Alfred Still.

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

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2 X ir(10 + 5) X 6.45
for path (2), P 2 = 5.5 x 2 .54 43.5

The flux through path (1) is

_ m.m.f.

^1 o XX J- 1

= 374,000
and the flux through path (2) is,

$2 = m.m.f. X P 2

= 0.4 TT X 8,260 X 43.5
= 450,000

The total leakage flux is $1 + 3> 2 = 824,000 maxwells. The
useful flux is 66 X 6.45 X 7,240 = 3,080,000 maxwells, and the
leakage factor is

3,080,000 X 824,000

The flux density in the cylindrical outer shell, near the yoke,
will be 7,240 X 1.27 = 9,200 gausses, and this will also be the
density in the central pole if the cross-section is the same, but a
higher density would be permissible. The section at every part
of the magnetic circuit can be calculated on the basis of an
assumed density. As an instance, if it is desired to have a density
of 11,000 gausses in the yoke at the section AB, the thickness of
the casting at this point, as indicated by the length of the line
AB, would be obtained from the equation

3,080,000 X 1.27
" A B X TT X 10 X 6.45

whence AB = \Y in.

In this manner the magnetic circuit may be proportioned.
The path of the magnetic flux is indicated by the dotted lines in


Fig. 22, and since the cross-section and length of each part of the
magnetic circuit are now known, the component of the total
ampere-turns necessary to overcome the reluctance of the iron
or steel casting can be calculated in the usual way. The re-
luctance of the steel sphere which constitutes the armature
would be considered negligible in these calculations.

In a well-designed magnet of this type, the reluctance of the
iron portions of the circuit is but a small percentage of the air-
gap reluctance, unless the specified air gap is very small. When
the armature is in contact with the pole faces, the total flux will
be greater than the amount necessary to produce the required
initial pull. It is interesting and instructive to calculate the
pull between magnet and armature when the air gap is practically
negligible. The limit is reached when all the exciting ampere-
turns are required to overcome the reluctance of the iron, and
the calculation has to be made by assuming probable values of the
flux density, and then calculating the loss of magnetic potential
across each portion of the circuit.

With reference to the important matter of cost; air-gap den-
sities other than the assumed density of 7,240 gausses may be
tried with a view to obtaining the design of lowest first cost. A
saving in copper may be effected by allowing the temperature
rise to approach as nearly as possible the specified limit; but in
this as in all economical designs of apparatus in which a saving
in first cost is accompanied by a loss of efficiency in working, the
interests of the user demand that proper attention be paid to the
cost of operation (in this case of the I 2 R losses) when considering
the expediency of lowering the manufacturing cost by econo-
mizing in materials.



18. Generation of E.m.f. It has been shown in previous
chapters how the strength and amount of the magnetic field
produced by an electric current may be calculated, and the next
step in the development of the dynamo is to consider how the
desired terminal voltage may be obtained by causing the armature
conductors to cut the magnetic flux which crosses the air gap from
pole to armature core.

The D.C. motor is merely a dynamo of which the action has
been reversed; that is to say, instead of providing mechanical
energy to drive the armature conductors through the magnetic
field, an electric current from an outside source is sent through
the armature winding which, by revolving in the magnetic field,
converts electrical energy into mechanical energy. In the design
of a D.C. motor, the procedure is exctly the same as for a D.C.
generator, and in the following pages the dynamo will be thought
of mainly as a generator.

Consider a flat coil of insulated wire of resistance R ohms,
consisting of S turns enclosing an area of A square centimeters.
Let this coil be thrust into, or withdrawn from, a magnetic
field of density B gausses, the direction of which is normal to the
plane of the coil. The quantity of electricity which will be set
in motion is expressed by the formula

a relation that can be proved experimentally.
But Q=/ m Xi = X*

where t = the time required to enclose or withdraw the flux

(* = BA),
I m = the average value of the current in the coil during

this period, and

E m the average value of the e.m.f. causing the flow of



Hence F (BA) X S

E m =

where all quantities are expressed in absolute C.G.S. units. If
we put < for the flux (BA) in maxwells, and express the e.m.f.
in the practical system of units, we have

