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is determined mainly by what is known as the whirling speed,
which, in turn, depends upon the deflection of the rotor considered
as a beam with the points of support at the centers of the two
bearings. There will be one or more critical speeds at which
the frequency of the- bending due to the weight of the rotating
part will correspond exactly with the natural frequency of vi-
bration of the shaft considered as a deflected spring. The vibra-
tion will then be excessive, causing chattering in the bearings
and abnormal stresses which may lead to fracture of the shaft.

The maximum deflection of the rotor due to its own weight
together with the unbalanced magnetic pull (if any) can be
calculated within a fair degree of accuracy when the position of
the bearings and the cross-section of the shaft are known. The
whirling speed of a rotor with steel shaft, in revolutions per min-
ute, can then be calculated, because it is approximately

190

V Deflection in inches

In turbo-alternators the whirling speed is generally higher
than the running speed; but this is not a necessary condition
of design; and in direct-current steam-turbine-driven dynamos,
where the provision of a commutator calls for the smallest possible
diameter of shaft, the whirling speed is commonly lower than the
normal running speed. In such cases it is necessary to pass
through the critical speed, causing vibration of the rotor, every
time the machine is started or stopped; but this is not a serious
objection. A good rule is to arrange for the whirling speed to
be either 25 per cent, above, or 25 per cent, below, the running
speed. Taking as an example the design under consideration,
the whirling speed should be either 1,800 + 25 per cent. = 2,250;
or 1,800 25 per cent. = 1,350. In the first place the permis-
sible deflection would be (o^n) = 0.00714 in.; and in the

(190 \ 2
J-OKQ) = 0.0198 in.

By making a very rough estimate of the rotor weight and the
span between bearings, it will be seen that the smaller deflection

328 PRINCIPLES OF ELECTRICAL DESIGN

(corresponding to the higher critical speed) is easily attainable,
the diameter of the shaft being of the order of 13 in. near the
rotor body, and 10^ in. in the bearings. It is not proposed to
go further into details of mechanical design; but attention may
be called to the fact that a rotor forged solid with the shaft,
or a solid rotor with the shaft projections bolted to the two ends,
(i.e., without a through shaft), is stiffer than a laminated rotor
with through shaft. We shall assume a solid rotor in this design,
although the length of the rotor body (about 49J^ in.), being
less than one and one-half times the diameter, would indicate
the feasibility of a rotor built up of steel plates.

Items (21) to (27). On account of our having an odd number
of conductors per slot, we shall decide upon a single layer winding
(see Art. 78, Chap. XII). The current density in the armature
windings cannot be determined by the empirical formula (96) of
Art. 81, because this is not applicable to speeds higher than 8,000
ft. per minute, and in any case, the conditions of cooling in an
enclosed machine with forced ventilation are not the same as
for a self-ventilating generator. In a turbo-alternator there is
usually plenty of room for the armature conductors, the chief
trouble being with the rotor winding, which may have to be
worked at a high current density. There is no definite rule for
the most suitable current density in the armature conductors,
the permissible copper cross-section being dependent on the
length of armature core, the position and area of the vent ducts,
and the supply of air that can economically be passed through the
on the allowable current density in the copper; and, as a guide
in making a preliminary estimate, we may use the formula

160,000

which gives us for item (22) a current density of 2,000 amp. per
square inch of armature copper.

It is well to laminate the conductors in a direction parallel
to the slot leakage flux (see Art. 88, page 267, and Art. Ill,
page 318), and we may build up each conductor of four flat
strips each % by 0.14 in., giving a total cross-section of 0.35
sq. in. per conductor.

There will be 12 copper strips in each slot, the total thickness,
including the cotton insulation, being about 1.92 in. The slot

EXAMPLE OF ALTERNATOR DESIGN

329

insulation should be about 0.16 in., or 5^2 m -> thick (see Art. 80,
page 259), and the total slot space for winding and insulation
will be 1 in. wide by 2^ in. deep. The thickness of wedge might

A \ B

r e '\

*r ^;

?'*

A u-.-!

,,

1.64

1.58

T

FIG. 135. "Developed" section through stator and rotor teeth of 8000
k.v.a. turbo-alternator.

be % in., and we shall, in this design, allow an extra slot depth
of 1^2 in- above the wedge, with a view to increasing the slot
inductance, and so limiting the instantaneous rush of current
in the event of a short-circuit. This increased armature indue-

330 PRINCIPLES OF ELECTRICAL DESIGN

tance might have been obtained by using a smaller width and
greater depth of copper conductor; but, seeing that the width
of tooth will probably be sufficient, the proposed design of slot
(as shown in Fig. 135) has the advantage that the eddy-current
loss in the armature inductors, from both causes referred to in
Art. Ill (page 318), will be very small.

