2036 lines
56 KiB
Plaintext
2036 lines
56 KiB
Plaintext
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],
|
|
"text": [
|
|
"All right. Good morning,",
|
|
"all.",
|
|
"You have two handouts, lecture notes and an article on",
|
|
"mixed signal chips. A mixed signal stands for",
|
|
"circuits that have both analog and digital components to them.",
|
|
"The reason I am giving you the handout is that Lab 4 and also",
|
|
"your last homework involve designing and building a mixed",
|
|
"signal circuit. It's a real fun exercise.",
|
|
"And I just wanted to tell you that from past experience people",
|
|
"who have taken 6.002 often view the last lab as the single most",
|
|
"fun thing they did in all of 6.002.",
|
|
"So, as you go into Lab 4, you should be telling yourself",
|
|
"I should be having fun, I should be having,",
|
|
"I should be having fun. You have to positively psych",
|
|
"yourself. Otherwise, it's going to go by.",
|
|
"And then you're going to say boy, that was fun,",
|
|
"I wish I had savored the moment as I was doing it.",
|
|
"All right. Let's see.",
|
|
"What do we do today? Today's lecture is actually",
|
|
"going to be a fair amount of fun.",
|
|
"We are going to blast through a bunch of fun things.",
|
|
"And some things that you will be quite unprepared for.",
|
|
"Until now, in the last two lectures with op amps we talked",
|
|
"about negative feedback. That is applying some portion",
|
|
"of the output voltage to the negative input so that I could",
|
|
"control this high strung device, my op amp.",
|
|
"Today, what we are going to do is try to get a handle on what",
|
|
"happens if we use positive feedback.",
|
|
"It's the usual curious child. You tell them to do this,",
|
|
"and of course they're going to try to do this as well.",
|
|
"And we are going to try to do that and see what happens and",
|
|
"look to see if we can build some useful circuits.",
|
|
"Today --",
|
|
"As motivation, let me do a quick review of a",
|
|
"circuit that should now become affixed in your brains in a",
|
|
"standard pattern. This is a circuit that gives",
|
|
"you negative feedback.",
|
|
"R1 and R2.",
|
|
"And I apply a vIN. By now you should be able to",
|
|
"look at this pattern. And this is your inverting",
|
|
"amplifier pattern. So, you should be able to write",
|
|
"down by inspection this is simply vIN or the minus vIN",
|
|
"times R2 divided by R1. This is an amplifier whose gain",
|
|
"is controlled by the ratio of R2 and R1.",
|
|
"This is a negative feedback circuit because it is always fun",
|
|
"to do the intuition thing and say that look,",
|
|
"if this voltage tends to go more positive than I care then",
|
|
"this negative input goes more positive than I care.",
|
|
"If that goes more positive then the negative input v minus",
|
|
"becomes more positive in the plus input which yanks the",
|
|
"output down. So, there is a nice",
|
|
"counteracting force that keeps the output stable.",
|
|
"Let's look at this circuit. Being curious engineers,",
|
|
"let's look at the opposite here where I give myself some",
|
|
"positive feedback in this op amp.",
|
|
"And it is going to be interesting to analyze this",
|
|
"because what we find out on the face of it is not quite actually",
|
|
"how it behaves. We are going to spend most of",
|
|
"the lecture today on understanding the dynamics of",
|
|
"circuits that look like this and to see if we can build some fun",
|
|
"and interesting circuits and systems based on this kind of",
|
|
"positive feedback. It is positive feedback because",
|
|
"I am feeding back a portion of the output to the positive",
|
|
"input. And you should be able to stare",
|
|
"at this and already begin to intuit what should happen to",
|
|
"this. Let's think about it.",
|
|
"This is zero. Remember, with positive",
|
|
"feedback, the famous v plus is equal to v minus method doesn't",
|
|
"apply anymore. Let's apply very simple",
|
|
"analyses. If this is zero,",
|
|
"let's say for example that this output tends to go a little bit",
|
|
"more positive. This output,",
|
|
"due to some noise or perturbation,",
|
|
"tends to go up a little bit. If that goes up a little bit",
|
|
"then because of feedback this node tends to go up a little",
|
|
"bit. If this node tends to go up a",
|
|
"little bit this exacerbates the positive input here and this one",
|
|
"goes cachunk, whacks into the positive rail.",
|
|
"Let's take the other point of view and look at it intuitively.",
|
|
"What if this one tries to droop a little bit?",
|
|
"If it droops a little bit then the input at the plus terminal",
|
|
"droops a little bit. If that tends to go down a",
|
|
"little bit, that makes the output droop further and it goes",
|
|
"and hits into the negative rail. I can see that this circuit",
|
|
