# Rise Time: The Role of RMS

A few months ago a personal computer trade magazine reviewed surge voltage protective power strips for home use. The article offered an explanation of what "120 volts RMS" power (what we typically have in our homes and offices) means. The problem was-the definition they was wrong!

The next issue gave a corrected definition. Problem was-it was wrong too! Two issues later they finally got it right in a very brief notice deep inside the magazine.

Actually, many people don't know what "120 volts RMS" truly means. And when they find out, they wonder why in the world we would use such a measure. So here is your opportunity to test your knowledge. We all know that the 120-volt power supplied to our homes is alternating current (AC) following a sinusoidal waveform. See Figure 1(a). When we say that the voltage is 120 V RMS, do we mean:

(a) The average value of the voltage is 120 volts?

(b) The peak value of the waveform is 120 volts?

(c) The peak-to-peak value of the waveform is 120 volts?

(d) Or none of the above?

In high school trigonometry we learned that a sine wave cycles between +1 and -1 around a horizontal axis whose value is zero. The waveform is symmetrical around that axis. Therefore, the average value of a sine wave is zero. So much for choice (a)! It turns out that choices (b) and (c) are not correct, either. The correct answer is (d), none of the above.

Let's start the explanation with a discussion of power. Power is defined as voltage*current (or P=V*I). By Ohm's Law, V=I*R, or I=V/R. Therefore, by substitution, power=V2/R. So if we apply a voltage sine wave (V*sin(Q)) across a resistor, the power applied to the resistor is given by the expression P=V2*sin2 (Q)/R (shown in Figure 1(b)). It can be shown that the average power delivered to the resistor is V2/2*R.

Now here is the key question. What DC (direct current) voltage will deliver the same power to the resistor as the AC waveform we have just been considering? The answer is:

Equivalent DC volts = (V2/2).5 = .707*V.

This is defined as the RMS voltage. It is the DC voltage that would deliver the same power to a resistive load as would the AC waveform.

Now, back to our home example. If our household voltage is 120 volts RMS, then 120 volts is .707*V, where V is the peak voltage. This means the peak voltage is 120/.707 or 169.7 volts. And the peak-to-peak voltage at our wall outlets is 2 times this or 339.5 volts! Expressed another way, the formula for the voltage at our wall outlets is169.7*sin(Q) where (Q) represents the angular position within one cycle.

Calculating the RMS value

So what does RMS actually mean? The term stands for "root mean square," which represents the method of calculation of the value. The calculation works like this:1

1. Take the waveform and divide it into a large number of individual increments.

2. For each increment, square the voltage value.

3. Sum these squared values over all increments and then calculate their mean value.

4. Take the square root of this mean.

That's how you get the root mean square value! Every waveform has a different RMS value. It is only a sine wave whose RMS value is .707*V. A DC waveform has an RMS value = V! RMS values for other common waveforms are published in standard reference texts.

Measuring RMS values

Measuring the true RMS value of a waveform is not easy. Inexpensive AC voltmeters simply rectify the waveform (by passing it through a diode, for example), measure an average value of the rectified waveform, and apply a correction factor (assuming a sine wave). Such meters, therefore, are only accurate for sinusoidal waveforms. They do not accurately measure any other waveform shape.

So-called "true" RMS meters in the past have depended on some sort of power measurement to derive the correct RMS value. Now that calculational power is so much more economically available, meters can analyze a waveform's shape and actually mathematically calculate the correct RMS value. In general, if an AC meter does not explicitly say it gives a "true" RMS reading, you can assume that it is accurate only for a sinusoidal waveform.

Footnote:

1.The formal equation for RMS is:

where V(t) is the AC variable, t is time, and T is the period of one complete cycle.

Doug Brooks is the president of UltraCAD Design Inc. (Bellevue, WA). His e-mail address is doug@eskimo.com

© 2001 *CMP Media LLC.*

7/1/01, Issue # 1807, page 42.

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*NOTE: The articles presented here contain only the text originally published in Printed Circuit Design magazine. Any accompanying graphics and illustrations have not been recreated here. You may view the article in its entirety in each printed issue of PCD.*

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