James K. Roberge: 6.302 Lecture 14

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JAMES K. ROBERGE: Hi, in the last session we looked at the possibility of compensating a feedback system by using minor-loop compensation, where we construct a system that has two loops. By adjusting the feedback element, or properly selecting the feedback element in an interim minor-loop, we're able to compensate the system.

In today's session and the following one, I'd like to look at how we do this with an actual existing operational amplifier. And how we choose compensation, in particular, minor-loop compensation, to tailor that operational amplifier to a variety of applications.

The amplifier that we're going to use is one that's been in existence for 10 or 15 years. It's the type 101A, or equivalently, the 301A, the difference between those two designation simply reflects the temperature range over which specifications are guaranteed and a minor grade out on certain parameters. So we'll use this readily available and quite inexpensive operational amplifier.

And there are several considerations that we have to factor in to our thinking that reflect the reality of that amplifier.

First of all, the amplifier is constructed using a commonly used integrated circuit process. One of the limitations of that process has to do with fabrication of PNP transistors. And a consequence of that limitation is that if we form a feedback loop using the 101A type operational amplifier, it's impossible to get crossover frequencies in the overall feedback loop much above a megahertz without the overall loop becoming unstable. And the reason for that, as I say, has to do with poor dynamic performance associated with the lateral PNPs.

In particular, above about a megahertz, they look very much like time delays, and we're not able to compensate, as we recall, for the phase shift associated with a pure time delay. So we have to bear in mind when we're doing our compensation, that any loop that includes this kind of an amplifier, the major-loop crossover is constrained to somewhere on the order of a megahertz. Actually, that's been getting a little bit better with time reflecting process improvements. We'll see in the amplifier we're using now, we can get about 2 megahertz crossovers without too much trouble.

The quantity g sub m over 2, which we mentioned last time enters into the approximation for the open-loop transfer function of the amplifier is about 2 times 10 to the minus fourth mho for this amplifier. That reflects the fact that the input transistors are running at 10 microamperes. Consequently, the corresponding transconductance is about 4 times 10 to the minus fourth mhos. Again, a manufacturing variation on that, but a nominal value for the transconductance. Or the quantity g sub m over 2 is about 2 times 10 to the minus fourth mhos.

A final thing that we didn't mention last time, but will become important actually in the session that follows this one, is that certain kinds of minor-loop compensation result in stability at problems associated with a minor-loop itself. And so we have to be careful when we do the compensation, in this system or any other system that uses minor-loop compensation, to make sure that the minor-loop remains acceptably stable.

One kind of compensation that we can implement very easily in minor-loop form is so-called single pole compensation. What we do there is use a single capacitor as the compensating element. That's sort of a degenerate case of a two port.

And if we go through the numbers, we find out that the transfer admittance for a capacitor, a single capacitor, is simply equal to the value of the capacitance. Let's call it c sub c times s.

Consequently, when we evaluated our approximation to the open-loop transfer function of the amplifier, it would be g sub m over 2 times c sub c s.

We can look at what we're trying to accomplish with this sort of compensation. Remember from last time, we mentioned that the way we determine the actual open-loop transfer function of the amplifier with compensation is to first draw the uncompensated transfer function of the amplifier.

Here I've assumed that our amplifier is actually three pole. Here's one pole, two poles, three poles. But the details of this aren't important. The point is, we do have the uncompensated open loop transfer function of the amplifier. That which would result if we disabled the compensating network, but included its loading. So that's this curve.

Then we draw the approximation. And in the case where we use a single capacitor, we of course get a one-pole approximation inversely proportional to s. So here's the approximation.

Then to a good degree of approximation, why our overall compensated open-loop transfer function for the amplifier is simply the lower of these two curves at all frequencies.

We've shown some dotted lines here over a range of frequencies where the open-loop transfer function of the operational amplifier is, in fact, controlled by minor-loop feedback. Remember, the inequalities between the uncompensated transfer function and the approximation involving the transfer admittance of the compensating element are such that in this region, the amplifier open-loop performance is very directly controlled by minor-loop compensation. And so over this range of frequencies, we can estimate the open-loop transfer function of the amplifier based only on the knowledge of the transconductance of its first stage and the compensating element that we're using. In this case, a capacitor.

The effect on the angle associated with the open-loop transfer function is shown below. Here we have the original uncompensated transfer function assumed to be three-pole. But again, that's not horribly important. The effect of adding the compensation, the single pole compensation, is to force a single pole roll-off between this frequency and some much higher frequency. And so we have a long range of frequencies here where the amplifier open-loop transfer function is, in fact, following a single pole roll-off. We get approximately 90 degrees of phase shift. So here we have a situation where the use of a single capacitor. And in fact, quite a small one as we see, controls the transfer function over a very wide range of frequencies. And in fact, gives us an approximately single pole roll-off over a very wide range of frequencies.

