[Illustration: FIG. 10.]
The drum coil may be applied either to the external surface of a
rotating armature, the field-magnet being external and stationary (fig.
9), or to the internal surface of a stationary armature (fig. 10), the
field-magnet being internal and rotating. While the former combination
is universally adopted in the continuous-current dynamo, the latter is
more usual in the modern alternator. In either case the iron armature
core must be "laminated"; the passage of the lines of the field across
its surface sets up E.M.F.'s which are in opposite directions under
poles of opposite sign, so that if the core were a solid mass a
current-sheet would flow along its surface opposite to a pole, and
complete its circuit by passing through the deeper layers of metal or by
returning in a sheet under a pole of opposite sign. Such "eddy-currents"
can be practically avoided by dividing the metal core into laminations
at right angles to the length of the active wires which are themselves
arranged to secure the greatest rate of line-cutting and maximum E.M.F.
The production of the eddy-current E.M.F. is not thereby prevented, but
the paths of the eddy-currents are so broken up that the comparatively
high resistance with which they meet reduces their amount very greatly.
The laminae must be lightly insulated from one another, right up to
their edges, so that the E.M.F.'s which still act across their thickness
will not be added up along the length of the core, but will only produce
extremely small currents circulating through the interior of the
separate laminations. Each thin iron plate is either coated with an
insulating varnish or has one of its sides covered with a sheet of very
thin paper; the thickness of the laminae is usually about one-fortieth
of an inch, and if this is not exceeded the rate at which energy is
dissipated by eddy-currents in the core is so far reduced that it does
not seriously impair the efficiency of the machine.
Lastly, the drum coils may be either attached to the surface of a smooth
armature core (fig. 9, I.), or may be wound through holes formed close
to the periphery of the core, or may be embedded in the slots between
projecting iron teeth (figs. 9 [II.] and 10). Originally employed by
Antonio Pacinotti in connexion with ring winding, the toothed armature
was after some considerable use largely discarded in favour of the
smooth core; it has, however, been reintroduced with a fuller
understanding of the special precautions necessitated in its design, and
it is now so commonly used that it may be said to have superseded the
smooth-surface armature.
Not only does the toothed armature reduce the length of the air-gap to
the minimum permitted by mechanical and magnetic considerations, and
furnish better mechanical protection to the armature coils, but it
also ensures the positive holding of the active wires against the
mechanical drag which they experience as they pass through the
magnetic field. Further, the active wires in the toothed armature are
relieved of a large proportion of this mechanical drag, which is
transferred to the iron teeth. The lines of the field, after passing
through the air-gap proper, divide between the teeth and the slots in
proportion to their relative permeances. Hence at any moment the
active wires are situated in a weak field, and for a given armature
current the force on them is only proportional to this weak field.
This important result is connected with the fact that when the
armature is giving current the distribution of the lines over the face
of each tooth is distorted, so that they become denser on the
"trailing" side than on the "leading" side;[13] the effect of the
non-uniform distribution acting on all the teeth is to produce a
magnetic drag on the armature core proportional to the current passing
through the wires, so that the total resisting force remains the same
as if the armature had a smooth core. The amount by which the stress
on the active wires is reduced entirely depends upon the degree to
which the teeth are saturated, but, since the relative permeability of
iron even at a flux density of 20,000 lines per sq. cm. is to that of
air approximately as 33:1, the embedded wires are very largely
relieved of the driving stress. An additional gain is that solid bars
of much greater width can be used in the toothed armature than on a
smooth core without appreciable loss from eddy-currents within their
mass.
A disadvantage of the slotted core is, however, that it usually
necessitates the lamination of the pole-pieces. If the top of the slot
is open, and its width of opening is considerably greater than the
length of the air-gap from the iron of the pole-face to the surface of
the teeth, the lines become unequally distributed not only at the
surface of the teeth, but also at the face of the pole-pieces; and
this massing of the lines into bands causes the density at the
pole-face to be rhythmically varied as the teeth pass under it. No
such variation can take place in a solid mass of metal without the
production of eddy-currents within it; hence if the width of the
slot-opening is equal to or exceeds twice the length of the single
air-gap, lamination of the pole-pieces in the same plane as that of
the armature core becomes advisable.
If the wires are threaded through holes or tunnels pierced close to
the periphery of the core, the same advantages are gained as with open
slots, and lamination of the pole-pieces is rendered unnecessary. But
on the other hand, the process of winding becomes laborious and
expensive, while the increase in the inductance of the coils owing to
their being surrounded by a closed iron circuit is prejudicial to
sparkless commutation in the continuous-current dynamo and to the
regulation of the voltage of the alternator. A compromise is found in
the half-closed slot, which is not uncommon in alternators, although
the open slot is more usual in continuous-current dynamos.
