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MIT6_685F13_chapter5

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MIT6_685F13_chapter5
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  Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.685 Electric MachineryClass Notes 5: Winding Inductances  c 2005 James L. Kirtley Jr. 1 Introduction The purpose of this document is to show how the inductances of windings in round- rotor machineswith narrow air gaps may be calculated. We deal only with the idealized air- gap magnetic fields,and do not consider slot, end winding, peripheral or skew reactances. We do, however, considerthe space harmonics of winding magneto-motive force (MMF). 2 Description of Stators Back IronSlotsTeethSlotDepression Figure 1: Stator Cross-SectionFigure 1 shows a cartoon view of an axial cross-section of a twelve-slot stator. Actually, whatis shown is the shape of a thin sheet of steel, or  lamination   that is used to make up the magneticcircuit. The iron is made of thin sheets to control eddy current losses. Thickness varies accordingto freuqency of operation, but in machines for 60 Hz (the vast bulk of machines made for industrial1  use), lamination thickness is typically .014” (.355 mm). These are stacked to make the magneticcircuit of the appropriate length. Windings are carried in the slots of this structure.Figure 1 shows trapezoidal slots with teeth of approximately uniform cross-section over most of their length but wider extent near the air-gap. The tooth ends, in combination with the relativelynarrow slot depression region, help control certain parasitic losses in the rotor of many machinesby improving uniformity of the air-gap fields, increase the air-gap permeance and help hold thewindings in the slots. It should be noted that large machines, with what are called “form wound”coils, have straight-sided rectangular slots and consequently teeth of non-uniform cross-section.The description that follows will hold for both types of machine. 1 2 3 4 5 6 7 8 9 10 11 12A A A’ A’B B B’ B’C’ C’ C C Figure 2: Full-Pitched WindingTo simplify the discussion, imagine the slot/tooth region to be “straightened out” as shown inFigure 2. This shows a three-phase, two-pole winding in the twelve slots. Such a winding wouldhave two slots per pole per phase. One of the two coils of phase A would be wound in slots 1 and 7(six slots apart). 1 2 3 4 5 6 7 8 9 10 11 12A A C’ C’ B B A’ A’ C C B’ B’A C’ C’ B B A’ A’ C C B’ B’ A Figure 3: Five-Sixths-Pitched WindingMachines are seldom wound as shown in Figure 2 for a variety of reasons. It is usually advanta-geous in reducing the length of the end turns and to reducing space harmonic effects in the machine(usually bad effects!) to wind the machine with “short-pitched” windings as shown in Figure 3.Each phase in this case consists of four coils (two per slot). The four coils of Phase A would spanbetween slots 1 and 6, slots 2 and 7, slots 7 and 12 and slots 8 and 1. Each of these coil spans isfive slots, so this choice of winding pattern is referred to as “Five-Sixths” pitch.So this cartoon-figure machine stator (which could represent either a synchronous or inductionmotor or generator) has both  breadth   because there are more than one slots per pole per phase,and it may have the need for accounting for winding  pitch  . What follows in this note is a simpleprotocol for estimating the important air-gap fields and inductances.2  3 Winding MMF To start, consider the MMF of a full- pitch, concentrated winding as shown in schematic form inFigure 4. Assuming that the winding has a total of   N   turns over  p  pole- pairs, and is carryingcurrent  I   the MMF is: ∞ F   = −   4  NI  sin npθ  (1) nπ  2  pn  = 1 nodd This distribution is shown, as a function of angle  θ  in Figure 5.This leads directly to magnetic flux density in the air- gap: ∞ µ 0  4  NI B r  = −   sin npθ  (2) g nπ  2  pn  = 1 nodd Note that a real winding, which will most likely not be full- pitched and concentrated, will have a winding factor   which is the product of pitch and breadth factors, to be discussed later. µµ NIpRgMagneticCircuit: Stator RotorAir-Gaprz θ Figure 4: Primitive Geometry ProblemNow, suppose that there is a polyphase winding, consisting of more than one phase (we will usethree phases), driven with one of two types of current. The first of these is  balanced  , current: I  a  =  I   cos( ωt )2 πI  b  =  I   cos( ωt −  )32 πI  c  =  I   cos( ωt + ) (3)3Conversely, we might consider  Zero Sequence   currents: I  a  =  I  b  =  I  c  =  I   cos ωt  (4)3  π 2 p π p π 32 p π p2F( ) θθ NIp Figure 5: Air-Gap MMFThen it is possible to express magnetic flux density for the two distinct cases. For the  balanced  case: ∞ B r  =   B rn  sin( npθ ∓ ωt ) (5) n =1 where ã  The upper sign holds for  n  = 1 , 7 ,... ã  The lower sign holds for  n  = 5 , 11 ,... ã  all other terms are zeroand3 µ 0  4  NI B rn  = (6)2  g nπ  2  p The zero- sequence case is simpler: it is nonzero only for the  triplen   harmonics: ∞ B r  =   µ 0  4  NI   3(sin( npθ − ωt ) + sin( npθ  + ωt )) (7) g nπ  2  p  2 n =3 , 9 ,... Next, consider the flux from a winding on the rotor: that will have the same form as the fluxproduced by a single armature winding, but will be referred to the rotor position: ∞ 4 B  =   µ 0  NI  sin npθ ′ rf   (8) g nπ  2  pn  = 1 nodd which is, substituting  θ ′ =  θ −  ωt p  , ∞ B rf   =   µ 0  4  NI  sin n (  pθ − ωt ) (9) g nπ  2  pn  = 1 nodd The next step here is to find the flux linked if we have some air- gap flux density of the form: ∞ B r  = n   B rn  sin( npθ ± ωt ) (10) =1 4

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Jul 23, 2017
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