James K. Roberge: 6.302 Lecture 02

[MUSIC PLAYING]

JAMES K. ROBERGE: Last time, we introduced the idea of a feedback system. And we spent some time looking at what's really the fundamental property of a feedback system, namely its ability to desensitize the closed loop transfer function to changes in the forward path gain. And we found out that that sort of desensitivity is obtained really only in exchange for starting with a larger amount of forward path gain than we actually need.

We then looked at the way feedback can modify impedance levels in a feedback system. We illustrated this with an electronic system, an operational amplifier. But the same kind of modification goes on in mechanical systems. If we ask for the output stiffness of a servomechanism, we get the same kinds of relationships. That's a mechanical impedance rather than an electrical one. But the same general kinds of things happen.

Today, I'd like to continue looking at basic properties of feedback systems. I'd like to look at how feedback systems modify noise injected at various points in the loop and also see how feedback can be used to moderate certain kinds of nonlinearities. Let's consider the system that we show here. And here we have a system with an input, output, of course, a feedback transfer function or a feedback element that monitors the output and applies it back to the summing point.

And then what we've shown in addition to our input, presumably the output is supposed to be some function of the input, we have three disturbances that are injected at various points in the system. Here we have a disturbance that's injected at the same point that the actual input signal is applied. We have a second disturbance that's injected after some gain in the forward path, after the gain a1. And we finally have a third disturbance that's injected after another amount of gain has been achieved in the forward path.

If we were looking at a physical system that this might represent, why we could consider for example some sort of an audio amplifier. We have an input signal. And a disturbance injected at the same point as the input signal. It might be noise associated with the pick up. For example if we have a audio reproduction system, noise injected simultaneously with the input signal or at the same point in the system, would be noise that was picked up by the cartridge that was tracking a record, for example.

Our second noise source might be noise that comes in, for example at the output of a preamplifying stage. We have an amplifier, maybe a transistor amplifier that's providing the package of gain, a1. If there's some noise on the supply voltage associated with that amplifier, the output stage of the amplifier is oft-times dependent on the supply voltage, as well as on its input signal. And so our second disturbance might reflect noise introduced because of an improperly regulated supply associated with a preamplifier or a noisy supply associated with a preamplifier.

And then finally, we might consider the third disturbance, which occurs near the output of the system as reflecting noise associated with the power handling stage of our final amplifier. So if we were thinking about an audio system, this might be the physical significance of the various disturbances that are shown.

Well, here we have a system that actually has four inputs, a desired input, and then the three noises that we talked about. And we can of course write the output as the superposition of the responses to those four inputs. When we do that, we get vi times the forward path over 1 minus the loop transmission. That's always the expression that we get for a closed loop gain. And we saw that last time. So we get vi times a1 a2. That's the forward path for the vi input.

The loop transmission for this particular system, again we can break the loop. We get a minus a1 a2 f for the loop transmission. Consequently, 1 minus the loop transmission is 1 plus a1 a2 f. So the relationship between the output voltage and the input voltage is the forward path a1 a2 over 1 plus a1 a2 f.

Since the first disturbance is injected at that same point, the transfer function relating the output to the first disturbance is the same as that which links the input to the first disturbance. Similarly, when we calculate the output in response to the second disturbance, we get its forward path, which is a2 over 1 minus the loop transmission. The loop transmission is the same for all cases. So we get a2 over 1 plus a1 a2 f. Finally when we look at the third disturbance, we get its forward path, simply 1 over 1 minus the loop transmission. We get 1 over 1 plus a1 a2 f.

I've repeated those expressions here. And simply repeating that the output voltage is a1 a2 over 1 plus a1 a2 f times Vi plus a1 a2 over 1 plus a1 a2 f times Vd1. The gain from the input and the first disturbance are exactly the same, since they appear at the same point or are applied to the system at the same point, plus a2 over 1 plus a1 a2 f times Vd2 plus 1 over 1 plus a1 a2 f times the third disturbance.

