,Amperes Circuit Law,H = Vector of Magnetic Field Intensity (Amp/meter) dl = Vector along the length of the path J = Vector field of current density enclosed by path dA = area enclosed by the path,The integration of flux density over a unit area gives total flux . For a torroid =BA Webers,Combining 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 the wire.,The reactance of the torroid is then,For Ferromagnetic Materials, a value of r o is read from a Saturation Curve,Effect of Saturation on Inductance,Equivalent Circuit of Induction Motor,R1 Primary resistance X1 Leakage reactance of primary winding XM Magnetizing Reactance gh = Resistance path to account for core loss X2 Secondary leakage reactance R2/s Resistance of squirrel cage winding,There 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) 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 XM,T=Reactance Factor D= Stator Bore Diameter (I.D.) L=Core Length in Inches KW = Stator Winding Factor Kd x Kp g = single air gap inches Kg = Carters AirGap Coefficient P = # Poles,T = Reactance Factor =,Where f= frequency m=# of phases Z = # stator conductors in series per phase,The 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 of 2 factors: Kp: Accounts for the pitch of the winding Kd: Accounts for the distribution of the winding,To 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=KpKd,Total Leakage Reactance X1 +X2,The Primary slot reactance The Secondary slot reactance The zig-zag reactance The belt-leakage reactance The Coil End Leakage Reactance The peripheral leakage,Because relatively few stators or rotors are skewed, the leakage reactance caused by either a skewed stator or skewed rotor 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 Leakage,Stator Slot Leakage Reactance,T = Reactance Factor L = Length of stator core S = # stator Slots Ks = Factor to account for coils of different phases being in the same slot 1 = Permeance Factor of Stator Slot Based on slot geometry,For Stator:,For Rotor:,Rotor Slot Leakage Reactance,XSR = Rotor Slot Leakage Reactance Referred to the Stator T = Reactance Factor L = Length of rotor core R = # rotor Slots Kw = stator winding factor 2 = 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 conditions, 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 cause the starting value of XSR to be significantly reduced.,Differential Leakage,Besides 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 that 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 Reactance S= # Stator Slots P = # Poles R + # Rotor Slots KgS Carters airgap coefficient for the stator KgR Carters airgap coefficient for the rotor KS Short Pitch Correction Factor KW Stator Winding Correction Factor,Coil end Leakage Reactance,Stator,Rotor,AS = Mean stator overhang AR = Mean Rotor Overhang,AS = Mean Stator overhang L.M.H.T. = Length of mean half turn stator coil L = Stator core length = per unit coil pitch = Pole Pitch KS = Short pitch correction factor,Peripheral Air Gap Leakage,Lines 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 removed, all the flux in the stator bore will be of this air leakage character.,D = Stator I.D. g = Single air gap,Peripheral 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, 800 frame with D=32.75 and g=.175: XP/XM = .00091 Consider a 4 pole synchronous motor, 800 frame with D=33.5 and g=1.0: XP/XM = .028,Summary of Induction Motor Leakage Reactance,XSS = Stator slot leakage reactance XRS = Rotor slot leakage reactance XZZ = differential leakage reactance (Zig-Zag + Belt Leakage) XES = Stator End Winding Leakage Reactance XES = Rotor End Winding Leakage Reactance,X2 = XSR +XZZ + XER,X1 = XSS +XZZ + XES,Leakage 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 coil In Order to decrease the leakage reactance do one or more of the following: Decrease the stator bore or core length. Increase the air gap I