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# Principles of electrical design; d. c. and a. c. generators online

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higher densities. In regard to the stator teeth, the mean value
of the tooth density may be used in determining the ampere-
turns required; the application of SIMPSON'S rule being in this

10000

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I 5000

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10000 12000 14000 16000 18000 20000 22000 24000 26000 28000

Ampere Turns per Pole

FIG. 139. Saturation curves for air-gap teeth, and slots 8000 k.v.a.

turbo-alternator.

case an unnecessary refinement. 1 In calculating the tooth
reluctance for plotting the saturation curve Fig. 138, no correc-
tion for leakage flux has been made. It is true that the total
flux in the body of the rotor is somewhat greater than the useful
flux entering the armature; but the omission of this correction
may be set against the fact that the tooth density calculations

1 In most cases, the practical designer who cannot afford to spend much
time on refinements of calculation calculates the density at a section
one-third of the tooth length measured from the narrowest end, and he
uses this value in getting an approximate average value of H from the B-H
curve.

EXAMPLE OF ALTERNATOR DESIGN 341

make no allowance for the flux lines which pass from the sides
of the tooth into the iron at the bottom of the slot, thus causing
the actual density at the narrowest part of the tooth to be some-
thing less than the calculated value.

The previously estimated depth of 5 in. for the rotor slot seems
rather large, as it leaves hardly sufficient section of iron at the
root of the tooth. We shall therefore reduce this depth to 4?4
in. as dimensioned in Fig. 135. The width of the tooth at the

bottom is therefore - - 1.625 = 1.2 in.

oZ

If a larger section of iron should be found necessary, it can
be obtained by reducing the size of the slots at the center of the
pole face (i.e., those which carry no field coils); but this question
can be settled later.

The curve marked "air gap, teeth, and slots" in Fig. 139,
shows what excitation is required to produce a particular density
in the air gap. The departure from the air-gap line (the dotted
straight line) is due almost entirely to saturation of the rotor
teeth, the reluctance of the stator teeth being negligible as com-
pared with that of the l>-in. air path.

Items (50) and (51). The upper curve of Fig. 140 shows the
ideal flux-distribution curve for open-circuit conditions. It is
a sine curve of which the average ordinate is

62.2 X 10 6
* 6.45 X 31.416 X 51 =

and of which the maximum value is therefore ~ X 6,010 = 9,450

i

gausses. The area of this curve is a measure of the total air-
gap flux on open circuit (item (17)). The pole pitch represented
by 180 electrical degrees has been divided into eight parts,
and the height of the vertical lines is a measure of the flux density
in the air gap over the center of a rotor tooth.

By providing a datum line and vertical scale of ampere-turns
immediately below the no-load flux curve, it becomes a simple
matter to plot an ideal curve of m.m.f. distribution over the
pole pitch, the shape of this curve being such as to produce the
desired flux distribution (see Art. 93, Chap. XIII). It is merely
necessary to read off the curve of Fig. 139 the ampere-turns
corresponding to the required air-gap density and to plot this
over the center of the corresponding tooth. In this manner
the lower curve of Fig. 140 is obtained. The practical approxi-

342

PRINCIPLES OF ELECTRICAL DESIGN

mation to this ideal m.m.f. distribution would be the arrange-
ment shown by the stepped curve, with 9,000 ampere-conductors
in each slot. This would produce a flat-topped flux-distribution
curve, a condition which might be remedied by putting 5,000
ampere-turns in the empty slots at the center of the pole face;

"0 22.5 U 45 U 67.5 f 90

Electrical Degrees

FIG. 140. Ideal flux-distribution and m.m.f. curves 8000
turbo-alternator.

k.v.a.

but such a procedure would be very uneconomical and unsatis-
factory. The best thing to do will be to increase the permeance
of the center tooth, either by reducing the width of the two slots
on each side of the center tooth, or by partly filling up these
slots with iron wedges so shaped as to produce the effect of a
tooth with parallel sides. These slots should, in any case, be

EXAMPLE OF ALTERNATOR DESIGN

343

filled with material equal in weight to the copper and insulation
in the wound slots, in order to improve the balance and equalize
the stresses at high speeds; and the proportion of magnetic to
non-magnetic metal can be so adjusted as to obtain any desired
tooth reluctance. If we provide wedges having a thickness of
% m - a ^ the bottom of the slot, we can get the equivalent of a
center tooth 2.2 in. wide with parallel sides. This calls for an
additional curve in Fig. 139, which can be calculated in the same
manner as the curve previously drawn, except that the correction
for taper of teeth (SIMPSON'S rule) has not to be applied.

