Chapter 15 introduction to the design of electric machinery

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

The reader is forewarned that the approach has been simplified Structural issues, thermal issues, and several loss mechanisms are neglected, and infinitely permeable magnetic steel is assumed, though saturation is considered Even so, the design problem

is nontrivial and provides an organized and systematic approach to machine design This approach may be readily extended to include a wide variety of design considerations

The machine design problem is made easier if given context To this end, our problem is to design a three-phase, wye-connected, permanent-magnet ac machine to produce a desired torque T e* at a desired mechanical speed ωrm* It is assumed that the

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.

INTRODUCTION TO THE DESIGN OF ELECTRIC

MACHINERY

15

inverter driving the machine is current controlled as discussed in Sections 12.8–12.11,

and operated from a dc bus voltage of v dc It is desired to minimize mass, to minimize loss, to restrict current density (since it is closely related to winding temperature), to avoid heavy magnetic saturation, and to avoid demagnetization of the magnet.Figure 15.1-1 illustrates a diagram of a two-pole version of the machine (we will

consider a P-pole design) The phase magnetic axes are shown, as well as the q- and d-axis The stator is broken into two regions, the stator backiron and the slot/tooth region The rotor includes a shaft, a magnetically inert region (which could be steel but need not be), a rotor backiron region, and permanent magnets Arrows within the per-manent magnet region indicate the direction of magnetization Also shown in Figure 15.1-1 is the electrical rotor position, θr, position measured from the stator, ϕs, and position measured relative to the rotor ϕr Since a two-pole machine is shown, these angles are identical to their mechanical counterparts, θrm, ϕsm, and ϕrm for the device shown

Again, our approach will be to formulate the design problem as a formal tion problem Hence, our goal will be to predict machine performance based on a geometrical machine description The work will proceed as follows First, Section 15.2 will set forth the details of the geometry Next, the winding configuration will be dis-cussed in Section 15.3 Needed material properties will be outlined in Section 15.4 The current control philosophy will be delineated in Section 15.5 At this point, atten-tion will turn to finding an expression for the radial flux density of the machine in Section 15.6, and a derivation of expressions for the electrical parameters of the

optimiza-figure15.1-1. surface-mountedpermanent-magnetsynchronousmachine.

q-axis

d-axis

as-axis bs-axis

MachInegeoMetry  585

machine in Section 15.7 The implications of the air-gap field on the field within the steel and permanent magnet is addressed in Section 15.8 As this point, the primary analytical results required for the machine design will have been put in place Thus, in Section 15.9, the formulation of the design problem is considered Section 15.10 pro-vides a case study in multiobjective optimization-based machine design Finally, Section 15.11 discusses extensions to the approach set forth herein

15.2.  MACHINE GEOMETRY

Figure 15.2-1 illustrates a cross-section of the machine As can be seen, the machine

is divided into regions Proceeding from the exterior of the machine to the interior, the outermost region of the machine is the stator backiron, which extends from a radius of

rsb to r ss from the center of the machine In this region, flux enters and leaves from the teeth and predominantly travels in the tangential direction The next region is the slot/tooth region, which contains the stator slots and teeth and stator conductors as discussed

in Chapter 2 The slot/tooth region extends from r st to r sb The next region is the air

gap, which includes radii from r rg to r st Proceeding inward, the permanent magnet

includes points with radii between r rb and r rg and consists of one of two types of rial, either a permanent magnet that will produce radial flux, or a magnetically inert

mate-spacer that may be air (as shown) The rotor backiron extends from r ri to r rb Flux enters and leaves the rotor backiron predominantly in the radial direction; but the majority of the flux flow through the rotor backiron will be tangential It serves a purpose similar

to the stator backiron The inert region (radii from r rs to r ri) mechanically transfers torque from the rotor backiron to the shaft It is often just a continuation of the rotor

