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All right. Good morning, all. So we take another big step forward today and get onto a new plane of understanding, if you will. In the last week and a half, our focus was on the storage element or storage elements called inductors and capacitors.

And capacitors stored change and inductors essentially stored energy in the field, the magnetic flux. And the state variable for an inductor was the current while that for a capacitor was the capacitor voltage.

We also looked at circuits containing a single storage element, we looked at RC circuits and we also looked at circuits containing a single inductor. And this was a single inductor with a resistor and a current source or a voltage source and so on.

What we are going to do today is do what are called "second-order systems." So they are on the next plane now. And with this second-order of systems, they are characterized by circuits containing two independent storage elements.

They could be an inductor and a capacitor or two independent capacitors. And you will see towards the end what I mean by two independent capacitors. If I have two capacitors in parallel, they can be represented as a single equivalent capacitor so that doesn't count.

It has to be two independent energy storage elements and resistors and voltage sources and so on. And what we end up getting is what is called "second-order dynamics." And much as first order circuits were represented using first order differential equations, this kind you end up getting second-order differential equations.

Before we go into this, I would like to start motivating this and give you one example of why this is important to study. There are many, many examples but I will give you one. What I would like to do is draw your attention to our good old inverter driving a second inverter.

The same circuit that we used to motivate RC studies, one inverter driving another. So let me draw the circuit. Here is one inverter. This is, let's say, 5 volts and this is, let's say, 2 kilo ohms.

And I connect the output of this inverter to a second inverter. And what we saw in the last few lectures was that in this specific example there was a parasitic capacitor or a capacitor associated with the gate of this MOSFET.

And that could be modeled by sticking a capacitor CGS between the gate of the MOSFET and ground. And we saw that the waveforms here, if I had some kind of step here. Let's say, for example, a step that went from high to low.

Then out here I would have a transition that instead of going up rapidly like this would transition a little bit more slowly. And this transition was characterized by an RC time constant. And this is what gave rise to a delay in the eventual output.

So that is what we saw previously, single energy storage element. Today what we are going to do is we are going to look at the same circuit, the exact same circuit, and have some fun with it. What we are going to say is look, this thing is pretty slow, so what I would like to do is — why don't we go ahead and put that up.

What we are going to see is that the yellow waveform is the waveform at the input here. And the green waveform here is the waveform at this intermediate node. And notice that this waveform here is characterized by the slowly rising characteristics that are typical of an RC circuit.

There are some other weirdnesses and so on going on here like a little bump and stuff like that. You can ignore all of that for now.

And capacitors stored change and inductors essentially stored energy in the field, the magnetic flux. And the state variable for an inductor was the current while that for a capacitor was the capacitor voltage.

We also looked at circuits containing a single storage element, we looked at RC circuits and we also looked at circuits containing a single inductor. And this was a single inductor with a resistor and a current source or a voltage source and so on.

What we are going to do today is do what are called "second-order systems." So they are on the next plane now. And with this second-order of systems, they are characterized by circuits containing two independent storage elements.

They could be an inductor and a capacitor or two independent capacitors. And you will see towards the end what I mean by two independent capacitors. If I have two capacitors in parallel, they can be represented as a single equivalent capacitor so that doesn't count.

It has to be two independent energy storage elements and resistors and voltage sources and so on. And what we end up getting is what is called "second-order dynamics." And much as first order circuits were represented using first order differential equations, this kind you end up getting second-order differential equations.

Before we go into this, I would like to start motivating this and give you one example of why this is important to study. There are many, many examples but I will give you one. What I would like to do is draw your attention to our good old inverter driving a second inverter.

The same circuit that we used to motivate RC studies, one inverter driving another. So let me draw the circuit. Here is one inverter. This is, let's say, 5 volts and this is, let's say, 2 kilo ohms.

And I connect the output of this inverter to a second inverter. And what we saw in the last few lectures was that in this specific example there was a parasitic capacitor or a capacitor associated with the gate of this MOSFET.

And that could be modeled by sticking a capacitor CGS between the gate of the MOSFET and ground. And we saw that the waveforms here, if I had some kind of step here. Let's say, for example, a step that went from high to low.

Then out here I would have a transition that instead of going up rapidly like this would transition a little bit more slowly. And this transition was characterized by an RC time constant. And this is what gave rise to a delay in the eventual output.

So that is what we saw previously, single energy storage element. Today what we are going to do is we are going to look at the same circuit, the exact same circuit, and have some fun with it. What we are going to say is look, this thing is pretty slow, so what I would like to do is — why don't we go ahead and put that up.

What we are going to see is that the yellow waveform is the waveform at the input here. And the green waveform here is the waveform at this intermediate node. And notice that this waveform here is characterized by the slowly rising characteristics that are typical of an RC circuit.

There are some other weirdnesses and so on going on here like a little bump and stuff like that. You can ignore all of that for now.

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