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Font size find their way into the core immediately below the bottom of
the slots, even if the iron in the teeth is practically saturated.
We shall therefore assume two equipotential surfaces, one being
the pole face and the other being the cylindrical surface passing
through the roots of the teeth. The flux density in the air ducts
and spaces not occupied by iron will therefore be the same as
the density, B s , in the slots, and the m.m.f required to overcome
the reluctance of air gap proper and slot will be the same as
the m.m.f. required to overcome the reluctance of air gap proper
and tooth; therefore

FLUX DISTRIBUTION 121

whence

R r> , Arv .

a = a (rtjr+ij)

Considering, now, the total flux entering the armature over
one slot pitch, this is made of two parts:

1. The flux in the iron of the teeth, of value B t tl n .

2. The flux in the slots and ducts, of value

B g [Sl a +t(l a ~ l n )]

or

B, (l a \ - l n t)

The total flux entering through one slot pitch can also be
expressed in terms of B , being:

<J>x = B a \l a
Thus

B a \l a = B t tl n + B,(\l a - tl n ) (61)

Substituting in (61) the value for B, given by formula (60) in
terms of B i} and solving for B , we get:

By assuming values of B t ranging between 20,000 and (say)
26,000 gausses, the corresponding values of B can be calculated
by formula (62), and a curve plotted from which values of B t can
be found when B is known.

The fact that this formula is based on assumptions justified
only if the value of B t is very high should not be lost sight of.
For very low values of B t it, may be assumed that all the flux
entering through one slot pitch passes through the iron of the
tooth. This leads to the expression:

B, = B. (63)

Curves may be plotted from the formulas (62) and (63) and a
working curve, which shall be a compromise between these two
extreme conditions, can then readily be drawn. This will be

38. Correction for Taper of Tooth. The assumption of parallel
sides to the tooth is justified only when the diameter of the
armature is large relatively to the slot pitch or when taper slots
are used in order to provide a uniform cross-section throughout
the whole length of the tooth. The dimension t in formula (62)

122 PRINCIPLES OF ELECTRICAL DESIGN

should, in the first place, be the width of the narrowest part of
the tooth, as it is important that the density at this point be
known; it rarely exceeds 25,000 gausses in continuous-current
machines, and is less in alternators. When the field system
revolves, as in most modern alternators, the armature teeth
will usually be wider at the root than at the top, and but little
error will be introduced by taking for t the average width, for
the purpose of calculating the average density B t and the ampere-
turns required for the teeth.

The case of a tooth with considerable taper, in which the
density at root is in excess of 10,000 gausses, may be dealt with
by the application of SIMPSON'S rule. Having determined the
density B t at the root of the tooth, by applying formula (62) or
(63) as the case may demand, the assumption is then made that
the total flux in the tooth remains unaltered through other
parallel sections. 1

FIG. 38. Taper tooth.

The value of the magnetizing force H (or the ampere-turns
required per unit length) can then be determined for any section
of the tooth by referring to the B-H curves for the iron used in
the armature. It is sufficient to determine H for three sections
only.
Let these values be:

H n at narrowest section
H w at widest section

(7> [ D
i.e., where the value of B m is - ^ ~

1 This is not a correct assumption when the root density is very high, be-
cause in that case flux will leak out from the sides of the tooth to the bottom
of the slot; and at some distance from the bottom of slot (the taper being
as indicated in Fig. 38) the total flux in the tooth will be greater than at the
root cross-section.

FLUX DISTRIBUTION

123

Then, on the assumption that the portion of the B-H curve
involved is a parabola, SIMPSON'S approximation is,

average H = %H n + %H m + %H W (64)

Referring to Fig. 38, it will be seen that H w is taken at the
section which would be the top of the tooth if the air gap were
increased from 5 to the " equivalent" value d e as calculated by
formula (58). This is recommended as a good practical com-
promise; and the m.m.f. in gilberts required to overcome the
reluctance of the tooth is H X d e where d e , the equivalent length
of tooth, must be expressed in centimeters. If preferred, the
formula (64) can be modified to give an average value of the
necessary ampere-turns per inch.