If we decide upon a rotor winding with 9,000 ampere-con-

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10 20 80 40 60 60 70 80 90 100 1 10 120 130 140 150 1GO 170 180 10 20 3C
Electrical Degrees
1. Air-gap flux-distribution curves 8000 k.v.a. turbo-generato

FIG

ductors per slot as indicated by the stepped curve in the lower
part of Fig. 140, we shall obtain an open-circuit flux curve (A)
as plotted in Fig. 141. This curve would be exactly similar in
shape to the flux curve of Fig. 140 if it were not for the fact that
the widening of the tooth at the center of the rotor pole face
has lowered the reluctance at this point rather more than would
have been necessary in order to obtain the perfect sine curve
of flux distribution. The slightly higher ordinate at the center
of the new flux curve adds so little to the area of this curve that
we shall not trouble to measure this. It is evident that the
proposed excitation with 9,000 ampere-conductors per slot will
generate the required open-circuit voltage.

344

PRINCIPLES OF ELECTRICAL DESIGN

The m.m.f. curve for flux curve A has been re-drawn in Fig.
142, the stepped curve being replaced by a smooth curve. In
this connection it should be noted that the " fringing" of flux at
the tooth tops tends to round off the sharp corners of the flux-
distribution curves, and so justifies the use of smooth curves in
any graphical method of study. At the same time, it will gen-
erally be possible to detect in oscillograph records of the e.m.f.
waves the irregularities or "ripples" due to the tufting of the
flux at the tooth tops; but these minor effects will not be con-
sidered, either here or later when calculating the form factor of
the e.m.f. wave.

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 10 20 30 40
.Electrical Degrees

FIG. 142. M.m.f. curves for 8000 k.v.a. turbo-generator.

Items (52) to (54). The area of the flux curve A of Fig. 141,
on the assumption that it is a true sine curve similar to the one of
Fig. 140, and on the basis of unit squares with sides equal to
1 cm., is 5.91 X 18 = 106.3 sq. cm., where 5.91 is the average
density in kilogausses (item (18)). The required area of the full-
load flux curve C, at the specified power factor of 0.8, is therefore

106.3 X

where the figure 4,000 is the length of the

vector OE'g of Fig. 137, as calculated under item (40), and the
figure 3,810 is the open-circuit star voltage (vector OEt).

In order to determine the field excitation necessary to provide
the required flux with full-load current taken from the machine

EXAMPLE OF ALTERNATOR DESIGN 345

on a power factor of 0.8, it is necessary to know the maximum
armature m.m.f. and also the position on the armature surface
(considered relatively to the field poles) at which this maximum
occurs. It was shown in Art. 94, Chap. XIII, that the armature
m.m.f. can be represented by a sine curve of which the maximum
value (by formula (100)) is

48 X 3 X 700 X -v/2
(SI) a = - v , - = 11,340 ampere-turns per pole.

7T X ^

The displacement of this m.m.f. curve relatively to the center of
the pole is obtained approximately by calculating the angle as
explained in Art. 98. The vectors representing the component
m.m.fs. have been drawn in Fig. 137, the angle \f/ f being calculated
from the previously ascertained values of the voltage vectors
(see calculations under item (40)). Thus

(0.8 X 3,800) + 8.4
COS * "4,000" ' 763

whence V = 40 40'.

Since 27,000 ampere-turns per pole are required to develop
3,810 volts per phase, and since the saturation curve does not
depart appreciably from a straight line, the m.m.f. vector OM
to develop OE' g (i.e., 4,000 volts) must represent approximately

27,000X4,000 ' . .

5~ Q1ft - = 28,400 ampere-turns, and the required angle is,

o,olU

t CM.
'-

where

CM = CM + MM = OM sin f + 11,340

= 29,740
and

OC = OM cos ^'
= 21,650

The angle is thus found to be 53 57' or (say) 54 degrees.