I think we can get a little better emphasis of the important feature of this expression by factoring out the a1 a2 over 1 plus a1 a2 f. When we do that, we get that factor out in front. Then we get Vi plus Vd1 one. And we get a Vd2 divided by this-- this should actually be a1. I'm sorry. This should be a1 and this should be a1 a2. So we get Vi plus Vd1 plus Vd2, the second disturbance, divided by a1.

The important point there is that the quantity a1 is the gain in the block diagram that precedes the point at which we inject noise. If we think back to our original block diagram, we had the disturbance injected after an amount of gain had been applied to the signal channel. And the amount of gain was a1. Similarly when we inject the third disturbance, the gain that precedes it, the gain that appears between the input signal and the third disturbance is the amount a1 a2. And so when we factor this term, we get an a1 a2 dividing the relative magnitude of the third disturbance.

And so what this relationship emphasizes is the fact that when we ask for the attenuation of disturbances that occur in the closed loop system further downstream, why we find that the attenuation of these disturbances relative to the input is dependent on the gain that separates the input and the point that the disturbance is applied. That's certainly reasonable. Because if we looked at the system, what happens is we apply an input, we then amplify it by a given amount, and then we inject the noise. And consequently, we boost the strength of the desired input by a certain amount, either a1 or a2, relative to the noise that's injected further downstream. Notice there's no moderation of a disturbance injected at point one relative to the input. That's reasonable since point one is the same point as the input.

The significance for an actual system, if we think back to the system I described where we talk about an audio amplifier, the significance for the actual system is that fine, noise that comes in at the point where the input signal is generated can't be removed. Feedback gives us no mechanism for improving that. That's really a fundamental signal to noise ratio kind of a thing or a detectability issue. And there's nothing we can do about that via feedback.

However, if we had some noise injected, as I had mentioned as a consequence of imperfect regulation associated with a power supply powering the preamplifier, we can attenuate that to a certain extent, depending on the signal path gain preceding that noise. We attenuate disturbances down here by an even greater amount. This may be very significant. It's oft-times fairly easy to get a well-regulated supply for a preamplifier. The preamplifier is run at very low power levels. And so it's not very hard to get a well-regulated supply for the preamplifier.

However, if you have a power handling stage, getting a very well-regulated power supply at the 100 watt level is a fairly difficult thing. And so this technique may give us a way of tolerating a more poorly regulated power supply associated with the power amplifier and effectively minimizing the effects of that disturbance or that noise associated with the power supply by preceding the point at which the noise is injected by sufficient gain. We still control the overall closed loop gain principally by means of f, providing the a1 a2 product is sufficiently large.

We can look at an open loop solution to this same problem. Consider the system shown, where we have a completely open loop system, a1 a2. We inject Vi in Vd1, Vd2, Vd3. And then we attenuate the resultant output by a factor 1 over 1 plus a1 a2 f. If we write the relationship between the output and the four inputs, or the one input and three disturbances in this case, we get exactly the same expression as we got here.

And so here we have an open loop solution that gives us the same input to output relationship as does the closed loop solution. My point being that the feedback isn't essential for the moderation of noise. The only thing that's important is the gain that precedes the point at which the noise is injected.

You recognize of course that usually the feedback solution will be a much more practical one. If we look back at our system, we have an attenuator here. Clearly, the quantity 1 over 1 plus a1 a2 f can be a very, very large number. It's not very practical to talk about a power amplifier with an output power level of 100 watts and then an attenuator just immediately preceding the output. We have to start with megawatts back in here, if we have a large attenuation ratio.

So the open loop solution may be a very impractical one. But my reason for bringing this up is just to indicate that there's nothing really fundamental about feedback. The important thing here is the gain that precedes the point at which we inject the noise. Or corresponding, the preferential gain we apply to the signal before we inject noise.

