Alfred Still.

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

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unit length of circuit) the value of the permeability, /*, varies
considerably with different kinds of iron; it also depends on the
past history of the particular sample of iron, and will not be the
same on the increasing as on the decreasing curve of magnetiza-
tion, as indicated by the curve known as the hysteresis loop.
For the use of the designer, careful tests are usually made by the
manufacturer on samples of iron used in the construction of
machines, and curves are then plotted, or tables compiled, based
on the average results of such tests. Curves of this kind have
been drawn in Figs. 2 and 3. The B-H curves of Fig. 2 should
be preferred when the C.G.S. system of units is used in the



16



PRINCIPLES OF ELECTRICAL DESIGN



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18000
17000
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11000
~ 10000

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Q 8000

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20 40 60 80 100 120 140 160 180 200 220 240
Magnetizing Force H

FIG. 2. B-H curves.



THE MAGNETIC CIRCUIT ELECTROMAGNETS 17



0*



20 40 60 80 100 120 140 160 180 200 220 240
Ampere-Turns per Inch

FIG. 3. Magnetization curves (inch units).



18



PRINCIPLES OF ELECTRICAL DESIGN



calculations; but so long as engineers persist in expressing linear
measurements in feet and inches, the curves of Fig. 3 will gen-
erally be preferred by the designer. Fig. 4 may be used for high
values of the induction in armature stampings of average
quality.

The value of /* is, of course, the ratio between B of Fig. 2 and
the corresponding value of H, and curves or tables giving the
relation between jj, and H could be used; but it is generally more
convenient to read directly off the curves of Figs. 2, 3, or 4, the



160



150




130

120
a

I

xllO

E



200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000
Ampere -Turns per Inch

FIG. 4. Magnetization curve for armature stampings (high values of

induction).

flux density in the iron coresponding to any known value of the
magnetizing force. As a matter of fact, it will be found that
the curves are more frequently used for the purpose of determin-
ing the necessary ampere-turns to produce a desired value of the
flux density.

4. Magnetic Circuits in Parallel. As an illustration of the
fundamental relations existing between magnetic flux and excit-
ing ampere-turns, it will be convenient to work out a numerical
example. A simple case will be chosen of magnetic paths in
series and in parallel, with small air gaps in a circuit consist-
ing mainly of iron, and the effect of any leakage flux through
air paths other than the gaps deliberately introduced will be
neglected.



THE MAGNETIC CIRCUIT ELECTROMAGNETS 19

The arrangement shown in Fig. 5 is supposed t'o represent a
steel casting consisting of the magnetic paths (1) and (2) in
parallel, with the common path (3) in series with them. It
will be seen that the paths (1) and (2) are provided with air gaps
and that the exciting coil is on the common limb (3) only.
Paths (1) and (3) have iron in them and the permeance of these
paths will depend upon the density B and therefore on the total
flux 4>i and 4> 3 in these portions of the circuit. In regard to path
(2), it also consists mainly of iron, but the cross-section of the
iron has purposely been made large, so that the reluctance of
this path is practically all in the gap; the value of B in the iron
will be very low, ju will be large, and the reluctance of this part
of the iron circuit will be considered negligible. The dimensions





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A 3


= 20 sq.in.


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"^




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t '


.






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8

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< ^2 = ^





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'' 1 IB I '




20 sq.in.








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FIG. 5. Typical magnetic circuit.

of the parts are indicated on the sketch, and the problem to be
solved is the calculation of the necessary ampere-turns in the coil
to produce a given total flux of, say, 1,000,000 maxwells through
the path (1).

The reluctance of path (1) alone consists of the air-gap reluc-
tance in series with the reluctance of the iron limb of length li
and cross-section AI. Thus,

p l *i

" A a X 1 "" Ai X MI

all dimensions being expressed in centimeters. The only
unknown quantity is MI, and this can be determined because the
flux density in the iron will be

B'\ = -~ - - = 100,000 maxwells per square inch,



20 PRINCIPLES OF ELECTRICAL DESIGN

where the index " is added to the symbol B to indicate that inch
units are used and that the density is not expressed in gausses.

