James K. Roberge: 6.302 Lecture 13

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JAMES K. ROBERGE: Hi, last time we had looked at how we might select the compensation for an externally compensatable operational amplifier as a function of its application. We were using an available integrated circuit operational amplifier, the type 101A that is compensated by means of a minor loop. And we discussed how that minor loop worked. And in fact, how we chose compensating admittances, or compensating two port networks as a function of the desired closed-loop, or as a function of the desired open-loop transfer function of the amplifier.

Today, I'd like to continue and look at several other types of compensation that we're able to use. If you recall, the comparison that we were going against was an amplifier that had fixed compensation, something like a 741 type operational amplifier.

Normally, when a manufacturer designs an operational amplifier with fixed compensation he chooses to use single pole compensation. And normally, the constant associated with that single pole compensation will be selected so that the amplifier goes through unity gain with adequate phase margin. As a consequence, when you have such a single pole type amplifier, it's unity gain stable. You can use it in a connection where the output is connected directly back to the inverting input, and it will remain stable. And that's a perfectly acceptable general purpose compensation for many applications. However, the difficulty is that it's really overly conservative for many other and more demanding types of applications.

We saw that one of the things we can do is stay with single pole compensation. That is, use a single capacitor to compensate the amplifier. But change the magnitude of the capacitor. We found that if we did that and, in fact, changed the capacitor size as we changed the desired closed-loop gain, we were able to get very dramatic improvements in the bandwidth of our externally compensated amplifier compared to that of a 741.

We then looked at another type of compensation, two pole compensation. Where the amplifier roll-off goes as 1 over s squared at frequencies well below crossover. And then we add a zero, so that crossover is accomplished with a slope of 1/s. And again, we found out that that sort of compensation could yield very greatly improved desensitivity over a wide range of frequencies compared to that which we could achieve with single pole compensation.

Let's consider another possible situation. Suppose we have our operational amplifier. And in fact, compensate it to have a single pole roll-off. So that gets us a 1/s kind of roll-off in our loop. And to the extent that we can ignore other energy storage mechanisms in the amplifier, why we'd get approximately 90 degrees of phase margin with this kind of compensation if we used purely resistive feedback.

However, consider what happens if we load the amplifier with a capacitor. And that's a fairly frequently occurring situation. We can certainly think, for example, of power supplies where a capacitive load is used routinely. We've already looked at one aspect of that problem. But if we're not able to force a dominant pole of the output as we suggested earlier, why here we have a situation where we have a capacitive load. Another possibility is a circuit known as a sample and hold, where one takes an amplifier, connects a capacitor to it, so that the capacitor can be charged to some input voltage applied to the amplifier. And then, at some later time, removes the capacitor, so that the value of the voltage is maintained. That sort of a circuit is called a sample and hold. But again, it involves a capacitive load on an operational amplifier.

The operational amplifier has finite output resistance. And consequently, adding a capacitor to its output gives us an additional pole. And so now we have a two pole kind of loop-transmission. And certainly, if the additional pole that we add occurs well below crossover, the stability of the amplifier is very severely compromised. I'd like to look at that.

We have the same demonstration set up that we used last time. Just a power supply, a signal generator, which in this case will be applying a square wave to the amplifier we're testing, an oscilloscope to look at the output of the amplifier. And this time, we're using this particular amplifier, and have it connected simply as a unity gain follower. In other words, we apply the input signal to the non-inverting input terminal of the operational amplifier. We take the output and bring it directly back to the inverting input terminal.

And at present, we have the amplifier compensated for single pole operation. And we see the resultant step response on the oscilloscope. We're using about a 20 picofarad capacitor with this particular amplifier, which is the same one we used in the last series of demonstrations. We get a rise time, 10% to 90% rise time of about 200 nanoseconds with that single pole-- with that value for single pole compensation. Implying that the crossover frequency for this configuration is somewhere in the order of 2 megahertz. And this is operating without any capacitive load.

Now let's see what happens when I throw this switch. And first, we'll connect a 0.01 microfarad capacitor to the output in the amplifier. And we notice that in addition to the-- we get a tremendous amount of overshoot. We get a very more poorly damped system than we had before. Comparing again, these two cases.

Here's no capacitive load. Here's an 0.01 microfarad capacitive load applied to the amplifier. We get a considerable overshoot, quite a poorly damped response. And the settling time is, of course, very, very dramatically increased. Before settling time was several hundred nanoseconds. Here we see that the amplifier is still ringing at 10 microseconds. One horizontal division corresponds to 2 microseconds. So we see that the amplifier is ringing for at least 10 microseconds.

