Alfred Still.

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large number of conductors, and a number of phases equal to

Lag of Current Behind Open Circuit E.M.F.
Corresponding to Brush Shift in D.C. Machines

FIG. 108. Method of obtaining armature m.m.f. curve from curve of
current distribution.

the number of conductors in the space of one pole pitch. Thus,
the ordinates of curve I of Fig. 108 (assumed to be a sine curve)
give the value of the current in the various conductors distributed
over the armature surface. It is understood that the current
in each individual conductor varies according to the simple
harmonic law; but it is constant in value for a given position on
the armature surface considered relatively to the poles. The
direction of the current in the conductors between the points
A and B may be considered as being downward, while the direc-
tion of the current in the adjoining section of width T would be
upward. The maximum value of the armature m.m.f., there-
fore, occurs at the point B, and we may write:

278 PRINCIPLES OF ELECTRICAL DESIGN

Maximum value of armature ampere-turns per pole = average
value of current in section OB X number of turns, or

(SI) a =*Ima X X ~ (100)

where Z' stands for the total number of inductors on the arma-
ture periphery. This fnay be compared directly with formula
66 (page 136), which applies to direct-current machines, by
putting it in the form
Max. m.m.f. (gilberts) per pole

0.47T X 2 X Ic V2Z'

1.11 X2p

where I c stands for the virtual or r.m.s. value of the current in
the armature conductors.

That the armature m.m.f. curve in Fig. 108 is also a sine curve
when the current follows the sine law is easily seen from the
general solution, thus:

The magnetizing effect of the conductors in the small space of
width dd is

z r de

I max sin X

V

rrt in

wherein l max sin 6 is the current per conductor, and is

the number of conductors in the space considered (the angle

The expression for the total ampere-conductors is therefore

7i'

I max S sin 6dd
pw

With an increase in the number of inductors (and phases) this
quantity approaches more and more nearly the definite integral,
i.e., the area of the current curve, as indicated by CC' in Fig. 108
being a measure of the shaded portion of the curve /; and we
can then write

Z f

Armature ampere-conductors = I max ""*/* s i n d6

Tip

Zi'

= - Imax COS + C

AIR-GAP FLUX DISTRIBUTION 279

the maximum value of which occurs when 6 = o and 6 = IT.
The constant of integration merely determines the position of
the datum line; and since we have symmetry and equal strength
of North and South poles, we can put C = o and write for the
maximum value of the armature ampere-turns per pole,

7' -\/^,7'

(SI) a = - I max = -I e which checks with formula (100).
wp irp

The angle of displacement (Fig. 108) of this curve relatively
to the center line of pole depends upon the " internal" power
factor, and also upon the displacement of the wave of developed
e.m.f., a displacement or distortion which is due to cross-magne-
tization. The angle is not very easily predetermined, but,
once known or assumed, the curve M can be drawn in the correct
position relatively to the curve of field m.m.f.; and the resultant
m.m.f. over the armature surface can be obtained exactly as for
the direct-current machine (see Fig. 53, page 137). An approxi-
mate method of predetermining the displacement angle /3 for
any load and power factor will be explained in Art. 98.

Armature M.M.F. Curve of Single-phase Alternator. When
single-phase currents are taken from an armature winding, the
m.m.f. due to this winding as a whole must necessarily be of zero
value at the instant of time when the current is changing from its
positive to its negative direction. This suggests that the mag-
netizing effect of the loaded armature will be pulsating; that is
to say, it cannot be of constant strength at any given point con-
sidered relatively to the poles, whatever may be the phase dis-
placement of the current relatively to the developed voltage. If
the change of current in any given conductor be considered over
a complete cycle, and if at the same time the position of this con-
ductor relatively to the poles be noted, it will be seen that, rela-
tively to the field magnet system, the armature windings produce
a pulsating field of double the normal frequency. The actual
flux component due to the armature currents will not, however,
pulsate to any appreciable extent, because the tendency to vary
in strength at comparatively high frequencies is checked by the
dampening effect of the field coils, even if the pole shoes and
poles are laminated.

No modern single-phase alternator, unless of very small size,
should be built without amortisseur windings, or damping grids.
These consist of copper conductors in holes or slots, running
parallel to the shaft, in the faces of the field poles. They are

280

PRINCIPLES OF ELECTRICAL DESIGN

joined together at both ends by heavy copper connections, and
form a " squirrel cage" of short-circuited bars which damp out
the flux pulsations, and also prevent the sweeping back and forth,
or "swinging," of the armature flux due to "hunting" when syn-
chronous alternating-current machines are coupled in parallel.

