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in the conductors outside the slots by the cutting of the flux lines
created by the currents in all the phase windings.

87. Calculation of Armature Inductance. The flux cut by
the conductors which project beyond the ends of the armature
slots is very difficult to calculate, and empirical formulas based

a

a

FIG. 101. Illustrating flux cut by end projections of armature coils.

on experimental data are generally used for predicting the prob-
able value of the inductance of the armature end connections.
In Fig. 101 let r represent the pole pitch of a three-phase generator
and V the average axial extension of the coils beyond the ends
of the armature core. 1 If the total flux produced by the arma-
ture currents, in the space r cm. wide by V cm. deep, as shown
cross-hatched in Fig. 101, can be estimated, the voltage developed
by the cutting of this flux can readily be calculated. Without
attempting to go into the niceties of mathematical analysis
which would apply only to one of the many different arrange-
ments of coils it can be shown that the flux produced through

1 The equivalent projection of a coil that is bent up to clear the coils of
other phases might be considered equal to the projection of the same coil if
flattened out. With r expressed in inches, an approximate value for l' t
based on the assumptions made in Art. 83, is

= 1.27 (k.v.

+3

7 cm.

ARMATURE WINDINGS 265

air paths by the currents in the axial prolongations of the slot
conductors will depend mainly upon the number of ampere-
conductors per pole on the armature, and on the amount of the
projection I'. There will be no exact proportionality between
ampere-conductors per pole and flux, the relation being a
logarithmic function of the pole pitch r and dependent on the
number of slots, i.e., whether the winding is concentrated or
distributed. The flux produced by the connections running
approximately parallel to the circumference of the armature will
depend not only on r but also on I'. Thus, the amount of the
projection I' beyond the ends of the slot would seem to be a
more important factor than the circumferential width of the
coils in determining the end flux, and for the calculation of the
total end flux per pole (both ends) in the case of a three-phase
generator the writer suggests the empirical formula

* e = kTJ c l e - '-= log* (12n.l') (98)

7l -f- O

where T s = the number of inductors in each slot.
n, = the number of slots per pole per phase.

V the projection of coil-ends beyond end of slots, in
centimeters.

l e = (2r -f 4Z') = approximately the total length in
centimeters per turn of wire in a coil, less the slot
portion.

I c = the armature current per conductor (r.m.s. value).

k = constant, approximately unity, depending upon the
design of the machine, the arrangement of the wind-
ings, and the proximity of masses of iron tending
to increase the induction.

The quantities 6/(n. + 5) and log (12n/) are factors in-
troduced mainly to correct for the increase of flux with a con-
centrated winding, and for the fact that the projection V of the
coils will influence the total flux to a greater extent than the end
length r which appears in the expression for the total length l e .

If p is the number of poles of the machine, the total number
of conductors per phase is pT 8 n s , and the average value of the
voltage developed in the end connections by the cutting of the
end flux will be 2f\$ e pT a n s X 10~ 8 . Assuming the form factor
to be 1.11, which would be correct if the flux distribution were
sinusoidal, and substituting for \$ e the value given by formula

266 PRINCIPLES OF ELECTRICAL DESIGN

(98), the voltage component developed per phase winding by the
cutting of the end flux is

E e = (2.22k)fpTfl. (^ps) logio (12w/) X J e X 10~ 8 (99)

This quantity is usually referred to as the reactive voltage
drop per phase due to the inductance of the end connections;
it appears as the vector PE g in Figs. 99 and 100. If the mul-
tiplier (2.22k) be taken as 2.4, the formula agrees well with
the average of tests on machines of normal design.

88. Total Losses to be Radiated from Armature Core. The
losses in the iron stampings teeth and core are calculated as
explained in Chap. VI (Art. 31). The flux to be carried at full
load by the core below the teeth is that which will develop the
necessary e.m.f . as obtained from the vector construction of
Fig. 100. The radial depth of the armature stampings is cal-
culated by assuming a reasonable flux density in the iron. This
will usually be between 7,000 and 8,500 gausses in 60-cycle
machines, increasing to 10,000 or even 11,000 in 25-cycle
generators.

