Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Electric Machinery

Transcription

1 Massachusetts Institute of Technoloy Deartment of Electrical Enineerin and Comuter Science Electric Machinery Class Notes 5: Windin Inductances Setember 5, 005 c 005 James L. Kirtley Jr. 1 Introduction The urose of this document is to show how the inductances of windins in round- rotor machines with narrow air as may be calculated. We deal only with the idealized air- a manetic fields, and do not consider slot, end windin, eriheral or skew reactances. We do, however, consider the sace harmonics of windin maneto-motive force (MMF). Descrition of Stators Back Iron Slots Slot Deression Teeth Fiure 1: Stator Cross-Section Fiure 1 shows a cartoon view of an axial cross-section of a twelve-slot stator. Actually, what is shown is the shae of a thin sheet of steel, or lamination that is used to make u the manetic circuit. The iron is made of thin sheets to control eddy current losses. Thickness varies accordin to freuqency of oeration, but in machines for 60 Hz (the vast bulk of machines made for industrial 1

2 use), lamination thickness is tyically.014 (.355 mm). These are stacked to make the manetic circuit of the aroriate lenth. Windins are carried in the slots of this structure. Fiure 1 shows traezoidal slots with teeth of aroximately uniform cross-section over most of their lenth but wider extent near the air-a. The tooth ends, in combination with the relatively narrow slot deression reion, hel control certain arasitic losses in the rotor of many machines by imrovin uniformity of the air-a fields, increase the air-a ermeance and hel hold the windins in the slots. It should be noted that lare machines, with what are called form wound coils, have straiht-sided rectanular slots and consequently teeth of non-uniform cross-section. The descrition that follows will hold for both tyes of machine. A A C C B B A A C C B B Fiure : Full-Pitched Windin To simlify the discussion, imaine the slot/tooth reion to be straihtened out as shown in Fiure. This shows a three-hase, two-ole windin in the twelve slots. Such a windin would have two slots er ole er hase. One of the two coils of hase A would be wound in slots 1 and 7 (six slots aart). A A C C B B A A C C B B A C C B B A A C C B B A Fiure 3: Five-Sixths-Pitched Windin Machines are seldom wound as shown in Fiure for a variety of reasons. It is usually advantaeous in reducin the lenth of the end turns and to reducin sace harmonic effects in the machine (usually bad effects!) to wind the machine with short-itched windins as shown in Fiure 3. Each hase in this case consists of four coils (two er slot). The four coils of Phase A would san between slots 1 and 6, slots and 7, slots 7 and 1 and slots 8 and 1. Each of these coil sans is five slots, so this choice of windin attern is referred to as Five-Sixths itch. So this cartoon-fiure machine stator (which could reresent either a synchronous or induction motor or enerator) has both breadth because there are more than one slots er ole er hase, and it may have the need for accountin for windin itch. What follows in this note is a simle rotocol for estimatin the imortant air-a fields and inductances.

3 3 Windin MMF To start, consider the MMF of a full- itch, concentrated windin as shown in schematic form in Fiure 4. Assumin that the windin has a total of N turns over ole- airs, and is carryin current I the MMF is: F = This distribution is shown, as a function of anle θ in Fiure 5. This leads directly to manetic flux density in the air- a: sin nθ (1) n B r = µ 0 sin nθ () n Note that a real windin, which will most likely not be full- itched and concentrated, will have a windin factor which is the roduct of itch and breadth factors, to be discussed later. Manetic Circuit: Stator Rotor µ NI θ R z r Air-Ga µ Fiure 4: Primitive Geometry Problem Now, suose that there is a olyhase windin, consistin of more than one hase (we will use three hases), driven with one of two tyes of current. The first of these is balanced, current: I a = I cos(ωt) I b = I cos(ωt 3 ) I c = I cos(ωt + 3 ) (3) Conversely, we miht consider Zero Sequence currents: I a = I b = I c = I cos ωt (4) 3

4 NI F( θ ) 3 θ Fiure 5: Air-Ga MMF Then it is ossible to exress manetic flux density for the two distinct cases. For the balanced case: B r = B rn sin(nθ ωt) (5) where The uer sin holds for,7,... The lower sin holds for n = 5, 11,... all other terms are zero and B rn = 3 µ 0 n The zero- sequence case is simler: it is nonzero only for the trilen harmonics: B r = n=3,9,... µ 0 n=1 n (6) 3 (sin(nθ ωt) + sin(nθ + ωt)) (7) Next, consider the flux from a windin on the rotor: that will have the same form as the flux roduced by a sinle armature windin, but will be referred to the rotor osition: B rf = µ 0 n sin nθ (8) which is, substitutin θ = θ ωt, B rf = µ 0 sin n(θ ωt) (9) n The next ste here is to find the flux linked if we have some air- a flux density of the form: B r = B rn sin(nθ ± ωt) (10) n=1 4

