Measurements of AC magnitude
So far we know that AC voltage alternates in
polarity and AC current alternates in direction. We also
know that AC can alternate in a variety of different ways,
and by tracing the alternation over time we can plot it as a
"waveform." We can measure the rate of alternation by
measuring the time it takes for a wave to evolve before it
repeats itself (the "period"), and express this as cycles
per unit time, or "frequency." In music, frequency is the
same as pitch, which is the essential property
distinguishing one note from another.
However, we encounter a measurement problem
if we try to express how large or small an AC quantity is.
With DC, where quantities of voltage and current are
generally stable, we have little trouble expressing how much
voltage or current we have in any part of a circuit. But how
do you grant a single measurement of magnitude to something
that is constantly changing?
One way to express the intensity, or
magnitude (also called the amplitude), of an AC
quantity is to measure its peak height on a waveform graph.
This is known as the peak or crest value of an
AC waveform:
Another way is to measure the total height
between opposite peaks. This is known as the peak-to-peak
(P-P) value of an AC waveform:
Unfortunately, either one of these
expressions of waveform amplitude can be misleading when
comparing two different types of waves. For example, a
square wave peaking at 10 volts is obviously a greater
amount of voltage for a greater amount of time than a
triangle wave peaking at 10 volts. The effects of these two
AC voltages powering a load would be quite different:
One way of expressing the amplitude of
different waveshapes in a more equivalent fashion is to
mathematically average the values of all the points on a
waveform's graph to a single, aggregate number. This
amplitude measure is known simply as the average
value of the waveform. If we average all the points on the
waveform algebraically (that is, to consider their sign,
either positive or negative), the average value for most
waveforms is technically zero, because all the positive
points cancel out all the negative points over a full cycle:
This, of course, will be true for any
waveform having equal-area portions above and below the
"zero" line of a plot. However, as a practical
measure of a waveform's aggregate value, "average" is
usually defined as the mathematical mean of all the points'
absolute values over a cycle. In other words, we
calculate the practical average value of the waveform by
considering all points on the wave as positive quantities,
as if the waveform looked like this:
Polarity-insensitive mechanical meter
movements (meters designed to respond equally to the
positive and negative half-cycles of an alternating voltage
or current) register in proportion to the waveform's
(practical) average value, because the inertia of the
pointer against the tension of the spring naturally averages
the force produced by the varying voltage/current values
over time. Conversely, polarity-sensitive meter movements
vibrate uselessly if exposed to AC voltage or current, their
needles oscillating rapidly about the zero mark, indicating
the true (algebraic) average value of zero for a symmetrical
waveform. When the "average" value of a waveform is
referenced in this text, it will be assumed that the
"practical" definition of average is intended unless
otherwise specified.
Another method of deriving an aggregate
value for waveform amplitude is based on the waveform's
ability to do useful work when applied to a load resistance.
Unfortunately, an AC measurement based on work performed by
a waveform is not the same as that waveform's "average"
value, because the power dissipated by a given load
(work performed per unit time) is not directly proportional
to the magnitude of either the voltage or current impressed
upon it. Rather, power is proportional to the square
of the voltage or current applied to a resistance (P = E2/R,
and P = I2R). Although the mathematics of such an
amplitude measurement might not be straightforward, the
utility of it is.
Consider a bandsaw and a jigsaw, two pieces
of modern woodworking equipment. Both types of saws cut with
a thin, toothed, motor-powered metal blade to cut wood. But
while the bandsaw uses a continuous motion of the blade to
cut, the jigsaw uses a back-and-forth motion. The comparison
of alternating current (AC) to direct current (DC) may be
likened to the comparison of these two saw types:
The problem of trying to describe the
changing quantities of AC voltage or current in a single,
aggregate measurement is also present in this saw analogy:
how might we express the speed of a jigsaw blade? A bandsaw
blade moves with a constant speed, similar to the way DC
voltage pushes or DC current moves with a constant
magnitude. A jigsaw blade, on the other hand, moves back and
forth, its blade speed constantly changing. What is more,
the back-and-forth motion of any two jigsaws may not be of
the same type, depending on the mechanical design of the
saws. One jigsaw might move its blade with a sine-wave
motion, while another with a triangle-wave motion. To rate a
jigsaw based on its peak blade speed would be quite
misleading when comparing one jigsaw to another (or a jigsaw
with a bandsaw!). Despite the fact that these different saws
move their blades in different manners, they are equal in
one respect: they all cut wood, and a quantitative
comparison of this common function can serve as a common
basis for which to rate blade speed.
