Q. Quick! What is the definition of V_RMS ?

a. The average DC voltage.
b. The peak voltage, times the square root of one half.
c. The peak voltage times the square root of two.
d. One over the square root of half the voltage 
e. None of the above. 

A. The answer is:

e. None of the above

If you said "b.", you were correct for a sinusoidal waveform, but the full definition of V_RMS is much more general.

In fact, most multimeters will make this error when the AC Volts button is pressed, based on the same assumption that a sinusoidal waveform is being measured.

Root Mean Squared Voltage (V_RMS)

Consider the "120 VAC" power line coming into your home. The average voltage in this signal is zero. (So, why pay the light bill?) In fact, electricity is billed in "kilowatt hours", or the time integral of power. The power is what is important to run electrical equipment.

In general, electric power varies as voltage squared, (P=V²/R) which is strictly non-negative. The average (mean) power can be computed using the "mean squared voltage". Specifically, power is found using the square of the "root mean squared voltage" (VRMS or V_RMS). This is the quantity which is important when shopping for AC signals. In fact, when talking about these signals, "120 VAC" refers to the RMS voltage of the cosine waveform delivered to your doorstep.

The concept of V_RMS may be examined graphically. Let s(t) be a cosine waveform.

s ( t ) =

The average value of cos(t) is zero. The peak value (amplitude) of the cosine is 1. The peak-to-peak value is 2. Using the trigonometric identity 2 cosA cosB = cos (A-B) + cos(A+B), the function s²(t) = cos²(t) = [1/2][ 1 + cos(2t) ].

s ²( t ) =

This is the squared version of the signal, and its mean value is 1/2, as shown. So the "mean squared" value is 1/2. By inspection, the "root mean squared" value is the square root of 1/2 (approximately 0.707).

If this was an electrical waveform, cos(t) would be called a 0.707 VAC signal.

The approximate parameters of a "120 VAC" waveform are summarized in the table below:

Average Voltage	RMS Voltage (VRMS)	Peak Voltage (V)	Peak-to-Peak Voltage (2V)
0	120	170	340

V_RMS Depends on the Shape of a Periodic Signal

The above argument can be generalized for any periodic signal s(t). Mathematically, the squared signal is s²(t) and the "mean squared" signal is the integral of s²(t) over an entire period, divided by the period. Finally, the "root mean squared" is the square root of this final result.

Let s(t) = cos(t), as before. The RMS voltage can be computed directly as :

Root Mean Squared Voltage for a cosine waveform.

Once again, the familiar factor of root two emerges. However, other waveforms will not have the simple factor of 1.414 to compute V_RMS.

Without further computations, consider a squarewave signal which goes peak-to-peak from +1 Volt to -1 Volt. When squared, this signal is +1 Volt everywhere. The mean squared voltage is therefore 1 V², and V_RMS = 1 V.

For Arbitrary Waveforms, Choose a "TRUE RMS" Voltmeter

Inexpensive multimeters measure the average positive voltage of a waveform and scale this value using the square root of two to produce a display value. Because this method only works for cosines, special purpose "TRUE RMS" voltmeters must be used for non-sinusoidal waveforms. One method of measuring true RMS voltage is to measure the heat generated when the signal passes through a resistor. Regardless of the shape of the input signal, the RMS voltage can be computed as the square root of the power dissipated in the test resistor.

"..only a TRUE RMS Voltmeter gives accurate results for non-sinusoidal waveforms..."

...which proves once again the value of knowing your instruments.

A True RMS Voltmeter