More on spectrum analysis
Computerized Fourier analysis,
particularly in the form of the FFT algorithm, is a powerful tool
for furthering our understanding of waveforms and their related spectral
components. This same mathematical routine programmed into the SPICE
simulator as the .fourier
option is also programmed into a variety of electronic test instruments
to perform real-time Fourier analysis on measured signals. This section
is devoted to the use of such tools and the analysis of several
different waveforms.
First we have a simple sine wave at a
frequency of 523.25 Hz. This particular frequency value is a "C" pitch
on a piano keyboard, one octave above "middle C". Actually, the signal
measured for this demonstration was created by an electronic keyboard
set to produce the tone of a panflute, the closest instrument "voice" I
could find resembling a perfect sine wave. The plot below was taken from
an oscilloscope display, showing signal amplitude (voltage) over time:
Viewed with an oscilloscope, a sine wave
looks like a wavy curve traced horizontally on the screen. The
horizontal axis of this oscilloscope display is marked with the word
"Time" and an arrow pointing in the direction of time's progression. The
curve itself, of course, represents the cyclic increase and decrease of
voltage over time.
Close observation reveals imperfections
in the sine-wave shape. This, unfortunately, is a result of the specific
equipment used to analyze the waveform. Characteristics like these due
to quirks of the test equipment are technically known as artifacts:
phenomena existing solely because of a peculiarity in the equipment used
to perform the experiment.
If we view this same AC voltage on a
spectrum analyzer, the result is quite different:
As you can see, the horizontal axis of
the display is marked with the word "Frequency," denoting the domain of
this measurement. The single peak on the curve represents the
predominance of a single frequency within the range of frequencies
covered by the width of the display. If the scale of this analyzer
instrument were marked with numbers, you would see that this peak occurs
at 523.25 Hz. The height of the peak represents the signal amplitude
(voltage).
If we mix three different sine-wave tones
together on the electronic keyboard (C-E-G, a C-major chord) and measure
the result, both the oscilloscope display and the spectrum analyzer
display reflect this increased complexity:
The oscilloscope display (time-domain)
shows a waveform with many more peaks and valleys than before, a direct
result of the mixing of these three frequencies. As you will notice,
some of these peaks are higher than the peaks of the original
single-pitch waveform, while others are lower. This is a result of the
three different waveforms alternately reinforcing and canceling each
other as their respective phase shifts change in time.
The spectrum display (frequency-domain)
is much easier to interpret: each pitch is represented by its own peak
on the curve. The difference in height between these three peaks is
another artifact of the test equipment: a consequence of limitations
within the equipment used to generate and analyze these waveforms, and
not a necessary characteristic of the musical chord itself.
As was stated before, the device used to
generate these waveforms is an electronic keyboard: a musical instrument
designed to mimic the tones of many different instruments. The panflute
"voice" was chosen for the first demonstrations because it most closely
resembled a pure sine wave (a single frequency on the spectrum analyzer
display). Other musical instrument "voices" are not as simple as this
one, though. In fact, the unique tone produced by any instrument
is a function of its waveshape (or spectrum of frequencies). For
example, let's view the signal for a trumpet tone:
The fundamental frequency of this tone is
the same as in the first panflute example: 523.25 Hz, one octave above
"middle C." The waveform itself is far from a pure and simple sine-wave
form. Knowing that any repeating, non-sinusoidal waveform is equivalent
to a series of sinusoidal waveforms at different amplitudes and
frequencies, we should expect to see multiple peaks on the spectrum
analyzer display:
Indeed we do! The fundamental frequency
component of 523.25 Hz is represented by the left-most peak, with each
successive harmonic represented as its own peak along the width of the
analyzer screen. The second harmonic is twice the frequency of the
fundamental (1046.5 Hz), the third harmonic three times the fundamental
(1569.75 Hz), and so on. This display only shows the first six
harmonics, but there are many more comprising this complex tone.
Trying a different instrument voice (the
accordion) on the keyboard, we obtain a similarly complex oscilloscope
(time-domain) plot and spectrum analyzer (frequency-domain) display:
Note the differences in relative harmonic
amplitudes (peak heights) on the spectrum displays for trumpet and
accordion. Both instrument tones contain harmonics all the way from 1st
(fundamental) to 6th (and beyond!), but the proportions aren't the same.
Each instrument has a unique harmonic "signature" to its tone. Bear in
mind that all this complexity is in reference to a single note
played with these two instrument "voices." Multiple notes played on an
accordion, for example, would create a much more complex mixture of
frequencies than what is seen here.
