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## AC circuit analysis

Current time:0:00Total duration:8:56

# Impedance vs frequency

## Video transcript

- [Voiceover] In this
video we're gonna continue talking about AC analysis
and the concept of impedance as the ratio of voltage to
current and an AC situation, and just as a reminder of
the assumptions we've made for AC analysis we've assumed
that all of our signals are of the form of some kind
of a sinusoid like that, cosine is the typical one we pick, cosine omega T plus some offset angle. And we're also going
to use Euler's equation which says we can decompose our cosine into two complex exponentials, so one of the complex
exponentials is either the plus J omega T plus fi and J multiplies both of
them, plus e to the minus J omega T plus fi. And when we use Euler's
theorem what we do is we love putting these things
through differential equations so we track each of these
one at a time through our circuits and typically
what I'll focus on is the plus term, but the
minus term all the math is the same except there's
that one little minus sign. And so when we made up
the concept of impedance, we assumed that V or I was of this form, this complex exponential, and
that may, when we put that through our components
we came up with the idea of impedance, and as a
review for a resistor, the impedance of a resistor
turned out to be just R, that's the ratio of V to I in a resistor. And for an inductor, we decided Z of an inductor
was equal to J omega L. And finally for a capacitor of value C, Z C equals one over J omega C. So those are the three forms of, those are the three impedences
of our three favorite passive components. And the units of these,
the units are in ohms. R is measured in ohms,
that's the normal ohms law, and we use the same units
for impedance of inductors and capacitors so this is
in ohms and this is in ohms. That may seem a little funny
that because there's this frequency term in here,
but, this is what it is. So in the rest of this
video I want to look at qualitatively look at the
value of these impedences and specifically look at
what happens when there's a range of values of the omega term, what happens at zero
frequency and low frequency and high frequency and infinite frequency. So let's go ahead and do that. So we're gonna look at the
impedance terms at different frequencies and we'll
measure frequency at radiant frequency in this, so, let's
talk about zero frequency and we'll talk about low frequency, these are qualitative boundaries, we'll talk about high frequency, and let's talk about infinite frequency. So we're gonna build a little chart here. Alright so these are
the value of omega here. And now we'll do our
components, so I'm gonna do, first we'll do our resistor, and we're gonna fill in the table for ZR, and we know that equals R. So, at any frequency, R is
just R, couldn't be simpler. At zero frequency which is
just called DC or battery, R is R, at any low frequency, R is R, R is R, at infinite frequency, so there's
no dependence on frequency in R, so now let's do the inductor, and we decided that Z of
an inductor was J omega L. And let me do something
very specific, I'm gonna do, I'm gonna get rid of this
J, I'm gonna basically say I wanna just look at the
magnitude of the impedance. If we just look at the magnitudes, the magnitude of Z inductor is omega L. So now let's fill our
table in for omega L. So when omega is zero, the magnitude of the impedance
is zero for an inductor. And when the frequency is
low, when omega is low, the impedance is going
to be relatively low. And as the frequency gets high, then omega L becomes a larger
number, so it becomes high. And if the frequency, if we let
the frequency go to infinity then omega becomes infinity
and this becomes infinity. So we see an inductor from low
frequency to high frequency, the impedance of the inductor
goes from zero up to infinity. Alright let's fill in the last one here, here's our capacitor, and the Z of a capacitor
equals one over J omega C, and the magnitude of the
impedance is just 1 over omega C. Alright so let's plot that
out, what is 1 over omega C when omega is zero? Well it's infinity. And what if omega is a small number? If omega is a small number
then this is a large number. This is a high number. And if the frequency gets very
high, the higher omega gets the smaller the magnitude of Z gets, so we get low here, you
see it's anti symmetric with the inductor, and finally,
if we let the frequency go to infinity, 1 over
infinity times C is what? Is about, is zero. With this chart I can show you
some of the words that we use some of the sort of slang
words or jargon words that we use in electrical
engineering to describe the behavior of L and C
at different frequencies. So as we said, R is R at any frequency, so resistance is constant. An inductor it changes over frequency, over this big range of
frequency, and at zero frequency or low frequencies the
impedance is very low. So this leads to the
expression we say an inductor looks like a short. An inductor looks like a
short at low frequency, at zero frequency or low frequency. And now let's go to high
frequency, the impedance gets high and it eventually goes to infinity. That's the impedance of an open circuit. So we say L equals an open at high F, high frequency, and it's
a short at low frequency. Let's do the same sort of
jargon investigation here for our capacitor. A capacitor at zero frequency
has an infinite impedance. So it looks like what? It looks like an open at low frequency. So let's go look over at high frequencies, high frequencies, the
impedance becomes very low. And at infinite frequency
it becomes zero and that's the impedance of a perfect short. So the expression, the
slang expression you'll hear is that a capacitor is a short
circuit at high frequency. So the reason I'm telling you about this, very often experienced
engineers will talk to beginners and use these kinds of terms,
that oh an inductor looks like a short at low frequency. And I wanted to show you
where these things come from, where these terms come
from and what they mean. And this is not a simple idea,
you saw how many assumptions we made and how much work
we did to get to this idea of these sort of common
or familiar expressions for how our components work. I hope that helped for you
to see how this comes about.