There's another situation where we might find that feedback would help us or at least might hold up hope for helping us. Suppose we have a system that includes a nonlinearity. And in particular, a nonlinearity in the forward path. Now, we said that feedback can be used to desensitize us to variations in forward path gain. We can look at nonlinearity as a variation in forward path gain that's dependent on signal level. That's what we mean by a nonlinearity. If we have at least a nonlinearity without energy storage, we find that that's simply an element whose gain is proportional to signal level.

If that nonlinearity is applied in the forward path or is located in the forward path of the feedback system, we might anticipate that the desensitivity associated with feedback systems would tend to moderate the effects of the nonlinearity. And I'd like to look at a system with that characteristic.

Here, we again have our input, the output. Notice, I've changed notation. I have the lowercase variable, the capital subscript, which is a total variable. And I can represent relationships in a nonlinear system that way. The earlier block diagram was a linear one. We have the input. We measure the output and scale it by a factor f of 0.1. Clearly, the objective here is to build an amplifier whose closed loop gain is 10. The ideal relationship between the output and the input providing the loop transmission magnitude is large enough. It would be 1 over f. We'd get an output that's 10 times the input.

We then, in the forward path, have an amplifier that provides a gain of 1,000, let's say. And then a nonlinear element. And that completes the forward path. So here we have a nonlinear amplifier in a forward path and a linear feedback element.

For purposes of analysis, let's assume some characteristics associated with the nonlinear element that are like this. We have a dead zone. In other words, the input for the nonlinearity shown on this axis, has to reach 1 volt before the output of the nonlinearity begins to change. That dead zone is typical, for example of many power handling stages.

Again, let me draw an analogy between this system and a power audio amplifier. One very common topology for an audio amplifier involves a complementary emitter follower, a push-pull pair of emitter followers using a PNP and an NPN transistor. That sort of topology has very nice characteristics in terms of maximum power output versus quiescent power consumption and that sort of thing. And so a very popular topology for a high power amplifier involves this sort of complementary emitter follower.

But at least in the simplest versions of that circuit, there's a crossover problem. In particular, what one has to do is exceed the forward threshold of the transistor's approximately 0.6 of a volt in order to get the output to change. And that's the sort of characteristic we've shown here. We implied that there's a region from minus 1 to plus 1 for which the output doesn't change.

We then go through a region for inputs between 1 and 3 where the slope is 1. We then go through a region of reduced slope in here. The slope is a 0.1, rather than 1. We go from 3 to 8 on the input, from 2 to 2.5 on the output. And finally, a region where the slope becomes 0. The nonlinearity is symmetrical.

Again, if we were trying to relate this to a power amplifier, we mentioned the crossover distortion. We then get a linear region, a region where the power handling element might be beginning to saturate. And then finally, a region where we've reached hard saturation. The output no longer changes in response to the input. In reality, if we look at the characteristics of power handling stages, they aren't quite as abrupt as this. But might curve a bit and do something like so and eventually saturate. So this might be a reasonable approximation for the nonlinear characteristics associated with many power handling stages.

We can analyze this particular system fairly easily. In general, the analysis of nonlinear systems is quite challenging. And we'll have more to say about that later in the subject. But this particular one this is quite easy. And the reason is that we speculated or hypothesized at the beginning, that the forward path had no dynamics. And that gives us an easy way to solve the system.

What we can do for this particular one is to look at the various signal levels. We recognize here that the nonlinear element is piecewise linear. And we can assume a value for the output. It's particularly convenient if we assume a value of it is at a breakpoint of the nonlinearity. Once we've made that assumption, we're able to determine all of the other signals in the system. We assume a value for the output voltage.

Once we know or have assumed that value, of course we determined vF immediately. The transfer characteristics of the nonlinear element determine vA, the vA that corresponds to that value of vB. Since vA is the output of a linear amplifier with a gain of 1,000, we immediately determine vE. And then we recognize that at this point, we have the relationship vI minus vF equals vE or vI equals vE plus vF.

