Well, in this video, we're going to break 
out of circuits consisting only of 
resistors and sources. 
We're going to talk about more generally 
circuits having capacitors and inductors. 
this will open the door so we can handle 
any circuit that we want. 
Now, if we go down the path we went 
through with resistive circuits, 
when we use the VR relations, KVL and 
KCL, it turns out, what you're going to 
wind up with, is, a differential equation 
to solve. 
I'll show you that. 
And that turns out out does not give us 
much insight into how the circuit is 
working, what they're doing, why were 
they designed this way, how do I change 
designs. We're going to introduce a new 
idea, something called impedance, 
which is going to allow us to think 
about,every element in the circuit as 
being a kind of resistor. 
That means we can use current divider, 
voltage divider all the techniques we've 
already learned with resistors. 
That's the point. Let's see how that 
works. 
So, here's a simple circuit. 
It's what's known in the trade as an RC 
circuit. 
And that terminology means it has one 
resistor and one capacitor usually. 
And here, they're connected in series. 
So, the source here is voltage of 
whatever it may be. 
And I've decided to take the output to be 
the voltage across the capacitor. 
Okay. 
So, what we want to do is find V out for 
any V in, that's the thing we have to do 
to solve the circuit. 
Well, let me remind you what the VR 
relation is for the capacitor. 
The current is proportional to the 
derivative of the voltage. 
So, we need to see how that comes in to 
solve the circuit. 
So remember, that current i has to be, 
for this VR relation, has to be going in 
the positive side of voltage, and V is 
now V out for this circuit, that's what 
we're playing with. 
So, what I'm going to do is write KVL. 
I'm going to start here and go around 
like this. 
And we know from KCL that the current 
through the resistor is also i. 
So, when I write the KVL equation, V out 
is equal to, 
but I want to go plus to minus this way. 
And that's voltage in that direction is 
-Ri, 
okay? From the VR relation. 
The current's going in the negative side 
not the positive side, 
we have to flip the sign. Plus the, 
well now, I'm going to substitute for i 
in there and you get -RC dV out, dt + V 
in. 
Okay. So, if I bring the derivative over 
to the other side, 
we wind up with the input/output 
relationship for this circuit. 
And it's what's known as an implicit 
relationship. 
This is the relationship you have to 
solve something. 
It isn't directly given to you like it 
was for the resistor circuits. 
So, RC times the derivative V out, 
plus V out has got to be V in. 
So now, what you have to do is you tell 
me what V in is. 
It could be a sinusoid, pulse, whatever. 
And now, I have to solve this 
differential equation for each case. 
Well, it turns out that's a little 
tedious and doesn't really tell us what's 
going on. 
So, I'm going to take a different 
approach. 
We're, so we're not going to solve 
differential equations in this course. 
We're going to sneak around them and 
figure out a better way to solve them. 
So, this is a very different approach 
which is based on a somewhat surprising 
assumption. 
Just suppose, just suppose the voltage 
source is a complex exponential. 
So, V in of t is equal to capital V in, 
this is known as the complex amplitude. 
Could be a complex number, that is just a 
number. 
The only pa, par, part it varies with 
time is the complex exponential. 
You do the j 2 pi ft. 
Okay. 
So, we're just going to assume that for 
right now. 
Well, here's the claim. 
That every voltage in the circuit, 
once you solve it for, for the complex 
exponential source, also turning out to 
be complex exponentials. 
Every voltage and every currents at that 
form. 
So, what I want to do now, is check to 
see if such a solution is consistent with 
the VI relations, 
and with KVL and KCL. 
So, let's do the VI relations. 
So let's start with our resistor with a 
VI relation whose voltage is equal to the 
resistance times the current. 
And I'm just going to plug in the voltage 
and the current in here. 
And it turns out it is consistent as long 
as the camp, complex samplitude of the 
voltage is relate, is R times the complex 
amplitude of the current. 
The other thing to notice is, I can 
cancel these complex exponentials. And 
there is explicitly relationship between 
the current and voltage complex 
amplitudes. 
This is an interesting point about this 
cancellation. 
So, let's talk about a capacitor, i = C 
dv / dt. 
So, here's the current i. 
And the derivative of this is just j 2 pi 
f times the same thing because of the 
properties of the exponential. 
And we need to plug in the capacitor 
values because that's what's over here. 
So again, with cancellation, we now find 
that this assumption is consistent as 
long as the amplitudes are related this 
way. 
The amplitude is current = c * j2pif 
times the amplitude of the voltage. 
And I'm pretty sure you can believe that 
for the inductors, some other kind of 
thing is going to happen. 
Amplitude is related by a different 
formula, but still they're related. 
Okay. This cancellation's really 
important. Because let's think about 
defining a quantity Z for each element 
that is the ratio of the complex 
amplitude of the voltage divided by the 
complex amplitude of the current. 
This is known as impedance. 
So, we're assuming that the source is a 
complex exponential at some frequency. 