5 Now, it is ossible to calculate flux linked by a sinle-turn, full-itched windin by: and, usin (10), this is: Φ = Φ = Rl 0 n=1 B r Rldθ (11) B rn n cos(ωt) (1) This allows us to comute self- and mutual- inductances, since windin flux is: λ = NΦ (13) The end of this is a set of exressions for various inductances. It should be noted that, in the real world, most windins are not full-itched nor concentrated. Fortunately, these shortcomins can be accommodated by the use of windin factors. The simlest and erhas best definition of a windin factor is the ratio of flux linked by an actual windin to flux that would have been linked by a full- itch, concentrated windin with the same number of turns. That is: k w = λ actual (14) λ full itch It is relatively easy to show, usin recirocity aruments, that the windin factors are also the ratio of effective MMF roduced by an actual windin to the MMF that would have been roduced by the same windin were it to be full- itched and concentrated. The arument oes as follows: mutual inductance between any air of windins is recirocal. That is, if the windins are desinated one and two, the mutual inductance is flux induced in windin one by current in windin two, and it is also flux induced in windin two by current in windin one. Since each windin has a windin factor that influences its linkin flux, and since the mutual inductance must be recirocal, the same windin factor must influence the MMF roduced by the windin. The windin factors are often exressed for each sace harmonic, althouh sometimes when a windin factor is referred to without reference to a harmonic number, what is meant is the sace factor for the sace fundamental. Two windin factors are commonly secified for ordinary, reular windins. These are usually called itch and breadth factors, reflectin the fact that often windins are not full itched, which means that individual turns do not san a full electrical radians and that the windins occuy a rane or breadth of slots within a hase belt. The breadth factors are ratios of flux linked by a iven windin to the flux that would be linked by that windin were it full- itched and concentrated. These two windin factors are discussed in a little more detail below. What is interestin to note, althouh we do not rove it here, is that the windin factor of any iven windin is the roduct of the itch and breadth factors: k w = k k b (15) With windin factors as defined by (14) and the sections below, it is ossible to define windin inductances. For examle, the synchronous inductance of a windin will be the aarent inductance of one hase when the olyhase windin is driven by a balanced set of currents as in (3). This is, aroximately: 3 4 µ 0 N Rlkwn L d = n (16) n=1,5,7,... 5

6 This exression is aroximate because it inores the asynchronous interactions between hiher order harmonics and the rotor of the machine. These are beyond the scoe of this note. Zero- sequence inductance is the ratio of flux to current if a windin is excited by zero sequence currents, as in (4): L 0 = 3 4 µ 0 N Rlkwn n (17) n=3,9,... And then mutual inductance, as between a field windin (f) and an armature windin (a), is: M(φ) = 4 µ 0 N f N a k fn k an Rl n cos(nφ) (18) 4 Windin Factors Now we turn our attention to comutin the windin factors for simle, reular windin atterns. We do not rove but only state that the windin factor can, for reular windin atterns, be exressed as the roduct of a itch factor and a breadth factor, each of which can be estimated searately. 4.1 Pitch Factor α θ z r Fiure 6: Short-Pitched Coils Pitch factor is found by considerin the flux linked by a less- than- full itched windin. Consider the situation in which radial manetic flux density is: A windin with itch α will link flux (see Fiure 6: λ = Nl B r = B n sin(nθ ωt) (19) + α α B n sin(nθ ωt)rdθ (0) 6

7 Pitch α refers to the anular dislacement between sides of the coil, exressed in electrical radians. For a full- itch coil α =. The flux linked is: λ = NlRB n n Usin the definition (14), the itch factor is seen to be: sin( n )sin(nα ) (1) k n = sin nα () 4. Breadth Factor Now for breadth factor. This describes the fact that a windin may consist of a number of coils, each linkin flux slihtly out of hase with the others. A reular windin will have a number (say m) coil elements, searated by electrical anle γ. (See Fiure 7 γ z θ r Fiure 7: Distributed Coils A full- itch coil with one side at anle ξ will, in the resence of manetic flux density as described by (19), link flux: λ = Nl ξ B ξ n sin(nθ ωt)rdθ (3) This is readily evaluated to be: λ = NlRB ( n Re e j(ωt nξ)) (4) n where in (4), comlex number notation has been used for convenience in carryin out the rest of this derivation. What haens here is that the coils link fluxes that differ in hase, so the addition of flux is as shown in vector form in Fiure 8. 7

8 Individual Flux Linkaes Total Flux Linkae Fiure 8: Vector Flux Addition Now: if the windin is distributed into m sets of slots and the slots are evenly saced, the anular osition of each slot will be: ξ i = iγ m 1 γ (5) and the number of turns in each slot will be N m, so that actual flux linked will be: The breadth factor is then simly: Note that (7) can be written as: λ = NlRB n n k b = 1 m k b = m 1 1 m m 1 m 1 ejnγ m ( ) Re e j(ωt nξ i) (6) m 1 jn(iγ γ) e (7) m e jniγ (8) Now, focus on that sum. We know that any coverin eometric sum has a simle sum: and that a truncated sum is: Then the sum in (8) can be written as: m 1 m 1 x i = 1 1 x = i=m e jniγ = (1 e jnmγ) (9) (30) e jniγ = 1 ejnmγ 1 e jnγ (31) Now, insertin the results of (31) into (8), and usin the definitions for sine, the breadth factor is found: nmγ sin k bn = m sin nγ (3) 8

9 4.3 Alternate Derivation of Breadth Factor Most textbooks, if they bother to rove the Breadth Factor, use a eometric roof as shown in Fiure 9. C A B γ m γ O Fiure 9: Alternate Proof of Breadth Factor The short vectors (e.. AC) reresent the voltaes induced in individual coils. In fact, what is shown in this fiure is the same as is shown in Fiure 8, but sread out to show the actual addition. Now, note that if each of the vectors is bisected by a line sement at riht anles, all of those line sements meet at oint O. The line sement that includes OB is one of these. Line sements that run from O to the ends of the vectors will have an anle γ from the bisectors of the vectors. Similarly, the line sement OA has an anle of mγ with resect to the bisector of the resultant voltae vector. Now, if we note F 1 as the lenth of each of the individual coil voltae vectors and F as the lenth of the resultant sum, the lenth of half of the bisector is: but then Then the resultant vector is: AB = F = OAsin mγ 1 AC = 1 F 1 = OAsin γ F = AB = mf 1 sin m γ m sin γ (33) (34) (35) 9