Series resistors | Circuit analysis | Electrical engineering | Khan Academy

- [Voiceover] Now that
we have our collection

of components, our favorite
batteries and resistors,

we can start to assemble
these into some circuits.

And here's a circuit shown here.

It has a battery and
it has three resistors,

and a configuration that's called

a series resistor configuration.

Series resistors is a familiar pattern,

and what you're looking for is resistors

that are connected head
to tail, to head to tail.

So these three resistors are in series

because their succession of nodes

are all connected, one after the other.

So that's the pattern that tells you

this is a series resistor connection.

So we're gonna label
these our resistors here.

We'll call this R1, R2, and R3.

And we'll label this as v.

And the unknown in this
is what is the current

that's flowing here, that's
what we want to know.

We know v, we want to know i.

Now one thing we know about i

is i flows down into resistor R1,

all of the current goes out of
the other end of resistor R1

because it has to, it
can't pile up inside there.

All that goes into here,

and all that comes out of R3.

And i returns to the place it came from,

which is the battery.

So that's a characteristic
of series resistors,

is in a series configuration
is they are head to tail,

and that means that all the components,

all the resistors share the same current.

Current.

That's the key thing.

The thing that we don't know

that's different between each resistors,

is the voltage here, and the voltage here,

let's call that v1,
this is v2, plus, minus,

and this is v3, plus, minus.

So in general, if these
resistors are different values

because they have the same
current going through them,

Ohm's Law tells us these
voltages will all be different.

So the question I want to
answer with series resistors

is could I replace all three of these

with a single resistor that
cause the same current to flow?

That's the question we have
on the table right now.

So we make some observations,

we have Ohm's Law, our friend, Ohm's Law.

And we know that means v equals i times R,

for any resistor.

That sets the ratio of voltage to current.

And this is another
thing we know about this,

which is that v3, plus v2, plus v1,

those are the voltages
across each resistor,

those three voltages have
to add up to this voltage

because of the way the
wires are connected.

So the main voltage from the battery

equals v1, plus v2, plus v3.

We know that's for sure,
and now what we're gonna do

is we're gonna write Ohm's Law

for each of these individual resistors.

v1 equals i, and i is
the same for everybody,

times R1.

v2, this voltage here, equals i times R2.

And v3 equals i times R3.

Now you can see, if I had four,
or five, or six resistors,

I would have four, or five, or
six equations just like this

for each resistor that was in series.

So now what I'm gonna do is
substitute these voltages

into here, and then we'll
make an observation.

So let's do that substitution.

I can say v equals i, R1,

plus i, R2, plus i, R3.

And because it's the
same i on every resistor,

I can write v equals i,
I'm gonna factor out the i.

R1, plus R2, plus R3.

Now what I want to do
is take a moment here

and compare this expression
to this one here,

the original Ohm's Law.

Alright, there's Ohm's Law.

So we have v equals i, some current,

times some resistor.

I can come up with a resistor value,

a single resistor that would
give me the same Ohm's Law.

And that is gonna be called,
let's draw it over here.

Here's our battery.

And I'm gonna say there's a
resistor that I can draw here,

R series, that's equivalent
to the three resistors here.

And it's equivalent in the sense that

it makes i flow here, that's
what we mean by equivalent.

So in our case, to get the
same current to flow there

I would say v equals i times R series,

in which case, what I've done
is I've said that R series

is what, is the sum of these three things,

R1 plus R2, plus R3.

This is how we think
about series resistors.

We can replace a set of series resistors

with a single resistor
that's equivalent to it

if we add the resistors up.

Let's just do a really fast
example to see how this works.

I'm gonna move this screen.

Here's an example with three resistors.

I have labeled them 100
ohms, 50 ohms, and 150 ohms.

And what I want to know
is the current here.

And we'll put in a voltage,

let's say it's 1.5 volts,

just a single small battery.

So what is the equivalent resistance here?

One way to figure this out
and to simplify the circuit

is to replace all three of those resistors

with a series resistor, RS,

and that is, as we said here, is the sum,

so it's 100, plus 50, plus 150.

And that adds up to 300 ohms.

So that's the value of the equivalent

series resistor right here.

And if I want to calculate the current, i,

i equals v over R, and
this case, it's R series,

and that equals 1.5 divided by 300.

And if I do my calculations right,

that comes out to .005 amperes.

Or an easier way to say it is
five milliamps, milliamperes.

So that's i.

And now that I know i, I can
go ahead and I can calculate

the voltage at each point
across each resistor

because I know i, I know R,

I can calculate v.

So v1, v1, which is the
voltage across that resistor,

v1 equals i, R1, as we said before.

So it's five milliamps
times 100 ohms, 0.5 volts.

Let's do it for the other one, v2,

equals i, same i, this time times R2,

five milliamps times 50 ohms,

and that equals 0.25 volts.

And finally, we do v3.

This is plus, minus v3.

And that equals the same
current again times 150 ohms,

which is equal to 0.75 volts.

So we've solved the
voltage and the current

on every resistor, so this
circuit is completely solved.

And let's do one final check.

Let's add this up.

Five, five (mumbles) is zero.

Carry the one, six, seven eight.

15, 1.5 volts, and that's very handy

because that is the same as that.

So indeed, the voltages across
the resistors did add up

to the full battery that was applied.

There's one more thing
I want to point out.

Here's an example of
some series resistors.

And that's a familiar pattern.

And you'll say, "Oh, those
are series resistors."

Now, be careful because if
there's a wire here going off

and there's, doing this,
or there's a wire here,

connected to this node here,

this still looks like they're in series,

but there might be current flowing

in these branches here.

If there's current flowing out

anywhere along a series branch,

anywhere along what looks
like a series branch,

then this i may or may
not be the same as this i.

And it might not be the same as this.

So you gotta be careful here.

If you see branches going
off your series resistors,

these are not in series
unless these are zero current.

If that's zero current, and
if that is zero current,

then you can consider these in series.

So that's just something to be careful of

when you are looking at a circuit

and you see things that
look like they're in series,

but they have little branches coming off.

So a little warning there.

So that's our series resistors.

If you have resistors and series,

you add them up to get
an equivalent resistance.