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 di®erent 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: Figure 1.14

Unfortunately, either one of these expressions of waveform amplitude can be misleading when
comparing two di®erent 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 e®ects of these two AC voltages powering a load would be quite di®erent: Figure 1.16

One way of expressing the amplitude of di®erent 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: Figure 1.17

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 de¯ned 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: Figure 1.18

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" de¯nition of average is intended unless otherwise speci¯ed.

Another method of deriving an aggregate value for waveform amplitude is based on the wave-
form's ability to do useful work when applied to a load resistance. Unfortunately, an AC mea-
surement 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: Figure 1.19

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 di®erent saws move their blades

In the two circuits above, we have the same amount of load resistance (2 ­) 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 speci¯cally, we would denote its voltage
value as being 10 volts RMS. The quali¯er \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 ¯nal 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 di®ering 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
\°ashover" 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 ¯gures. Because the mechanical inertia and dampening e®ects of an electromechanical meter
movement makes the de°ection of the needle naturally proportional to the average value of the AC,
not the true RMS value, analog meters must be speci¯cally 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 speci¯cally designed for RMS measurement are best for the task. Some in-
strument 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 ele-
ment powered by a voltage proportional to that being measured. The heating e®ect of that resistance
element is measured thermally to give a true RMS value with no mathematical calculations whatso-
ever, just the laws of physics in action in ful¯llment of the de¯nition of RMS. The accuracy of this
type of RMS measurement is independent of waveshape.

For \pure" waveforms, simple conversion coe±cients exist for equating Peak, Peak-to-Peak, Av-
erage (practical, not algebraic), and RMS measurements to one another: Figure 1.21

  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 RMS 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.