Alfred Still.

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10. Calculation of Magnet Windings. The calculation of the
ampere-turns necessary to produce a given flux of magnetism
has already been explained (see Arts. 3 and 4), and it is a fairly
simple matter to determine the
exciting force approximately,
provided the magnetic circuit
consists mainly of iron of known
magnetic characteristics, and
that the air gaps are short.
These calculations will be more
fully illustrated when working

out one or two numerical exam- t

. . FIG. 16. Cylindrical magnet coil,
pies; but for the present it is

assumed that a definite number of ampere-turns, SI, have to be
wound on a bobbin or former, and that the applied D.C. potential
difference, E, is known.

If SI = the total ampere-turns in the coil shown in Fig. 16,
then, whatever may be the number of the turns S, the total
ampere-wires in the cross-section d X I is (*S/). For a first ap-
proximation of the area required, it is well to assume a certain
current density in the windings, which is not likely to cause an
excessive heat loss and therefore an unsafe rise of temperature.
The following figures may be used :

For large magnets try A = 700, or (M) = 1,800

For medium-sized magnets try A = 900, or (M) = 1,400

For small magnets try A = 1,100, or (M) = 1,1 50]

where A = current density in amperes per square inch, and (M)
= number of circular mils per ampere. The relation between
these quantities, as previously explained (Art. 9) is,

A X (M) = 1.273 X 10 6

By assuming the current density, it is then easy to calculate the
probable cross-section of the copper in the coil. This, however,
is not equal to the product d X I because the winding space factor

42 PRINCIPLES OF ELECTRICAL DESIGN

must be taken into account. A little practice will enable the
designer to form a rough idea as to the size of wire that will
be required, this being of small diameter for high voltages and of
large diameter for low voltages. He can select a probable value
of the space factor from the curves of Fig. 15. The cross-section
of the coil can now be calculated because,

O T

JXd = A3<tf (25)

Any convenient relation between I and d may be chosen, but
the value of one of these dimensions is usually decided upon in
the first instance. It is well to avoid making the depth of wind-
ing, d, more than 3 in., even in large magnets, because the internal
temperature is then liable to become excessive.

The size of the wire will depend upon the length of the mean
turn; and, with a known value for d, and a core of circular cross-
section, we have:

Mean length per turn = ir(D + d) in. (See Fig. 16.)

Applying formula (21) for the resistance of a copper wire at a
temperature of about 60C., we may write,

. , length in inches E

resistance = - -, x - = y

(ra) /

whence

ir(D + d)S E
(m) = I
and

(m) = *

til

In this manner the size of the wire can be determined. It
should be noted that, for a given excitation, its cross-section
depends only upon the applied potential difference and the
average length per turn; it is quite independent of the number
of turns of wire, S. That this must necessarily be the case is
seen when it is realized that for every increase in S, the resist-
ance increases in like manner, causing the current I to decrease
by a proportional amount.

By referring to a wire table such as those on pages 34 and 35,
the standard gage size nearest to the calculated cross-section
can be chosen. If it does not seem close enough to the required
size for practical purposes, the coil can be wound with two sizes

THE MAGNETIC CIRCUIT ELECTROMAGNETS 43

of standard wire, as will be explained shortly, but it is generally
possible to modify the average length per turn and so obtain the
desired result. The formula (26) can be written,

( + d} - _^L

" irXSI
whence

d- E(m) D

-*xsr

Now, if (A) is the area in circular mils of the standard size of
wire it is proposed to use instead of the previously calculated
cross-section (m) the required ampere-turns can be obtained
by making the depth of winding,

* - -

Now estimate (by using the space factor curves, or by calculation)
the number of turns required to fill the spool to the required
depth, and calculate the total resistance, R, and the current,

A convenient rule, which usually provides sufficient winding
space to prevent excessive temperature rise, is to allow 1 sq.
in. of winding space cross-section for every 500 ampere-turns
required on the coil. This simply means that the product
A X sf of formula (25) is taken as 500.

