Alfred Still.

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

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it will be well to consider exactly how the resultant flux cut by
the armature windings varies when load is put on the machine.

Considering first the flux cut by the active belt of conductors
under the pole face, this is not usually the same under load con-
ditions as on open circuit (the field excitation remaining con-
stant), for the following reasons. The current in the armature
windings produces a magnetizing effect which, together with the
field-pole magnetomotive force, determines the resultant mag-
netomotive force and the actual distribution of the flux in the air
gap. When the power factor of the load is approximately unity,
the armature current produces cross-magnetization and dis-
tortion of the resultant field, accompanied usually by a re-
duction of the total flux owing to increased flux density in the


armature teeth where the air-gap density is greatest. The effect
is, however, less marked in alternating-current than in con-
tinuous-current generators, because in the former the tooth
density is rarely so high as to approach saturation. On low
power factor, with lagging current, the armature magnetomotive
force tends to oppose the field magnetomotive force, and on zero
power factor its effect is wholly demagnetizing, thus greatly re-
ducing the resultant air-gap flux. With a leading current the
well-known effect of an increased flux and a higher voltage is
obtained. The effect known as armature reaction, as distin-
guished from armature reactance, is therefore dependent not
only on the amount of the armature current but also largely
upon the power factor.

The effect of the individual conductors in producing slot
leakage was discussed in Art. 95 of Chap. XIII, and illustrated
by Figs. 110 and 111, wherein it is clearly shown that, as current
is taken out of the armature, the total flux cut by the active
conductors is less than at no load (with the same field excitation)
by the amount of the slot flux or equivalent slot flux which
passes from tooth to tooth in the neutral zone.

Turning now to the flux cut by the end connections, i.e., by
those portions of the armature winding which project beyond
the ends of the slots, this flux is set up almost entirely by the
magnetomotive force of the armature windings, and is negligible
on open circuit. For a given output and power factor, the end
flux in a polyphase generator is fixed in position relatively to the
field poles, being stationary in space if the armature revolves.
The maximum value of the armature magnetomotive force occurs
at the point where the current in the conductors is zero, and on
the assumption of a sinusoidal flux distribution, the electromotive
force generated by the cutting of these end fluxes may be repre-
sented correctly as a vector drawn 90 degrees behind the current
vector. It is therefore permissible to consider this e.m.f. com-
ponent as a reactive voltage such as would be obtained by con-
necting a choking coil in series with the " active" portion of the
armature windings; and if the inductance, L e , of the end windings
is known, and a sinusoidal flux distribution assumed, the electro-
motive force developed by the cutting of the end fluxes under
load conditions is given by the well-known expression 2irfL e I c ,
where I c is the virtual value of the current in the armature wind-
ings, and / is the frequency.


This quantity was calculated in Art. 87, Chap. XII, and ex-
pressed in formula (99), the calculation being based upon an
amount of end flux per pole ($ f ) given by the empirical formula
(98). Although the writer likes to think of the cutting of the
end flux by the conductors projecting beyond the ends of the
slots, the idea of flux-linkages and a coefficient of self-induction,
L f , expressed in henrys, may be preferred by others. If it is de-
sired to substitute the terms of the formulas (98) and (99) in the
expression 2irfL e I c , the value of the coefficient of self-induction,
in henrys, will be

X 10*

106. Regulation on Zero Power Factor. In practice, any
power factor below 20 per cent, is usually considered to be equiva-
lent to zero, so that the calculations can be checked when the
machine is built, by providing as a load for the generator a suit-
able number of induction motors running light. On these low
power factors with lagging current the phase displacement of the
armature current causes the armature magnetomotive force to
be almost wholly demagnetizing, that is to say, it directly opposes
the magnetomotive force due to the field windings, the distor-
tional or cross-magnetizing effect being negligible. Its maximum
value per pole is given by formulas (100) and 101) of Art. 94,
Chap. XIII, and its effect in reducing the flux in the air gap is
readily compensated (on zero power factor) by increasing the
field excitation so that the resultant ampere-turns remain un-
changed. This statement is not strictly correct because the in-
creased ampere-turns on the field poles give rise to a greater leak-
age flux, and this alteration should not be overlooked, especially
when working with high flux densities in the iron of the magnetic
circuit. If the estimated leakage flux for a given developed
voltage on open circuit is fy maxwells, then, for the same voltage
with full-load current on zero power factor, the leakage flux would

be approximately &i = <* (~~~/fr ; where M is the number of

field ampere-turns on open circuit, and (M + M a ) is the number
of field ampere-turns with full-load current in the armature, the
power factor being zero. The quantity M a is the demagnetizing
ampere-turns per pole due to the armature current.