Knowing B for a given sample of iron, the value of the per-
meability />t can be found, and Ri calculated by putting the
numerical values in the above equation. The necessary m.m.f.
for this portion of the magnetic circuit (i.e., path (1) only) is
$1 X Ri gilberts.

The actual procedure would be simplified by using the curves
of Fig. 3 thus:

Referring to the upper curve (for steel), the ampere-turns per
inch required to produce a flux density of 100,000 lines per square
inch is seen to be 80, and since the iron portion of path (l)is
25 in. long, the ampere-turns required to overcome the reluctance
of iron only are 80 X 25 = 2,000.

For the air portion of path (1), we have,

m.m.f. = $1 X reluctance of air gap
or

0.4rS7 - 1,000,000 X 4 p * g'f g

whence

SI = 1,560

The total SI for path (1) are therefore 2,000 + 1,560 = 3,560
or,

m.m.f. = 0.47T X 3,560 = 4,470 gilberts.

Observe now that this' m.m.f. is the total force which sets up
the magnetic flux in path (2), or, in other words, it is the differ-
ence of magnetic potential which produces the flux of induction
in the two parallel paths (1) and (2). To calculate the total flux
in path (2) we have,

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

20 V fi 45
= 4,470 X i CO'K>I = 227,000 maxwells

I /\ Z.OT:

The total flux in limb (3) under the exciting coil is, therefore,
$ 3 = ^ -f <|> 2 = 1,227,000. The density in this core is,



The necessary ampere-turns per inch (from Fig. 3) are 8, and
the SI for path (3) are 8 X 50 = 400.



THE MAGNETIC CIRCUIT ELECTROMAGNETS 21

The total ampere-turns required in the exciting coil to produce
a flux of 1,000,000 lines across the air gap in path (1) are there-
fore 3,560 -f 400 = 3,960, which is the answer to the problem.

In all cases when there is no iron in the magnetic path, i.e.,
when n, = 1, as in the air gaps of dynamo-electric machines, the
required ampere-turns depend merely on the density B, and the
length I of the air gap. The fundamental relation, m.m.f. = HI
can tfren be written QAirSI = Bl whence,

T>

SI per centimeter (in air) =
similarly

SI per inch (in air) = f _ B = 2.02 B

(5)
= 2B approximately

If the density is expressed in lines per square inch,

B"

SI per inch (in air) = s~s

O.Z

These formulas are easily remembered and are useful for making
rapid calculations.

With a view to the more thorough understanding of electro-
magnetic problems likely to arise in the design of electrical
machinery, it should be observed that path (2) of the magnetic
system shown in Fig. 5 may be thought of as a shunt or leakage
path, the useful flux being the 1,000,000 maxwells in the air
gap of path (1). It is important to note that this surplus or
leakage magnetism has cost nothing to produce, except in so
far as the total flux < 3 is increased in the common limb (3),
calling for a slight increase in the necessary exciting ampere-
turns. Even this small extra I 2 R loss could be avoided by in-
creasing the cross-section A z of the common limb; but this would
generally add to the cost, not only because of the greater weight
of iron, but also because the length per turn of the exciting coil
would usually be greater, thus increasing the weight and cost of
copper if the PR losses are to remain unaltered. For these
reasons alone it is well to keep down the value of the leakage flux
in nearly all designs of electrical machinery; but the point here
made is that the existence of a magnetic flux, whether it be useful
or leakage magnetism, does not involve the idea of loss of energy
in the sense of an PR loss which must always be associated with
the electric current. Attention is called to this matter in order



22 PRINCIPLES OF ELECTRICAL DESIGN

to emphasize the danger of carrying too far the analogy between
the magnetic and electric circuits. The product PR in the
electric circuit is always associated with loss of energy; but
$ 2 X reluctance does not represent a continuous loss of energy in
the magnetic circuit. If the energy wasted in the exciting coil
is ignored, it may be said that the magnetic condition costs
nothing to maintain. It does, however, represent a store of
energy which has not been created without cost; but, with the
extinction of the magnetic field, the whole of this stored energy is
given back to the electric circuit with which the magnetic circuit
is linked. This may be illustrated by the analogy of a frictionless
flywheel which dissipates no energy while running, but which,
on being brought to rest, gives up all the energy that was put
into it while being brought up to speed.

