Alfred Still.

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

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Referring again to the type of generator to operate at moderate
speeds, it is not -necessary for the field to rotate, and small units,
especially when the voltage is low, may be built generally on the
same lines as D.C. dynamos, i.e., with rotating armatures and
an external crown of poles. In this case the commutator is re-
placed by two or more slip rings connected to the proper points
on the armature winding. For a three-phase generator three
slip rings are required, and since two rings only are necessary if
the field rotates, the design with stationary armature is the more
common. It should also be observed that the insulation of the
slip rings for the comparatively low voltage of the exciting circuit
offers no difficulty, whereas the insulation of the alternating-
current circuit may have to withstand fairly high pressures.

The field magnet windings of salient pole alternators are gen-
erally similar to those of D.C. dynamos, that is to say, all the
poles are provided with exciting coils. Machines have been
built in the past with windings on alternate poles only; with a
single exciting coil (as in the "MORDEY" flat-coil alternator);
and, again, without any windings on the rotating part. The
latter type is known as the inductor alternator, and the field
winding is then put on the stationary armature rings, thus dis-
pensing with slip rings for the collection of either alternating or
continuous current. Iron projections on the rotating part so
modify the reluctance of the magnetic circuit through the arma-
ture coils that alternating e.m.fs. are generated therein; but these
machines, together with those having single exciting coils, have
the disadvantage that the magnetic leakage is very great and
the design therefore uneconomical.

67. Number of Phases. Whether a machine is to supply
single-phase, two-phase, or three-phase, currents does not ap-
preciably affect the design. The calculations on a machine for
a large number of phases are not more difficult than when the
number of phases is small. The theory of the single-phase gen-
erator is, in fact, somewhat less simple than that of the polyphase
machine. In the succeeding articles it is the three-phase gen-
erator that we shall mainly have in mind, because it is the most
commonly met with, but the points of difference in the electrical
design of single-phase and polyphase generators will be pointed
out as they arise.

Whatever may be the type of machine, or number of poles,
we may consider the armature conductors to be cut by the mag-



netic lines in the manner indicated in Fig. 89. Here we have a
diagrammatic representation of single-phase, two-phase, and
three-phase, windings. In each case, the system of alternate
pole pieces is supposed to move across the armature conductors
in the direction indicated by the arrow. It will be noted that
the conductors of each phase are shown connected up to form a
simple wave winding; but this is only done to simplify the dia-

FIG. 89. Single-phase, two-phase, and three-phase, armature windings.

gram, and it will be readily understood that each coil may con-
tain a number of turns, attention being paid to the manner of
its connection to the succeeding coil, in order that the e.m.fs.
generated in the various coils shall not oppose each other.

The upper diagram shows a single winding, in which an alter-
nating e.m.f. will be generated. In the middle diagram there
are two distinct windings, A and B, so placed on the armature
surface that the complete cycle of e.m.f. variations induced in


A will also be induced in B, but after an interval of time repre-
senting a quarter of a period. This diagram shows the positions
of the poles at the instant when the e.m.f. in A is at its maximum,
while in B it is passing through zero value. From these two
windings we can, therefore, obtain two-phase currents with a
phase displacement of 90 electrical degrees.

In the bottom diagram the arrangement of three windings is
shown, from which three-phase currents can be obtained, with a
phase angle between them of 120 degrees, or one-third of a cycle.
It will be seen that, at the instant corresponding to the relative
positions of coils and poles as indicated on the diagram, the e.m.f.
in A is at its maximum, while in B and C it is of a smaller value
and in the opposite direction.

68. Number of Poles. Frequency. For a given frequency
the number of poles will necessarily depend upon the speed.

Thus p = -T^I where N stands for the speed in revolutions per

minute. Since / is usually either 25 or 60, it follows that N must
be some definite multiple of the number of poles p.

69. Usual Speeds of A.C. Generators. The speed at which a
machine of a given kilowatt output should be driven will depend
upon the prime mover. The speed may be very low, as when
the generator is direct-coupled to a slow-speed steam engine or
low-head waterwheel. Higher speeds are obtained when the
generator is belt-driven or direct-coupled to high-speed steam or
oil engines. Very high speeds are necessary when the generator
is direct-connected to a steam turbine.