Let curve A of Fig. 125 be the open-circuit saturation curve



of the machine, referred to in Art. 103 ; it is the curve for the com-
plete machine, and can be plotted only after the magnetic circuit
external to the armature has been designed.

Knowing the increase of flux in pole and frame the magneto-
motive force absorbed in overcoming the increased reluctance of
these parts can be calculated, and in this way the dotted curve
A' of Fig. 125 can be drawn. This is merely the open-circuit
saturation curve corrected for increased leakage flux due to the
additional field current required to balance the demagnetizing
effect of a given armature current.

Assuming all the alternating quantities to be simple harmonic

functions, the vector diagram
(Fig. 126) can be drawn as ex-
plained in Art. 96. It shows
the voltage components for one
.phase of the winding; the vector
PE t being the IR drop, while
E P and E' E g stand for the re-
actance voltage drops due to
end flux and slot flux respec-
tively. The numerical value of
E g P can be obtained from for-
mula (99) page 266, and oiE' E g
from formula (106) of page 288.
The apparent developed volt-

FIG. 125. Methods of construct- age E' g obtained from Fig. 126
ing suturation curve for zero power . ,, . , ,, ,,

factor. gives the point M on the cor-

rected no-load saturation curve

A' in Fig. 125, and the distance E' g M or OM' shows what exciting
ampere-turns are required to develop this electromotive force.
The terminal voltage is, however, only E t , which gives the point
N of the triangle MNR. Now draw NR parallel to the horizontal
axis to represent the total number of ampere-turns per pole due
to the^armature current, which, as previously explained, will be
entirely demagnetizing, and must therefore be compensated by
an equal number of ampere-turns on the field pole. Thus E t R
or OR' is the field excitation necessary to produce Et volts at the
terminals of the machine. If the load is now thrown off, the
terminal pressure will rise to E and the percentage regulation
for this particular current output on zero power factor will there-

fore be 100 p p, This simple construction enables the de-


Ampere-turns per pole


signer to predetermine with but little error the regulation on zero
power factor provided he can correctly calculate the reactances
required for the vector quantities of Fig. J.26. The complete
load characteristic O'R is quickly obtained by sliding the triangle
MNR along the corrected no-load saturation curve. 1 The dif-
ference of pressure, SR, corresponding to any particular value
OR' of field excitation (Fig. 125) is called the synchronous react-
ance drop because, although it is made up partly of real reactance
drop and partly of armature reaction, it may conveniently be
treated as if it were due to an equivalent or fictitious reactance
capable of producing the same total loss of pressure if the magneto-
motive force of the armature had no demagnetizing or distortional
effect. Thus, by producing the line PE' g to E in Fig. 126, so
that PE is equal to RS of Fig. 125, the vector diagram shows the
difference between the open-circuit pressure OE and the terminal



FIG. 126. Vector diagram for zero power factor.

pressure OE t under load conditions at zero power factor when the
field excitation is maintained constant. The additional (ficti-
tious) reactance drop E E' g is correctly drawn at right angles to
the current vector because on zero power factor the effect of the
armature magnetomotive force is wholly demagnetizing; in other
words, it tends to set up a magnetic field displaced exactly 90
degrees (electrical space) behind the current producing it; hence
when the load is thrown off, the balancing m.m.f. component on
the field poles will generate the additional voltage in the phase
OE . It should be realized that the fictitious reactance drop,
E E'g of Fig. 126, cannot be predetermined until the whole of the

