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OK. Good morning. In the last lecture I did a little demonstration for you where I showed you a pair of inverters. And showed you that the output of the first inverter looked weird, certainly not like anything we have seen thus far.

It looked like a slow rising transition like this. And using that motivation we have begun our study of RC circuits. And in particular for today the lecture is titled "Digital Circuit Speed." We are going to look at the fundamentals of digital circuit speed.

And it all boils down to an RC delay. By the end of the lecture, I am going to show you two numbers that you can look at a circuit and obtain by observation, multiply them out and you will get a good idea of the speed at which a circuit will run.

It is pretty amazing. So as a quick review — The relevant section for this is Chapter 10.4. As a review, we said to understand things like this we need to develop the foundations for RC circuits.

And the example I covered was that of a very simple circuit that looked like this — An RC circuit of this form. And I also showed you that for an input of the form, input that steps from zero to VI at time T equal to zero.

And assuming that the capacitor state at time T equals zero was zero. What this means is that the capacitor starts from rest, so at time T=0, oops, this is VI, I'm sorry. So we assume that the capacitor starts from rest.

At time T=0 I apply a VI step, capital VI. And then I want to look at how the voltage across the capacitor behaves. And we did a bunch of analysis. And at the end of the day, in the final demo in the lecture last time I showed you that the capacitor would behave like this.

It would start off at, oops. I am sorry. This should be, let's assume that started off at VO. We get a different equation for zero. So let's say the capacitor started off at VO, in which case VC at time T=0 is VO as expected.

And we showed that the output would look something like this. After a long period of time this would come up to VI and this rise had a time constant of tau=RC. So we wrote the equation for this waveform.

And this is the case when VI is greater than VO. I would like you to stare at the circuit and this result here to get more intuition on what is going on. At time T=0, VC starts off at VO as expected because I am telling you that is the case, that is initial condition.

It starts off at VO. Then this one steps to VI. There is no infinite transition anywhere here, and so the capacitor holds its voltage at VO, at time T=0. And then the VI here, which is greater than VO, begins to charge the capacitor up, charge it through this resistor.

And so therefore the capacitor charges up. After a long period of time, from the basic foundations of capacitors, we know that the capacitor appears like a long-term open circuit to DC. This is a DC voltage VI.

So it appears like an open circuit. So after a long period of time VI appears at the end. And from here to here I have an exponential rise that is typified by an equation of the form -t/RC. This kind of waveform rising from a smaller value to a higher value is typified by this expression.

We saw the expression when we developed the equations last time.

It looked like a slow rising transition like this. And using that motivation we have begun our study of RC circuits. And in particular for today the lecture is titled "Digital Circuit Speed." We are going to look at the fundamentals of digital circuit speed.

And it all boils down to an RC delay. By the end of the lecture, I am going to show you two numbers that you can look at a circuit and obtain by observation, multiply them out and you will get a good idea of the speed at which a circuit will run.

It is pretty amazing. So as a quick review — The relevant section for this is Chapter 10.4. As a review, we said to understand things like this we need to develop the foundations for RC circuits.

And the example I covered was that of a very simple circuit that looked like this — An RC circuit of this form. And I also showed you that for an input of the form, input that steps from zero to VI at time T equal to zero.

And assuming that the capacitor state at time T equals zero was zero. What this means is that the capacitor starts from rest, so at time T=0, oops, this is VI, I'm sorry. So we assume that the capacitor starts from rest.

At time T=0 I apply a VI step, capital VI. And then I want to look at how the voltage across the capacitor behaves. And we did a bunch of analysis. And at the end of the day, in the final demo in the lecture last time I showed you that the capacitor would behave like this.

It would start off at, oops. I am sorry. This should be, let's assume that started off at VO. We get a different equation for zero. So let's say the capacitor started off at VO, in which case VC at time T=0 is VO as expected.

And we showed that the output would look something like this. After a long period of time this would come up to VI and this rise had a time constant of tau=RC. So we wrote the equation for this waveform.

And this is the case when VI is greater than VO. I would like you to stare at the circuit and this result here to get more intuition on what is going on. At time T=0, VC starts off at VO as expected because I am telling you that is the case, that is initial condition.

It starts off at VO. Then this one steps to VI. There is no infinite transition anywhere here, and so the capacitor holds its voltage at VO, at time T=0. And then the VI here, which is greater than VO, begins to charge the capacitor up, charge it through this resistor.

And so therefore the capacitor charges up. After a long period of time, from the basic foundations of capacitors, we know that the capacitor appears like a long-term open circuit to DC. This is a DC voltage VI.

So it appears like an open circuit. So after a long period of time VI appears at the end. And from here to here I have an exponential rise that is typified by an equation of the form -t/RC. This kind of waveform rising from a smaller value to a higher value is typified by this expression.

We saw the expression when we developed the equations last time.

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