Introduction
In our study of AC circuits thus far, we've
explored circuits powered by a single-frequency sine voltage
waveform. In many applications of electronics, though,
single-frequency signals are the exception rather than the
rule. Quite often we may encounter circuits where multiple
frequencies of voltage coexist simultaneously. Also, circuit
waveforms may be something other than sine-wave shaped, in
which case we call them non-sinusoidal waveforms.
Additionally, we may encounter situations
where DC is mixed with AC: where a waveform is superimposed
on a steady (DC) signal. The result of such a mix is a
signal varying in intensity, but never changing polarity, or
changing polarity asymmetrically (spending more time
positive than negative, for example). Since DC does not
alternate as AC does, its "frequency" is said to be zero,
and any signal containing DC along with a signal of varying
intensity (AC) may be rightly called a mixed-frequency
signal as well. In any of these cases where there is a mix
of frequencies in the same circuit, analysis is more complex
than what we've seen up to this point.
Sometimes mixed-frequency voltage and
current signals are created accidentally. This may be the
result of unintended connections between circuits -- called
coupling -- made possible by stray capacitance and/or
inductance between the conductors of those circuits. A
classic example of coupling phenomenon is seen frequently in
industry where DC signal wiring is placed in close proximity
to AC power wiring. The nearby presence of high AC voltages
and currents may cause "foreign" voltages to be impressed
upon the length of the signal wiring. Stray capacitance
formed by the electrical insulation separating power
conductors from signal conductors may cause voltage (with
respect to earth ground) from the power conductors to be
impressed upon the signal conductors, while stray inductance
formed by parallel runs of wire in conduit may cause current
from the power conductors to electromagnetically induce
voltage along the signal conductors. The result is a mix of
DC and AC at the signal load. The following schematic shows
how an AC "noise" source may "couple" to a DC circuit
through mutual inductance (Mstray) and
capacitance (Cstray) along the length of the
conductors.
When stray AC voltages from a "noise" source
mix with DC signals conducted along signal wiring, the
results are usually undesirable. For this reason, power
wiring and low-level signal wiring should always be
routed through separated, dedicated metal conduit, and
signals should be conducted via 2-conductor "twisted pair"
cable rather than through a single wire and ground
connection:
The grounded cable shield -- a wire braid or
metal foil wrapped around the two insulated conductors --
isolates both conductors from electrostatic (capacitive)
coupling by blocking any external electric fields, while the
parallal proximity of the two conductors effectively cancels
any electromagnetic (mutually inductive) coupling because
any induced noise voltage will be approximately equal in
magnitude and opposite in phase along both conductors,
canceling each other at the receiving end for a net
(differential) noise voltage of almost zero. Polarity marks
placed near each inductive portion of signal conductor
length shows how the induced voltages are phased in such a
way as to cancel one another.
Coupling may also occur between two sets of
conductors carrying AC signals, in which case both signals
may become "mixed" with each other:
Coupling is but one example of how signals
of different frequencies may become mixed. Whether it be AC
mixed with DC, or two AC signals mixing with each other,
signal coupling via stray inductance and capacitance is
usually accidental and undesired. In other cases,
mixed-frequency signals are the result of intentional design
or they may be an intrinsic quality of a signal. It is
generally quite easy to create mixed-frequency signal
sources. Perhaps the easiest way is to simply connect
voltage sources in series:
Some computer communications networks
operate on the principle of superimposing high-frequency
voltage signals along 60 Hz power-line conductors, so as to
convey computer data along existing lengths of power
cabling. This technique has been used for years in electric
power distribution networks to communicate load data along
high-voltage power lines. Certainly these are examples of
mixed-frequency AC voltages, under conditions that are
deliberately established.
In some cases, mixed-frequency signals may
be produced by a single voltage source. Such is the case
with microphones, which convert audio-frequency air pressure
waves into corresponding voltage waveforms. The particular
mix of frequencies in the voltage signal output by the
microphone is dependent on the sound being reproduced. If
the sound waves consist of a single, pure note or tone, the
voltage waveform will likewise be a sine wave at a single
frequency. If the sound wave is a chord or other harmony of
several notes, the resulting voltage waveform produced by
the microphone will consist of those frequencies mixed
together. Very few natural sounds consist of single, pure
sine wave vibrations but rather are a mix of different
frequency vibrations at different amplitudes.