"wants to hammer into the positive rail or hammer into the",
|
|
"negative rail because of the positive feedback.",
|
|
"It is like if you give incredibly positive feedback all",
|
|
"the time, and by positive feedback I mean feedback",
|
|
"encouraging the child to do whatever the child is doing.",
|
|
"It could be if he does bad stuff you give a lot of positive",
|
|
"feedback or good stuff you give a lot of positive feedback then",
|
|
"you are guaranteed to have a very good child or a very bad",
|
|
"child. You are not going to have",
|
|
"anybody in the middle. Same way here.",
|
|
"By giving positive feedback you're going to drive this into",
|
|
"the positive rail or drive this into the negative rail.",
|
|
"Now, I am going to analyze this in two steps.",
|
|
"First I am going to analyze this using a method you've seen",
|
|
"before which is replace the op amp with its equivalent circuit",
|
|
"and analyze it statically. And by analyzing it statically",
|
|
"we are going to show that the simple static analysis will",
|
|
"yield the following expression. I put this in quotes,",
|
|
"well, for a reason you will see shortly.",
|
|
"When I apply a plain and simple static analysis here is what I",
|
|
"find. Let's go ahead with the",
|
|
"analysis and see what is basically different about these",
|
|
"two. And, first of all,",
|
|
"I will confirm for you that our naive analysis we have seen so",
|
|
"far will give rise to that expression.",
|
|
"So, let's go ahead and analyze that circuit.",
|
|
"And to analyze that circuit what I will do is replace the op",
|
|
"amp with its equivalent circuit. If you remember the op amp is",
|
|
"characterized by the following circuit, A times v+ minus v-,",
|
|
"vOUT. This is the equivalent circuit",
|
|
"of my op amp. And let me just impose that",
|
|
"external circuit on this op amp. I have grounded my v- terminal.",
|
|
"My v+ terminal goes through a resistor and a supply,",
|
|
"the v into ground, it's the resistance R1.",
|
|
"This terminal goes to the output through a resistor R2.",
|
|
"So, this is the equivalent circuit.",
|
|
"And I can apply the same good-old techniques I have",
|
|
"learned about all through this course to this circuit and see",
|
|
"what vOUT looks like. Very simply,",
|
|
"vOUT is this expression here A times v+ minus v-.",
|
|
"And because of my ground connection v- is zero.",
|
|
"Then let me go ahead and replace v+ with the voltage that",
|
|
"relates vOUT and vIN. What is v+?",
|
|
"v+ is simply the current through this part of the",
|
|
"circuit, the current flowing here times the resistance R1.",
|
|
"That gives me the drop across R1.",
|
|
"And to that I add vIN and that will give me V+.",
|
|
"And then of course I multiply this by the gain here.",
|
|
"So, let me write down that expression.",
|
|
"The current through this is simply vOUT minus vIN.",
|
|
"That is the voltage drop between these two points.",
|
|
"I divide that by the resistance R1 plus R2.",
|
|
"That gives me the current flowing through here.",
|
|
"That times R1 is the drop across resistor R1.",
|
|
"And to that I add vIN and that gives me the voltage v+.",
|
|
"So, this is v+. That is simply vIN plus the",
|
|
"drop across the resistance R1. Let me shuffle things around",
|
|
"and put all the vOUT terms on this side here.",
|
|
"I get a 1+ for that vOUT and let me move AR1 divided by R1",
|
|
"plus R2 to the left-hand side. And I pick up a minus sign.",
|
|
"So, I get AR1 divide by R1 plus R2.",
|
|
"I pick up that. And on the left-hand sign I end",
|
|
"up with vIN, and my vIN here is a function of the vIN that I",
|
|
"have here. I have an A multiplying both",
|
|
"the vINs. And then I get a one for this",
|
|
"vIN here and there is a minus sign, so I get a minus R1",
|
|
"divided by R1+R2. That is the expression that I",
|
|
"have. Let me go ahead and simplify",
|
|
"that a little further and move this whole thing down here.",
|
|
"That gives me my expression as a function of vIN.",
|
|
"What I will do is, let me continue here.",
|
|
"vOUT=vIN A(1-R1/(R1+R2)). By the way, you may be",
|
|
"wondering why I am going through so laboriously what is seemingly",
|
|
"a very simple exercise. The reason I want to do is it I",
|
|
"want to very carefully show you that the result produced by this",
|
|
"exercise is exactly that. No magic here.",
|
|
"No cheating. We are going to get exactly",
|
|
"that. And then stare at it and say",
|
|
"huh, how did that happen? And then we are going to try to",
|
|
"figure out how it actually behaves following that.",
|
|
"I divide this by 1-AR1/(R1+R2). And by now you should be",
|
|
"familiar with the technique of ignoring small numbers when I",
|
|
"have a big number next to it. So, AR1/(R1+R2) can be very",
|
|
"much larger than one because A is very large.",
