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160 PRINCIPLES OF ELECTRICAL DESIGN

The numerical value of k in multipolar machines of modern
design will usually lie between 1.3 and 3.5, the high value being
taken when the end connections lie on a steel or iron supporting
cylinder against which they are held by bands of steel wire.

For first approximations the formula may be put into simpler
form.

Let Ac stand for the ampere-conductors per pole pitch of arma-
ture periphery (A c = 2TnI c ). Let k 2.4 (being an average
value), and for the quantity (log e 2n 1) put the numerical
value 2.2 (the assumption here being that there are 12 to 14
slots per pole), then

3TA V

E e (approximately) = 1Q8 C (75)

where V is, as before, the peripheral velocity of the armature
in centimeters per second.

The above calculations and conclusions are based on the
assumption of a full-pitch winding. With a chorded or short-
pitch winding, the average flux density, in the commutating zone
will be slightly reduced; and there will be a further gain due to
the shortening of the end connections (ABC and A'B'C' in
Fig. 59). Thus the voltage generated by the cutting of the end
fluxes, with a short pitch winding, will be slightly less than the
value calculated by formula (74) , or by the approximate formula
(75), which applies to a full-pitch winding.

49. Calculation of "Slot Flux" Cut by Coil during Commuta-
tion. A reference to the diagrams of flux distribution in the
commutating zone (Figs. 57 and 58) will make clear the fact that,
even when the effect of the end connections is neglected, the
center of the neutral commutating zone is not the point on the
armature periphery where flux neither enters nor leaves the teeth ;
because in order that the short-circuited conductors shall not
cut the slot leakage flux, this flux must be provided by the
main field pole toward which the brushes are shifted to obtain per-
fect commutation. The point on the armature periphery where
flux neither enters nor leaves the teeth may be found by drawing
curves representing the magnetomotive forces exerted by field
poles and armature windings at every point on the armature
periphery, and where the sum of the ordinates of such curves is
zero the surface flux density must also be zero. The brushes
must, however, be moved forward beyond this point until the
reversing flux entering the teeth comprised in the brush arc has

'COMMUTATION 161

the value 3> e as given by the formula (72), to compensate for the
end fluxes, plus the total slot flux < s , which is twice the leakage
from tooth to tooth in one slot when the conductors are carrying
the full armature current. 1 In practice, when we wish to cal-
culate the volts generated in the coil of T turns by this slot
leakage flux, it is the equivalent slot flux that must be considered,
because the total number of lines crossing between the sides of
adjacent teeth does not link with all the wires in the coil.

It will be convenient to assume the same number of slots
as there are commutator bars, and the whole of the slot space
to be filled with 2T conductors, each carrying a current of I c
amp. (this follows from the assumption of a full-pitch winding).
Thus no account will be taken of the fact that a small space
occurs between upper and lower coils, where the slot flux will
not pass through the material of the conductors. The lines of
the slot flux will be supposed to take the shortest path from tooth
to tooth; the small amount of flux that may follow a curved
path from corner to corner of tooth at the top of the slot will be
neglected. Refinements of this nature may be introduced, if
desired, when solving the problem for a concrete case. 2 If the
usual assumption is made that the reluctance of the iron in the
path of the magnetic lines is negligible in comparison with the
slot reluctance, the small portion of slot flux in the space dx
(Fig. 62) considered 1 cm. long axially (i.e., in a direction per-
pendicular to the plane of the paper) is

d\$ = m.m.f. X dP
where dP is the permeance of the air path. Thus,

d\$ = 0.47T (277 C ) -, X

CL S

1 In the case of short-pitch windings, or when there are more commutator
bars than there are slots on the armature, the amount of the slot flux must
be calculated for the instant when the coil enters or leaves the commutating
zone. This flux will depend not only upon the dimensions of the slot but
also upon the current carried by each conductor and the position of the
latter in the slot.

2 With short-pitch windings, or when there are several coils per slot, all
the conductors in the slot may not be carrying the full armature current
when the coil enters or leaves the commutating zone. In such cases the
actual conditions must be studied, an average value for the slot flux being
readily arrived at. If necessary, the straight-line law of commutation
may be assumed in order to estimate the current values in the coils passing
through the intermediate stages of commutation.

