Alfred Still.

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Font size the unknowns, and then checking results by assuming a larger
and a smaller value for the unknown quantity.

Since very low values of B will lead to great weight of iron,
and very high values will lead to an increased weight of copper,
it is safe to assume that B will not be less than 5,000 or more than

THE DESIGN OF ELECTROMAGNETS 55

10,000 gausses. We shall select the value of 7,000 for a trial
design, and carry this through to completion, although it would
usually be preferable to carry at least three designs through to
the point where it becomes clear that one of these is distinctly
preferable to the others from the point of view of economy; this
being the main consideration which the designer must always bear
in mind.

Putting for B, in formula (34), the value 7,000 gausses, the side
w of the (square) pole shoe is found to be 1.88 in. Let us make

13 150
this 2 in.; whence B = = 6,570. The thickness 1 2 of the

pole shoe must be small in order to keep down the magnetic
leakage; a value of Y in. for 1 2 should be satisfactory.

The diameter, d, of the magnet cores under the winding is
obtained by assuming a leakage factor and a suitable flux density
in the iron. The leakage factor (refer Art. 7, page 28, for defi-
nition) might be about 1.5 in a magnet of this type and size; and
the density in the core of magnet steel or wrought iron, may be
as high as 90,000 lines per square inch. Thus;

TT 1.5X4X 6,570 X 6.45

4 90,000

whence d = 1.9 or (say) d = 1.875 in.

Summing up the quantities so far determined, we have:
w = 2 in.

h = K in.
d = 1% in.

B (air-gap density) = 6,750 gausses.

4> (useful or effective flux per pole) = 6,750 X 6.45 X 2 X 2

= 174,000 maxwells.

Depth of Winding. The length I of the winding space, and the
thickness of winding, t, will depend upon the ampere-turns neces-
sary to produce the desired flux density in the air-gap, and also
upon the allowable current density, A, in the copper of the
exciting coils. The relation of I to t is not determined by the
amount of the ampere-turns, since this only calls for a sufficient
cross-section, or product I X t. The thickness t should not exceed
3 in., because a greater thickness may lead to excessive tem-
peratures inside the coil ; but the most suitable dimensions of the
coil are really determined by the current density, the winding
space factor, and the cooling surface necessary to prevent

56 PRINCIPLES OF ELECTRICAL DESIGN

excessive temperature rise. It is usual to assume a value for the
thickness, t, which may be something more than one-third of the
diameter of the core, with the previously mentioned limit of
about 3 in. Thus, even if d were greater than 9 in., the depth of
winding should, preferably, not exceed 3 in. In the present
case t should be about 1.875 -5- 3 = 0.625 in. Let us try t =
%in.

Current Density in Windings. If a suitable value for the
current density in the windings can be chosen, it will be an easy
matter to determine the length, I, of the winding space, and so
complete the preliminary design.

Let A = the current density (amperes per square inch) .
R" = the resistance, in ohms, between opposite faces of an
inch cube of copper. By formula (29) of Art. 9,

*" - at 60 c -

sf = the winding space factor, as given in Fig. 15, page 40.

As the size of wire is not yet known, a probable value of

0.5 will be chosen for this factor, in the preliminary

calculations.
T = the allowable temperature rise, being 40C. in this

example.
k = the cooling coefficient, being denned in Art. 9 formula

cooling surface

(30), as T X u i. - ? An average value

watts to be dissipated

of 180 may be taken for k.

Equating the I 2 R losses with the watts that can be dissipated
without exceeding the temperature limit, we can write,

T
total surface of coil X IT = watts lost

= R" A 2 X cubic inches of copper
or

[2lr(d + 0X2 + 4fcr(d + 0] = #"A 2 X 2Ur(d + t) X sf

whence

.
~

In order to eliminate I, we may consider t in the numerator to
be negligible, since, in this particular design, with an air gap of

THE DESIGN OF ELECTROMAGNETS 57

considerable reluctance, t will be small in comparison with I.
The approximate value of A will then be,

<36)

2 X 40 X 1,273,000
180X 1 X0.75 X 0.5
= (say) 1,250

Length of Winding Space. The ampere-turns required for the
double air-gap only, i.e., not including those required to overcome
the reluctance of the iron portions of the magnetic circuit, will be,
by formula (5) of Art. 4,

(SI) g = 2.025 X 21,

= 2.02 X 6,570 X 2 X 0.35
= 9,300.

The ampere-turns for the iron part of the magnetic circuit cannot
be calculated accurately until the length / and the actual leakage
factor have been determined; but, since the air-gap, in this case,
offers far more reluctance than the remaining portions of the
magnetic circuit, we shall assume the iron portions to require
only one-tenth of the air-gap ampere-turns. Thus,

total SI (both spools) == 10,230 approx.

