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.