The width of copper strip was selected to fit into the 1-in.
slot, because this seems to provide a suitable cross-section for
the stator tooth. Thus, a section halfway down the tooth, or
(say) 2 in. from the top, will have a diameter of 44 in., and the

average width of tooth will be v . g - 1 = 1.88 in. On the

basis of B g = 6,000 gausses, and a sinusoidal flux distribution
over the pole pitch, the " apparent" tooth density (item (27))

u i_ v n \la TT X 6,000 X 2.62 X 51

would be 2 B aWn = - 2 xi.88X46^8- 14 ' 3 gaUSS6S

which is not too high (see Art. 76, page 251).

Items (28) to (32). Assuming a flux density of 8,500 gausses
in the armature core (see Art. 88, Chap. XII), the net radial depth
of stampings below the slots will be

62.2 X 10 6

= 11.6 in.

2 X 8,500 X 6.45 X 48.8

The actual radial depth should be greater than this to allow for
the reduction of section due to the presence of axial vent ducts.
In this particular machine it is proposed to ventilate, if possible,
with axial ducts only, and a fairly large cross-section of air
passages must therefore be allowed. An adequate supply of air
will probably be obtained if the total cross-section of air duct
through the body of the stampings (in square inches) is not less
than 0.005 X cubic inches of iron in stator below slots. In this
case the volume of iron in the stator ring will be approximately
7r(48.25 + 11.6) X 11.6 X 46.8 = 102,000 cu. in.; and the total
cross-section of air ducts in the stampings should be 0.005
X 102,000 = 510 sq. in.

The actual radial depth of stamping below the teeth can be
calculated by assuming that the air ducts reduce the gross depth

510 f . 510

by an amount equal to : 7 ' or (say) ft0 =

average circumference TT X 62

2% in. approximately. Let us make the depth R d (item (29)) = 14
in., and provide vent ducts arranged generally as shown in Fig.

EXAMPLE OF ALTERNATOR DESIGN

331

136, where there are 10 holes per slot, each \Y in. irTdiameter,
making a total of ^ (1.25) 2 X 10 X 48 = 589 sq. in.
The weight of iron in core (item 30) is

0.28 X 46.8 X [7r(38.T25 2 - 2fl25 J ) - 589] = 28,000 lb., approxi-
mately.

The weight of the iron in the teeth (item (31)) is
0.28 X 46.8 X [7r(24.l25 2 - 20 2 ) - (48 X 1 X 4.125)] = 4,900 lb.

Taking the approximate flux densities as previously calculated
(items 27 and 28), and referring to the iron-loss curve, Fig. 34,

Holes (Total) U* Diam. .-

FIG. 136. Armature stamping of 8000 k.v.a. turbo-alternator.

page 102, the iron loss per pound for carefully assembled high-
grade armature stampings is found to be 6.1 and 3.2 watts in
teeth and core respectively. The loss in the teeth is therefore

6.1 X 4,900 = 30,000 watts, and in the core below the teeth,

3.2 X 28,000 = 90,000 watts, making a total of 120 kw., or 1.5
per cent, of the rated output, which is not excessive although
quite high enough for a machine of 8,000 kw. capacity.

Items (33) to (36). In a machine of so large an output as the
one under consideration, the weight and cost of copper should be

332 PRINCIPLES OF ELECTRICAL DESIGN

determined by making a drawing of the armature coils and care-
fully measuring the length required. Since this design is being
worked out for the purpose of illustration only, we shall use the
formula (97) of page 261, and assume the length per turn of
armature winding (item (33)) to be

(2 X 51) + (2.5 X 31.42) + (2 X 6.6) + 6 = 199.8 in.

It will be safer to use the figure 210 in. for this mean length ;
because all the coils will probably be bent back and secured in
position by insulated clamps in order to resist the mechanical
forces which tend to displace or bend the coils when a short-
circuit occurs.

The cross-section of the conductor (four strips in parallel) is
0.35 sq. in., or 445,000 circular mils. The number of turns per
phase is 24, and the resistance per phase at 60C. is, by formula

210 V 24
(21), page 36, 445^ = 0.01135 ohm. The IR drop per

phase (item (35)) is 6.01135 X 700 = 7.95, or (say) 8.4 volts in
order to include the effect of eddy currents in the conductors.
The PR loss in armature winding (item 36) is 3 X 0.01135 X
(700) 2 = 16,700 watts, which should be increased about 25 to 30
per cent, (see Art. Ill, Chap. XIV) to cover sundry indeterminate
fore be estimated at 21 kw.,.or 0.26 per cent, of the rated full-
load output; which is about what this loss usually amounts to in
a turbo-generator of 8,000 k.v.a. capacity.