We are now able to solve for the length of the winding space,
which is,

SI total
= X /'

10,230

2 X 0.75 X,250 X 0.5

= 10.9 or (say) 11 in.

Before proceeding further with the design, it will be well to see
whether the long magnet limbs will not be the cause of too great a
leakage flux. If the leakage factor is much in excess of the
assumed value (1.5) there is danger of saturating the cores under
the windings, and so limiting the useful flux available for drawing
up the armature Let the distance between the windings be 2 in.,
as indicated on Fig. 20; this gives all necessary dimensions for
calculating the permanence of the leakage paths.

Calculation of Leakage Flux. The total leakage flux between

58 PRINCIPLES OF ELECTRICAL DESIGN

the two magnet limbs should be considered as made-up of two
parts :

(a) The flux leakage from pole shoe to pole shoe, which is due
to the total m.m.f., available for the air gap,

(6) The flux leakage between the circular cores under the
windings, which increases in density from the yoke to the pole
pieces and is equal to the average m.m.f. X the permeance of the
air paths between the two iron cylinders. This average m.m.f.
will be approximately one-half the total m.m.f. of the exciting
coils.

For the permeance of the paths comprised under (a), we have:

1. Between opposing rectangular faces,

6.45 X 2 X 0.25
2.54 X 3.375

2. Between the two pairs of faces parallel to the plane of the
paper (Fig. 20), by formula (10) Art. 5,

A

0.25 X 2.54 v /TTX 2 + 3.375 \

7T

= 0.425.

X2.31og lo

Neglecting the flux lines that may leak out from the pole faces
farthest removed from each other, and also those between the
small ledges caused by the change from square pole piece to
circular section under the coil, the total leakage flux between pole
pieces is

3> p = m.m.f. X CPi + P 2 )
= 0.47T X 9,300 X 0.801
= 9,350 maxwells.

The permeance of the air paths comprised under (b) may be
calculated by applying formula (13) of Art. 5. Thus,

T X 11 X 2.54

2.31og 10 f h875

= 51

,3.5 + 1.875- V(3^) 2 +(2 X 3.5 X 1.875) /

The leakage flux between magnet cores under the windings is,
^ c= m.m.f. x ^

0.47T X 10,230 x
o

= 330,000 maxwells

THE DESIGN OF ELECTROMAGNETS 59

and the total leakage flux is

\$i = 9,350 + 330,000 = 339,350 maxwells.

The leakage factor is

170,000 + 339,350

170,000

which is greatly in excess of the permissible value, unless the
cross-section of the core under the windings is increased to keep
the flux density within reasonable limits. The simplest way to
reduce the amount of the leakage flux is to shorten the magnet
limbs, and although the long limbs with no great depth of winding
may lead to economy of copper, it is seen to be necessary in this
design to increase the depth of winding, t, in order to reduce the
length, I, of the exciting coils. The dimension t will have to be
more than doubled. Let us make this 1% in. and at the same
time retain the full section of 2 in. square under the windings;
that is to say, the square section bar will be carried up through the
coils without being turned down to a smaller section as in the trial
design.

Using formula (36) l to calculate the current density, we have,

80 X 1,273,000
: \180X 1.75 X0.5

= (say) 800
whence

10,230

2 X 1.75 X 800 X 0.5
= 7.3 in.

Let us try I = 7 in.

Allowing still a separation of 2 in. between the outside surfaces
of the windings, the distance between the two parallel magnet
cores of square section will now be 5.5. in. The permeance
between the opposite faces is

7.25 X 2 X 6.45
r\ - K~*;~v'~9~A"~

and between the sides of the magnet cores (by formula 10, page

25)

7.25 X 2.54 /TX2+5.5\_
" 2 = 2 X - - X 2.6 logio I - ) o.y

7T \ O.O /

1 This formula and also the correct formula (35) are applicable to rec-
tangular as well as to circular coils.

60 PRINCIPLES OF ELECTRICAL DESIGN

The total leakage flux will be approximately

= 100,000 maxwells.

This calculated value of the leakage flux should be slightly
increased because the total permeance between the two magnet
limbs will actually be greater than as calculated by the con-
ventional formulas. Let us assume the total flux to be 125,000
maxwells. This makes the value of the leakage factor.

_ 170,000 + 125,000 _
170,000

The maximum value of the density in the iron cores will be

170,000 + 125,000

7 - = 73,600 lines per square inch.