So once we've assumed the value for vO, that pegs vF. The nonlinearity already pegs vA, consequently vE. We can determine the value of vI that gives us that value of vO. If we do this for values of vO at the breakpoints, why we're basically able to connect the dots, follow the dots, and get the overall input to output transfer characteristic for the system.

I've indicated the important relationships, other than those associated with the nonlinearity for this system. In particular, the fact that the error voltage is simply 10 to the minus third times the voltage at the input to the nonlinearity. That reflects the gain of a 1,000 in the forward path. That the relationship at the summing quite tells us that the error voltage is the input voltage minus vF, solving for vI, so that we can determine the input voltage that ultimately gives rise to a given output. From this equation, we simply get vI is equal to vE plus vF.

Well, we can do those manipulations. And they are summarized in this table. And as I say, we can do is assume some value for vO. If we recall the original nonlinearity, the maximum negative value of vO is minus 2.5. So we can consider two cases. One where we're right at that breakpoint, with an input voltage, the nonlinearity, of minus 8 volts, which gives rise to an output of minus 2.5. And we can then chase through the relationships that we indicated earlier of vF will be one quarter of vO. vE will be 1/1,000 of vA. And we can determine the corresponding input voltage by adding vE and vF.

And we then continue that. We go to the next breakpoint and transfer characteristics at minus 2. We find out that vF is of course 0.2 of a volt. The input to the nonlinearity is minus 3 volts. That tells us we need 0.0003 millivolts at the input to the gain of 1,000 amplifier. The corresponding value for vI is 0.203 volts, and so forth.

We now go to the point where the input is 0. And there's this 2, a range of values there, that the output of a system is 0. There's a range of values at the input to the nonlinearity. In particular, values between minus 1 and plus 1 at the input to the nonlinearity, give us 0 at the output of the nonlinearity. 0 for vF correspondingly. And we can calculate the corresponding vE's to these two points, the corresponding vI. And then we could do the same kind of things for positive inputs.

And so now we have the relationship between the input and the output at all breakpoints. And as I indicated, we simply connect the dots. When we do that, we get the following closed loop transfer characteristic. One of the things-- this is grossly not to scale-- one of the things we notice is that the relative size of the dead zone has been very, very dramatically reduced. Originally, we had a dead zone whose width was one unit. Here all the way we're all the way down to 10 to the minus third for the half-width of the dead zone, along the input axis.

We then get into the region, recall in here, that the incremental forward path gain of our system was 1,000. We had the amplifier gain of 1,000. And we have the attenuation of 0.1 in the feedback path. But the nonlinearity here has a gain of 1. Consequently, the incremental forward path gain is 1,000. f is 0.1. If we calculate on an incremental basis, the closed loop gain, we find out it's 1,000 over 1 plus 100. 1 minus the loop transmission in that region. You get a slope of 9.9. That's same result of course we get by connecting the dots.

If we do the same thing when the gain of the forward path, the incremental gain of the forward path has dropped by a factor of 10 because of the partial saturation of the nonlinearity, we again find out that we get an incremental gain in that region that now is 9.1, 100 over 1 plus 10.

And then finally, we get to a region of 0 slope. Here, the loop transmission in an incremental sense has finally gotten to 0, because the forward path is now hard saturated. And if we calculated an incremental value for a over 1 plus aF, it of course is 0 also, since a is 0.

But again, the thing of value here is the moderation of the dead zone, the very dramatic reduction of the effective width of the dead zone. And again, as the forward path gain, the incremental forward path gain changes by a factor of 10, we get a very mild change in the closed loop incremental gain. So we've done a very good job of moderating the effects of the nonlinearity.

