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103
13
Alternating
Current
Objectives
After studying this chapter, you will be
able to answer these questions:
1. How does an ac generator differ from
the dc generator?
2. What is a sinusoidal (sine) waveform?
3. What is meant by phase relationships?
4. How can dc current be compared with
ac current?
Important
Words and Terms
The following words and terms are key
concepts in this chapter. Look for them as
you read this chapter.
average value
cycles per second
effective value
frequency
hertz (Hz)
peak-to-peak value
peak value
period
phase displacement
root-mean-square
(rms) value
sine wave
single-phase
generator
sinusoidal waveform
slip ring
three-phase
generator
vector
In the chapter on direct current generators, we learned an alternating current
was generated in the rotating armature.
However, due to the commutator, the
alternating current (ac) appeared as
direct current (dc) in the output of the
generator.
The most common form of electrical
current used is alternating current. It is
used in industry to run motors that operate
machines. It is used in the home for
lighting, heating, cooling, cooking, and
entertainment purposes.
Many forms of electrical power plants
are located throughout the United States. A
power generating plant converts various
forms of energy into electrical energy. A
steam plant burns coal to make steam.
Many power generation facilities create
steam to drive turbines (a fan-like assembly used to catch moving steam), which
then rotates a generator. Steam can be
created by any fossil fuel: coal, oil, and
natural gas.
The energy of falling water is captured
in a hydroelectric plant and changed via
turbines and generators into electrical
energy. Nuclear power plants heat water
104 Electricity
using the heat generated by the fission of
uranium fuel, Figure 13-1. The steam from
the heated water drives a turbine that is
joined to a generator and rotates the
armature.
The ac generator is similar to the
simple dc generator with one exception.
The commutator has been replaced with
slip rings, Figure 13-2.
A slip ring is connected to each end of
the coil. The brushes are in constant sliding
contact with the rings providing a way to
pick up the electrical energy generated in
the rotating armature. Note that slip rings
remain in contact with each coil connection—no switching (commutation) takes
place.
The changing polarity and changing
direction of current in the rotating armature
has already been explained in the unit on
the principles of the generator. The two
important things to remember in the case of
the ac generator is that:
1. The current in an external circuit
connected to the brushes of the
generator is an alternating current
periodically rising and falling
and changing direction.
Figure 13-1.
The Calvert Cliffs nuclear power plant.
(Constellation Nuclear LLC)
Magnetic field
N
S
Brush
Slip rings
Brush
Figure 13-2.
The alternating current generator has slip rings
instead of a commutator.
2. The same rising and falling
and changing of current is
also found in the armature.
Vectors and Sinusoidal
Waveforms
Engineers and technicians have a
convenient way of representing a force. It is
represented by means of a vector. A vector
is simply an arrow. The length of the arrow
represents the magnitude (level or amount)
of the force. The direction that arrow is
pointing is the direction in which the force
is acting. Sometimes this technique, used to
combine variables, is called graphical
mathematics.
The vector can be used to represent the
magnitude of a generated voltage and the
direction in which electromotive force is
acting or causing a current to flow. To
better understand the use of vectors, refer
to Figure 13-3 and the generation of the
sinusoidal or sine waveform.
For example, let’s assume the generator
can produce a voltage of 50 volts. (Scale
Chapter 13
105
Alternating Current
Position 2
50 volts
2
X
X
Z
+
Z
135°
45°
Position 1
Position 3
Base
line
Position 1
1
3
1/8
1 1/4″ = 50 V
3/8
50 volts
4
Position 4
Time for one revolution = 1/60 sec.
Figure 13-3.
The development of a sinusoidal, or sine, waveform.
Position l: 0 Degrees of Rotation. Conductor
not cutting through the magnetic field.
No voltage generated.
Position 2: 90 Degrees of Rotation. Conductor
in center of the magnetic field.
Maximum voltage. Positive polarity
generated.
1/4 inch = 10 volts.) The vector rotates
around point 0 in a counterclockwise direction. This can be compared to the rotating
armature of the generator. Figure 13-4
shows the rotating armature in four positions. Now follow the explanation using
both Figures 13-3 and 13-4.
E
N
0°
N
S
+
0
–
N
–
+
Time
0
180°
–
Position 2
Position 1
Position 3
E
N
270°
S
+
0
90°
Time
E
S
E
S
+
0
–
Position 4
N
Time
360°
S
+
0
Time
–
Position 1
Figure 13-4.
Compare the pictures of the vector and the rotating armature coil of the ac generator.
Time
106 Electricity
Position 3: 180 Degrees of Rotation.
Conductor not cutting through the
magnetic field.
No voltage generated.