The dotted lines on the right-hand side of the magnetic circuit
shown in Fig. 5 indicate two extra iron paths for the magnetic
flux. It should particularly be observed that the closed iron
ring D can be linked with the exciting coil as indicated without
modifying the amount of the useful flux through path (1):
there may obviously be a large amount of flux in this closed
iron ring, but it has cost nothing to produce because the exciting
ampere-turns have not been increased. The same might be
said of the circuit C except that the flux in this circuit has to
go through the common core (3), and in so far as extra ampere-
turns would be necessary to overcome the increased reluctance
of the iron under the coil, the m.m.f. available for sending flux
through paths (1) and (2) would be reduced, and if path C
were of high-grade iron of large cross-section relatively to A 3,
the useful flux in path (1) might be appreciably reduced. A
proper understanding of the points brought out in the study of
Fig. 5 will greatly facilitate the solution of practical problems
arising in the design of electrical machinery.

5. Calculation of Leakage Paths. The total amount of the
magnetic leakage cannot be calculated accurately except by
making certain assumptions which are rarely strictly permissible
in the design of practical apparatus. Whether the machine is
an electric generator or an electromagnet of the simplest design,
the useful magnetic flux is always accompanied by stray magnetic
lines which do not follow the prescribed path. This leakage
flux will always be so distributed that its amount is a maxi-
mum; that is to say, the paths that it will follow will always



THE MAGNETIC CIRCUIT ELECTROMAGNETS 23

be such that the total permeance of these leakage paths has
the greatest possible value. It is well to bear this fact in mind,
because it enables the experienced designer to make sketches of
various probable distributions of the leakage flux, and base his
calculations on the arrangement of flux lines which has the
greatest permeance. The fact that the leakage flux usually
follows air paths means that the permeance of these paths
does not depend upon the flux density B\ this simplifies the
problem because it is not necessary to take into account
values of the permeability, /z, other than unity; the difficulty
lies in the fact that with the exception of very short air
gaps between relatively large polar surfaces it is rarely possi-
ble to predetermine the distribution of the stray flux, except
by making certain convenient assumptions of questionable
value. .A designer of experience will frequently be able to
estimate flux leakage even in new and complicated designs
with but little error, and it is surprising how the intelligent
application of empirical or approximate formulas and rules will
often conduce to excellent results. The errors introduced are
some on the high side and some on the low side, and the averages
are fairly accurate; but the estimation of leakage flux like
many other problems to be solved by the designer or practical
engineer savors somewhat of scientific guesswork; it calls for
a combination of common sense and engineering judgment
based on previous experience. The following examples and
formulas cover some of the simplest cases of flux paths in air; the
usual assumptions are made regarding the paths followed by the
magnetic lines, but it may safely be stated that, when all possible
leakage paths have been considered, and these formulas applied
to the calculation of the leakage flux, the calculated value will
almost invariably be something less than the actual stray flux
as subsequently ascertained by experimental means.

Case (a). Parallel Flat Surfaces. If the length of air gap
between the parallel iron surfaces is small relatively to the
cross-section, and if the two surfaces are approximately of the
same shape and size, the average cross-section (see Fig. 6) is



and the permeance is



p = f ()



24



PRINCIPLES OF ELECTRICAL DESIGN



all dimensions in this and subsequent examples being expressed
in centimeters.

Case (6). Flat Surfaces of Equal Area Subtending an Angle 6.
The assumption here made is that the lines of induction in
the air gap are circles described from a center on the axis
where the planes of the two polar surfaces meet. Let I = length
of polar surface at right angles to the plane of the section shown
in Fig. 7. The sum of the permeances of all the small paths such
as dr is then,

p = C" I X dr

J'i '.