Returning to the magnetizing effect of the single-phase arma-
ture, it is, therefore, the average or resultant armature m.m.f.
considered relatively to the poles with which we are mainly con-
cerned. The most satisfactory way of studying an effect of this
kind is to draw the actual m.m.f. curves at definite intervals of

FIG. 109. Instantaneous and average values of armature m.m.f. in single-
phase alternator.

time, and then average the values so obtained for different points
on the armature surface, the position of these points being
considered fixed relatively to the field poles.

This has been done in Fig. 109, where the windings are shown
covering 60 per cent, of the armature surface, and the distance r
is one pole pitch. This distance is divided into 10 equal parts,
each corresponding to 18 electrical degrees. The thick line repre-
sents the armature m.m.f. when the current has reached its
maximum value. The armature is then supposed to move 18
degrees to the right of this position, and a second m.m.f. curve is

AIR-GAP FLUX DISTRIBUTION 281

drawn, corresponding to this position of the windings. Its
maximum ordinate is, of course, less than in the case of the first
curve, because the current (which is supposed to follow the sine
law) now has a smaller value. This process is repeated for the
other positions of the coil throughout a complete cycle, and the
resultant m.m.f. for any point in space (i.e., relatively to the
poles, considered stationary) is found by averaging the ordinates
of the various m.m.f. curves at the point considered. In this
manner the curve M of Fig. 109 is obtained. It is seen to
be a sine .curve, of which the maximum ordinate is half the
instantaneous maximum m.m.f. per pole of the single-phase
winding, and it may be used exactly in the same way as the
curve M in Fig. 108 (representing armature m.m.f. of a poly-
phase machine) ; that is to say, it can be combined with the field
pole m.m.f. curve to obtain the resultant m.m.f. at armature
surface from which can be derived the flux distribution curves

The maximum value of the resultant ampere-turns per pole is,
therefore,

*/., X j p = I max X ^ (102)

where Z is the total number of armature face conductors. Ex-
pressed in gilberts the formula is,

Maximum ordinate of armature m.m.f. } QAw\/'2l c Z

curve in single-phase alternator. 2 X 2p

0.4rrZ/ c

which, together with formula (102), may be compared with the
formula (100) and (101) for polyphase generators.

95. Slot Leakage Flux. Referrring again to Fig. 108, if we
wish to derive a curve of resultant m.m.f. over the armature
before combining the curves of armature and field pole m.m.f.,
to determine the relative positions of these two curves. In the
direct-current machine, the position of maximum armature
m.m.f. coincides with the brush position; but the point B in
Fig. 108 is not so easily determined. Its distance from the
center of the pole is + 90, a displacement which depends
not only on the power factor of the load (i.e., on the lag of the
current behind the terminal potential difference), but also on the
strength of the field relatively to the armature, because this

282

PRINCIPLES OF ELECTRICAL DESIGN

relation determines the position (relatively to the center of the
pole) of the maximum e.m.f. developed in the conductors.

The field m.m.f. will depend upon the flux in the air gap, and
since this includes the slot leakage flux, it will be necessary to
consider the meaning, and determine the value, of the slot flux
before attempting to calculate the angle of Fig. 108.

Apart from the action of the armature winding as a whole,
causing a reduction of the total flux crossing the air gap from
pole face to armature teeth, the current in the individual con-
ductors, by producing a leakage of flux in the slots themselves,
still further reduces the useful flux when the machine is loaded.
The whole of the flux entering the tops of the teeth is not cut

j

''i! >','! i: 1 ,', : :!i !!!!! ; taiill 1!i ;.'; : '<, . \$m

nW hrr W^ Mtff, \$M ,w- /\$w tm , l tH J - RM

T-^K/,//, vv^fpuuuf4uffla^ \\>::^

Tti:

11
li

^-

FIG. 110. Flux entering armature of A.C. generator under open circuit

conditions.

by the conductors buried in the slots, and the voltage actually
developed in the " active " portion of the armature windings will
be reduced in proportion to the amount of flux which, instead of
entering the armature core, is diverted from tooth to tooth.
This loss of voltage is usually attributed to the reactance of the
embedded portion of the windings, and is referred to as a react-
ance voltage. This term, however, although very convenient, is
liable to lead to confusion when an attempt is made to realize
the physical meaning of armature reactance. It suggests that a
certain electromotive force is generated in the conductors, thus
causing a flow of current which, in turn, produces the flux of
self-induction and a reactive electromotive force. This is
incorrect and leads to a mistaken estimate of the actual amount of

AIR-GAP FLUX DISTRIBUTION

283

flux in the armature core a mistake of little practical import
yet tending to obscure the issue when considering the problem
of regulation, and standing in the way of a clear conception of
the flux distribution in the air gap.

