Alfred Still.

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Font size Arts. 92 and 93. Now draw OM a exactly 90 degrees behind 01,
to represent the maximum value of the armature m.m.f. (formula

19

290 PRINCIPLES OF ELECTRICAL DESIGN

100). This must be balanced by the field component MM ,
giving OM as the required field excitation at full load. If the
load is now thrown off, the developed voltage will be OE , where
the point E is the intersection of OM and the prolongation of

the line PE' g , because this satisfies the condition ~, = ~Qjnr'

The maximum value of the e.m.f. OE will be generated in the
conductors immediately opposite the center of the pole face.
The required angle of displacement, ft, between center line of
pole and position of conductor carrying the maximum current
may thus be calculated, and the full-load flux curves plotted as
in the case of the D.C. machine, where the displacement of the
curve of armature m.m.f. is determined by the movement of the
brushes. It must not be overlooked that this method is not
strictly accurate, since it is based on assumptions that are rarely
justified in practice.

99. Air-gap Flux Distribution under Load. Having deter-
mined the value of the angle ft (Fig. 108), the curve of resultant

FIG. 116. Vector diagram of alternator m.m.fs.

the ordinates of the field and armature m.m.f. curves. The
procedure is then the same as was followed in the D.C. design
to obtain the load flux curve C (Art. 43, Chap. VII), except that
the drawing of the flux curve B as an intermediate step will
not now be necessary, seeing that the effect of armature distor-
tion and demagnetization has been taken account of in the vector
construction of Fig. 116. The final check is obtained by measur-
ing the area of flux curve C, which must satisfy the condition

Area of full-load flux curve C _ OE' g
Area of open-circuit flux curve A ~ OE t

AIR-GAP FLUX DISTRIBUTION 291

If the approximate value of the field ampere-turns, as given
by the vector OM of Fig. 116, does not produce the proper
amount of flux in the air gap, a correction must be made, and a
new curve of resultant m.m.f. obtained, from which the correct

100. Form of Developed E.m.f. Wave. Having plotted the
curve of air-gap flux distribution for any given condition of
loading, it is an easy matter to obtain a curve of e.m.f. due to the
cutting of the flux by the armature conductors. It may be
argued that it is not quite correct to derive the e.m.f. wave from
the curve of air-gap flux distribution, because the flux actually
cut by each armature conductor at a given instant depends not
only upon the value of the air-gap density, but also on the amount
of the slot leakage flux which is not cut by the conductor. By
referring to Fig. Ill (page 283) it will be seen that, although the
slot leakage appears at first sight to pass between the pole and
the conductor, it actually enters the armature core through the
teeth, and, with the exception of the slot flux in the neutral zone,
it all links with the armature winding. The shape of the e.m.f.
wave is therefore not modified to any great extent by the slot
leakage flux; but, unless the armature current is zero in the con-
ductors passing through the neutral zone, the average value of
the developed voltage must be less than it would be if all the
flux entering the tops of the teeth were cut by the conductors.
This is shown in the diagram, Fig. 115, where QE' g is the " appar-
ent" developed voltage (assuming all the flux lines in the air
gap to be cut), and OE g is the actual developed voltage. It is
unnecessary to introduce refinements with a view to determining
the exact wave shape of the e.m.f. actually developed in the con-
ductors because, by using the flux curve C of air-gap distribution,
the wave shape of the ''apparent" developed e.m.f. is obtained,
and with the aid of equivalent sine-waves (to be explained later)
the terminal voltage can be calculated with sufficient accuracy
for practical purposes. It is important to bear in mind that the
e.m.f. wave-shape obtained at the terminals of a Y-connected
three-phrase generator is not necessarily the same as the wave
shape developed in each phase-winding by the cutting of the
flux in the air gap. This was explained in Art. 71 (page 246),
and in order to obtain the wave-form of e.m.f. at the terminals
of a Y-connected generator, it is necessary to add the corres-
ponding ordinates of two star-voltage waves plotted with a

292

PRINCIPLES OF ELECTRICAL DESIGN

phase displacement of one-third of a pole pitch (i.e. 60 electrical
degrees).