Position 4: 270 Degrees of Rotation.
Conductor in center of the magnetic field.
Maximum voltage generated.
Negative polarity generated.
Position 1: 360 Degrees of Rotation
(0 Degrees). Conductor not cutting
through the magnetic field.
No voltage generated.
The positive voltage is a force in one
direction and the negative voltage is a force in
the opposite direction. Current also corresponds to the direction of the force or voltage.
Assuming that the armature is revolving at a speed of 60 revolutions per second,
the elapsed time for one revolution would
be one-sixtieth (1/60) of a second. Refer to
Figure 13-3 again. Note that one complete
waveform occurs in 1/60 of a second.
Therefore, the voltage starting at zero rises
to maximum force in one direction (Position
2) then at the end of 180° rotation returns to
zero, (Position 3). In the next 90° of rotation
the voltage rises to maximum but in the
opposite direction (Position 4). In the last
90° the voltage returns to zero (Position 1).
The waveform in Figure 13-3 is a graph
of the instantaneous voltages and represents the rise and fall of the voltage during
one revolution. Instantaneous values are
found when voltage (or current) levels are
determined at precise instances of the
waveform. Points X and Z have been
plotted. The waveform is the result of a line
drawn through several plotted points. Point
X is taken at 45° revolution and at 1/8 of
the time base distance. Point 2 is taken at
90° revolution and 1/4 of the time base.
Point Z at 135° revolution and 3/8 of the
time base. Many points can be plotted to
get a very accurate curve. The curve is
known as the sinusoidal waveform and
History Hit!
Heinrich Rudolph Hertz (1857–1894)
Hertz was a German physicist who
became a university professor. Much of
his research centered along radio waves.
In fact, he was the first to demonstrate the
existence of such waves. Unfortunately, he
died before Guglielmo Marconi turned
radio wave transmission into global communication. Hertz is, however, honored for
his contribution to electronic communication by having the unit of frequency named
in his honor.
represents an alternating current or voltage.
We simplify the word sinusoidal by calling
the waveform a sine wave.
The term sine wave will give you an
introduction to trigonometry. It shows that
the instantaneous voltage developed at any
point on the sine wave depends upon the
degrees of rotation of the vector.
Frequency
The chain of events described in the
previous examples represents one cycle of
operation. You will notice that from position 4 (270 degrees of rotation), the vector
returns to position 1 (0 degrees of rotation)
and the whole sequence of events repeats
over and over for each revolution of the
vector. The sine wave is a graphic picture of
the instantaneous voltages during one cycle
of generation. If the time duration of one
cycle is 1/60 of a second, then the frequency
of the generated voltage is 60 cycles per
second. The measurement of cycles per
second has been replaced by the term hertz,
abbreviated Hz. One hertz equals one cycle
per second. Electrical power generation in
the United States is based upon the 60 Hz
frequency standard.
Chapter 13
107
Alternating Current
Period
Voltage A
It is important to become familiar with
the time duration of a wave, which is called
its period. The period of a wave is the
inverse of the frequency. A 60 Hz wave has
a period of 1/60 of a second. Therefore an
equation can be written which tells us that:
period (in seconds) =
270°
0°
90°
360°
180°
cannot be considered as equal to its peak
voltage, because the peak is reached at only
one point during the half cycle. One can find
an average value (voltage, current, or power)
represented by a sine wave by the formula:
Eaverage = 0.637 × Epeak
1
frequency (in hertz)
Math Manipulation!
Find the period of a 1000 Hz wave.
Its period would be:
period =
Voltage B
108 Electricity
1
1000 Hz
or 0.001 second (1 ms)
The generator discussed in Chapter 12
would be known as a single-phase
generator. It produces electricity of only
one phase.
It is possible to place a second coil in
the generator that would produce another
voltage. Usually this coil is so spaced that
the second voltage is 90° out of step (out of
phase) with the first voltage. The waveform
output of this generator is shown in Figure
13-5. Voltage A is 90° out of phase with
voltage B, or there is a 90° phase
displacement (difference) between the two
voltages. Such a generator is a two-phase
generator. The output circuits can be connected to use voltage A, voltage B, or both
voltages in series. This type of generator is
not very common.
A third very common type of alternator
or generator is three-phase. A three-phase
generator has coils spaced 120° apart and
three separate voltages are generated. Each
voltage has a phase displacement of 120°.
or
Figure 13-5.
The waveform of a two-phase alternating
current.
The output wave is shown in Figure 13-6.
Voltage B is 120° out of phase with voltage
A, and voltage C is 120° out of phase with
voltage B. The term phase is sometimes
written with the symbol φ. Examples of such
use are: 1φ equals single phase and 3φ equals
three phase.
