Welcome back to our course on linear circuits.
And today, we're going to be talking about impedance.
And trying to apply the things that we learned last time about phasers into a
practical application. Something that we're doing for circuit
analysis, and seeing how useful it can be.
The aim for this lesson is to identify impedances, which is this mathematical
tool for analyzing linear circuits. That have have sinusoidal.
So again, in our previous lesson, we talked about sinusoids.
I reviewed some of the properties of those, as well as defined phasors.
And found how to calculate them. So, now we get to move onto impedance.
Impedance is as I mentioned before, somewhat analogous to resistance.
And it will become a major point that going forward and doing search analysis
and techniques in AC systems. For this lesson, our objective is to be
able to describe impedance. Calculate the impedance of resistors,
capacitors, and conductors and identify the relationship between the voltage and
the current based upon an impedance value.
So, first of all, the defining impedance. Impedance is, and you can see how similar
it is to resistance here. It's a way of relating the voltage and
the current. When our voltage or current sources are
sinusoidal. And so, here I've showed, and whenever I
use these block letters, this is corresponding to phasor.
So, now I have a phasor for my voltage, and a phasor for my current.
Impedance is not a phasor, but it relates the phasers to one another.
So remember, the phasers are complex numbers.
And so, if v is a complex number, and i is a complex number, and I know what they
are. Then, it's possible for me to find some
complex number z, that makes this equation work.
So then, we think about the ratio. And so, you'll notice that this is very
similar to Ohm's law, which Ohm's law stated that E is equal to I times R.
And so, that's really kind of what it is, is R impedance is going to be a relation
of our phasor voltage, to our phasor current.
So, let's replace this arbitrary impedance, this arbitrary z device with a
resistor, and see what we get. Well, if v is equal to some magnitude Vm
times the cosine of omega t plus theta, and I know what my R is.
Then, we can calculate my current i using i is equal to v divided by R.
So, i's then going to be just Vm divided by R times the cosine of omega t plus
theta. because it really doesn't care what's
going on in time or any of this type of changing stuff.
All it really needs to know is that this function is going to be scaled by R, and
then I get my current out. So, very simple.
Well, if I put these into phasor term, then V is going to be Vm arc theta, and I
is going to be im arc theta which is Vm over R arc theta.
So taking the ratio, I discovered that V over I, which is z, is simply R, no
resistance. So, let's clear all of this up.
And here we see the phasor for the voltage, the phasor for the current, and
that z sub R is equal to R. Just the resistance.
Very simple. In this case, our impedance.
Impedance has a complex number, but in this case it just happens to be real,
with a resistance, resistant element. It's just simply R.
So, pretty simple to calculate. Let's look at one that's a little bit
more tricky. What's the impedance of an inductor?
So again, we're going to be using some of these ideas of the behavior that we see
in the circuits, so we know that with the inductor, v is L di/dt, right?
And so again, we're going to assume that we have phasers for inputs.
And then, we're going to see if our voltage, for example, is a sinusoid, and
our current's a sinusoid, we have some known inductance.
We want to see the behavior between them and put them into phaser form.
So, this is what we see. If i is this function here, it has a
phaser like this. If V is a function, but we can calculate
these based on L. And so, if you plug in the i here for the
function i, we discover that V is equal to this.
For our phaser, it's going to be LI sub m omega arc theta plus pi halves.
Now, I'm not going to go through the derivation of how exactly this came out.
In order to do it, you need to apply a few trigonometric entities.
But I encourage you to try it as an exercise.
And if you're having trouble understanding where this came from go
ahead and post to the forums. If I take the ratio of those two
impedances, I discover that we get this L omega arc pi halves, which turns out is
equal to j omega L. So that means if I have an impedance, if
I want to calculate an impedence for an inductor, it's going to be this j omega
L. So a few important things to note, first
of all, this frequency is important. With resistors there was no frequency, it
was completely invarient of the frequency.
We also notice that it's purely imaginary.
There's no real term to this, that's all an imaginary thing.
So, what does this kind of mean? Well, if you think about it this way, as
I put a voltage onto this inductor here, it's going to cause current to start to
want to flow more. As I increase my voltage, my current's
going to try and match it. As I start to decrease my voltage, then
the current's going to kind of resist that change, but is eventually kind of
going to go down. And so again, if we go back and remember
from the phasers how we saw kind of the space difference between the current and
the voltage and a capacitor. The same thing happens with an inductor.
And so, you can go back and take a look and see the behavior.
And identify that when we have these complex terms that leads us to these
phase shifts. And we also have attenuations based upon
these frequencies for capacitors and inductors.
So to recap, inductors are purely imaginative impedances.
They scale based on frequency, and they're positive-imaginary, and
positive-imaginary values means that the current lags the voltage.
If it were negative-imaginary, current would lead the voltage, like a capacitor.
This puts all of them together. So, if you've gone through and you've
tried to derive them, this is kind of your little go-to page to make sure that
you got the right answers. We already did the, the derivation for
the resistors, and discovered that the impedance of a resistor is simply R.
And because of this behavior, we see that the voltages and the currents are
in-phase. And in all these pictures, I know it's a
bit small but red corresponds to currents and blue corresponds to voltages.
So for resistors, voltages and currents are zero in the same places and there is
maximum and minimum values in the same places.
And so is the frequency invariant. If I make this frequency much, much
higher, it doesn't matter. The red curve is going to have the same
amplitude as it does here. that's not, not an issue.
Looking, looking at a capacitor, we see that, as we saw in the previous lesson,
our current leads voltage. There's been a bit of a phase shift,
turns out that the impedance for capacitors with one over j omega C.
And this is a negative imaginary number. If you don't understand why it's
negative, go back and look at complex numbers.
But, it's important that you kind of understand.
Also as the frequency increases, the voltage attenuate which means the voltage
gets smaller. If I have a very very high frequency
system my voltages are going to be smaller in the capacitors.
And why is this? We know that the voltage can't
instanteously change on a capacitor, and the voltage basically is desribing the
amount of charge tha'ts on the plates. So, if we want to keep a high voltage
with a high frequency, we need a lot of current to put charge on to this place
very very quickly. And so, it's sometimes good to go through
and try and get an intuitive understanding of why this behavior
occurs. Now, I'm just going to go and quickly
recap for the inductors. The inductance gives us this j omega L
for our impedance. The current now lags the voltage in an
inductor, and current attenuates for high frequency.
And you can see why it's kind of analogous to the voltage attenuating for
high frequencies with capacitive circuits.
so this is kind of a good table, good point of reference.
I encourage you to go through and do some of these analyses yourself following the
structure that's been presented here. And see if you can understand where all
of these things are coming from. It's a good exercise.
To summarize, we defined impedance and we calculated impedance of linear devices
and even define, derive where it comes from.
Well, I must say it's kind of a tryst summary and inductor.
And then, we describe the relationships between the current and the voltage given
from the known impedance. In the next lesson, we'll start using all
of the things we've learned here. And look at our analysis techniques that
we learned in previous modules, and put them together and see what works.
What doesn't and what we can, what we can do going forward.
So, look forward to seeing you next time and seeing you on the forums.
Because you'll probably have some questions on the material that we covered
in these previous two lessons. Until then.