For usual speeds the reader is referred to the table on page
81, the values there given being applicable to both D.C. and
A.C. machines. In hydro-electric work the generator is usually
direct-coupled to the waterwheel, the speed of which will be
high in the case of impulse wheels working under a high head.
As an example, a PELTON waterwheel to develop 1,500 hp. under
a head of 1,000 ft. would have a wheel about 5 ft. in diameter,
running at 500 revolutions per minute. This would be suitable
for direct coupling to a six-pole 25-cycle generator.

In the case of steam turbines with a blade velocity of about
5 miles per minute the speeds are always very high. The actual
speed best suited to a given size of unit will depend upon the
make of the turbine, but the following table gives the approxi-
mate range of speeds covered by modern steam turbines.



Output, kilowatts Approximate range of speed,

revolutions per minute

2,000 3,000 to 6,000

5,000 2,000 to 4,000

10,000 1,200 to 2,500

20,000 900 to 1,800

70. E.m.f. Developed in Windings.

Let $ = flux per pole (maxwells).
N = revolutions per minute.
p = number of poles.

The flux cut per revolution is then ^ X p and the flux cut per

second is $p g. The average value of the e.m.f. developed in

each armature conductor must therefore be


E c (mean) = 6Q x 1Q8 volts.

If the space distribution of the flux density over the pole pitch
follows the sine law, the virtual value of the e.m.f. is 1.11 times
the mean value. In other words, the form factor, or ratio

r.m.s. value . TT

= . is ,-. or 1.11. in the case of a sine wave.

mean value 2v2

Concentrated and Distributed Windings. If each coil-side may
be thought of as occupying a very small width on the armature
periphery, and if the coil-sides of each phase winding are spaced
exactly one pole pitch apart, the arrangement would constitute
what is usually referred to as a concentrated winding. With a
winding of this kind, all conductors in series in one phase would
always be similarly situated relatively to the center lines of the
poles, and the curve of induced e.m.f. would necessarily be of
the same shape as the curve of flux distribution over the armature
surface. In practice, a winding with only one slot per pole per
phase would be thought of as a concentrated winding. When
there are two or more slots per pole per phase, the winding is
said to be distributed; and since the conductors of any one
phase cover an appreciable space on the armature periphery,
all the wires that are connected in series will not be moving in
a field of the same density at the same instant of time. Except
in the case of a sine-wave flux distribution, the form factor may
depend largely upon whether the winding is concentrated or
distributed. The wave shape of the developed voltage can
always be determined when the flux distribution is known; but,


in the preliminary stages of a design, it is usual to assume that
the pole shoes are so shaped as to give a sinusoidal distribution
of flux over the armature surface. The calculation of a correct-
ing factor for distributed windings is then very simple. Thus,
if there are two slots per pole per phase in a three-phase machine,
there will be six slots per pole pitch, the angular distance between

them being ~- = 30 electrical degrees. It is therefore merely

necessary to add together, vectorially, two quantities having a
phase displacement of 30 degrees, each representing the e.m.f.
developed in a single conductor. The result, divided by 2, will
be the average voltage per conductor of the distributed winding.
As an example, with three slots per pole per phase, the graphic
construction would be as indicated in Fig. 90 where length
AB = length BC = length CD, and what [may |be called the


distribution factor is k = 040- The value of this distribu-

FIG. 90. Vector construction to determine distribution factor.

tion factor is therefore always either equal to, or less than, unity.
If Z = the total number of inductors in series per phase, the
final formula for the developed voltage is:

E (per phase) = 6Q *^ Qg X form factor (93)

On the sine wave assumption, the form factor is 1.11, and the
formula may, if preferred, be used in the form

E (per phase) = - 1fi8 '- volts (on sine-wave assumption) (94)

Values of k can easily be calculated for any arrangement of
slots and windings. With a full-pitch three-phase winding,
the distribution factor, k, will have the following values:

Number of slots per pole Distribution factor,

per phase k

1 1.0

2 0.966

3 0.960

4 0.958

Infinite. . .0.955



71. Star and Mesh Connections. Consider the armature
winding of an ordinary continuous-current two-pole dynamo.
If we imagine the commutator of such a machine to be entirely
removed, the winding whether the armature be of the drum
or ring type will be continuous, and closed upon itself. If the
armature be revolved between the poles of separately excited
field magnets, there will be no circulating current in the windings,
because the magnetism which passes out of the armature core
induces an e.m.f. in the conductors exactly opposite, but equal

in amount, to that induced by the
entering magnetism.

