The first time I was in school for electrical engineering (long story), I had a professor who had never worked in the industry. I was in her class and the topic of the day was measuring AC waveforms. We got to see some sine waves centered on zero volts and were taught that the peak voltage was the magnitude of the voltage above zero. The peak to peak was the voltage from–surprise–the top peak to the bottom peak, which was double the peak voltage. Then there was root-mean-square (RMS) voltage. For those nice sine waves, you took the peak voltage and divided by the square root of two, 1.414 or so.
You know that kid in the front of the class? They were in your class, too. Always raising their hand with some question. That kid raised his hand and asked the simple question: why do we care about RMS voltage? I was stunned when I heard the professor answer, “I think it is because it is so easy to divide by the square root of two.”
So What’s the Right Answer?
This made me really angry. I was paying good money to be there and that was the answer? Even at that young age I knew better. There are two things wrong with the professor’s answer. First, the dividing by the square root of two is only valid on the pretty sine waves we were studying. Any more complex waveform required calculus to get the right answer. For example, a triangle wave’s RMS voltage is the peak value divided by the square root of three.
However, the biggest problem is that the answer is nonsense. It is even easier to divide by ten, but that’s of no value. The reason you want to measure RMS voltage is simple: 1 V RMS does the same amount of work in a load as 1 V DC. So a resistor subjected to 1 V DC and 1 V RMS will generate the same amount of heat, for example.
Not only is this practical, but it makes Ohm’s law continue to work. We know that power is I times E. Since I is also equal to E/R, you can deduce that power is E (the voltage) squared over the resistance. But in an AC circuit, what is that voltage? It isn’t peak or peak-to-peak–that would give a wrong answer. It is, instead, the RMS voltage.
Take another example. If you look up common RMS values for different waveforms, you’ll see the RMS voltage of a square wave with peak voltage V is just V. That’s because a real square wave goes in equal amounts positive and negative. So a 5 V square wave, for example, is always at either 5 V or -5 V and, either way, the same amount of work gets done.
But what about a pulse train from 0 V to 5 V with a duty cycle of 50%? Now the RMS value is the peak voltage (5 V) time the square root of the duty cycle (0.5) or about 3.5 V. The calculus can get hairy, but if you have a set of discrete measurements (as you probably do in any real-life situation) you simply apply the name backward. Square each sample. Find the mean (average). Then take the square root.
Consider the pulse train example. With eight samples taken at twice the frequency of the PWM, you’d expect to get four 5 V readings and four 0 V readings. If you square these samples, you get four values of 25 and four that are still 0. So the average will be 100/8 or 12.5 and the square root of 12.5 is about 3.5. That matches the answer from the table (that is, five times the square root of 0.5).
You can compute RMS voltage using an oscilloscope. However, with a meter, it can be tricky depending on what the meter measures. Most meters that don’t claim to measure RMS, read the average value of the voltage. Some meters measure RMS but only for a sine wave. Older true-RMS meters used thermal or electrodynamic methods to measure the RMS value. However, modern meters are usually adept at measuring RMS, at least for pure AC signals. A very few meters will have an option to measure RMS voltage for signals with a DC component, like an offset.
According to Fluke, an average-reading meter will read 10% high on a square wave, 40% low on the output of a single-phase diode rectifier, and anywhere from 5% to 30% low on the output of a three-phase diode rectifier. Big difference.
A Mad Professor (as in Angry)
In those days, I wasn’t smart enough to hold my tongue, so I raised my hand from the back of the class and explained the above (perhaps a little more succinctly). The response from the professor: “Oh yeah. That too.” I won’t mention the name of the school or the professor, but it did prompt me to find a new school.
With the increased use of digital instrumentation and calculators, there is a propensity for spewing off numbers without thinking about what they mean. In this case, saying that an AC voltage is 20 volts isn’t really a complete answer. We need to know what kind of voltage measurement the 20 represents and we also need to understand what that means and how it fits for the question at hand.
If you want to get more into the calculus, you might enjoy [Darryl Morrell’s] video, below.
Photo credit: sine wave graph by [AlanM1] (CC BY-SA 3.0 )]