With the addition of more turns to the elementary drum loop or of
several complete coils, new questions arise, and in connexion therewith
the two great classes of machines, viz. alternators and
continuous-current dynamos, which have above been treated side by side,
diverge considerably, so that they are best considered separately. The
electromotive-force equation of the alternator will be first deduced,
and subsequently that of the continuous-current machine.
[Illustration: FIG. 11.]
Corresponding to the number of pairs of poles in the multipolar
alternator, it is evident that there may also be an equal number of
coils as shown diagrammatically in fig. 11. The additional coils, being
similarly situated in respect to other pairs of poles, will exactly
reproduce the E.M.F. of the original coil in phase and magnitude, so
that when they are connected in series the total E.M.F. will be
proportional to the number of coils in series; or if they are connected
in parallel, while not adding to the E.M.F., they will proportionately
increase the current-carrying capacity of the combination. But within
each coil the addition of more loops will not cause an equal increase in
the total E.M.F., unless the phases of the component E.M.F.'s due to the
several turns are identical, and on this account it becomes necessary to
consider the effect of the width of the coil-side.
[Illustration: FIG. 12.]
If the additional loops are wound within the same slots as the original
loop, the winding is "concentrated," and each turn will then add the
same E.M.F. But if the coil-side is divided between two or more slots,
the phase of the E.M.F. yielded by the wires in one slot being different
from that of the wires in another neighbouring slot, the sum of all the
E.M.F.'s will be less than the E.M.F. of one component loop multiplied
by the number of loops or turns in the coil. The percentage reduction in
the E.M.F. will depend upon the number of the slots in a coil-side and
their distance apart, i.e. on the virtual width of the coil-side
expressed as a fraction of the "pole-pitch" or the distance measured
along the pitch-line from the centre of one pole to the centre of a
neighbouring pole of opposite sign (fig. 12). The winding is now to be
regarded as "grouped," since a small number of distinct phases
corresponding to the groups within the two, three or four slots have to
be compounded together. As the number of slots per coil-side is
increased, an approach is gradually made to the case of "uniform
distribution," such as would obtain in a smooth-core armature in which
the turns of the coil are wound closely side by side. Thus in the
six-turn coil of fig. 12 A, which represents the development of a
two-pole armature when the core is cut down to the shaft and opened out
flat, there are in effect six phases compounded together, each of which
differs but little from that of its next neighbour. With numerous wires
lying still closer together a large number of phases are compounded
until the distribution becomes practically uniform; the decrease in the
E.M.F., as compared with that of a single turn multiplied by the number
in series, is then immediately dependent upon the width of the coil-side
relatively to the pole-pitch.
[Illustration: FIG. 13.]
If the width of the inner loop of fig. 12 A is less than that of the
pole-face, its two sides will for some portion of each period be moving
under the same pole, and "differential action" results, the net E.M.F.
being only that due to the difference between the E.M.F.'s of the two
sides. The loop of smallest width must therefore exceed the width of
pole-face, if direct differential action is to be avoided. The same
consideration also determines the width of the outer loop; if this be
deducted from twice the pole-pitch, the difference should not be less
than the width of the pole-face, so that, e.g., in a bipolar machine the
outer loop may stand to the S. pole exactly as the inner loop stands to
the N. pole (fig. 13). In other words, the width of the coil-side must
not exceed the width of the interpolar gap between two fields. Evidently
then if the ratio of the pole-width to the pole-pitch approaches unity,
the width of the coil-side must be very small, and vice versa. A
compromise between these conflicting considerations is found if the pole
is made not much more than half the pole-pitch, and the width of the
coil-side is similarly about half the pole-pitch and therefore equal in
width to the pole (fig. 13). A single large coil, such as that of fig.
12 A, can, however, equally well be divided into two halves by taking
the end-connexions of one half of the turns round the opposite side of
the shaft (fig. 12 B), as indeed has already been done in fig. 13. Each
sheaf or band of active wires corresponding to a pole is thereby
unaffected, but the advantages are gained that the axial length of the
end-connexions is halved, and that they have less inductance. Thus if in
fig. 11 there are four turns per coil, fig. 14 is electrically
equivalent to it (save that the coils are here shown divided into two
parallel paths, each carrying half the total current). When the large
coils are divided as above described, it results that there are as many
coils as there are poles, the outer loop of the small coil having a
width equal to the pole-pitch, and the inner a width equal to the
pole-face.