39. Variation of Permeance over Pole Pitch Permeance
Curve. The permeance per square centimeter of the air gap
when the armature is slotted may be calculated for the center of
the pole face, by using formula (57). This value will not change
appreciably for other points under the pole shoe if the bore of
the field magnets is concentric with the armature; but near the
pole tips, and in the interpolar space, it will decrease at a more
or less rapid rate, depending on the geometric configuration of

A'*B E F

FIG. 39. Flux lines in air gap of dynamo. (One pole acting alone.)

the pole pieces, and their circumferential width relatively to pole
pitch and air-gap length. In considering the reluctance of the
air paths between pole shoe and armature, it is convenient to
think of an equivalent air gap of length 8 e as calculated by formula
(58) of Art. 36; and in the following investigation the actual
toothed armature must be thought of as being replaced by
an imaginary smooth-core armature of the proper diameter
to insure that the reluctance of the air gap per unit area at any

124

PRINCIPLES OF ELECTRICAL DESIGN

point on the periphery shall be the same as the average reluctance
per unit area taken over the slot pitch.

In Figs. 39 and 40 an attempt has been made to represent
the actual distribution of flux lines (1) for the condition of one
pole acting alone without interference from neighboring poles,
and (2) for the practical condition of neighboring poles of equal
strength and opposite polarity. The machine to which these
diagrams apply is a continuous-current dynamo of pole pitch
37 cm., pole arc 27 cm., and equivalent air gap of 0.8 cm. at
center of pole face. The air gap is of uniform length except
near the pole tips, where it is slightly increased, as indicated

A N

FIG. 40. Flux lines in air gap of dynamo. (Effect of neighboring poles.)

on the drawings. With a little practice, unlimited patience, and
ample time in which to perform the work, diagrams of flux dis-
tribution such as those of Figs. 39 and 40 can be drawn, and
they will indicate accurately the actual arrangement of the
flux lines. The method is one of trial and gradual elimination
of errors, based on the well-known principle that the space
distribution of the flux lines will be such as to correspond with
maximum total permeance, or, in other words, such as will
produce the maximum flux with a given m.m.f.

Probably one of the most practical and at the same time
most accurate methods of procedure is that proposed by DR.
LEHMANN 1 and followed in preparing the flux diagrams (Figs.
39, 40, 42 and 43) . A section perpendicular to the shaft through
the pole shoe and armature is considered, and all flux lines in

1 "Graphische Methode zur Bestimmung des Kraftlinienverlaufes in der
Luft," Elektrotechnische Zeitschrift, vol. 30 (1909), p. 995.

FLUX DISTRIBUTION 125

the air gap are supposed to lie in planes parallel to this section.
Equipotential lines are drawn in directions which seem reason-
able to the draughtsman, and tubes of flux, all having the same
permeance, are then drawn with their boundary lines perpen-
dicular at all points to the equipotential lines. At the first trial
it will generally be found that these conditions cannot be fulfilled,
but by altering the direction of the tentative equipotential
lines the work is repeated until the correct arrangement of
lines is obtained. The tube of induction A BCD (Fig. 39) is the
first to be drawn. Its permeance in the particular case con-
sidered is 0.25 because it consists of four portions in series, each
one of which is exactly as wide as it is long (a thickness of 1 cm.
measured axially is assumed). Proceeding outward from left
to right, and making each section of the individual tube of
induction exactly as wide as it is long, the permeance of every
one of the component areas in the diagram is always unity, and
any complete tube, such as EFGH, has the same permeance (in
this example 0.25) as every other tube. The computation of
the total permeance between the pole shoe and armature over
any given area is thus rendered exceedingly simple. Although
the armature surface is represented as a straight line in the
accompanying illustrations, the actual curvature of the armature
may be taken into account if preferred; but the error introduced
by substituting the developed armature surface for the actual
circle is generally negligible.

In Fig. 40 the flux lines have been drawn to ascertain the
effect of the neighboring pole in altering the distribution over
the armature surface in the interpolar space. The perpendicular
AW has been erected at the geometric neutral point, and may
be considered as the surface of an iron plate forming a con-
tinuation of the armature surface AN. Thus ANN' will be an
equipotential surface between which and the polar surface
the intermediate equipotential surfaces must lie.

It may be mentioned tha,t in Figs. 39 and 40, and also in
the other flux-line diagrams, the pole core under the windings
cannot properly be considered as being at the same magnetic
potential as the pole shoe, relatively to the armature. The
proper correction can be introduced in calculating the flux in
each tube of induction; but since the present investigation is
confined to the flux entering the armature from the pole shoe, it
will not be necessary to make this correction.

