Chapter 2 distributed windings in ac machinery

Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek

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 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

flux-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

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

Arbitrary Position

Stator Rotor

Rotor Reference Axis

frm

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

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 quency Figure 2.2-3 illustrates the operation of 2-, 4-, and 6-pole machines Therein

fre-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

θ φr+ = r φs (2.2-6)Finally, it is also useful to define a generic position as

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 conducconduc-tors 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)

DescribingDistributeDWinDings  57

Generalizing this notation, N x ,i is the number of conductors in slot iof 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

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 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

where “x” designates the winding (e.g., “as”) The total number of turns associated with

the winding may be expressed

N x N x i N x i i

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 xto 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 = Nbs = Ncs, 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

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(ϕm ), 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

nas(φsm)=N s1sin(Pφsm/ )2 −N s3sin(3Pφsm/ )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 x =∫nx( ) ( ( ))φm u nx φm dφm

π 0

DescribingDistributeDWinDings  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 x i S P, + y/ = −N x i, (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

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

i

S N

a j N x i j ys i i

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/Ss of the center of the i’th slot and to consider them to be associated with the i’th slot This yields

S S

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,idenote 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

DescribingDistributeDWinDings  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

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

The total number of end conductors is defined as

E x E x i i

Tooth 4

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

Nas1 18− =N[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)

Mas1 18− =M as, 36+N[0 0 0 0 1 3 5 6 6 6 6 6 5 31 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

Figure2.2-7  statorwindingforafour-pole36-slotmachine.

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

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

32 33 34 35 36

DescribingDistributeDWinDings  63

Figure2.2-8  Windingarrangements.(a)concentricwindingarrangement;(b)consequent polewindingarrangement;(c)lapwindingarrangement;(d)wavewindingarrangement.

Figure 2.2-8a depicts a concentric winding arrangement wherein the a-phase

con-ductors are organized in 12 coils, with three coils per set Each coil is centered over a

magnetic axis or pole associated with that phase For this arrangement M as ,36 = −3N,

C as,i = 0∀i, and E as = 88N.

In Figure 2.2-8b, a consequent pole winding arrangement is shown In this ment, the windings are only wrapped around every other pole From Figure 2.2-8b, we

arrange-have M as,36= 0, C as,i = 0∀i, and E as = 108N The increase in E as will cause this ment to have a higher stator resistance than the concentric pole winding

arrange-A lap winding is shown in Figure 2.2-8c Each coil of this winding is identical

For this arrangement M as,36= −3N, as in the case of the concentric winding However,

in this winding C as,6= C as,15= C as,24= C as,34= 2N, and all other C as ,i= 0 The total number

of end turn segments is 96N, which is better than the consequent pole winding, but not

as good as the concentric winding

Figure 2.2-8d depicts a wave winding, in which the winding is comprised of six

coil groups For this case, M as,36= −6N, C as ,i = 0, and E as = 108N Like the consequent

pole winding, a relatively high stator resistance is expected; however, the reduced number of coil groups (and the use of identical groups) offers a certain manufacturing benefit

2.3.  WINDING FUNCTIONS

Our first goal for this chapter was to set forth methods to describe distributed windings Our next goal is to begin to analyze distributed winding devices To this end, a valuable concept is that of the winding function discussed in Reference 1 The winding function has three important uses First, it will be useful in determining the MMF caused by dis-tributed windings Second, it will be used to determine how much flux links a winding Third, the winding function will be instrumental in calculating winding inductances.The winding function is a description of how many times a winding links flux density at any given position It may be viewed as the number of turns associated with

a distributed winding However, unlike the number of turns in a simple coil, we will find that the number of turns associated with a distributed winding is a function of position Using this notion will allow us to formulate the mathematical definition of the winding function

Let us now consider the discrete description of the winding function Figure 2.3-1 illustrates a portion of the developed diagram of a machine, wherein it is arbitrarily

WinDingFunctions  65

assumed that the winding of interest is on the stator Let W x ,i denote the number of times

winding “x” links flux traveling through the i’th tooth, where the direction for positive

flux and flux density is taken to be from the rotor to the stator

Now, let us assume that we know W x ,i for some i It can be shown that

To understand (2.3-1), suppose N x ,i is positive Of the N x ,i conductors, suppose α of

these conductors go to the right (where they turn back into other slots), and β of these conductors go to the left (where again they turn back into other slots) The α conductors

form turns that link flux in tooth i but not flux in tooth i+1 since they close the loop to

the right The β conductors form turns that are directed toward the left before closing the loop, and so do not link tooth i, but do link tooth i+1, albeit in the negative direction