Whatever may be the number and spacing of the slots on the
armature surface, the usual e.m.f. wave shapes can always be
plotted in the manner indicated in Fig. 117 where the full line
curve may be any one of the flux curves previously obtained,
the ordinates of which are a measure of the flux density in the
air space near the surface of the armature. Draw A and B
properly spaced to represent the slots, e.g., two in this illustra-
tion, of one phase of the armature winding. At the instant when
the center line C of this phase winding occupies the position
shown in Fig. 117, the conductors in slot A are moving in a
field of density A A', while the conductors in slot B are moving

rve

FIG. 117. Method of deriving e.m.f. wave from curve of air-gap flux

distribution.

in a field of density BB f . These conductors are all in series, and
the instantaneous value of the voltage per phase winding will be

60 X 10*

where B a average value of flux density, in gausses;

AA' + BB' . . ' .
= - ^ - in this instance.

2i

N = revolutions per minute.
D = armature diameter in centimeters.
l a = armature length (axial) in centimeters.
Z = total number of conductors in series per phase.

This value of e is plotted as CC' to a suitable scale, and the process
is repeated for other positions of the armature slots, thus pro-
ducing the dotted curve 7, representing the electromotive force

AIR-GAP FLUX DISTRIBUTION

293

that would be developed in the windings if all the flux in the air
gap were cut by the conductors in the slots.

The general solution, which includes fractional pitch windings,
is illustrated in Fig. 118. The instantaneous value of the aver-
age flux density for n slots per pole per phase is

Bn =

(a + b + c +

) - (a' + V + c'+

2n

The relative positions of the slots and the center of the coil (P)
may be marked on a separate strip of paper that can be moved
to any desired position under the flux curve; and the instan-

Curve of Volts ( 6 ) Plotted

Relatively to Position of

Center of Armature

A

\

\

\

\

FIG. 118. Flux curve and resulting e.m.f. wave fractional pitch
armature winding.

taneous values of the voltage can then conveniently be plotted
over the point P. For this instantaneous voltage we may write

Average instantaneous
e.m.f. per conductor

= flux cut per centimeter of travel
X centimeters per second X 10~ 8

= (Bah) X v X 10~ 8

where v

60

cm. per second. The instantaneous voltage

per phase is therefore

e = e c X z =
as stated in formula (107).

60 X10 8

294

PRINCIPLES OF ELECTRICAL DESIGN

This step-by-step method of drawing the e.m.f . waves will yield
surprisingly accurate results, with the one exception that the
ripples known as "tooth harmonics" which are generally present
in oscillograph records, will not appear in the graphical work.
The effect of the distributed winding in smoothing out the irregu-
larities of the flux-distribution curve is very clearly shown by the
shape of th.e e.m.f. wave in Fig. 118.

101. Form Factor. The ratio of the r.m.s. or virtual value to
the mean value of an alternating e.m.f. or current is the form
factor. The average ordinate of an irregular wave such as may
be obtained by the process represented in Figs. 117 and 118, is
readily obtained by measuring its area with a planimeter and

FIG. 119. Illustrating calculation FIG. 120. Wave of alternating
of r.m.s. value of variable quantity e.m.f. plotted to polar coordinates,
plotted to polar coordinates.

then dividing this area by the length of the base line, i.e., the pole
pitch. If another curve is plotted by squaring the ordinates of
the original curve, it is merely necessary to take the square root
of the average ordinate of this new curve in order to obtain the
virtual value of the alternating quantity. It will, however,
be more convenient to re-plot the original curve to polar co-
ordinates. The general case of a variable quantity plotted to
polar coordinates is illustrated in Fig. 119, where the radial
distance from the point represents the instantaneous value of
the variable quantity, while time (or distance of travel) is
measured by the angular distance between the vector considered
and the axis OX.

