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α
θ
π
t
s t
S
=
2
(15.2-1)
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
α
θ
π
tt
s tt
S
=
2
(15.2-2)
As previously noted, not all the variables in Figure 15.2-1 and Figure 15.2-2 are inde-
pendent, being related by geometry. One choice of variables sufficient to define the
geometry is given by
G
x
rs i rb m tb ttc tte t tt sb pm s ss
T
r d d d g d d d d l P S
=
[ ]
α α α φ
1
(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 G
x
, namely ϕ
ss1
, in this chapter; it is the center location of the first slot as
discussed in Chapter 2. Given G
x
, 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
r r d
ri rs i
= +
(15.2-4)
r r d
rb ri rb
= +
(15.2-5)
r r d
rg rb m
= +
(15.2-6)
r r g
st rg
= +
(15.2-7)
r r d
si st ttc
= +
(15.2-8)
θ πα
t t s
S= 2 /
(15.2-9)
θ πα
tt tt s
S= 2 /
(15.2-10)
θ π θ
st s tt
S= −2 /
(15.2-11)
w r
tb st t
= 2 2sin( / )
θ
(15.2-12)
w r
tt st tt
= 2 2sin( / )
θ
(15.2-13)
r w r d d
sb tb st t tb ttc
= + + +( / ) ( cos( / ) )2 2
2 2
θ
(15.2-14)
θ
tb
tb
sb
w
r
=
2
2
a sin
(15.2-15)
θ
ti
tb
si
w
r
=
2
2
a sin
(15.2-16)
d r r
st sb st
= −
(15.2-17)
588 INTRODUCTION TO THE DESIGN OF ELECTRIC MACHINERY
r r d
ss sb 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
w r
so st
st
=
2
2
sin
θ
(15.2-19)
In addition to the computing the dependent geometrical variables, there are several
other quantities of interest that will prove useful in the design of the machine. The first
of these is the area of a tooth base, which is the portion of the tooth that falls within
r
si
≤ r ≤ r
sb
and is given by
a w d
r
r w
r
r w
tb tb tb
sb
sb tb tb
tb
si
si ti tb
ti
= + −
(
)
)
− −
2
2
2
θ
θ
θ
θ
cos cos
22
(
)
)
(15.2-20)
The area of a tooth tip, which is the material at a radius r
st
≤ r ≤ r
si
from the center of
the machine, may be expressed as
a
w d w w r r d
tt
tt tte tb tt si ti st tt tte
=
+ + − −
1
2
2 2 2( )( cos( / ) cos( / ) )
θ θ
−− +
(
)
+ −
(
)
r r w r r w
st tt st tt
tt
si ti si tb
ti
2 2
2 2
θ
θ
θ
θ
cos
/
cos
/
(15.2-21)
The slot area is defined as the cross-sectional area of the slot between radii r
sb
and r
si
.
This area is calculated as
a
S
r r
a
slt
s
sb si
tb
=
−
( )
−
π
2 2
(15.2-22)
The total volume of all stator teeth, v
st
, the back iron, v
sb
, and the stator laminations,
v
sl
, may be formulated as
v S a a l
st s tt tb
= +( )
(15.2-23)
v r r l
sb ss sb
= −
π
( )
2 2
(15.2-24)
v v v
sl st sb
= +
(15.2-25)
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 r r l
rb rb ri
= −
π
( )
2 2
(15.2-26)
v
r r
l
ri
ri rs
=
−
( )
π
2 2
(15.2-27)
v
r r
l
pm
rg rb
pm
=
−
( )
π α
2 2
(15.2-28)
MACHINE GEOMETRY 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
w r
ttR st t t
=
θ
(15.2-29)
The depth of the rectangular approximation to the tooth tip is set so that the tooth tip
has the same cross sectional area. In particular,
d
a
w
ttR
tt
ttR
=
(15.2-30)
Next, the width of the slot between the stator tooth tips is approximated by circumfer-
ential distance between the tooth. Thus
w r
stR st st
=
θ
(15.2-31)
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
w r
S
r
S
siR si
s
ti
sb
s
tb
=
−
+
−
sin sin
π θ π θ
2 2
(15.2-32)
Maintaining the area of the slot and the area of the tooth base, the depth of the slot
(exclusive of the tooth tip) and width of the tooth base are set in accordance with
d
a
w
siR
slt
siR
=
(15.2-33)
w
a
d
tbR
tb
siR
=
(15.2-34)
Figure 15.2-3. Rectangular slot approximation.
