Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek
53 2.1. INTRODUCTION Many ac machines are designed based on the concept of a distributed winding. In these machines, the goal is to establish a continuously rotating set of north and south poles on the stator (the stationary part of the machine), which interact with an equal number of north and south poles on the rotor (the rotating part of the machine), to produce uniform torque. There are several concepts that are needed to study this type of electric machinery. These concepts include distributed windings, winding functions, rotating MMF waves, and inductances and resistances of distributed windings. These principles are presented in this chapter and used to develop the voltage and flux-linkage equations of synchronous and induction machines. The voltage and flux linkage equations for permanent magnet ac machines, which are also considered in this text, will be set forth in Chapter 4 and derived in Chapter 15. In each case, it will be shown that the flux- linkage equations of these machines are rather complicated because they contain rotor position-dependent terms. Recall from Chapter 1 that rotor position dependence is necessary if energy conversion is to take place. In Chapter 3, we will see that the com- plexity of the flux-linkage equations can be greatly reduced by introducing a change of variables that eliminates the rotor position-dependent terms. Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk, Scott Sudhoff, and Steven Pekarek. © 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc. DISTRIBUTED WINDINGS IN AC MACHINERY 2 54 DISTRIBUTED WINDINGS IN AC MACHINERY 2.2. DESCRIBING DISTRIBUTED WINDINGS A photograph of a stator of a 3.7-kW 1800-rpm induction motor is shown in Figure 2.2-1, where the stator core can be seen inside the stator housing. The core includes the stator slots in between the stator teeth. The slots are filled with slot conductors which, along with the end turns, form complete coils. The windings of the machine are termed distributed because they are not wound as simple coils, but are rather wound in a spatially distributed fashion. To begin our development, consider Figure 2.2-2, which depicts a generic electrical machine. The stationary stator and rotating rotor are labeled, but details such as the stator slots, windings, and rotor construction are omitted. The stator reference axis may be considered to be mechanically attached to the stator, and the rotor reference axis to Figure 2.2-1. Distributed winding stator. Figure 2.2-2. Definition of position measurements. f sm q rm Stator Reference Axis Arbitrary Position Stator Rotor Rotor Reference Axis f rm DESCRIBING DISTRIBUTED WINDINGS 55 the rotor. Angles defined in Figure 2.2-2 include position measured relative to the stator, denoted by ϕ sm , position measured relative to the rotor, denoted by ϕ rm , and the position of the rotor relative to the stator, denoted by θ rm . The mechanical rotor speed is the time derivative of θ rm and is denoted by ω rm . The position of a given feature can be described using either ϕ sm or ϕ rm ; however, if we are describing the same feature using both of these quantities, then these two measures of angular position are related by θ φ φ rm rm sm + = (2.2-1) Much of our analysis may be expressed either in terms of ϕ sm or ϕ rm . As such, we will use ϕ m as a generic symbol to stand for either quantity, as appropriate. The goal of a distributed winding is to create a set of uniformly rotating poles on the stator that interact with an equal number of poles on the rotor. The number of poles on the stator will be designated P, and must be an even number. The number of poles largely determines the relationship between the rotor speed and the ac electrical fre- quency. Figure 2.2-3 illustrates the operation of 2-, 4-, and 6-pole machines. Therein N s , S s , N r , and S r denote north stator, south stator, north rotor, and south rotor poles, respectively. A north pole is where positive flux leaves a magnetic material and a south pole is where flux enters a magnetic material. Electromagnetic torque production results from the interaction between the stator and rotor poles. When analyzing machines with more than two poles, it is convenient to define equivalent “electrical” angles of the positions and speed. In particular, define φ φ s sm P / 2 (2.2-2) φ φ r rm P / 2 (2.2-3) θ θ r rm P / 2 (2.2-4) ω ω r rm P / 2 (2.2-5) In terms of electrical position, (2.2-1) becomes