|
|
"So, I can ignore my one there. And then what I am going to do",
|
|
"is multiply the numerator and denominator by R1+R2.",
|
|
"Oh, this A and this A is going to cancel out.",
|
|
"This A and this A will then cancel out.",
|
|
"And then I multiply the numerator and denominator by",
|
|
"R1+R2, so this R1+R2 vanishes. I get R1+R2 here.",
|
|
"R1+R2 minus R1 is simply R2. And then down here I get a R1",
|
|
"and then I have a minus sign out there.",
|
|
"Notice that vOUT we have found to be equal to vIN R2 divided by",
|
|
"R1. That is not wrong.",
|
|
"That is correct. Technically that is correct.",
|
|
"But you will see in a few seconds that in practice that",
|
|
"that's rarely what you are going to see happen.",
|
|
"And we will try to understand why that is so.",
|
|
"What we have done so far, if you stare at these two",
|
|
"panels here, first of all, we know that the inverting",
|
|
"amplifier has the expression for vOUT up there.",
|
|
"And through this laborious exercise we have also shown that",
|
|
"even with positive feedback, if I take a static view of the",
|
|
"circuit -- If I take a snapshot of the",
|
|
"circuit and simply analyze it as a static circuit,",
|
|
"I get the same expression vOUT. But what we are going to do is",
|
|
"when I explain to you that look, a small perturbation in vOUT is",
|
|
"going to drive the op amp to the positive and negative rail,",
|
|
"that is where the insight begins to show.",
|
|
"That if everything were magical and I could somehow exactly keep",
|
|
"things just so that will be true.",
|
|
"I will be able to build that positive feedback circuit where",
|
|
"the output is equal to R2/R1 vIN.",
|
|
"But remember even the slightly amount of perturbation is going",
|
|
"to send the op amp scurrying off to the positive rail or the",
|
|
"negative rail. How do we analyze that?",
|
|
"How do we analyze the behavior of a circuit that based on a",
|
|
"small perturbation begins to move one place or another?",
|
|
"We want to analyze the dynamics of the op amp.",
|
|
"And to analyze the dynamics what I need to do is give you a",
|
|
"slightly more detailed view of the operational amplifier.",
|
|
"If the operational amplifier is not moving instantaneously",
|
|
"between the plus and minus rail, I need to give you a more",
|
|
"detailed model that encapsulates the behavior of the op amp.",
|
|
"And so let me do that. If you want to study the",
|
|
"dynamics of an op amp -- By dynamics I mean how an op",
|
|
"amp moves as I perturb the input or the output and so on.",
|
|
"To capture the dynamics of the op amp we build a slightly more",
|
|
"involved circuit, so v+ and v-.",
|
|
"This is what we've seen before, two terminals and dependent",
|
|
"source that amplifies the difference input here by a large",
|
|
"amount. Instead what we are going to do",
|
|
"here is something slightly different and interpose the",
|
|
"following circuit in the middle here.",
|
|
"This is a model of the dynamics of an op amp.",
|
|
"We are going to impose a small RC circuit in here.",
|
|
"This is R. This is C.",
|
|
"And I am going to call the voltage across the capacitor v*.",
|
|
"Notice what I have done is rather than say this is Av+",
|
|
"minus v- I am breaking it apart in two dependent sources,",
|
|
"the first dependent source, which is simply v+ minus v-,",
|
|
"and there is a RC time constant surrounding it and then here I",
|
|
"simply add on my gain Av*. Notice that if it turned out",
|
|
"that the resistance here, for example,",
|
|
"was zero then v+ minus v- would appear across v* and this would",
|
|
"be A(v+ - v-), what you have seen before.",
|
|
"It is always good to take a look at circuits and look at",
|
|
"what happens when some component goes to an extreme value.",
|
|
"This would give you your basic op amp circuit.",
|
|
"What I would like to do next is analyze the following circuit to",
|
|
"understand how positive and negative feedback work together.",
|
|
"And by understanding that then be able to explain how a",
|
|
"positive feedback circuit works or a negative feedback circuit",
|
|
"works. Here is what I will do.",
|
|
"This part simply corresponds to my positive feedback circuit,",
|
|
"R2, R1. So, that is my positive",
|
|
"feedback circuit. And I will do the same thing on",
|
|
"this side.",
|
|
"All I am doing is applying both a positive feedback through R2",
|
|
"and R1 and negative feedback through R4 and R3 and",
|
|
"representing the dynamics of the op amp and then standing back",
|
|
"and ee, all right, let's see what happens to you.",
|
|
"So, I am sticking positive feedback, negative feedback,",
|
|
"the dynamics of the op amp here and let's see what happens.",
|
|
"What I would like to do is impose this circuit on top of",
|