The effect of the current in the buried conductors will be under-
stood by comparing Figs. 110 and 111, where the dotted lines
indicate roughly the paths taken by the magnetic flux under
open-circuit conditions (Fig. 110) and under load conditions
(Fig. 111). In the first case, when no current flows in the arma-
ture conductors, the whole of the flux entering the tops of the
teeth passes into the armature core and is cut by all the conduc-
tors. In the second case the magnetomotive force due to the

Direction of travel of poles

FIG. 111. Flux entering armature of A.C. generator when the conductors
are carrying current.

armature current diverts a certain amount of flux from tooth to
tooth, which since it does not enter the armature core is not cut
by all the conductors. This conception of the slot flux, as that
portion of the total flux leaving the pole shoe, which crosses the
air gap but does not enter the armature core below the teeth,
disposes of the difficulties encountered by many engineers when
faced with the necessity of calculating the slot inductance. It
is unnecessary to consider the leakage flux in the slots under the
pole face, but it is important to know the amount of flux in the
neutral zone, * which passes from tooth to tooth and generates no

l By neutral zone is meant the space between poles on the armature
surface where the lines of magnetic flux are parallel to the direction of travel
of the conductors.

284

PRINCIPLES OF ELECTRICAL DESIGN

electromotive force in the conductors. If this leakage slot flux
(in the neutral zone) were actually cut by the conductors, it
would generate a component of electromotive force lagging one-
quarter period behind the main component (on the assumption of
sine-wave form), and it can therefore conveniently be represented
in vector diagrams as if it were an electromotive force of self-
induction. The quantitative calculation of this electromotive
force will be considered in Art. 97. Although only brief
mention has been made of the leakage flux in slots other than
those in the neutral zone, it is not suggested that this flux is
negligible in amount; but the flux distribution under the pole
face, whatever may be its distortion, affects only the wave shape
(and form factor) of the generated electromotive force, and in no
way alters the average value of the developed voltage. The
difference between the total flux entering the armature teeth
from each pole face and the amount of the slot flux (or the
equivalent slot flux) in the neutral zone represents the flux
actually cut by all the conductors on the armature.

96. Calculation of Slot Leakage Flux. The effect of slot in-
ductance being generally to reduce 1 the amount of the total air-

' '''

<A> (B)

FIG. 112. Flux entering teeth in neutral zone; showing effect of armature
current in producing slot leakage.

gap flux which is actually cut by the conductors, the simplest
way to obtain a quantitative value for the slot reactance is to
calculate the total flux which leaks from tooth to tooth in the
neutral zone. Diagrams (A) and (B) in Fig. 112 indicate the
approximate paths of the magnetic lines in the neutral zone, (A)
when the current in the slot conductors is zero, and (B) when it
has an appreciable value. The amount of flux diverted from
the armature core into the leakage paths referred to may be

1 Except in the case of a condenser load and leading current, in which
case the tendency would be to increase the total useful flux.

AIR-GAP FLUX DISTRIBUTION

285

calculated by assuming the current in the slot conductors to be
acting independently of the field magnetomotive force. Thus, in
Fig. 113 the total slot flux is the sum of three component
fluxes: 3> r passing through the space occupied by the copper, a
portion of which will be cut by some of the conductors; <J> 2
crossing the space above the windings, usually occupied by the
wedge; and <J> 3 which leaks from tooth top to tooth top. If the
conductors were concentrated as a thin layer at the bottom of
the slot, the loss of voltage due to reduction of core flux (see
Fig. 112) could be calculated by assuming that the useful flux is
reduced by an amount equal

to the slot flux. The portion , - *'* ""*"*'*..