Picture a jigsaw and bandsaw side-by-side,
equipped with identical blades (same tooth pitch, angle,
etc.), equally capable of cutting the same thickness of the
same type of wood at the same rate. We might say that the
two saws were equivalent or equal in their cutting capacity.
Might this comparison be used to assign a "bandsaw
equivalent" blade speed to the jigsaw's back-and-forth blade
motion; to relate the wood-cutting effectiveness of one to
the other? This is the general idea used to assign a "DC
equivalent" measurement to any AC voltage or current:
whatever magnitude of DC voltage or current would produce
the same amount of heat energy dissipation through an equal
resistance:
In the two circuits above, we have the same
amount of load resistance (2 Ω) dissipating the same amount
of power in the form of heat (50 watts), one powered by AC
and the other by DC. Because the AC voltage source pictured
above is equivalent (in terms of power delivered to a load)
to a 10 volt DC battery, we would call this a "10 volt" AC
source. More specifically, we would denote its voltage value
as being 10 volts RMS. The qualifier "RMS" stands for
Root Mean Square, the algorithm used to obtain the DC
equivalent value from points on a graph (essentially, the
procedure consists of squaring all the positive and negative
points on a waveform graph, averaging those squared values,
then taking the square root of that average to obtain the
final answer). Sometimes the alternative terms equivalent
or DC equivalent are used instead of "RMS," but the
quantity and principle are both the same.
RMS amplitude measurement is the best way to
relate AC quantities to DC quantities, or other AC
quantities of differing waveform shapes, when dealing with
measurements of electric power. For other considerations,
peak or peak-to-peak measurements may be the best to employ.
For instance, when determining the proper size of wire (ampacity)
to conduct electric power from a source to a load, RMS
current measurement is the best to use, because the
principal concern with current is overheating of the wire,
which is a function of power dissipation caused by current
through the resistance of the wire. However, when rating
insulators for service in high-voltage AC applications, peak
voltage measurements are the most appropriate, because the
principal concern here is insulator "flashover" caused by
brief spikes of voltage, irrespective of time.
Peak and peak-to-peak measurements are best
performed with an oscilloscope, which can capture the crests
of the waveform with a high degree of accuracy due to the
fast action of the cathode-ray-tube in response to changes
in voltage. For RMS measurements, analog meter movements (D'Arsonval,
Weston, iron vane, electrodynamometer) will work so long as
they have been calibrated in RMS figures. Because the
mechanical inertia and dampening effects of an
electromechanical meter movement makes the deflection of the
needle naturally proportional to the average value of
the AC, not the true RMS value, analog meters must be
specifically calibrated (or mis-calibrated, depending on how
you look at it) to indicate voltage or current in RMS units.
The accuracy of this calibration depends on an assumed
waveshape, usually a sine wave.
Electronic meters specifically designed for
RMS measurement are best for the task. Some instrument
manufacturers have designed ingenious methods for
determining the RMS value of any waveform. One such
manufacturer produces "True-RMS" meters with a tiny
resistive heating element powered by a voltage proportional
to that being measured. The heating effect of that
resistance element is measured thermally to give a true RMS
value with no mathematical calculations whatsoever, just the
laws of physics in action in fulfillment of the definition
of RMS. The accuracy of this type of RMS measurement is
independent of waveshape.