|
"this op amp model. To save myself some effort,",
|
|
"let me just go ahead and modify this circuit directly.",
|
|
"I get an R2 here, an R1 here, and then up here I",
|
|
"get an R4, R3 here. The math is going to be just a",
|
|
"little bit grubby but the result is actually pretty spectacular.",
|
|
"So, all I have done is replace the op amp with its internal",
|
|
"circuit out here. And now we are going to take a",
|
|
"look at what happens to op amp dynamics when there is a small",
|
|
"perturbation. Let's develop an equation of",
|
|
"this circuit containing a capacitor using techniques that",
|
|
"we already know. Just to give you some insight",
|
|
"into what you're going to see, notice that if I make a small",
|
|
"perturbation in the voltage across the capacitor,",
|
|
"let's say I make a small perturbation to the capacitor",
|
|
"voltage let's say by applying some initial condition kind of",
|
|
"thing onto the capacitor. Then let's say that the output",
|
|
"changes to some value K. So, the change on the capacitor",
|
|
"must have been K divided by A. And what you are going to see",
|
|
"is what happens to the op amp when the initial condition on",
|
|
"the capacitor is such that this output gets perturbed to the",
|
|
"value K. Let's write an equation for",
|
|
"this little circuit and see what happens.",
|
|
"Recall our goal was to understand what happens when I",
|
|
"perturbed the output a little bit.",
|
|
"Here I perturbed the output such that its value goes to K.",
|
|
"And I can perturb the output by changing what happens at the",
|
|
"capacitor. Let me write the equation for",
|
|
"this circuit now and then to understand what happens to this",
|
|
"capacitor circuit if I let go after giving it a small",
|
|
"perturbation. What I am going to do is let me",
|
|
"start by writing the good old equation for this little circuit",
|
|
"here. And that equation is simply the",
|
|
"voltage here v+ minus v- equals the voltage across the RC.",
|
|
"So, v+ minus v- will be equal to the voltage drop across the",
|
|
"resistor plus that across the capacitor.",
|
|
"The voltage across the capacitor is v*.",
|
|
"The voltage across the resistor is the current through the",
|
|
"capacitor C dv*/dt times R. So, v* plus RC dv/dt is equal",
|
|
"to v+ minus v-. RC dv*/dt plus v* is v+ minus",
|
|
"v-. You have done this millions of",
|
|
"times before, but yet again.",
|
|
"This voltage here is equal to the drop across these two,",
|
|
"and the drop across these two is v*, the drop across C,",
|
|
"plus the current through the capacitor C dv/dt times the",
|
|
"resistance R. Or you can apply the node",
|
|
"method as well and get the same expression.",
|
|
"Now, we also know here that vO divided by A is v*.",
|
|
"I can go ahead and replace this guy here, v* by vO divided by A.",
|
|
"RC/A dvO/dt. Recall, I want the dynamics of",
|
|
"vO so let me just get an expression in vO.",
|
|
"So, I get vO divided by A plus v+ minus v- equals.",
|
|
"Now, I want an expression in vO, an equation in vO,",
|
|
"so I need to express v+ and v- in terms of vO.",
|
|
"What are these expressions? The expression for v- is vO and",
|
|
"this voltage divider, so it's vOR3/(R3+R4).",
|
|
"And just for simplicity, let me call this some constant",
|
|
"gamma minus. This is some fraction",
|
|
"R3/(R3+R4). And let me call that fraction",
|
|
"gamma minus. Similarly, v+ is vO R1/(R1+R2).",
|
|
"And let me call that gamma plus.",
|
|
"All I am doing is replacing v+ and v- in terms of vO.",
|
|
"So, effectively, what I have here is v+ is some",
|
|
"fraction of vO. That's the best intuitive way",
|
|
"of thinking about it, some fraction of vO.",
|
|
"And v- is some fraction of vO as well.",
|
|
"And I just stick these. I now have an expression in vO.",
|
|
"Don't get psyched by gamma plus and gamma minus.",
|
|
"Simply read this as if it is an F1 and F2 if you would like.",
|
|
"So, vO times some fraction minus vO times some other",
|
|
"fraction. I am feeding back some fraction",
|
|
"of the output to the positive and to the negative terminals.",
|
|
"Then, just moving things around a little bit,",
|
|
"dividing throughout by A divided by RC.",
|
|
"So, I divided by A divided by RC.",
|
|
"Plus vO divided by RC. And what I am going to do here",
|
|
"in a second, vO gamma plus minus gamma minus.",
|
|
"And I have multiplied by A divided by RC throughout.",
|
|
"Finally, collecting all the vO terms I get vO times one divided",
|
|
"by RC plus A divided by RC. I got a plus sign here so I",
|
|
"will just reverse these two guys in there, gamma minus minus",
|
|
"gamma plus equals zero. All I have done here is simply",
|
|
"grunged through some math to express this equation in terms",
|
|
"of vO. And just to make it even",
|
|