Winding Shunt Coils With Two Sizes of Wire. For a definite
mean length per turn, the exact ampere-turns required on a
magnet can always be obtained with standard sizes of wire by

N 3 Feet of A Ohms *+*-y Feet of B Ohms - H

i - ! - 1

One Foot R Ohms

FIG. 17. Two sizes of wire in series.

using, if necessary, two wires of different diameter connected in
series or in parallel. The series connection is most usual for
magnets or field coils to be connected across a definite voltage.
Let R stand for the ohms per foot length of wire to give the
required excitation at the proper temperature; A = ohms per
foot, at the same temperature, of the standard wire of larger size;

44 PRINCIPLES OF ELECTRICAL DESIGN

and B = the corresponding resistance of the smaller standard
size of wire. It is proposed to make up the resistance R by
connecting x feet of A ohms per foot in series with y feet of B
ohms per foot, as indicated in Fig. 17. Thus,

xA + yB = R
also

x + y = 1 or, y = I - x

whence

xA + B - xB = R
and

. = (28)

If it is preferred to work with cross-sections in circular mils,
instead of resistances in ohms per foot, we can put the relation of
formula (28) in the form

(A) (B)
_ (A) (B) - (m)
~ (m) X (B) - (A)
where (m) = calculated circular mils,

(A) = circular mils of larger standard wire,

(B) = circular mils of smaller standard wire.

When winding with two sizes of wire in series, it is usual to put
the smaller wire on the outside where the heat will be most readily
dissipated.

11. Heat Dissipation Temperature Rise. The winding, if
calculated as explained in the preceding article, will furnish the
required excitation; but it is possible that the estimated value for
the current density may result in a temperature so high as to
injure the insulation, or so low as to render the cost of the magnet
owing to excess of copper commercially prohibitive. The
highest temperature will be attained somewhere inside the coil,
and it is not easily calculated; the temperature as measured by a
thermometer on the outside of the coil is only a rough guide to
that of the hottest part. The average temperature is also higher
than the outside temperature; it can be ascertained by meas-
uring the resistance of the coil hot and cold. The maximum
temperature can be measured only by burying thermometers or

THE MAGNETIC CIRCUIT ELECTROMAGNETS 45

test resistances in the center of the coil when it is being wound.
The depth of winding has much to do with the relation between
outside and inside temperatures. This depth should rarely
exceed 3 in., and a long coil of small thickness will, obviously,
have a much more uniform temperature than a short thick coil
of the same number of turns.

As a rough indication of what may be expected in the matter
of internal temperatures, it may be stated that, in magnet coils
of average size, the mean temperature might be 1.4 times, and
the maximum temperature 1.65 times, the external temperature.
The maximum allowable safe temperature for cotton-covered
wires is 95C., and as this may be reached when the outside tem-
perature is 40C. above that of the surrounding medium, a
maximum rise of temperature of 40 or 45C., as measured at
the hottest accessible part of the finished coil, is usually specified.
If the calculated temperature rise is in excess of this, the coil
must be re-designed in order to increase the cooling surface or
reduce the PR loss.

The calculation of temperature rise is based largely upon
coefficients which are the result of tests, preferably conducted on
coils of the same type and size as the one considered. The cool-
ing surface of a magnet winding of the type shown in Fig. 16, page
41, may be taken as the outside cylindrical surface only; or this
outside surface plus the area of the two ends; or, again, the whole
surface, not omitting the inside portion in proximity to the iron
of the magnet core. This is largely a matter of individual choice
based on experience gained with similar types of coil, and the
heating coefficient will necessarily have a different value in
each case.

E 2
The watts lost amount to PR, or El, or p . The heating

coefficient is the cooling surface necessary to dissipate one watt
per degree difference of temperature between the outside of the
winding and the surrounding air. Thus

k-^
~ PR

and

(30)

where T is the temperature rise in degrees Centigrade; fcis the
heating coefficient, which can, if preferred, be properly defined

46 PRINCIPLES OF ELECTRICAL DESIGN

as the degrees Centigrade rise in temperature when the loss in
watts is equal to the cooling surface in square inches; and A is
the actual cooling surface expressed in square inches. The area
of this cooling surface will be reckoned as the sum of the outside
and inside perimeters multiplied by the length of the coil, plus
the area of both ends of the coil. The temperature rise is
found to differ very little whether the coil is surrounded entirely
by air, or provided with an iron core, and for this reason the
writer prefers to consider the total external area of the coil as
the cooling surface. 1

. The heating coefficient k is not a constant, even for a given size
and shape of magnet. It is a function of the difference of
temperature between the coil surface and the surrounding
medium; it also depends upon the material of the spools or
bobbins, on the insulating varnish and wrappings (if any), and
other details of construction. Assuming a surface temperature
rise of about 40C. and open type coils that is to say, coils
with ends and outside surface exposed to the air finished with
a coat or two of varnish over the cotton-covered wire, the coeffi-
cient k might lie between 160 and 200, with an average value of
180. With a temperature rise of only 20C. the average value
of k should be taken as 190.