E m = f~ volts (37)

For the condition S = unity, this formula is clearly seen to
express the well-known relation between rate of change of flux
and resulting e.m.f., namely that one hundred million maxwells
cut per second generate one volt. This is the fundamental law upon
which all quantitative work in dynamo design is based. The
procedure for obtaining a given amount of flux was explained
in previous chapters, and we now see that the voltage of any
dynamo-electric generator may be calculated by applying formula
(37). For the rest, the electrical part of the designer's work
consists in providing a sufficient cross-section of copper to carry
the required current, and a sufficient cross-section of iron to carry
the required flux, in order that the machine shall not heat ab-
normally under working conditions. There are other matters of
importance such as regulation, efficiency, economy of material,
and in D.C. machines commutation, which require careful
study; but it is hardly an exaggeration to say that apart from
mechanical considerations, which are not dealt with in this book
the work of the designer of electrical machinery is based on two
fundamental laws: (1) the law of the magnetic circuit, namely,
that the flux is equal to the ratio of magnetomotive force to reluc-
tance, and (2) the law of the generation of an e.m.f., namely, that
one hundred million lines cut per second generate one volt.

At any particular moment it is the rate of change of the flux in
the circuit that determines the instantaneous value of the
voltage, or, in symbols,

instantaneous volts in circuit of one turn = -^ X 10~ 8

where the negative sign is introduced because the developed
e.m.f. always tends to set up a current the magnetizing effect of
which opposes the change of flux.

Consider 'a dynamo with any number of poles p. Fig. 23
shows a four-pole machine with one face-conductor driven


mechanically at a speed of N revolutions per minute through the
flux produced by the field poles. If $ stands for the amount of
flux entering or leaving the armature surface per pole, we may

volts generated per conductor = 1<1

1<18 V- an

.LU /\ OU

It is not at present necessary to discuss the different methods of
winding armatures, but let there be a total of Z conductors
counted on the face of the armature. Then, if the connections
of the individual coils are so made that there are p\ electrical
circuits in parallel in the armature, the generated volts will be

= 60 X Pl X 10 8

This is the fundamental voltage equation for the dynamo; it
gives the average value of the e.m.f. developed in the armature

conductors, and since the virtual and
average values are the same in the
case of continuous currents, the
formula gives the actual potential
difference as measured by a volt-
meter across the terminals when no
current is taken out of the armature.
Under loaded conditions, the e.m.f.
as calculated by formula (38) is the
terminal voltage plus the internal IR
FlG 23 pressure drop.

The expression " face conductors "

may be used to define the conductors the number of which is
represented by Z in the voltage formula. It is evident that this
number includes not only the top conductors, but also those that
may be buried in the armature slots. The word "inductor" is
sometimes used in the place of "face conductor," and where
either word is used in the following pages it must be under-
stood to refer to the so-called "active" conductor lying parallel
to the axis of rotation whether on a smooth core or slotted

19. The Output Formula. The part of the dynamo to be
designed first is the armature. After the preliminary dimensions
of the armature have been determined, it is a comparatively


simple matter to design a field system to furnish the necessary
magnetic flux. The designer is usually given the following data:

Kw. output,

Terminal voltage,

Speed revolutions per minute. .

Sometimes the proper speed has to be determined by the de-
signer, as in getting out a line of stock sizes of some particular
type of machine; in that case he will be guided by the practice of
manufacturers and the safe limits of peripheral speed. Other
conditions such as temperature rise, pressure compounding,
sparkless commutation of current, may be imposed by specifica-
tions, but if the designer can evolve a formula which will give him
an approximate idea of the weight or volume of the armature,
this will be of great assistance to him in determining the leading
dimensions for a preliminary design. Modifications or correc-
tions can easily be made later, after all the influencing factors
have been studied in detail. Many forms of the output formula
are used by designers. The formula is based on certain broad
assumptions, and is used for obtaining approximate dimen-
sions only. Attempts to develop exact output formulas of
universal application should not be encouraged because it is not
possible to include all the influencing factors. The art of
designing will always demand individual skill and judgment,
which cannot be embodied in mathematical formulas.

