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.