Musical chords are produced by
blending one frequency with other frequencies of particular
fractional multiples of the first. However, investigating a
little further, we find that even a single piano note
(produced by a plucked string) consists of one predominant
frequency mixed with several other frequencies, each
frequency a whole-number multiple of the first (called
harmonics, while the first frequency is called the
fundamental). An illustration of these terms is shown
below with a fundamental frequency of 1000 Hz (an arbitrary
figure chosen for this example), each of the frequency
multiples appropriately labeled:
FOR A "BASE" FREQUENCY OF 1000 Hz:
Frequency (Hz) Term
-------------------------------------------
1000 --------- 1st harmonic, or fundamental
2000 --------- 2nd harmonic
3000 --------- 3rd harmonic
4000 --------- 4th harmonic
5000 --------- 5th harmonic
6000 --------- 6th harmonic
7000 --------- 7th harmonic
ad infinitum
Sometimes the term "overtone" is used to
describe the a harmonic frequency produced by a musical
instrument. The "first" overtone is the first harmonic
frequency greater than the fundamental. If we had an
instrument producing the entire range of harmonic
frequencies shown in the table above, the first overtone
would be 2000 Hz (the 2nd harmonic), while the second
overtone would be 3000 Hz (the 3rd harmonic), etc. However,
this application of the term "overtone" is specific to
particular instruments.
It so happens that certain instruments are
incapable of producing certain types of harmonic
frequencies. For example, an instrument made from a tube
that is open on one end and closed on the other (such as a
bottle, which produces sound when air is blown across the
opening) is incapable of producing even-numbered harmonics.
Such an instrument set up to produce a fundamental frequency
of 1000 Hz would also produce frequencies of 3000 Hz, 5000
Hz, 7000 Hz, etc, but would not produce 2000 Hz, 4000
Hz, 6000 Hz, or any other even-multiple frequencies of the
fundamental. As such, we would say that the first overtone
(the first frequency greater than the fundamental) in such
an instrument would be 3000 Hz (the 3rd harmonic), while the
second overtone would be 5000 Hz (the 5th harmonic), and so
on.
A pure sine wave (single frequency), being
entirely devoid of any harmonics, sounds very "flat" and
"featureless" to the human ear. Most musical instruments are
incapable of producing sounds this simple. What gives each
instrument its distinctive tone is the same phenomenon that
gives each person a distinctive voice: the unique blending
of harmonic waveforms with each fundamental note, described
by the physics of motion for each unique object producing
the sound.
Brass instruments do not possess the same
"harmonic content" as woodwind instruments, and neither
produce the same harmonic content as stringed instruments. A
distinctive blend of frequencies is what gives a musical
instrument its characteristic tone. As anyone who has played
guitar can tell you, steel strings have a different sound
than nylon strings. Also, the tone produced by a guitar
string changes depending on where along its length it is
plucked. These differences in tone, as well, are a result of
different harmonic content produced by differences in the
mechanical vibrations of an instrument's parts. All these
instruments produce harmonic frequencies (whole-number
multiples of the fundamental frequency) when a single note
is played, but the relative amplitudes of those harmonic
frequencies are different for different instruments. In
musical terms, the measure of a tone's harmonic content is
called timbre or color.
Musical tones become even more complex when
the resonating element of an instrument is a two-dimensional
surface rather than a one-dimensional string. Instruments
based on the vibration of a string (guitar, piano, banjo,
lute, dulcimer, etc.) or of a column of air in a tube
(trumpet, flute, clarinet, tuba, pipe organ, etc.) tend to
produce sounds composed of a single frequency (the
"fundamental") and a mix of harmonics. Instruments based on
the vibration of a flat plate (steel drums, and some types
of bells), however, produce a much broader range of
frequencies, not limited to whole-number multiples of the
fundamental. The result is a distinctive tone that some
people find acoustically offensive.
As you can see, music provides a rich field
of study for mixed frequencies and their effects. Later
sections of this chapter will refer to musical instruments
as sources of waveforms for analysis in more detail.
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REVIEW:
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A sinusoidal waveform is one shaped
exactly like a sine wave.
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A non-sinusoidal waveform can be
anything from a distorted sine-wave shape to something
completely different like a square wave.
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Mixed-frequency waveforms can be
accidently created, purposely created, or simply exist out
of necessity. Most musical tones, for instance, are not
composed of a single frequency sine-wave, but are rich
blends of different frequencies.
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When multiple sine waveforms are mixed
together (as is often the case in music), the lowest
frequency sine-wave is called the fundamental, and
the other sine-waves whose frequencies are whole-number
multiples of the fundamental wave are called harmonics.
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An overtone is a harmonic produced
by a particular device. The "first" overtone is the first
frequency greater than the fundamental, while the "second"
overtone is the next greater frequency produced.
Successive overtones may or may not correspond to
incremental harmonics, depending on the device producing
the mixed frequencies. Some devices and systems do not
permit the establishment of certain harmonics, and so
their overtones would only include some (not all) harmonic
frequencies.
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