Figure 2.2-3. P-pole machines. P = 2 P = 4 N s N s N s S s S s S s P = 6 S r S r N r N r N r S r S s S s S r S r N s N s N r N r N r N s S s S r 56 DISTRIBUTED WINDINGS IN AC MACHINERY θ φ φ r r s + = . (2.2-6) Finally, it is also useful to define a generic position as φ φ P m / 2 (2.2-7) The reason for the introduction of these electrical angles is that it will allow our analysis to be expressed so that all machines mathematically appear to be two-pole machines, thereby providing considerable simplification. Discrete Description of Distributed Windings Distributed windings, such as those shown in Figure 2.2-1, may be described using either a discrete or continuous formulation. The discrete description is based on the number of conductors in each slot; the continuous description is an abstraction based on an ideal distribution. A continuously distributed winding is desirable in order to achieve uniform torque. However, the conductors that make up the winding are not placed continuously around the stator, but are rather placed into slots in the machine’s stator and rotor structures, thereby leaving room for the stator and rotor teeth, which are needed to conduct magnetic flux. Thus, a discrete winding distribution is used to approximate a continuous ideal winding. In reality, the situation is more subtle than this. Since the slots and conductors have physical size, all distributions are continuous when viewed with sufficient resolution. Thus, the primary difference between these two descriptions is one of how we describe the winding mathematically. We will find that both descriptions have advantages in different situations, and so we will consider both. Figure 2.2-4 illustrates the stator of a machine in which the stator windings are located in eight slots. The notation N as,i in Figure 2.2-4 indicates the number of conduc- tors in the i’th slot of the “as” stator winding. These conductors are shown as open circles, as conductors may be positive (coming out of the page or towards the front of the machine) or negative (going into the page or toward the back of the machine). Figure 2.2-4. Slot structure. N as,1 N as,2 N as,8 f sm N as,3 N as,4 N as,5 N as,6 N as,7 DESCRIBING DISTRIBUTED WINDINGS 57 Generalizing this notation, N x,i is the number of conductors in slot i of winding (or phase) x coming out of the page (or towards the front of the machine). In this example, x = “as.” Often, a slot will contain conductors from multiple windings (phases). It is important to note that N x,i is a signed quantity—and that half the N as,i values will be negative since for every conductor that comes out of the page, a conductor goes into the page. The center of the i’th slot and i’th tooth are located at φ π φ ys i y ys i S , , ( ) /= − +2 2 1 (2.2-8) φ π φ yt i y ys i S , , ( ) /= − +2 3 1 (2.2-9) respectively, where S y is the number of slots, “y” = “s” for the stator (in which case ϕ ys,i and ϕ yt,i are relative to the stator) and “y” = “r” for the rotor (in which case ϕ ys,i and ϕ yt,i are relative to the rotor), and ϕ ys,1 is the position of slot 1. Since the number of conductors going into the page must be equal to the number of conductors out of the page (the conductor is formed into closed loops), we have that N x i i S y , = ∑ = 1 0 (2.2-10) where “x” designates the winding (e.g., “as”). The total number of turns associated with the winding may be expressed N N N x x i x i i S y = = ∑ , , u( ) 1 (2.2-11) where u(·) is the unit step function, which is one if its argument is greater or equal to zero, and zero otherwise. In (2.2-11) and throughout this work, we will use N x to represent the total number of conductors associated with winding “x,” N x,i to be the number of conductors in the i’th slot, and N x to be a vector whose elements correspond to the number of conductors in each slot. In addition, if all the windings of stator or rotor have the same number of conductors, we will use the notation N y to denote the number of conductors in the stator or rotor windings. For example, if N as = N bs = N cs , then we will denote the number of conductors in these windings as N s . It is sometimes convenient to illustrate features of a machine using a developed diagram. In the developed diagram, spatial features (such as the location of the conduc- tors) are depicted against a linear axis. In essence, the machine becomes “unrolled.” This process is best illustrated by example; Figure 2.2-5 is the developed diagram cor- responding to Figure 2.2-4. Note the independent axis is directed to the left rather than to the right. This is a convention that has been traditionally adopted in order to avoid the need to “flip” the diagram in three-dimensions. 