Tooth Slot

Permanent Magnet

m d

backiron (possibly with areas removed to reduce mass) or could be a lightweight posite material Material in this region does not serve a magnetic purpose, even if it is

com-a mcom-agnetic mcom-atericom-al

Variables depicted in Figure 15.2-1 include: d sb —the stator backiron depth, d st—the

stator tooth depth, g—the air-gap depth, d m —the permanent magnet depth, d rb—the

rotor backiron depth, d i —the magnetically inert region depth, and r rs—the rotor shaft radius The active length of the machine (the depth of the magnetic steel into the page)

is denoted as l The quantity αpm is the angular fraction of a magnetic pole occupied by the permanent magnet All of these variables, with the exception of the radius of the

rotor shaft, r rs, which is assumed to be known, will be determined as part of the design process

In terms of the parameters identified in the previous paragraph, the following may

be readily calculated: r ri —the rotor inert region radius, r rb—the rotor backiron radius,

rrg —the rotor air-gap radius, r st —the stator tooth inner radius, r sb—the stator backiron

inner radius, and r ss—the stator shell radius A stator shell, if present, is used for tion, mechanical strength, and thermal transfer It will not be considered in our design.Figure 15.2-2 depicts a portion of the stator consisting of one tooth and one slot (with ½ of a slot on either side of the tooth) Variables depicted therein which have not

protec-been previously defined include: S s—the number of stator slots, θtt—the angle spanned

by the tooth tip at radius r st, θst —the angle spanned by the slot at radius r st , r si—the radius to the inside tooth tip, θti —the angle spanned by the tooth at radius r si, θtb—the

angle spanned by the tooth at radius r sb , w tb —the width of the tooth base, d tb—the depth

of the tooth base, d tte —the depth of the tooth tip edge, and d ttc—the depth of the tooth tip center at θt/2

For the purposes of design, it will be convenient to introduce the tooth fraction αt

and tooth tip fraction αtt The tooth fraction is defined as the angular fraction of the

slot/tooth region occupied by the tooth at radius r st Hence,

MachInegeoMetry  587

π

t s t S

=

The tooth tip fraction is herein defined as the angular fraction of the slot/tooth region

occupied by the tooth tip at radius r st It is defined as

inde-Gx =[r d d d g d d d rs i rb m tb ttc tteα αt tt d sbαpm l P S sφss1]T (15.2-3)

where a “G” is used to denote geometry and the subscript “x” serves as a reminder that

these variables are considered independent Note that we have not discussed the last

element of Gx, namely ϕss1, in this chapter; it is the center location of the first slot as

discussed in Chapter 2 Given Gx, the locations of the slots and teeth may be calculated using (2.2-8) and (2.2-9); next, the remaining quantities in Figure 15.2-1 and Figure 15.2-2 can be readily calculated as

=2  

2

r ss=r sb+d sb (15.2-18)Another geometrical variable of interest, although not shown in Figure 15.2-2, is the slot opening, that is, the distance between teeth This is readily expressed as

of these is the area of a tooth base, which is the portion of the tooth that falls within

rsi ≤ r ≤ r sb and is given by

The total volume of the rotor backiron, denoted as v rb , rotor inert region, v ri, and

per-manent magnet, v pm, are readily found from

v rb =π(r rb2−r l ri2) (15.2-26)

v ri =π(r ri2−r l rs2) (15.2-27)

v pm =π(r rg2−r rb2)αpm l (15.2-28)

MachInegeoMetry  589

For purposes of leakage inductance calculations, it is convenient to approximate the slot geometry as being rectangular, as depicted in Figure 15.2-3

There are many ways such an approximation can be accomplished One approach

is as follows First, the width of the tooth tip is approximated as the circumferential length of the actual tooth tip

Next, the width of the slot between the stator tooth tips is approximated by ential distance between the tooth Thus

The width of slot between the base of the tips is taken as the average of the distance

of the chord length of the inner corners of the tooth tips at the top of the tooth and the chord distance between the bottom corners of the teeth This yields