The sine curve M a representing armature m.m.f. can now be
drawn in Fig. 142, with its maximum value displaced (54 + 90)
degrees beyond the center of the pole. The required field
ampere-turns are given approximately by the length of the vector
OM (Fig. 137), except that the increased tooth saturation has

OC
not been taken into account. The length OM is - ^ =

346 PRINCIPLES OF ELECTRICAL DESIGN

Q Igg = 36,800. An excitation slightly in excess of this amount
will probably suffice, 1 because if the average density over the pole

pitch is raised from 6,010 gausses to 6,010 X ^?^ = 6,300

o,olU

gausses, the average effect of increased tooth reluctance, as shown
by Fig. 139, is small, and we shall try 37,000 ampere-turns on the
field. This full-load field excitation is represented by the curve
Mo of Fig. 142. Now add the ordinates of M and M aj and ob-
tain the resultant m.m.f . curve M. Using this new m.m.f . curve,
we can obtain from Fig. 139 the corresponding values of air-gap
flux density, and plot in Fig. 141 the full-load flux curve C of
which the area, as measured by planimeter, is found to be 112.3
sq. cm. This checks closely with the calculated area (112 sq.
cm.) and it follows that a field excitation of 37,000 ampere-turns
will provide the right amount of flux to give the required terminal
voltage when the machine is delivering its rated full-load current
at 80 per cent, power factor.

Items (55) to (57). When the shape of the flux curves of Fig.
141 is considered in connection with the fact that a distributed
armature winding tends to smooth out irregularities in the result-
ing e.m.f. wave (see Fig. 118, page 293), it is evident that we need
not expect any great departure from the ideal sine curve in the
e.m.f. waves of this particular machine either on open circuit or
at full load. At the same time it will be well to illustrate the
procedure explained in Arts. 100, 101, and 102, by plotting the
actual e.m.f. wave resulting from the full-load flux distribution
curve, C, of Fig. 141.

The average flux density corresponding to any given position
of the four slots constituting one phase-belt is obtained as ex-
plained in Art. 100, and the instantaneous values of the " ap-
parent" developed e.m.f. are calculated by formula (107). The
results of these calculations have been plotted in Fig. 143 to
rectangular coordinates, and in Fig. 144 to polar coordinates.
The mean ordinate of Fig. 143 is 3,610 volts, and the r.m.s., or
virtual value of the e.m.f., is the square root of the ratio, twice

1 This method of estimating the full-load field ampere turns is not scientifically
sound, especially in the case of salient-pole machines, because the m.m.f.
distribution over the armature surface, due to the field-pole excitation, is
rarely sinusoidal as here assumed. The correct increase of field excitation
to obtain a given full-load flux must, therefore, be obtained by trial; but the
method here used indicates the approximate increase of excitation required.

EXAMPLE OF ALTERNATOR DESIGN

347

area of the curve Fig. 144 -s- T, which is 3,975 volts. The form

Q in* C%

factor (Art. 101) is ~~^ = 1.10. It must not be forgotten that

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10 20 80 40 60 60 70 80 90 100 110 120 130 140 150 160 170 180 10 20
Position of Center of Phase Belt on Pole Pitch.fElectrical Degrees)

Fio. 143. Wave-shape of e.m.f. developed in armature windings of 8000
k.v.a. turbo-generator on 80 per cent, power factor full-load output.

CO

150

30

15

FIG. 144. E.m.f. wave of Fig. 143 re-plotted to polar coordinates.

the machine under consideration is Y-connected, and the wave-
shape of the potential difference between terminals will be

348

PRINCIPLES OF ELECTRICAL DESIGN

very closely represented by the addition of two curves similar to
Fig. 143 with a phase displacement of 60 degrees, as pointed
out on page 291.

As a check on the work, it should be noted that if the e.m.f.
wave shape (Fig. 143) had been a true sine-wave, the virtual
value of the apparent developed voltage would have been 3,605 X
1.11 = 4,000, which proves the accuracy of the graphical work.
It is not, however, suggested that a difference of 1 per cent, in
the form factor is a matter of practical importance; but in salient-

FIG. 145. Diagrammatic representation of flux lines in turbo-alternator.

pole machines with incorrectly shaped pole faces and a con-
centrated armature winding, the wave shape may depart very
considerably from the sine curve, and it is under such condi-
tions that the methods here illustrated will be of the greatest
value.