58 DISTRIBUTED WINDINGS IN AC MACHINERY Continuous Description of Distributed Windings Machine windings are placed into slots in order to provide room for stator teeth and rotor teeth, which together form a low reluctance path for magnetic flux between the stator and rotor. The use of a large number of slots allows the winding to be distributed, albeit in a discretized fashion. The continuous description of the distributed winding describes the winding in terms of what it is desired to approximate—a truly distributed winding. The continuous description is based on conductor density, which is a measure of the number of conductors per radian as a function of position. As an example, we would describe winding “x” of a machine with the turns density n x (ϕ m ), where “x” again denotes the winding (such as “as”). The conductor density may be positive or negative; positive conductors are considered herein to be out of the page (toward the front of the machine). The conductor density is often a sinusoidal function of position. A common choice for the a-phase stator conductor density in three-phase ac machinery is n as sm s sm s sm N P N P( ) sin( / ) sin( / ) φ φ φ = − 1 3 2 3 2 (2.2-12) In this function, the first term represents the desired distribution; the second term allows for more effective slot utilization. This is explored in Problem 6 at the end of the chapter. It will often be of interest to determine the total number of conductors associated with a winding. This number is readily found by integrating the conductor density over all regions of positive conductors, so that the total number of conductors may be expressed N d x x m x m m = ∫ n u n( ) ( ( )) φ φ φ π 0 2 (2.2-13) Symmetry Conditions on Conductor Distributions Throughout this work, it is assumed that the conductor distribution obeys certain sym- metry conditions. The first of these is that the distribution of conductors is periodic in a number of slots corresponding to the number of pole pairs. In particular, it is assumed that N N x i S P x i y , / ,+ = 2 (2.2-14) Figure 2.2-5. Developed diagram. f sm N as,6 N as,5 N as,4 N as,3 N as,2 N as,1 N as,8 N as,7 DESCRIBING DISTRIBUTED WINDINGS 59 Second, it is assumed that the distribution of conductors is odd-half wave symmetric over a number of slots corresponding to one pole. This is to say N N x i S P x i y , / ,+ = − (2.2-15) While it is possible to construct an electric machine where these conditions are not met, the vast majority of electric machines satisfy these conditions. In the case of the continuous winding distribution, the conditions corresponding to (2.2-14) and (2.2-15) may be expressed as n n x m x m P( / ) ( ) φ π φ + =4 (2.2-16) n n x m x m P( / ) ( ) φ π φ + = −2 (2.2-17) Converting Between Discrete and Continuous Descriptions of Distributed Windings Suppose that we have a discrete description of a winding consisting of the number of conductors of each phase in the slots. The conductor density could be expressed n x m x i m ys i i S N y ( ) ( ) , , φ δ φ φ = − = ∑ 1 (2.2-18) where δ(·) is the unit impulse function and ϕ m is relative to the stator or rotor reference axis for a stator or rotor winding, respectively. Although (2.2-18) is in a sense a continuous description, normally we desire an idealized representation of the conductor distribution. To this end, we may represent the conductor distribution as a single-sided Fourier series of the form n x m j m j m j J a j b j( ) cos( ) sin( ) φ φ φ = + = ∑ 1 (2.2-19) where J is the number of terms used in the series, and where a j d j x m m m = ( ) ∫ 1 0 2 π φ φ φ π n ( )cos (2.2-20) b j d j x m m m = ( ) ∫ 1 0 2 π φ φ φ π n ( )sin (2.2-21) Substitution of (2.2-18) into (2.2-20) and (2.2-21) yields 60 DISTRIBUTED WINDINGS IN AC MACHINERY a N j j x i ys i i S y = ( ) = ∑ 1 1 π φ , , cos (2.2-22) b N j j x i ys i i S y = ( ) = ∑ 1 1 π φ , , sin (2.2-23) Thus (2.2-19), along with (2.2-22) and (2.2-23), can be used to convert a discrete winding description to a continuous one. It is also possible to translate a continuous winding description to a discrete one. To this end, one approach is to lump all conductors into the closest slot. This entails adding (or integrating, since we are dealing with a continuous function) all conductors within π/S s of the center