Closer Estimate of Exciting Ampere Turns. The modified
magnet will now be generally as shown in Fig. 21. The ampere
turns required to overcome the reluctance of the two air gaps
have already been calculated; the remaining parts of the mag-
netic circuit consist of the two magnet limbs under the windings,
together with the yoke and the armature. If we know the
amount of the flux through the iron portions of the circuit we can
readily calculate the flux density, and then ascertain the neces-
sary m.m.f. to produce this density, by referring to the B-H
curves of the material used in the magnet.

In the magnet cores under the coils, the flux density varies
from a minimum value near the poles to a maximum value near
the yoke; and as the leakage flux is not uniformly distributed
over the length l c (Fig. 21), it would not be correct to base re-
luctance calculations upon the arithmetical average of the two
extreme densities, even if the flux density were below the "knee"
of the B-H curve, with the permeability, /*, approximately
constant. With high values of B } the length of the magnetic
core should be divided into a number of sections, and each section
treated separately in calculating the required ampere-turns.
With comparatively low densities, as in this example, the
calculation can be made on the assumption of an average density
in the magnet cores, the value of which is

D V ' p

D c = o

THE DESIGN OF ELECTROMAGNETS

61

where B y = flux density at end near yoke,

and B p = flux density at end near pole pieces.

The total ampere-turns for the magnet of Fig. 21 can now be
calculated by using the B-H curve of Fig. 3 which is supposed to
apply to the particular quality of magnet steel or iron which it
is proposed to use. The calculation can conveniently be put in
tabular form as shown below.

TZ

-0.36

FIG. 21. Horseshoe magnet. (Modified design.)

Part of circuit

Length,
inches

Cross-
section,
square
inch

Total flux,
maxwells

Density,
lines per
square inch

SI per
inch

SI,
total

Air gaps

0.7

4.0

170,000

42,500

9,300

Armature
Magnet cores
Yoke

8.3

14.5
12

2.5
4.0
4

170,000
220,000
295,000

68,000
55,000
73,600

9.0
7.5
9 5

75
110
115

Total

9,600

Thus, it is necessary to have not less than 9,600 ampere-turns
on the two bobbins.

Calculation of Windings. The formula (26) of Art. 10 can be
written,

(m) =

mean length of turn (inches) X SI
E

62 PRINCIPLES OF ELECTRICAL DESIGN

The mean length of turn is approximately 4(u> + ), or 4 X
3.75 = 15 in.; and the potential difference across the two coils
is 110 volts; thus,

15 X 9,600

Referring to the wire table on page 34, the wire of cross-section
nearest to the required value is No. 18 B. & S. gage, because
No. 19 will be too small to provide the necessary excitation.
This larger wire will provide a factor of safety, and it may be used
if the watts lost and the temperature rise are not excessive.

Calculation of Temperature Rise. The space factor for No.
18 B.& S. D.C.C. wire, as taken off the curve of Fig. 15, is 0.54.
The cross-section of the copper in the coil is therefore 7 X
1.75 X 0.54 = 6.62, and the number of turns per coil will be
6.62/0.001276 = 5,180. Other required values are:

Length of wire in one coil = 5,180 X TS = 6,500 ft.

\.z

Resistance at 60C. = 6.5 X 7.42 = 48.2 ohms.

Current = 75-^ =1.14 amp.

4o.Z

Total PR loss = 1.14 X 110 = 125 watts.

Outer surface of both coils = 2X7 X 4 X 5.5 =308
Inner surface of both coils = 2X7 X4X2 =112
End surfaces of both coils = 4 X 1.75 X 4 X 3.75 = 105

Total cooling surface ................... .... = 525 sq. in.

The rise of temperature, by formula (30) Art. 11 taking
k = 180, is

1OK

T = 180 X = 43C.

which is only slightly in excess of the specified temperature rise
(40C.). Another layer or two of winding would bring the
temperature down to the required limit; or, if preferred, the
length of the coil may be increased by a small amount without
appreciably adding to the reluctance of the magnetic circuit.

It should be mentioned that the design of magnet as shown in
Fig. 21, is probably larger than would be necessary to fulfil
practical requirements, because it is not likely that the full
pressure of 110 volts would be maintained across the terminals

THE DESIGN OF ELECTROMAGNETS 63

for many hours. The magnet would be designed either for inter-
mittent operation, in which case the temperature rise might be
calculated as in Case (a) of Art. 12, page 47, or, if left con-
tinuously in circuit, a resistance would automatically be thrown
in series with the coil windings in order to reduce the PR loss
and effect a saving of copper while still maintaining the required
pull of 200 Ib. through the reduced air gap.

