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.
- REVIEW:
- A sinusoidal waveform is one
shaped exactly like a sine wave.
- A non-sinusoidal waveform can
be anything from a distorted sine-wave shape to something completely
different like a square wave.
- 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.
- 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.
- 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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