In developing an output formula it is not necessary to enter into
details of the armature winding, provided the total number of
conductors, together with the current and e.m.f. in each, are

Let 3> = maxwells per pole.
p = number of poles.
N re volutions 'per minute.
Z = total number of armature inductors.
EC = volts per conductor.
I c = amperes per conductor.

The output of the armature, expressed in watts, will be

W = ZEJc (39)

where I c should include the exciting current in the shunt coils
of the field winding; a refinement which need not, however,
enter in the preliminary work.


The voltage per conductor is

*-e^i (40)

where the unknown quantity $ may be expressed in terms of flux
density and armature dimensions. Thus

$p = QA5B g l a wDr (41)

where B g = average flux density in the air gap under the pole

face. (Gausses.)

l a = gross length of armature core, in inches.
D = diameter of armature core, in inches.

pole arc

r = the ratio ^ - r r-
pole pitch

It will be seen that the quantity l a X irDr is the area in square
inches of the armature surface covered by the pole shoes; while the flux in the air gap per square inch of polar surface.

The pole pitch is usually thought of as the distance from center
to center of pole measured on the armature surface; and the
ratio r is therefore a factor by which the total cylindrical surface
of the armature must be multiplied to obtain the area covered
by the pole shoes the effect of "fringing" at the pole tips being

Substituting for $p in equation (40) its value as given by equa-
tion (41), and putting this value of E c in equation (39), we

_ gac

60 X 10 8

from which it is necessary to eliminate Z and I c if the formula
is to have any practical value.

A quantity which does not vary very much, whatever the
number of poles or diameter of armature, is the specific loading,
which is defined as the ampere-conductors per inch of armature
periphery. It will be represented by the symbol q. Thus


q = *D

ZI C = qwD

Substituting in equation (42), we have
W =



This is not an empirical formula since it is based on fundamental
scientific principles, and it is capable of giving valuable infor-
mation regarding the size of the armature core, provided the
quantities B gj q, and r, can be correctly determined.

The quantity B g will depend somewhat upon whether the
pole shoe is of cast iron or steel, also upon the flux density in the
armature teeth, which, in turn, depends upon the proportions
of the teeth and slots. If the flux density in the teeth is very
high, this may lead to (1) an excessive number of ampere-turns
on the field poles to overcome tooth reluctance, and (2) excessive
power loss in the teeth through hysteresis and eddy currents.

As a guide in selecting a suitable gap density for the preliminary
calculations, the accompanying table may be used. The
column headed B g is the apparent air-gap density in gausses,
while E" g is the same quantity expressed approximately in lines
per square inch.


Output, kw.

B, (gausses)

B' g (lines per sq. in.)






















500 and larger



The expression " apparent gap density" means that the flux is
supposed to be distributed uniformly over the face of the pole
and the effect of "fringing" is neglected. Thus

total flux per pole
area of pole face

It is customary to think of this as the average density over the
armature surface covered by the pole face, in which case

B a =

" la XT X r

where 3>, l a) and r, have the same meaning as in formula (41),
and T is the pole pitch or length of arc from center to center of
pole measured on the armature periphery. The lower values of



B g corresponding to the smaller outputs are required because
the increased taper of the teeth with the smaller armature
diameters would lead to abnormally high densities at the root
of the teeth if the air-gap density were not reduced. The figures
given in the table are applicable to machines with pole shoes of
steel or wrought iron. If the pole shoes are of cast iron, these
values should be reduced about 20 per cent. Cast-iron pole
shoes are rarely used except in very small machines.

The quantity q in formula (43) is determined in the first place
by the heating limits; but armature reaction and sparkless
commutation have some bearing upon its value. Suitable
values of specific loading for use in formula (43) may be taken
from the accompanying table.

