IEEE Transactions on Energy Conversion, Vol. EC-2, No. 1, March 1987
116
SIZING EQUATIONS FOR ELECTRICAL MACHINERY
V.B. Honsinger, Life Fellow IEEE
Consultant
6 Woodstead Rd., Ballston Lake, N.Y., 12019
Abstract - Sizing equations for electrical machinery are developed from basic principles. The technique provides new insights
into:
1. The effect of stator inner and outer diameters.
2. The amount of copper and steel used.
3. A maximizing function.
4.dequivltydslobt
density distribution.
dimensions.in terms of diameters andflux
5. Pole number effects.
While the treatment is analytical, the scope is broad and intended to assist in the design of electrical machinery. Examples
are given showing how the machine's internal geometry can assume extreme proportions through changes in basic variables.
nous machines, the procedure can be modified for application to
other machine types such as dc machines, homopolar, claw-pole
and also to transformers.
GEOMETRICAL CONDITIONS
As prescribed by Fig. 1, the stator slot shape is constructed using circular and straight line segments. These conditions allow for
semi-enclosed or open slots; round bottom, flat bottom and intermediate configurations of the usual slot shape. The area of the
slot is given very nearly by
(1)
As= (wI + w2)d,2 + (wI - c1rI)rI + (w2 - clr2)r2
This equation assumes that the arc of each circle given by the radii
r, and r2, extends over 90 degrees whereas, in fact, two arcs cover
slightly more and two slightly less than 90 degrees; the error in
most cases is negligible. The constant cl is (4 - 7r)/2. Fig. 1 also
INTRODUCTION
prescribes that the tooth width is constant along a distance d,. For
most
part, the
equations
will
contain
this
th
*been used for decades to size
. electrical
.
ot
~~~~~~the
te
eutin
il
cnan
ti
pr,
The Dr2L equation 2has
e
.
condition.
However,
the
case
of
rectangular
slots
with
tramachinery. This equation written for motor operation, iS hp/ n~teeth
a subcase and is treated herein. Flux
does
form
equation
pezoidal
reltes
por
D
.
r 2,L. The equation relates power output hp and synchronous
densities are maximum, sine-wave densities. Such densities for
t
t
a output ccoefficient
an
speed n, to the rotor volume through
(, Itt thaigp,tehndcr(yk)reivnb
has been found worthwhile to re-examine the sizing procedure t
since the D 2L equation does not consider several key factors. For
Bg= (ir/2)X/ (r Di L/P)
example, machine size is affected by the complete stator geometry
(2)
B,= (r12)01(tS1LksP)
which includes the relative proportions of the stator inner and
outer diameters, slot and tooth dimensions, flux densities in the
B, = (0/2)/(dcLk,)
iron parts and the actual current densities in the conductors. The From these equations, flux density ratios referred to the air gap
output coefficient , in the D2L equation contains only the air gap
quantities "flux density Bg" and "surface current density KI".
There are no relationships connecting these air gap quantities with
the flux and current densities existing in the machine's interior.
A new sizing equation is developed in this report which eliminates the deficiencies just mentioned. It has the form hp/n, =
(0D3L. The equation is based primarily on stator geometry. The
output coefficient (O considers the iron flux densities and conducD
d
c
tor current densities; it contains a maximizing function fo(x)
/
which is extremely important to the theory described here and is
discussed in the body of the report.
A wide difference exists in the amount of geometry used to
derive the two equations. The D3L approach provides many rela2
tions between physical dimensions and density-like quantities and,
2
hence, is well adapted to produce a design that is geometrically
t
compatible from the start. From these two equations two other
equations are derived.
d
s
Sizing equations, by themselves, cannot predict machine performance which still has to be satisfied. Consequently, the sizing
t
procedure must be accompanied by a calculation of the machine's
electrical, thermal and mechanical parameters followed by an estimation of performance. Only the sizing procedure is reported
here. While the method applies directly to induction and synchro-
,i.eyThe
moutp hpeatn isyhrono
86 WM 212-5
A paper recommended and approved/
by the IEEE Rotating Machinery Committee of the/
IEEE Power Engineering Society for presentation at//
the IEEE/PES 1986 Winter Meeting, New York, New//
York, February 2 - 7, 1986. Manuscript submitted
August 29, 1984; made available for printing/
November 7, 1985.
\
Dr
Dj
/
/
/
Figure 1. A stator slot whose geometric shape is constructed using
only circular and straight line segments.
