Alfred Still.

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Pull between Inclined Surfaces. Conical Plungers. Sketch
(a) of Fig. 12 shows a portion of an electromagnet of rectangular
cross-section, with air gap (of length I) normal to the direction
of movement. The sketch (b) shows a similar bar of iron, but
with the air gap inclined at an angle 6 with the normal cross-
section. The total movement, which is supposed to be confined
to the direction parallel to the length of the bar, is the same in
both cases; that is to say, the air gap measured in the direction

<*) v,

FIG. 12. Magnet with inclined air gap.

of motion, has the same value, I, although the actual air gap
measured normally to the polar surfaces is smaller in (b) than
in (a) . The magnetic pull will actually be exerted in a direction
normal to the opposing surfaces, that is to say, in the direction
OF 8 in case (b), although it is the mechanical force exerted in
the direction OFz which it is proposed to calculate.

For the perpendicular gap (sketch a) we can write,

total longitudinal force F l = kB l 2 A l (18)

where k is a constant.

For the inclined gap (sketch 6) ;

total longitudinal force F 2 = fc# 2 2 A 2 X cos 6 (19)
Now express equation (19) in terms of AI and Bi. With the
ampere-turns of constant value, and considering the reluctance
of the air gap only, the flux density will be inversely proportional
to the shortest distance between the two parallel surfaces.
Thus,

1
Zcos 6

THE MAGNETIC CIRCUIT ELECTROMAGNETS 31

therefore
Also, since

B l

cos

A 2 =

cos

A!

cos

X cos0

cos 2 "0" ~ cos 2 (20)

The same relation holds good for cone-shaped pole pieces.
Thus, referring to Fig. 13, in which the magnet core is supposed
to have a circular cross-section of radius r;

FI = kBi 2 Ai = kBi 2 irr 2 for normal gap

and

FIG. 13. Magnet with conical pole faces.

F 2 = kB z z Az X cos

for conical gap; where the factor cos is introduced as before
to obtain the axial component of the magnetic forces.

The conical surface, which corresponds to the cross-sectional
area of the air gap, is,

A* = y 2 X2wrX ^ = e = ^~
Also, for the same exciting ampere-turns, we have as before,

I cos

and,

COS 2

which is identical with formula (20).

32 PRINCIPLES OF ELECTRICAL DESIGN

Thus in both cases the inclined gap has the effect of increasing
the initial pull for the same total length of travel. This is
sometimes an advantage, and conical poles are occasionally
introduced in designs of electromagnets when the total travel is
small relatively to the diameter of the plunger. In this manner
it is possible to obtain increased length of travel without adding
to the weight of the magnet. For the same initial pull, the
length of travel obtainable by providing conical surfaces is,

li
2 ~ cos 2

where l\ is the length of the normal gap which corresponds to
the required initial pull. This formula is easily derived from
the expressions previously developed.

There is a limit to the amount of taper that can be put on the
conical pole pieces, and a large amount of taper will prove to be
of little use. It should be noted that a limit of usefulness is
reached when the flux density in the iron core approaches satura-
tion limits, because the air-gap density which determines the
magnetic pull cannot then be carried up to high values, even
with greatly increased exciting ampere-turns. A reference to the
curves of Figs. 2 or 3 (pages 16, 17) will enable the designer to
judge when the density in the magnet is approaching uneconom-
ical values. Thus, in the case of cast iron it will rarely pay to
carry the induction above 11,000 gausses, while, in wrought iron,
or cast steel as used for electronlagnets, 1 the upper limit may be
placed at about 19,000 gausses, although, as will be explained
later, it is often advantageous to force the density up to higher
values in the teeth of laminated armature cores.

9. Materials Wire and Insulation. Before going further into
the design of electromagnets it will be advisable to consider
briefly the qualities of the materials used in their construction.
The most important of these materials is the iron, which con-
centrates the magnetic flux and so provides the necessary dis-
tribution and density in the air gap where it performs the duty
required of it. The effect of iron in the magnetic circuit has
however already been discussed at some length, and as its various
properties will be considered further in the course of subsequent
articles, it is proposed to confine the remarks immediately fol-

1 This is practically pure iron, with magnetic characteristics very similar
to those of soft annealed iron of good quality.

THE MAGNETIC CIRCUIT ELECTROMAGNETS 33

lowing to the only other materials of consequence in the design of
magnets or dynamos, namely, the copper wire, which is the
material universally used for the windings, and the insulating
materials, which prevent electrical contact between neighboring
turns of wire, and also between the winding as a whole and the
iron of the magnetic circuit or supporting framework.