The advantage of course, or at least one potential advantage for this sort of thing, is that nowhere in this discussion does power level appear. And so again, let's think back to the audio amplifier that we might be trying to build. It's awfully expensive to make a very high quality power handling stage. And let's say, a popular topology for a power handling stage is a complementary emitter follower. You find out if you try to get that stage built so you have a very small dead zone or no dead zone, you run a very real risk of thermal runaway in the transistors. The design becomes unreliable. And in fact, there's been a great deal of effort expended trying to get very good and clever biasing schemes for those sorts of output stages.

Things are greatly simplified if you're willing to allow yourself some nonlinearity in the output stage. And one of the hopes might be then that we can compromise the design of the power handling stage, make it a little less perfect if you will, or a little less ideal than we might like to have it. And make up for that at a lower power level in the amplifier, by adding a gain at low level signals or low power signal levels. Put in an operational amplifier that has a gain of 100,000 or something like that and use its gain to help moderate the effects of the nonlinearities in the output stage.

And there may be a very substantial economic advantage in that approach. You can save considerable money in the output stage by making it have a little bit poorer characteristic. You can make up for that by adding a package of gain at a very low power level, where it costs you a quarter or something to put in another operational amplifier. And that may be a very advantageous trade-off from an economic point of view.

There's another way that we can use nonlinearities in connection with feedback systems. And that's where we're really trying to get a nonlinear closed loop transfer function. This is an extension of the idea that the input to output closed loop transfer function is sort of the reciprocal of the feedback element in the case where the feedback element is linear. If we have a nonlinear element, we get an inverse relationship.

And let's look quickly at an example of that. Here we have a connection that involves an operational amplifier and uses a bipolar transistor as the feedback element. And the objective of this connection is to build a logarithmic amplifier or built an amplifier whose output is proportional to the log of the input voltage. And the reason that we can do that with this connection is because the feedback element, in this case the transistor, has exponential characteristics between base to emitter voltage and its collector current.

And let's model this thing. This circuit only works for positive input currents, which gave rise to negative output voltage. If the output voltage goes positive, the transistors cut off and the circuit doesn't behave very well. However, a positive input current will give us a negative output voltage. That turns on the transistor.

And so we can write that the negative of the output voltage, which I think is a little easier way to represent things, the negative of the output voltage is equal to some multiple. Rather than representing the operational amplifier as a voltage gain, let's represent it as a transfer resistance. Since the variables of interest at the input are actually currents, we could recognize there's some input resistance to the operational amplifier and some voltage gain. The product of those two will be a transresistance that tells us how the output voltage is related to current at this node.

So the negative of the output voltage is then this transfer resistance times the difference between the current iI and the current iF, which is actually the collector current of the transistor. That difference of course is injected into the input. We multiply that by the transresistance of the amplifier to get the output voltage.

However to a very good degree of approximation, and actually over decades and decades of operating current, you can typically verify this relationship over something like eight decades for a good quality silicon transistor. We find out that the collector current, or iF in on our diagram, is approximately equal to some constant times e to the q over kT, where q, k, and T are physical constants, the charge on the electron, Boltzmann's constant, and absolute temperature, times, in this case, minus vO, since vO is the emitter voltage on the transistor.

But as I say, that exponential relationship holds over many, many decades. That allows us to make a nonlinear block diagram, if you will, an input current or a fedback current, a transresistance, giving us the negative of the output voltage, and then the exponential characteristics, giving us the fedback current as a function of output voltage.

I'm sure most of you recognize, it's actually a minus 1 in this expression. But at least we're running at current levels, typically above about 10 to the minus 11th amps for a good, small geometry silicon transistor, why the minus 1 is inconsequential compared to the exponential. And so we get this kind of a relationship, which allows us to draw this sort of block diagram.

