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Font size COMMUTATION 147

is constant over the surface of contact, the current entering the

q

brush through any surface of width S is 2/ c X yy' To cal-

culate the volts e that must be developed in the coil of resistance
R when the distance yet to be travelled before the end of com-
mutation is w, consider the sum of the potential differences in
the local circuit AabB which is closed through the material of
the brush. This leads to the equation

e = iR + i b Rb - iaRa (67)

where R a and Rb are contact resistances depending upon the
areas of the surfaces through which the current enters the brush.
Under the conditions shown in Fig. 56, the contact surfaces S a
and Sb are equal, and the currents i a and 4 are therefore also
equal. It follows that the voltage drops i a R a and ibRb are equal
and cancel out from equation (67). The same is true in the
later stages of commutation when S a is no longer equal to Sb
but to the portion w of the brush which remains in contact with
the segment A. The relations between the currents and the
surface resistances are then obtained by expressing these quan-
tities in terms of the contact surface, thus:

i a = w X k\

ii = Sb X &i

R. = ~ X k>

Rb = ~cT X &2
06

where k\ and k 2 are constants, and the voltage drop i a R a is seen
to be still equal to the drop ibRb. It follows that the only e.m.f.
to be developed in the short-circuited coil when uniform current
distribution is required will be e = iR.

The instantaneous value (i) of the current in the coil under-
going commutation can be expressed in terms of the brush width
W and the distance (w) through which the coil still has to travel
before completion of commutation, because,

i = I c - 2I C X
lc

(68)

W
and

148 PRINCIPLES OF ELECTRICAL DESIGN

At the beginning and end of commutation, when w is equal to
W or to zero, the maximum value of the required voltage is

6 = I C R.

In this study of the voltage to be developed in the coil under-
going commutation in order to produce a uniform current
distribution over the brush surface, the resistance of the brush
itself has been considered negligible; but with the assumption
of a uniform current distribution over the cross-section of the
brush the actual resistance of the brush material, even if it
is relatively high, will not appear as a modifying factor in the
general formula (68).

Referring again to Fig. 55, if the flux curve EB' may be con-
sidered a straight line, the current \i = ^} will also obey a
straight-line law. It will fall from the value +7 C to zero in
the time ^ and rise again to the value I c at the end of the

period t c , according to the simple law expressed by the straight
line in Fig. 55. If the change of current actually occurs in this
manner, we have what is called " straight-line " or ideal com-
mutation. The commutation is then ideal or perfect, not only
because it relieves the designer of much intricate and discouraging
mathematical work, but because it is the only means by which
the current density can be maintained constant over the brush
surface of the usual rectangular form. It is generally the aim of
the designer to maintain this current density as nearly constant
as possible, because unequal current density leads to local varia-
tions of temperature and resistance in the carbon brush, and in
those parts where the density attains very high values the ex-
cessive heating leads to pitting of the commutator surface even
if visible sparking does not occur. Whatever method of study-
ing commutation phenomena is followed, it is usual to assume
some law connecting the variable current i with the time t
condition. The straight-line law will therefore be assumed, but
the thing of immediate moment being in fact the whole problem
of commutation in its broader aspect is the location, or the
creation, of a neutral zone where the actual resultant flux cut by
the coil undergoing commutation will be zero.

Although the assumption of a smooth-core armature very
greatly simplifies the problem, especially when an effort is made

COMMUTATION 149

to picture the actual distribution of the magnetic flux, it seems
preferable to consider a machine with toothed armature because
this is the case which has generally to be dealt with by the
practical designer, and moreover it is exactly this question of
teeth, or what is known as the slot flux, which sometimes leads
to confusion of ideas, if not to inaccurate conclusions, and it
should therefore not be disregarded in any modern theory of
commutation.