"simpler, I will just replace this thing by one divided by T,",
|
|
"much as we did for first order equations.",
|
|
"What I end up with is dvO/dt+vO/T=0.",
|
|
"Despite all the grubbiness, I end up with something that is",
|
|
"very, very familiar to all of us.",
|
|
"I went through a bunch of gyrations to substitute for v+,",
|
|
"v- and v*, but at the end of the day I got the simple",
|
|
"expression which was dvO/dt+vO/T=0.",
|
|
"Where capital T is the time constant of the circuit,",
|
|
"and the time constant of the circuit relates to the",
|
|
"expression in there 1/RC+A/RC(gamma minus - gamma",
|
|
"plus). The gamma minus and gamma plus",
|
|
"are the respective portions of the output fed back to the",
|
|
"negative input and the positive input.",
|
|
"Now, as we all know, based on very simple intuition",
|
|
"that we can completely predict the behavior of a first order of",
|
|
"an RC circuit once we know what the initial condition of the",
|
|
"capacitor is and once you know the time constant.",
|
|
"That's it. We know, we are masters at the",
|
|
"fact that the capacitor is going to behave like this.",
|
|
"It is going to be exponential. And I do know that the time",
|
|
"constant capital T. What's here?",
|
|
"It is simply the initial condition.",
|
|
"There is no drive input. I am not driving this with any",
|
|
"input here. There is no input drive",
|
|
"anywhere here. This is simply the natural",
|
|
"dynamics of the system. And, recall,",
|
|
"I start off with bumping the capacitor voltage such that the",
|
|
"output starts off being K. That is it.",
|
|
"You should be able to write down this expression and the",
|
|
"form of the response simply based on this.",
|
|
"So, this is what I bumped up the output to be by perturbing",
|
|
"the capacitor voltage. My output response based on",
|
|
"this equation is going to look like that.",
|
|
"Let's try to understand what that means.",
|
|
"It is actually quite a lot of fun.",
|
|
"How do we plot that response? You all learned that the way to",
|
|
"plot the response is plot the initial value,",
|
|
"plot the final value, and go cachoock,",
|
|
"right? It's pretty simple.",
|
|
"I am going to start at K. I know that.",
|
|
"I am going to start at K and I am going to go and find out what",
|
|
"the steady state value is. Here is where the interesting",
|
|
"stuff comes in. The final value on the",
|
|
"capacitor depends a lot on whether T is positive or",
|
|
"negative. In my RC circuits that I looked",
|
|
"at what was T? In the very simple RC circuit",
|
|
"we looked at what was capital T? What was the time constant?",
|
|
"RC. This was RC.",
|
|
"This was a positive quantity. When capital T is positive my",
|
|
"output is going to look like this.",
|
|
"When T is positive. And T is positive when this",
|
|
"expression is positive. And if A is so large that I can",
|
|
"ignore the 1/RC term, if A is very,",
|
|
"very large and I can ignore the left-hand term here then T is",
|
|
"positive when gamma minus is greater than gamma plus.",
|
|
"So, when gamma minus is greater than gamma plus,",
|
|
"I have a stable circuit, this is the good-old stuff we",
|
|
"have seen before. Now things begin to make sense.",
|
|
"Intuitively, what am I saying here?",
|
|
"All the gammas and other pieces of crapola aside,",
|
|
"what am I really saying here in English?",
|
|
"What I am saying here is that if the portion of the output fed",
|
|
"to the negative input is greater than that fed to the positive",
|
|
"input then I have net negative feedback.",
|
|
"I have net negative feedback. I am feeding the output back to",
|
|
"both the positive and negative inputs.",
|
|
"And if my negative input has a stronger effect then I am going",
|
|
"to see the op amp output decay down to a value that I expect",
|
|
"which is going to be zero. Notice that since I am not",
|
|
"applying any input here, I expect the stable point for",
|
|
"this to be output going to zero. I don't have any input there.",
|
|
"Let's take a look at another situation.",
|
|
"What happens when the opposite is true?",
|
|
"What happens when gamma minus is less than gamma plus?",
|
|
"When I feedback more, what happens when I do this,",
|
|
"when gamma plus is greater than gamma minus?",
|
|
"The opposite is true. This means that I am feeding",
|
|
"back more to the positive input. A bigger proportion goes to the",
|
|
"positive than the negative. What happens then?",
|
|
"Then what happens is capital T becomes negative.",
|
|
"We cannot see this happening on the RC circuit because capital T",
|
|
"is equal to RC, but here we have a more",
|
|
"complicated circuit and capital T can go negative.",
|
|
"If capital T goes negative then this whole thing in the exponent",
|
|
"there goes positive. If that goes positive what",