(7)






FIG. 6. Permeance between FIG. 7. Permeance between non-
parallel surfaces. parallel plane surfaces.

For the special case when 6 = 90 degrees,



(8)



For the special case when = 180 degrees, and the two surfaces
lie in the same plane,

Case (c). Equal Rectangular Polar Surfaces in Same Plane.
This is a case similar to the one last considered, but the formula
(9) is not applicable when TI is large relatively to r 2 because the
actual flux lines would probably be shorter than the assumed
semicircular paths. With a greater separation between the polar



THE MAGNETIC CIRCUIT ELECTROMAGNETS 25



surfaces, the lines of flux are supposed to follow the path indicated
in Fig. 8. Let I stand, as before, for the length measured per-
pendicularly to the plane of the section shown, then,

IXdr



('

Jo



dr



I . TTt + 8

log t

7T 6< S



(10)




FIG. 8. Permeance between surfaces in same plane.

Case (d). Iron-clad Cylindrical Magnet. Fig. 9 shows a
section through a circular magnet such as might be used for
lifting purposes. The exciting coil is supposed to occupy a
comparatively small portion of the total depth, and in order to
calculate the total flux between the inner core and the outer
cylinder forming the return path for the useful flux we may
consider the reluctance of the air path as being made up of a
number of concentric cylindrical shells of height h and thickness
dx. Thus,

R /TJ

reluctance = 2 ~ T

M io r R

The reciprocal of this quantity is the permeance, whence,

P = -^J> (ID



When the radial depth of the winding space (R r) is not



26



PRINCIPLES OF ELECTRICAL DESIGN



greater than the radius of the iron core (r), the permeance may
be expressed with sufficient accuracy for practical purposes as

_ mean cross-sectional area

length of flux path
_ TT (R + r)h

' (R ~ r)

Case (e). Same as Case (d) Except that Coils Occupy the
Whole of the Available Space. This is the more usual case, and
it is illustrated by Fig. 10. The leakage flux in the annular
space occupied by the windings will depend not only upon the





FIG. 9. Leakage paths in circu-
lar magnet (space not occupied by
copper).



r >|< -



FIG. 10. Leakage paths in circu-
lar magnet (space entirely fitted by
exciting coils).



permeance of the air path, but also upon the m.m.f. tending to
establish a magnetic flux. This m.m.f. has no longer a constant
value, but, on the assumption that the reluctance of the iron paths
is negligible, it will increase according to a straight-line law from
zero when x = (see Fig. 10) to a maximum when x = h.

Starting with the fundamental formula, $ = m.m.f. X P,
we have,

~ 2irdx



= OAwSI X X



R



whence



or



1 27T

0.4mS7Xr X



loge
r



v rh

R \ xdx

t-Jo



QAirSI 2nh

T\ /\ 7



(12)



THE MAGNETIC CIRCUITELECTROMAGNETS 27

which, if the dimensions are in centimeters, will be the leakage
flux in maxwells; and this is seen to be merely the product,
average value of m.m.f. X permeance.

It is evident that this formula can be applied to case (d) in
order to calculate the leakage flux in the space occupied by the
coil, and so obtain the total leakage flux inside a magnet of the
type illustrated, where the coils do not occupy the whole of the
annular air space between the core and the cylinder forming the
return path.

Case (/). Parallel Cylinders. The permeance of the air paths
between the sides of two parallel cylinders of diameter d and




FIG. 11. Permeance between parallel cylinders.

length I which are shown in section in Fig. 11 cannot be
calculated so easily as in the examples previously considered;
but the following formula may be used, 1

P = ~ . (13)



loge



b + d - VV + 26d



It will be observed that the logarithm in this and previous
equations is to the base e, and although the formula could be
rewritten to permit of the direct use of tables of logarithms to
the base 10, there appears to be no good reason for doing so. If
a table of hyperbolic logarithms is not available, the quantity
log ( can always be obtained by using a table of common logarithms
and multiplying the result by the constant 2.303.