(which can be seen using the right hand rule and recalling that flux is considered positive from the rotor to the stator) Thus we have

Since N x ,i = α + β, (2.3-2) reduces to (2.3-1) Manipulation of (2.3-1) yields an

expression for the winding function In particular,

12

=

=

Using (2.3-5) and (2.3-1), the winding function can be computed for each tooth It

should be noted that it is assumed that S y /P is an integer for the desired symmetry

conditions to be met In addition, for a three-phase machine to have electrically identical

phases while ensuring symmetry of each winding, it is further required that S y /(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,

the winding function is a continuous function of position, which can be position relative

to the stator (ϕ m = ϕ sm) for stator windings or position relative to the rotor (ϕ m = ϕ rm) for rotor windings Let us assume that we know the value of the winding function at position ϕ m, and desire to calculate the value of the winding function at position

ϕ m + Δϕ m The number of conductors between these two positions is n x(ϕ m)Δϕ m, assuming

Δϕ mis small Using arguments identical to the derivation of (2.3-1), we have that

wx(φm+∆φm)=wx( )φm −nx( )φm ∆φm (2.3-6)Taking the limit as Δϕ m→0,

d d

x m m

In order to utilize (2.3-8), we must establish w x(0) As in the discrete case for computing

W x,1, we require that the winding function obeys the same symmetry conditions as the conductor distribution, namely (2.2-17) Thus

0

(2.3-11)

In summary, (2.3-1), along with (2.3-5) and (2.3-11), provide a means to calculate the winding function for discrete and continuous winding descriptions, respectively The winding function is a physical measure of the number of times a winding links the flux

in a particular tooth (discrete winding description) or a particular position (continuous winding description) It is the number of turns going around a given tooth (discrete description) or given position (continuous description)

air-gapMagnetoMotiveForce  67

Figure 2A-1 depicts the conductor distributions and winding function for the winding The discrete description of the winding function is shown as a series of arrows, sug-gesting a delta function representation The corresponding continuous distribution (which is divided by 4) can be seen to have a relatively high peak It is somewhat dif-ficult to compare the discrete winding description to the continuous winding description since it is difficult to compare a delta function with a continuous function The discrete representation of the winding function is shown as a set of horizontal lines spanning one tooth and one slot, and centered on the tooth These lines are connected to form a contiguous trace The continuous representation of the winding function can be seen to

be very consistent with the discrete representation at the tooth locations Had we chosen

to include the next two harmonics, the error between the continuous winding function representation and the discrete winding function representation at the tooth centers would be further reduced

The next step of this development will be the calculation of the MMF associated with

a winding As it turns out, this calculation is very straightforward using the winding function The connection between the winding function and the MMF is explored in the next section

2.4.  AIR-GAP MAGNETOMOTIVE FORCE

In this section, we consider the air-gap magnetomotive force (MMF) and the ship of this MMF to the stator currents We will find that the winding function is

relation-EXAMPLE 2A We will now consider the winding function for the machine shown

in Figure 2.2-7 Recall that for this machine, P = 4and S s= 36 From Figure 2.2-7, observe that the first slot is at ϕ sm = 0, hence ϕ ss = 0 The conductor distribution is

given by (2.2-30), where Nwas the number of conductors in a group Applying (2.3-5),

we have W x,1 = 3N Using (2.3-1), we obtain

W as 1 18− =N[3 3 3 2 0 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 22) and 23) where the slot positions are given by (2.2-8) Truncating the series (2.2-19) after the first two nonzero harmonics yields

(2.2-n as=N(7 221 sin(2φsm)−4 4106 sin(6φsm)) (2A-2)

Comparing (2A-2) with (2.2-12), we see that N s1= 7.221N and N s3= 4.4106N From

instrumental in establishing this relationship In doing this, we will concentrate our efforts on the continuous winding description.

Let us begin by applying Ampere’s law to the path shown in Figure 2.4-1 In ticular, we have