- [Voiceover] Alright, now we're ready
to learn how to do circuit analysis.
This is what we've been shooting for
as we've learned our Fundamental Laws.
And the fundamental laws are, Ohm's law,
and Kirchhoff's laws,
which we learned was Kirchhoff's Current Law
and Kirchhoff's Voltage Law.
This says that the sum of all the currents going into a node
adds up to zero.
And the Voltage Law says the sum of the voltages
going around a loop adds up to zero.
And the other thing we learned was the sign convention.
So, we had a passive sign convention,
which was, if I label the voltage like that,
then I label the current flowing into the positive end,
the positive end and the current go together.
So, these are the tools that we have
for analyzing a circuit and I've drawn a circuit here.
So, what we see is, we have a voltage source,
we'll call that VS, and we'll give it a value of 15 volts.
We have a resistor, R1,
that has a value, we'll give that a value of four K ohms.
And another resistor, R2 is sitting right here,
and we'll give it a value of 2 K ohms, 2,000 ohms.
And finally, there's a current source sitting over here.
And this has a value of three milliamps
flowing downwards.
So, the point of circuit analysis means what we wanna do is,
we wanna find all the currents and all the voltages
in this circuit using our Fundamental Laws.
So first, let's identify the nodes in the circuit.
We'll give them letter names.
We'll call this node A,
and that's the junction between R1 and Vs.
We'll give this one a name,
this is a node that's distributed,
all three of these components joined right about here,
we'll call that node B.
And that's one of the nodes.
And down at the bottom here, there's the third node,
which is this distributed node here.
It connects the voltage source, the resistor,
and the current source, and this is all,
we'll call that node C.
Now, we could also label some voltages on here.
We'll label the voltages, we'll do that in orange.
We'll call this V1 and we'll give it a plus sign here.
And the minus sign here, that's the voltage across R1.
We'll give R2 a name, we'll get the voltage here,
we'll call this plus, minus V2.
That's the voltage across R2,
it's also the voltage across the current source.
B, it's the voltage between node B and node C.
And now let's label the currents.
Let's label the currents in this circuit.
We have a current
coming through R1, we'll call that i1.
There's another current going through R2
and we'll call that i2.
There's a current going here through Is
and we actually already know that, we'll just call that Is.
And that's all the currents,
that's all the unknown currents in our circuit.
If we stare at this, we're not gonna be able to tell,
on just inspecting this, what's going on,
we have to come up with some solution technique.
And we're going to solve this by, what's called,
Application of the Fundamental Laws.
And here's our laws over here.
And the whole technique is based on solving simultaneous
equations.
That's the skill of circuit analysis,
circuit analysis means setting up
and solving simultaneous equations.
So, let's write down some things we know about the circuit.
And first thing we're gonna apply is,
we'll apply Ohm's law to the two resistors.
So, we can say
V1 equals i1 times R1.
And we can say that V2 equals i2 times R2.
Alright, that's not a bad start, we have two equations,
how many unknowns do we have?
I2 and V2 are unknown.
I1 and R1 are unknown.
So, we have four unknowns and we have two equations,
so, we're halfway there.
And now, we need to come up with some more equations.
And I'm just gonna stare at the circuit
and what I'm gonna do is,
I'm gonna apply KVL around this loop here,
that's gonna be my KVL loop.
And after we get that equation, what we're gonna do is,
we're gonna write KCL,
we're gonna write the Current Law at node B.
That's the two equations we're gonna develop here.
So, let's do KVL.
Let's start at this node, at this corner of the circuit
and go around clockwise around.
And what we'll see first is a voltage rise,
plus Vs.
Next, I see a voltage drop, so it's minus V1,
going around this way, minus V1.
I'm at this point, now I'm gonna go through R2
and I see a voltage drop of V2, minus V2 equals zero.
That's Kirchhoff's Voltage Law around this loop.
One rise, two drops, one rise, two drops.
Now, we're gonna move up here,
and do Kirchhoff's Current Law at this node right here.
So let me draw a few more arrows.
This is i1, it comes right through that resistor.
And this is Is.
So Kirchhoff's Current Law,
let's count the currents going into node B,
current going into node B is i1.
Currents go out, we'll set that equal
to the sum of the currents going out of node B.
So, that's equal to i2 plus Is.
So look, now I have my four equations,
here's my four simultaneous equations.
And this is what we have to solve.
So let me move the screen up and we'll start doing that.
Let's do a double check,
we have four unknowns, V1, i1,
V2, i2.
Here's V1 and V2, and here's i1 and i2.
And I have four equations, so we oughta be able to do this.