11

162

PRINCIPLES OF ELECTRICAL DESIGN

but this flux links with only 2T X - , conductors, and the equiva-

Ci

lent slot flux, which would generate the same e.m.f. if cut by all
the conductors in the slot, is therefore

(equivalent) =

X

X j

and

\J \JI\ JL J. c

*** '(equivalent) = ^2^

_ Q.Sird
~ 3s

(76)

FIG. 62. Illustraing slot-flux calculations.

This is the equivalent flux in maxwells per centimeter of axial
length of armature slot. If l a = axial length of armature core
expressed in centimeters, 1 the total slot flux cut by each coil-side
during commutation, being twice the flux per slot, as shown in
Fig. 57, is

*..^ r /j. . (77)

The voltage component due to the cutting by both coil-sides 2
of the slot flux considered separately from other fluxes would be

tc =

where the time of commutation

brush arc in centimeters of armature periphery
peripheral velocity in centimeters per second

Wa

V

Thus,

3 X 10W a

The factor W a in the denominator of this formula indicates that
the slot flux is of less importance with a wide than with a narrow

1 It is well to let l a stand for the gross length of armature core, although
in slot flux calculations the total width of vent ducts is sometimes deducted.

2 The 2,T conductors in the one slot are here considered as equivalent to
the two coil-sides, each of T wires, in separate slots one pole pitch apart.

COMMUTATION 163

brush. This is generally true, although it must be remembered
that the formula (78), in common with other formulas pre-
viously derived, is not of general application. Within the limits
of this chapter, it is not possible to consider all special cases;
and commutation formulas of general applicability cannot be
developed. When there is more than one coil per slot, and when
there are "dead coils," inequalities occur which complicate the
problem and make it impossible to obtain ideal commutation
with every coil on the armature. In such cases the slot flux must
be calculated for the coils that are differently situated in regard
to the brush position and an average value selected for use in the
calculations.

Knowing the slot flux \$ e and the previously calculated end flux
\$ cut by the conductors of the short-circuited coil while travel-
ling over the distance W a , the correct brush position is found when
the reversing flux entering the teeth comprised in the commu-
tating zone of width W a , is approximately 3> c , + \$ e maxwells.
The flux actually cut by the one coil-side is, however, only \$ e ;
the component 3? e , of the total flux entering the commutating
zone merely supplies the leakage from tooth to tooth across the
slot. The presence of the slot flux undoubtedly tends to com-
plicate the problem of commutation. It should be noted that
the slot flux \$ e g, if calculated by formula (77), is what has been
referred to as the equivalent slot flux; that is to say, a flux of
this value, if cut by an imaginary concentrated winding of T
turns, would develop the same voltage in the coil as the actual
slot flux develops in the actual winding. The condition of
importance to be fulfilled is simply that the "equivalent" flux
cut by the coil-side in the reversing field shall have the value \$<,.
The flux cut by the coil-side may be separated into two parts: (1)
the flux passing through the teeth into the armature, which links
with all the conductors in the slot, and (2) the equivalent slot
flux. It is important to note that although the total or actual
slot flux is the same whether it enters the top or the root of the
tooth, the flux linkage and therefore the developed voltage have
not the same value in the two cases. The total slot flux, on the
basis of the assumptions previously made, is

=

as

Jo

164 PRINCIPLES OF ELECTRICAL DESIGN

being one and one-half times the equivalent slot flux given by
formula (77). The equivalent flux when the total slot flux enters
the tooth top instead of passing through root of tooth is no longer
expressed by formula (77) ; it may be calculated thus :

The magnetic lines represented by the expression d\$ =

f fJ^r i*

0.4?r (2TI C ) -j no longer link with 2T -, conductors, but with
d s ct

2T- -j - conductors (see Fig. 62). The equivalent flux, when

no part of this flux passes into the armature core below the teeth,
is therefore

x

I

d

x(d x)dx

-^HA (80)

or just half the equivalent slot flux as given by formula (77).
The question now arises : What is the necessary total flux enter-
ing the tops of the teeth comprised in the commutating zone to
develop the proper voltage component in the short-circuited
coil?