In the case of iron-clad coils such as those found in many
designs of lifting magnets and magnetic clutches, the final
internal temperature will depend largely on the shape and thick-
ness of the surrounding iron, and on the total radiating surface;
but, for approximate calculations, the same coefficient may be
used as for the open coils, bearing in mind that, in all cases,
the temperature rise T of formula (30) is that of the outside layer
of wire, and the area A is that of the total external surface of the
copper coil.

12. Intermittent Heating. Without attempting to discuss
exhaustively the effects of intermittent service, the two extreme
cases may be considered : (a) the apparatus is alternately carrying
the full current, and carrying no current, during short periods of
time extending over many hours, so that the total cooling surface
is the factor of importance; and (6), the apparatus is in use at

1 This is the recommendation of MR. G. A. LISTER in his excellent paper
published in the British Journal, Inst. E. E., vol. 38, p. 402, to which the
reader is referred if he wishes to pursue further the subject of *magnet-
coil heating.

THE MAGNETIC CIRCUIT ELECTROMAGNETS 47

only rare intervals of time, with long periods allowed for cooling,
so that the factor of importance is the capacity for heat.

Case (a). During a period of 1 hr., the current is passing
through the magnet coil for a known short interval of time,
and is then switched off for another known period, so that out of
a total of 60 min., the current flows through the coil during h
mm. only; the temperature rise can then be caculated, as
previously explained, by making the assumption that the watts

to be dissipated are not W = PR; but W h = - ~

This method cannot safely be used if the "on" and "off"
periods are long; but no general rule can be formulated in this
connection because the size of the magnet is an important
factor.

Case (b). If used only at rare intervals of time, with long
periods allowed for cooling down, a magnet coil can be worked
at very high current densities. The temperature rise is then
determined solely by the specific heat of the copper, and its total
weight or volume.

The specific heat of a substance is the number of calories re-
quired to raise the temperature of 1 gram, 1C. The specific
heat of water at ordinary temperatures being taken as unity,
that of copper is about 0.09. One calorie will raise 1 gram of
water 1C.; and since 1 calorie is equivalent to 42 X 10 8 ergs
(or dyne-centimeters), it follows that, to raise 1 gram of copper
1C. in 1 sec., work must be done at the rate of 0.09 X 42 X 10*
ergs per second. But 1 watt is the rate of doing work equal to
10 7 ergs per second; and 1 Ib. = 453.6 grams; this leads to the
conclusion that the power to be expended to raise 1 Ib. of copper
1C. in 1 sec. is

0.09 X 42 X 10 6 X 453.6

T7y7 - = 171.5 watts.

A cubic inch of copper weighs 0.32 Ib., and (173 X 0.32) or
55 watts will therefore raise the temperature of 1 cu. in. of copper
1C. in 1 sec. assuming no heat to be radiated or conducted
away from the surface of the coil.

In this manner it is possible to calculate how long an electro-
magnet for occasional use can be left in circuit without damage
to insulation. A temperature rise of 50 to 55C. is generally
permissible in making calculations on the heat-capacity basis.

CHAPTER III
THE DESIGN OF ELECTROMAGNETS

13. Introductory. The object of this chapter is partly to
summarize and coordinate what has already been discussed;
but mainly to familiarize the reader with the laws of the magnetic
circuit and the simple computations which will enable him to
proportion the iron cores and calculate the field windings of
electric generators. A little practice in the design of the simple
forms of lifting magnet, or magnetic brake, will be of the greatest
value in illustrating the practical application of the fundamental
principles underlying the design of all electromagnetic machinery.
The designer who wishes to specialize in lifting magnets, magnetic
clutches, and electromagnetic mechanisms generally, must pursue
his studies elsewhere : he is referred to other sources of information
such as MR. C. R. UNDERBILL'S book on electromagnets. 1
There are many matters of interest, such as the means of ob-
taining quick, or slow, action in magnets; equalizing the pull
over long distances; special features of alternating-current electro-
magnets; and the mechanical devices introduced to attain specific
ends, but none of these can receive adequate attention here.