For the operation of electromagnets, high voltages are rarely
used, and the provision of appropriate insulation presents no
serious difficulties; but it must not be overlooked that, when
the inductance is great i.e., when the flux links with a large
number of turns and the product maxwells X number of turns
is large there may be, at the instant of switching off the current,
differences of potential between neighboring turns of wire,
considerably in excess of the normal potential difference cal-
culated on the assumption of a steady impressed voltage between
the terminals of the coil. In the design of continuous-current
machines, pressures up to 5,000 volts may have to be considered,
and in alternating-current generators, the pressure may be as high
as, but rarely in excess of, 16,000 volts. The higher pressures,
as used for transmission of energy to great distances, are obtained
by means of static transformers, and the question of insulation
then becomes of such great importance that it has to be very
thoroughly studied by experts. Pressures of 100,000 volts are
now common for step-up transformers, and there are many
transformers actually in operation at 150,000 volts and even
higher pressures; so that the provision of the requisite insulation
for machines working at pressures not exceeding 16,000 volts
(which is the limit for any of the designs dealt with in this book)
offers no insuperable difficulties. It is, therefore, proposed to
devote but little space to the discussion of insulation problems;
although, as occasion arises, data and information of a practical
nature will be given.

Copper Wire. With silver as the one exception, copper is
the metal with the highest electrical conductivity; it is also
mechanically strong, easy to handle, and generally the most
suitable material for electrical windings. The resistance of a
given size and length of wire is usually obtained by reference
to a wire table, similar to the accompanying tables, which con-
tain such information as the designer of electrical apparatus
requires. The very large, and the very small, sizes of wire are
omitted; but wire tables for the use of electrical engineers are

3

34

PRINCIPLES OF ELECTRICAL DESIGN

so common in electrical handbooks and textbooks, that par-
ticulars of sizes not here included can generally be obtained
without difficulty.

The Brown and Sharp gage is commonly used in America,
while the legal standard gage (S.W.G.) which has been adopted
by the Engineering Standards Committe is used almost with-
out exception by electrical engineers in England. In using the
accompanying tables, reference should always be made to the
heading, to ensure that the figures relate to the required wire
gage.

WIRE TABLE, BROWN AND SHARP GAGE, COPPER

Gage
No.,
B. &S.

Diameter,
inches
(bare)

Area of cross-
section

Weight,
Ib. per
1,000 ft.
(bare)

Approx.
diame-
ter
D.C.C.

(mils)

Approx.
number
of turns
per inch
D.C.C.

Resistance, ohms
per 1,000 ft.i

Gage
No.,
B. &S.

Square
inches

Circu-
lar mils

15C.

(59F.)

60C.

(140F.)