We can make matters even worse. Let me load the amplifier with a 1/10 of a microfarad capacitor. When we do that, we'll have to slow down the signal generator just a bit.

Here again, we notice a further deterioration of relative stability, considerable overshoot. We're now at 10 microseconds per division. And we notice that the response still, is wiggling 100 microseconds later. So the settling time of this amplifier has been deteriorated to more than 100 microseconds with the addition of the 1/10 microfarad load capacitor.

We can also notice the ring frequency. We're now at 2 microseconds per division. So the ring frequency is somewhere in the order of 200 kilohertz for this particular amplifier.

We can explain why that happens. Let's look at the amplifier open-loop transfer function in the absence of a capacitive load.

Here, we have chosen single pole compensation. For this particular amplifier with the 20 picofarad compensating capacitor that we're using, crossover occurs at about 2 megahertz, or 1.2 times 10 to the seventh radians per second. So this is the open-loop transfer function of the amplifier, a single pole roll-off over a wide range of frequencies when we have no capacitive load.

The output resistance of the amplifier is somewhere on the order of 100 ohms. And consequently, when we load the amplifier with a 1/10 of a microfarad capacitor, we'll put in another pole with a time constant of 10 microseconds. In other words, we'll add a pole at 10 to the fifth radians per second. Back about in here.

If we add one additional pole to the open loop transfer function of the amplifier, our resultant a of s will be this. We notice that the magnitude of the open-loop gain at the frequency where the pole is located is about 120. It's back a factor of 120 from the unity gain frequency that results with a single pole roll-off. We cross over at the geometric mean between this frequency and the original frequency since we're-- or the original crossover frequency since we're falling off as 1 over s squared. And so we cross over somewhere on the order of 10 to the sixth radians per second, or maybe a little bit higher than that. And that corresponds to the roughly 200 kilohertz ring frequency that we saw on the oscilloscope trace.

Furthermore, notice that we cross over after we've been rolling off as 1 over s squared for about a factor of 10, from this frequency to this one. There's been a long 1 over s squared region. And if you consider the implications of that, why we would anticipate a phase margin of somewhere on the order of a 1/10 of a radian. Roughly 5 degrees because we're crossing over a factor of 10 beyond the second pole. We'd certainly expect that that sort of phase margin would result in very marginally stable performance, and that's what we see in the step response.

Well, how might we improve the situation? The thing to do is to attempt to make the open-loop transfer function of the amplifier in the absence of load include a zero. In other words, start out with an open-loop transfer function that rolls off as 1/s at low frequencies. But then at some higher frequency, flattens out.

If we can do that, if we can add a zero, we presumably could use that zero to somehow offset the effects of the pole that we get when we add capacitive loading. We might, for example, attempt to line up the zero in the unloaded open-loop transfer function with the pole. Actually, we find out that there may a little better thing to do than that. but effectively, what we can do is use this zero in an attempt to offset the effect of the pole that results from capacitive loading.

If we can choose compensating admittance or a compensating network to give us this sort of ideal open-loop transfer function, we can look at the result in a little bit more detail here.

Here again, we have the assumed uncompensated transfer function of the amplifier. We have the approximation, the ideal open-loop transfer function that's simply related to the transconductance of the first stage and the compensating admittance, or the short circuit transfer admittance of the compensating network. That approximation we'd like to have a 1/s roll-off at low frequencies, and then flatten out at higher frequencies.

The intersection of that with the uncompensated curve, choosing the lower of the curves at all frequencies, gives us a good estimation of the actual open loop gain of the amplifier. And so between possibly this frequency and this frequency, we find out that we are able to control the transfer function of the amplifier by the compensating network. We get the single pole roll-off at low frequencies, the zero that we hope to use to compensate the pole resulting from capacitive loading.

Again, we see the effect on phase. The uncompensated phase, of course, rolling off-- the approximation indicating we'd have minus 90 degrees of phase shift at low frequencies corresponding to the 1/s roll-off. And increasing actually to 0 degrees. In reality, we don't achieve that, but we got an actual compensated phase curve that looks something like so.

At very low frequencies, it's 0. It then dips down close to minus 90 degrees over a wide range of frequencies. Heads back more positive in the vicinity of the 0. And eventually, falls off again at higher frequencies. So this then is the kind of an open-loop transfer function we'd expect from the amplifier.