In the design of electromagnets with movable armatures or
plungers, the work to be done is usually reckoned as the initial
or starting pull, in pounds, multiplied by the travel, in inches.
Many designs, of varying sizes and costs, can be made to comply
with the terms of a given specification, and the main object
of the designer should be to put forward the design of lowest
cost which will fulfil the conditions satisfactorily. It is not
proposed to devote much space, either here or elsewhere, to the
detailed discussion of the commercial aspects of design; but it
is well to emphasize the fact that a designer who does not con-
stantly bear in mind the factors of first cost and cost of upkeep,
is of little or no value to the manufacturer. In the design of
electromagnets, especially of the larger sizes, the material

1 "Solenoids, Electromagnets and Electromagnetic Windings:" D. VAN
NOSTRAND Co.

48

THE DESIGN OF ELECTROMAGNETS

49

cost is the main item, and the total cost of iron and copper is
a good guide to the cost of the finished magnet, when it is
desired merely to compare alternative designs based on a given
specification.

It is an easy matter to estimate the volume and weight of
materials in so simple a design as a lifting magnet, and although
formulas can be developed which aim to give the proportions
and sizes for the most economical design, these are usually of
doubtful value, and it is generally simpler to apply a little
common sense and the engineering judgment which will come
with practice, and try two or three designs with different pro-
portions before selecting the one that seems most suitable in all
respects, not omitting the important item of initial cost.

14. Short-stroke Tractive Magnet. With a design of plunger
magnet as shown in Fig. 18, there is not much magnetic leakage,

Brass Ring

FIG. 18. Plunger or iron clad magnet.

because the travel of the plunger, or length of air gap, is small in
comparison with the area of the pole faces. Given a definite total
amount of flux to produce the required pull, the cross-section of
the various parts of the magnetic circuit is readily calculated.

50 PRINCIPLES OF ELECTRICAL DESIGN

Various proportions can be tried, also different values of the
magnetic density in the air gap. The pull per square inch de-
pends upon B 2 ; but, by forcing the density up to high values, the
ampere-turns required become excessive, and the weight and cost
of the copper coils, prohibitive. More will be learned by trying
various proportions and roughly estimating the cost, than by a
lengthy discussion of the manner in-which the various dimensions
are dependent upon each other. It will probably be found that
the most economical initial density will not exceed 11,000
gausses; and (by formula 16, Art. 8) the pull, in pounds per
square inch, is

(11, OOP) 2 _ 7n
1,730,000 ~

thus, with the usual cylindrical core,

ird 2
total force, in pounds = F 70 -

whence

d = 0.135 \/F (31)

The magnet can now be sketched approximately to scale, and the
necessary ampere-turns computed, all- as previously explained in
Art. 4. Although Fig. 18 shows a very short air gap, the same
methods apply to the calculation of magnets with longer air gap,
provided this is not so great as to cause excessive magnetic
leakage. A practical rule which determines the minimum length
of the winding space is that this length, h, should never be less
than twice the air-gap length, I.

15. Magnetic Clutch. The design of a magnetic clutch to
transmit power between a shaft and pulley or any piece of rotat-
ing machinery, is generally similar to that of the circular type
of lifting magnet. Fig. 19 shows a common type of magnetic
clutch with conical bearing surfaces, although the conical shape
is not essential, and the wedge action of the cone-shaped rings is
not relied upon to increase the pressure between the surfaces in
contact. When the two iron surfaces are held together by the
action of the exciting coil, the flux density over the area between
the two annular pole faces must be such as to produce a force
that will prevent slipping between these faces. A factor of
safety of 2.5 to 3 is generally allowed.

THE DESIGN OF ELECTROMAGNETS

51

Let R = mean radius, in feet.

A = area of all North, or of all South, polar surfaces in

contact (square inches).

