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.