AIR-GAP FLUX DISTRIBUTION 295

If r be the length of the vector, and 8 the angular distance from
the reference axis, we may write

Area of triangle OSP = Y^r X rdB

and the area included between any given angular limits ft and
a is

Area OAB (shaded) == 2^ %r*dO

= % (average value of r 2 ) X 03 )
twice area of curve

whence average value of r 2

Applying this rule to the case of a periodically varying e.m.f.
or current, we have in Fig. 120 a representation of an e.m.f.
wave plotted to polar coordnates. This may be thought of as the
actual e.m.f. wave obtained by the graphical method previously
outlined, but transferred from rectangular to polar coordinates.
The radius vector (moving in a counter-clockwise direction,
covers the complete area of one lobe when it has moved through
an angle of 180 degrees; because, in this diagram, the electrical
degrees are correctly represented by the actual space degrees.'
The angle moved through during the half period is TT radians, and
the virtual value of the alternating e.m.f. is therefore

T-J /2 (area of one lobe)

E '-'- ~ -

The area of the curve is easily measured with a planimeter,
and the value of E thus obtained has merely to be divided by the
previously calculated average value in order to obtain the form
factor of the irregular wave.

102. Equivalent Sine -waves. Equivalent sine-waves are a
great convenience in power calculations because they permit the
use of vectors, and enable us to express the power factor as the
cosine of a definite angle. Whenever vector diagrams are used,
the alternating quantities must be sine functions of time; and
when applied to practical calculations involving irregular (i.e.,
non-sine) wave shapes, they must be thought of as representing
"equivalent" sine functions. It will, of cour e be understood
that, in many cases, the substitution of a sine curve for the actual
wave form is not permissible; the effects, for instance, of the
higher harmonics on a condensive load cannot be annulled by

296

PRINCIPLES OF ELECTRICAL DESIGN

imagining the actual wave to be replaced by a so-called equiva-
lent smooth wave; but the use of equivalent sine- waves for
power calculations on practical A.C. circuits, can generally be
justified. An equivalent sine- wave may be defined as a sine-
wave of the same periodicity and the same virtual value as the
irregular wave which it is supposed to replace. An equivalent
sine-wave of current would produce the same heating effects as
the irregular wave; but its mean value, and therefore its form
factor, may be different.

A sine-wave plotted to polar coordinates will be a circle, of
which the diameter representing the maximum value of the
sine-wave is easily calculated since the equivalent wave must

FIG. 121. Irregular wave and equivalent sine wave plotted to polar

coordinates.

have the same root-mean-square value as the non-sinusoidal
wave, and therefore also the same area when plotted to polar
coordinates.

Let d = the diameter of the equivalent circle (or maximum

value of the equivalent sine -wave)
and let A = the area of one lobe of the irregular wave plotted to

polar coordinates,
then

whence

d =

(108)

AIR-GAP FLUX DISTRIBUTION

297

The next point to consider is the position of the equivalent
sine-wave of maximum ordinate d, relatively to some particular
value of the irregular wave. It is obvious that neither the
maximum nor the zero value of the two waves must necessarily
coincide; but by so placing the equivalent sine-wave relatively to
the irregular wave that each quarter wave of the one has the
same virtual value as the corresponding quarter wave of the
other, the proper position of the equivalent wave may be
determined. This will be better understood by referring to
Fig. 121.

The irregular wave is plotted to polar coordinates, and its
area measured with the aid of a planimeter. The line OM is then
drawn, dividing this area in two equal parts. This is easily
done with the help of the planimeter, the proper position of the
dividing line being found when the shaded area of the irregular

Actual E.M.F. Wave

Equivalent Sine
Wave

Geometric

Neutral f~

FIG. 122. Curves of Fig. 121 re-plotted to rectangular coordinates.

wave is exactly equal to the unshaded area. It is upon this line
(OM) that the center of the equivalent circle of diameter d (see
formula 108) must be placed. If the irregular wave has been
correctly plotted relatively to some reference axis, such as the
geometrical neutral line, or the pole center, the angle a can be
measured. This angle represents the displacement of the
maximum value of the equivalent wave beyond the pole center,
and when used in connection with the irregular wave, it may be
thought of as the average displacement of the distorted e.m.f.
behind the position of open-circuit e.m.f., which will be sym-
metrically placed about the center line of the pole. This angle a
has the same meaning as the angle M OM of Fig. 116; but it has
now been determined with greater accuracy than could be

298 PRINCIPLES OF ELECTRICAL DESIGN

expected of the vector construction, in which the loss of pressure
due to armature distortion was assumed to be in accordance with
the sine law. Fig. 122 illustrates the same condition as Fig.
121 except that the e.m.f. waves have been re-plotted to rec-
tangular coordinates. The practical application of equivalent
sine -waves in predetermining the regulation of an alternating-
current generator will be taken up in the following chapter, and
again in Chap. XV, when working out a numerical example.