1 S. H. MORTENSEN. "Regulation of Definite Pole Alternators." Trans.
A. I. E. E., vol. 32, p. 789, 1913. Also B. A. BEHREND. " The Experimental
Basis for the Theory of the Regulation of Alternators." Trans. A. I. E. E.,
vol. 21, p. 497, 1903; and B. T. McCoRMicx in discussion on "A Con-
tribution to the Theory of the Regulation of Alternators" (H. M. HOBART
and F. PUNGA). Ibid., vol. 23, p. 330, 1904.





magnetic circuit of the machine has been designed. The distance
DR in Fig. 125 is the loss of voltage corresponding to E' g P of
Fig. 126, and is approximately constant for a -given armature
current. The portion SD, however, of the total difference of
voltage depends on the slope of the line MS, and is thus some
function of the degree of saturation of the iron in the magnetic
circuit. It is far from being constant (except over the linear part
of the open-circuit saturation curve) and must be measured off
the diagram for each different value of the field excitation. This
diagram (Fig. 125) shows very clearly the advantage of high flux

densities (magnetic saturation) in some
portion of the magnetic circuit, if good
regulation is aimed at.

)4X(slobs) 107. Short-circuit Current. The
amount of the short-circuit current is in-
timately connected with the regulating
qualities of a machine, and in large gen-
erators becomes a matter of importance.
The maximum value of the armature
current at the instant a short-circuit
occurs depends mainly on the induc-
tance of the armature windings; but

when the armature magnetomotive force has had time to react
on the field and has actually reduced the flux of induction in the
air gap, the resulting current may be fairly accurately calculated
by using the construction indicated in Figs. 127 and 128.

The vector triangle Fig. 127 is constructed for any assumed
value, I c , of the armature current. It shows that when the
terminal voltage is zero, the machine being short-circuited, the
flux in the air gap must be such that the pressure OE' g would be
developed in the armature conductors on open circuit. The
value OF (Fig. 128) of the ampere-turns necessary to produce this
flux in the air gap is thus obtained, the ordinate OE' g being the
generated voltage as determined by the vector diagram. Now
since the magnetomotive force of the armature windings will be
almost wholly demagnetizing, it is correct to assume that the field
excitation must be increased by an amount equal to the maxi-
mum armature ampere-turns per pole in order that the resultant
excitation may be OF. Thus FG in Fig. 128 is made equal to the
maximum armature ampere-turns, and by drawing, to a suitable
scale, the ordinate GJ equal to the assumed armature current I c ,



the point / on the short-circuit current curve is obtained. By
repeating the construction for any other assumed value of the
current it will be seen that so long as E' g lies on the linear portion
of the no-load characteristic, the relation between the short-
circuit current and the field-pole excitation is also linear. When
the field excitation is OL, giving a pressure OE on open circuit,
the short-circuit current will be LK.

FIG. 128. Method of constructing curve of armature current on short-circuit.

108. Regulation on any Power Factor. Unless the effects of
cross-magnetization are taken into account, it is impossible to
predetermine the regulation accurately when the power factor
differs appreciably from zero, but by the intelligent use of vector
quantities (involving as they do the assumption of simple
harmonic curves) very satisfactory results can be obtained. The
best method known to the author by which the load saturation
curve for any power factor may be drawn, without resorting to
flux distribution and wave-shape analysis, is that given by
PROFESSOR ALEXANDER GRAY/ and recently embodied in the
Standardization rules of the American Institute of Electrial En-
gineers. A. E. CLAYTON 2 has also suggested a similar method.

*A. GRAY. "Electrical Machine Design."
2 Electrician, vol. 73, p. 90, 1914.