The permissible density in the teeth, as previously mentioned,
rarely exceeds 16,000 gausses at 60 cycles and 18,000 gausses at
25 cycles. Higher densities may have to be used occasionally,
but special attention must then be paid to the methods of cool-
ing, in order to avoid excessive temperatures. The tooth density
being appreciably lower than in D.C. machines, the apparent
flux density at the middle of the tooth may be used for estimating
the watts lost per pound. The maximum value of the tooth
density will depend upon the maximum value of the air-gap
density, and this, in turn, is modified by armature distortion and
slot leakage. The flux that must enter the core and be cut by
the armature inductors is known, but the amount of flux enter-
ing the teeth under each pole face is greater, since it includes the
slot leakage flux in the neutral zone, the .amount of which de-
pends not only upon the current in the armature, but also upon
its phase displacement, i.e., upon the power factor of the load.
Then, again, the maximum value of the air-gap flux density de-
pends not only upon the average density, but also on the shape
of the flux distribution over the pole pitch. It will not . be
necessary to go into details of this nature for the purpose of
estimating the temperature rise of the armature, and a sinusoidal

ARMATUR^ WINDINGS 267

flux distribution may be assumed, making the maximum air-

7T

gap density times the average value over the pole pitch. The

Zt

calculation of slot leakage flux will be explained later, and its
effect may for the present be neglected. 1

As a check on the calculated core loss, the figures of Art. 32
(page 104) may be used; but these values will depend upon
whether the copper or the iron losses are the more important,
i.e., on the relative proportions of iron and copper in the machine.
Iron losses 50 per cent, in excess of the average values given on
page 104 would not necessarily betoken inefficiency or a high
temperature rise.

When computing the total losses to be carried away in the
form of heat from the surface of the armature core, the whole of
the copper loss should not be added to the iron loss, but only
the portion of the total PR loss which occurs in the buried part
of the winding, i.e., in the " active" conductors of length l a .
In the case of large machines, it may be necessary to make some
allowance for eddy-current loss in the armature conductors.
This loss might be considerable if solid conductors were used;
but it is usual to laminate the copper in the slot so that the
eddy-current loss due to the slot flux is very small. This point
must not, however, be overlooked in large units ; . and special
means may have to be adopted to avoid eddy-current loss in the
armature conductors.

89. Temperature Rise of Armature. The probable tem-
perature rise of the armature is estimated as explained in Art.
34 of Chap VI, in connection with the design of D.C. dynamos.
The cooling surfaces are calculated in a similar manner; but with
the stationary armature and internal rotating field magnets, the
belt of active conductors is the inside cylindrical surface of the
armature; and this is cooled by the air thrown against it by the
fanning action of the rotor. The cooling coefficient, contain-
ing the factor v (the peripheral velocity), may be used, just as if
the armature were rotating instead of the field magnets. The
external cylindrical surface of the armature core will have no
air blown against it (in the self -ventilating machine), and the
value of v in the formula will be zero. In regard to the radial

1 The amount of the slot leakage flux, expressed as a percentage of the
total flux per pole, becomes of importance in well-designed machines only
when the pole pitch is very small.

268 PRINCIPLES OF ELECTRICAL DESIGN

vent ducts, the cooling is not quite so good as when the armature
rotates, but a blast of air is driven through the ducts, and this
is effective in carrying off the heat. The difficulty in deter-
mining cooling coefficients that shall be applicable to all sizes
and types of machine stands in the way of obtaining great
accuracy in the calculation of temperature rise. It is, however,
suggested that the formulas (53) and (55) of Art. 34 (page 110)
be used, and that the temperature rise as calculated by the
application of these formulas be increased 20 per cent. A tem-
perature rise of 45 is usually permissible.

In designing steam-turbine-driven machines with forced venti-
lation, the quantity of air required to carry off the heat losses
must be estimated (see Art. 34, page 112) and the size and con-
figuration of the various air passages must be carefully studied
with a view to preventing very high air velocities and consequent
increase of loss by friction. The average velocity of the air
through the ducts of machines provided with forced ventilation
is usually between 1,500 and 4,000 ft. per minute. This velocity
should preferably not exceed 5,000 ft. per minute; it is usually
possible to keep within this limit by carefully designing the
system of ventilation.