ttR
w
tbR
w
siR
w
siR
d
wR
d
ttR
d
stR
w
590 INTRODUCTION TO THE DESIGN OF ELECTRIC MACHINERY
Note that this approach is not consistent in that it does not require w
siR
+ w
tbR
=
w
ttR
+ w
stR
. However, this does not matter in the primary use of the model—the cal-
culation of the slot leakage permeance as discussed in Appendix C. The final param-
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 G
x
. 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
G
y ri rb rg st ri t tt tb tt sb tb ti st ss so
slt tt t
r r r r r w w r d r w
a a a
= [
θ θ θ θ
bb st sb sl rb ri pm ttR ttR st R siR siR t bR
T
v v v v v v w d w w d w ]
(15.2-35)
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 F
G
such that
G G
y G x
= F ( )
(15.2-36)
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 G
x
and G
y
; it will therefore be convenient
to define
G
G G
=
[ ]
x
T
y
T
T
(15.2-37)
15.3. STATOR WINDINGS
It is assumed herein that the conductor distribution is sinusoidal with the addition of a
third harmonic term as discussed in Section 2.2, and given, in a slightly different but
equivalent form, by (2.2-12). In particular, the assumed conductor density is given by
n N
P P
as sm s sm sm
( ) sin sin
* *
φ φ α φ
=
−
1 3
2
3
2
(15.3-1)
n N
P P
bs sm s sm sm
( ) sin sin
* *
φ φ
π
α φ
= −
−
1 3
2
2
3
3
2
(15.3-2)
n N
P P
cs sm s sm sm
( ) sin sin
* *
φ φ
π
α φ
= +
−
1 3
2
2
3
3
2
(15.3-3)
STATOR WINDINGS 591
where
N
s
1
*
and
α
3
*
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
N
N
P
P P
S
P
as i
s
ss i
s
,
*
,
*
sin sin sin=
−round
4
2 2 3
3
2
1 3
φ
π α
φφ
π
ss i
s
P
S
,
sin
3
2
(15.3-4)
N
N
P
P P
S
bs i
s
ss i
s
,
*
,
*
sin sin si= −
−round
4
2
2
3 2 3
1 3
φ
π π α
nn sin
,
3
2
3
2
P P
S
ss i
s
φ
π
(15.3-5)
N
N
P
P P
S
cs i
s
ss i
s
,
*
,
*
sin sin si= +
−round
4
2
2
3 2 3
1 3
φ
π π α
nn sin
,
3
2
3
2
P P
S
ss i
s
φ
π
(15.3-6)
where N
as,i
, N
bs,i
, and N
cs,i
are the number of conductors of the respective phase in the
i’th slot and where ϕ
ss,i
denotes the mechanical location of the center of the i’th stator
slot, which is given by (2.2-8) in terms of ϕ
ss,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/S
s
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 N N N
s i as i bs i 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 α
3
are given by
N N
s as i ss i
i
S
s
1
1
1
=
( )
=
∑
π
φ
, ,
cos
(15.3-8)
α
π
φ
3
1
1
1
3= −
( )
=
∑
N
N
s
as i ss i
i
S
s
, ,
cos
(15.3-9)
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
592 INTRODUCTION TO THE DESIGN OF ELECTRIC MACHINERY
M M N
x i x i x i, , ,
= +
− −1 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 M
x i x S P i
s
, , /
= −
+
(15.3-11)
From (15.3-10) and (15.3-11), it can be shown that (Problem 4)
M N
x x i
i
S P
s
, ,
/
1
1
1
2
= −
=
∑
(15.3-12)
Thus, once the slot conductor distribution is known, (15.3-12) and (15.3-10) can be
used to find an end conductor distribution. The distribution chosen herein corresponds
to a concentric winding. Note that (15.3-12) can yield a noninteger result. In this case,
minor alterations to the end conductor arrangement can be used to provide proper con-
nectivity with an integer number of conductors.
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 α
pf
.
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
a
c
slt pf
s
=
α
N
max
(15.3-13)
d
a
c
c
=
4
π
(15.3-14)
where ‖N
S
‖
max
denotes the maximum element of the vector N
S
. 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
d
a
w
wR
s
c
pf siR
=
N
max
α
(15.3-15)
MATERIAL PARAMETERS 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
l
M M M
a
d
ew
as bs cs
c
pf wR
=
+ +
max
α
(15.3-16)
Another variable of interest is the total volume of stator conductor per phase, v
cd
. From
(2.7-1)–(2.7-3),
v l l a N
S
r r
a M
cd eo c as i
i
S
s
st sb
c as i
i
S
s
y
= + +
+
(
)
= =
∑ ∑
( )
, ,
2
2
1 1
π
(15.3-17)
where l
eo
is the end winding offset, which is the amount of overhang of the end winding
between the end of the stator stack and the end winding bundle. The end winding offset
is a function of the manufacturing process. In general, it is desirable to make this as
small as possible, though extremely small values may increase leakage inductance and
core loss somewhat.