If we could select a compensating or compensating network that had a short circuit transfer admittance such that g sub m over 2Yc had this form. How do we accomplish that?

Well, let's see. We'd like to have gm over 2Yc c of s have the general form K over s times a zero. We can do that without even resorting to a true two port. We can do it as simply a series combination of a resistor and a capacitor.

If we look at the simple RC connection and write its admittance, we find that the admittance of that network is simply CS over RCS plus 1. Consequently, g sub m over 2Yc has the general form that we'd like.

And what we might do then is, for example, choose a zero location as I've shown it, somewhere out in here. If we did that, we'd use that zero from the compensated amplifier open-loop transfer function in the absence of load. We'd use that zero to offset the effects of the pole that resulted from capacitive loading. As a result, our overall loop-transmission, or the negative of the loop-transmission, might look something like that.

The hope is that, of course, we greatly improve the phase margin by virtue of having added this zero to the loop-transmission. Let's see how that works.

Here again, we have the amplifier as we left it, with the 1/10 microfarad capacitive load and considerable ringing. And what I'm going to do is add a network that consists of a series connection of a resistor and a capacitor. Actually, there's a third little capacitor here. And I'll apologize for that. We'll see in a moment why we want to have that third element, the second capacitor. We'll see what we do with that in a moment.

But starting out, we simply have the series resistor capacitor combination that I mentioned. And what I'd like to do is look back at the oscilloscope trace. And recall that the settling time with 1/s compensation, is in excess of 100 microseconds. And now let's go to the compensation that includes a zero.

I'll remove the single capacitor that was doing the compensation before, and I'll put in the series RC connection. And when we do that, we notice a very dramatic improvement in performance. We're now able to go to a much faster time scale. And we notice, for example, that the trace has certainly, virtually completely settled three or four divisions after the start of the transient. We're 2 microseconds per division this time. And consequently, we're settling somewhere on the order of 6 to 8 microseconds. The basic transient is a very good one, a very pleasant one. The overshoot is-- oh, I don't know-- 20% possibly of the amplitude of the step, or something like that. So we've very dramatically improved the phase margin of the system as evidenced at least by its step response. But there's one bad thing remaining.

We notice that there's a series of little teeth on the trailing edge of the transient. Actually, they exist on the leading edge too, if you look at it carefully. And so the transient response of this setup, or of this configuration, is almost right. It just doesn't quite make it. And I'd like to think a little bit about why we might have that problem. Why do we get the teeth in this kind of compensation?

Well, what we're doing is using a network of the general type shown of course. And the important part of this network is that at high frequencies, or at least as far as the teeth are concerned, at high frequencies this network looks resistive. In contrast to either of the two other kinds of compensating networks we've looked at, either the single capacitor or the T network that had two capacitors across the top of the T, both of those networks looked capacitive at high frequencies. One looks resistive between the terminals.

In other words, when we use this for compensation around the second stage, that local feedback path, the minor loop feedback path, looks resistive at high frequencies. Let's see what that does.

Here I've showed a simplified model for the second stage of our operational amplifier. The input of the second stage, which if you recall, is driven by a differential first stage in the operational amplifier, generally looks like a parallel combination of an input resistance and an input capacitance. And then we have a generator that gives us an output voltage from the second stage proportional to the voltage at the input to the second stage. But there's also typically, a single pole associated with that transfer function. The physical reason for this pole is normally capacitive loading, or at least one possibility is capacitive loading at the output of the second stage itself. Not at the output of the overall operational amplify, but rather at the output of the second stage itself. So that gives us a pole.

And now we put a minor loop from the output of the second stage back to its input. Well, if at crossover in the minor loop, this network looks principally capacitive from here to here. In other words, if it looks like a single capacitor or if it looks like a T that has two capacitors across the top and a single resistor down on the vertical portion of the T, then at high frequencies we get simply an attenuation between this voltage and this one. We basically have a capacitive divider at high frequencies that involves the effective capacitance from this point to this point in the compensating network working with the input capacitance to the second stage. So we get simply, some fraction of the voltage at the output of the second stage applied back to its input. That's fine.

If we look at the minor loop, the minor loop is quite well-behaved under those conditions. Because we have effectively, a single pole roll-off-- this pole-- in the minor loop, in the vicinity of the minor loop crossover. However, consider what happens if at the crossover frequency of the minor loop, this network looks basically resistive.