(Ampere Conductors per Inch of Armature Periphery)

Kw. output




















800 and upward

760 to 850

The quantity r in formula (43) usually has a value between 0.60
and 0.80, a common value being 0.70. When the machine is

provided with commutating interpoles the ratio j r r must

be small in order to make room for the interpole. In this
case the lower figure of 0.60 would probably be selected as a
suitable value for r.

Approximate Constants for Use in Output Formula. For a
first approximation, the output formula (43) may be simplified
by substituting average values for the quantities B g , q, and r.
Thus, if B g = 7,500; q = 500; and r = 0.7, the output formula

kw. output =


If the speed of rotation (N) is not specified, it is necessary
to make some assumptions regarding the peripheral velocity of
the armature. This velocity lies between 1,200 and 6,000
ft. per minute; the lower values corresponding to machines
of which the speed of rotation is low, while the higher values
would be applicable to belt-driven dynamos, or to direct-coupled
sets of which the prime mover is a high-speed engine or high-head
waterwheel. When the generator is coupled to a steam turbine,
the speed is always exceptionally high, and the surface velocity of
the armature may then attain 2 or 3 miles per minute. The
discussion of steam-turbine-driven generators, in so far as the
electrical problems differ from those of the lower-speed machines,
will be taken up in connection with alternator design.

The peripheral velocity in feet per minute is,


whence A r ^ v


Inserting this value of N in formula (44) we get

kw. output = (45)

Relation of l a to D. The output equation (43) shows that there
is a definite relation between the volume of the armature and the
output, provided the quantities represented by the symbols
B g , q, and r, can be estimated. In order to determine the rela-
tion between the length l a and the diameter D, certain further
assumptions must be made. Thus,

I. = - pT ^ (46)

where p = the number of poles, and k is the ratio -TT-.

armature length

It is desirable to have the pole face as nearly square as possible
because this will lead to the most efficient field winding. If the
section of the pole core departs considerably from the circular or
square section, the length per turn of field winding increases
without a proportionate increase of the flux carried by the pole.
For a square pole face, k = 1 and

D p

= -*- = 0.45p (approximately)

l a Tit

The ratio D/l a usually lies between the limits of 0.35p and 0.65p.


It is not always possible or desirable to provide a square pole
face, and indeed it is necessary to check the dimensions of the
armature core by calculating the peripheral velocity. If a
suitable value for the peripheral velocity can be assumed, the
diameter is readily calculated because


XX - TT-


20. Number of Poles Pole Pitch Frequency. For calcula-
ting the relation between the length and diameter of armature core
by formula (46), the number of poles p must be known. The
selection of a suitable number of poles will be influenced by
considerations of frequency and pole pitch.

Frequency of D.C. Machines. The frequency of currents in the
armature conductors and of flux reversals in the armature core
generally lies between 10 and 40 cycles per second in continuous-
current generators. Higher frequencies are allowable, but should
be avoided, if possible, because on account of increased losses
in the iron, or increased weight to limit these losses the use of

p N
high frequencies is uneconomical. The frequency is / = ^ X


P - . (47)

This relation is useful for determining the probable number of
poles when the diameter, and therefore the peripheral velocity,
are not known.

Pole Pitch. The width of the pole pitch is limited by armature
reaction. It will readily be understood that the armature
ampere-turns per pole will be proportional to the pole pitch,
except for variations in the specific loading (q) . With a large
number of ampere-turns per pole on the armature, it is necessary
to provide a correspondingly strong exciting field in order that the
armature shall not overpower the field and produce excessive
distortion of the air-gap flux, resulting in poor regulation and
sparking at the brushes with changes of load. A good practical
rule is that the ampere-conductors on the armature shall not
exceed 15,000 per pole; i.e. } in the space of one pole pitch.

Ampere-turns on Armature. Exactly what is meant by the
expression "ampere-turns per pole" when applied to the arma-
ture winding should be clearly understood. In a two-pole



machine, the total number of ampere-turns on the armature is


SI 2 X I c because the current I c in -~ conductors on one-
half of the armature surface is balanced by an equal but opposite

FIG. 24. Current distribution bi-polar armature.

current in -~ conductors on the other half of the armature

surface, as indicated in Fig. 24. Now, since there are two poles,
we may say the ampere-turns per pole are

1*7 T
J 1 c

2 ~2~


or just half the number of ampere-conductors in a pole pitch.
This rule applies also to the multipolar machines. Thus, in Fig.

