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.
- REVIEW:
- 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.
- 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.
- 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".
- 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.
- "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.
- The crest factor of an AC
waveform is the ratio of its peak (crest) to its RMS value.
- The form factor of an AC
waveform is the ratio of its peak (crest) value to its average value.
- 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.
|
