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Series resistors | Circuit analysis | Electrical engineering | Khan Academy

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    - [Voiceover] Now that
    we have our collection
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    of components, our favorite
    batteries and resistors,
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    we can start to assemble
    these into some circuits.
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    And here's a circuit shown here.
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    It has a battery and
    it has three resistors,
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    and a configuration that's called
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    a series resistor configuration.
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    Series resistors is a familiar pattern,
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    and what you're looking for is resistors
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    that are connected head
    to tail, to head to tail.
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    So these three resistors are in series
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    because their succession of nodes
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    are all connected, one after the other.
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    So that's the pattern that tells you
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    this is a series resistor connection.
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    So we're gonna label
    these our resistors here.
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    We'll call this R1, R2, and R3.
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    And we'll label this as v.
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    And the unknown in this
    is what is the current
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    that's flowing here, that's
    what we want to know.
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    We know v, we want to know i.
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    Now one thing we know about i
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    is i flows down into resistor R1,
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    all of the current goes out of
    the other end of resistor R1
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    because it has to, it
    can't pile up inside there.
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    All that goes into here,
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    and all that comes out of R3.
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    And i returns to the place it came from,
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    which is the battery.
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    So that's a characteristic
    of series resistors,
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    is in a series configuration
    is they are head to tail,
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    and that means that all the components,
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    all the resistors share the same current.
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    Current.
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    That's the key thing.
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    The thing that we don't know
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    that's different between each resistors,
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    is the voltage here, and the voltage here,
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    let's call that v1,
    this is v2, plus, minus,
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    and this is v3, plus, minus.
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    So in general, if these
    resistors are different values
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    because they have the same
    current going through them,
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    Ohm's Law tells us these
    voltages will all be different.
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    So the question I want to
    answer with series resistors
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    is could I replace all three of these
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    with a single resistor that
    cause the same current to flow?
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    That's the question we have
    on the table right now.
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    So we make some observations,
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    we have Ohm's Law, our friend, Ohm's Law.
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    And we know that means v equals i times R,
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    for any resistor.
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    That sets the ratio of voltage to current.
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    And this is another
    thing we know about this,
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    which is that v3, plus v2, plus v1,
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    those are the voltages
    across each resistor,
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    those three voltages have
    to add up to this voltage
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    because of the way the
    wires are connected.
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    So the main voltage from the battery
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    equals v1, plus v2, plus v3.
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    We know that's for sure,
    and now what we're gonna do
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    is we're gonna write Ohm's Law
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    for each of these individual resistors.
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    v1 equals i, and i is
    the same for everybody,
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    times R1.
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    v2, this voltage here, equals i times R2.
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    And v3 equals i times R3.
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    Now you can see, if I had four,
    or five, or six resistors,
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    I would have four, or five, or
    six equations just like this
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    for each resistor that was in series.
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    So now what I'm gonna do is
    substitute these voltages
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    into here, and then we'll
    make an observation.
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    So let's do that substitution.
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    I can say v equals i, R1,
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    plus i, R2, plus i, R3.
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    And because it's the
    same i on every resistor,
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    I can write v equals i,
    I'm gonna factor out the i.
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    R1, plus R2, plus R3.
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    Now what I want to do
    is take a moment here
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    and compare this expression
    to this one here,
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    the original Ohm's Law.
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    Alright, there's Ohm's Law.
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    So we have v equals i, some current,
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    times some resistor.
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    I can come up with a resistor value,
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    a single resistor that would
    give me the same Ohm's Law.
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    And that is gonna be called,
    let's draw it over here.
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    Here's our battery.
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    And I'm gonna say there's a
    resistor that I can draw here,
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    R series, that's equivalent
    to the three resistors here.
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    And it's equivalent in the sense that
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    it makes i flow here, that's
    what we mean by equivalent.
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    So in our case, to get the
    same current to flow there
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    I would say v equals i times R series,
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    in which case, what I've done
    is I've said that R series
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    is what, is the sum of these three things,
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    R1 plus R2, plus R3.
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    This is how we think
    about series resistors.
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    We can replace a set of series resistors
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    with a single resistor
    that's equivalent to it
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    if we add the resistors up.
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    Let's just do a really fast
    example to see how this works.
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    I'm gonna move this screen.
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    Here's an example with three resistors.
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    I have labeled them 100
    ohms, 50 ohms, and 150 ohms.
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    And what I want to know
    is the current here.
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    And we'll put in a voltage,
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    let's say it's 1.5 volts,
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    just a single small battery.
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    So what is the equivalent resistance here?
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    One way to figure this out
    and to simplify the circuit
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    is to replace all three of those resistors
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    with a series resistor, RS,
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    and that is, as we said here, is the sum,
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    so it's 100, plus 50, plus 150.
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    And that adds up to 300 ohms.
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    So that's the value of the equivalent
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    series resistor right here.
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    And if I want to calculate the current, i,
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    i equals v over R, and
    this case, it's R series,
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    and that equals 1.5 divided by 300.
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    And if I do my calculations right,
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    that comes out to .005 amperes.
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    Or an easier way to say it is
    five milliamps, milliamperes.
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    So that's i.
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    And now that I know i, I can
    go ahead and I can calculate
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    the voltage at each point
    across each resistor
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    because I know i, I know R,
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    I can calculate v.
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    So v1, v1, which is the
    voltage across that resistor,
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    v1 equals i, R1, as we said before.
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    So it's five milliamps
    times 100 ohms, 0.5 volts.
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    Let's do it for the other one, v2,
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    equals i, same i, this time times R2,
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    five milliamps times 50 ohms,
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    and that equals 0.25 volts.
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    And finally, we do v3.
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    This is plus, minus v3.
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    And that equals the same
    current again times 150 ohms,
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    which is equal to 0.75 volts.
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    So we've solved the
    voltage and the current
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    on every resistor, so this
    circuit is completely solved.
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    And let's do one final check.
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    Let's add this up.
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    Five, five (mumbles) is zero.
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    Carry the one, six, seven eight.
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    15, 1.5 volts, and that's very handy
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    because that is the same as that.
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    So indeed, the voltages across
    the resistors did add up
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    to the full battery that was applied.
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    There's one more thing
    I want to point out.
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    Here's an example of
    some series resistors.
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    And that's a familiar pattern.
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    And you'll say, "Oh, those
    are series resistors."
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    Now, be careful because if
    there's a wire here going off
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    and there's, doing this,
    or there's a wire here,
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    connected to this node here,
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    this still looks like they're in series,
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    but there might be current flowing
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    in these branches here.
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    If there's current flowing out
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    anywhere along a series branch,
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    anywhere along what looks
    like a series branch,
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    then this i may or may
    not be the same as this i.
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    And it might not be the same as this.
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    So you gotta be careful here.
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    If you see branches going
    off your series resistors,
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    these are not in series
    unless these are zero current.
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    If that's zero current, and
    if that is zero current,
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    then you can consider these in series.
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    So that's just something to be careful of
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    when you are looking at a circuit
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    and you see things that
    look like they're in series,
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    but they have little branches coming off.
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    So a little warning there.
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    So that's our series resistors.
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    If you have resistors and series,
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    you add them up to get
    an equivalent resistance.
Title:
Series resistors | Circuit analysis | Electrical engineering | Khan Academy
Description:

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Video Language:
English
Team:
Khan Academy
Duration:
11:57

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