0885-8969/87/0300-01 16$01 .00©)1987 IEEE
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117
Bg/Bt = tSIkj(7rD1)
The first term on the right of (10) is the ratio copper volume
Bg/Bc = Pdks/D
(3) (core only) =to steel volume modified by a space factor k,,.
Defining V, SIAJ,L and Vt = ir D2L/4, (11) becomes
The ratio of the conductor area in each slot to the slot area is
() = V ( V k,) + 8I/D]
defined, for round wire, as
fu (
The steel volume ignores the portion removed
k, - A,u/A, - (v/4) d11.
d2 CsA
rotor and
cu
s- T
cu (4)sstator slots and is therefore related to the "purchasedforsteel"
D 2L
Designers often use other criteria related to the windability or easy rather than the active steel.
insertion of wire in slots. One such windability space factor for a
slot is
k'p = di2 CI(A5 - Ai)
(5) The DoL Equation Derived from Magnetic and Electric Loading
The equation purposely ignores the nesting effect of round wires. Constraints
It follows from (4) and (5) that kCu and k5, are related by k,u= ksp
Equation (10) is now revised to include the effect of magnetic
[(v/4) (duldiQs)2 (1- AIAs)]
and electric loading. This is achieved by directing the a,b flux
density coefficients and the conductor area ACU to follow certain
constraints. These are:
THE D3L
SIZING
EQUATION
0
Using the geometry of Fig. 1, five equations are formed. The
= 2 ml N1 h/Si Acu
first two contain the slot widths wl, W2 and the tooth width t. The
E = 4.44 f N1 kw (2 DiL/P) Bg x 10-8
third relates the stator outer diameter to the inner
diameter.
The last two equations define the tooth width t and
hp = ml V11 sj cos H/746
(12)
core depth dc in terms of the flux density ratios given by (3).
f = PnsI120
[DrDj+ 2 (do + r1)] = (t + w1) S1
The copper area is constrained to follow restrictions placed on
ii [Do - 2 (d" + r2)] = (t + w2) SI
the current density J1. The air gap flux density is directed to take
on values compatible with voltage, number of turns, etc. Solving
=
+
+
+
+
+
2
(7) the set of equations (12) for SIA,, obtains the total copper crossDo Di
(do r, ds r2 d,)
rD,Bg/ S1B,k5
t=
sectional area
SIA,, = 20.16 (hp/ns) x 101
d = DiBgIPB,ks
The principal slot dimensions w1, w2 and ds are found by simultaneous solution of the equation set (7). This obtains:
w, = (rIS1) [D (1 - G,) + 2 (do + rl)]
W2 = OTISO) [Do -Di ( Gt + G,) -2r2]
(8)
(W2i
- D (G5 + G~)
ds S WI)(SD0
d5= (W2- w1)
(51/277)/2
-
(8)
The slot area has been given by (1). By successively eliminating wl, W2 and ds in (1) through use of (8), a quadratic equation is
obtained which relates the stator geometry to the a,b coefficients:
aD,2-2b DiDo + D2= (4S1/7r) (A,l/k,,) + 81
(9)
(13)
Using (13) to eliminate SIA,, in (10) and after rearrangement,
the D 3L sizing equation is obtained
h
hpln, = 60 D 3L
(14)
where the output coefficient f0 is
0.0390 Bg Jlfo(X) k,, X I[ - ji/u
The function fo(x), like
LU
x 1011
fJx(X), is important to the theory which
follows
fo(X) = X f, (X)
where
The Importance of the Function f0 ( x )
The importance of the function fO(x) is two fold: 1) it directly
G,
G,
b = G, + G,
i = 4 [r2 (CS1 +1) + r? (cS1 -1) ± d2 + D,d0 (1- G,)]
+
c = (4 - f)/27T = 0.1366
The a, b coefficients in (9) are numerics that depend on flux
density ratios G, and G, which describe the flux distribution in the
machine referred to the air gap density. The quantity al accounts
for the variation in slot shape.