Average Values
A closer look at the sine wave for
one-half cycle or one alternation discloses
that the voltage is continuously varying. It
starts at zero, rises to a maximum or peak
value, and then returns to zero, Figure 13-7.
The value here is a voltage, but it could be
a voltage, current, or power value. The
overall value of this voltage waveform
A
B
C
A
B
C
Epeak = 1.57 × Eaverage
Math Manipulation!
What is the average value of an alternating voltage that has a peak voltage of
50 volts?
Eaverage = 0.637 × Epeak
Eaverage = 0.637 × 50 V
Eaverage = 31.85 V
Carefully note that the average value is
computed as the average value of one-half
cycle. The average value of a complete
cycle would be zero. Remember, one
complete cycle has an equal amount of
waveform above the zero reference line
(base line) and below it. In this way, the
Effective Values
A more meaningful value of an
alternating current is its effective value. To
determine the effective value, the alternating
current is compared to a direct current. In
other words, if a direct current will produce
a certain power, what equivalent alternating
current will produce the same power or
work performed in an electrical circuit?
If a load, such as a resistor, were connected across an ac circuit, the instantaneous currents between zero and peak
could be taken and the effective power
could be found. As power equals:
P = I 2R
all instantaneous current values could be
squared and added together. Dividing the
sum by the number of current values used
gives the average of the squared currents.
The square root of this value is known as
the root-mean-square (rms) value. Effective
values and rms values are the same thing.
Either term can be used. It is found to be
0.707 times the peak value.
Eeff (rms) = 0.707 × Epeak
or
Epeak = 1.414 × Eeff (rms)
Peak
value
Peaktopeak
value
2 × Epeak = Ep-p
Epeak =
Figure 13-6.
The waveform of a three-phase alternating
current.
positive alternation cancels out the negative
alternation and the net result is zero.
Ep-p
2
Figure 13-7.
A waveform showing the instantaneous peak
voltage (or current) value.
Math Manipulation!
Find the effective value of a 20 ampere
peak current.
Eeff = 0.707 × Epeak
Eeff = 0.707 × 20 A
Chapter 13
109
Alternating Current
110 Electricity
Eeff = 14.14 A
A 20 ampere peak alternating current
has an effective value equal to 14.14 amps
of direct current.
Remember the importance of the rootmean-square (rms), or effective, value of the
voltage or current. These values provide the
same work (heating effect) in a resistive circuit as dc. For this reason, most meters are
designed to provide you rms, or effective,
values for all ac measurements (voltage and
current). These values are used for all home
and most industrial applications of ac.
A summary of these formulas for
average and effective values is as follows:
Eeff = 0.707 × Epeak
Epeak = 1.414 × Eeff
Eavg = 0.637 × Epeak
Epeak = 1.57 × Eavg
Eeff = 1.11 × Eavg
Eavg = 0.9 × Eeff
Peak-to-Peak Values
Another value of an ac wave you
should recognize is its peak-to-peak value.
Study Figure 13-7 and note these values.
The peak-to-peak value of a waveform
always appears on an oscilloscope, which is
one of the basic instruments used in
observing and measuring ac waves.
The oscilloscope, generally referred to
as “the ‘scope,” is a graphical voltmeter.
Basically, the oscilloscope will display
waveforms of both dc and ac over a variable time base. This means the waveform
can be spread out horizontally for very fast
or very slow signals. Additionally, the
waveform displayed can be adjusted
vertically (magnitude) for very small or
very large signals. Not a basic test instrument for the beginner, an oscilloscope is
widely used in phase relationships,
Figure 13-8.
An oscilloscope used in many industrial
applications. (Knight Electronics)
electronic communication circuits, and
other specialized industrial applications.
An oscilloscope is shown in Figure 13-8.
Quiz—Chapter 13
Write your answers to these questions on a
separate sheet of paper. Do not write in this
book.
1. On graph paper, construct a sine wave.
Show the rotating vector and plot
points at 30 degree intervals of rotation.
2. A dc generator uses a commutator, but
the ac generator has ______.
3. A vector represents the ______ and
______ of a force.
4. Frequency of an alternating waveform
is measured in ______.
5. A certain electric current has a frequency of 400 Hz. What is the time
interval or period of one cycle?
6. What is the effective value of a 150
peak voltage?
7. What is the peak voltage of the electricity
in your home that is rated at 120 volts?
8. Meters usually are read in ______ voltage and current values.
9. On graph paper draw a single-phase
wave, a two-phase wave, and a threephase wave.
10. What is the rms of an average voltage
of 90 volts?
Many modern test instruments, like this power analyzer, can
display their results graphically. (Fluke Corp.)