CHAPTER XIV
REGULATION AND EFFICIENCY OF ALTERNATORS

103. The Magnetic Circuit. Except for the fact that the field
magnets usually rotate, the design of the complete magnetic cir-
cuit of an alternating-current generator differs little from that of
a D.C. dynamo. Given the ampere-turns required per pole, and
the voltage of the continuous-current circuit from which the ex-
citing current is obtained (usually about 125 volts), the procedure
for calculating the size of wire required is the same as would be
followed in designing any other shunt coils (see Art. 10, Chap. II
and Art. 58, Chap IX). When estimating the voltage per pole
across the field winding, a suitable allowance must be made for
the pressure absorbed by the rheostat in series with the field
windings. The exact amount of excitation required under any
after the complete magnetic circuit has been designed.

A higher current density may be allowed in the copper of rotat-
ing field coils than in stationary coils against which air is thrown
by the rotation of the armature, because the cooling is more ef-
fective, the difference being especially noticeable at the higher
peripheral speeds. In the absence of reliable data on any par-
ticular type and size of machine, the curve of Fig. 123 may be
used for selecting a suitable cooling coefficient. The cooling sur-
face considered includes, as before, the inside surface near the
pole core and the two ends, in addition to the outside surface of
the coil. It is to be understood that the cooling coefficient ob-
tained from Fig. 123 is approximate only, being an average of
many tests on different sizes and shapes of coils on rotating field
magnets.

In determining the amount by which the pole must project
from the yoke ring, it is well to allow about 1 in. of radial length
of winding space for every 1,500 ampere-turns per pole required
at full load (i.e., estimated maximum excitation). An effort
should be made to keep the radial projection of the poles as small
as possible in order to prevent excessive magnetic leakage. A

299

300

PRINCIPLES OF ELECTRICAL DESIGN

radial length of pole greater than two or two and one-half
times the width of pole core (measured circumferentially) would
be a poor design, 'because the gain in winding space due to increase
of radial length would be largely neutralized by the greater amount
of flux per pole due to leakage.

The required useful flux per pole being known, the flux to be
carried by pole core and yoke may be calculated if the leakage
factor is known. The calculation of the permeance of the air

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3 0.020
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8 0.019

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S 0.016

f ' 015

P.

0.014
^ 0.013

4J

| 0.012
J 0.011
J3 0.010
^ 0.009
0.008

/

/

/

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/

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>

/

/

/

/

/

/

/

/

/

/

/

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1000 2000 3000 4000 5000 6000 7000 8000

Peripheral Speed of (Salient Pole) Rotor- Feet per Minute

FIG. 123. Cooling coefficient for field windings of rotating-field alternators.

paths between poles is tedious and somewhat unsatisfactory. It
seems therefore best, in designs of normal types, to assume a
leakage factor based on measurements made on existing machines.
The leakage coefficient will be low in machines with large pole
pitch, and high in the case of slow-speed engine-driven genera-
tors with a large number of closely spaced poles. The following
approximate values may be used for estimating the flux in poles
and yoke ring.

REGULATION OF ALTERNATORS 301

High values; to be selected when pole pitch is small,
and radial length of pole core great in proportion to

1.32 to 1.42

1.22 to 1.32

width.
Average values; for pole pitch 8 to 12 in., and length

of winding space about equal to width of pole core.
Low values; for large pole pitch and small radial \

i .1 e i ( 1 1O tO 1 . A*

length of pole core.

These leakage coefficients apply to the case of alternators with
field excitation to give approximately normal voltage at terminals
on open circuit.

Provided a reasonably high leakage factor has been used, the
cross-section of the poles and yoke of good dynamo steel may be
calculated for a flux density up to 15,500 gausses. Although
the flux density in the pole core (of uniform cross-section) will
fall off in value as the distance from the yoke ring increases, the
effect of the distributed leakage may be taken care of by calcu-
lating the ampere-turns for the pole core on the assumption that
the total leakage flux is carried by the pole core, but that the
length of the pole is reduced to half its actual value.