The Function fcu ( x ) Copper to Steel Volume Ratio
Equation (9) has a valuable interpretation which is obtained by
dividing through by Dl to form a function of the ratio DI/D0:
2
aI{l D I-2bI|'|+1l CrO/4 +l D
(10)
The function on the left side of (10) is of great importance in the
theory which follows. Hereafter, it will be called f A(X where A =
DID0.
fcu (A) = aX2_2bA+ 1
(16)
In machines where the slot radii (r1 and r2) are relatively small,
the output coefficient can be approximated by
= 0.0390 Bg Jil fo(A) X k,, x 10-11
(17)
G,= Bg/B,ks
G= (2/P) (Bg/Bcks)
a = (G, + GC)2 - (1-G)
b
(15)
influences machine size and 2) its locus passes through a maximum at intermediate values of X = DIDo. Actually the function
f0(X)
represents a family of curves (see Fig. 2) with each
defined by the parameters a and b. Each curve in the familycurve
has
its own maximum given by
=0
dfb(u)Idf
X(max) = [2b - (4b2 - 3a)'21/3a
(18)
To find an extremum of all functions, it is necessary not just to
find a maximum; rather, it is necessary to find a large member of
the family given by a specific a,b pair and then find the maximum
of that large member. The procedure will be to find a large
of the family of curves and then find a design region near
to the maximum using the equation
~~~~~~~~~~~~~~member
ThcofienQprvdsaotonomdfyheontXo
it wil be nert u o eesrl ttemxmmgvnb
(18). This will then define a design region as shown in Fig. 2.
(11)
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_~ ~ ~ ~ ~ ~ ~ ~ ~ -
118
fo (A)
Figure
2. Ilutrtn
th
faiyofcre
b
a
DESIGN
REGION
0
C >1
.8
NX
the
ag maximum
bvalu.2
f
)ad
2 t)2
A =Di /D00
1.0
~~~~~~~~~~~~~~~~~~~A Di /
Figure 3. The copper function fA (x) versus X = DiDDC plotted for
the special case where B, 0.8 B1 and P =4.
Figure 2. Illustrating the family of curves f0 (X) and a large
member of that family. The design region is near to or at
the maximum value of the large member.
_____
Simplification of the Functions fcu ( x ) and fo (OCUS)
To simplify the determination of f,,(x) and fW(x), attention will
be focused on only one set of poles at any one time. Since it
would not be unusual to relate Bc to Bt by some fixed proportion,
BC will be made equal to 0.8 B, for the time being. With this convention, a and b become functions solely of G, with the result that
the functions f,,(X) and fW(X) can be definitely specified. Figs. 3
and 4 illustrate the copper function f,,(X) and the output function
f0(X) including the extreme cases G, = 0 and G, = 1. Chart 1
shows the data associated with these plots.
.8
.6
.4
.3
____
____
_________
(MAX)
Gt=O
2
+
D
Gt .25|
.1
Chart 1: Data Associated with Figs. 3 and 4
Gt=35
x (zero)
G,
G,
a
b
x (max)
0
0.25
0
0.16
0.31
-1.0
-0.40
+0.41
0.0
0.41
0.81
0.58
0.46
0.36
0.39
0.25
0.17
1
0.87
0.76
Figure 4. The output function f0(X) versus X = DiDo plotted for
1.0
0.63
+2.64
1.63
0.21
0.09
0.62
the special case where B, = 0.8 B, and P = 4.
0.50
f0
(max)
A = Di / Do
Alternatively, one can fix the variable G, and vary the pole
numbers to ascertain their broad effects. Fig. 5 shows the rise of
the function fo(x) as the pole numbers increase with G, fixed at
0.5 and B, = 0.8 Bt.
The Flux Density Ratios: Gt and G c_
0.4
Several interpretations of G, and GC are now given. The quantities G, and GC are expressed by
0.3
G,= Bg/B, k5 = t/(7r D|/S,)
THE VALUE OF A Di /Do
WHERE fo (A) IS MAXIMUM
4
|l
/
MAXIMUM VALUE OF THE
0.2
Gt= (2/)(Bg/Bct ) = 2 dD
While G, is the ratio of the air gap flux density to tooth densi- -°
/
ty, it is also the ratio of tooth width to slot pitch. Similarly, G0 is
0.1
an air gap to core flux density ratio modified by the number of
poles but it is also the ratio of back iron depth x 2 to the stator 1
inner diameter. G, indirectly represents the relative proportions of
6
8
2
4
16
12
14
tooth and slot widths. The size of G, has much to do with wheth10)
oL
er the machine will be a "copper machine"~ or an "iron machine".
NUMBER OF' POLES
In fact, specification of the size of G, (and A) will guarantee that
the machine will take on the features shown by Fig. 6 which is deFigure 5. Effect of the number of poles on the output function.
scribed by the general behavior of the functions fCu(X) and f0(X).
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119
ALL COPPER
NO STEEL
NO44AL COPPER
NO COPPER
from which wI/w22= X (1 + G,) = 1. Since w1 and w2 are equal, it
follows from (8) that d, = 0 because d, (W2 - wI) (S1/21T) = 0.