of the i’th slot and to consider them to be associated with the i’th slot. This yields N d x i x m m S S ys i y ys i y , / / ( ) , , = − + ∫ round n φ φ φ π φ π (2.2-24) where round( ) denotes a function which rounds the result to the next nearest integer. End Conductors The conductor segments that make up the windings of a machine can be broken into two classes—slot conductors and end conductors. These are shown in Figure 2.2-1. Normally, our focus in describing a winding is on the slot conductors, which are the portions of the conductors in the slots and which are oriented in the axial direction. The reason for this focus is that slot conductors establish the field in the machine and are involved in torque production. However, the portions of the conductors outside of the slots, referred to as end conductors, are also important, because they impact the winding resistance and inductance. Therefore, it is important to be able to describe the number of conductor segments on the front and back ends of the machine connecting the slot conductor segments together. In this section, we will consider the calculation of the number of end conductor segments. Herein, we will focus our discussion on a discrete winding description. Consider Figure 2.2-6, which is a version of a developed diagram of the machine, except that instead of looking into the front of the machine, we are looking from the center of the machine outward in the radial direction. Therein, N x,i denotes the number of winding x conductors in the i’th slot. Variables L x,i and R x,i denote the number of positive end conductor in front of the i’th tooth directed to the left or right, respectively. These variables are required to be greater than or equal to zero. The net number of conductors directed in the counterclockwise direction when viewed from the front of the i’th tooth is denoted M x,i . In particular M L R x i x i x i, , , = − (2.2-25) DESCRIBING DISTRIBUTED WINDINGS 61 Unlike L x,i and R x,i , M x,i can be positive or negative. The number of canceled conductors in front of the i’th tooth is denoted C x,i . This quantity is defined as C L R x i x i x i, , , min( , )= (2.2-26) Canceled conductors are undesirable in that they add to losses; however, some winding arrangements use them for manufacturing reasons. It is possible to relate M x,i to the number of conductors in the slots. From Figure 2.2-6, it is apparent that M M N x i x i x i, , , = + − −1 1 (2.2-27) where the index operations are ring mapped (i.e., S y + 1→1,1−1→S y ). The total number of (unsigned) end conductors between slots i − 1 and i is E M C x i x i x i, , , = + 2 (2.2-28) The total number of end conductors is defined as E E x x i i S y = = ∑ , . 1 (2.2-29) Figure 2.2-6. End conductors. R x,4 2 1 Tooth 3 Tooth 4 Front of Machine Back of Machine Tooth Tooth R x,3 R x,2 R x,1 N x,2 N x,1 N x,3 L x,4 L x,3 L x,2 L x,1 M x,1 M x,2 M x,3 M x,4 62 DISTRIBUTED WINDINGS IN AC MACHINERY Common Winding Arrangements Before proceeding, it is convenient to consider a practical machine winding scheme. Consider the four-pole 3.7-kW 1800-rpm induction machine shown in Figure 2.2-1. As can be seen, the stator has 36 slots, which corresponds to three slots per pole per phase. Figure 2.2-7 illustrates a common winding pattern for such a machine. Therein, each conductor symbol represents N conductors, going in or coming out as indicated. This is a double layer winding, with each slot containing two groups of conductors. Both single- and double-layer winding arrangements are common in electric machinery. The number of a-phase conductors for the first 18 slots may be expressed as N as N 1 18 0 0 0 1 2 2 1 0 0 0 0 0 1 2 2 1 0 0 − = − − − − [ ] . (2.2-30) From (2.2-27) M as as M N 1 18 36 0 0 0 0 1 3 5 6 6 6 6 6 5 3 1 0 0 − = + [ ] , . (2.2-31) To proceed further, more details on the winding arrangement are needed. Figure 2.2-8 illustrates some possible winding arrangements. In each case, the figure depicts the stator of a machine in an “unrolled” fashion similar to a developed diagram. However, the vantage point is that of an observer looking at the teeth from the center of the machine. Thus, each shaded area represents a tooth of the machine. Figure 2.2-7. Stator winding for a four-pole 36-slot machine. a a a a a a a a a a a a a a a a a a a a a a a a b b b b b b b b b b b b b b b b b b b b b b b b c c c c c c c c c c c c c c c c c c c c c c c c 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 [...]