Now, if the loop transmission magnitude, and there's a little bit of hedge as to how do we define loop transmission magnitude for a nonlinear system, we'll have more to say about that further downstream. But if the loop transmission magnitude is much, much larger than one, the implication is that the current fed back is about equal to the input current or i sub I being equal to iF is Ke to the q minus vO over kT. We can invert that equation and solve for vO. And when we do that, we get the output voltage is kT over q times the natural log of iI over K-- we actually should have minus vO-- being equal to kT over q times the natural log of iI over K.

So here we have a way of taking a log of an input current by putting an exponentiating element in a feedback path. And this kind of a connection is actually used. There's some minor modifications. I talk about the topology, I think in Chapter 11 or 12 of the book. There's some minor modifications that are really made to get around some temperature dependencies associated with this expression. But the basic technique is the one that I've shown.

What I'd now like to do is illustrate some of the ideas we've been talking about, in particular the effects of feedback on noise and on nonlinearities, with a demonstration. And what I have is a demonstration that's something like the nonlinear system we talked about before, with the aid of the viewgraphs. What we have is a system as follows. We have an input signal, which we apply directly to an operational amplifier. This amplifier models the linear portion of the forward path gain associated with the system we mentioned before. In fact, the open loop gain of the operational amplifier is much, much larger than 1,000, much larger than the number that we used in the earlier example.

We then have our nonlinear element. And again, I picked the nonlinear element to have features similar to those that we talked about in our system. I'm not trying to pretend there's a one to one relationship. But we have selected the nonlinear elements so that it has a dead zone and two different degrees of compression, a linear region and then some compression associated with it. We'll see those characteristics in a moment. So the general nature of this nonlinearity is similar to the one we had talked about. We also have the ability to inject noise to the nonlinearity.

And what I have done basically is put a potentiometer between the output of the nonlinearity and the input of the nonlinearity. And what that allows us to do is in sort of a smooth way, incorporate the nonlinearity inside the forward path of the loop. Hopefully, then the feedback will improve the performance of the system or moderate the effects of the nonlinearity.

Or if I move the POT to this end, all I've done is configure the operational amplifier as a follower. Then the input to output characteristics should reflect the nonlinearity, since we have basically unity gain between here and here. And then have the nonlinearity working outside the loop, the overall input and output characteristics would be simply those with the nonlinearity. If I move the POT to the other end, the nonlinearity is in the forward path. And the feedback should moderate the effects of the nonlinearity.

The test set up that allows us to do this is shown here. We have first of all, simply a power supply. Here is the little nonlinear amplifier or nonlinear element. We have here, the operational amplifier and a couple of transistors. Actually, we are using emitter followers to get us the crossover distortion. And there's some diode networks on the input to those transistors that give us the shaping, aside from the dead zone. So the emitter followers give us the dead zone. The diodes give us some shaping in front of that, some compression.

Here's the potentiometer that changes the feedback from the output of the nonlinearity to its input. We apply an input signal. We get an output signal, which we observe on an oscilloscope. We also have the ability to listen to that output signal.

Now, this whole board runs at a very, very low power level. It's output emitter followers are these little epoxy case transistors. So it's running at a very, very low power level. We take its output signal and any distortion that it might be applying to that output signal and apply that to a power amplifier, this amplifier here. And we're then able to listen to the signal that has the same characteristics as those that come through our little amplifier or our nonlinear amplifier.

These boxes are signal generators, of course. We can apply a sinusoid as the input to our nonlinear element. We can also apply a disturbance from the bottom signal generator. The oscilloscope of course, let's us look at various signals of interest. And finally, we have a tape recorder. And we can substitute the output of the tape recorder for the sinusoidal input. And so we get a little bit more interesting signal sources to exercise the system.

Well, let's go ahead and look at some of this. What we now have is simply a plot of the overall input versus the output of the system. And we see the general kind of characteristic on the oscilloscope that I had mentioned earlier. We have the dead zone in the middle, and then a region with one slope, and then finally a region with a second slope. We don't quite drive this out far enough to get into the eventual flat region of the gain characteristic.