If it looks effectively like a resistor between here and here, then it's entirely possible that we get an additional pole in the minor loop loop-transmission that reflects the loading of the input capacitance of the second stage working on that resistance. We'd simply get an RC transfer function from here to here. We combine that with one additional pole in the open-loop transfer function of the second stage, and we find out that we end up with a two pole roll-off. If that two pole roll-off occurs, or is in progress in the vicinity of the minor loop crossover, why we can have very low phase margin in the minor loop. And I think that's what gives us the teeth that we see in the transient response. I think that's a sign of impending minor loop instability.

I think that what's happening is that if we could physically reach into the amplifier and look at the actual output of the second stage, it'd be oscillating quite violently, an awful lot of overshoot, and so forth. We see some remain of that because we're looking at this through a low pass filter. We go ahead and remember we have a 1/10 of a microfarad capacitive load. The ring frequency at present is several megahertz. We're at 2 microseconds per division. The frequency of the teeth is several megahertz, and we're looking at that through a very low pass filter because of the output resistance of the operational amplifier and the 1/10 microfarad capacitive load that we've seen. And so we see just a little remnant of this large oscillation that's occurring at the actual output of the second stage before the final buffer amplifier associated with the op amp. And there's some nonlinearity to some crossover problems and so forth. But I believe that the teeth reflect an impending minor loop instability.

The way to get around that, the way to improve the situation, is to consider what would happen if we added one more capacitor here.

Now, what we'd like to do is maintain the general feature associated with the transfer function we had mentioned earlier. And what that means is that this capacitor must be much smaller than our original capacitor. So we might call this some big capacitor and this some small, or s is probably a poor choice C little. And the ratio between C big and C little should be quite great. The reason being that when we add C little, the effect on the ideal open-loop transfer function of the operational amplifier is to add an additional pole, maybe out here. We'd like to make sure that pole occurs well after crossover in the overall loop, so that we don't compromise the stability of the overall loop. We only want to use this capacitor to improve the stability of the minor loop.

And in this particular setup, we've chosen our small capacitor here to be roughly 1/10 of the value of the big capacitor. And that results in a very good compromise between maintaining stability in the minor loop and yet, not compromising the stability of the overall loop. So let's now add that capacitor.

When we do that, we notice the basic transient response remains the same. We have about the 20% overshoot, or thereabouts. But the teeth that completely gone away. We've eliminated the minor loop problem by making the compensating network look capacitive at the crossover frequency of the minor loop. That frequency, of course, is much higher than the overall crossover frequency.

Let's do it once more. Let's take out that capacitor. The teeth come. We add the capacitor back in. Simply move this lead over. The minor loop-- whoops. Well. It's really a matter of placement. The minor loop gets much better behaved, and we really haven't significantly changed the overall features. We haven't changed things significantly in the vicinity of the major loop crossover.

What's the difficulty with this kind of compensation? Well, one of the things is it's very, very special purpose. We have already looked at 1 over s squared compensation and found out that it was somewhat special purpose. Here we're tailoring compensation to a specific load. In particular, a capacitive load. And we're choosing the zero associated with this form of compensation to bear some relationship to the pole caused by the capacitive load.

What happens if we take the capacitive load away? Well, the system goes unstable. We can see that. Let me go back to zero capacitive load. And we notice even if we eliminate the input, why the amplifier is oscillating. I'm not sure whether we can sync on that or not. Sure.

Here the amplifier is oscillating at quite a high frequency, several megahertz, reflecting the fact that we've taken away the capacitive load, which is really necessary for stabilization when we use this sort of compensation.

Let's put the capacitive load back on. The amplifier is now well-behaved. And we can get our well damped response back.

The amplifier will oscillate even if we load it with a capacitor other than the intended one. If we go to the 01 microfarad capacitive load, once again the amplifier is oscillating. Presumably at a different frequency. But once again, oscillating.

So if we go back to the capacitive load we designed the compensation for, things are fine. But if we change the magnitude of that capacitive load or we eliminate it entirely, things get much worse. So that's one of the problems with this form of compensation.

I'd like to look at one final form of compensation. We found out that one way to offset the effects of an additional pole in the loop is to add a zero. That's what we were doing here. But now let me pose a different problem.

