Welcome back to Linear Circuits.
Today we are going to be talking about the root mean square statistic.
So we will introduce it, and show how to calculate it.
This begins Module Five, which covers the concepts of power.
And so today, we'll talk about root mean squared, and then the statistics we
calculate today will be then used our
power factor, and power calculations in 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 to motivate us, first we'll look at a sinusoid.
Suppose that 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 the t aught. we could start at anywhere we'd like.
Start here maybe, and so one period is
from this point till we reach that point again.
And so the period here is going to be t. And as we integrate
over this, filling in all of 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 irr-,
it really doesn't matter, whether this is, five or whether this is a million,
because it 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, where the
root means square statistic. And here it is defined FRMS.
So, the RMS of the function is equal to the square root, of this one over T
and integrating from tnot to tnot plus T of f squared of tdt.
So, it's called the root mean square because here you have the square root.
And then this one over T integration is the average or 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 wanted to look at here
is if we take a square of this sinusoid, what will happen?
Well, this point
in going to go to VM squared.
When we get to zero we're going to be back at zero.
When we get down here to negative Vm if
you square that, again, your back up to Vm squared.
And, if we take this 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.
So here's cosine.
It's at twice the frequency of the original function.
So you can see that it starts up, it has a
peak here, has a peak here, and goes from high to low.
That's time it comes back to a PQ, or the third peak on the square 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 to 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 max value of VM squared, at a base at zero.
And since it's just a 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 that 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,
or our amplitude increases, so does the RMS value.
And, if you double the RM, the the amplitude, 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 its 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 will 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 root of that.
So, first of all we'll square it.
So if I square this triangle function it looks like this series of quadratic
functions is quadratic curves. And if you average, and
to kind of help with illustration. What we're going to be doing is taking
an average of one period, where
we're integrating underneath that curve.
And if you do that you find that the average value is VM squared over three.
When you take the square root of that, you get v m over the square root of 3.
Again, the frequency doesn't matter.
As amplitude increases, so does the amplitude of this this result.
This RMS result.
But the shape matters, before with the sinusoid, it was VM
over root two, with the triangle, it's VM 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 120 volts at 60 hertz.
However, this is 120 volts RMS. So that's not the peak amplitude.
So suppose we want to calculate the peak amplitude.
How would 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 in sinusoidal.
Which mean that, if I want to calculate VRMS.
Is going to be equal to the amplitude, divided by the square root of two.
We know here, VRMS 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.
So summarize, we define the root mean square calculation.
We calculated the RMS values for
sinusoidal functions, and triangular fun, 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 you use,
and you can use that same calculation to find the peak, the peak values.
Turns 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 help us to start describing how AC power works.
It has a couple of little nuances that you don't see with
DC power, but that are very useful, and important for us to understand.
Until next time.