The area of one lobe (Fig. 144), using 1 volt as the unit radius
vector, is 24,700,000, and the maximum ordinate of the equiva-
lent sine-wave, as given by formula (108), is

, 4X24,700,000 = 5)600yolts

EXAMPLE OF ALTERNATOR DESIGN 349

This is the diameter of the equivalent circle in Fig. 144, and the
angle a, obtained as explained in Art. 102, is found to be 11
degrees.

It is obvious that the wave shape under no-load conditions
will be very nearly a true sine-wave, and the form factor will
therefore be approximately 1.11.

Item (59). No reference has been made to item (58) because
this applies mainly to the salient-pole type of machine; the pro-
cedure in proportioning the field poles and yoke ring being then
similar to that followed in D.C. design. The depth of iron below
the slots in the rotor of a turbo-alternator is usually more than
sufficient to carry the flux, including the leakage lines. In the
particular design under consideration we shall be able to provide
air ducts at the bottom of the rotor slots as shown in Figs. 135
and 145, and still leave enough section of iron to carry the
flux.

The amount of the leakage flux at the two ends of the rotor
is not easily estimated; but when expressed as a percentage of
the useful flux it is never large in extra high-speed machines with
wide pole pitch; the greater the axial length of rotor in respect
to the diameter, the smaller will be the percentage of the flux
leaking from pole to pole at the ends. The rotor leakage which
occurs from tooth to tooth, and in the air gap, over the whole
length of the machine is shown diagrammatically in Fig. 145.
This sketch shows a total of four lines of leakage flux which pass
through the body of the rotor, but do not enter the armature.
This leakage flux will not appreciably affect the flux density in
the rotor teeth near the neutral zone, because it will follow the
path of least reluctance and be distributed between several
teeth.

The calculation of the rotor slot flux may be carried out as for
the stator windings. Thus, at full load, with 12,300 ampere-
conductors per slot, the flux passing from tooth to tooth below
the wedge is

0.47T X 12,300 (3.5 X 49.5) X 2.54

~ - = 2,100,000 maxwells.

Average m.m.f. Permeance

The flux in the space occupied by the wedge and insulation
above the copper, including an allowance for the spreading of the

350 PRINCIPLES OF ELECTRICAL DESIGN

flux lines into the air gap above the wedge (tooth top leakage) , is
approximately,

0.47T X 12,300 X (L75 X * 9 ' 5) X 2<54 = 1,900,000 maxwells,

l.o

where the figure 1.75 is the assumed radial depth, in inches, of
the flux path, and 1.8 is the assumed average length of the flux
lines (somewhat greater than the width of slot below the wedge).
The sum of the two flux components is 4,000,000 maxwells,
making the total slot flux for both sides of the pole face equal to
twice this amount, or 8,000,000 maxwells. As a rough estimate,
we may assume the end leakage to be about one-sixth of this,
making a total of 9,300,000 maxwells. The full-load leakage

65 21 93

coefficient is therefore - ' '- = 1.142 or (say) 1.15.

oo.J

The maximum number of flux lines in the rotor, which cross
the section below the slots (represented by 28 lines in Fig. 145)

x LIB - 87,600,000

The cross-section of iron below the vent ducts is 12% X 49^ =
600 sq. in.; which makes the average flux density 62,500 lines
per square inch. The section of iron below the slots is therefore
sufficient, and the reluctance of the body of the rotor is a neg-
ligible quantity in comparison with that of the teeth and air gap.

Items (62) to (68). With a density of 8,500 gausses in the
stator core (item 28), and ample iron section in the rotor, the
additional ampere-turns required to overcome reluctance of
armature and field cores will probably not exceed 200; and since
this is a very small percentage of the excitation for air gap and
teeth, we shall not need to draw a new curve for item (62):
the curves of Fig. 139 may be thought of as applying to the
machine as a whole. The ampere-turns at no load and at full
load (items 63 and 64) will therefore be taken at 27,000 and
37,000 respectively, as previously calculated.

The slot insulation should be about % in. thick, and the field
winding might be in the form of copper strip 1J4 in. wide laid
flat in the slot. Allowing % in. total depth of insulation prefer-
ably of mica or asbestos fabric between the layers of the wind-
ing, the cross-section of copper will be 2% X 1M = 3.6 in.

12 300
making the current density at full load, A = = 3,430

EXAMPLE OF ALTERNATOR DESIGN 351

amp. per square inch. This is a high, but not necessarily an
impossible figure.