46. Effect of Slot Flux. In Fig. 57 an attempt has been made
to represent, by the usual convention of magnetic lines, the flux
due to the armature current alone, which enters or leaves the
armature periphery in the interpolar space when the field mag-
nets are unexcited. The position chosen for the brushes is the
geometric neutral i.e., the point midway between two (sym-
metrical) poles and the magnetic lines leaving the teeth will
cross the air spaces between armature surface and field poles
and so close the magnetic circuit. The brush is supposed to
cover an angle equal to twice the slot pitch: the current in the
conductor just entering the left-hand end of the brush is -f-/ c ,
the current in the conductor under the center of brush is zero,
and the current in the conductor just leaving the brush on the
right-hand side is I c . The armature is supposed to be rotat-
ing, and it will be seen that the conductors in which the current
is being commutated are cutting the flux set up by the armature
as a whole. It is important to note that the flux cut by a con-
ductor while travelling between the two extreme positions during
which the short-circuit obtains is not only the flux passing into
the air gaps from the tops of the teeth included between these
extreme positions of the conductor, but includes also the flux
due to the currents in the short-circuited conductors, which
crosses the slot above the conductor 1 and leaves the armature
surface by teeth which are not included in what at first sight may
seem to be the commutating zone. In other words, the portion
of the armature flux cut by a conductor undergoing commuta-
tion when no reversing flux is provided from outside is that
which passes up through the roots of the teeth included between
the two extreme positions of the short-circuited coil. This
picture of the conductor cutting the field set up by the armature

1 For the sake of simplicity, a single conductor is shown at the bottom of
each slot and the whole of the slot flux is supposed to link with it. The
calculation of the "equivalent" slot flux will be taken up later.

150

PRINCIPLES OF ELECTRICAL DESIGN

currents is especially useful when calculations are made, as will
frequently be found convenient, by considering the separate

FIG. 57. Flux in commutating zone due to armature m.m.f. only.

component fluxes due to distinct causes, all combining to pro-
duce the actual or resultant flux. It is not difficult to see that
the flux shown in Fig. 57 is never such as to generate an e.m.f.

FIG. 58. Flux in commutating zone near leading pole tip.

tending to reverse the current in the short-circuited conductor.

Consider now Fig. 58. The main poles have been excited

and the brushes moved forward until a satisfactory commutating

COMMUTATION 151

zone has been found where the fringe of flux from the leading
pole tip is sufficient to neutralize the flux due to the armature
windings. l With the excitation of the leading main-pole tending
to send flux through the armature core from right to left, and the
armature e.m.f. tending to produce a flux distribution gener-
ally as indicated in Fig. 57, the resulting flux distribution in the
commutating zone will be somewhat as shown in Fig. 58. Here
the flux cut by the conductors during cpmmutation is represented
by eight lines only, the direction of this commutating flux
being such as to maintain the current during the earlier stages
of commutation and reverse it during the later stages. At a
point midway between the two extreme positions the conductor
is cutting no flux, and the current is therefore zero. It should be
observed that the correct position for the brush is in advance of
the "apparent" neutral zone; that is to say, the position of the
neutral field on the surface of the armature does not correspond
with the correct position for the center of the brush. That is
because the slot flux must enter through the upper part of the
teeth if it is not to be cut by the conductors during commutation.

Thus the conductor, which at the instant of time t = ^ must be

a

in a neutral field, is actually below a point on the armature
periphery where flux is entering or leaving the teeth, and this
condition occurs even when, as in the present instance, the effect
-of the end connections is entirely negligible. This flux, which
enters the teeth comprised between the two extreme positions
of the short-circuited conductors, is neither more nor less than
the slot flux (or equivalent slot flux, as the case may be). It is
represented by 12 lines in the diagram Fig. 58, and it must
be provided by the leading pole shoe if brush shift is resorted
to, or by the commutating interpole when this method of
cancelling the armature flux is adopted.

47. Effect of End Flux. When the effect of the end connec-
tions of the armature coils cannot be neglected, the armature
flux cut by this portion of the short-circuited coil must be
cancelled in the same manner as the slot flux; that is to say, an

1 Credit is given the reader for the ability to read in the expression "a
flux neutralizing a flux" the more scientific but less convenient expression
"the magnetic force due to one magnetizing source being of such magnitude
and direction as to neutralize the magnetic force due to a second magnetizing
source, thus causing the resultant flux of induction to be of zero value."

