Welcome back to Linear Circuits. Today we were going to be talking about the root means squares statistics. So we will introduce it and show how to calculate it. Lets begins module 5 which covers the concepts of power. And so today we get to talk about root-mean square and then the statistics we calculate today will be then used in our power factor and power calculations and subsequent lessons. The objectives for this lesson are to identify the equation for calculating root mean square values. To calculate root mean square values using this formula for simple periodic functions and then to find the peak value given an RMS statistic. So the motivate this first we'll look at a sinusoid, suppose we want to know the average voltage of this sinusoid. Well to calculate the average we just use this equation right here where the average value is equal to one over a period, so this is a periodic function. So the width of the period and then we integrate over one period of the function. And so we can start anywhere we'd like, this probably should be T plus a TI. We can start anywhere we'd like, start here maybe and so one period is the, this point tells you reach that point again. And so the period here is going to be T and if we integrate over this, filling all this area, we recognize that there's as many low regions as high regions and so our average happens to be zero. And this turns out to be, it really doesn't matter whether this is five or whether this is a million, because it's a sinusoid. It's centered, and the average is always going to be zero. Consequently, just taking the average of a sinusoid isn't particularly useful information. So there's some better statistics that we can use when we're dealing with sinusoidal functions to give us a little bit more information about the system and the signal. One of these methods is to use what's known as the quadratic mean or the root-mean square statistic. And here it is to find the frms so the rms of the function is equal to the square root of this 1 over T and integrating from to not to T not plus T of f squared of tdt. So it's called the roots means square because here you have the square root and then this one over t integration is the average for the mean, and then you're going to take the square of the function. To make it a little bit easier to see how to calculate this, we'll do a little bit of animation. But first of all the thing we want to look at here is if we take a square of this sinusoid what will happen? Well this point is going to go to Vm squared, when we get to zero we're going to be back at zero. When we get down to here to negative Vm but if you square than again, you're back to Vm squared. And if we take this function and we square it, it turns out that there's a useful trig identity which is this. It's not particularly pretty. It allows us to basically do this kind of calculation exactly, but we're going to kind of give an illustrative example to show what this actually means. What you end up having is this function which has a max value at Vm squared. It again is a sinusoidal curve. Here's the cosine. It's at twice the frequency of the original function. So you can see that it starts out, has a peak here and has a peak here and this goes from high to low. By the time this comes back up to a peak you are then to the third peak on the squared value. So it's a useful identity, but it's kind of hard to remember. So if you just kind of remember this illustration hopefully that will help you remember how to calculate it. But if I take the average of that, it's going to be one half of Vm squared. Because this curve has a maximum value Vm squared at a base of zero and since this is just normal cosine function, the value would be at half of that for the average. So when we take the square root of that, we get the rms value of the voltage is Vm over the square root of two. It's important to notice a couple things. First of all, as now our voltage maximum value our amplitude increases so does the rms value. And if you double the amplitude you double the rms value. This is not based upon anything about the frequency, you can double the frequency, triple the frequency it doesn't matter. As long as it's a sinusoidal function the shape is all that really matters and you'll get the same result. So, it doesn't matter what kind of sinusoid you have, the rms value would just be taking the root two of the amplitude and that will tell you the rms value. Let's look at what happens if we're using a different function, for example, a triangle wave. Again, we are going to do the same thing. We're going to start by squaring it, then we're going to take the average of that squared thing, and then take the square route of that. So first of all, we'll square it. So if I square this triangle function it looks like this series of quadratics functions as quadratic curves. And if you average, and to kind of help with the illustration, what we're going to be doing is taking an average of one period over integrating underneath that curve. And if you do that you find that the average value is Vm squared over three. And you take the square root of that, you get Vm over the square root of three. Again the frequency doesn't matter, as amplitude increases so does the amplitude of this result, this rms result. But the shape matters before with a sinusoid it was Vm over two with triangles over root three. Let's see how to use this. As an example, consider the voltage that goes into your home. Now this varies from country to country but in the United States the voltage specified is 120V at 60 Hz. However, this is 120V RMS so that's not the peak amplitude. So supposed we want to calculate the peak amplitude. How can we do that? Well remember again frequency doesn't matter. It's just an extra little piece of information. And we know that the voltage coming into homes is sinusoidal which means that if I want to calculate the RMS it's going to be equal to the amplitude divided by the square root of two. We know here the RMS is 120, so we get 120 times the square root of two is equal to Vm. So consequently, the peak voltage or the highest voltage that you will see if you measure the voltage coming out of your home is 169.7 volts, with a minimal value of minus 169.7 volts. To summarize, we defined the root mean square calculation, we calculated the RMS values for sinusoidal functions and triangular wave functions. And then we applied this to find the peak value in residential home power. You can do the same thing no matter what country you live in, just look in, look up what the voltages are for the power that you use and you can use that same calculation to find the peak values. Tuns out that RMS calculations are very useful in power calculations and we will see the reason for that in subsequent lessons. In the next lesson we will be talking about the concept of power factor and power triangles. And this helps us to start describing how AC power works, it has a couple little nuances you don't see with DC power but are very useful and important for us to understand. Until next time.
