In the 2017 Manila Mini Maker Fare while demo’ing my Automatic Test Equipment (ATE), one kid asked me what my project is all about, and I told him, its a 24/7 automatic tester without human intervention, it read logic status, the power level, heat, the AC voltage swing at the load in RMS value, and he stopped me and asked me to explain what is RMS value……. I do not know how to explain to him.
Back to basic of the BASICS
I though that (jokingly) explain the Voltage RMS like the picture below :
The RMS is acronym for ROOT MEAN SQUARE, see the above representation hahahaha. Many Engineer will say and explained it as
” equivalent dc voltage which would deliver the same power in the same time”.
“a square root the mean of the squared function”
If I am going to explain this to a kid, I am an idot! After all, Albert Einstein said that “If You Can’t Explain it to a Six Year Old, You Don’t Understand it Yourself”.
Let me Explain then
Lets do something interesting, by UNRAR or UNZIP the mathematical explanation of the popular sinewave signal V(rms) = EM / 1.4142 volts.
The figure below is an Alternating Current (AC) Sine wave signal transverse in the positive to the negative direction in one cycle, this is the kind of signal that is impressed in our household to power our TV, Refrigerator, PC, Audio Amplifier —- sine wave signal. Mathematically, it is represented by +EMSinwt for the positive waveform and -EMsinwt for the negative waveform.
From the above representation of the Sine wave signal, I wanted to understand what other say as “square root the mean of the squared function“. Let me make some clarifications :
we can see now by using mathematics that the squared function, the T (period) and the “mean” meaning. What we have not seen yet is the Square Root like this
The most disturbing fact is “an equivalent dc voltage which would deliver the same power in the same time”. Some says, it is the equivalent DC voltage when passes to a load with an ohmic value will produce the same heat as DC voltage. Oh no! I am not making progress to clarify, instead I am introducing more variables.
A Direct Durrent (DC) like the voltage on the batteries (Smartphone batteries, solar power batteries or your 1.2V cell batteries) have a straight line and it is not varying with time like the sine wave signal.
So, how is that possible that a straight line waveform can relate to a varying sinewave signal? This is the key — so I would like to unzip how to make that happened. Let’s go back to Mathematics aided by graphical Representation.
If you squared the function (the sinewave) you will have the waveform in red color above, a negative value when square will result in a positive value. If you cannot imagine the red one, imagine Mayon Volcano for the first half cycle, is looks like …… let me re-draw it on the positive quadrant, and adding the peak of the waveform, see below.
There are 2 mountains with each peak, I called them A1 and A2 and I wanted to get the Area of A1 and A2, which is similar. I want to make it as DC waveform, a straight line. To make that a straight line as same as DC to fulfill the “an equivalent dc voltage which would deliver the same power in the same time”, I will need to re-distribute the waveform in this way , looked at what I did to the left side waveform represented by A1, I pressed the top downward so ti will filled-in the left and right sides.
Ok, so we have almost the same straight line now (almost) which now resembled a DC waveform.
To get the Value, I need to go back to find the total Area = A1 + A2 of the squared function, such that I am going to use an Intregral Calculus by using a “dwt” to sample the sinwave signal.
From the definition, the square root of the squared function is set mathematically, and let’s solved now the values of E(RMS) in terms of EM or the Peak Value of the sinewave,
Here you go!!! The Sine wave Voltage and Current RMS value.
Well, what do you think?