- [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.