The analytical power of the oscilloscope
and spectrum analyzer permit us to derive general rules about waveforms
and their harmonic spectra from real waveform examples. We already know
that any deviation from a pure sine-wave results in the equivalent of a
mixture of multiple sine-wave waveforms at different amplitudes and
frequencies. However, close observation allows us to be more specific
than this. Note, for example, the time- and frequency-domain plots for a
waveform approximating a square wave:
According to the spectrum analysis, this
waveform contains no even harmonics, only odd. Although this
display doesn't show frequencies past the sixth harmonic, the pattern of
odd-only harmonics in descending amplitude continues indefinitely. This
should come as no surprise, as we've already seen with SPICE that a
square wave is comprised of an infinitude of odd harmonics. The trumpet
and accordion tones, however, contained both even and odd
harmonics. This difference in harmonic content is noteworthy. Let's
continue our investigation with an analysis of a triangle wave:
In this waveform there are practically no
even harmonics: the only significant frequency peaks on the spectrum
analyzer display belong to odd-numbered multiples of the fundamental
frequency. Tiny peaks can be seen for the second, fourth, and sixth
harmonics, but this is due to imperfections in this particular triangle
waveshape (once again, artifacts of the test equipment used in this
analysis). A perfect triangle waveshape produces no even harmonics, just
like a perfect square wave. It should be obvious from inspection that
the harmonic spectrum of the triangle wave is not identical to the
spectrum of the square wave: the respective harmonic peaks are of
different heights. However, the two different waveforms are common in
their lack of even harmonics.
Let's examine another waveform, this one
very similar to the triangle wave, except that its rise-time is not the
same as its fall-time. Known as a sawtooth wave, its oscilloscope
plot reveals it to be aptly named:
When the spectrum analysis of this
waveform is plotted, we see a result that is quite different from that
of the regular triangle wave, for this analysis shows the strong
presence of even-numbered harmonics (second and fourth):
The distinction between a waveform having
even harmonics versus no even harmonics resides in the difference
between a triangle waveshape and a sawtooth waveshape. That difference
is symmetry above and below the horizontal centerline of the
wave. A waveform that is symmetrical above and below its centerline (the
shape on both sides mirror each other precisely) will contain no
even-numbered harmonics.
Square waves, triangle waves, and pure
sine waves all exhibit this symmetry, and all are devoid of even
harmonics. Waveforms like the trumpet tone, the accordion tone, and the
sawtooth wave are unsymmetrical around their centerlines and therefore
do contain even harmonics.
This principle of centerline symmetry
should not be confused with symmetry around the zero line. In the
examples shown, the horizontal centerline of the waveform happens to be
zero volts on the time-domain graph, but this has nothing to do with
harmonic content. This rule of harmonic content (even harmonics only
with unsymmetrical waveforms) applies whether or not the waveform is
shifted above or below zero volts with a "DC component." For further
clarification, I will show the same sets of waveforms, shifted with DC
voltage, and note that their harmonic contents are unchanged.
Again, the amount of DC voltage present
in a waveform has nothing to do with that waveform's harmonic frequency
content.
Why is this harmonic rule-of-thumb an
important rule to know? It can help us comprehend the relationship
between harmonics in AC circuits and specific circuit components. Since
most sources of sine-wave distortion in AC power circuits tend to be
symmetrical, even-numbered harmonics are rarely seen in those
applications. This is good to know if you're a power system designer and
are planning ahead for harmonic reduction: you only have to concern
yourself with mitigating the odd harmonic frequencies, even harmonics
being practically nonexistent. Also, if you happen to measure even
harmonics in an AC circuit with a spectrum analyzer or frequency meter,
you know that something in that circuit must be unsymmetrically
distorting the sine-wave voltage or current, and that clue may be
helpful in locating the source of a problem (look for components or
conditions more likely to distort one half-cycle of the AC waveform more
than the other).
Now that we have this rule to guide our
interpretation of nonsinusoidal waveforms, it makes more sense that a
waveform like that produced by a rectifier circuit should contain such
strong even harmonics, there being no symmetry at all above and below
center.
- REVIEW:
- Waveforms that are symmetrical above
and below their horizontal centerlines contain no even-numbered
harmonics.
- The amount of DC "bias" voltage
present (a waveform's "DC component") has no impact on that wave's
harmonic frequency content.
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