Note that this approach is not consistent in that it does not require w siR + w tbR =

wttR + w stR However, this does not matter in the primary use of the model—the culation of the slot leakage permeance as discussed in Appendix C The final param-

cal-eter shown in Figure 15.2-3 is the depth of the winding within the slot, d wR This parameter will not be considered a part of the stator geometry, but rather as part of the winding

Before concluding this section, it is appropriate to organize our calculations in order to support our design efforts In (15.2-3), we defined a list of “independent”

variables that define the machine geometry, and organized them into a vector Gx Based

on this, we found a host of related variables which will also be of use It is convenient

to define these dependent variables as a vector

where again “G” denotes geometry and the “y” indicates dependent variables We may

summarize our calculations (from 15.2-4 to 15.2-34) as a vector-valued function FG

such that

This view of our geometrical calculations will be useful as we develop computer codes

to support machine design, directly suggesting the inputs and outputs of a subroutine/function calls to make geometrical calculations Finally, other calculations we will need

to perform will require knowledge of both Gx and Gy; it will therefore be convenient

statorWIndIngs  591

where N s1 and α3 are the desired fundamental amplitude of the conductor density, and ratio between the third harmonic component and fundamental component, respectively

The goal of this chapter is to design a machine that can be constructed, which means that we need to specify the specific number of conductors of each phase to be placed in each slot To this end, we can use the results from Section 2.2 Using (2.2-24)

in conjunction with (15.3-1)–(15.3-3) yields

s

P S

32

where N as,i , N bs,i , and N cs,iare the number of conductors of the respective phase in the

i’th slot and where ϕss,i denotes the mechanical location of the center of the i’th stator

slot, which is given by (2.2-8) in terms of ϕss,1, which is the location of the center of

the first slot This angle takes on a value of 0 if the a-phase magnetic axis is aligned

with the first slot or π/Ss if it desired to align the a-phase magnetic axis with the first

tooth

The total number of conductors in the ith slot is given by

N s i, = N as i, + N bs i, + N cs i, (15.3-7)For some of our magnetic analysis, we will use the continuous rather than discrete description of the winding Once the number of conductors in each slot are computed using (15.3-4)–(15.3-6), then from (2.2-12), (15.3-1), and (2.2-20), the effective value

of N s1 and α3 are given by

i

S s

1 1

S s

It is also necessary to establish an expression to describe the end conductor distribution The end conductor distribution for each winding may be calculated in terms of the slot conductor distribution using the methods of Section 2.2 In particular, repeating (2.2-25) for convenience, the net end conductor distribution for winding “x” is expressed

M x i, =M x i, − 1+N x i, − 1 (15.3-10)

Using (15.3-10) requires knowledge of the net number of end conductors M x,1 on the

end of tooth 1 This, and the number of cancelled conductors in each slot, C x,i (see Section 2.2), determines the type of winding (lap, wave, or concentric) For the purposes

of this chapter, let us take the number of canceled conductors to be zero and require the end winding conductor arrangement to be symmetric in the sense that for any end

conductor count over tooth i, the end conductor count over the diametrically opposed

tooth (in an electrical sense) has the opposite value Mathematically,

M x i, = −M x S P i, /s + (15.3-11)From (15.3-10) and (15.3-11), it can be shown that (Problem 4)

1

12

In addition to the distribution of the wire, it is also necessary to compute the wire cross-sectional area To this end, the concept of packing factor is useful The packing factor is defined as the maximum (over all slots) of the ratio of the total conductor cross-sectional area within the slot to the total slot area, and will be denoted by αpf Typical packing factors for round wire range from 0.4 to 0.7 Assuming that it is advan-tageous not to waste the slot area, the conductor cross-sectional area and diameter may

be expressed as

a c a slt pf s

d c= 4a c

where ‖NS‖max denotes the maximum element of the vector NS If desired, a c and d c

can be adjusted to match a standard wire gauge In this case, the gauge selected should

be the one with the largest conductor area that is smaller than that calculated using (15.3-13)