|
|
"should the output look like? It should take off into",
|
|
"never-never land. There we go.",
|
|
"I start off at zero and a make a small perturbation,",
|
|
"and the output should go as t divided by capital T.",
|
|
"The dynamics of this it goes berserk, so it is net positive",
|
|
"feedback. This is called a stable",
|
|
"situation. This is unstable.",
|
|
"What happens when capital T goes to infinity?",
|
|
"When capital T goes to infinity, spend five seconds",
|
|
"thinking about what it means physically.",
|
|
"What does it mean for the time constant of an RC circuit to go",
|
|
"to infinity? That means that your R and C",
|
|
"are very, very, very large.",
|
|
"That means that circuit is going to be very,",
|
|
"very sluggish. Think elephant.",
|
|
"A big time constant. I want to move a leg.",
|
|
"It takes a while to do that. Think big.",
|
|
"Big time constant. So, everything is going to",
|
|
"happen really slowly. It's like moving in molasses.",
|
|
"Big time constant. Everything is going to happen",
|
|
"really, really slowly. If gamma minus is greater than",
|
|
"gamma plus with a huge time constant it is going to look",
|
|
"like this. And the output is going to look",
|
|
"like this. I make T even larger.",
|
|
"All right. It is going to like this.",
|
|
"I make these so large that T tends to zero,",
|
|
"T tends to infinity in which case I get this situation.",
|
|
"The output goes dah. OK?",
|
|
"Very slow. Very lethargic.",
|
|
"Big time constant. T tends to infinity.",
|
|
"And so if this is stable, this is unstable,",
|
|
"this is called corresponding neutral.",
|
|
"And there is a mechanical analog to all of this.",
|
|
"You can show that this situation is akin to let's say I",
|
|
"had a physical well of the sort and I had a ball in there.",
|
|
"I let the ball go. Then the ball will come down",
|
|
"here and settle down in a stable state.",
|
|
"Any small perturbation of the ball will get it to come down",
|
|
"and settle down here. The unstable situation is this",
|
|
"situation where I have a ball sitting up here where any small",
|
|
"perturbation will get it to zip down to a positive rail or to a",
|
|
"negative rail. So, this is an unstable",
|
|
"equilibrium situation. And exactly the reason we got",
|
|
"this analysis in the static situation is that this can",
|
|
"happen. If I do this circuit here and",
|
|
"don't perturb it then I could get the output sitting at zero,",
|
|
"but the slightest perturbation, boom, it is going to fall down",
|
|
"or go up. What about the neutral",
|
|
"equilibrium state? That can be modeled like a",
|
|
"table top and the ball is here. It doesn't matter where you go.",
|
|
"There you are. How many people saw the",
|
|
"Buckaroo Bonzi thing? Possibly well before your time.",
|
|
"OK. I have this table here.",
|
|
"No matter what I do to it, it just goes and settles down",
|
|
"where it is, and that is neutral equilibrium.",
|
|
"But what this gives you is a fun view of the dynamics of the",
|
|
"operational amplifier as I make small perturbations to it.",
|
|
"And the even more interesting thing here is you have the tools",
|
|
"based on your first order RC analysis to analyze the dynamics",
|
|
"of a simple op amp circuit. OK, so much for theory.",
|
|
"Now let's get to some action here.",
|
|
"All right. Fine.",
|
|
"That is really pretty, good and so on,",
|
|
"but what can you do for me? What good does this property do",
|
|
"for me? What can I build?",
|
|
"What we will do is look at the op amp circuit and focus on the",
|
|
"situation where I have net positive feedback.",
|
|
"In particular just look at this circuit with R1 and R2 and send",
|
|
"both to infinity. So, I have no negative feedback",
|
|
"and I ground this terminal here and take a look at what happens",
|
|
"to a circuit with positive feedback and see if I can build",
|
|
"some interesting circuits. What you are going to do is",
|
|
"build on a circuit called the basic comparator.",
|
|
"What is that? If I have an op amp that looks",
|
|
"like this, and remember a VS rail and minus VS supply there,",
|
|
"this is v+, this is v-, I can build a very basic",
|
|
"comparator by doing the following.",
|
|
"All the circuits I am going to show you are going to build on",
|
|
"this basic little circuit. What I am going to do is",
|
|
"consider applying an input to the v- terminal,",
|
|
"applying some sort of an input and taking a look at how the",
|
|
"output behaves. So, I apply some input vIN.",
|
|
"And if I just do that, if this is v+ minus v- here",
|
|
"then I am going to get something that goes like this.",
|
|
"That is when this is positive here then this guy is going to",
|
|
"go to the VS rail and this guy is going to go to the minus VS",
|
|
"rail. In terms of the,",
|
|