Let the external power factor be cos 6, and OR' (Fig. 129) the
constant field excitation which would, on open circuit, develop
the pressure E represented by R'S. The full-load-current zero-
power-factor saturation curve O'R has been drawn as previously
described. If then it is possible to determine the point Q on
the full-load saturation curve for power factor cos 6, the required

r\ o

percentage regulation may be expressed as 100^,-

In Fig. 130 draw the right-angled triangle E t PE such that
PE t represents the armature resistance drop per phase with full-
load current, and EoP the corresponding synchronous reactance
drop, as given by SR in Fig. 129. From E t draw the line E t m

O 0' R'

FIG. 129. Method of constructing saturation curve for any load and

power factor.

of indefinite length and so that mE t P is the required power-factor
angle 6. From E as center describe the arc of a circle of radius
R f S (Fig. 129) equal to the open-circuit voltage, and cutting
mE t produced at 0. Then OE t will be the required terminal vol-
tage, which may be plotted as R'Q in Fig. 129. This construc-
tion provides for the proper angle 8 between terminal voltage
and current; and in regard to the relation between the terminal
voltage E t and the open-circuit voltage E when load is thrown
off, it will be seen that the total synchronous reactive drop has


been used in the impedance triangle E t PE of I^ig. 130. This
virtually assumes the demagnetizing and distortional effects of
the armature current to be equivalent to a fictitious reactance
drop capable of being treated vectorially like any other reactance
drop, and of which the direct effect on regulation is proportional
to the sine of the angle of lag a not unreasonable assump-
tion, though scientifically inaccurate. This gives good results
in machines of normal design. It is when departures are made
from standard practice that such approximations are liable to be

Armature current

FIG. 130. Vector diagram showing construction to obtain terminal voltage
for any load and power factor.

109. Influence of Flux Distribution on Regulation. So long
as a sinusoidal air-gap flux distribution can be assumed both on
open circuit and under load conditions, the previously described
methods of predetermining regulation are satisfactory; but in the
case of new or abnormal designs of machines, correct results can
be obtained only by taking into account the alteration in the
amount of the useful flux due to cross-magnetization and the
changes in the e.m.f. wave-shapes due to flux distortion. An
attempt will be made to outline as briefly as possible a method of
study which, although it has been elaborated by the writer, is not
essentially new; indeed, it is probably used in a modified form by
some practical designers when aiming at a closer degree of
accuracy than can be expected from methods based on the usual
sine-wave assumptions.

The method about to be described is based on the fact that for
salient-pole machines approximately correct flux-distribution
curves can be drawn when the width and shape of the pole shoe
have been decided upon; and for high-speed generators with air
gap of constant length, when the disposition and windings of the
slots in the rotor have been determined. . From these flux curves,


whether representing open-circuit or loaded conditions, the e.m.f .
waves and their form factors can be obtained, all as explained in
Arts. 100 and 101 of Chap. XIII, and the problem of regulation
may be summed up as follows : Plot the open-circuit saturation
curve for the complete magnetic circuit, correcting for the form
factor of the developed voltage if this departs appreciably from
the assumed value of 1.11. Now obtain the actual flux distribu-
tion and corresponding full-load "apparent" developed voltage for
a given power factor, and correct for the internal pressure losses
ohmic and reactive. Let E t be full-load terminal voltage obtained
by this method. The field excitation for air gap and teeth is
known for the particular condition considered, and the ampere-
turns required to overcome the reluctance of the remaining parts
of the magnetic circuit are also readily ascertained since the
total flux per pole (the area of full-load flux curve C) is known.
It is therefore merely necessary to read off the open-circuit
characteristic the voltage E corresponding to the ascertained
value of the total field excitation in order to determine the

T|T Tjl

regulation, which is *

The actual working out of the problem is not quite so simple
as this statement may suggest, the chief difficulty being that a
knowledge of the external power-factor angle is insufficient to
determine the exact position of the armature m.m.f. curve rela-
tively to the center line of the pole. The position of this curve
depends upon the internal power-factor angle and also upon the
phase displacement of the generated electromotive force under
load conditions, i.e., on the degree of distortion of the resultant
air-gap flux which, on open circuit, was distributed symmetric-
ally about the center line of the pole face. The manner in which
the displacement of the armature m.m.f. curve may be determined
approximately, for any given external power factor, was ex-
plained in Art. 98 and illustrated by the vector diagram, Fig. 116.