Suppose we anticipate an additional pole in our loop-transmission, but we don't know the frequency at which that pole will be located. So what we're looking here for now is really compensation for those who don't know what they're doing, possibly. Or something where we have a system that has certain uncertain parameters. Some uncertain parameters. How might we compensate in that case?

Incidentally, this is a more than academic interest. There are certain elements which, in fact, have dynamics that vary as a function of conditions. For example, a very simple example is that of a power supply.

Again, if you build a power supply into some system, why people go ahead and connect loads to the power supply. Very frequently those loads include their own decoupling capacitors. Consequently, you can add a great deal of capacitance to the output of the power supply. You don't necessarily know the exact amount. You can put a lower bound on it by including some capacitance internally. But you're never sure what the total amount of capacitance, how large the load capacitor can be.

There's another possibility. There's an element that's quite frequently used as a gain control element that consists of a photo resistor illuminated by a lamp. And by changing the illumination, you change the value of the resistor. You're able to use that as some sort of a gain control mechanism. Build a feedback loop that adjusts lamp intensity in order to get a desired signal level out of an overall system, for example.

Well, the nature of the photo resistors used in those devices is such that as you change illumination, the time constant, the pole associated with that transfer function, the transfer function relating light to resistance value, changes. The time constant is actually lower at low illumination levels than it is at high level illumination levels.

Another problem exists in that kind of a loop, in that the gain, the loop transmission in these sort of gain control loops, is always a function of operating conditions. It's a function of the amount of gain control that's present at a given moment. And so here you have a loop that has two uncertain variables. One the location of a pole, another one being the loop-transmission magnitude. And what I'd like to look at now is a fourth and final form of compensation that will allow us to maintain a good degree of stability in view of these sorts of uncertain parameters.

The test configuration that we're going to look at is as follows. Here we build up a loop that allows us to very easily add a pole to the open-loop transfer function of the operational amplifier that's used. And furthermore, vary its location over a very wide range of frequencies.

Here we have the operational amplifier that we're using with its compensating network. At its output, we add an R and a C, and we're able to adjust the values of the R and C, change the values so that we can add an additional pole to loop-transmission over a very wide range of frequencies by appropriately selecting the R1 C1 product. We then have a buffer amplifier, a little unity gain follower, that has a bandwidth much, much greater than the actual operational amplifier we're using. And we just use this to prevent any loading on the RC network. A little resistor that's recommended by the manufacturer when one uses this amplifier. Then we close the overall feedback loop.

The dynamics of this amplifier are such that it's basically transparent. Consequently, we have a loop-transmission in this loop that's that of the operational amplifier times 1 over R1 C1 s plus 1, the low pass characteristic of this network. And as I say, what we're going to do is attempt to come up with a compensation where we maintain adequate stability over a very wide range of R1 C1 products.

The difficulty, of course, is that if we use a single pole compensation with a fixed value of capacitor, why certain values of the R1 C1 product will cause us to cross over in a 1 over s squared region. In fact, it's certainly possible to have very, very long regions of 1 over s squared roll-off with appropriate choices of the R1 C1 product. And so we can get very, very low phase margins in that sort of a system.

If we choose compensation the includes a zero, we're again in trouble. Because we saw in the last demonstration that we have to tailor the location of a zero in that kind of compensation specifically to the load capacitance we're going to have. And if we used it in this configuration, why what we'd have to do is tailor the zero location to the precise value of the R1 C1 product that's used. At least within a factor of 2 or 3.

And if we state as one of the rules of the game, or one of the uncertainties here that we don't know the R1 C1 product beforehand, well, we're not able to use that sort of compensation.

Let me suggest another approach. Suppose we could come up with a way of compensating the amplifier such that it's open-loop transfer function rolled off proportional to 1 over the square root of s. If we did that, let's see. What's the characteristics that's associated with a 1 over the square root of s roll-off?

Well, the phase shift associated with the amplifier itself, with its open-loop transfer function, would always be minus 45 degrees. That's the angle associated with a 1 over radical s transfer function.

However, the magnitude would drop off. And that means that if we designed our 1 over radical s amplifier properly, we could arrange to have it go through unity gain at a sufficiently low frequency, so that it would be stable without an additional pole added to its loop-transmission. Yet if we added another pole to loop-transmission in addition to the 1 over radical s roll-off, why we could be assured of at least 45 degrees of phase margin. Regardless of the location of the additional pole. The pole could give us, at most, minus 90 degrees of phase shift. That combined with the 45 degrees of negative phase shift associated with the 1 over radical s roll-off would give us, at most, 135 degrees of negative phase shift associated with the af product. And so we'd have at least 45 degrees of phase margin. We could have more depending on the exact value of the R1 C1 product. But we're guaranteed to have at least 45 degrees of phase margin.