Factor of Safety. Seeing that the coils are actually wound with
a wire of greater cross-section than the calculated value, the
initial pull will be somewhat greater than the specified 200 Ib.
The actual ampere-turns are 5,180 X 2 X 1.14 == 11,800, and
since the density in the iron is not carried above the "knee"
of the B-H curve, the actual flux density in the air gap, instead

1 1 800
of being 6,570 gausses, will be approximately 6,570 X o

8,070 or (say) 8,000 gausses. The initial pull will actually be

(Q 000") 2
200 X / 6 ' 570 r2 = 30 lb - nearly. This factor of safety of 1.5

may seem excessive, and if the strictest economy of material is
necessary, the coils should be wound with a wire of the calcu-
lated size, or, if standard gage numbers must be used, as would
generally be the case, the mean length of turn may be modified by
providing a greater or smaller depth of winding space. As an
alternative, two sizes of wire may be used as explained in Art. 10.

Most Economical Design. The cost of materials is easily
estimated by calculating the weight of iron and copper sepa-
rately. For the purpose of comparing alternative designs, it
is usual to take the cost of copper as five times that of the iron
parts of the magnet. If actual costs are required, the figures
would be about 20c. per pound for copper wire, and 4c. per
pound for the magnet iron.

The reader will recollect that this design has been worked
through on the assumption that about 6,500 gausses would be a
suitable density in the air gap. If many magnets are to be made
to the one design, or in any case if the magnet is large and costly,
the designer should now try alternative designs, using air-gap
densities of (say) 4,000 and 8,000 gausses respectively. By com-
paring the three designs, all of which will comply with the terms
of the specification, he will be able to select the one which can
be constructed at the least cost. This method of working may
seem slow and tedious, but it is sure, and if actually tried

64 PRINCIPLES OF ELECTRICAL DESIGN

will be found to involve less time and labor than might be
supposed. The student following the courses at an engineering
college does not unless he has had outside experience ap-
preciate the value of his time. Time may be used, abused, or
wasted; and when a concrete and definite piece of work has to
be done, the time spent upon it, not only by the workman, but
also by the designer, may be of no less, or even of greater, im-
portance than the cost of the materials. The case in point
exemplifies this. If the required magnets are -small, and but
two or three are likely to be wanted, the designer should not
spend much time on refinements of calculation and in endeavor-
ing to reduce the cost of manufacture to the lowest limit; but if
the magnets are of large size and several hundred will be re-
quired, then time spent by the designer in comparing alternative
designs and in striving to reduce material and labor cost, would
be amply justified. These considerations and conclusions may,
to many, appear elementary and obvious; but they emphasize
the importance of what is generally understood by " engineering
judgment" which is rarely acquired or rightly valued until after
the student has left school.

Before taking up the design of another form of magnet, it
may be well to state that the method of procedure here followed
in the case of a horseshoe magnet is not put forward as being
necessarily the best, or such as would generally be adopted by
an experienced designer. It serves to illustrate much that has
gone before, and emphasizes the fact that, even if the designer
must make some assumptions and do a certain amount of guess-
work at the beginning, and during the course, of his design, he
can always check his results when the work is completed, and
satisfy himself that his design complies with all the terms of the
specification.

17. Circular Lifting Magnet. The electromagnet of which
Fig. 22 is a sectional view is circular in form. Its function is to
lift a ball of steel weighing, say, 4,000 lb., which, on the opening
of the electric circuit, will fall upon a heap of scrap iron. This
device is referred to colloquially as a "skull cracker." The
diameter of a solid steel sphere weighing 4,000 lb. is approxi-
mately 30 in. If the outer cylindrical sheel forming one of the
poles of the magnet has an average diameter of 21 in., it will
include an angle of 90 degrees, as indicated in Fig. 22, and lead
to a design of reasonable dimensions. If the required width of

THE DESIGN OF ELECTROMAGNETS

65

the annular surface forming the outer pole of the magnet should
be less than 1 in., it might be necessary, for mechanical reasons,
to reduce the diameter in order to obtain a practical design.
The total pull required is 4,000 lb., or 2,000 Ib. per pole. The
pull per square inch of polar surface is, by formula (16) page 29.

B 2
Pounds per square inch = Y~73QQQ

whence the area of each pole face is

2,000 X 1,730,000
B 2

If B = 6,000 gausses, A == 96 sq. in.; and if B = 8,000 gausses,
A = 54 sq. in. Either of these flux densities would probably

Magnetising Coil

Retaining Plate
I of (Non-magnetic)
\ Manganese Steel,

with Stiffening

Ribs

FIG. 22. Circular lifting magnet.

be suitable for a magnet of the type considered. Assuming a
minimum width of 1 in. for the pole face on the outer shell, we
have,

Area of outer ring = 1 X TT X 21
= 66 sq. in.

which would provide the required pull if B = 7,240 gausses.