High-speed machines such as turbo-alternators, when pro-
vided with forced ventilation, are usually totally enclosed,
the air passages being suitably arranged to prevent the out-
going (hot) air being mixed with the incoming (cool) air. Large
ducts must be provided for conveying the air to and from the
machine. A safe rule is to provide ducts or pipes of such a
cross-section that the mean velocity of the air will not exceed
2,000 ft. per minute.

CHAPTER XIII
AIR-GAP FLUX DISTRIBUTION WAVE SHAPES

90. Shape of Pole Face. When an alternator is provided
with salient poles, the open-circuit flux distribution over the
pole pitch can be made to approximate to a sine curve by suit-
ably shaping the pole face. One method of increasing the air-
gap reluctance from the center outward is illustrated in Fig. 102.
The " equivalent" air gap, 5 e , at center of pole face is calculated
as explained in Art. 36 of Chap. VII (formula 58), and the

(Electrical)

cos or

FIG. 102. Method of shaping pole face of salient pole alternator,
equivalent gap at any other point under the pole is made equal
where a is the angle (electrical space degrees) between

to

cos a

the center of pole and the point considered. The pole face would
extend about 56 degrees on each side of the center, the pole tips
being rounded off with a small radius. In practice the curve
of the pole face would probably not conform exactly with this
cosine law; it would generally be a circular arc, not concentric
with the bore of the armature, but with the center displaced so

269

270

PRINCIPLES OF ELECTRICAL DESIGN

that the air gap near the pole tip would be approximately as
determined by the method here described.

With the cylindrical field magnet it is not usual to shape the
pole face. The clearance between the tops of the teeth on
armature and rotor would have a constant value, the proper
distribution of flux over the armature surface being obtained
by spreading the field coils over the periphery of the cylindrical
rotor. From 15 to 25 per cent, of the pole pitch is left unwound
at the center of the pole. This unwound portion is usually
slotted, but it can be left solid if it is desired to reduce the re-

Alternative Three-Part
Wedge, which allows of
Central Wedge being
preased down while the
Steel Liners are driven In

Detail of Slot and
Wedge

FIG. 103. Rotor of four-pole turbo-alternator, with radial slots.

luctance of the air gap at the center of the pole face while yet
retaining the cylindrical form of rotor. One advantage of
equally spaced slots over the surface of the rotor is that there
are no sudden changes in the air-gap permeance the average
value of which is then the same at all points on the periphery
and another argument in favor of slotting the unwound portion
of the pole face is that the field may be " stiffened" by using
high flux densities in the teeth.

Fig. 103 is a section through part of the rotor of a four-
pole turbo-alternator. The rotor of a two-pole machine may

AIR-GAP FLUX DISTRIBUTION

271

be constructed in the same way, with radial slots, but parallel
slots as shown in Fig. 104 are sometimes used. The calculation
of windings for the cylindrical type of field magnet will be taken
up in Art. 93.

FIG. 104. Rotor of two-pole turbo-alternator, with parallel slots.

91. Variation of Permeance over Pole Pitch Salient-pole
Machines. The curve representing the variations of per-
meance between pole face and armature core over the entire
pole pitch, can be drawn exactly as in the case of the D.C.
design. This was explained in Arts. 39 and 41 of Chap. VII.
The pole shoe in Figs. 42 and 43 is shaped in accordance with the
cosine law as explained in the preceding article, and the
flux lines in Fig. 43 are therefore such as would enter the
armature of a salient-pole alternator on open circuit. The
practical construction of Fig. 45 has been carried out on a pole
of similar shape, and nothing more need be added here concern-
ing the manner of plotting a permeance curve similar to the one
shown in Fig. 44.

The effect of tooth saturation may be dealt with also as in the
D.C. design (Art. 42), and a set of curves such as those of Fig.
49 should be drawn. On account of the higher frequencies,
the tooth densities are lower in A.C. than in D.C. machines,
and the effect of saturation of the armature teeth is therefore
less noticeable.

272 PRINCIPLES OF ELECTRICAL DESIGN

92. M.m.f. and Flux Distribution on Open Circuit Salient-
pole Machines. The procedure here is still the same as in D.C.
design, and the reader is referred to Arts. 40, 41, and 42, of
Chap. VII.