A total slot flux as given by formula (79) has the "equivalent"
value as given by formula (80) ; that is to say, it is on the basis
of the assumptions previously made three times as great as
the equivalent flux. The total flux entering the teeth comprised
in the commutating zone should therefore be

3> c = 3<'e S + flux passing directly into armature core through
the teeth.

but \$'es + \$d = & e , where 3> e is, in this particular instance, the
equivalent value of the total flux to be cut by the "active"
portion of the short-circuited conductors.
Thus

^Ac ' ^Mr s I JT Q ^O-Ly

or, if preferred,

*c = * + *. (82)

where \$ e8 is the equivalent slot flux as originally calculated and
expressed by formula (77).

COMMUTATION 165

The flux actually entering the armature teeth in the com-
mutation zone should therefore be equal to the sum of the end
flux \$ e and the equivalent slot flux \$ es . It is because this con-
clusion is not obvious that it has been deduced from the fore-
going arguments.

Having determined the value of the flux \$ c which must enter
the teeth comprised in the commutating zone of width W a ,
it is evident that the average air-gap density in this zone, to
produce perfect commutation, must be

*'-wr*i. (83)

By referring to the final flux distribution curve, C, obtained by
the method outlined in Art. 43, Chap. VII, it may easily be
seen whether or not the desired field can be obtained in the
fringe of the leading pole tip. If the required field is greater than
that obtainable in the interpolar space, commutating poles must
be provided, or the machine must be re-designed. If the flux-
distribution curves have not been drawn, the calculated density
B c required in the commutating zone, as expressed by formula
(83), may be compared with the average air-gap density under
the main poles. If the required density does not exceed one-half
of the average density of the main flux taken over the pole pitch,
it will usually be possible to obtain satisfactory commutation in
the fringe of the leading pole tip, provided carbon brushes are
used. In the event of B c being in excess of this value, interpoles
will probably be necessary.

50. Commutating Interpoles. Assuming the same number of
interpoles as there are main poles, and an axial length of interpole
equal to that of the main pole, the flux from each interpole
which enters the armature teeth included in the commutating
zone of width W a is, as before, \$ ea + \$ e .

If, as is usually the case, the interpole face does not cover
the whole length of the armature core, then some flux due to the
total m.m.f. of the armature windings will enter or leave the
armature core by the teeth included in the commutating zone,
and this flux will be cut by that portion of the slot conductors
which is not covered by the interpole. With the brushes on
the geometric neutral line, this armature flux is unaffected by
the excitation of the main poles ; its value depends only upon the
armature ampere-turns and the reluctance of the air paths

166

PRINCIPLES OF ELECTRICAL DESIGN

between the armature surface and neighboring masses of iron.
It can be predetermined within reasonably close limits by plot-
ting the full-load flux curve, C, as indicated in Art. 43, Chap.
VII. The flux entering and leaving the surface of the armature
in the commutating zone, when the axial length l p of the interpole
is appreciably less than the gross length l a of the armature core,
is indicated in Fig. 63. Before calculating the flux which must
enter the armature teeth from the commutating pole, it will
be advisable to define clearly the various flux components to be
considered.

FIG. 63. Commutating pole : showing direction of flux at armature surface.

Many of the symbols used in the previous calculations will be
employed, but it is proposed to alter the meaning of some of
these because it will be more convenient to think of the slot
flux per centimeter length of slot instead of the flux over the
whole length of slot as in the previously developed formulas.
This slight change will probably lead to less confusion than if
a complete new set of symbols were to be introduced here.

< c = total flux entering armature teeth from interpole, over

area of width W a and length l p .
& e = total end flux (one end of armature).
\$>, = total slot flux per centimeter of armature length (two

slots).
&e8 = equivalent slot flux per centimeter, if magnetic lines

pass outward from armature core through root of teeth

(two slots).
\$'es = equivalent slot flux per centimeter, if magnetic lines pass

inward from air gap through top of teeth (two slots).

COMMUTATION 167

<l> d = portion of interpole flux per centimeter length, which
enters armature core through root of teeth.

\$a = armature flux per centimeter length, which leaves
teeth over the commutating zone of width W a and length
l a - IP (Fig. 63).