For "pure" waveforms, simple conversion
coefficients exist for equating Peak, Peak-to-Peak, Average
(practical, not algebraic), and RMS measurements to one
another:
In addition to RMS, average, peak (crest),
and peak-to-peak measures of an AC waveform, there are
ratios expressing the proportionality between some of these
fundamental measurements. The crest factor of an AC
waveform, for instance, is the ratio of its peak (crest)
value divided by its RMS value. The form factor of an
AC waveform is the ratio of its peak value divided by its
average value. Square-shaped waveforms always have crest and
form factors equal to 1, since the peak is the same as the
RMS and average values. Sinusoidal waveforms have crest
factors of 1.414 (the square root of 2) and form factors of
1.571 (π/2). Triangle- and sawtooth-shaped waveforms have
crest values of 1.732 (the square root of 3) and form
factors of 2.
Bear in mind that the conversion constants
shown here for peak, RMS, and average amplitudes of sine
waves, square waves, and triangle waves hold true only for
pure forms of these waveshapes. The RMS and average
values of distorted waveshapes are not related by the same
ratios:
This is a very important concept to
understand when using an analog meter movement to measure AC
voltage or current. An analog movement, calibrated to
indicate sine-wave RMS amplitude, will only be accurate when
measuring pure sine waves. If the waveform of the voltage or
current being measured is anything but a pure sine wave, the
indication given by the meter will not be the true RMS value
of the waveform, because the degree of needle deflection in
an analog meter movement is proportional to the average
value of the waveform, not the RMS. RMS meter calibration is
obtained by "skewing" the span of the meter so that it
displays a small multiple of the average value, which will
be equal to be the RMS value for a particular waveshape and
a particular waveshape only.
Since the sine-wave shape is most common in
electrical measurements, it is the waveshape assumed for
analog meter calibration, and the small multiple used in the
calibration of the meter is 1.1107 (the form factor π/2
divided by the crest factor 1.414: the ratio of RMS divided
by average for a sinusoidal waveform). Any waveshape other
than a pure sine wave will have a different ratio of RMS and
average values, and thus a meter calibrated for sine-wave
voltage or current will not indicate true RMS when reading a
non-sinusoidal wave. Bear in mind that this limitation
applies only to simple, analog AC meters not employing
"True-RMS" technology.
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REVIEW:
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The amplitude of an AC waveform is
its height as depicted on a graph over time. An amplitude
measurement can take the form of peak, peak-to-peak,
average, or RMS quantity.
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Peak amplitude is the height of an
AC waveform as measured from the zero mark to the highest
positive or lowest negative point on a graph. Also known
as the crest amplitude of a wave.
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Peak-to-peak amplitude is the total
height of an AC waveform as measured from maximum positive
to maximum negative peaks on a graph. Often abbreviated as
"P-P".
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Average amplitude is the
mathematical "mean" of all a waveform's points over the
period of one cycle. Technically, the average amplitude of
any waveform with equal-area portions above and below the
"zero" line on a graph is zero. However, as a practical
measure of amplitude, a waveform's average value is often
calculated as the mathematical mean of all the points'
absolute values (taking all the negative values and
considering them as positive). For a sine wave, the
average value so calculated is approximately 0.637 of its
peak value.
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"RMS" stands for Root Mean Square,
and is a way of expressing an AC quantity of voltage or
current in terms functionally equivalent to DC. For
example, 10 volts AC RMS is the amount of voltage that
would produce the same amount of heat dissipation across a
resistor of given value as a 10 volt DC power supply. Also
known as the "equivalent" or "DC equivalent" value of an
AC voltage or current. For a sine wave, the RMS value is
approximately 0.707 of its peak value.
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The crest factor of an AC waveform
is the ratio of its peak (crest) to its RMS value.
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The form factor of an AC waveform
is the ratio of its peak (crest) value to its average
value.
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Analog, electromechanical meter movements
respond proportionally to the average value of an
AC voltage or current. When RMS indication is desired, the
meter's calibration must be "skewed" accordingly. This
means that the accuracy of an electromechanical meter's
RMS indication is dependent on the purity of the waveform:
whether it is the exact same waveshape as the waveform
used in calibrating.
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