6. Flux Leakage in Similar Designs. In all the above formulas
it will be seen that the permeance, P, remains unaltered per
unit length measured perpendicularly to the cross-section shown

1 This formula can be developed mathematically in the same manner as
the better-known formulas giving the electrostatic capacity between parallel
wires.



28 PRINCIPLES OF ELECTRICAL DESIGN

in the sketches 1 provided the cross-sections are similar, apart
from the actual magnitude of the dimensions. Thus, if the
exciting ampere-turns were to remain constant, the leakage flux in
similar designs of apparatus would be proportional to the first
power of the linear dimension I; but since the cross-section of
the winding space is proportional to I 2 , the exciting ampere-
turns would not remain of constant value, but would also vary
approximately as I 2 . Given the same size of wire which
obviates the necessity of considering changes in the winding space
factor the number of turns, S, will be proportional to I 2 ,
and the resistance, R, will vary as I 3 . For the same rise of
temperature on the outside of the windings, the watts lost in
heating the coil must be proportional to the cooling surface.
Thus,

PR oc ja



whence
and



81



The total leakage flux in similar designs of magnets will be pro-
portional to I X I 1 - 5 or I 2 - 5 , and as a rough approximation it
may be assumed that, with a proportional change in all linear
dimensions, the leakage flux will vary as the third power of the
linear dimension, or as the volume of the magnet.

7. Leakage Coefficient. The leakage coefficient, or leakage

,. useful flux + leakage flux
factor, is the ratio - . 1 a

useful flux

or



where <J>j is the total number of leakage lines calculated for every
path where an appreciable amoun-t of leakage is likely to occur.
When designing electromagnets or the frames of dynamo
machines, a fairly close estimate of the probable leakage factor
is necessary in order to be sure that sufficient iron section will be

1 The sections shown in Figs. 9 and 10 have, for convenience, been taken
through the axis of length instead of at right angles to this axis as in the
other examples.



THE MAGNETIC CIRCUIT ELECTROMAGNETS 29

provided under the magnetizing coils and in the yoke to carry the
leakage flux in addition to the useful flux. The leakage factor
is always greater than unity, and the product of the useful flux
by the leakage factor will be the total flux to be carried by
certain portions of the magnetic circuit enclosed by the exciting
coils.

8. Tractive Force. The tractive effort, or the tension which
exists along the magnetic lines of force, is one of the effects of
magnetism which it is necessary to calculate, not only in electro-
magnets for lifting purposes, or in magnetic clutches or brakes,
where this is the most important function of the magnetism;
but also in rotating electric machinery, where decentralization of
the rotating parts may lead to very serious results owing to the
unbalancing of the magnetic pull.

MAXWELL'S formula is,

B 2

Force in dynes = -A (14)

O7T

where A is the cross-section in square centimeters of a given area
over which the flux density, B, is assumed to have a constant
value.

This formula can be used to calculate the pull between two
parallel polar surfaces when the air gap between them is small
relatively to the area of the surfaces. The engineer desires to
know the pull in pounds exerted between the two surfaces, and
since 1 Ib. = 444,800 dynes, the above formula can be written,

B*

putt, in pounds per square centimeter = 1 1 1 Qn -__ (15)

11,



or

B 2
pull, in pounds per square inch = Y73Q 000

In both of these formulas the density, B, is expressed in gausses
(i.e., in C.G.S. lines per square centimeter).
If B )f stands for lines per square inch, then,



pull, in pounds per square inch



B 2



72 v 10

If the density is not constant over the whole surface considered,
the area must be divided into small sections, after which a
summation of the component forces can be made. In averaging
the density to get a mean result, it is obviously not the square of
the average density that must be taken, but the average of the



30



PRINCIPLES OF ELECTRICAL DESIGN



squares of the densities taken over the various component areas
of the cross-section considered. This is briefly summed up in
the general expression,

pull, in dynes = ^- I B 2 dA



Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 3 of 30)