Finally, it will be necessary to compute the depth of the winding within the slot for the rectangular slot approximation This may be readily expressed as

w

wR= Ns max c

MaterIalParaMeters  593

Another variable of interest is the dimension of the end winding bundle in the direction parallel to the rotor shaft Assuming the same depth as calculated by (15.3-15), this dimension may be approximated as

As in the case of the stator geometry, it is convenient to organize the variables discussed into independent and dependent variables, which will show the relationship

of the variables from a programming point of view To this end, it is convenient to organize the independent variables of the winding description as

Wx=[N s*1α α3* pf l eo]T (15.3-18)The output of our winding calculations are encapsulated by the vector

As part of the design process, we will also need to select materials for the stator steel,

the rotor steel, the conductor, and the permanent magnet We will use s t , r t , c t , and m t

as integer variables denoting the stator steel type, the rotor steel type, the conductor

type, and the permanent magnet type Based on these variables, the material parameters can be established using tabulated functions in accordance with

where “sc,” “cc,” and “mc” denote “steel catalog,” “conductor catalog,” and “magnet

catalog,”, and where S, R, C, and M are vectors of material parameters for the stator

steel, rotor steel, conductor, and magnet, and may be expressed as

The permanent magnet parameters are illustrated in terms of the magnet B–H and

M –H characteristic in Figure 15.4-1, where M is magnetization In any material, B, H, and M are related by

The B–H relationship is often referred to as the material’s normal characteristic, while the M–H relationship is referred to as the intrinsic characteristic In Figure 15.4-1, B r

is the residual flux density of the material (the flux density or magnetization when the

field intensity is zero), H c is the coercive force (the point where the flux density goes

to zero), and H ci is the intrinsic coercive force (the point where the magnetization goes

to zero), and χm is the susceptibility of the material Permanent magnet material is generally operated in the second quadrant if it is positively magnetized or fourth quad-rant if it is negatively magnetized It is important to make sure that

H≥H lim (positively magnetized) (15.4-10)

H≤H lim (negatively magnetized) (15.4-11)

in order to avoid demagnetization, where H lim is a minimum allowed field intensity to avoid demagnetization, which is a negative number whose magnitude is less than that

MaterIalParaMeters  595

of H ci, and which is a function of magnet material and often of operating temperature

It should also be noted that while the shape of the M–H characteristic is fairly consistent between materials, the shape of the B–H curve is not; indeed B–H may take on the

slanted shape shown in Figure 15.4-1, or appear relatively square

Material data for a limited number of steels, conductors, and permanent magnets

is given in Table 15.4-1, Table 15.4-2, and Table 15.4-3, respectively Note that

recom-mended saturation flux density limits are a “soft” recommendation since the B–H

TABLE 15.4-3 Permanent Magnets

characteristic of magnetic materials is a continuous function Magnetic steel properties vary not only with grade, but also manufacturer Further, the recommendation on maximum current density is a soft recommendation All material parameters are a func-tion of temperature though this aspect of the design is not treated in this introduction

to the topic of machine design Temperature dependence can have a particularly strong impact on permanent magnet characteristics

15.5.  STATOR CURRENTS AND CONTROL PHILOSOPHY

In our design, we will consider a machine connected to a current-regulated inverter, and that through the action of the inverter controls, the machine currents are regulated

to be equal to the commanded q- and d-axis currents This is reasonable, assuming the

use of a synchronous current regulator (see Section 12.11) or a similar technique The

corresponding abc currents are readily found from the inverse rotor reference-frame

transformation; alternately, they may be expressed

Ix qs r

ds r T

i i

Iy=[I sφi]T (15.5-7)and

respectively The amalgamation of variables associated with the currents is

I=[I IT x y T T] (15.5-9)

radIalfIeldanalysIs  597

15.6.  RADIAL FIELD ANALYSIS

The objective of this section is a magnetic analysis of the machine A key assumption

is that the MMF drop across the steel portions of the machine is negligible Unless the steel becomes highly saturated, this is a reasonable assumption because relative perme-ability of most permanent magnet materials is very low compared with steel, and so the MMF drop across the permanent magnet and air-gap dominate that of the steel