Items (37) and (38). The reactive voltage drop per phase due
to the cutting of the end flux cannot be predetermined accurately;
but we may use the empirical formula (99) page 266, wherein the
symbols have the following numerical values.

The constant k will be fairly high in turbo-alternators, and we
shall assume the value k = 1.5. For the other symbols we have:

/ = 60

p = 4
T s = 3
n s = 4
I c = 700

In regard to l e and V , the mean length per turn (item (33)) was
assumed to be 210 in. The length l e is therefore 210 2l a or,
l e = (210 - 102) X 2.54 = 275 cm.

EXAMPLE OF ALTERNATOR DESIGN 333

The average projection beyond ends of slots measured along
the side of the coil, whether straight or bent, is

The reactive voltage due to cutting of end flux, with full-load
current per phase, is then

2.22 X 1.5 X 60 X 4 X 9 X 275 X ( g) logio (12 X 4 X 28.8)

X 700 X 1C- 8 == 116 volts.

The loss of pressure due to the slot flux can be calculated as
explained in Art. 97 using formula (106) on page 288, wherein
the quantities /, p, T t , n t , and I c have the same numerical values
as in the formula for end-flux reactive voltage. The remaining
quantities are: the gross core length l a = 51 X 2.54 == 129.5 cm.,
d\ = 2% in., and s = 1 in. The permeances P 2 and P 8 of the
flux paths in the air spaces above the conductors can be calcu-
lated as follows. Neglecting the widening of the slot to accom-
modate the wedge, the permeance of the slot above the winding

2X1

(see Fig. 135) is PI = ^ = 2, per centimeter length of arma-
ture core measured parallel to the shaft. The permeance of the
path from tooth top to tooth top may be calculated by assum-
ing the m.m.f. of one armature slot to set up the flux in an air
space of radial depth d = % in., and of length X = 2.62 in.

7X1
Thus PS = Q g = 0.334. This last quantity cannot have

o X ^.D^ (

a numerical value smaller than as calculated by this method.
If the rotor teeth were built up of thin plates like the stator, the
numerical value of PS would be greater than 0.334; but we are
assuming a solid steel rotor (which is customary), and for this
reason it will be best to neglect the flux paths through the iron
of the rotor teeth. We are concerned mainly in providing enough
armature reactance to keep the short-circuit current within
reasonable limits, and as a sudden growth of leakage flux is im-
possible in solid iron owing to the demagnetizing effect of the
eddy currents produced, the value for P 3 as here calculated will

By inserting all these numerical values in formula (106), we
get for the volts lost by slot leakage when the armature conduc-
tors are carrying full-load current, E. = 189, or (say) 190 volts.

334 PRINCIPLES OF ELECTRICAL DESIGN

Items (39) and (40). We are now in a position to draw a
vector diagram similar to Fig. 115 for any power-factor angle 6,
the calculated numerical values of the component vectors being:

6,600

V 3
E t P = 8.4
PE g = 116
E E' Q = 190

For a given terminal voltage of 6,600 (or 3,810 volts as meas-
ured between terminal and neutral point) the required developed
voltage will be a maximum when the external power-factor angle
is equal to the angle E g E t P because the additional voltage to be
generated will then be E t E a , which, in this particular example,

c
FIG. 137. Vector diagram for 8000 k.v.a. turbo-alternator.

amounts to V(8.4) 2 + (116) 2 = 116.4 volts, the effect of the
armature resistance being negligible as compared with the react-
ance of the end connections.

For 80 per cent, power factor, as mentioned in the specification,
the angle 6 will be 36 52', and the developed voltage per phase
winding (see Fig. 137) is

OE g = \/(OB)*+(BE g ) 2
where OB = OA + AB = OE t cos 6 + E t P
and BE g = BP + PE g = OE t sin + PE g .

Similarly, for calculating the full-load flux per pole (see Art. 99)
we have:

"Apparent" developed voltage = OE' g = V(OB) 2 + (BE' g ) 2

EXAMPLE OF ALTERNATOR DESIGN 335

Solving for the numerical values, we get:

OE g = 3,875 volts (item (39))
and

OE' a == 4,000 volts.

The full-load flux per pole (item (40)) is therefore,

Item (17) X OE' g _ 62.2 X 10 6 X 4,000
OE a 3,810

= 65.2 X 10 6 maxwells.