If we now connect two points of
the winding from the opposite ends
of a diameter to a pair of slip rings,
the machine will be capable of de-
livering an alternating current. If
we provide three slip rings, and
connect them respectively to three
points on the armature winding dis-
tant from each other by 120 de-
grees, the machine will become a
three-phase generator.

In this manner polyphase cur-
rents of any number of phases can
be obtained, and if the windings
and field poles are symmetrically
arranged, there will be no circu-
lating current.

This method of connecting up the various armature circuits
of a polyphase generator is known as the mesh connection. In
the case of three-phase currents it is usually referred to as the
delta connection.

The diagram, Fig. 91, shows the three equidistant tappings
from armature winding to slip rings, required to obtain three-
phase currents. It is evident that the potential difference be-
tween any two of the three rings will be the same, since each
section of the winding has the same number of turns, and occupies
the same amount of space on the periphery of the armature core.
Moreover, the variations in the induced e.m.f. will occur suc-
cessively in the three sections at intervals corresponding to
one-third of a complete period.

FIG. 91. Collection of three-
phase currents from bi-polar ring



The load may be connected across one, two, or three, phases;
but in practice, especially in the case of power circuits, the
three-phase load is usually balanced', i.e., each phase winding of
the machine provides one-third of the total output.

FIG. 92. Diagram of connections for delta-connected three-phase

Fig. 92 is a diagram of connections referring to a delta-
connected three-phase generator, and Fig. 93 is the correspond-
ing vector diagram, showing how the current in the external
circuit may be expressed in terms of the armature current. The
current leaving the terminal
A (Fig. 92) is /i - / 2 , and
since there will be a difference
of 120 electrical degrees be-
tween the currents I\ and 1 2,
the vector construction of Fig.
93 gives 01 as the line cur-
rent. Its value is / = 27 a
cos 30, or x/3/a, where I a is
the current in the armature
conductors. The assumptions
here made are that the load
is balanced and that the cur-
rent variations follow the sim-
ple harmonic law. It is well

FIG. 93. Vector diagram of current
relations in delta-connected three-phase

to bear in mind that vectors and vector calculations can be used
only when the variable quantities follow the sine law; when used
in connection with irregular wave shapes, they must be supposed
to represent the " equivalent" sine function, because under no
other condition can the phase angle have any definite meaning.


Star Connection of Three-phase Armature Windings. If the
starting ends of all the phase windings of a polyphase generator
are" connected to a common junction, or neutral point, the arma-
ture windings are said to be star-connected. In the three-
phase machine this is also referred to as the Y connection.
The outgoing lines being merely a continuation of the phase
windings, it follows that, with a star-connected machine, the
line current is exactly the same as the current in the armature
windings. The voltage between terminals is, however, no longer
the same as the phase voltage, as in the case of the previously
considered mesh-connected machine. Referring to the vector
diagram Fig. 94, we see that the voltage between lines 1 and 2
is EI E 2 , which leads to the relation E = \/3E a where E is

FIG. 94. Vector diagram showing voltage relations in Y-connected three-
phase generator.

the terminal voltage, and E a the phase voltage as measured
between any one terminal and the neutral point.

There is little to be said in regard to the choice of armature
connections in a three-phase generator, except that, for the
higher pressures, the Y connection has the advantage of a lower
voltage per phase winding, and, for heavy current outputs, the
A connection has the advantage of a smaller current per phase

Effect of Star and Delta Connections on Third Harmonic.
There is one difference resulting from the method of connecting
the phase windings of a three-phase generator which should be
mentioned. This has reference to the wave shape of the e.m.f.
The wave shape of the terminal voltage is not necessarily the


same as that of the e.m.f. developed in the armature windings.
Thus, what is known as the third harmonic, and all multiples
of the third harmonic, are absent from the voltage measured
across the terminals of a star-connected three-phase generator.
By the third harmonic is meant a sine wave of three times the
periodicity of the fundamental sine wave, which, when superim-
posed on this fundamental wave, produces distortion of the
wave shape.