\$1, however, in Fig. 113, be-
ing cut by some of the con-
ductors, requires the calcu-
lations to be based on an
equivalent slot flux which, if
cut by all the conductors,
would develop an electromo-

FIG.

113. Illustrating method of cal-
culating slot leakage flux.

tive force equal to the actual

loss of pressure. This flux

may be calculated as follows:

The amount of flux in the small strip dx deep (Fig. 113) of 1 cm.

axial length, i.e., perpendicularly to the plane of the paper, is d\$ t

= m.m.f . X dP, where dP is the permeance of the air path the

reluctance of any iron in the path of the lines being neglected.

Whence

(**! = (0.4ir!r.J.)Jxy

where T 8 is the number of conductors per slot; 7, is the current
per conductor in amperes; and the dimensions d l and s (see Fig.
113) are in centimeters. Since, however, this flux element (see
Fig. 112, B) is cut by T 8 (d l - x)/d l conductors, the loss of
pressure is due to the fact that it is not cut by T s xld v conductors.
The " equivalent" flux to cause the same loss of pressure would,
if it did not link with any of the conductors, therefore be

(equivalent) ~~ C^Pj X ?

Thus

(equivalent)

QAirTJ, f dl
~d?T~) X * dx

0.4 ird,
3s

TJ 8

286 PRINCIPLES OF ELECTRICAL DESIGN

The permeances of the air paths of the component fluxes
\$2 and \$3 can be calculated fairly accurately (See Art. 5, Chap.
II). Let them beP2 and P 3 respectively. Then, if l a is the axial
length of the armature core in centimeters, the total " equivalent "
slot flux in the neutral zone is

(104)

97. Effect of Slot Leakage on Full -load Air-gap Flux. Before
considering a method of drawing the curve of air-gap flux dis-
should be the area of this curve. The area of the required flux
distribution curve is a measure of the total flux per pole in the air
gap, and it would be possible to express this in terms of the open-
circuit flux distribution curve if we knew the e.m.f. that would
have to be developed in the armature windings on the assumption
of all the flux passing from pole face to armature teeth being
actually cut by all the conductors. It is therefore proposed to
determine what may be called the " apparent" developed e.m.f.,
that is to say, the e.m.f. that would be developed in the armature
windings under load conditions if it were not for the fact that
some of the flux in the air gap leaks across from tooth to tooth in
the neutral zone, and is not actually cut by the conductors.

Consider first the condition of zero power factor. The current
then lags 90 degrees behind the e.m.f., and reaches its maximum
value in the conductors situated midway between poles. The
slot leakage flux, and the demagnetizing effect of the armature
winding, will then both have reached their maximum value.

In the vector diagram, Fig. 1 14, let OE t represent the required
terminal voltage if the machine is mesh-connected, or the cor-
responding potential difference per phase winding if the machine
is star-connected. (In a three-phase Y-connected generator OE t

would be /-= times the terminal voltage.) The vector 01, drawn

V 3

90 degrees behind OE t , is the armature current on zero power
factor. The impedance-drop triangle is constructed by drawing
EtP parallel to 01 of such a length as to represent the IR drop
per phase, and PE g at right angles to 01 to represent the reactive
pressure drop per phase in the end connections. OE g is therefore
the voltage actually developed in the slot conductors, because it
contains the component PE g to balance the voltage generated by

AIR-GAP FLUX DISTRIBUTION 287

the cutting of the end flux, and the component EtP to overcome
the ohmic resistance of the windings. Now produce PE g to E a r
so that EgE'g represents the voltage that would be developed by
the slot flux if this were cut by the conductors. OE' g , which
may be called the apparent developed voltage, is then the elec-
tromotive force that would have been developed in the armature
windings if the slot flux had actually entered the core instead of
being diverted from tooth to tooth by the action of the current
in the conductors. It is therefore also a measure of the total
flux passing through the air gap into the armature teeth, and the
magnetizing ampere-turns necessary to produce this flux would,
on open circuit, actually develop this electromotive force in the
armature. Thus when the resultant magnetomotive force in the
magnetic circuit is such that E' a volts would be developed on
open circuit, the terminal voltage under the assumed load con-

(5lot)

I

FIG. 114. Vector diagram of alternator operating at zero power factor.

ditions would be E t . It is usual, when the power factor is zero,
to consider this loss of pressure as equal to the total reactive
drop (E'gP] because, owing to the relative smallness of PE t and
the fact that its direction is such as to have little effect on
the pressure drop, the error introduced by this assumption is
negligible.