The equivalent flux to be cut by conductors under the interpole
must equal the total of all the flux components that have to be
neutralized. This leads to the equation

\$ d lp + Veslp =\$e+ *a(la ~ lp) + \$>es(la ~ lp)

from which a value for <f>d can be calculated. The total flux
leaving interpole is

* c = \$ s l p + \$> d l p

Inserting for \$<* in this last equation the value derived from the
previous equation, we get

\$c = \$.lp + <*>,+ \$> a (la - IP) + Mk - IP) - *'.Jp (84)

This equation may be simplified by expressing to total slot flux
\$, and the equivalent slot flux \$' M in terms of the equivalent slot
flux \$ e - The relation between these quantities is obtained by
comparing the previously developed equations (79), (80) and

(77). Thus,

\$>, = % *e,

and

*'.. = 1 A *e,

Inserting these values in equation (84) we get

\$ e = *. + \$ e , la + ^a ~ IP) (85)

wherein the symbols & e , and \$ a stand for flux components per unit
length of armature core, as previously mentioned.

Knowing the amount of the flux to be provided by each inter-
pole, its cross-section can be decided upon and the necessary
exciting ampere-turns calculated, bearing in mind the following
requirements :

(a) The average air-gap density should be low (about 6,000
gausses corresponding to full-load current), to allow of increase

(b) The leakage factor should be as small as possible. This
involves keeping the width and axial length of interpole small,

168 PRINCIPLES OF ELECTRICAL DESIGN

thus conflicting with condition (a) and presenting one of the
difficulties of commutating-pole design.

(c) the minimum width of pole face must be such that the equi-
valent pole arc (which must include an allowance for fringing)
shall cover the commutating zone of width W a .

(d) The equivalent pole arc should, if possible, be an exact
multiple of the slot pitch (either once or twice the slot pitch)
as this tends to reduce the magnitude of the flux pulsations in
the interpole. The effects of these flux pulsations, caused by
variations in the reluctance of the interpole air gap, are, however,
usually of no great practical importance, but the width of brush
should not be determined independently of the interpole design.

(e) In order to keep down the PR losses in the series turns on
the interpole (usually amounting to less than 1 per cent, of the
total output), the ampere- turns and the length per turn should
be as small as possible. The gain resulting from a small air
gap is, however, not great, because the ampere-turns required
to overcome the air-gap reluctance rarely exceed 25 per cent,
of the total, the balance being required to oppose the armature
m.m.f. A reasonably large air gap has the advantage of re-
ducing the flux pulsations referred to under (d).

(f) The effect of the interpole being to increase the flux in
that portion of the yoke which lies between the interpole and
the main pole of opposite polarity, it is important to see that
the resulting flux density in this part of the magnetic circuit is
not excessive. A similar condition exists in the armature core,
but this does not usually determine the limits of the allowable
average flux density below the teeth.

(g) Series- or wave-wound armatures are to be preferred on
machines with commutating poles, especially when the air gap
under the main poles is made smaller than it would have to be
if interpoles were not used.

(h) The total line current should, if possible, pass through
all the interpole windings in series; that is to say, parallel
circuits should be avoided because of the possibility that the
current may not be equally divided. If the total current is
too great, a portion may be shunted through a diverter. 1 The
diverter should be partly inductive, the resistance being wound

1 The use of a resistance as a shunt to the series field winding usually
known as a diverter will be referred to again later when considering the
design of the field magnets.

COMMUTATION 169

on an iron core in order that the time constants of the main and
shunt circuits may be approximately equal. If this is not done,
the interpole winding will not take its proper share of the total
current when the change of load is sudden, and this may lead to
momentary destructive sparking.