"if I plot the same thing, in terms of vIN,",
|
|
"and this is vOUT, if I plot the thing in terms of",
|
|
"vIN then notice that as vIN increases this guy should go to",
|
|
"a negative rail. So, in terms of vIN it looks",
|
|
"like this. What this says is that as the",
|
|
"input becomes more and more positive applied to v- then the",
|
|
"output goes to minus VS, and if the input becomes more",
|
|
"and more negative then the output goes to VS.",
|
|
"This is what is called a very basic comparator circuit.",
|
|
"It compares the two inputs and goes up if the input is in one",
|
|
"direction and goes to the other rail if the input is in the",
|
|
"opposite direction. So supposing I feed this- I can",
|
|
"plot this is a function of time. Let's say I plot vIN.",
|
|
"Let's say I feed some vIN here. Let me just call this.",
|
|
"I feed some vIN to this circuit here, then what do you expect",
|
|
"the output to look like, the output wave form?",
|
|
"For all positive vINs the output is negative.",
|
|
"So, my output vO is going to be negative as long as vIN is",
|
|
"positive. And when vIN becomes negative",
|
|
"this one shoots up and behaves like this.",
|
|
"This is minus VS. That is plus VS.",
|
|
"This is my input vIN. Then this guy is going to be my",
|
|
"output. As vIN is positive output slams",
|
|
"to the negative rail. When vIN becomes negative the",
|
|
"output slams to the positive rail.",
|
|
"So, that is quite nice. And so such a circuit is pretty",
|
|
"useful to me. Let's say, for example,",
|
|
"I want to build a little digital circuit that is fed ones",
|
|
"and zeros. I can use a comparator to turn",
|
|
"my vIN voltage into a sequence of ones and zeros.",
|
|
"When vIN is positive I produce a zero and when vIN is negative",
|
|
"I produce a one. I can get this one,",
|
|
"zero, one, zero sequence coming out corresponding to the values",
|
|
"of vIN being greater or less than zero.",
|
|
"Now, one problem with something like this is that this circuit",
|
|
"can be quite messy in the following situation.",
|
|
"Suppose I superimpose a small amount of noise in vIN.",
|
|
"In particular, let's say that I have some",
|
|
"amount of noise on vIN. I get a bunch of noise sitting",
|
|
"around here. What happens is that at this",
|
|
"point where the value goes negative, I do bump up.",
|
|
"But when for a second I have my input going above zero again --",
|
|
"-- this output comes down again and out here it goes up again.",
|
|
"I get this nasty behavior at the point where the input is",
|
|
"around zero. When the input is around zero,",
|
|
"the input is meandering around zero because of noise,",
|
|
"I get a huge amount of up and down glitches on the output.",
|
|
"That's not very nice. And we will do a little circuit",
|
|
"that attempts to fix that little problem.",
|
|
"What we are going to do is use positive feedback.",
|
|
"And I am going to build you a circuit that shows that we can",
|
|
"eliminate this for small noise on the input.",
|
|
"So, let's build the following circuit.",
|
|
"So I still feed vi to the negative input,",
|
|
"but this time around I give it some positive feedback.",
|
|
"So, I give it some positive feedback.",
|
|
"And what I am going to do is feedback a portion of vO to the",
|
|
"positive input. This is positive feedback.",
|
|
"And, in particular, let's assume that VS equals 12",
|
|
"volts. And to the negative one I",
|
|
"connect -VS. This guy is going to go between",
|
|
"12 and -12. And correspondingly because",
|
|
"these two are equal this one is going to go between 6 and -6.",
|
|
"This is going to be a 12 or -12.",
|
|
"Remember, the top rail and the bottom rail.",
|
|
"And this one is going to be a +6 or -6.",
|
|
"And let's understand how this circuit works when I apply an",
|
|
"input vIN. Let's start by saying that",
|
|
"assume my input is zero for a moment.",
|
|
"And let's say my output starts off being 12 volts.",
|
|
"The output is 12 volts then the input here is going to be 6",
|
|
"volts. In this case v+ is going to be",
|
|
"6 volts. The output is 12,",
|
|
"v+ is going to be 6 volts. And my circuit is sitting out",
|
|
"there doing nothing. Now, this started off being",
|
|
"zero. Let's say vIN increases.",
|
|
"As vIN begins to increase what happens?",
|
|
"Well, nothing until vIN reaches 6 volts.",
|
|
"Since this is 6, vIN has to go up to 6 volts,",
|
|
"has to equal this voltage before I can flip the circuit.",
|
|
"What happens when vIN is greater than 6 volts,",
|
|
"if vIN goes above 6 then I have more voltage on a negative",
|
|
"terminal than the positive so the op amp flips its state.",
|
|
"And vO gets to -12 volts. When vi goes above 6,",
|
|
"vO gets to 12 volts. And what does v+ go to?",
|
|
"In this state v+ goes to half of -12 which is -6 volts.",