5

66 PRINCIPLES OF ELECTRICAL DESIGN

Let us therefore decide upon this dimension. The diameter
of the inner core is obtained from the equation

j D 2 = 66

whence D 9.16 in. It will be better to. provide a 2-in. hole
through the center of the magnet, and have a conical face to the
core, as shown in sketch. The diameter of the central pole core
may be 10 in., and the edges can be slightly bevelled off so that
the polar surface shall not exceed 66 sq. in.

In order to introduce a factor of safety, and permit of the iron
ball being lifted even when the contact between magnet and
armature is imperfect, the specification would probably call for
a magnet powerful enough to attract the ball through a distance
of, say, J4 m - Let us further assume that, the action being
intermittent, the current will flow through the exciting coil dur-
ing only half the time that the magnet is in action. This will
probably permit the use of a current density of 1,000 amp. per
square inch of copper section. Thus, if the winding space factor
may be taken as 0.5, it will be necessary to provide 2 sq. in. of
cross-section of coil for every 1,000 ampere-turns of excitation
required.

The ampere-turns necessary to overcome the reluctance of the
double air gap are

l (SI) a = 2.025 X l" g

= 2.02 X 7,240 X }4
= (say) 8,000, which includes a small

allowance for the reluctance of the iron in the circuit. The
required section of coil is therefore about 16 sq. in. One of the
dimensions should, if possible, be kept within the limit of 3 in.
in order to avoid excessive internal temperatures. A cross-
section of 5 in. by 3 in. = 15 sq. in. will probably be large enough
to accomodate the winding.

The average length per turn of wire is 7r(10 + 5) = 47.2 in.,
and (by formula 26, Art. 10, page 42) the cross-section of
the wire, in circular mils, will be

47.2 X 8,000
(m) = - - ir

where E is the voltage across the terminals of the magnet.
Assuming this to be 120 volts, the value of (m) will be 3,140.

1 Art. 4, formula (5).

THE DESIGN OF ELECTROMAGNETS 67

Referring to the wire table on page 34, the calculated size is seen
to be only slightly less than the cross-section of No. 15, B. & S.
gage. Using this wire, and allowing }- in. for insulation be-
tween the iron and the coil, there will be about 68 layers of 41
turns, making a total of, say, 2,800 turns in the coil. The length
of wire will therefore be 2,800 X 47.2 -v- 12 = 11,000 ft., and
the resistance hot, i.e., at a temperature of 60C., will be 3.702
X 11 = 40.6 ohms. The current = 120/40.6 = 2.95 amp. and
the actual ampere-turns = 2.95 X 2,800 = 8,260.

Rise of Temperature. The watts lost in the field when the
current is flowing are El = 120 X 2.95 = 354; but since the
current is supposed to be passing through the windings during
only one-half the time that the magnet is in operation, we can
apply the rule referred to in Art. 12, and assume that the power to
be dissipated amounts to only 354/2 = 177 watts. The total
surface of the coil is 47.2(10 + 6) = 755 sq. in. ; and if we use the
average value of 180 for the heating coefficient fc, as suggested
in Art. 11 page 46, the temperature rise will be

180 X = 42.2C.

above the temperature of the air. This figure is a safe one, and,
since the iron shell offers a large cooling surface in contact with
the air, it is probable that the value of the coefficient k in this
particular design might be about 250. The temperature of
the windings will therefore not be excessive, and the amount of
copper might even be slightly reduced if the greatest economy in
manufacturing cost is to be attained. Exact data for the cal-
culation of temperatures in coils entirely surounded by iron are
not available, because the thickness and radiating surface of
the external shell are factors which will have an appreciable
influence on the value of the heating coefficient.

Calculation of Leakage Flux. In order to provide sufficient
cross-section in the magnet, and ensure that the flux density in
the iron shall not be carried too near the saturation limit, it is
necessary to estimate the amount of the leakage flux.

The permeance of the leakage paths may be calculated by
considering two separate components of the leakage flux: (1)
the flux which passes between the core and the cylindrical shell
through the space occupied by the windings, and (2) the flux
which passes between the uncovered portions of the central core

68 PRINCIPLES OF ELECTRICAL DESIGN

and the outer shell. Referring to Fig. 22, upon which the
approximate dimensions of the leakage paths have been marked,
the numerical, values of the two permeances are seen to be,
approximately,

3X7r(10 + 5) X6.45
for the path (1), Pi = - 5 v 9 54 " =

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