What we're doing to obtain this plot is of course simply plotting the input and the output of the system with the potentiometer change moved to its leftmost position. What that does is precede the nonlinearity with simply a unity gain amplifier. We plot the overall input/output characteristics of the system. And what that does is of course give us the characteristics of the nonlinearity itself. So this is simply an XY plot of the nonlinearity.

If we change the position of the potentiometer, we now are moving our feedback toward the output of the nonlinearity. We include the nonlinearity inside the feedback path, in particular in the forward path of our feedback system. Because feedback desensitizes our system to changes in forward path gain, we end up with what appears to be a perfectly linear input to output relationship. And that reflects the fact that we have so much gain preceding the nonlinearity. The open loop gain of the operational amplifier is possibly 100,000. And that moderates the nonlinearity to a remarkable degree.

Even when we're out in the second slope of the nonlinearity, why the incremental loop transmission is still very, very large compared to 1. The closed loop gain is 1 over f. f in this case is 1. And so we get very nearly linear performance. We certainly can't see any remnants at all of the crossover distortion or the saturation.

Let's look at the same thing in the time domain. This is an input/output plot. Let's look at the output signal that we get with sinusoidal excitation. Here's the output signal of the system when we have a feedback around the nonlinearity. As we move the feedback back to the other side, we notice the distortion.

We have the crossover distortion evident as the flat portions at the zero-crossings of the sine wave. We also see the flattening as a consequence of the saturation of the nonlinearity on the tops of the sine wave. So that's the sort of picture we get. If we again look at the output now, changing the feedback to the output of the nonlinearity, we get a very, very nice sinusoid in response to that.

Well, let's consider what going on here. We have our nonlinearity. And we're applying a signal from the preamplifier, from our operational amplifier with a gain of 1,000, or in this case, maybe 100,000, to the input of the nonlinearity. If the output of the nonlinearity is sinusoidal or very nearly appearing sinusoidal, the input to the nonlinearity must be sort of predistorted in the right way to make up for the characteristics of the nonlinearity. Let's look at that.

Here's the output. Let me just lower the scale on the output a bit. And then let's look at the output from the operational amplifier. And now what we see is that the output, the top trace, is a very nice sine wave. But in order to make that happen, the input to the nonlinearity has to have fairly peculiar characteristics.

For example, because of the dead zone associated with the nonlinearity, why the signal at the input of the nonlinearity has to move very rapidly through the zero-crossings. We notice the essentially vertical region. And then there's the corresponding kinds of distortion up at the peak values of the sine wave that provides the right signal to the nonlinearity so that it's output is very nearly sinusoidal. Now, in reality what's happening of course is there's a very little remnant of the distortion left in the output signal.

When we complete the feedback loop and subtract the output signal from the input signal, there is again this very tiny piece of distortion if you will, left. The output is very nearly a sine wave. We subtract that from the input sine wave. And we get the very little residual distortion. That residual distortion is amplified to a large extent, amplified by the op-amp. And in fact, then what happens is that that's highly distorted signal is applied to the input of the nonlinearity. And that gives us a very nearly sinusoidal output.

Again, we can look at the change. This is when we're feeding back from the output of the nonlinearity. We get a very nice output signal with top signal. The signal at the input to the nonlinearity, but that's not a signal of interest particularly, is terrible.

If we go the other way, we get a distorted output signal. We're now feeding back right around the op-amp. So the op-amp output is a faithful reproduction of the sinusoidal input. This is not the case of interest, of course. We have a terrible output from the system. Although the internal signals look good. We can go back to the more useful case, where the output is clean. But in order to get that, the signal at the input to the nonlinearity has to be distorted.

We could also listen to this, which I think gives us a good indication of what's going on. Here we have the sinusoid. And this is when we have good quality, if you will, when we have the feedback from the output of the nonlinearity. Now, let's change and see what happens. We notice the tinny sound.