Among the advantages of commutating poles may be men-
tioned the fixed position of the brushes and the fact that fairly
heavy overloads can be taken from the machine without de-
structive sparking, because of the building up of the commutat-
ing flux with increase of load. The limiting factor in this con-
nection is the saturation of the iron (mainly of the interpole
itself) in the local circuit, and this is aggravated by the large
percentage of leakage flux due to the proximity of main and com-
mutating poles. Deeper armature slots may be used than in
the case of machines without interpoles, and the specific load-
ing (ampere-conductors per unit length of armature periphery)
may be increased, thus allowing of greater output notwithstand-
ing the slight reduction in width of main poles necessary to
accommodate the interpoles. The maximum output of the
machine with commutating poles is usually determined by the
heating limits, the ventilation being less effective than in the
case of non-interpole machines. The PR loss in interpole
windings is to some extent compensated for by a reduction of
the ampere-turns on the main poles when shorter air gaps are
used. With the brushes on the geometric neutral and an air
gap which is small relatively to the space between pole tips,
field distortion per se has nothing to do with commutation,
whether interpoles are used or not; if the fringe from the leading
pole tip is not to be used for counteracting the effects of end flux
and slot flux on the coil undergoing commutation, the unequal
flux distribution under the main poles due to cross-magnetization
does not affect the field at a point midway between two main
poles. It is not suggested that field distortion in unobjectionable
when the brushes are on the geometric neutral or when inter-
poles are used. The concentration of flux at one side of the main
pole may lead to flashing over the commutator surface (an effect
often attributed to unsatisfactory commutation, though rarely
due to this cause); but the chief objection to a large number of
armature ampere-turns per pole is the fact that the flux in the
zone of commutation due to this m.m.f. must be compensated for
somehow if satisfactory commutation is to be obtained. It

170

PRINCIPLES OF ELECTRICAL DESIGN

is exactly in the zone corresponding to the brush position that
the armature m.m.f. has its maximum value. In the case of
the interpole machine, the windings necessary to compensate
for armature cross-magnetization are an objectionable feature,
and, except for the added cost and tendency to interfere with
ventilation, there is much to be said in favor of pole-face wind-
ings, the function of which is to neutralize the magnetizing
effect of the armature winding and maintain an approximately
constant flux density over the pole faces. The writer has in
mind machines such as those which, for the last 20 years, have
been constructed under the THOMPSON-RYAN patents. One

\

Center Line of Pole

Compensating Coi
Main Exciting Coi]
lutorpole Winding

FIG. 64. Compensating pole-face winding.

of the attractive features of such designs is the fact that the
winding on the commutating poles need be no greater than that
required to overcome the reluctance of the air gap and send the
requisite flux into the armature teeth comprised in the commu-
tating zone. A pole-face compensating winding is shown in
Fig. 64. The balancing coils pass through slots in the pole
face and carry the full current of the machine; that is to say,
they are connected in series with the commutating-pole windings
and the compounding series turns (if any) on the main poles.
The connections between the pole-face compensating coils are
so made that the current in these will always tend to neutralize
the magnetic effect of the currents in the armature coils, and so
prevent distortion of the flux over the pole face.

COMMUTATION 171

51. Example of Interpole Design. The numbers and dimen-
sions used to illustrate the method of calculation outlined above
will be chosen without reference to an actual design of interpole
dynamo, and they must not be considered representative of
modern practice. Assume the leading particulars of the machine
to be as follows:

Output = 200 kw.

Volts = 440.

R.p.m. = 500.

Number of main poles p = 6.

Armature core diameter D = 30 in.

Armature core length l a = 11 in. = 28 cm.

Total number of slots = 120.

Number of slots per pole n = 20.

Slot pitch X = 0.785 in.

Slot width s = 0.39 in.

Slot depth d = 1.5 in.

Style of winding: full-pitch, multiple.

Current per armature path I c 76 amp.

Number of conductors per slot = 8.

Total number of conductors Z = 120 X 8 = 960.

Number of commutator bars = 240. (There are four coil-
sides per slot, or two coils, giving two turns between adjacent
commutator bars.)

Diameter of commutator = 20 in.

Pitch of commutator bars = ^940 = O- 2 ^ 2 in.

Number of bars covered by brush = 3.5.

Thickness of brush (brush arc) W = 0.262 X 3.5 = 0.916 in.

Brush arc referred to armature periphery

Wm ,

Assuming the same number of commutating poles as there are
main poles, the flux entering the armature teeth in the commutat-
ing zone of width W a should have the value given by formula (85) .
The end flux cut by the short-circuited coils is given approxi-
mately by formula (72) in which the coefficient k may be given
the value 2. Thus:

3> e = 0.4V2fcr/cnTF.[(lo& 2n) - 1]

= 0.4\/2 X 2 X 4 X 76 X 20 X 3.5 X (3.69 - 1)
= 64,800 maxwells.

172

PRINCIPLES OF ELECTRICAL DESIGN

In regard to the slot flux, at the beginning or end of commuta-
tion (depending upon the position of the coil in the slot) the con-

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