Welcome back to Electronics.
This Dr. Ferri.
This lesson will be a review of impedance.
In the previous lesson, we did a review of Kirchhoff's laws.
In this lesson, we will look at impedances.
In particular, impedance is a,
a method that we use for steady-state sinusoidal inputs.
In other words, what we call AC, or alternating current, inputs.
So we can define the impedance for our basic elements in R, C, and in L.
With a resistor, if I look at the voltage current behavior.
So, say,
this is my voltage across a resistor, and then my current is going in this way.
Then if I have it a sinusoidal input or sinusoidal voltage across this resistor,
my current's going to be sinusoidal, and in particular it's in-phase.
The zero crossings are at the same time, the peak happens at the same time.
We often say that, that and that doesn't matter what frequency the input is.
So we can say that this is frequency invariant that output is, or
the evolved current is always in face with the voltage.
Now there's a couple of things that we're going to be needing here so
I want to define them.
One here is the period.
That's how, when it, how long it takes to repeat itself, and we define the frequency
as be in hertz, being cycles per second,
a cycle being one time that it, it repeats itself.
That is one over T, which we're going to define as F.
My variable F, as in hertz, is a frequency.
There's also a frequency variable, in radians per second.
It's a different unit, and it is defined as 2 pi F.
Or in terms of T, it's 2 pi over T radians per second.
And we use the symbol omega to represent frequency in radians per second.
And I need all that once I go to my other variables, my, my C impedance and
my L impedance, or I'll call it Z sub C.
And I'm going to define it this way.
That is, 1 over J omega, omega being the frequency and radiance per second.
J being my symbol for the square root of minus 1, and then C is the capacitance.
So if I define my voltage this way and my current going in this way,
I can represent for a sinusoidal voltage, my steady state current looks like this.
Notice that the current, this being the current I and
the voltage V, the current leads the voltage.
Leads means it comes before.
So it comes before this.
So that's my impedance for a capacitor.
For an inductor, it's J times omega times L.
And again, if I plot the voltage current relationship for
a sinusoidal voltage, if this is the voltage in this plot, and
the current is in this plot, then what we say is that the current lags the voltage,
it comes after it, looks like it's delayed.
So these are the three impudence's that I'm going to need a great deal when I'm
analyzing circuits with an alternating current, another with a sinusoidal input.
Let's look at impudence's in series.
Here's the good thing about impedances.
They're messy because they involve complex numbers,
but they use all the same rules that we've already defined for resistors.
So, impedances in series, I treat just like they were resistors in series.
So, this circuit is equivalent to a circuit.
With one impedance, and that impedance
is equal to the sum of the other ones, Z1, plus Z2, plus Z3.
So, impedances in series, they just add.
So it could be 3, it could be a lot more than 3, it's just a summation of
all impedances in the series, become our, our equivalent impedance.
Impedances in parallel.
Well, I treat them just like resistors in parallel,
which I get by inverting 1 over each one of them.
And it's usually easier to think of it in terms of, if I have two resist,
resistors in parallel, in other words if, if this resistance isn't there, if it's
if I only have these two impedances, so I'm going to write Z1 in parallel with Z2.
If I look at this expression and Z3 is gone, then all I've got is two terms.
I can simplify it, to Z1 times Z2,
over Z1 plus Z2.
So, that's two impedance in parallel with one another.
Kirchhoff's Laws still work.
Kirchhoff's current law says that all the currents going into
a node, have to equal all the current leaving the node.
So the currents going in I1, is equal to the leaving, I1 plus or I2 plus I3.
And in terms of Kirchhoff's Voltage Law, I can look at a,
going around in a loop this way and sum up all my voltages.
And in my case, when I sum up voltages around a loop, I use a trick and
I just say, in order to get my polarity right,
whenever I hit a minus sign first, I subtract that minus VS.
And going around, I hit a plus sign, this way, plus V1.
Going around this way, I hit a plus sign, plus V2.
Going around this way, I hit a plus so plus V3 is equal to 0.
In other words, I can rewrite it down this way, right here.
So Kirchhoff's Laws still work.
The only thing different about all these laws is,
now we're working with complex numbers.
So again, impedances treat just as if they were resistors.
The only thing is, they're going to be complex, complex numbers.
Let's look at an example of this.
How to analyze this circuit with impedances.
The first thing I'm going to do,
is replace this circuit with impedances, Z sub R.
And 1 over Z sub C, representing the capacitance.
I've got my input voltage, and my output voltage.
I'm going to use the voltage divider law here.
The voltage divider law says that V out is equal to Z sub C,
over ZR plus Z sub C, times the input.
If I plug in for Z sub C, I get 1 over J omega C.
If I plug in for Z sub R, I get R,
plus 1 over J omega C times Vi.
If I clear my fractions, I will get 1 over RC,
J omega plus 1 times Vi, and that gives me V out.
So once I know what Vi is, then I can solve for V out.
Now let's take a look at a series RLC circuit.
First thing I do is redraw the circuit in terms of its impedances.
So I get Z sub R, in series with the Z sub L,
in series with the Z sub C.
And I'm looking for V out, V sub 0, in terms of V sub i.
I can use a voltage divider log in.
V out is equal to Z sub C over ZR,
plus ZL plus ZC times Vi.
If I substitute in for Z sub C, I substitute in 1 over J omega C.
R is for Z sub R, and J omega L, for Z sub L.
And then I've got my other Z sub C.
If I clear my fraction here, I'm going to get a one in the numerator.
RC, J omega, plus J omega squared, LC,
plus 1 in the denominator, times Vi.
And remember that J is the square root of minus 1, so J squared is minus 1.
So that's 1 over RC J omega minus omega squared LC plus 1,
all times Vi is equal to V out.
If I know what V sub i is, then I can solve for V out.
And the other thing about this,
it's implicit in here that I know what omega is.
Omega is the input voltage, so
if we say Vi is equal to Ai cosine of omega T,
that omega is input frequency.
Maybe 100 hertz, in which case that would be 2 pi times 100 radians per second.
It could be whatever frequency we want, but
it is a particular frequency to be able to solve for a particular value of V out.
So in summary, we've introduced the KVL and KCL,
applied the KVL to parallel elements and series elements.
And we solve simple circuits using Kirchhoff's Laws.
All of this is applied to circuits with impedances.
The bottom line is, an impedance is, acts like a complex resistance.
All of our standard laws apply,
the only thing is it's messier because it's complex numbers.
In our next lesson, we will do a review of transfer function.
Thank you.