SLT ,T S T
SLOT
SLOT
SLOT
In this limit, the slots have degenerated to infinitesimally small
rectangles existing on the stator inner surface as shown by the
V A X 1 S ,k r
r
upper right figure of Fig. 6. The slots have vanished and the machine has become all steel. The output function and output
1.0
w\\
coefficient
again are zero.
Case 2, G, = 1 and O: as G, becomes zero at intermediate
.8 \K-\ \ r
values of X, the tooth width t and core depth d, (d, is assumed to
>
P
\\
\
\;
>
>
be a fixed proportion of t) vanish. The machine now contains all
G,
O
+
copper and no steel as shown by the lower right figure of Fig. 6.
X
\\ X
\I
r\l
i
Similarly as G, approaches unity, the slots vanish and the ma.6-chine, so to speak, becomes all steel as shown by the lower left
Gt-.5
figure of Fig. 6. In direct contrast with case 1, however, is that
A
Athe output function and output coefficient do not vanish at the extremes Gt = 1 and 0 as they did for f,,(x) = 1 and 0. They do
CSG,-1 \
\
\
\
/
3
\ \
\
>
\ /
vanish, however, when x = 1 and x 1/(1 + Gc).
.2To summarize:, the proportion of copper to steel and the features of the machine together with the extremum of the output
_____\__\
function are affected in an intricate manner by the specification of
o
.2
.4
.6
A = DiD, and of G,. It now appears that one must not just find a
.8
1.0
large member of the family of curves f0(X) as previously specified
A? Di /Do
in Fig. 2; one must, instead, find a feasible and practical large
SLOT
member which probably is not the largest member.
S L, O < SLOTt
m
w
SLOTSL
_
_
EQUIVALENT SLOT DEPTH AND WIDTH
INCLUDING RECTANGULAR SLOTS
>w
>
s
\
>>
__Equation (9) can be written as two factors representing an
.8equivalent slot depth multiplied by a slot width. This product will
give the precise area of the slot. First (9) is rewritten as
+
68
&
t\1
1\
1
(7r/4S1) (A- 81) = A,
(19)
\
\\ 12 .5 \
\
04
where Al (an area roughly equal to the area of all slots) is
C).Nt> Al = a D,2_ 2b DIDo + DI
The area AI can be factored into a length times a width
.2- _ xX
A1 = [Do- Di (1 + GC)] [Do- Di (2Gt + GC - 1)]
______\_\_L
Consequently, using the factored A1 and after rearrangement of
(19), the area of one slot is precisely given by diw' = A, where
.
.
.
d'= (Do/2) [1-A (1 + G)]I
A=Di /Do
w'= (7r Do/2S1) [1- x (2G, + GC - 1)] (1- 81/AI)
Figure 6. The machine geometry caused first, by varying, A =
D/D0 (upper figure) and second, by varying Gt (lower
In the form just given, the slot depth d' has been defined as a
distance starting from the stator bore (see Fig. 1) and extending to
figure).
the outermost part of the slot or a distance d' = do + ri + d, + r2.
The
equivalent slot width, which is compatible with the depth d',
GenralBehviooftheFunctions General
f ( A ) and f( A )
Behavior of the ~ cu ~o is w'. The selection of an equivalent slot depth is arbitrary providThe proportion of copper to steel and the general features of
ed the product of the equivalent slot depth and width gives the
the machine are affected by the specification of A = DiDo and G,=
precise slot area. An improved version, which can simulate the
t/(7T D,/S1) which appear in the function f00(A). Changes in maslot reactance, is given by dow = d'w' where
chine geometry can be traced by examining the trend of f0"(A) as
e e
it proceeds from one extreme to another. The two basic extremes
de d' (1 - do/d')
to be considered are: Case 1, f00(A) = 1 and 0; Case 2 G, =
we- w' (1 + do/de)
1 and 0. Fig. 6 traces the geometry at and between these extremes.
In this case de has been defined as a distance, in Fig. 1, of deri + d, + r2. This model retains the tooth tip which has the actual
Case 1, f00(x) = 1, occurs when A = DI/D. = 0. This obtains
depth do- The equivalent slot, shown by Fig. 7, simulates both the
slot area and the slot reactance by adjusting the slot opening W0 to
ASS1= vD2/4orhat he otalslo cros sctioal rea ccuies
the entire space vr D2/4 as shown in the upper left figure of Fig. 6.
be Woe such that the slot reactances are equal. The permeance of
The steel has vanished. Although the copper function f00(A) is unthe slot is proportional to: P0 = d0/3W0 + d0/W0, in which
ity, the output function f0(A) and output coefficient (0 are zero.