A frequently used value when we design an amplifier that includes internal compensation, rather than one where we can add the compensation ourselves-- an amplifier such as a 741 as I mentioned last time, is one that uses this general topology, the general two-stage topology. But it has an internal capacitor. And the value of that capacitor is typically 30 picofarads. The transconductance of a 741 input transistor is the same as for the 101a. And consequently, the quantity g sub m over 2 times c sub c s is equal to g sub m over 2, or 2 times 10 to the minus fourth mhos divided by 3 times 10 to the minus 11th farads in the case of a 30 picofarad capacitor.

And then, thus our approximation to the open-loop transfer function is about 6.7 times 10 to the sixth over s. That transfer function actually holds over a range of frequencies that typically extends from about 10 hertz to roughly a megahertz. So here a 30 picofarad capacitor compensates the amplifier, dominates its performance over about a factor of 10 to the fifth to 1 in frequency. That's very significant actually, for the fabrication of integrated circuits.

Because using this minor-loop technique in the topology we've discussed, we're able to compensate the amplifier with a size capacitor that can be fitted on to the chip. It doesn't consume excessive chip area. And so one of the things that made internal compensation of operational amplifiers possible was exploitation of this sort of minor-loop topology.

The reason for the choice of a 30 picofarad capacitor in fixed compensation amplifiers is that that forces crossover. In other words, we go through unity gain in the amplifier for the amplifier open-loop transfer function, at a frequency of course, of 6.7 times 10 to the sixth radians per second. Or about a megahertz.

And we mentioned that the phase shift problems associated with the dynamics of the PNP transistors that are typically used in this topology, are such that we have to constrain crossover to about a megahertz, or else we'll get stability problems. So this amplifier, one that uses a 30 picofarad capacitor, is an example of one that will be unity gain stable. In other words, if we form a feedback loop with f equal to 1, by connecting the output of the amplifier directly back to its input, the resulting configuration should be stable since the major-loop thus formed crosses over at about a megahertz.

The problem is that that sort of compensation is overly conservative in many applications. It's fine if we have direct feedback from the output to the input. But it really deteriorates the dynamic performance in many, many other feedback configurations. Let's consider the following.

Here we have an operational amplifier connected as a non-inverting amplifier. Of course we've seen this configuration before. We have attenuation from the output back to the inverting input via the R1 R2 network. We have a single capacitor that's used for compensation. And I'd like to investigate that configuration.

The block diagram, assuming that we can represent the amplifier by our approximation, has a forward path that will be simply the approximation g sub m over 2 times c sub c s. The feedback path reflects the attenuation provided by the R1 R1 plus R2 network.

If we estimate the closed-loop gain based on this expression for the forward path gain, we get the expression shown, g sub m over 2 c sub c s 1 plus g sub m over 2 cc s times f. 1 minus the loop-transmission, assuming that the loop-transmission can use this estimate for the forward path gain.

And if we rearrange terms, basically multiply numerator and denominator by the reciprocal of the second term in the denominator, we find out that the closed-loop gain is single pole, has a value 1/f, the ideal closed-loop gain, which results at low frequencies. And then, a single pole roll-off where the time constant associated with that roll-off is related to the ratio of c sub c and f.

Now, the problem is that as we go to larger and larger closed-loop gains-- in other words, if we go back to our original amplifier. And in order to get higher closed-loop gains, of course, we'd made R1 smaller relative to R2. Thereby, lowering f. And since the ideal closed-loop gain for this non-inverting configuration is, of course, simply 1/f, why as we lower that ratio, we make f smaller. We get progressively higher closed-loop gains.

The problem is that for a fixed value of compensating capacitor, if c sub c is fixed, why as we lower f, the time constant associated with the closed-loop pole gets progressively larger. In other words, the amplifier gets slower as we go to larger and larger closed-loop gains.

However, consider what happens if as we change f, we modify c sub c. We change c sub c in about a proportionate way. And if we do that, why we can keep the dynamics, the closed-loop dynamics, fairly constant. We can again, look at the effect of that on one of our earlier view graphs.

This is the curve that is g sub m over 2 times c sub c s. As we change the value of c sub c to larger values, the curve moves down. As we decrease the value of c sub c, the curve moves up. Implying wider bandwidth out of the amplifier itself.

Now, as we decrease c sub c, the unity gain frequency of the amplifier itself will increase. And in fact, may very easily go above a megahertz for certain values of c sub c. In fact, if c sub c is less than about 30 picofarads, the unity gain frequency of the amplifier itself extends beyond a megahertz.

That's no problem though, if f gets correspondingly smaller. Since the crossover frequency of the major-loop is the one we have to worry about in terms of loop stability problem. So here, as we lower f, we can make corresponding changes in c sub c. The overall major-loop still crosses over at frequencies of about a megahertz or so. And so we don't get into stability problems because of the lateral PNP transistors.

Let's see how this works in practice. Here we have a board that includes an amplifier where we can change the compensating element. And the rest of the equipment then includes a power supply, which simply powers the amplifier in question. A signal generator that allows us to put steps or other test signals into the amplifier. And an oscilloscope to look at the resulting responses.

