So in this video, we're going to figure 
out how to solve circuits in a much more 
quicker, much more efficient way, than we 
have by writing lots of equations and 
lots of anomalies. 
And, the way to see how that could be is 
by asking a rather interesting question, 
what does a source see? And of course 
multi-sources and current sources don't 
have eyes, 
but I'll show you what that means. 
When you go down the path, they're asking 
what the source sees, that leads to the 
concept of equivalent circuits, one of 
the most powerful concepts and circuit 
theory. And we'll explore series in 
parallel combinations and learn how to 
think about circuits, in terms of these 
structural components. 
Let's go back to our example that we did 
last time. 
the we have a series combination of 
resistors and a voltage source. 
And what we calculated last time was the 
output voltage was proportional to the 
input voltage with a constant 
proportionality, that's the resistor R2 / 
R1 + R2. 
And this as I pointed out last time is 
called voltage divider, and that applies 
any time, you have a series combination 
of resistors and you're applying a 
voltage across them. 
A voltage across one of them, is that 
resistance divided by the sum of the 
resistance as a rule that you can use to 
simplify your life. 
I asked you last time, what was the 
voltage across R1, and I think it's 
pretty clear, but if you plot a voltage 
divider, it's the resistance of the 
resistor. 
I'm talking about divided by sum times v 
in. 
very easy to see, and of course, by KVL, 
this and this have to add up to be v in, 
so everything checks out. 
Well, what does the source see? And what 
do I mean by that? The voltage source, in 
this case, certainly notice what it's 
voltage is. 
The question is, when it gets to see 
through the current. 
So, the idea is, I'm going to put the 
resistors in a box, 
so that the voltage source can't cheat. 
The only thing you can do is figure out 
it's current, the current going into the 
box by one and based on what that, how 
that's related to the voltage source, 
figure out what's inside the box. 
We know the V out is equal to this, 
and we want to know the current i1, but 
KCL says that i1 = i out, 
that's the way it always works out for a 
series combination, 
the current goes straight through them, 
and so it's easy to see if the currents 
were all the same. 
Well, I can figure out i out from v out 
by simply dividing by R2, 
and, that's what I get. 
So, what does the voltage source see? Put 
in a box, 
for all the world, the relationship 
between the source voltage and the 
current, that it's supplying to the 
circuit looks like that of a resistor. 
A single resistor having a value equal to 
the sum and that's the idea of what's 
called an equivalent circuit. 
So, what does the source see? It looks in 
here, through its current and it sees 
something that behaves just like a single 
resistor, whose value is the sum of the 
resistances. 
Of course, it doesn't know that there is 
a sum. It might see a resistor that looks 
like 1 kilo ohm. 
It doesn't know how many resistors or if 
there is only one resistor, 
that can equally be, 
well be the case. 
But the equivalent resistance of a series 
combination is the sum of the 
resistances. 
You can really use that to simplify your 
calculations for circuits. 
And by the way, this result all comes 
from the vi relations KVL and KCL. 
Putting them in a much more concise, 
understandable form when we think about 
equal answers. 
So, like I said, it's a single resistor. 
Now, let's try another simple circuit and 
this one has what's called a parallel 
structure. 
We have three circuit elements in 
parallel beside each other, connected 
together with wires, 
and that's called a parallel circuit. 
So we have two resistors in parallel and 
a current source has been placed in 
parallel with that. 
And what I want to do is I want to find 
out how the output current, the currents 
of R2 is related to the input current. 
Well, how many nodes, how many loops are 
there for writing KVL? And you should 
see, there are two nodes, 
there's a big node here, and a big node 
down here. 
We are only going to write KCL at that 
node, 
and here are two loops, 
so we get exactly the number of equations 
we need. 
We have two vi relations and the third 
one is the, where the current source. 
We have the KCL equation, one KCL 
equation, we have two KVL equations. 
One thing to point out is that when you 
have resistors in parallel, it's pretty 
clear that the voltage across the them, 
each of them is the same. 
And since the current source is in 
parallel with that, all the voltage is 
here, all the voltages are equal to each 
other, so that's not really a question. 
The question is what is this current and 
how it's related to in? Well, 
all we have to do is put it all together. 
it's not hard to show that this is the 
result and the way you do that is by 
looking at this set of equations here, 
the vi relations, because these two 
voltages are equal, 
these two quantities are equal. 
And I can substitute for i1 in this 
equation by solving this set for i1. 
What I get is i1 = R2 / R1 times i out. 
And if I substitute that in there and 
solve, I will get this result, and this 
is known as current divider as you might 
expect, so in a parallel circuit, if you 
have a current going in, it splits 
between the two resistors. So, that the, 
the part through this resistor is the 
other, the value of the other resistor 
divided by the sum, 
and so, it works in a similar way, 
the voltage divider, except it's not the 
resistance of the one that we're talking 
about, it's the other resistor. Okay. 