0.3249

0.08291

105,560

319.5

338

2.95

0.0964

0.1142

1
2

0.2893
0.2576

0.06573
0.05212

83,690
66,370

253.3
200.9

302
270

3.30
3.69

0.1217
0.1534

0.1440
0.1816

1
2

3

0.2294

0.04133

52,630

159.3

242

4*12

0.1934

. 2290

3

4

0.2043

0.03278

41,740

126.4

216

4.60

0.2439

. 2888

4

5

0.1819

. 02600

33,090

101.2

194

5.13

0.3076

0.3642

5

6

0.1620

0.02061

26,250

79.5

174

5.70

0.388

0.459

6

7

0.1443

0.01635

20,820

63.0

156

6.36

0.489

0.579

7

8

0.1285

0.01297

16,510

50.0

140

7.10

0.617

0.730

8

9

0.1144

0.01028

13,090

39.6

126

7.88

0.778

0.921

9

10

0.1019

0.00815

10,380

31.4

114

8.70

0.981

1.161

10

11

0.0907

0.00646

8,230

24.9

103

9.60

1.237

1.464

11

12

. 0808

0.00513

6,530

19.8

93

10.65

1.559

1.846

12

13

0.0720

. 00407

5,178

15.7

84

11.80

1.966

2.328

13

14

0.0641

0.00323

4,107

12.43

76

13.0

2.480

2.936

14

15

0.0571

0.00256

3,260

9.86

68

14.5

3.127

3.702

15

16

. 0508

0.00203

2,583

7.82

62

15.9

3.942

4.667

16

17

0.0453

0.00161

2,048

6.20

56

17.5

4.973

5.887

17

18

. 0403

0.001276

1,624

4.92

51

19.2

6.27

7.42

18

19

0.0359

0.001012

1,288

3.90

46

21.3

7.90

9.36

19

20

0.0320

. 000802

1,022

3.09

42

23.3

9.97

11.80

20

21

0.0285

. 000636

810

2.45

38

25.6

12.57

14.88

21

22

0.0253

. 000503

642

1.945

35

27.8

15.86

18.77

22

23

0.0226

0.000401

510

1.542

32

30.3

20.00

23.66

23

24

0.0201

0.000317

404

1.223

30

32.3

25.20

29.84

24

25

0.0179

0.000252

320

0.970

27

35.7

31.80

37.60

25

26

0.0159

0.0001985

254

0.769

24

40.0

40.20

47.50

26

27

0.0142

0.0001584

202

0.610

22

43.5

50.60

60.00

27

28

0.0126

0.0001247

159

0.484

21

45.5

63.80

75.40

28

1 A variation in resistance up to 2 per cent, increase on the calculated values for pure
copper is generally allowed.

THE MAGNETIC CIRCUIT ELECTROMAGNETS 35

WIRE TABLE, STANDARD WIRE GAGE, COPPER

Gage
No.,
S.W.G.

Diameter,
inches,
(bare)

Area of cross-
section

Weight,
Ib. per
1,000ft.
(bare)

Appro*. | Appro*,
diame- number
ter of turns
D.C.C. per inch
(mils) D.C.C.

Resistance, ohms
per 1,000ft. 1

Gage
No.,
S.W.G.

Square
inches

Circu-
lar ' mils

15C.
(59F.)

60C.
(140F.)

0.324

0.08245

105,000

318.0

340

2.93

0.097

0.1148

1

0.300

0.07070

90,000272.0

316

3.15

0.1131

0.1339

1

2

0.276

. 05980

76,180231.0

292

3.41

0.1337

0.1583

2

3

0.252

. 05000

63,500 192.0

268

3.72

0.1604

0.1900

3

4

0.232

0.04230

57,820

166.0

248

4.01

0.1892

0.2240

4

5

0.212

0.03530

44,940

136.0

228

4.37

. 2260

ii _><;s:;

5

6

0.192

. 02895

36,860

111.5

208

4.78

0.2767

0.3276

6

7

0.176

0.02433

30,980

93.8

192

5.18

0.3291

. 3896

7

8

0.160

0.02010

25.600

77.5

174

5.72

0.398

0.471

8

9

0.144

0.01630

20,740

62.8

158

6.28

0.491

0.581

9

10

0.128

0.01287

16,380

49.6

140

7.10

0.625

0.740

10

11

0.116

0.01057

13,460

40.7

128

7.75

0.759

0.902

11

12

0.104

0.00850

10,820

32.7

116

8.55

0.941

1.114

12

13

092

0.00665

8,465

25.6

104

9.52

1.203

1.424

13

14

0.080

0.00503

6.400

19.4

92

10.75

1.591

1.884

14

15

0.072

0.00407

5,185

15.7

84

11.75

1.964

2.325

15

16

0.064

0.00322

4,095

12.4

76

13.0

2.486

2.943

16

17

0.056

0.00246

3,135

9.5

68

14.5

3.246

3.844

17

18

0.048

0.00181

2,305

7.0-

58

17.0

4.420

5.234

18

19

0.040

0.001257

1,600

4.84

50

19.6

6.37

7.54

19

20

0.036

0.001018

1,296

3.92

46

21.3

7.85

9.30

20

21

0.032

0.000804

1,024

3.10

42

23.2

9.94

11.77

21

22

0.028

0.000616

784

2.37

38

25.6

12.99

15.38

22

23

0.024

0.000452

576

1.74

34

28.6

17.67

20.93

23

24

0.022

0.000380

484

1.47

31

31.0

21.08

24.91

24

25

0.020

0.000314

400

1.21

29

33.0

25.50

30.20

25

26

0.018

0.000255

324

0.98

27

36.0

31.40

37.20

26

27

0.0164

0.000211

270

0.81

25

38.0

37.80

44.70

27

28

0.0148

0.000172

219

0.665

24

40.0

46.50

55.00

28

1 A variation in resistance up to 2 per cent, increase on the calculated values for pure
copper is generally allowed.

The cross-section of a wire may be expressed in square inches
or in square mils (1 mil = 1/1000 in.) ; the metric system is rarely
used in English-speaking countries.

Circular Mils. The cross-section of a wire or conductor may
also be expressed in " circular mils." This is the unit of area
commonly used in America when the cross-section of electrical
conductors is referred to. The confusion of ideas resulting from
the conception of the circular mil as a unit of area may be com-
are not obvious. The circular mil is the area of a circle 1 mil

36 PRINCIPLES OF ELECTRICAL DESIGN

in diameter, and the number of circular mils in a given area

is therefore greater than the number of square mils. Thus, in 1

4
sq. in. there are 1,000,000 square mils; but 10 6 X = 1,273,237

7T

circular mils. The cross-section of a cylindrical wire in circular
mils is,

(m) = (diameter in mils) 2

true area in square mils
0.7854

The area of any conductor expressed in circular mils is always
greater than the true area expressed in square mils.