P = pressure in pounds per square inch of contact surface.
N = revolutions per minute.

c = coefficient of friction, the meaning of which is that
c X PA is the tangential force which will just produce
slipping.
then

, PAX cX 2irR XN

hp ' = -33,000

FIG. 19. Magnetic clutch.

which gives the horsepower that the clutch will transmit just
before slipping occurs. If k is the safety factor, and (hp.) is
the horsepower which has to be transmitted, the value of hp.
to insert in formula (32) should be (hp.) X k. Note also that

B 2
P = T-s>r and different values of B can be tried; these

52

PRINCIPLES OF ELECTRICAL DESIGN

values should be fairly high, but they will depend upon whether
the magnetic circuit is of cast iron or steel. The unknown
quantities are then R and A, and equation (32) can be put in the
form

33,000 X (hp.) X k X 1,730,000

R X A =

2wNCB 2

91 X 10 8 X (hp.) X k
NCB 2

(33)

where B is in gausses, and the coefficient of friction, c (for dry
surfaces), may be taken from the accompanying table. If the
surfaces are lubricated with oil or grease, the friction coefficient
may be lowered as much as 40 per cent. One reason for using a
large factor of safety is to allow for the possibility of dirt or oil
getting between the surfaces in contact.

TABLE GIVING APPROXIMATE VALUES OF THE COEFFICIENT OF FRICTION, c

Pressure, Ib.
per sq. in.

Wrought iron on
wrought iron

Wrought iron on
cast iron

Cast iron on steel

Cast iron on cast
iron

50

0.151

0.182

0.188

0.113

100

0.187

0.222

0.237

0.140

150

0.217

0.254

0.275

0.165

200

0.259

0.280

0.304

0.190

The formula (33) permits of either quantity R or A being
calculated when one of them is known or assumed. It is the
business of the designer to determine usually by trial the
dimensions which will give the best results. So far as cost is
concerned, a large diameter may show a saving in materials; but
the labor cost not omitting the cost of patterns when but few
castings are required should also be considered.

At times when slipping occurs, as in a magnetic brake, or
when throwing in a magnetic clutch while there is some relative
movement between the two parts, there is a powerful retarding,
or driving, action as the case may be, due not to the direct
magnetic pull between the surfaces in contact, but to the fact that
eddy currents are produced in the polar faces on account of
the cutting of the magnetic lines. This cutting of flux is similar
to what occurs in the unipolar, or so-called homopolar, type of
D.C. generator, where the currents are confined to certain paths
and collected by means of sliding brushes.

THE DESIGN OF ELECTROMAGNETS
\

NUMERICAL EXAMPLES

53

16. Horseshoe Lifting Magnet. Assume the specified condi-
tions to be as follows:

Initial pull = 200 Ib.

Travel of armature (being the length of the single air gap) =
0.35 in.

B.C. voltage = 110.

Allowable temperature rise = 40C.

K - -K; -H

Pole Face 2 Square

FIG. 20. Horseshoe magnet.

The temperature to be taken on the outside surface of the
exciting coils, after a sufficient time has elapsed for the final tem-
perature to be reached.

The required magnet might be generally as shown in Fig. 20,
where the iron limbs are of circular section with square pole pieces.
These limbs may be steel castings, or they can be turned down
from square bar iron. Cast iron would not be a suitable material,

54 PRINCIPLES OF ELECTRICAL DESIGN

because the large cross-section necessary to keep the flux density
within reasonable limits would lead to an unnecessary and
wasteful increase in the weight of the copper coils.

Applying formula (16) of Art. 8, page 29, the pull, expressed
in terms of the air-gap density is

2AB*
~ 1,730,000

where B is in gausses, and A is the area of one polar face, expressed
in square inches. Thus,

200 *****

~ 1,730,000
whence,

13,150
"-.-]&- (34)

This relation between size of pole face and the air-gap density
must exist if the pull of 200 Ib. is to be obtained, but the density B
can be varied within wide limits. It is obvious that high values
of B are advantageous in so far as they reduce the weight and
cost of the iron in the magnet; but since the initial air-gap length
remains constant, the necessary ampere-turns will increase
almost in direct proportion to any increase of B. The economical
value of the flux density, B, cannot be immediately determined;
and although formulas for minimum cost can be developed, they
become unwieldy and unpractical when all the important
factors are taken into account. On the other hand, if all deter-
mining factors including such items as cooling surfaces and
magnetic leakage are not taken into consideration, the formulas
are very inaccurate and not of general application. It is very
interesting to develop approximate formulas for use in arriving at
the economical dimensions of any particular type of electro-
magnetic apparatus, and the reader may learn much by trying
to put the various, and frequently conflicting, requirements in the
form of a mathematical equation; but we shall follow here the
method adopted by a large number of experienced designers,
which consists in trying what seems a probably value for one of

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 5 of 30)