Direction of Motion of Armature

FIG. 25. Armature m.m.f. multipolar dynamo.

25, the horizontal datum line may be thought of as the developed
surface of a four-pole dynamo armature. The brushes are so
placed on the commutator that they short-circuit the coils when
these are approximately halfway between the pole tips. It is,
therefore, permissible to show the brushes in this diagram as if


they were actually in contact with the conductors on the " geo-
metric neutral" line. The armature m.m.f. will always be a
maximum at the point where the brushes are placed, because the
direction of the current in the conductors changes at this point,
producing between the brushes belts of ampere-conductors of
opposite magnetizing effect, as indicated in Fig. 25. The broken
straight line indicates the distribution of the armature m.m.f.
over the surface. Its maximum positive value occurs at A
and its maximum negative value at B. These maximum ordi-
nates are of the same height, and equal to one-half the ampere-
conductors per pole pitch as will be readily understood by inspect-
ing the diagram. Thus, whether the machine is bipolar or
multipolar, the armature ampere-turns per pole are

(SI) a = ~ ampere-conductors per pole pitch

- 1 <

In practice, a safe limit for the pole pitch is r = -

If q = 750, the maximum allowable pole pitch is r = 20 in.,
which dimension is rarely exceeded in ordinary types of dynamos.
Number of Poles. The formula (45) gives the output in terms
of the peripheral velocity. In its complete form it would be

kw. output = l a Dv X B g qr X 4 X 1Q- 11 (49)

By eliminating l a and D from the equation, it is possible to
arrive at an expression for the number of poles in terms of the
peripheral velocity and other quantities for which values can
be assumed. Thus, by (46)

_ TirD
' ~pk~

In order to eliminate D, let (SI) a be the armature ampere-
turns per pole. Then

= *D = 2(SI) a

p q

whence ^ _ 2p(SI) a


By substituting these values for l a and D in equation (49) we

. vy irkq X 10 11 ,_ n ,

P = kw * X (50)



By assuming values for the quantities in the right-hand side
of the equation, a reasonable figure for the number of poles can
be obtained. As an example, let the assumed values be as


q = 650
r = 0.7
v == 4,500
B a = 7,500
(SI) a = 7,500

The number of poles will then be

p = 0.0138 kw.

If the machine is to have an output of 500 kw., the estimated
number of poles is

p = 7

and since an even number of poles is necessary, the required
figure is 6 or 8. This, of course, would mean a change in one
or more of the assumed quantities.

The proper number of poles is determined partly by the amount
of the current to be collected from each brush set. This will
influence the selection of suitable values for k and (SI) a in
formula (50). Values as high* as 1,000 amp. per brush arm are
used in connection with low-voltage machines; but, on machines
wound for 250 to 500 volts, the current collected per brush arm
usually lies between the limits of 700 and 300 amp.

As a guide in selecting a suitable number of poles for a pre-
liminary design, the accompanying table may be of use. It is
based on the usual practice of manufacturers.


Output, kw.

No. of poles

Speed, rev. per min.

Oto 10


2,400 to 600

10 to 50


1,300 to 350

50 to 100

4 or 6

1,100 to 230

100 to 300

6 or 8

700 to 160

300 to 600

6 to 10

500 to 120

600 to 1,000

8 to 12

400 to 100

1,000 to 3,000

10 to 20

200 to 70


When using this or any other table or data intended to assist
the designer with approximate values, it is necessary to exercise
judgment, or at least be guided by common sense. For instance
it may be necessary to depart from the values given in the table
in the case of machines direct-coupled to slow-running engines.
This is especially worth noting in the case of the smaller
machines, which may require more than four poles in order to
give the best results on very low speeds.


Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 7 of 30)