Simple Formulas for Resistance of Wires. A very convenient
and easily remembered rule is that the resistance of any copper
wire 1 is 1 ohm per circular mil per inch length, or

I"

R = T^T (21)

(m)

at a temperature of about 60C. (or 140F.). This formula is
therefore applicable to the calculation of coil resistances under
operating conditions, when they are hot.

The system on which the B. & S. (Brown and Sharp) gage is
based, exactly halves the cross-section with an increase of three
sizes. It will also be found that a No. 10 B. & S. copper wire
has a cross-section of about 10,000 circular mils (diameter = 0.1
in. approx.) and its resistance at normal temperatures (about
20C.) is 1 ohm per 1,000 ft. Thus, for approximate calculations,
sizes of wire on the B. & S. gage can be determined if necessary
without reference to tables.

The weight of any size of round copper wire may be calculated
by the formula:

d 2

Weight in pounds per 1,000 ft. = ^7^ (22)

ooU

where d = diameter in mils.

Variation of Resistance with Temperature. If the resistance of
a wire is known for any given temperature it can readily be
calculated for any other temperature by remembering that the
resistance of all pure metals tends to become zero at the absolute
zero of temperature, and that the variations in resistance follow

1 The specific resistance of commercial wires can be, and usually is, equal
to that of pure electrolytic copper of 100 per cent, conductivity by
Matthiesson's standard.

THE MAGNETIC CIRCUIT ELECTROMAGNETS 37

a straight-line law, all as indicated in Fig. 14. Thus, if R t and
R Q stand respectively for the resistances at temperatures of t
degrees and zero degrees, the relation is,

R t = R Q (1 + at)

(23)

If it is desired to calculate the
change in resistance which occurs
when the temperature is raised
from ti to t 2 degrees, we have,

R 2 = R Q (l -f at 2 )

Dividing the first equation by
the second, in order to eliminate
R 0j we get,

(1 + at,)
(1 + at,) Kl

If,

(24) -^3 c -

Resistance

(Absolute Zero)

FIG. 14. Diagram illustrating
by which the resistance R 2 at the variation of resistance with temper-

temperature tz can be calculated

when the resistance R\ at the temperature t\ is known.

The coefficient a = 0.004 if the temperatures are expressed in
scale, a = 0.0024.

Numerical Example Change of Resistance with Temperature.
The resistance of copper per circular mil per foot is 12 ohms at
60C. Calculate the temperature at which the resistance will
be 10 ohms per circular mil per foot.

60a)

R t = R (l + ta)

Divide the first equation by the second, and solve for t, the value
of which is found to be,

R t (l + 60a) - fleo

t =

X a

Substitute the numerical values, R 6 o = 12; R t = 10; and a =
0.004 which will give the answer 8.33C.

Insulating Materials. The covering on the copper wires may
consist of one, two, or three layers of cotton or silk. Silk cover-
ings are used only on the smaller sizes, especially when it is
important to economize space, that is to say, where the space

38

PRINCIPLES OF ELECTRICAL DESIGN

taken up by the cotton covering would be excessive in propor-
tion to the cross-section of copper. When great economy of
space is necessary, enamelled wire may be used. This is simply
bare copper wire on which a thin coating of flexible enamel has
been applied by a special process. Enamelled wire may often
be used to advantage, especially in connection with the very
small diameters; but there is always the possibility of contact
between adjacent wires at abnormally high temperatures, and in