In performing our analysis, we will take the rotor position to be fixed This may strike the reader as overly restrictive However, as stated in the introduction, it will be our objective to produce a design that yields a constant torque T e* In such a machine, except for the perturbation caused by slot effects, the flux and current densities are traveling waves that rotate but do not change in magnitude under steady-state condi-tions Thus, ideally, it is only necessary to consider a single position, which could, for example, be taken to be zero In practice, slot effects are a factor so we will consider several fixed rotor positions, although all positions will be within one slot/tooth pitch

of zero

In order to analyze the field in the machine, we note that from Section 2.4 that in the absence of rotor currents the air-gap MMF drop is equal to the stator MMF In particular, from (2.4-20)

Fg(φsm)=Fs(φsm) (15.6-1)

In the definition of air gap MMF used in defining (2.4-20), it is important to recall that the definition of air gap MMF drop given by (2.4-3) extended from the rotor steel to the stator steel In particular, in terms of the dimensions of Figure 15.2-1,

where H(r, ϕsm) denotes the radial component of field intensity

For the purposes at hand, it will be convenient to define

Fa(φsm)+Fpm(φsm)=Fs(φsm) (15.6-5)

In the following subsections, we will establish expression for each term in (15.6-5), and then use these to establish an expression for the radial flux density in the machine.

Expressions of the b- and c-phases are similarly derived.

From (2.5-7), the stator MMF may be expressed as

Fs=w i as as+w i bs bs+w i cs cs (15.6-8)Substitution of the winding functions and the expressions for currents (15.5-1)– (15.5-3) into (15.6-8) and simplifying yields the expression for stator MMF, namely

N I P

to take the radial variation of the flux density into account when determining MMF components

radIalfIeldanalysIs  599

In order to establish the radial variation in the field, consider Figure 15.6-1 Therein, a cross section of an angular slice of the machine is shown Assuming that the flux density is entirely radial for radii between the rotor backiron and the stator teeth,

then the flux through surface S rb at the rotor backiron radius must, by Gauss’s law, be

equal to the flux through the surface S r at an arbitrary radius, whereupon it follows that the flux density of an arbitrary radius is given by

design constraint), the relationship between flux density and field intensity for

11

positively magnetizednegatively maggnetizedinert region between magnets

where B m(ϕrm) is due to the residual flux density in the permanent magnet and μrm(ϕrm)

is the relative permeability of the permanent magnet region (including the inert rial), and ϕrm denotes position as measured from the q-axis of the rotor, and both B and

mate-H refer to the radial component of the field directed from the rotor to the stator These functions are illustrated in Figure 15.6-2 in developed diagram form

The spatial dependence of B m(ϕrm) and μrm(ϕrm) is illustrated in the second and third traces of Figure 15.6-2 From this figure, we may express μrm(ϕrm ) and B m(ϕrm) as

2

Active Material (North Pole) Active Material (South Pole)

r B

rg r r

1+cm m f rm ( )rm

a

radIalfIeldanalysIs  601

where sqw s(⋅) is the square wave function with sine symmetry defined as

sqw s( , )

sin( ) sin( ( ) / )sin( ) sin( ( ) / )sin

It should be observed that R m(ϕrm) is not a reluctance; however, it plays a similar role

It takes on two values depending upon stator and rotor position For positions under the permanent magnet,

We will use this result extensively in the sections to follow It should be noted that θrm

is an implicit argument of Fm(φsm), Fs(φsm), and R m(ϕsm) Because of this, we will

some-times denote the radial flux density given by (15.6-25) as B(r, ϕsm,θrm) when it is tant to remember this functional dependence

is denoted as R s rather than r s This is done to avoid confusion with a machine radius.Let us start with the calculation of the stator resistance From (2.7-4) and (2.7-5),

λdm =L i dm ds r +λm (15.7-5)where