1x 1+G)
a*S
The second part of this case, f0,(x) = 0, occurs when A =d=
[- (1+G)
1/(1 + G0). Substituting A = 11(1 + G0) in (8) obtains a slot
3WF 37r- [1 -A(2 G;, + G0 -1)] (1 - 8/A1)
geometry of
The semi-enclosed, rectangular slot just described is a
(1 - G)mathematical slot. The actual tooth width t is still constant. Howwl=
the equations do define the case where the actual slot shape
'
~~~~~~~~~~~~~ever,
iS open and rectangular such that, in Fig. 1, if w1 = w2 and r1 W2= (&rD,/S1) (1 - G,)/ (1 + G0)
NORMAL STEEL
ALL STEEL
=
yS
D
1.0-I
0O2f4
(rD,/S1)
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120
We
do
-
/~~~~~~~~~~~~~~~~~~~~~~
(/C\X
)
/
ede
lWoe
~~~~~~~~~~~~~~~~~~~~Dii
:- t
do
Figure 8. The rotor slot geometry.
Figure 7. An equivalent slot with tooth tips.
= 0 (the tooth now is trapezoidal), then al = 0 and A, = d'
w' where
2
td'-=
(L,/2:) [1 - Ak (1 ± G9]
(2G, + G,, - 11
provided that 2G, is defined to be
2G,= Bg/ks B, + Bg/k, B,2
where B,, and B,2 are the actual tooth flux densities at the extreme
ends of the trapezoidal tooth.
Finally, but of considerable importance, is that the slot area can
also be written as
A, = (rr D,/418) jt,,(x) (1 -- 61/A1)
(20)
THE ROTOR GEOMETRY
Although the rotor geometry used here (see Fig. 8) is typical of
an induction motor, the geometry can readily be changed to that
of either a salient pole or round rotor synchronous machine. Regardless of machine type, the rotors are of two kinds: 1) those
built on a rim with a large inner diameter; 2) those built on a shaft
which, itself, carries flux and where the effective inner diameter is
zero. Geometrical equations for the rotor use the same procedure
employed for the stator. From the geometry of Fig. 8:
[Dr - 2 (r1 + do)] = (WI + t) S2
(D,, + 2d,.) = (w2 + t) S2
Dr Dii + 2 (d. + d, + r1 + d,)
(21)
DrB2/S2 B, k,
d(. Dr Bg¢/PBc k,
Simultaneous solution of (21) gives
a'- 2b' (DIt-{fD,2 =
I D, J tDr J
82A',,
(1T Dr2/4) k'(,
where
a' = (I - G',)2_- G', -G',)2
b'= G',- G'r
62 = (r1 ± do) Ii + 4 Dr (1 -G',)]
.
DijIDr, the function fCU(X2) for the rotor is:
2a' - 2b' A-
-
For those rotors which use the rotor shaft as flux carrying member
and where, effectively, the rotor inner diameter can be set to zero
fc,(X2) = a
The slot area for an equivalent semi-enclosed rectangular slot is
given by A,2 = de2 We2 where de2= d' (1 - do/d') and We2 = W' (1 +
' = (irD,j2Sl) [1I -
t
Defining X2
do/de):
d'= (1/2)[Dr(l - G',) - Di,]
w'= (r/2S2) [Dr (1 + 2G', - 3G',) - Di] (1 - 8A2)
A2= a'D21
bDD
A r=aD
The definition of J1 and J2 are
2b'D,Dji- Di2
J, = 2m, N1 I1/SIA,,
J2= 2M2N2 12S2A 'c,
(23)
hence
J2
Jl
S1 A,,
S2 A'cu
m2 N2 12 l
ml N, I,
SI Ac, 1"2
S2 A'c, 1,
(24)
THE D2L SIZING EQUATION
r
An equation which commonly appears in the literature is the
Dr2L sizing equation. This equation makes use of a current sheet
which is obtained by removing all current from the stator slots and
spreading this uniformly over the air gap interface. The surface
current density K, A/in of the sheet is used instead of the volume
current density J1 A/in2 in the conductors. The Dr2L sizing equation is found by simultaneous solution of five equations:
E = 4.44 f N1 kw x 10-8
Bg= (17/2) hk/(ITDrL/P)]
+
62
Dr2
(22)
m=
hp mV 11 7 cos o/746
K1 =2tn1N1 Ii! TDr
(25)
f = PnJ/120
which gives
hp/n5= (r Dr2L
(26)
where (r iS the output coefficient equal to
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121
r=0.1558 Bg K1 x x 101
THE D,25L AND D2s5L SIZING EQUATIONS
Multiplication of the two sizing equations for Do3L and Dr2L or,
(14) x (26) and taking the square root yields a basic equation.