Bearing in mind the above-mentioned points, the open-circuit
saturation curve connecting ampere-turns per pole and resulting
terminal voltage can be calculated and plotted exactly as in the
case of a continuous-current dynamo with rotating armature and
stationary poles (see Art. 57, Chap. IX, and item 128 of Art. 63,
Chap. X).

of Chap. XI to the regulation of alternating-current generators,
and it was pointed out that the designer does not always aim at
producing a machine with a high percentage inherent regulation,
because it is recognized that automatic field regulation or some
equivalent means of varying the field ampere-turns is necessary
in order to maintain the proper terminal voltage under varying
conditions of load and power factor. The large modern units
driven at high speeds by steam turbines are, indeed, purposely
designed to have large armature reactance in order to limit the
short-circuit current, the maximum value of which at the
instant the short-circuit occurs depends rather upon the arma-
ture reactance than upon the demagnetizing or distortional effect
of the armature current. These considerations tend to empha-
size the importance of correctly predetermining the inherent
regulation of machines, and it is important to know exactly
what the term " armature reactance" should include, in order

302

PRINCIPLES OF ELECTRICAL DESIGN

that the probable short-circuit current may be estimated, not
only after the armature ampere-turns have had time to react
upon the exciting field, but also at the instant when the impedance
of the armature windings alone limits the current.

The usual methods of predetermining the regulation of alter-
nators involve almost invariably the use of vectors or vector
algebra. This is convenient, and to some extent helpful, be-
cause the problem is thus presented in its simplest aspect: the
very fact that vectors are used assumes the sinusoidal variation
of the alternating quantities, or the substitution of so-called

equiyalent sine-wave forms for
the actual wave shapes, thus
eliminating the less easily cal-
culated effects caused by cross-
magnetization and the conse-
quent distortion of the wave
shapes. On the other hand, the
omission of these factors, espe-
to incorrect conclusions, and in
any case the plotting of the ac-
tual flux and e.m.f. curves is
of great value to the designer.

Ampere- turns on field H . It is therefore proposed to con-
FIQ. 124. Inherent regulation ob- sider in the first place what is

the best that Can be done with
the aid of vectors on the usual

assumption of sine-wave form, and afterward show how a greater
degree of accuracy can be attained by using curves representing
the actual flux distribution in the air gap, corresponding to the

The curves in Fig. 124 may be considered as having been
plotted from actual test data. The upper curve is the open-
circuit saturation characteristic, giving the relation between the
number of ampere-turns of field excitation per pole and the pres-
sure at the terminals, which in this case is the same as the electro-
motive force actually developed in the armature windings. The
lower curve is the load characteristic corresponding to a given
armature current and a given external power factor. The in-
herent regulation when the field excitation is of the constant

REGULATION OF ALTERNATORS 303

value OP is therefore the difference of terminal voltage E Ei
divided by the load voltage EI, or, expressed as a percentage of
the lower voltage,

OE - OE t
OEi

Thus the error in predetermining the inherent regulation de-
pends upon the degree of accuracy within which curves such as
those shown in Fig. 124 can be drawn before the machine has
been built and tested.

105. Factors Influencing the Inherent Regulation of Alter-
nators. By enumerating all the factors which influence the
terminal voltage of a generator driven at constant speed with
constant field excitation, it will be possible to judge how nearly
the methods about to be considered approximate to the ideal
solution of the problem. These factors are:

(a) The total or resultant flux actually cut by the armature
windings (this involves the flux linkages producing armature
reactance).

(6) The ohmic resistance of the armature windings.

(c) The alteration in wave shape of the generated electro-
motive force, due to changes in air-gap flux distribution. This
means that the measured terminal voltage depends not only
upon the amount of flux cut by the conductors but also upon the
distribution of flux over the pole pitch, because the amount of
flux cut.determines the average value of the developed voltage,
while the form of the e.m.f . wave determines the relation between
the mean value and the virtual or r.m.s. value.

By far the most important items are included under (a), and

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