Welcome back, and we'll get back into some linear circuits.
Today we're going to be talking about something called Linearity.
And linearity is probably just really fundamental to this whole course.
It's actually why this course is Linear Circuits.
And so today we'll be describing some, some of these mathematical concepts.
Linearity, super position and something called homogeneity.
So, in the previous lesson, we covered some things about Ohm's Law and leading
to parallel and series combining resistors.
And then there were also some labs identifying the root, the behavior that
we seen mathematically does hold true to physical devices.
Now we're going to get a little bit more into the analytical side of things, and
some analysis techniques to help us to work with some more complicated
electrical systems. In context of the course, we finished
talking about resistors, now we move on to a topic called, Superposition.
And this is going to be broken into two parts.
And this is the first part. where we talk about linearity.
And then the second part is superposition itself.
Then after that we'll look at some systematic ways of getting systems of
equations using the resistor circuits. And the Ohms law and Kirchoff's law that
we've already covered. The objectives for this lesson are to
allow you to identify the ideas of linearity, superposition and homogeneity,
and to be able to describe what they mean.
And then identify whether a system or a function exhibits the property of
linearity. So, first of all, let's talk about why
this course is called Linear Circuits. When we talk about circuits, there's a
lot of different types of things that go into it.
And you might be familiar with things like semiconductors or, or some of these
concepts of transistors. And we don't cover any of those things in
this class, because those types of devices are non-linear.
There's also devices like resistors which are linear.
The reason we covered them separately is that linear systems are a lot simpler to
deal with. But they give us a good practice area to
try using these analysis techniques that can then be applied to the more
complicated systems. In order to also understand the more
complicated systems with the non-linear devices, having a good foundation in
linear devices is important. Since the behavior that we observe in the
non-linear devices is highly dependent on the linear devices.
In the devices that we cover in this class, resistors is the ones that we've
covered so far. Also capacitors and inductors which we
will be covering in subsequent lessons. All of these devices are linear devices.
And so, that gives us some neat mathematical tricks that we can use in
doing our circular analysis, and evaluating our systems.
Any system that is composed of these linear devices and linear elements is
said to be a linear system. These technique that we're going to be
talking about today, apply to all of those type of circuits.
So, lets define what linearity means. Linearity is a concept that requires two
different properties to hold true. And so those two properties are here,
superposition and homogeneity. For superposition, what that means is, if
I have two inputs x1 and x2, and then I combine them by adding them together and
then give those two inputs sum together as an input to some device.
Some circuit, for example, I get this output.
Where it's, whatever my device does, operated on the sum of those two inputs.
Another way I could try and get an output is to take each input individually, and
send it through the system. And then after I get my two results, this
is going to be f of x1. Then here this is f of x2.
I add my two results together to get another result.
If these two values are the same for all inputs, then we say that the property of
superposition holds. For homogeneity, what that means is that
if I have an input x, and I multiply it by some constant, K.
And then give that as an input to my system, my output looks like this.
Another way I could get data is, if I take my input and send it through my
system, and then after I go to results here, this is going to be f of x, I then
multiply it by that same constant. I get another function.
If these two values are equal, then we say that this function has the property
of homogeneity. If both of these properties hold, we say
that this system, this f, is linear. So, superposition to review is, if I add
the two inputs, and then get my output, or if I get my two outputs and add them
together afterwards. I get the same answers, the
superposition. If I multiply by a constant before, or if
I multiply it by a constant after, the same constant for any input.
Then it's said to be, homogeneous. Or it has the property of homogeneity.
Let's look at an example of something that is homogeneous.
And that has the property of superposition.
And that's Ohm's law. So, for this, for Ohm's law, we say that
v, which I will let be a function of the current, i, is equal to i times R.
So, if I want to check the property of superposition, that would be like saying,
I have some i, so let's say that i is equal to 1 amp plus 2 amps.
And I want to see if, if I make this 1 amp plus 2 amps, if that's the same as
doing this one time with a 1 amp source, and again with a 2 amp source.
The way that I can show that is if v of i1 plus i2 is given as an input, then
that gives me i1 plus i2 times R, but I can distribute that out as R times i1
plus R times i2, which is equal to v of i1 plus v of i2.
And, so, it doesn't matter if I add my two currents before, and give those as my
input to my function v, or if I give them each individually to my function v, and
then add up afterwards, I get the same results.
And so it doesn't matter if I add I get the same result.
So, that means the superposition holds. Another thing I could do, is I could do
the exact same thing for homogeneity, but instead of using i, 1 plus i2, we would
say take this current and double it. And so if homogeneity holds then doubling
my current here would mean that my voltage should double.
And so, if you go through the analysis and follow kind of the same example,
you'll see that homogeneity also holds. And so here is kind of an overview of the
work that is used to do that. And so, you can always go through it in
more detail and make sure that you understand all of the steps.
Let's take a look at some things that are linear, and some things which are
non-linear. Somethings that are kind of counter
examples to linearity. So, the first one we looked at is f of x
is equal to 0. And that's almost a trivial case, and so
all of these by the way, I'm not going to derive them.
But it's an excellent exercise to make sure that you understand the material by
checking each of these things to see if you understand why these are linear, and
why these are non-linear. So, f of x is equal to 0 is kind of the,
the trivial example. This multiplying by some constant k, this
is Ohm's Law, essentially. Of course Ohm's Law is actually an
example of this specific case of this constant multiple.
It turns out that if you take the differential, and if you take the
derivative, that also works, is, is also linear.
As is taking an integral. But for the integral, it has to be a
definite integral, has to have these limits of integration.
And I list here the indefinite integral as being a counter example, because if
you do this, there's always this plus C term.
Since this is the class of things, and you have to have that plus C, if you
remember from calculus, you always have to have the plus C.
And so, that kind of makes it non-linear, because x plus c is also non-linear.
Also, things like x squared and sine of x are going to be non-linear.
And so we go through as an example, as an exercise.
And make sure that you understand why all these are linear and non-linear.
And if you get stuck, it's a good, a, good time to ask question on the forms,
to make sure you understand these concepts.
So, summarized we introduced linear operators, superposition and homogeneity,
and identified a little bit of both of those properties hold, then linearity
holds. We identified if the operator is linear
by defining those, or verifying those properties.
And then finally we used linear operators.
You can generate new linear operators by using other linear operators.
I wanted to just remind you that there are quizzes placed at the end of the
lectures. And so, make sure to stay for those
quizzes. And give you a little bit of practice
before you go and, and do the homework. And then finally be the graded quizzes.
In the next lesson, we will be applying all the principles that we covered today
to circuit analysis. And see how superposition can be used as
an analysis technique. Until then.