In deriving the curve of m.m.f. from the open-circuit flux
distribution curve (A), the modified method, as explained in
Chap. X under items (72) to (76), may be used. This short
cut is permissible in predetermining the air-gap flux distribution
of almost any alternating-current generator, because, as pre-
viously mentioned, the effect of low flux densities in the teeth
is to discount their influence on the distribution of the flux
density over the armature surface.

By carefully shaping the pole face, a sinusoidal distribution
of flux density over the armature surface can be obtained on
open circuit; but the design of a salient-pole machine to give
other factors, and is by no means a simple matter. The effect
of the armature m.m.f. will be considered after taking up the
special case of the cylindrical field magnet.

93. Special Case of Cylindrical Field Magnet with Distributed
Winding. In the case of a slotted rotor carrying the field coils,
and an air gap of constant length due to the fact that the bore
of the stator or armature is concentric with the (cylindrical)
rotor the shape of the m.m.f. curve due to field excitation alone
can readily be found without resorting to the somewhat tedious
process of getting the permeance between pole and armature
points, as described and recommended for salient-pole machines.

If the whole surface of the rotor is provided with equally
spaced slots, the average permeance of the air gap between
stator and rotor will have a constant value for all points on the
armature periphery. This condition is represented in Fig. 105,
if the center portion of the pole, of width W, is slotted as indicated
by the dotted lines. This constant average air-gap permeance
can be calculated within a close degree of approximation by mak-
ing conventional assumptions in regard to the path of the mag-
netic lines, as was done in the case of the salient-pole machines
when deriving formula (57) (page 117) giving the permeance at
center of pole. The flux lines can be supposed to be made up
of straight lines and quadrants of circles; and if the permeance
over one tooth pitch is worked out for different relative positions
of field and armature, very satisfactory results can be obtained

AIR-GAP FLUX DISTRIBUTION

273

by this method. It will usually suffice to make the calculations
for one tooth pitch in 'the position of greatest permeance, and
again in the position of least permeance. The average of these
calculated values, divided by the area of the tooth pitch in square
centimeters, will give the average value of the air-gap permeance
per square centimeter.

Under this condition of constant air-gap permeance, the flux
distribution on open circuit will follow the shape of the m.m.f.
curve; but, in any case, since B = m.m.f. X permeance per
square centimeter, the flux curve can always be obtained when
the m.m.f. distribution is known. Thus, if the portion W

M.M.F. Curve

for Field Winding only \ \

FIG. 105. M.M.F. over pole pitch, due to distributed field winding.

of the pole (Fig. 105) is not slotted, the permeance curve, in-
stead of being a straight line of which the ordinates are of con-
stant value, would be generally as shown in Fig. 106. From
these values of the permeance per square centimeter of armature
surface, curves such as those of Fig. 49 (page 133) can be drawn,
so as to include the reluctance of the armature teeth. From all
points on the armature between A and B, and C and D (Fig.
106), the average air-gap permeance would have the value A A'.
Over the central portion W it would have the value EE', while,
in the neighborhood of the points B and C, it may be assumed
to have an intermediate value as indicated by the ordinate BB'.

18

274

PRINCIPLES OF ELECTRICAL DESIGN

If the depth of the slots in the rotor has been decided upon
and the number of ampere-conductors in each slot determined,
the distribution of m.m.f. over armature surface due to the
field winding can readily be plotted as in the lower sketch of
Fig. 105. Thus the ampere-turns in the coil nearest the neutral
zone are represented by the height AB, those in the middle coil
by BC, and those in the smallest coil by CD. The broken straight
line so obtained is best replaced by the dotted curve, which takes
care of fringing, and represents the average effect. This curve,
being the open-circuit distribution of m.m.f. over armature
surface for a given value of exciting current, may be combined
with the curve of armature m.m.f. to obtain the resultant
m.m.f. under loaded conditions. The flux curves for open-
circuit and loaded conditions can then be derived by proceed-
ing exactly as in the case of the salient-pole designs.

A'

B E

D

FIG. 106. Distribution of air-gap permeance in turbo-alternator
when unwound portion of pole face is not slotted.