...Describing Distributed Windings 63 Front 34 33 32 25 24 23 16 15 14 7 6 5 16 15 14 7 6 5 16 15 14 7 6 5 16 15 14 7 6 5 (a) Front 34 33 32 25 24 23 (b) Front 34 33 32 25 24 23 34 33 32 25 24 23 (c) Front (d) Figure 2. 2-8. Winding arrangements (a) Concentric winding arrangement; (b) consequent pole winding arrangement; (c) lap winding arrangement; (d) wave winding arrangement 64 Distributed Windings in. .. − 2 / 3) (2. 8 -22 ) Lcsc s = Llsp + L A − LB cos 2 (θ r + 2 / 3) (2. 8 -23 ) L fdfd = Llfd + Lmfd (2. 8 -24 ) 1 Lasbs = Llsm − L A − LB cos 2 (θ r − π / 3) 2 (2. 8 -25 ) 1 Lascs = Llsm − L A − LB cos 2 (θ r + π / 3) 2 (2. 8 -26 ) 1 Lbscs = Llsm − L A − LB cos 2 (θ r + π ) 2 (2. 8 -27 ) Lasfd = Lsfd sin θ r (2. 8 -28 ) Lbsfd = Lsfd sin (θ r − 2 / 3) (2. 8 -29 ) Lcsfd = Lsfd sin (θ r + 2 / 3) (2. 8-30) In (2. 8 -21 )– (2. 8-30),... ( 2 s − 2 r )) dφs P (2. 8-19) Evaluating (2. 8-19) and substituting into (2. 8-17), we obtain 2 α N Lasas = Llsp + πµ0rl s α1 − 2 cos 2 r P 2 (2. 8 -20 ) Applying this procedure to the other inductances, the inductance elements in (2. 8-5)– (2. 8-8) may be expressed as 80 Distributed Windings in ac Machinery Lasas = Llsp + L A − LB cos2θ r (2. 8 -21 ) Lbsbs = Llsp + L A − LB cos 2. .. calculated using the method of Section 2. 7 Synchronous Machine We will now consider an elementary synchronous machine In this machine, in addition to the three-phase stator windings, there is a field winding on the rotor In a practical machine, there would also be damper windings on the rotor; these will be considered 78 Distributed Windings in ac Machinery in Chapter 5, but will not be treated in this... − 2 − 3 − 3 − 3 − 3 − 3 − 3 − 2 0 2 3 3 3] (2A-1) The winding function is only given for the first 18 slots since the pattern is repetitive In order to obtain the continuous winding function, let us apply (2. 2 -22 ) and (2. 223 ) where the slot positions are given by (2. 2-8) Truncating the series (2. 2-19) after the first two nonzero harmonics yields nas = N ( 7 .22 1sin (2 sm ) − 4.4106 sin(6φsm )) (2A -2) ... ⋅ [ 2 −1 1 2 2 1 −1 − 2 − 2 −1 1 2 2 1 −1 − 2 ] T How much flux is linking the winding? 2. Suppose the turns (conductor) density of a winding function is given by nbr = 26 2 cos(8φrm − 2 / 3) Compute the rotor b-phase winding function in terms of position measured from the rotor 3. Consider (2. 2 -22 ) and (2. 2 -23 ) of the text, which are used to convert a discrete winding description to a continuous... given by n ar (φrm ) = N r1 sin( Pφrm / 2) (2. 8-44) n br (φrm ) = N r1 sin( Pφrm / 2 − 2 / 3) (2. 8-45) n cr (φrm ) = N r1 sin( Pφrm / 2 + 2 / 3), (2. 8-46) for which, making use of the relation (2. 3-11) w ar (φr ) = N r cos(φr ) / P (2. 8-47) wbr (φr ) = N r cos(φr − 2 / 3) / P (2. 8-48) wcr (φr ) = N r cos(φr + 2 / 3) / P (2. 8-49) 82 Distributed Windings in ac Machinery where Nr is the total... York, 1965 84 Distributed Windings in ac Machinery PROBLEMS 1. The number of conductors in each slot of the a-phase of the stator of the machine are as follows: N as = [10 20 20 10 − 10 − 20 − 20 − 10 10 20 20 10 − 10 − 20 − 20 − 10 ] T Compute and graph the winding function associated with this winding versus tooth number Suppose the flux traveling from the rotor to the stator in each tooth is given... far in regard to the operation of a machine Let us consider a three-phase stator winding We will assume that the conductor distribution for the stator windings may be expressed as n as (φsm ) = N s1 sin( Pφsm / 2) − N s 3 sin(3Pφsm / 2) (2. 5-1) n bs (φsm ) = N s1 sin( Pφsm / 2 − 2 / 3) − N s 3 sin(3Pφsm / 2) (2. 5 -2) n cs (φsm ) = N s1 sin( Pφsm / 2 + 2 / 3) − N s 3 sin(3Pφsm / 2) (2. 5-3) In (2. 5-1)– (2. 5-3),... each winding, it is further required that Sy/(3P) is an integer Let us now consider the calculation of the winding function using a continuous description of the winding In this case, instead of being a function of the tooth number, 66 Distributed Windings in ac Machinery the winding function is a continuous function of position, which can be position relative to the stator (ϕm = ϕsm) for stator windings . diagram in three-dimensions. 58 DISTRIBUTED WINDINGS IN AC MACHINERY Continuous Description of Distributed Windings Machine windings are placed into slots in. machine. a a a a a a a a a a a a a a a a a a a a a a a a b b b b b b b b b b b b b b b b b b b b b b b b c c c c c c c c c c c c c c c c c c c c c c c c 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 DESCRIBING DISTRIBUTED WINDINGS 63 Figure 2. 2-8. Winding