A voltmeter placed across the terminals of a star-connected
generator measures the sum of two vector quantities with a
phase difference of 60 degrees (see Fig. 94). Since a phase dis-
placement of 60 degrees of the fundamental wave is equivalent
to a phase displacement of 60 X 3 = 180 degrees of the third
harmonic, it follows that the third harmonics cancel out so far
as their effect on the terminal voltage is concerned. The general
rule is that the nth harmonic and its multiples cannot appear
in the terminal voltage of a star-connected polyphase generator
of n phases. The same arguments apply to the line current of a
raes/i-connected polyphase generator; the nth harmonic of the
current wave can circulate only in the armature windings; it
cannot make its appearance in the current leaving the terminals
of the machine.

72. Power Output of Three-phase Generator. Let E and /
be the line voltage and line current, and let E a and I a stand for
the phase voltage and armature current, respectively; then, in
the A-connected machine,

E = E a

/ = \/3/

while, in the Y-connected machine,

E = V3E a

/ = /a

Assuming unity power factor, we may write:
Output of A-connected machine = 3(E a I a )

= 3EX



and, similarly,


Output of Y-connected machine = 3(E a I a )

= 3-^X7


The total output is, of course, the same in both cases. If the
power factor is not unity, the output, in watts, of the three-
phase generator on a balanced load is:

W = \/3EI cos 6.

where 8 is the angle of lag between terminal e.m.f . of a phase
winding and current in the winding. The quantity cos 6 is the
power factor when both current and e.m.f. waves are sinusoidal.

Since the magnetic circuit of an alternating-current generator
has to be designed for a certain flux to develop a given voltage,
while the copper windings must be of sufficient cross-section to
carry a given current, the size of the machine will depend upon
the product of volts and amperes, and not upon the actual
power output. Alternating-current generators are therefore
rated in kilovolt-amperes (k.v.a.), the actual output, in kilowatts,
being dependent upon the power factor of the external circuit.

73. Usual Voltages. Owing to the absence of the commu-
tator, A.C. machines can be wound for higher voltages than
D.C. machines. Large A.C. generators may be wound to give
as high a pressure as 16,000 volts at the terminals, but it is
rarely economical to develop much above 13,000 volts in the
generator; when higher pressures are required, as for long dis-
tance power transmission, step-up transformers are used. In
this country, a very common terminal voltage for three-phase
generators to be used in connection with step-up transformers,
is either 2,200 or 6,600 volts, the higher voltage being adopted
for the larger outputs, in order to avoid heavy currents in the
machine and between the machine and the primary terminals
of the transformer.

74. Pole Pitch and Pole Arc. Although there can be no
sparking at the sliding contacts, as in D.C. designs, with their
commutation difficulties, the effects of field distortion and demag-
netization are apparent in the voltage regulation of alternating-
current generators. A very large pole pitch, involving as it
does a large number of ampere-conductors per pole, is objec-
tionable, and should be avoided if possible. Where it is un-
avoidable, a large air gap must be provided in order to prevent


the armature m.m.f. overpowering the field excitation; but this
leads to increased cost.

The pole pitch, r, is a function of the peripheral speed and

the frequency, thus:



where p = number of poles, and D = diameter of armature




f =

ipheral veloc
It follows that

where v = peripheral velocity of armature in feet per minute.


which explains why the pole pitch is always large in steam-
turbine-driven alternators.