And we're going to, we now know that 
every voltage and current in the circuit 
is also a complex exponential. 
Well, let's sort of hide the complex 
exponential for a while and just look at 
the complex amplitudes. 
We're going to define Z, the impedance to 
be V/I. 
I want to point out that Volt, V as in 
Volts, and I as in Ampere, so the ratio 
is Ohms, 
okay? So, impedence has got units of 
ohms. 
However, for the capacity and the 
inductor, they're complex valued. 
So, the impedience of a resistor is just 
the resistance, 
but the impedance of the capacitor is 
1/j2pifc, impedance of the inductor is 
j2pifl. 
Alright? So, what this means is that we 
can think about every element in the 
circuit, resistors, capacitors, 
inductors, as being complex valued 
resistors. 
Which is going to be the key thing. 
But we've now just checked the VI 
relations. 
We've got to check KVL and KCL before we 
get to figure out the consequences of 
that. 
So, it looks, it's looking good. 
But when you look at KVL and KCL, you see 
that everything hangs together again. 
Because if every voltage in the circuit 
is a complex exponential always at the 
same frequency, 
but they have different amplitudes. 
you can cancel that common complex 
exponential to find the result that the 
complex amplitude of the voltage around 
any loop equal zero, 
which means it obeys KVL. 
And same thing for the currents, 
the complex amplitudes of the currents 
obey KCL. So, we now have a very 
consistent picture. 
When the sources are complex 
exponentials, so they look like the V in 
we've just had, it turns out all the that 
we had just discovered that all currents 
and voltages are complex exponentials. 
If you had solved that differential 
equation for the input voltage being a 
complex exponential, you would have 
discovered that if you assumed V out was 
complex exponential, 
yeah, you get, you get that answer. 
It's a consistent solution. 
Once you have a solution to a 
differential equation. 
No matter how you got it, you know it's 
the only solution. 
That's the way differential equations 
work. 
So, 
all sources and elements, and element 
voltages and currents, can be considered 
just to be complex amplitudes because 
we're going to assume the source 
frequency is understood. 
We're going to assume the complex 
exponential is setting there, 
that's the cancellation I mentioned. 
And when we do that, all elements can be 
considered as complex value resistors the 
technical word is they are impedance. 
That is really makes our life a whole lot 
easier as I can, let's go back to our RC 
circuit. 
So, we're going to talk about our RC 
circuit. 
And, instead of it being out of real 
world, we're going to consider it as 
being built out of impedances. 
So, 
be sure you notice something in the 
notation. 
Over here, this is a lowercase v in of t. 
We're assuming that's v in e^j 2pift. 
This is a lowercase b which is going to 
be something which we know is going to be 
an uppercase V out e^j 2pift. 
So these are uppercase, and these are 
lowercase. 
So, what you do is you think about 
everything as being a impedance of the 
resistors, capacitors, inductors, are 
impedance's. 
The source is a constant given by the 
complex amplitude of the complex 
exponential back here. 
So, time is gone. 
There is no time variation in this 
circuit, 
and we have a V out. And now, we want to 
figure out what the amplitude of the 
output is given the amplitude of the 
input. 
Well, now there are things resistor we 
can use voltage dividers. 
What could be simpler? You're going to 
use the, instead of resistor values, 
we're going to use impedence values in 
voltage divider. 
So, the voltage divider says that the 
output voltage is equal to the impedance 
of the capacitor divided by the sum of 
the impedances of the resistor and the 
capacitor. 
Just like we had before with the 
resistors. 
Plug in what your impedances are and 
simplify a little bit, we now know what V 
out is. 
It is equal to the amplitude of source 
times a complex number, 1/j2pifRC + 1. 
So, what could be simplier? Now to 
complete the story, we now know that this 
is true. We put back in the complex 
exponentials, we will uncancel the 
complex exponential that was lurking in 
the setup from the very beginning. And 
now, we know that when the input is a 
complex exponential, the output is also a 
complex exponential where V out is equal 
to this quantity. 
We have solved the circuit for the case 
of a complex exponential source. 
That's the way the impedance gain works 
if you will. 
So, when the circuit consists of sources 
and any number of resistors, capacitors, 
inductors, it doesn't really matter. 
you can pretend the sources are complex 
exponentials having some frequency, 
f. Many consider each element to be an 
impedance, a complex value resistor. 
And now you can use all the tools we've 
already had, 
voltage divider, current divider, 
series/parallel rules, to figure out how 
the output's complex amplitude is related 
to the complex amplitude of the source. 
Okay, so I'm going to show you the 
consequences of this. You may be a little 
concerned that what about this pretend 
part? suppose my source isn't a complex 
exponential, which is usually the case. 
You're not going to, you're going to have 
a hard time generating a complex 
exponential with any real world source. 
Well, it turns out we're not really going 
down a blind alley. 
This is going to open the door to a whole 
new way of thinking about things, and 
we'll get that in succeeding videos.