BELDENAMEL WIRE DATA

Nos. B. & S.
gage

B. & S. sizes, bare
(inches)

Increase thickness
of enamel insula-
tion

Allowable varia-
tion in thickness

Average diameter
over enamel

13

0.0720

0.002

0.0005

0.074

14

. 0641

0.002

0.0005

0.0661

15

0.0571

0.002

0.0005

0.0591

16

0.0508

0.002

0.0005

0.0528

17

0.0452

0.0018

0.0004

0.047

18

0.0403

0.0018

0.0004

0.0421

19

0.0359

0.0018

0.0004

0.0377

20

0.0320

0.0018

0.0004

0.0338

21

0.0284

0.0017

0.0004

0.0301

22

0.0253

0.0016

0.0004

0.0269

23

0.0225

0.0015

0.0004

0.0240

24

0.0201

0.0014

0.0003

0.0215

25

0.0179

0.0013

0.0003

0.0192

26

0.0159

0.0012

0.0003

0.0171

27

0.0142

0.0011

0.0003

0.0153

28

0.0126

0.0010

0.0003

0.0136

29

0.0112

0.0009

0.0003

0.0121

30

0.0100

0.0008

0.0002

0.0108

31

0.0089

0.0008

0.0002

0.0097

32

0.0079

0.0007

. 0002

0.0086

33

0.0071

0.0007

0.0002

0.0078

34

0.0063

0.0006

0.0002

0.0069

35

0.0056

0.0006

0.0001

0.0062

36

0.005

0.0005

0.0001

0.0055

37

0.0044

0.0005

0.0001

0.0049

38

0.004

0.0004

0.0001

0.0044

39

0.0035

0.0004

0.0001

0.0039

40

0.0031

0.0004

0.0001

0.0035

THE MAGNETIC CIRCUIT ELECTROMAGNETS 39

many cases enamelled wire protected by a single covering of
cotton has been used with very satisfactory results. In all cases
when it is desired to save space by reducing the thickness of
insulation on wires, the points to be considered are: (1) insulation;
(2) durability; and (3) cost. The price of the silk covering is of
course much higher than that of the cotton covering.

Enamel insulation does not add much to the diameter of the
wire as will be seen by reference to the accompanying table based
on data kindly furnished by the Belden Manufacturing Co. of
Chicago. This wire will not suffer injury with the temperature
maintained at 200F. continuously, and it will withstand without
breakdown a pressure of 900 volts per mil thickness of enamel;
but on account of the possibility of abrasion during winding, a
large factor of safety (not less than four) should be used, and
indeed it is always advisable to place paper between the layers of
enamelled wire, unless a careful study of the conditions appears
to justify its omission.

Triple cotton covering can be used with advantage on the
larger sizes of wire when the working pressure between adjacent
turns exceeds 20 volts. When extra insulation is required be-
tween the layers of the winding, this is usually provided in the
form of one or more thicknesses of paper or varnished cloth.
It is the insulation between the finishing turns of a layer of wire
and the winding immediately below which requires special at-
tention, because this is where the difference of potential is great-
est. One advantage of the ordinary cotton covering is that it
lends itself admirably to treatment with oil or varnish, either
before or after winding.

Space Factor. The amount of space taken up by the insulation
and the air pockets between wires of circular cross-section is
important, because it reduces the cross-section of copper in the

coil. If A is the cross-section of the copper, and A' the total area

j^

of cross-section through the winding, the ratio -p is called the

space factor. The calculated space factor, based on the assump-
tion of a known diameter over the insulation, and a close packing
of the wires, does not always agree with the value obtained in
practice, but the curves of Fig. 15 will be found to give good
average values. It will be understood that the space factors of
Fig. 15 include no allowance for extra insulation between layers
of wire or for the necessary lining of the spool upon which the coil

40

PRINCIPLES OF ELECTRICAL DESIGN

is wound. If the number of turns per layer is very small, there
will be an appreciable loss of space due to the turning back of the
wire at the end of each layer.

Insulation on Spools or Metal Forms. The materials used for
insulating between the winding as a whole and any grounded
metal by which it is supported include mica, micanite paper and
cloth, pressboard, "presspahn," varnished cambric, oiled linen

U.72

70

\

\

s v v ^

fl fiR

\

\

66

\

s x

^

>n

64

\

^

*r*

fl fi9

\

rt

^

9>

\

n fin

fc

\

\

Jo *y?

*

s

\

n "y?

s

\

\

g-

'~ n KA

\

.

\

^Q

\

48

\

4fi

\

44

\

6 8

32 34

10 12 14 16 18 20 22 24 26 28
Size of Wire, B & S Gauge

FIG. 15. Space factors for wires of circular cross-section.

or cotton (empire cloth), cotton tape, etc. The voltage that
some of these materials will withstand before breakdown is
approximately as follows : empire cloth (usually 7 to 8 mils thick)
will rarely puncture with less than 600 volts per mil; mica will
withstand about 800 volts per mil; and micanite paper or cloth
which affords also an excellent mechanical protection can
generally be relied on to withstand 400 volts per mil. A large
factor of safety is usually allowed, especially on the lower
voltages. With a good quality of insulation, the total thickness
between the cotton-covered wires and the supporting metal work
should have the following values:

THE MAGNETIC CIRCUIT ELECTROMAGNETS 41

Up to 500 volts 0.045 in.

For 1,000 volts 0.060 in.

For 2,000 volts 0.080 in.

For 3,000 volts 0.10 in.

For higher pressures, up to 12,000 volts, add 0.03 in. per 1,000
volts increase.

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 4 of 30)