D1
(28)
hp/n5 =
Equation (28) can be expressed in terms of Dr or else Do. In
terms of D, (28) becomes after elimination of Do using
Di= Dr(1+2g/Dr) = DrCg and Do = Di/X = DrCglx.
C2 /X3 D25L
(29)
liplns=
In terms of Do, (28) becomes after replacing Dr in (25) with Di
and using Di = xDo.
X2D5L
hlp/n5 = 1o Dr
The factor
,
(30)
3. A maximizing function, either fi(x) or X2f/(X), which is used
to maximize the output coefficient. While this is a desirable
goal, other criteria could favor a design not at the maximum
output coefficient.
4. Equivalent slot dimensions in terms of the broad variables
Si,D,,x,G,, and GC. Hence, an equivalent slot (a slot with an
area equal to the actual slot) can be found early in the design
process.
5. Pole number effects as depicted by Fig. 5.
The design process begins with a sizing procedure but must be
accompanied by a computation of the machine parameters fol
lowed by an estimation of machine performance. This involves a
search technique of considerable complexity. It is believed that
the equations developed here will serve to unify and simplify the
many relations and constraints needed for that technique.
/x2 in (30) with k8 = 1-81/[DO2 fcu(X)I is
-O =/ 2 = 0.0779 BgX J/ KI kc5 A 2 f0 (AX) ka x I10- I
(31)
Equations (29) and (30) add to the knowledge as follows:
1. Common to both equations is the term '/-Ki in which the
product J1K, is proportional to the stator I2R loss (core only)
per unit of stator core area and is therefore closely related to
the temperature rise of the machine.
pi .Ji piK1Ji=Kiml mih1 I?r1/7TDIL
rilaDiL
=
where pi is the resistivity of the stator conductors.
2. In the output coefficient for the D,25L equation, the function
x2f0(X) like fo(x) displays a maximum given by
X (max) = 0.8(b-J
b-0.9375a) /a
The location of this x(max) falls to the right of (is larger than)
the A(max) given by the D3L equation shown by Fig. 4. In
contrast, the output coefficient for the Dr2_5L equation continually decreases with respect to X; it displays no maximum.
CONCLUSIONS
Four sizing equations have been developed to assist in the
design of electrical machinery. These are:
hp/ns= (0 DI3L
hp/n, = r Dr2L
hp/ ns = ( DU, L
hp/ n = 'r D 2.5L
o
The equations are derived using algebraic and geometrical relationships. The D,3L and Dj25L equations are preferred because
they give so much more information than do the other two. The
information is contained within the output coefficient (% or f.o
which includes:
1. Useful information regarding the ratio x = Dl/Do.
2. Insights into the amount of copper and steel used.
LIST OF PRINCIPAL SYMBOLS
Do, Di = Stator outer and inner diameters
Dr, Dii = Rotor outer and inner diameters
=
Di/Do,Dii/Dr
~LX,X2 = Core
length
Si, S2 = Number of stator slots and rotor slots
Gt
G,
a, b
81
Acu
= Bg/B5k, = t/(7Di/Sj)
kS
= l-81/[DDo
=
= (2/p) (Bg/Bc
= See Equationks)(9) 2dc/Di
= See Equation (9)
= Area of copper per slot
= Area of slot
As
= aD2 - 2bDiD0 + D 2
Ai
= Area of slot insulation
Ai
= AcuAs
kc
= Steel stacking factor
COC Sks
=kw Stator pitch x distribution factors
Cs
= Conductors per slot
Ni
= Turns per phase
mlI
= Number phases
= Diameter of bare conductor
dcu
= Diameter of conductor over insulation
dins
= Flux per pole
0
= Volume current density, A/lin2
Ji
= Surface current density, A/in
K1
11r
= Stator line current
II
= Number of poles
p
V
= Voltage at machine terminals (per phase)
0
cos
=
Product of efficiency and power factor
-q
x
= k, (VIE) -q cos o
f
= Frequency, hz
fcu(X)I
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