126

PRINCIPLES OF ELECTRICAL DESIGN .

The flux density at all points on the armature periphery is
easily calculated when the flux lines have been drawn. Thus,
since each tube of induction encloses the same number of mag-
netic lines, exactly the same amount of flux will enter the arma-
ture in the space EF (Fig. 39) as in the space A B. If B a b is
the flux density in the tube CDAB at the center of the pole face,
the average density over the space EF will be

B e f = Bab X Wp

Thus curves of flux distribution such as Fig. 41 can readily
be drawn. It will be seen that the dotted curve, giving actual
distribution of flux for the case of Fig. 40, does not differ from

Flux Distribution,
One Pole only

O Surf ace of Armature Core N

FIG. 41. Curve of flux distribution over armature surface.

the full-line curve (case of Fig. 39, with no interference from
neighboring poles) except in the interpolar. space where the de-
magnetizing effect of the opposite polarity is appreciable, and
causes the flux to diminish rapidly until it reaches zero value on
the geometric neutral (the point N), where its direction re-
verses. This is what one would expect to find, because, although
the magnetic action of any one pole considered alone will ex-
tend far beyond each pole tip, this action will not be appreciable
beyond the interpolar space, on account of the shading effect
of the neighboring poles. In order to ascertain how far the
demagnetizing effect of neighboring poles is likely to extend
when the air gap is not constant but increases appreciably in

FLUX DISTRIBUTION

127

length as the distance from the center of the pole increases, the
case of a salient pole alternator has been considered. The flux
lines and equipotential surfaces for an alternator with shaped
poles are shown in Figs, 42 and 43. The object of shaping the
poles by gradually increasing the air gap from the center out-
ward is to obtain over the pole pitch a distribution of flux which

FIG. 42. Flux lines in air gap of alternator. (One pole acting alone.)

FIG. 43. Flux lines in air gap of alternator. (Effect of neighboring poles.)

shall approximate to a sine curve. The data of the machine
under consideration are as follows:

Pole pitch = 22 cm.

Pole arc = 14.3 cm.

Equivalent air gap at center of pole face 5 = 1 cm.

Air gap at other points on armature surface = -

COS v

where

is the angle (in electrical degrees) between the center of the
pole and the point considered.

In Fig. 44 the curve marked "permeance" has been plotted
from Fig. 42. Its shape indicates the flux distribution over the
armature surface on the assumption that the effect of neighbor-

128

PRINCIPLES OF ELECTRICAL DESIGN

ing poles is negligible. The m.m.f. between pole shoe and arma-
ture core being the same at all points on the armature surface,
it is evident that this curve of flux distribution will correctly
represent the variations of air-gap permeance per unit area of
the armature surface. Thus, the permeance per square centi-
meter at the point 7 cm. from center of pole is the permeance
of the tube GHEF in Fig. 42 divided by the area of the surface
EF. The permeance of the tube GHEF is exactly the same
as that of the tube CDAB, i.e., 0.25 per centimeter of depth
measured axially. The area of the surface EF is 0.5, and the
permeance per square centimeter at the point considered is,
therefore,

* - is -

40. Open-circuit Flux Distribution and M.m.f. Curves. The
dotted curve marked "flux" in Fig. 44 has been plotted from

1 2 3 4 5 6 7 8 9 10 11 12 13
Surface of Armature Core

FIG. 44. Curves of permeance, flux, and m.m.f.

Fig. 43, and shows the effect of the neighboring pole in reducing
the air gap flux and causing it to pass through zero value at a
point exactly halfway between the poles. A comparison of the
full line and dotted flux curves shows that, even with the greatly
increased air gap, the influence of the neighboring poles is not
appreciable except in the uncovered spaces between the pole
tips. The vertical dotted line in Fig. 44 shows the limit of the
pole arc.

In order to find a scale for the flux curve it is necessary to
know either the total flux entering the armature in the space

FLUX DISTRIBUTION

129

of a pole pitch or the m.m.f. between pole shoe and armature.
If the resultant m.m.f. tending to send flux from the pole to any
point on the armature is known, the flux density can be calculated
because, B = flux per square centimeter = (m.m.f.) X permeance
per square centimeter.