The mean length per turn of the rotor winding should be meas-
ured off a drawing showing the method of bending and securing the
end connections. We shall assume this length to be 156 in. All
the turns will be in series, and the mean length per turn for the
four poles in series will be 156 X 4 = 624 in. Assuming the
potential difference at the slip rings to be 120 volts, the cross-
section of the winding, by formula (26), is

(m) = - == 192,500 or 0.1512 sq. in.

If we use a copper strip 0.12 in. thick, the number of conductors

2 875
in each slot will be -/rro" := 24, making the turns per pole 24 X 3

= 72.

37 000
The current per conductor at full load must be ~n^~ =514

514

amp., whence the current density is A = roK~\ To = 3,430

i.zo /s u.i &

amp. per square inch.

72 X 4 X 156
The total length of copper strip is - ;- - = 3,740 ft.

The resistance (hot) will be about 0.250 ohm, and the required
pressure at slip rings will be 0.25 X 514 = 128.5 volts. The
PR loss is therefore 128.5 X 514 = 66 kw., or 0.825 per cent.
of the rated output. This is rather on the high side for so large
a machine, and it may be accounted for by the fact that the air
gap is perhaps somewhat greater than it need be; but the ef-
ficiency will not be affected appreciably.

Item (69). The cooling air, which enters at one end of the
machine, is supposed to travel through the longitudinal vent
ducts to the other end of the machine, no radial ducts being
provided. Such an arrangement leads to the temperature of
one end of the machine being higher than the other end; but
systems of ventilation designed to obviate this are usually less
simple, and the straight-through arrangement of ducts has much
to recommend it. In machines larger than the one under con-
sideration, it might be necessary to have the cold air enter at
both ends, in which case one or more radial outlets would be
provided at the center.

352 PRINCIPLES OF ELECTRICAL DESIGN

made, we may provide a number of spaces between the stator
iron and the casing to allow of air being passed over the outside of
the armature core. Let us suppose that there are twelve such
ducts, each 10 in. wide by 1 in. deep; the total cross-section of
the air ducts is then made up as follows:

Outside stator stampings 12 X 10 X 1 = 120 sq. in.

Holes punched in stator stampings (Fig. 136) == 590

Spaces above wedge in stator slots (Fig. 135)

48 X 1.5 X 1 = 72

Clearance between stator and rotor % X TT X 39^ =108
Spaces in rotor forging below slots 32 X 1.94 = 62

Total = 952 sq. in.

or 6.6 sq. ft.

The total losses in the machine, without including windage and
sundry small losses, are

Total core loss (item (32)) 120 kw.

Stator PR loss (item (36)) 21 kw.

Rotor PR loss 66 kw.

Total 207 kw.

If we allow 100 cu. ft. of air per minute for each kilowatt dissi-
pated (see Art. 33, Chap. VI), it will be necessary to pass 20,700
cu. ft. of air through the machine per minute. This makes the

20 700
velocity in the vent ducts ' = 3,140 ft. per minute, which

is well below the permissible limit.

Item (70). With varying degrees of tooth saturation espe-
cially when, as in this design, all the teeth are not of the same
cross-section the only correct method of predetermining the
open-circuit saturation curve (similar to Fig. 124, Art. 104), is to
plot the flux distribution for different values of the exciting
ampere-turns, and calculate the e.m.f. developed in each case.
It is not necessary to calculate a large number of values in this
manner; two or three points taken with fairly high values of the
exciting current will show how the tooth saturation affects the
resulting flux; and a curve can be drawn connecting the known
straight part of the saturation curve with these ascertained values
for the higher densities.

The saturation curve for zero power factor can be drawn as
explained in Art. 106 (Fig. 125), and the construction of Figs.

EXAMPLE OF ALTERNATOR DESIGN 353

129 and 130 can be applied for obtaining curves giving the ap-
proximate connection between terminal volts and exciting current
for any other power factor. We shall confine ourselves here to
calculating the inherent regulation by the more correct method as
outlined in Art. 110, and since much of the work has already
been done in connection with full-load current on 80 per cent,
power factor, this is the condition which we shall choose for the
purpose of illustration.

We know that although 27,000 ampere-turns per pole will
develop the specified terminal voltage when no current is taken

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 28 of 30)