|
|
"Now, this guy is sitting at -6 and this guy is sitting at -12.",
|
|
"If this one keeps rising nothing happens,",
|
|
"so output can stay at -12. So I am pretty safe.",
|
|
"Then let's say v begins to come down.",
|
|
"As v begins to come down, does anything happen when v",
|
|
"gets to 6 again? If v is equal to 6 what",
|
|
"happens? Nothing because this is at -6",
|
|
"now. So, there is still a huge net",
|
|
"negative voltage here from v+ to v-.",
|
|
"And so therefore I sit at -12. Oh, well, I keep coming down",
|
|
"until I reach -6. When I reach -6 here these two",
|
|
"become equal. And what happens when this",
|
|
"becomes less than -6? v- becomes less than -6.",
|
|
"If this one goes below this voltage, this is -6 and this is",
|
|
"-7. There is a net positive voltage",
|
|
"between v+ and v-, so this output swings to the",
|
|
"positive rail like so. We will spend a lot more time",
|
|
"on this in the next few minutes to really hammer the point home.",
|
|
"What is interesting about this is that even though the moment",
|
|
"vi became more than 6, I swung to the positive rail,",
|
|
"and then I had to go all the way back down to -6 before I",
|
|
"could change state. I had to go way down before it",
|
|
"could flip again. How can we make use of that?",
|
|
"Well, let me draw you a little vi versus vO diagram and then",
|
|
"talk about how that can be useful to us.",
|
|
"This is vi, this is vO, this is zero.",
|
|
"Let's say this is 12, -12, -6, +6.",
|
|
"Let's plot that on the screen and see what it looks like.",
|
|
"As I told you, the output was at 12 volts to",
|
|
"begin with and my input was at zero.",
|
|
"So, my input kept increasing. When the input hit +6 what",
|
|
"happened to my output? My output swung down to -12.",
|
|
"As the input kept increasing nothing happened.",
|
|
"This was step one, this was step two,",
|
|
"step three. My input kept increasing and",
|
|
"output stayed at -12 volts. Then what I said was well,",
|
|
"let's bring the input down. So, my input began to go down,",
|
|
"step four, became more and more negative.",
|
|
"Nothing happened until I reached -6.",
|
|
"When I reached -6 I swung positive, step five.",
|
|
"Again, one, two, three, four,",
|
|
"five. I am going up here.",
|
|
"It came up here. And nothing happens until I",
|
|
"reach -6, but at -6 boom, I switch to the positive rail.",
|
|
"And as I get more and more negative I stay there.",
|
|
"Then again, as I start increasing again,",
|
|
"nothing happens until I reach +6.",
|
|
"Think of that as your seventh step.",
|
|
"What is spectacular about this is that I seem to have a circuit",
|
|
"that now has some knowledge of where it came.",
|
|
"If it is coming from here it switches at +6,",
|
|
"but if it is coming from here it switches at -6.",
|
|
"So, there seems to be sort of a lag in the behavior of the",
|
|
"circuit or some memory property in the circuit.",
|
|
"This kind of behavior is called hysteresis.",
|
|
"The word comes from magnetic circuits where,",
|
|
"or rather elements that you're trying to magnetize.",
|
|
"Where if you take a magnet and move it over a piece of metal it",
|
|
"may leave some residual magnetism in it.",
|
|
"And, in the same way, that is called hysteresis.",
|
|
"Same way here. As the voltage increases it",
|
|
"seems to leave some residual in the circuit so that it effects",
|
|
"when it shifts. The good news with this is that",
|
|
"now, if I take the same kind of noisy wave form that I had",
|
|
"before and do this -- If this is vi then what is",
|
|
"going to happen is for vO I am going to be negative at this",
|
|
"point. Nothing happens here because I",
|
|
"have to get to -6 or +6 before something happens.",
|
|
"Out here I get to -6 and I switch state and go up to +12.",
|
|
"And then this one comes up above -6 very slightly out",
|
|
"there. Nothing happens because the",
|
|
"next change will happen only when the input goes to +6.",
|
|
"So, if eventually the input gets to +6 and then I am going",
|
|
"to change state again. It is actually a really cool",
|
|
"property and something that is completely non-obvious.",
|
|
"In the last 30 seconds let me show you a quick demo.",
|
|
"And, based on this property of hysteresis, I have actually",
|
|
"built a little circuit. Let me do that first.",
|
|
"Notice here that I am showing you the input on the X axis vi",
|
|
"and vO on the Y axis. Notice how the output switches",
|
|
"at +6 volts and switches at a -6 volts to +12 or -12.",
|
|
"That's the hysteresis property. And we can actually use this",
|
|
"property to build a clock circuit, which is on page 9,",
|
|
"build an oscillator that sits there and oscillates by itself.",
|
|
"And you will see details of that in recitation tomorrow."
|
|
]
|
|
} |