Thus, if the m.m.f. necessary to overcome air-gap reluctance
at center of the pole is known, the curve of resultant m.m.f.
at all points on the armature can be plotted. This has been

done in Fig. 44, where the ordinate of the m.m.f. curve at any

i n R

point, such as 10, is simply 10 M - p> where the or-
dinate 10 B of the flux curve must be expressed in gausses.

41. Practical Method of Predetermining Flux Distribution.
Although the method outlined above, gives excellent results
in the hands of an experienced designer who can afford the time
required to map out the actual paths of the flux lines, it is not
suitable for general use in actual designing work. By adopting
a simple construction which assumes a certain flux distribution
and avoids the drawing of equipotential lines, results of sufficient
accuracy for practical purposes can very quickly be obtained.

A BC ODEF G H

FIG. 45. Approximate flux paths between pole and armature.

This construction is indicated in Fig. 45. A section through
one-half of the pole shoe, showing the air gap in its proper pro-
portion, is drawn to a sufficiently large scale, preferably full
size. The " developed" armature surface may be used, in
which case the construction is a little simpler because radial
lines may be shown as perpendiculars erected on the horizontal
datum line representing the armature surface. The distance
AM is the " equivalent" air gap, as calculated for the center
of the pole face by using formula (58). Draw the perpendicular
OR tangent to the pole tip, and, through the first poiht of
tangency Q, draw the semicircle BQF with its center at 0.
Bisect QO at the point P and draw PD at an angle of 30 degrees

9

130

PRINCIPLES OF ELECTRICAL DESIGN

with OQ . Produce DP to 0' where it meets the perpendicular
erected on AD at the point B. Flux lines of which the length
and direction will be approximately correct can now be drawn.
All the lines from points on armature surface lying between A
and B will be considered perpendicular to the armature surface,
i.e., verticals erected on the datum line. Between the points
B and D the lines will be considered straight, but with a slope
determined by the position of the point 0' through which they
will pass if produced beyond the pole face. From the point F
the equivalent flux line will be the arc of the circle described
through the point Q with the radius OQ, and lines from points
between D and F must be put in by the eye so that their curva-
ture shall be something between the circle through F and the
straight line through D. One of these intermediate lines has
been drawn from the point E. Over the region beyond F all flux

FIG. 46. Effect of neighboring pole in modifying flux distribution.

lines will be drawn as circles described from the center and
continued beyond the vertical OR until they meet the pole in a
direction normal to the surface of the iron. Thus the line from
the point G is completed by an arc of circle with its center at
the junction of the line OR and the flat surface of the polar
projection, while the flux line from H. is continued as a straight
line (the shortest distance) until it meets the pole perpendicularly
to the surface.

Any desired number of lines can very quickly be drawn in
this manner, and they may be thought of as the center lines of
" equivalent" tubes of flux of uniform cross-section over their
entire length. If, now, the length of any one of these imaginary'
flux lines be measured in centimeters, the reciprocal of this
length will be the permeance per square centimeter between pole
shoe and armature at the point considered. It is, therefore,
an easy matter to plot a permeance curve similar to the one

FLUX DISTRIBUTION

131

shown in Fig. 44. This curve, which represents the permeance
per square centimeter of armature surface between pole and
armature, can evidently be thought of as a flux-distribution curve
on the assumption that one pole acts alone without interference
from neighboring poles.

In regard to the actual flux distribution for no-load conditions,
it may be argued that if two neighboring poles each acting alone
would produce a flux distribution as shown respectively by the
full-line and dotted curves of Fig. 46, then the flux at any point
p will be pm pn. This method of plotting the resulting flux-
distribution curve should give satisfactory results in the space

/

\

>

/

|
Eg (Average)

V

VVlCurve j

*

/

Curve B N^K^^

X

1

A

s

^ T_t_ T

\N

:_u T ^J

FIG. 47. Practical construction for deriving flux curve from permeance

curve.

between pole tips, but it does not provide for the gradual change
in the flux distribution near the pole tips where the shading effect
of the masses of iron becomes important. For the practical
designer the writer recommends the approximation indicated in
Fig. 47 where P is the permeance curve previously obtained.
The flux curve is derived therefrom by drawing the straight
line RS, connecting the point on the permeance curve directly
over the geometric neutral to the point S immediately under the
pole tip. By subtracting from the ordinates of the curve P
the corresponding ordinates of the triangle ORS, the curve OA'N
is obtained, representing the flux distribution on open circuit.
This curve has yet to be calibrated, because the value of its
ordinates cannot be determined unless either the m.m.f. or the
total flux per pole is known. In designing a machine, the total

