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.