Well, that's fine. The only bad news is that one can't realize a radical s kind or 1 over radical s kind of roll-off as a collection of lumped elements. There are distributed systems that can give us that sort of a transfer function or transfer admittance. But you can't do that as a collection of lumped elements. But you can approximate it quite well. And the way you do the approximation is as follows.

We've already looked at what happens if you take a single RC. And we found out that in that case, 1 over Y sub c looks something like so.

Well now what I can do is add another parallel branch. We might start out with this being a relatively large capacitor. That gives us the 1/s roll-off here. We then get to the frequency where the admittance of the capacitor equals the conductance of this element, and then things level off. The admittance gets no lower.

We now could add a smaller capacitor, and maybe another resistor. And now what would happen is that at some higher frequency, this capacitor would become important. It would now shunt this path. Both this capacitor and this resistor are smaller in value than the corresponding elements in the top rung. And the net result there might be to give us another region of 1/s roll-off. And then, another flat region. And I think now you can see what we might do. We continue to build up our network with parallel RC chunks. And the result on 1 over Yc is to have a 1/s roll-off, a flat region, of 1/s roll-off, a flat region, and so forth.

And if we spaced the poles and the zeroes associated with this transfer function closely enough, we begin to approximate of 1 over radical s roll-off. We'd roll-off as 1/s half the time on logarithmic coordinates, be flat half the time, 1/s and so forth. And begin to approximate a 1 over radical s roll-off.

So what we would chose to do is use a network that looks something like this. If alpha is a number larger than 1-- if we start in the middle, for example, both resistors and capacitors get larger as we go this way in the ladder. As we go to higher rungs in the ladder, resistors and capacitors gets smaller.

And you can chase that through and analyze that sort of a network. The thing we're really interested in getting out of this network is 45 degrees of phase margin in our eventual system. In other words, we want the transfer admittance of that thing to have 45 degrees of actually positive phase shift.

If you look at the actual phase shift associated with a transfer admittance, of course, it ripples. When the transfer admittance is going as s, why you have 90 degrees of phase shift or approaching 90 degrees of phase shift. When the transfer admittance is constant with frequency, you're approaching 0 degrees of phase shift. And in general, just as the roll-off of that network ripples, so does the phase. And you can control the amount of ripple, really by the closeness of spacing of pole and zero pairs. Or equivalently, by the choice of the quantity alpha in this network.

Well, it turns out that if you use only one pole and one zero per decade, you get a very, very good approximation to 45 degrees of phase shift associated with the transfer admittance. You can choose that or you can look at that situation. Use, as I say, a value of alpha that's the square root of 10, which gives you one pole and one zero in the transfer admittance of the network per decade and frequency. And you find out that the peak to peak ripple associated with the phase shift of that network is only 3 degrees. So we can approximate 45 degrees, the desired value, and the peak to peak ripple associated with that approximation, assuming that all of the elements had the specified values-- there was no component tolerance-- would only be 3 degrees.

Well, in order to see how that works, we build up such a network. And here we have-- they look a little bit unwieldy. This happens to be a 11 rung ladder. And what we have are, of course, simply 11 parallel RC sections. This network gives us a 1 over radical s roll-off from several megahertz to fractions of hertz. Again, using one RC section per decade, we get that sort of performance with this kind of a network.

What I'd like to do is investigate how that sort of a network stabilizes a system, pretty much independent of where we locate an additional pole in the loop.

Here we have our experimental setup. This is a little bit different from the past one. What I have, once again, is the power supply powering everything. A generator that let's us put steps into our circuit. We changed the board in this case.

Here's the circuit, itself. We have the 301A or the 101A type amplifier and its buffer amplifier. And a series of switches on the front that, first of all, let us select from various types of compensation. Also, let us select the R1 C1 product. So that's what these three switches do.

And what I'd like to initially look at is the behavior of the system when we have a single pole roll-off associated with the amplifier. And we get that by having the right-hand switch in the upward position. And we now have removed the additional pole in the loop, the R is quite small and the C's an open circuit. So we've now removed the additional pole.

