Chapter5rev[1]

Chapter5rev[1] - 1 5.3 VOLTAGE EQUATIONS AND WINDING...

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ECE 321, © Oleg Wasynczuk, December 3, 2009 1 5.3 VOLTAGE EQUATIONS AND WINDING INDUCTANCES The voltage equations for the induction machine depicted in Fig. 5.3-1 may be expressed as The voltage equations may be expressed (5.3-1) (5.3-2) (5.3-3) (5.3-4) where is the resistance of the stator windings and is the resistance of the rotor windings. It is convenient, for future derivations, to write (5.3-1) through (5.3-4) in matrix form as (5.3-5) (5.3-6) FIGURE 5.3-1 A two-pole two-phase symmetrical induction machine. v as r s i p λ + = v bs r s i p λ + = v ar r r i p λ + = v br r r i p λ + = r s r r v abs r s i p λ + = v abr r r i p λ + =

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2 Electromechanical Motion Devices ECE 321, © Oleg Wasynczuk, December 3, 2009 where is the shorthand notation for the operator . We will assume that the magnetic system is linear, whereupon the flux linkages may be expressed as linear functions of inductances and currents. In particular, we can write (5.3-7) (5.3-8) Symbolically, (5.3-9) (5.3-10) Our job now is to express the self- and mutual- inductances of all windings. As in the case of a two-winding transformer, the self-inductance of each winding is made up of a leakage inductance caused by the flux which fails to cross the air gap and a magnetizing inductance caused by the flux that traverses the air gap and circulates through the stator and rotor steel. For symmetrical, identical stator windings, the self-inductances and are equal and will be denoted where (5.3-11) In (5.3-11), is the leakage inductance and the magnetizing induc- tance. The machine is typically designed to minimize the leakage inductance; it generally makes up approximately 10 percent of the self-inductance. The self-inductance of the symmetrical rotor windings may be expressed similarly, (5.3-12) The magnetizing inductances and may be expressed in terms of turns and reluctance. In particular, p dd t λ as λ bs L asas L asbs L bsas L bsbs i i L asar L asbr L bsar L bsbr i ar i br + = λ λ L aras L arbs L bras L brbs i i L arar L arbr L brar L brbr i i + = λ abs L s i L sr i abr + = λ L rs i L r i + = L L L ss L L ls L ms + = L L L rr L lr L mr + = L L
ECE 321, © Oleg Wasynczuk, December 3, 2009 3 (5.3-13) (5.3-14) The magnetizing reluctance is due primarily to the air gap and, since the winding is assumed to be an equivalent sinusoidally distributed winding, per- haps should be considered an equivalent magnetizing reluctance. Never- theless, expressions for , , and thus , as defined in (5.3-13) and (5.3-14), may be derived. In particular, it can be shown that [1] (5.3-15) (5.3-16) where is the permeability of free space, is the mean radius of the air gap, is the axial length of the air gap (rotor), and is the radial length of the air gap. One must perform a rather involved and lengthy derivation to obtain (5.3- 15) and (5.3-16). We will not conduct this derivation; instead, the use of an equivalent magnetizing reluctance without evaluation will be sufficient for our purposes.

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Chapter5rev[1] - 1 5.3 VOLTAGE EQUATIONS AND WINDING...

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