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
W
x
s pf eo
T
N l
=
[ ]
1 3
* *
α α
(15.3-18)
The output of our winding calculations are encapsulated by the vector
W
N N N N M M M
y
s as bs cs s as bs cs c wR ew cd
T
N a d l v
=
[ ]
1 3
α
(15.3-19)
Functionally, we have
W F W G
y W x
= ( , )
(15.3-20)
It will also prove convenient to define
W
W W
=
[ ]
x
T
y
T
T
(15.3-21)
15.4. MATERIAL PARAMETERS
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
594 INTRODUCTION TO THE DESIGN OF ELECTRIC MACHINERY
type, and the permanent magnet type. Based on these variables, the material parameters
can be established using tabulated functions in accordance with
S
=
F s
sc t
( )
(15.4-1)
R
=
F r
sc t
( )
(15.4-2)
C
=
F c
cc t
( )
(15.4-3)
M
=
F m
mc t
( )
(15.4-4)
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
S =
[ ]
ρ
s s lim
T
B
,
(15.4-5)
R =
[ ]
ρ
r r lim
T
B
,
(15.4-6)
C =
[ ]
ρ σ
c c lim
T
J
(15.4-7)
M =
[ ]
ρ χ
m r m lim
T
B H
(15.4-8)
In (15.4-5)–(15.4-8), ρ denotes volumetric mass density, B
s,lim
and B
r,lim
denote flux
density limits on the stator and rotor steel so as to avoid saturation, σ
c
is the conductor
conductivity, J
lim
is a recommended limit on current density, and B
r
, χ
m
, and H
lim
are
the permanent magnet parameters.
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
B H M= +
µ
0
(15.4-9)
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 χ
m
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
MATERIAL PARAMETERS 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
Figure 15.4-1. B–H and M–H characteristics of PM material.
H
H
c
H
ci
B
B
r
M
dB
dH
= +
0
(1 )
m
dB
dH
m
c
TABLE 15.4-1. Magnetic Steels
Material M19 M36 M43 M47
s
t
/r
t
1 2 3 4
B
s,lim
/B
r,lim
(T)
1.39 1.34 1.39 1.49
ρ
s
/ρ
r
(kg/m
3
)
7400 7020 7290 7590
TABLE 15.4-2. Conductors
Material Copper Aluminum
c
t
1 2
J
c,lim
(MA/m
2
)
7.60 6.65
σ
C
(MΩ
−1
/m)
59.6 37.7
ρ
c
(kg/m
3
)
8890 2710
TABLE 15.4-3. Permanent Magnets
Material SmCo5-R20 SmCo5-R25 SmCo17-R28 SmCo17-R32
m
p
1 2 3 4
B
r
(T)
0.9 1.0 1.1 1.15
χ
m
0.023 0.027 0.094 0.096
H
lim
(kA/m)
−1200 −1200 −1000 −675
ρ
m
(kg/m
3
)
8400 8400 8300 8300
596 INTRODUCTION TO THE DESIGN OF ELECTRIC MACHINERY
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
i I
as s r i
= +
(
)
2 cos
θ φ
(15.5-1)
i I
bs s r i
= + −2 2 3cos( / )
θ φ π
(15.5-2)
i I
cs s r i
= + +2 2 3cos( / )
θ φ π
(15.5-3)
where I
S
is the rms current, and ϕ
i
is the current phase advance. From our work in
Chapter 3, these quantities are readily expressed as
I
i i
s
qs
r
qs
r
=
( )
+
( )
1
2
2 2
(15.5-4)
φ
i qs
r
ds
r
i ji= −angle( )
(15.5-5)
Although the calculations of this section are very straightforward, for the sake of con-
sistency, they will be organized as in previous sections. We will define an input vector,
and output vector, and a functional relationship as
I
x
qs
r
ds
r
T
i i
=
[ ]
(15.5-6)
I
y
s i
T
I
=
[ ]
φ
(15.5-7)
and
I I
y I x
= F ( )
(15.5-8)
respectively. The amalgamation of variables associated with the currents is
I
I I
=
[ ]
x
T
y
T
T
(15.5-9)
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