Circuit effects
The principle of non-sinusoidal,
repeating waveforms being equivalent to a series of sine waves at
different frequencies is a fundamental property of waves in general and
it has great practical import in the study of AC circuits. It means that
any time we have a waveform that isn't perfectly sine-wave-shaped, the
circuit in question will react as though it's having an array of
different frequency voltages imposed on it at once.
When an AC circuit is subjected to a
source voltage consisting of a mixture of frequencies, the components in
that circuit respond to each constituent frequency in a different way.
Any reactive component such as a capacitor or an inductor will
simultaneously present a unique amount of impedance to each and every
frequency present in a circuit. Thankfully, the analysis of such
circuits is made relatively easy by applying the Superposition
Theorem, regarding the multiple-frequency source as a set of
single-frequency voltage sources connected in series, and analyzing the
circuit for one source at a time, summing the results at the end to
determine the aggregate total:
Analyzing circuit for 60 Hz source alone:
Analyzing the circuit for 90 Hz source
alone:
Superimposing the voltage drops across R
and C, we get:
Because the two voltages across each
component are at different frequencies, we cannot consolidate them into
a single voltage figure as we could if we were adding together two
voltages of different amplitude and/or phase angle at the same
frequency. Complex number notation give us the ability to represent
waveform amplitude (polar magnitude) and phase angle (polar angle), but
not frequency.
What we can tell from this application of
the superposition theorem is that there will be a greater 60 Hz voltage
dropped across the capacitor than a 90 Hz voltage. Just the opposite is
true for the resistor's voltage drop. This is worthy to note, especially
in light of the fact that the two source voltages are equal. It is this
kind of unequal circuit response to signals of differing frequency that
will be our specific focus in the next chapter.
We can also apply the superposition
theorem to the analysis of a circuit powered by a non-sinusoidal
voltage, such as a square wave. If we know the Fourier series (multiple
sine/cosine wave equivalent) of that wave, we can regard it as
originating from a series-connected string of multiple sinusoidal
voltage sources at the appropriate amplitudes, frequencies, and phase
shifts. Needless to say, this can be a laborious task for some waveforms
(an accurate square-wave Fourier Series is considered to be expressed
out to the ninth harmonic, or five sine waves in all!), but it is
possible. I mention this not to scare you, but to inform you of the
potential complexity lurking behind seemingly simple waveforms. A
real-life circuit will respond just the same to being powered by a
square wave as being powered by an infinite series of sine waves
of odd-multiple frequencies and diminishing amplitudes. This has been
known to translate into unexpected circuit resonances, transformer and
inductor core overheating due to eddy currents, electromagnetic noise
over broad ranges of the frequency spectrum, and the like. Technicians
and engineers need to be made aware of the potential effects of
non-sinusoidal waveforms in reactive circuits.
Harmonics are known to manifest their
effects in the form of electromagnetic radiation as well. Studies have
been performed on the potential hazards of using portable computers
aboard passenger aircraft, citing the fact that computers' high
frequency square-wave "clock" voltage signals are capable of generating
radio waves that could interfere with the operation of the aircraft's
electronic navigation equipment. It's bad enough that typical
microprocessor clock signal frequencies are within the range of aircraft
radio frequency bands, but worse yet is the fact that the harmonic
multiples of those fundamental frequencies span an even larger range,
due to the fact that clock signal voltages are square-wave in shape and
not sine-wave.
Electromagnetic "emissions" of this
nature can be a problem in industrial applications, too, with harmonics
abounding in very large quantities due to (nonlinear) electronic control
of motor and electric furnace power. The fundamental power line
frequency may only be 60 Hz, but those harmonic frequency multiples
theoretically extend into infinitely high frequency ranges. Low
frequency power line voltage and current doesn't radiate into space very
well as electromagnetic energy, but high frequencies do.
Also, capacitive and inductive "coupling"
caused by close-proximity conductors is usually more severe at high
frequencies. Signal wiring nearby power wiring will tend to "pick up"
harmonic interference from the power wiring to a far greater extent than
pure sine-wave interference. This problem can manifest itself in
industry when old motor controls are replaced with new, solid-state
electronic motor controls providing greater energy efficiency. Suddenly
there may be weird electrical noise being impressed upon signal wiring
that never used to be there, because the old controls never generated
harmonics, and those high-frequency harmonic voltages and currents tend
to inductively and capacitively "couple" better to nearby conductors
than any 60 Hz signals from the old controls used to.
- REVIEW:
- Any regular (repeating),
non-sinusoidal waveform is equivalent to a particular series of
sine/cosine waves of different frequencies, phases, and amplitudes,
plus a DC offset voltage if necessary. The mathematical process for
determining the sinusoidal waveform equivalent for any waveform is
called Fourier analysis.
- Multiple-frequency voltage sources can
be simulated for analysis by connecting several single-frequency
voltage sources in series. Analysis of voltages and currents is
accomplished by using the superposition theorem. NOTE: superimposed
voltages and currents of different frequencies cannot be added
together in complex number form, since complex numbers only account
for amplitude and phase shift, not frequency!
- Harmonics can cause problems by
impressing unwanted ("noise") voltage signals upon nearby circuits.
These unwanted signals may come by way of capacitive coupling,
inductive coupling, electromagnetic radiation, or a combination
thereof.
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