Specifications
9
While the amplitude A shown in Figure 2 is used in the mathematical expression of the sine wave, it's
rarely used in practical measurements. Instead, two other measures are used. We've already discussed
the peak-to-peak amplitude and can thus provide the relationship:
= 2
(1)
where
is the peak-to-peak amplitude. The amplitude A is sometimes called the zero-to-peak
amplitude.
Another amplitude measure is RMS, which stands for "root mean square". If you have a sequence of n
discrete voltage measurements
(like you would in the above digital oscilloscope measurement) over
one period of the waveform, you can calculate the RMS value of this sequence by a formula hinted at
by the name:
=
In other words, square each voltage, add them all together, and divide by n to get their mean. Then
take the square root. For continuous functions, there's a corresponding definition of the RMS value in
terms of an integral.
In this document, we'll often use the units of Vpp and Vrms to denote a peak-to-peak voltage
measurement and an RMS voltage measurement. These are read "volts peak-to-peak" and "volts
RMS".
One can derive for a sine wave that:
=
2
(2)
From equations (1) and (2), one can derive the often-used relationship:
= 2
2
= 2.82
(3)
Remember, this is only true for a sine wave.
In the use of oscilloscopes, the peak-to-peak voltage is often used because it is usually the easiest type
of amplitude to measure for a waveform (we're assuming the user has to measure it manually from the
screen). The RMS voltage is important because it is used to quantify the electrical power in time-
varying waveforms. For a DC circuit, the power dissipated in a resistor is defined as:
=
where V is the voltage across the resistor in volts, i is the current through the resistor in amperes, and P
is the power being dissipated (i.e., turned into heat) in watts. The RMS measures of the current and
voltage are used in the analogous relationship for a time-varying waveform:
=
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