And the result of that, of course, is to give us effectively an exponential kind of response we'd expect when our loop was rolling off as 1/s or proportional to 1/s. We'd have a first-order system. We've chosen numbers so the crossover in the loop occurs well below any additional phase shifts associated with the amplifier itself. So we get a response that looks very, very nearly first-order.

Now let's begin to add additional poles. Let me start out putting in a pole at a megahertz. Well, that doesn't really have much effect on the response. Here's a pole going in at a megahertz. That's actually beyond crossover in this loop, and so there's very little change in the response. We see just a hint of overshoot. But really, nothing very dramatic.

Let me move the additional pole back to 100 kilohertz by appropriate choice of the R1 C1 product. I do that by throwing this middle switch down.

Well, now things are getting worse. Now the pole actually is occurring in the vicinity of crossover in the overall loop. And so now we're down to something on the order of 40 degrees, 30 degrees of phase margin possibly. As I say, the pole is probably actually a little bit above crossover. And so we get some significant overshoot.

Let me move the pole down another decade to 10 kilohertz. This is again, lowering or increasing the value of the R1 C1 product. Things get much worse.

So now we're not settling by the time square wave comes along. Let me slow the generator. But now we have considerable overshoot. We're putting in a three division step and we're getting 60% overshoot, or something like that. Very prolonged transient. Very poorly damped.

Let's try it just one more time. That was with a 10 kilohertz pole location. Here's one at 1 kilohertz. Again, things haven't quite settled out. Let me slow the generator even further.

We can see, once again, very, very nearly 100% overshoot, something like 90% overshoot. A very, very prolonged settling. Let's see if we can do it just one more time.

Move the pole down to 100 hertz. And I don't know whether we can slow things enough to observe that or not. I'm just going to lower drive frequencies at present.

And here we see a very, very poorly damped transient. Let's go back to one that's a little bit easier to observe.

Here we are again with the transient response that corresponds to a 1 kilohertz pole location associated with R1 C1. So we see that when we use that sort of compensation, the 1/s compensation in the loop where we can add an additional pole, we're able to get into very, very severe stability problems.

We have another version of the 1 over radical s network here, a somewhat better behaved network. Or at least, easier to use rather than have it spread out in the Jacob's Ladder kind of configuration. This is a little bit easier to use in our test setup. Let's go back to the situation where we have the actual-- the R1 C1 pole located at very high frequencies. And then let me add the network that compensates our loop for 1 over radical s behavior. And let's look at that response.

What happens here is that we get a very long tail, if you will. We spoke a little bit about this prolonged settling transient that results when we have a pole zero doublet. We looked at that in conjunction with the 1 over s squared compensation last time. I mentioned the same kind of thing happens with a lag network.

Well, here, we have an extreme example of this kind of thing. The effect of using the approximation to the 1 over radical s roll-off is that we end up with an open-loop transfer function that has poles and zeroes alternated along the negative real axis.

And if we draw a root locus diagram for that, we end up with a very large collection of pole zero doublets. In this case, 11. And we get a very, very long tortured settling to final value. And we can see that final value has not even been reached here yet. We go halfway or so to final value quite quickly. We go halfway in a few microseconds. We're at 5 microseconds per division. 1 and 1/2 divisions vertically is half of final value. But 500 microseconds later, we're still settling.

And in fact, if we examine things very carefully, half a millisecond or 5 milliseconds later, the transient response is still settling. And that reflects the collection of closely spaced pole zero doublets that result when this sort of compensation is used.

The good news is that we get very dramatically improved stability as we add capacitive loads, or an additional pole in the loop transmission. Let me, first of all, add the pole at 100 or at 1 megahertz. Again, not much happens. That is at sufficiently high frequency, so that not much changes.

But now let's go to 100 kilohertz. And this was the case where we were beginning to get measurable significant overshoot of when we use 1/s compensation.

Here things are still very well behaved. In fact, we really can't discern any overshoot. A little ripple, but we really don't see any overshoot.

Let's move the pole back to 10 kilohertz. Getting a little bit of overshoot, but a very, very well-behaved response.

Now let's go to a kilohertz rather than 10 kilohertz. Do this. OK, there we are with the network giving us a pole at 1 kilohertz. Very well-behaved response. Reflecting the approximately 45 degrees of phase margin.

Let's go to 100 hertz. Things get slower. In fact, much slower. But the basic shape of the response now remains almost invariant.