So what does this look like? Well, again, 
because we get that for the input-output 
relationship, 
well, the only way the, the current 
source can see what's going on is to look 
at its voltage. 
Well, that voltage is just that. 
Well, that's again the vi relationship 
for a resistor. 
It looks like a single resistor whose 
resistance is the product over the sum 
and I use this shorthand notation for 
that result. 
So R1 parallel R2 means R1 R2 divided by 
the sum of R1 and R2. 
So, again, what the source sees is a 
single resistor it does not know, 
there are two resistors in there. All it 
can tell it looks like one resistor. 
So, if the series in parallel combination 
structures are very important to 
recognize to help you simplify and see 
what's going on. 
So let's go through this little exercise 
here. 
I'm not going to tell you if I'm, what 
kind of a source I'm going to attach out 
here, It doesn't really matter, 
the question is what does the source see? 
Well, we have to figure out the structure 
first. 
It's clear that R1 is in parallel with 
all of this, or what's all of this. I see 
R2 in parallel with R3 at there, 
and then that's in series with R4, and 
that's going to allow us to figure out 
what the equivalent resistance is. 
So, let's start the parallel combination 
here, 
that looks like a single resistor whose 
value is R2 parallel R3. 
This set now looks like a single resistor 
which, whose value is R2 parallel R3 plus 
R4, because this parallel combination is 
in series with that. 
And then, this single resistor, is what 
that looks like there, is in parallel 
with R1, 
and so our final result is it's a little 
complicated, but very simple to write. 
And that is that R1 is in parallel with 
the combination R2 R2 and R3, and then, 
which is in series with R4. 
Well, 
I'm going to use those results to solve a 
circuit the easy way and this is actually 
a pretty practical circuit. 
So what I want to point out here is that 
this is the system we've already been 
talking about. 
'Kay? And what usually happens is when 
you try to pass that signal on to the 
sync, that it has an input resistance. 
In some sense, its equivalent resistance 
is something which we're going to call R 
sub L and L means load. And the question 
is, does the relationship between v in 
and v out change, because there is a load 
resistance out there? 
You might say, well, geez, this voltage 
here is the same as that voltage there, 
so it probably doesn't change. 
Well, that's not quite true, 
and that's because these are in parallel 
and these voltages are actually across 
the parallel combination. 
There's something to figure out there. 
I've essentially given away how you 
analyze it. 
What we discover is that the output 
voltage is using our series of parallel 
combination levels. 
This looks like a single resistor, which 
is R2 in parallel with RL, and then 
that's in series with R1, 
so we simply use voltage divider to 
figure it out. 
And is the concise expression, and when 
you simplify everything, you get this for 
a final answer. 
Well, that doesn't much look like the 
result that we had before, which was R2 / 
R1 + R2 * v in. 
So, when is, is there a situation as we 
change RL, that this result looks like 
that? If you think about that for a 
second, there are sort of two extremes to 
look at, 
RL going to zero, RL going to v in. 
That's the easy cases to look at, 
which is pretty clear if you had set RL 
to zero, the output is zero voltage, 
right? Because there's an RL [INAUDIBLE] 
More interesting is RL going to infinity. 
When RL goes to infinity this term down 
here essentially disappears, because 
these terms are getting big, and then the 
RLs cancel, and sure enough, 
you get this result. 
So, if you want to do develop the system 
such that when you attach it to the sync, 
it has input resistance RL. 
It better, it better be that RL is big 
and it's going to take until the 
succeeding video we define what being in 
comparison what is. 
That's in the next video. 
Alright. 
So, in summarize, the resistor 
equivalence. 
So, if you have resistors in series, the 
equivalent resistance looks like the sum. 
And voltage divider, what it looks like, 
the voltage across any one of them is 
that resistance divided by the sum of all 
of the resistances. 
If you have a lot of resistors in 
parallel, the equivalent conductance is 
the sum of the conductances and the 
current through any one of them is that 
conductance divided by the sum of the 
conductances. 
So, it turns out writing the current 
divider rule, which is what this is over 
here. 
In terms of conductances, it makes it 
look a lot like the voltage divider up 
here, 
is when you write in terms of 
resistances, the expressions get 
complicated. 
It's a whole lot easier to write it in 
terms of the conductances in this case. 
By the way, there's a, a minor error on 
this sign. 
It should be a voltage v here and an i 
there. 
Okay. 
[SOUND] So, that's what we've learned 
about solving circuits. 
The equivalent resistance idea comes up 
because you asked the question, what does 
the source see? And it turns out as you 
can see from all these examples, when you 
use the idea of an equivalent resistance 
along with the ideas of voltage divider 
and current divider, you can solve 
circuits rather quickly. 
And notice, we solved that circuit having 
a load resistor on it without ever 
writing KVL, KCL, and V averages. 
We just wrote down the answer. 
That's really very, very nice. 
So, the whole idea about being able to 
solve circuits easily means seeing the 
structure, 
what's in series with what, what's in 
parallel with what, and finding the 
equivalent resistances accordingly.