* With salient-pole machines variations in the open-circuit
flux distribution can be obtained only by shaping the pole shoes
so as to alter the air gap over the pole pitch in a manner which
is not easily determined except by trial, but with the field wind-
ing distributed in slots it is not difficult to arrange the coils to
give any desired distribution of m.m.f. over the pole pitch.
Thus, if the desired flux curve is known (a sinusoidal distribu-
tion is usually best for alternating-current machines), and if the
average permeance per square centimeter of the air gap has been
calculated, an ideal m.m.f. curve can easily be drawn, since, at
every point, the m.m.f. is the ratio of the flux density to the
permeance per square centimeter. The spacing and depth of
slots can then be arranged to produce a magnetizing effect as
nearly as possible the same as that of the ideal curve.

94. Armature M.M.F. in Alternating-current Generators.
In a continuous-current machine the current has the same value

AIR-GAP FLUX DISTRIBUTION 275

at all times in all the armature conductors, and equation 65
(page 136) shows how the armature m.m.f . follows a straight-line
law over a zone equal to the pole pitch, this being the distance
between brushes referred to armature surface. In alternating-
current and polyphase generators the curve of armature m.m.f.
can no longer be represented graphically by straight lines as in
Fig. 53, because the value of the current will not be the same in
all the conductors included in the space of a pole pitch.

Considering first the polyphase synchronous generator, and
assuming a sinusoidal current wave, it is an easy matter to draw
a curve representing the armature m.m.f. at any particular in-
stant of time, provided the phase displacement or position of
the conductors carrying the maximum current relatively to
center line of pole is known. If this be done for different time
values, a number of curves will be obtained, all consisting of
straight lines of varying slopes, the length of which relatively
to the pole pitch will depend on the number of phases for which
the machine is wound. The average of all these curves will be
a sine curve of which the position in space relatively to the
poles is constant, and exactly 90 electrical space degrees behind
the position of maximum current.

The method of drawing the curve of armature m.m.f. for any
instant of time, is illustrated in Fig. 107, where the upper diagram
shows the distribution of m.m.f. over the armature periphery
of a three-phase generator at the instant when the current in
phase (2) has reached its maximum value. If the power factor
is unity (load non-inductive), the current maximum will occur
simultaneously with the voltage maximum, i.e., when the belt of
conductors is under the center of the pole face, as shown in the
diagram. A low power factor would cause the current to attain
its maximum value only after the center of the pole has travelled
an appreciable distance beyond the center of the belt of conduc-
tors, and this effect will be explained later; at present we are
concerned merely with the distribution, and magnitude, of the
armature m.m.f. The vector diagram on the right-hand side
of the (upper) figure shows how* the value of the current in
phases (1) and (3), at the instant considered, will be exactly half
the maximum value; and the magnetizing effect of phase (1)
or (3) is therefore exactly half that of phase (2). The angle of
60 degrees between vectors representing three-phase currents
with a phase displacement at terminals of 120 degrees is

276

PRINCIPLES OF" ELECTRICAL DESIGN

accounted for by the fact that the angular displacement between

180
adjoining belts of conductors is only -^ 60 electrical degrees;

o

but the connections between the phase windings are so made as
to obtain the phase difference of 120 degrees between the re-
spective e.m.fs. Thus, the reversal of the vector (1) in the

_ 180

loU 5^1

Direction of Travel of Poles

Current=

V

(2) | (3)

I
Current !=0.866 /max. !

/ cos 30 = 0.866 /

FIG. 107. Instantaneous values of armature m.m.f. in three-phase
v generator.

diagram would cause it to lead vector (2) by 120 degrees instead
of lagging behind by 60 degrees; and the reversal of the vector (3)
would cause it to be 120 degrees behind (2) instead of 60 degrees

The lower diagram of Fig. 107 shows the armature m.m.f.
one-twelfth of a period later, i.e., when the poles have moved to
the left (or the conductors to the right) 30 electrical space de-

AIR-GAP FLUX DISTRIBUTION

277

grees. The current in phases (1) and (2) now has the instan-
taneous value i = I max cos 30 = 0.866/ max ; while the current
in (3) is zero. If several curves of this kind are drawn, it will
be found that the instantaneous values of m.m.f . at any point on
the armature periphery (considered relatively to the poles) differs
very little from the average value; in other words, the pulsations
of flux due to cyclic changes in the m.m.f. will, in a three-phase
machine, be negligibly small. For this reason, and also in order
to shorten and simplify the work, the armature m.m.f. of a
polyphase generator may conveniently be studied by assuming a

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