152

PRINCIPLES OF ELECTRICAL DESIGN

equal amount of flux, but of opposite sign, must enter the arma-
ture from the leading pole tip or interpole, and this component
of the compensating flux will actually be cut by the conductors
in the slot, thus neutralizing the e.m.f. developed by the cutting
of the end fluxes. This will be made clearer by reference to Fig.
59, which is generally similar to Fig. 54, except that, instead of

FIG. 59. Diagram showing component fluxes cut by coil during
commutation.

the actual interpolar flux, two distinct curves have been drawn,
the one, F, representing flux distribution over armature peri-
phery due to the field coils acting alone, and the other, Z, repre-
senting the flux distribution due to the armature windings acting
alone. The addition of these two fluxes at every point will not
always reproduce the actual flux curve of Fig. 54, because of
possible saturation of portions of the iron circuit such as the
armature teeth and pole tips; but, in the commutating zone, the
method of adding the several imaginary components of the

COMMUTATION 153

actual flux is not objectionable, and the active conductors AA'
in Fig. 59 may be considered as moving in a field of which the
density is represented by the length MN, since the portion of
the field flux represented by the distance between the point
N and the datum line is neutralized by the armature flux at
this point. Let ABC represent the position of the end connec-
tions of the coil undergoing commutation, then the portion AB is
cutting end flux due to the armature currents in all the end con ?
nections, and the direction of this flux will be the same as that
represented by the curve Z, all as indicated by the direction of the
shading lines. The portion BC of the short-circuited coil will
be cutting flux of the same nature as the armature flux cut by the
slot conductors CC', and the e.m.f. due to the cutting of the end
fluxes will be of the same sign as that due to the cutting of the
armature flux Z; that is to say, it will tend to oppose the reversal
of current and must therefore be compensated for by a greater
brush lead or a stronger commutating pole. Similar arguments
apply to the end connections A'B'C' at the other end of the
armature. A means of calculating the probable value of the
effective end flux will be considered later; but for the present
it may be assumed that the average value of the density B e
of the field cut by the end connections is known. It may, there-
fore, be used for correcting the ordinate of the curve Z at the
point 0. Thus, the flux cut by the portion ABC of the end
connections (see Fig. 59) in the time t c is

\$, = B e X x X length of ABC
or

\$>* = BeW a sm a X length of ABC

where a is the angle between the lay of the end connections and
the direction of travel, and W a is the arc covered by the brush,
expressed in centimeters of armature periphery. The equivalent
flux density B a which has to be cut by the slot conductors A A'
to develop the same average voltage is obtained from the relation
B a Wa X length AA' = B a W a sin a X length ABC
which gives

i,,,,,,-i i, A Jin

(69)

or, if preferred,

> length AA'
. 2(BH)

154 PRINCIPLES OF ELECTRICAL DESIGN

This may be plotted in Fig. 59 as the ordinate OE, making NE
represent the armature flux, on the assumption that the whole
of this flux component is cut by the "active" portion of the
coil; and this suggests a graphical method of locating the correct
brush position when commutating poles are not used, because
what may be called the equivalent neutral zone is found when
the conductor A A' occupies a position such that the length
NE is exactly equal to OM. If this position cannot be found
without passing under the tip of the pole shoe (represented by
the heavy dotted rectangle), the machine will not commutate
perfectly without the addition of a commutating interpole.

The question of relative magnitude of these end flux e.m.fs.
deserves some attention, because it would be foolish to compli-
cate the problem of commutation if the correction, when made,
is of little practical moment. It is claimed by some writers
that refinement of analysis and calculation is always commend-
able even when built upon a foundation that is admittedly a
mere approximation. With this attitude of mind the present
writer has no sympathy; it appears to lack the sense of propor-
tion. Apart from any considerations of a mechanical nature,
the practical problem of commutation, from whatever point of
view it is approached, is, and always will be, the correct determi-
nation of the field in which the short-circuited coil is moving,
whether this conception of the magnetic condition is buried
in the symbols L and M , and referred to as inductance, expressed
in henrys, or considered merely as any other magnetic field;
and it would surely be a waste of time and mental effort to intro-
duce refinements if the percentage correction, when made, is of
a small order of magnitude. The question of end fluxes, how-
ever, is one of real practical importance; the end flux in actual
machines is not of negligible amount, and although it cannot
be calculated exactly, it is a factor which should not be left
out of consideration . It is true that we do not concern ourselves
with the end fluxes when calculating the useful voltage developed
in the active coils; but, apart from the fact that in this connec-
tion the amount of the end flux is relatively small, it is not difficult
to see that the e.m.f.s generated in the end connections as they
cut through the end fluxes due to the armature currents balance
or counteract each other and have no effect on the terminal
voltage. The conception of the end connections cutting through
the flux due to the armature as a whole, as indicated in Fig. 59,