Here's the response with the RC pole located at 100 hertz. Here is the response with it located at a kilohertz. Comparable kinds of responses in terms of stability. A little bit more overshoot here. I think that reflects the ripple in the phase curve and probably the ripple is more than 3 degrees peak to peak because of component tolerances.

Let's locate the pole at 10 hertz. We weren't even able to do that with 1/s compensation. Things do get slower. But we have still, a very well-behaved response. Not excessive overshoot. And so we're able to stabilize the system.

Notice that as we did this, I've been moving the pole over 5 decades. I started at a megahertz, got all the way down to 10 hertz. And we still get very, very acceptable transient response.

So as the pole moves, over in this case, 10 decades in frequency, we still get very acceptable performance.

Well, one may question the practicality of using this sort of a network to compensate an operational amplifier. Fortunately there's a middle ground.

We found out that an approximation that uses one pole and one zero per decade did a pretty good job. Let's consider a very crude approximation to the ladder, just a two rung ladder. And in fact, let's choose things such that the capacitors have a ratio of 10:1.

By doing that, we can end up with a very crude approximation to the 1 over radical s roll-off. Let's see.

What I've shown here normalized to a 10 to the sixth radian per second crossover frequency is, first of all, a transfer function that's falling off as 1/s. That has minus 90 degrees of phase shift at all frequencies.

I then show a network, or an amplifier that's rolling off as 1 over radical s. That has minus 45 degrees of phase shift at all frequencies.

Here I show a transfer function that rolls off as 1/s from the unity gain frequency back to a decade below the unity gain frequency. It then flattens out for another decade from here to here. And then, once again, rolls off as 1/s below that. So this is a very crude approximation to the 1 over radical s kind of roll-off. The corresponding phase characteristics are like this. But that's a very achievable network. We can do that, as I say, with just two capacitors and a resistor as I've shown. And let's look at the results of that.

Let me remove the 1 over radical s kind of network. This one. Let's go back to the very high frequency location for the RC pole. And let's speed up things now.

Well, what we find is again, a delayed settling time. . This sort of compensation, again, gives us a pole zero doublet if we look at it. We get to about 90% of final value very quickly on the leading edge. But it takes is much, much longer. The response is basically two exponentials, one of which has a much longer time constant than the other one. And we see the quite prolonged settling to final value.

Let's begin to add the pole. This is with the additional pole in the loop at infinite frequency effectively. Now let's go ahead and add the pole at 1 megahertz.

Again, not much effect. We see just a little peaking. We can begin to lower the location of the additional pole in the loop. Here's 100 kilohertz. Well behaved.

10 kilohertz. Notice this is actually better damped than the one corresponding to the pole location at 100 kilohertz. That's not surprising. The phase of this network, this very simple approximation to a 1 over radical s network changes far more violently as a function of frequency than does the 1 over radical s approximation using the much better, or ladder with a considerably more-- larger number of rungs. So it's not surprising that relative stability is more dependent on the exact location of the additional pole.

Here's one of the kilohertz. Have to slow things down. Again, relative to the pole location of 10 kilohertz, we have poorer stability.

At 100 hertz, considerably poorer stability once again.

And beyond this, we begin to lose things quite dramatically. But again, at this point, the 1/s compensation had just completely lost. In fact, we had a lot of trouble with 1/s compensation when we located the pole at 1 kilohertz. Here the system is very well-behaved with this very simple compensating network when the poles of 1 kilohertz.

If you look at this in detail, you find that you can maintain something like 30 degrees of phase margin as a minimum, until you move the additional pole back to somewhere on the order of a kilohertz. Or actually, below that. So you're able to move the pole from crossover, back by approximately 3 decades before you begin to get into serious stability problems.

The implication is that we can design for a system with an uncertain pole location where the pole location varies over as much as 3 decades, and still get acceptable stability with a very, very simple compensating network independent of the actual location of the pole.

To summarize very quickly, we've compared special purpose compensation, compensation tailored to the specific application. And we've compared that with what happens when we live with sort of the manufacturer's recommendation in an internally compensated operational amplifier.

We've found that we can either use single pole compensation where we tailor the constant associated with the compensation to the specific feedback attenuation used. We improve performance compared to fixed single pole compensation. We can use two pole compensation, which gets us greatly improved desensitivity compared to single pole compensation over a wide range of frequencies.

We can use compensation that includes a zero, and that allows us to make up for one additional pole in the loop. But it's dependent on a good knowledge of where the additional pole is located.