Let's see. I think probably we just have to look at the output trace here. And we hear the tinny sound. And that's a very characteristic kind of distorting sound. The principal feature giving us that audible distortion is actually the crossover distortion, the flat portion in the middle of the sine wave.

Here we have the nearly harmonically pure output, free of harmonic distortion because we're feeding back around the nonlinearity. And here we have the distorted output, very obvious audible distortion. We can also, in addition that signal, we can add a disturbance. And remember, we added the disturbance at the input or somewhere in the middle of the nonlinearity. And I can do that with this second generator.

Here is the noise added to the signal. And I think we can now hear the effects of that. Let's go back in here. And you hear the low frequency component. That's the effect of the second generator. There we have a disturbing frequency that's at a very low frequency.

Here, I'm increasing the frequency of the disturbance. And we see the effect on the output as the wave forms are non-synchronous, the time wave forms or non-synchronous. But we can see the distortion applied to the output signal.

And now, let's go back to the position where we feedback from the output of the nonlinear element. And we also have noise being injected in the middle of that, of course. Now, we have the pure output. The feedback not only moderates the nonlinearity, but because the noise is injected downstream from a large package of gain, why we get the signal as shown. We very largely eliminate the effects of the noise.

We could again, simultaneously, look at the signal at the input to the nonlinear element under these conditions. And when we do that, why here we see the noise running through the signal at the input to the nonlinearity. The phase of that noise is counter to the actual input signals, the actual disturbing signal, so that it cancels it.

When we're in the other condition, here we see the noise appearing in the output signal. But the signal at the input to the nonlinearity under these conditions is clean. And it's not, as I mentioned, the useful case. Again, here we have the useful case. The signal at the input to the nonlinearity is both distorted and noisy. But in spite of all of that, it's distorted and noisy in precisely the right way so that the output of the nonlinearity is very nearly a pure signal.

I'd like to finish up this demonstration by showing how this all works when we listen to an actual signal source. The purely sinusoidal source becomes rather boring in a hurry. But let's go ahead and see what happens when we actually play music through this system.

So here we'll use the tape recorder. And what I will do is change the input signal to be that from the tape recorder. And we'll keep the ability to inject both the noise and the signal. Let's look at the XY plot, which I think is probably the more interesting one in this case.

So here we have the output versus the input. And we notice that this is the case where we have good reproduction. Let's change our feedback. Temporarily, let me get rid of the noise. And we notice very soon, an increase in audible audio distortion.

It turns out that crossover distortion is a particularly disturbing kind of distortion. And the reason is that it tends to suppress small signal levels you see. The crossover distortion eliminates small amplitude components. So precisely when you have a soft passage, a low level signal, you get the maximum fractional distortion. When you have a large signal and could better tolerate it, you get less distortion. Here, we have a very small signal. And we're never exceeding the threshold. You don't get anything out of the system.

If you contrast a signal that's distorted by crossover distortion with one that's distorted by clipping, even if they have the same sort of average distortion, the same sort of average harmonic content, what you find out is that the signal distorted by crossover distortion sounds far worse. That's something audio people have know for a long time. And consequently, they're very conscious, very sensitive, to crossover distortion in their audio amplifiers.

We hear the very characteristic grainy behavior. Let's go ahead and once again add noise in, as well. Here we see the sort of asynchronous characteristic of the noise. Things are very well buried in all of this. And we can make life much better by means of feedback. We got ahead and move the feedback to the output of the nonlinearity and sweep all the noise and distortion under a rug.

Finally, I'd like to change the single source and go to an alternative tape. Now, I think the interesting thing about this tape is that as we go from the nondistorted condition to the distorted condition, we notice virtually no change whatsoever. And so the indication is that we're really very sensitive to signal source, as well as the characteristic of the system. And I think there's a moral here if this is the kind of thing you choose to listen to, you really don't need a very good quality system. I defy you to tell me where the POT is set at present. Thank you.