COMMUTATION 155

seems more natural, and is more helpful to the understanding of
commutation phenomena, than what might be termed the
academic method, in which more or less reasonable assumptions
are made in respect to self- and mutual inductances; but it is
not suggested that the one method is necessarily superior to the
other so far as practical results are concerned. 1 While moving
from the position at the commencement of the commutation
where the current is -f-/ c to the position at the end of commuta-
tion where the current is I t , the short-circuited coil has cut
through the flux of self- and mutual induction through the
whole of it, not merely through certain components of the total
flux in the particular region considered. This is well expressed
by MR. MENGES when he says 2 : ". . . Self-induction is in
no way distinguishable from other coexistent electromagnetic
induction. Therefore, when the real magnetic flux resulting
from all causes, and its changes relative to a given circuit, are
taken into account, the self-induction is already included, and
it would be erroneous to add an e.m.f. of self-induction."

48. Calculation of End Flux. With a view to calculating
within a reasonable degree of accuracy the flux density in the
zone ABC of Fig. 59, it is necessary to make certain assumptions
and to use judgment in applying the calculated results, because
it is not possible to determine this value with scientific accuracy
even when the exact shape of the armature coils and the con-
figuration of the surrounding masses of iron are known.

In the first place, the angle a of Fig. 59, wil) be taken as 45
degrees, which, although perhaps slightly greater than the
average on modern multipolar machines, has the advantage
that it permits us to treat the wires AB and BC as being at right
angles to each other. The further assumption will be made
that the armature is of large diameter, and the developed view
of the end connections, as shown in Fig. 60, can therefore be
considered as lying in the plane of the paper. The flux in the
zone occupied by the portion A B of the coil undergoing commu-

1 As a matter of fact, a careful study of the problem will show that the
total armature flux cut by one commutating coil during the period of short-
circuit being the difference between the number of lines threading the coil
immediately before and immediately after commutation is almost
entirely due to the changes of current that have taken place in the coils
under the brushes during the period of commutation.

2 C. L. R. E. MENGES in the London Electrician, Feb. 28, 1913.

156

PRINCIPLES OF ELECTRICAL DESIGN

tation is due to the currents in all the neighboring parallel con-
ductors comprised in the parallelogram A DEC. The direction
of flow of current in these parallel conductors is indicated by the
arrows, being outward (i.e., from A to B) on the left-hand side
of the commutating zone, and inward (from B to A) on the right-
hand side. The intensity of the field produced on AB by any
one of these wires, if the lines of force are assumed to be circles
in a path consisting entirely of air (the proximity of masses of
iron being for the present ignored), will be inversely proportional

FIG. 60. Developed view of armature end connections.

to the distance of the wire considered; and the extent to which
this wire will be effective in producing a field along AB will
depend upon its length. Thus, a conductor such as EF will pro-
duce not only a stronger component of the resulting field in the
commutating zone than the wire GH, but a field of which the
extent is measured by the length EF, while the more distant
wire will produce a weaker field over a length equal to GH only.
Thus the effect of the more distant wires in building up the flux
over the commutating zone decreases very rapidly with increase
of distance. Fig. 61 represents a section perpendicular to the
conductor AB of Fig. 60. It is assumed that the brush covers

COMMUTATION

157

two bars (a reasonable, but not a necessary, assumption), 1
and the condition shown in Fig. 61, corresponds to the middle
of the commutation period, with zero current in the short-cir-
cuited coil and full armature currents of +/ c and I c respectively
in the neighboring conductors.