132 PRINCIPLES OF ELECTRICAL DESIGN

flux per pole will be known at this stage of the work, and the
unknown factor will be the ampere-turns on the poles necessary
to produce this flux. Measure the area of the curve OA'N and
construct the rectangle OO'N'N of exactly the same area. The
height of this rectangle will be a measure of the average density
over the pole pitch. This is known to be

<

B g (average) =

where 3> = total flux per pole in the air gap.
T = pole pitch in centimeters.
l a = gross length of armature in centimeters.

In this manner a scale is provided for the flux curve OA'N,
which should preferably be replotted. The curve of resultant
m.m.f. over armature surface can now be derived as explained
in connection with Fig. 44.

Since the permeance curve as obtained by either of the methods
here explained does not take into account the reluctance of the
armature teeth, or indeed the reluctance of any part of the
magnetic circuit other than the air gap, the actual ampere-turns
necessary to produce the required flux will be greater than the
amount indicated by the maximum ordinate of the m.m.f. curve
of Fig. 44. The fact that the reluctance of the teeth at different
points under the pole face is dependent upon the flux distribu-
tion tends to complicate the problem, but a method of accounting
for this variation will be explained in the following article.

42. Open-circuit Flux-distribution Curves, as Influenced by
Tooth Saturation. Before considering the effect of the armature
current in altering the distribution -of magnetizing force over the
armature periphery, it will be necessary to examine briefly how
the degree of saturation of the teeth may be taken into account
and a correct flux-distribution curve plotted. The method about
to be explained is due to PROFESSOR C. R. MOORE, it is probably
more easily applied and less tedious than an equally scientific
method more recently proposed by DR. ALFRED HAY. 1

In Fig. 48 a permeance curve has been drawn. It has the
same meaning as the curve marked " Permeance" in Fig. 44
(Art. 39), and it may be obtained for any given machine in the
manner described in Art. 41. If this curve (Fig. 48) has been

1 A. HAY: "Predetermining Field Distortion in Continuous-current
Generators," Electrician, 72, pp. 283-285, Nov. 21, 1913.

FLUX DISTRIBUTION

133

calibrated to read air-gap permeance per square centimeter of
armature surface, it follows that, for a given value of m.m.f.
between pole and armature, the flux density at any point
as for instance d will be

Bd = (m.m.f.)d X value of ordinate of Fig. 48 at d,
but in order to get the flux into the armature core the reluctance
of the teeth must be considered.

FIG. 48. Permeance curve.

O P b P a P e

Ampere -Turns Required for Air-Gap, Teeth and Slots

FIG. 49. Saturation curves for air gap, teeth, and slots.

For any value of the air-gap density B g there is a correspond-
ing value of the tooth density, B t , which can be calculated by
formula (62) or (63), as the case demands; and the ampere-turns
required to overcome the reluctance of the tooth can be found, all
as explained in Art. 38. This value can be plotted in Fig. 49

134

PRINCIPLES OF ELECTRICAL DESIGN

against the corresponding values of B g , and the resulting curve
shows the excitation required to overcome tooth reluctance for
all values of the air-gap density. The curves for the air gap
proper will all be straight lines when plotted in Fig. 49. Let
OR be the curve for the point a at center of pole. Add the
ampere-turns required for the teeth, and obtain curve (a),
which gives directly the ampere-turns required to overcome
reluctance of air gap, teeth, and slots, for all values of the air-
gap density. It will be understood that the ordinates represent
the average value of the air-gap density at armature surface

\a b c d e
FIG. 50. Open-circuit m.m.f. curve.

\

over a slot pitch. Any other curve, such as (d), is obtained
by first drawing a straight line OQ such that PQ bears to PR
the same relation as the ordinate at d in Fig. 48 bears to the
ordinate at a, and then adding thereto the ampere-turns for the
teeth, as already obtained. The curves of Fig. 49 should in-
clude a sufficient number of points on the armature surface; and
when the resultant m.m.f. between armature points and pole is also

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 11 of 30)