110. Outline of Procedure in Calculating Regulation from
Study of E.m.f. Waves. In Fig. 131 let the curve F represent the
distribution of magnetomotive force over the armature surface
tending to send flux from pole to armature on open circuit.
Let BD be the magnetomotive force due to armature current
only. If the load current be sinusoidal (an almost essential as-
sumption, since its exact shape cannot be predetermined), BD
will also be a sine curve, the maximum ordinate CD of which will



be displaced beyond the center line of the pole by an amount
depending upon the power factor of the load and the distortion
of the resulting air-gap flux distribution. This maximum value
will occur where the current in the conductors is zero, and the
maximum armature current will be carried by the conductor
displaced exactly 90 degrees (electrical space) from the point C.
The point B is therefore the position on the armature surface,
considered relatively to the poles, where tlje current is a maxi-
mum, the length AB or 0, which depends largely on the power
factor, being determined approximately as explained in Art. 98.
Add the ordinates of curves F and D to get the curve M which
gives the resultant magnetomotive force under the assumed
conditions of load. Having calculated the permeance of the

FIG. 131. Distribution of m.m.f. over armature surface.

magnetic circuit for various points on the armature surface, the
flux distribution curves A and C of Fig. 132 can be plotted.
The first, which represents open-circuit conditions, is plotted
from the m.m.f. curve F, while curve C, showing the flux distribu-
tion under load, is derived from the m.m.f. curve M. The
respective areas of these curves are a measure of the total air-gap
flux under the two conditions, but we cannot say that the actual
ampere-turns on the field will be the same in both cases, be-
cause the component of the total m.m.f. required to overcome
the reluctance of the pole-core and yoke ring has not been
taken into account.

The correct solution of the problem involves the actual
wave shapes of the developed electromotive forces. Assuming



all the flux in the air gap to be cut by the armature conductors,
the wave shapes of the " apparent" developed e.m.fs. can be
drawn, and their form factors calculated as explained in Arts. 100
and 101. The terminal voltage which must be known before
the regulation can be calculated is most readily obtained by
using vector diagrams; but this involves the substitution of
" equivalent sine curves " for the irregular waves. The maximum
value of a so-called equivalent sine wave is \/2 times the r.m.s.
value of the irregular wave; but its time phase relatively to any
defined instantaneous value of the irregular wave is not so easily
determined. It can be obtained from the irregular curve when
plotted to polar coordinates as explained in Art. 102, Chap. XIII ;

FIG. 132. Flux distribution, (C) under load, and (A ) when load is

thrown off.

but a method to be preferred for purposes of explanation, although
more tedious, consists in obtaining the average value of the true
power and making the displacement between electromotive force

/ true power \ ,

and current vectors equal to cos" 1 I- ). The

\apparent power/

current wave (assumed to be a sine curve) from which the m.m.f .
curve BD of Fig. 131 is derived would have its maximum value
at the point B, displaced /3 electrical degrees beyond the center,
A, of the pole. The actual full-load e.m.f. wave, can also be
drawn in the correct position relatively to the center line of the
pole; and, by multiplying the corresponding instantaneous
values of electromotive force and current, the power curve can be
drawn and the average value of its or dinat es calculated . The ratio


of this quantity to the volt-amperes is equal to the cosine of the
angle \j/' in Fig. 133. This vector diagram can be constructed as
follows :

Draw OE representing the phase of the open-circuit voltage,
i. e., the center of the pole, to be used as a reference line
from which the phase angles can be plotted. Make the angle
E OI C equal to ft of Fig. 131. This is the estimated lag of
current behind the open-circuit electromotive force. Draw
OE' a equal in length to the calculated e.m.f. value of the "ap-
parent" developed voltage under load conditions, and so that \f/ f =

I watts \

cos" 1 1 ) , where the watts referred to are calculated

\ volt-amperes/

by multiplying the corresponding instantaneous values of E' g
and I c . From E' g drop a perpendicular on to 0/ c , and set off

IX ( slots)

FIQ. 133 Vector diagram for determining the inherent regulation of an
alternating-current generator.

E' g E and E g P to represent the reactance drops per phase in slots
and end connections respectively. Draw PE t parallel to OI e to
represent the resistance pressure drop per phase, and join the
point with E g and E t respectively. The angle# ( 0/ c and the
length of the vector OE t may not correspond with the exact
values of external power factor and terminal voltage assumed

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 25 of 30)