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— will try it again at the end of this lecture and you show you that stuff hopefully next time. For today we are going to start with nonlinear analysis. Before we do that I wanted to do a little bit of review.

I wanted to give you the past three weeks in perspective and show you how all of these things fit into the grand scheme of things. We began by building a great little playground, and within that playground we said that by enforcing upon ourselves the lumped matter discipline we created the lumped circuit abstraction.

So within that playfield we assumed that we had dq by dt and d phi by dt to be 0 so that gave us as the lumped circuit abstraction. And within that lumped circuit abstraction, within this playground we looked at several methods of analyzing circuits, including the KVL, KCL method.

We also learned the method involving composing resistors, the voltage dividers and so on and solving circuits intuitively. And we also looked at the node method, which is kind of the workhorse of the circuits industry.

So when in doubt apply the node method and it will get you where you want to go. Now, we also said that this is good, here is our playground. We said hey, if we focus on those circuits that are linear we come to the left part of our playground.

And we said that for linear circuits in this part of the playground we can further use a couple of techniques, a few techniques, superposition, Thevenin, Norton and so on. So these techniques allow you to very quickly analyze complicated circuits, especially when you're looking to find a single current, or voltage or some parameter of interest.

Whenever you see, if you see a circuit containing multiple voltage sources or two or more voltage sources or current sources, as a first step think superposition. And so these are very powerful techniques that let you analyze very complicated circuits very effectively.

After we did this we said, oh, let me draw another playground here. This is another piece of our playground. And if these are linear circuit then this half of the playground is nonlinear circuits. And we said that if you further focus on discretized values, if you discretized values and focused only on circuits that dealt with binary signals, highs and lows, then we came into this small regime of the playground.

And notice that digital circuits are, by their very nature, nonlinear. Remember the circuit, A, B, this was one of our NOR gate circuits? And if you look at transfer functions, that is if I plot, let's say for example, for some combination of input values.

Let's say I plot v in verses v out. Let's say, for example, I turned this guy off by setting B to 0 and then I simply apply a low to high transition at v in, then what I would see at the output is a transfer function of the following sort where as v in changes the output switches at some point and then stays at a low value.

So when v in is low v out is high and v in and high v out is low. So that's kind of the v out versus v in when B is set at 0. So notice that this is a nonlinear curve. This is not a straight line.

It's a nonlinear curve. And so therefore in the digital domain we see highly nonlinear functions that look like this and so on. However, take a look at this circuit. Suppose I focus on the circuit for a given set of switch settings.

Let's say, for example, I focus on the circuit when A and B are both 1s. For a given set of switch settings, notice that I'm going to be either in this region or in this region.

I wanted to give you the past three weeks in perspective and show you how all of these things fit into the grand scheme of things. We began by building a great little playground, and within that playground we said that by enforcing upon ourselves the lumped matter discipline we created the lumped circuit abstraction.

So within that playfield we assumed that we had dq by dt and d phi by dt to be 0 so that gave us as the lumped circuit abstraction. And within that lumped circuit abstraction, within this playground we looked at several methods of analyzing circuits, including the KVL, KCL method.

We also learned the method involving composing resistors, the voltage dividers and so on and solving circuits intuitively. And we also looked at the node method, which is kind of the workhorse of the circuits industry.

So when in doubt apply the node method and it will get you where you want to go. Now, we also said that this is good, here is our playground. We said hey, if we focus on those circuits that are linear we come to the left part of our playground.

And we said that for linear circuits in this part of the playground we can further use a couple of techniques, a few techniques, superposition, Thevenin, Norton and so on. So these techniques allow you to very quickly analyze complicated circuits, especially when you're looking to find a single current, or voltage or some parameter of interest.

Whenever you see, if you see a circuit containing multiple voltage sources or two or more voltage sources or current sources, as a first step think superposition. And so these are very powerful techniques that let you analyze very complicated circuits very effectively.

After we did this we said, oh, let me draw another playground here. This is another piece of our playground. And if these are linear circuit then this half of the playground is nonlinear circuits. And we said that if you further focus on discretized values, if you discretized values and focused only on circuits that dealt with binary signals, highs and lows, then we came into this small regime of the playground.

And notice that digital circuits are, by their very nature, nonlinear. Remember the circuit, A, B, this was one of our NOR gate circuits? And if you look at transfer functions, that is if I plot, let's say for example, for some combination of input values.

Let's say I plot v in verses v out. Let's say, for example, I turned this guy off by setting B to 0 and then I simply apply a low to high transition at v in, then what I would see at the output is a transfer function of the following sort where as v in changes the output switches at some point and then stays at a low value.

So when v in is low v out is high and v in and high v out is low. So that's kind of the v out versus v in when B is set at 0. So notice that this is a nonlinear curve. This is not a straight line.

It's a nonlinear curve. And so therefore in the digital domain we see highly nonlinear functions that look like this and so on. However, take a look at this circuit. Suppose I focus on the circuit for a given set of switch settings.

Let's say, for example, I focus on the circuit when A and B are both 1s. For a given set of switch settings, notice that I'm going to be either in this region or in this region.

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