Item 41. Since we have decided upon a cylindrical rotor, the
variation of flux density over the pole pitch must be obtained by
distributing the field winding in slots on the rotor surface; but
in the design of salient-pole machines, the pole face should be
shaped as explained in Art. 90 (page 269), and the approximate
dimensions of the pole core should be decided upon with a view
to providing sufficient space for the exciting coils. The cross-
section of the pole cores would be determined as in the case of
continuous-current dynamos by calculating or assuming a leakage
factor (see Art. 103, Chap. XIV) and deciding upon a flux density
in the iron (about 14,000 or 15,500 gausses).

Items (42) to (44). Let us try a rotor as shown in Fig. 103,
with eight Blots per pole, only six of which are wound, leaving
two slots without winding at the center of each pole. The slot

vx OO OC

pitch (item (43)) is therefore ~~ " = 3-76 in. This dimen-

X X 12
sion expressed in terms of the stator diameter is g = 3.93 in.

The slot width may be decided upon by arranging for a fairly
high density in the rotor teeth. Thus, the open-circuit flux
(item (17)) which passes through a total of eight teeth is 62.2 X 10 6
maxwells. If t r is the average width of rotor tooth, in inches, the
average tooth density in maxwells per square inch is B" =

8 XL V 4Q V wn ^ cn mus t be multiplied by to obtain the ap-
proximate maximum density in the teeth near the center of the
pole. Neglecting leakage flux, and assuming B" max . = 120,000,
the tooth width t r will be 2.06 in., which indicates that a slot 1%
in. wide will probably be suitable.

Before deciding upon the depth of rotor slot, it will be advisable
to calculate the equivalent air gap in order that the field ampere-
turns and necessary cross-section of copper may be determined.

336 PRINCIPLES OF ELECTRICAL DESIGN

The thickness of wedge for keeping the field windings in posi-
tion might be about 1J^ in. as shown in Fig. 135; but as the cen-
trifugal force exerted upon it by the copper in the slot may be
very great on account of the high peripheral velocity, careful
calculations should be made to determine the compression and
bending stresses in the wedge. The allowable working stress for
manganese-bronze or phosphor-bronze wedges is about 14,000 Ib.
per square inch.

Although we are not designing a single-phase turbo-alternator,
it may be stated here that a convenient means of providing
ammortisseur or damping windings on the rotors of single-phase
machines (see Art. 113) is to use copper wedges in the slots and
connect them all together at the ends by means of substantial
copper end rings.

While discussing the matter of rotor slot design, the question
of stresses in the rotor teeth should be mentioned. After the
slot depth has been decided upon, the centrifugal pull on the
rotor tooth should be calculated and the maximum stress in the
steel determined, the slot proportions being modified if this stress
exceeds 14,000 Ib. per square inch for cast steel or 16,000 Ib. per
square inch for mild steel. The total centrifugal pull at the root
of one tooth is due to the weight of the tooth plus the contents
of one slot, including the wedge, while the pull at the narrow
section near the top of tooth (the width W in Fig. 135) is due to
the contents of one slot plus the wedge and the portion of the
tooth above the section considered.

Items (45) and (46) . The calculation of the average permeance
of the air gap between rotor and stator is carried out as explained
in Art. 93 of Chap. XIII. The approximate paths of the flux
lines are shown in Fig. 135 which is a " developed" section through
the stator and rotor teeth; that is to say, no account is taken of
the curvature of the air gap, the tooth pitch on the rotor being
made exactly equal to 1.5 times X, namely 3.93 in., or 10 cm.
The actual air gap from tooth top to tooth top (item (9)) is d =
0.875 in., and if we neglect the slightly increased reluctance due
to the angle of the tooth sides under the wedge, the component
flux paths may be thought of as made up of straight lines, or 1 of
straight lines terminating in quadrants of circles. The perme-
ance of each section of the flux path between stator and rotor
over a space equal to the rotor slot pitch is easily calculated as ex-
plained in Art. 5, Chap. II (cases a and c). The calculated nu-

EXAMPLE OF ALTERNATOR DESIGN 337

merical values of the permeances per centimeter of air gap
measured axially are:

Path A = 0.5725

Path B = 0.408

Path B = 0.408

Path C - = 1.063

Path D = 0.2845

Path E == 0.323

Total = 3.059

The permeance per square centimeter cross-section of air gap

3 059
is therefore -' = 0.3059, and the equivalent air gap is d e =

SOW = ^'^ cm ' J or l'^85 in. If great accuracy is required,
a similar set of calculations should now be made with the relative
position of rotor and stator teeth slightly changed so as to bring
the center lines of two teeth to coincide, instead of the center
lines of two slots as shown in Fig. 135, and the mean of the two
calculated values will more nearly correspond with the average
air-gap permeance. The actual permeance is always somewhat
greater than the value obtained from calculations based on cer-
tain conventional assumptions regarding the flux paths, and we
may take the equivalent air gap to be 6 e = 1.25 in.