.-Connection EF

<8> 8

\M ! I

FIG. 61. Mangetic flux due to end connections.

Considering the full-pitch coil of pitch
over the armature surface, we can write

T = nX

centimeters measured

where n = number of slots per pole, and X = slot pitch in
centimeters. Then the length AB (Fig. 60), which is approxi-
mately one-quarter of the total length of " inactive" copper
per coil, is

T nX

=

and the pitch of the end windings (which are supposed to lie
in the same plane as the conductors in the slots) will be,

X I
a =

Let T number of conductors or turns per coil (which is not
necessarily the same as the number of turns between com-
mutator bars, because there may be more commutator bars than
there are slots on the armature), and let I c = the amperes of

1 Within practical limits, the width of the brush does not appreciably
affect the average density of the armature flux cut by the commutated coil.
A wide brush, by short-circuiting several coils, reduces the number of con-
ductors carrying the full armature current, and to this small extent the total
m.m.f. producing the flux in the commutating zone is less with a wide
brush than with a narrow brush.

158 PRINCIPLES OF ELECTRICAL DESIGN

current per conductor; then, since the field intensity at a distance
y cm. from a straight conductor is

0.2 X current in amperes
ti =

y

we may write, for field intensity due to the group of conductors
EF,

0.2 X TI C

H " '- ~W

The flux produced in the zone AB by the same group of wires
(EF) will be proportional to the value of H multiplied by the
length EF. Thus, in a zone 1 cm. wide, of which the center line
is AB, the flux of induction due to the conductor EF is

= 0.277Q _

The sum of all such elements of the total flux, taken for all the
parallel conductors on both sides of AB will be

QATI C [, I - 2a I - 3a I - i

which, bearing in mind the relation I = an, simplifies into

(70)

The total flux (maxwells) cut by the end connections ABC,
being one-half of the length of " inactive" copper in the corn-
mutated coil, is given by the expression

\$ e = 2\$ X W a sin a

= VZ\$ X W a

= OAV2TI c nW a g + J + 1 + J) (71)

in which W a is, as before, the brush arc expressed in centimeters
of armature periphery.

The value of the series in the brackets is readily computed
with the aid of a table of reciprocals, but if preferred this series
can be put in the form (log. 2n) 1, which is really more ac-
curate, since it assumes a current uniformly distributed through
the copper section of the conductors instead of being concen-
trated at the center of each coil as assumed in deriving formula
(70).

COMMUTATION 159

The assumption has been made that the paths of the flux lines
are air paths only; but on account of the proximity of the pole
shoes to the points where the end connections leave the slots,
and also because the conductors actually remain parallel to the
shaft for a short distance beyond the core, the value of 3> e would
be larger than as given by formula (71). A constant should
therefore be included, and if the convergent series is replaced by
the logarithmic function, the formula becomes,

\$ e = OA<^2kTI e nW a [(lo& 2n) - l] (72)

The average flux density over the zone considered is

n j-. 1 ] (73 )

If the end coils lie on a cylindrical support of iron or steel, the
reluctance of the flux paths is very nearly halved, and the
value of k in formulas (72 and (73) should therefore be doubled.
If it is desired to consider the increased flux due to the use of
steel binding wires or bands, this can be done by making a suit-
able correction to the factor k. 1

Should it be desired to calculate separately the average value
of the total e.m.f. generated in the end connections of the short-
circuited coil, we have

10 8
where l e = total length of coil outside armature slots (both

ends) in centimeters,
= 2*\/2\n, if we keep to the assumption of angle a =

45 degrees,
and V = speed of cutting, in centimeters per second,

= 7^, in which D is the armature diameter in

V2

centimeters, and N, is the speed in revolutions per second.
The factor \/2 is the necessary correction to give the component
of the velocity at right angles to the conductor. Inserting the
value of B e given by formula (73), and making the required
simplifications, the formula for voltage becomes

k X 0.8V2nI c T* frZW.) (log, 2n - 1)

1 This is discussed by MR. LAMME in his Institution paper previously

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 13 of 30)