The length of the vector E E' g in Fig. 114 can be calculated
from the known slot flux \$ es as given by formula (104) of the
preceding article. If <i> a is the' flux per pole actually cut by the
conductors, the total flux per pole in the air gap under load con-
ditions will be <i> = 3>a + 2< es .

This total flux, if actually cut by the armature conductors,
would generate the electromotive force referred to as the "ap-
parent" developed voltage, and represented by OE' a in Fig. 114.

The flux 2\$ 8 maxwells is the portion of the total air-gap flux
which, under load conditions, is no longer cut by the armature

288 PRINCIPLES OF ELECTRICAL DESIGN

conductors. The average value of the voltage lost per phase
winding is therefore

2 \$ T)N

& (average) = 1 Q8 \X QC\ ' s n sP)

or, since Np = 120/

E (average)

Assuming the sinusoidal wave shape, it is necessary to multiply
by ,- to obtain the r.m.s. value. Thus

M ..

V2X10-

The slot flux in the neutral zone will be a maximum on zero
power factor when the current I s producing it is approximately
equal to the maximum value of the armature current, or to \/2l c >
Inserting this value of I s in formula (104) and substituting in
formula (105) we get

E 8 = 2irf X OA7rT s z n s pl a I c ~ -f P 2 + P 3 X 10~ 8 (106)

This quantity is usually referred to as the reactive voltage drop
per phase due to the slot inductance. It appears as the vector
E' a E g in Fig. 114. .

If it were permissible to assume the alternating quantities
and the flux distribution in the air gap to be sinusoidal, the con-
struction of Fig. 114 might be repeated.for conditions other than
zero power factor. These assumptions involve the idea of a
slot leakage flux diminishing with increasing power factor, the
actual change with varying angle of lag being in accordance with
the sine law. This does not take into account tooth saturation
and distortion of the current wave; but as a practical and ap-
proximate method it is permissible. The vector diagram for any
power-factor angle is then as shown in Fig. 115. Here \I/ is
the angle of lag between the current and the e.m.f. actually
developed in the armature conductors; and cos ^ is the " internal"
power factor. The angle \I/' shows the lag of the current behind
the "apparent" developed voltage, OE' g , and it will be seen that
the combined effect of end flux and slot flux is to reduce this
voltage by an amount approximately equal to PE' g sin \l/'.

98. Method of Determining Position of Armature M.m.f.
Turning again to Fig. 108 (page 277), we are still unable to de-
termine the angle 0, or the displacement (@ + 90) of the

AIR-GAP FLUX DISTRIBUTION 289

maximum armature m.m.f . beyond the center line of the pole, be-
cause the angle \j/ f of Fig. 115 shows merely the lag of the current
behind the apparent developed e.m.f.; but, owing to armature dis-
tortion, the full-load flux distribution curve (from which the voltage
OE'g is derived) will not be symmetrically placed relatively to the
center line of the pole; it will be displaced in the direction of
motion of the conductors, i.e., to the right in Fig. 108. With the
aid of the vector diagram Fig. 115 we can, however, obtain a
value for the angle /3 of Fig. 108 which will enable us to place
the curve of armature m.m.f. in a position relatively to the field
m.m.f. which will be approximately correct for any given power
factor of the external load. The construction is shown in Fig.
116, and since vectors are used, the assumption of sine-wave
functions must still be made. This is where an error is intro-
duced, because the distortion of the flux curves, especially with

IR Drop

FIG. 115. Vector diagram of alternator on lagging power factor.

salie.nt-pole machines, is not actually in accordance with this
simple law; but the final check on the work will be made later
when the flux distribution curves are plotted.

The vectors 01, OE t and OE' g have the same meaning as in
Fig. 115, the component PE' g being the total reactive voltage
drop both of end connections (formula 99) and slot leakage
(formula 106). Draw the vector OM in phase with OE' g to
represent the resultant m.m.f. necessary to overcome air gap
and tooth reluctance when the air-gap flux is such as would
develop OE' g volts per phase in the armature if it were cut by all
the conductors. If we neglect the effect of increased tooth
saturation, this m.m.f. can be expressed as

riTpf

OM = (open circuit SI per pole) X ~^ET

\JEjt

the open-circuit field excitation being calculated as explained in

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 23 of 30)