In 60-cycle machines running at moderate speeds, the pole
pitch usually lies between 6 and 12 in., a pitch of 8 to 10 in.
being very common. The speed of the machine which is
usually a factor in determining the peripheral velocity has
an appreciable influence upon the choice of pole pitch; pole cores
of approximately circular or square section are not always feasible
or economical, and the designer must make some sort of a com-
promise to get the best proportions. The output formula, as
used for determining the proportions of dynamos, is not so
readily applicable to the design of alternators, because the arma-
ture diameter in A.C. machines will be settled largely by con-
siderations of peripheral velocity and pole pitch, the propor-
tions of the pole face being a secondary matter. The value of

k. or ratio - P , 6 5 rr, is therefore determined largely by
armature length'

the limits of peripheral velocity, which may lead to a smaller
diameter and greater axial length than would be strictly eco-
nomical if the weight of copper in the field coils were the only

It is usual, when possible, to limit the armature ampere-turns
per pole to 10,000, which will determine the maximum permis-


sible value of the pole pitch; but this limit must sometimes
be exceeded, as in the case of steam-turbine-driven machines,
in which a pole pitch of 3 to 4 ft. is by no means uncommon.

The pole arc is a smaller portion of the total pitch than in
continuous-current machines; the value of r (i.e., 'the ratio

~ rarely exceeds 0.65. A very common value is 0.6,

while it may frequently be as low as 0.55. The reason for
the smaller circumferential space occupied by the pole face is
partly to avoid excessive magnetic leakage, but mainly to pro-
vide a proper distribution of flux over the pole pitch. An at-
tempt is usually made to obtain sinusoidal distribution; but the
means of obtaining this will be explained later.

75. Specific Loading. As in the case of D.C. machines, the
specific loading, q, is defined as the number of ampere-conductors
per inch of armature periphery. The conductors of all phases
are counted, and the current considered is the virtual, or r.m.s.,
value of the armature current. The magnetizing effect of the
armature as a whole will, at any moment, depend upon the
instantaneous value of the currents in the individual conductors,
but this matter will be taken up later.

The following are average values of q, as found in commercial
machines :

Outpjit of A.C. gener- Average value

ator (k.v.a.) of q

50 400

100 430

200 470

500 520

1,000 570

5,000 . 625

10,000 670

The proper value of q to be used in a given design will depend
on several factors. Apart from the fact that its value increases
with the size of the machine, it will depend somewhat upon the
following factors:

(a) Number of poles.

(6) Frequency.

(c) Voltage.

(a) Machines with a small number of poles usually have a
small armature diameter and a large pole pitch; calling for a
small value of q. In modern steam-turbine-driven generators


there is, however, a tendency to use high values of q in order to
limit the length of the armature (and increase the critical speed)
and also to increase the armature m.m.f. with a view to lowering
the short-circuit current.

(6) With low frequency it is easy to keep the iron loss small,
and more copper, or a greater current density in the conductors,
is permissible.

(c) If the e.m.f. is low, the insulation occupies less space,
and there is more room for copper without unduly reducing the
cross-section of the armature teeth.

The approximate figures given in the above table may be
increased or reduced about 20 per cent., the highest values being
used only when there is a combination of low voltage, low fre-
quency, and large number of poles.

76. Flux Density in Air Gap. Since the pole shoe is shaped
to give as nearly as possible a sinusoidal distribution of flux
density over the pole pitch, it is convenient to think of the
maximum value of the air-gap density, because this will determine
the maximum density in the iron of the teeth. The frequency
being usually higher than in B.C. machines, lower tooth densities
must be used in order to avoid excessive loss. The allowable
flux density in the air gap will depend upon the proportions of
tooth and slot; but the following values may be used for pre-
liminary calculations.

For a frequency of 25, B a = 4,000 to 5,800 gausses.
For a frequency of 60, B = 3,500 to 5,000 gausses.

These values of B g stand for the average density over ihe pole
pitch. If $ = the total number of maxwells per pole; and the
shape of the flux distribution over the pole pitch is assumed to
be a sine wave, we have:

Area of pole pitch = TT

> a

which determines the axial length of the armature core. The
maximum air-gap density, on the above assumption, is ^ B g , and

after deciding upon the tooth and slot proportions, it is advisable
to see that this density will not lead to an unreasonable value

Online LibraryAlfred StillPrinciples of electrical design; d. c. and a. c. generators → online text (page 20 of 30)