Should any difficulty be experienced in calculating the per-
meance of a flux path such as E, with curved flux lines at both
ends, it is always permissible to divide it in two parts as indicated
by the letters EI and E 2 in Fig. 135, the permeance of each part
being calculated separately. Thus, in the example which has
just been worked out, the permeance of E\ is 0.686, and the per-
meance of E 2 is 0.612. The total of 0.323 is obtained by taking
the reciprocal of the sum of the reluctances.

Item (47). The calculations for the curves of Fig. 138 have
been carried out in the same way as for Fig. 81 (item (70), Art.
63, Chap. X), using formula (62) of Art. 37 for obtaining the ap-
proximate relation between air-gap and tooth densities when the
iron is nearly saturated. A curve must be plotted for the rotor
teeth as well as for the stator teeth; indeed it is in the rotor teeth
that the difference between "apparent" and actual density in the
iron will be most marked, since it is there that the flux density
will attain the highest values. In practice it will rarely be neces-

338

PRINCIPLES OF ELECTRICAL DESIGN

sary to take account of the effects of saturation in the stator
teeth of alternators, because with the comparatively low flux
densities especially in 60-cycle machines no serious error will

ZDOUO

24000
28000
22000
21000
20000
19000
~ 18000

5 T7fW^

/

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/

y

4

/

/

J

/

/

/ 1 .

/

/

/

/

/

/

/

/

/ /

/

i
i

/

/

/

y

1

/

/'/

'/

//

/

Tooth Density B t (Ga

L

/

/
/

y

//

/

/

//

/

/

/

\$

/

f

/

/

/

/

/

-

/

/

/

f

/

6000 8000 10000 12000 14000 16000 18000

Air Gap Density B g CGausses)

FIG. 138. Tooth densities in terms of air-gap density 8000 k.v.a.

turb o-alternat or.

be introduced by using the " apparent" tooth density in the,
calculations.

The depth of slot in rotor can be approximately determined as
follows :

EXAMPLE OF ALTERNATOR DESIGN 339

With an equivalent air gap d e = 1.25 in. and an assumed
sinusoidal flux distribution, the field ampere-turns on open cir-
cuit to overcome the reluctance of air gap only will be

or. 62.2X10'X,r 1.25X2.54

" (51 X 31.42 X 6.45) X 2 * 0.4ir

24 000

which makes the ampere-conductors per slot - = 8,000.

o

This does not include the. ampere-turns to overcome the reluc-
tance of the teeth and the remainder of the magnetic circuit,
neither does it take into account the considerable increase of ex-
citation with full-load current on a power factor less than unity.
The current density in the copper may, however, be carried up
to 2,500 or even 3,000 amp. per square inch of copper cross-sec-
tion, and it is probable that a slot 5 in. deep will provide sufficient
space for the field winding.

The numerical value of the symbol d in formula (62) should be
something greater than the actual clearance of % in. between
the tops of the teeth on armature and field magnet, and since the
difference between the equivalent and actual air gap is % in.,
we may suppose the effect of slotting the surfaces to be equivalent
to removing %\$ in. from both stator and rotor. Thus the numer-
ical value of 6 for use in formula (62) will be % in. -f % e m - =
iHe m - If we take the tooth width at a point halfway down
the tooth, the symbols in formulas (62) and (63) have the follow-
ing values:

d =* .0625 in.

t = 1.89 in.

d = 4. 125 in.

t == 1.64 in.

d = 5.0 in.

Stator
Rotor

Since we are considering the air-gap density over the surface of
the stator, the slot pitch for the rotor has been taken as X = 3.93,
or one and one-half times the stator slot pitch. As there are no

radial vent ducts in the rotor, the ratio T- in formula (62) may be

la

taken as

Axial length of rotor _ 49.5 _
Gross length of armature ~ 51

Item (48). With equally spaced slots around the rotor
including the unwound portions of the pole face only one satura-

340

PRINCIPLES OF ELECTRICAL DESIGN

tion curve for air gap, teeth and slots need be drawn. This
curve (Fig. 139) is constructed as explained in Art. 42, the con-
struction having previously been illustrated in connection with
the example in dynamo design (item (71), Art. 63, Chap. X).
The ampere-turns per inch length of tooth are read off Figs.
3 and 4; and SIMPSON'S rule (see Art. 38, formula 64) is used in
calculating the ampere-turns required for the rotor teeth at the

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 27 of 30)