[Illustration: FIG. 14.]
Such is the form which the "single-phase alternator" takes, but since
only one-half of the armature core is now covered with winding, an
entirely distinct but similar set of coils may be wound to form a second
armature circuit between the coils of the first circuit. The phase of
this second circuit will differ by 90 deg. or a quarter of a period from
that of the first, and it may either be used to feed an entirely
separate external circuit possibly at a different pressure or, if it be
composed of the same number of turns and therefore gives the same
voltage, it may be interconnected with the first circuit to form a
"quarter-phase alternator," as will be more fully described later. By an
extension of the same process, if the width of each side of a coil is
reduced to one-sixth of the pole-pitch, three armature circuits can be
wound on the same core, and a "three-phase alternator," giving waves of
E.M.F. differing in phase by 120 deg., is obtained.
The fundamental "electromotive-force equation" of the heteropolar
alternator can now be given a more definite form. Let Z_a be the
number of C. G. S. lines or the total flux, which issuing from any
one pole flows through the armature core, to leave it by another pole
of opposite sign. Since each active wire cuts these lines, first as
they enter the armature core and then as they emerge from it to enter
another pole, the total number of lines cut in one revolution by any
one active wire is 2pZ_a. The time in seconds taken by one revolution
is 60/N. The average E.M.F. induced in each active wire in one
revolution being proportional to the number of lines cut divided by
the time taken to cut them is therefore 2Z_a(pN/60) x 10^(-8) volts.
The active wires which are in series and form one distinct phase may
be divided into as many bands as there are poles; let each such band
contain t active wires, which as before explained may either form one
side of a single large coil or the adjacent sides of two coils when
the large coil is divided into two halves. Since the wires are joined
up into loops, two bands are best considered together, which with
either arrangement yield in effect a single coil of t turns. The
average E.M.F.'s of all the wires in the two bands when added together
will therefore be 4Z_a(pN/60)t x 10^(-8). But unless each band is
concentrated within a single slot, there must be some differential
action as they cross the neutral line between the poles, so that the
last expression is virtually the _gross_ average E.M.F. of the loops
on the assumption that the component E.M.F.'s always act in agreement
round the coil and do not at times partially neutralize one another.
The _net_ average E.M.F. of the coil as a whole, or the arithmetical
mean of all the instantaneous values of a half-wave of the actual
E.M.F. curve, is therefore reduced to an extent depending upon the
amount of differential action and so upon the width of the coil-side
when this is not concentrated. Let k' = the coefficient by which the
gross average E.M.F. must be multiplied to give the net average
E.M.F.; then k' may be called the "width-factor," and will have some
value less than unity when the wires of each band are spread over a
number of slots. The net average E.M.F. of the two bands corresponding
to a pair of poles is thus e_(av) = 4k'Z_a(pN/60)t x 10^(-8).
The shape of the curve of instantaneous E.M.F. of the coil must
further be taken into account. The "effective" value of an alternating
E.M.F. is equal to the square root of the mean square of its
instantaneous values, since this is the value of the equivalent
unidirectional and unvarying E.M.F., which when applied to a given
resistance develops energy at the same rate as the alternating E.M.F.,
when the effect of the latter is averaged over one or any whole number
of periods. Let k" = the ratio of the square root of the mean square
to the average E.M.F. of the coil, i.e. = effective E.M.F./average
E.M.F. Since it depends upon the shape of the E.M.F. curve, k" is also
known as the "form-factor"; thus if the length of gap between
pole-face and armature core and the spacing of the wires were so
graduated as to give a curve of E.M.F. varying after a sine law, the
form-factor would have the particular value of [pi]/2[root]2 = 1.11,
and to this condition practical alternators more or less conform. The
effective E.M.F. of the two bands corresponding to a pair of poles is
thus e_(eff) = 4k'k"Z_a(pN/60)t x 10^(-8).
In any one phase there are p pairs of bands, and these may be divided
into q parallel paths, where q is one or any whole number of which p
is a multiple. The effective E.M.F. of a complete phase is therefore
pe{eff}/q. Lastly, if m = the number of phases into which the armature
winding is divided, and [tau] = the total number of active wires on
the armature counted all round its periphery, t = [tau]/2pm, and the
effective E.M.F. per phase is E_a = 2k'k"Z_a(pN[tau]/60mq) x 10^(-8).
The two factors k' and k" may be united into one coefficient, and the
equation then takes its final form