If we examine this board in detail, it has more than we're actually going to use in this demonstration. We're actually just using this little corner of it. Here is a 301A type operational amplifier right here. There are two terminals on that amplifier, here and here, that are connected to the compensating or that allow us to connect the compensating element to the amplifier. We're presently using a single capacitor for compensation. As I mentioned, single pole compensation. Here is that compensating capacitor.

In this particular amplifier, or with this particular amplifier, we found that we get very nice, stable responses with just a small amount of overshoot as presently shown on the oscilloscope. In unity feedback configurations, when we use actually a 20 picofarad capacitor. Also, if we look at the transconductance of this amplifier, we find it's a little bit higher than the nominal value. Crossover with a 20 picofarad capacitor or the unity gain frequency with a 20 picofarad capacitor is actually about 2 megahertz in this particular amplifier.

The switches on the front of the box, in particular, these two switches, allow us to change f. Thereby, establishing different values for the ideal closed-loop gain. Right now we're set up for a closed-loop gain of 1 with f equals 1. As I mentioned earlier, we have the 20 picofarad capacitor. And we're looking at the resultant step response. The horizontal scale is 100 nanoseconds per division.

And one very good measure of speed of response of course, which is easily interpreted in terms of bandwidth, is the 10% to 90% rise time for the system. So let's see.

Here is the 10% point. Up here is the 90% point. We have just about 1 and 1/2 divisions horizontally between those two points. We're at 100 nanoseconds per division. And so we have a rise time of about 150 nanoseconds.

So let's tabulate values. Here we're running at an ideal closed-loop gain, or a low frequency gain, a0, which is equal to 1/f. We're running at a closed-loop gain of 1 for this particular part of the demonstration. And the rise time that we measure is about 150 nanoseconds. We're using a 20 picofarad capacitor when we make that measurement.

For unity feedback, for an f of 1, the optimum value of the capacitor is close to the 20 picofarad value. Going to smaller values of capacitor with this value of f, would result in progressively more overshoot, progressively poorer stability. Because we'd be pushing the crossover of the major-loop out toward higher frequencies.

If we wish to, we could slow the overall system. We can certainly use larger than 20 picofarad capacitors. That would result in somewhat improved stability, although the stability of this one certainly looks good for many applications. But would also generally slow the system. And in fact, the crossover of the major-loop would drop just about in proportion to that capacitor size.

Now let's change the gain of the amplifier, so that it's running at a closed-loop gain of 10 rather than 1. In other words, we'll change f by means of this switch to a 1/10. And when we do that, we'll notice that the oscilloscope trace gets considerably larger, reflecting the factor of 10 increase in gain.

Let me change the scale, the vertical scale on the oscilloscope, by again a factor of 10. The important thing though, for our purposes, is that the step response has gotten much slower.

Let's change the time scale. And in fact, here we are at 1 microsecond per division horizontally. And we can now measure the 10% to 90% rise time, now between here and here. We notice again, it's very close to 1 and 1/2 divisions. This time we're at 1 microsecond per division. So we have a rise time with a 20 picofarad capacitor of 1 and 1/2 microseconds when we have f a 1/10 or 1/f equal to 10.

Notice that the ratio is 10:1, which is exactly what we'd expect. We found out that the time constant associated with the closed-loop gain was proportional to the ratio c sub c over f. Here we've kept c sub c fixed. We've changed f by a factor of a 1/10. We'd anticipate that the closed-loop response would slow by just about that factor of 10. And, in fact, it does.

We also notice a little change in the character of the response. The original response-- this one-- had just a little bit of overshoot. Once we are observing this response, we notice it looked very, very much first-order.

However, we have lost the factor of 10 in response time, or in bandwidth. And we can buy that back by lowering the size of the capacitor by about a factor of 10. There's a little bit of departure from the ideal case here. There are a couple of effects in the amplifier that actually force us to use a capacitor that's somewhat larger than 1/10 of the previous value. But we've made up a capacitor in that range, made up a capacitor of about 3 picofarads, or thereabouts. And we'll replace the compensating element with this lower value compensating capacitor. And we find out that we get a response that's very similar to the original one.

Here we have just a little bit of overshoot. So we're back to corresponding to stability. Actually, probably a little bit better than the original one. This capacitor might be a little larger than the one that would give us exactly the same amount of overshoot as we got in the first experiment.

We can measure the 10% to 90% rise time in this case. And let's see. Let me go back to 100 nanoseconds per division.

Here is 10%, 90%. Let's see, we have 1, 2-- about 2 and 1/2 divisions. And so we're able to, by choosing an appropriate capacitor, reduce the rise time. Not quite to its original value, but we got about 250 nanoseconds. So rather than losing a factor of 10:1 in bandwidth, or rise time, we've lost a factor of maybe 6:1. Well, let's continue.

Let's, first of all, go back to the original 20 picofarad capacitor. And now, up the gain by another factor of 10. If I throw this switch down, we get a closed-loop gain of 100. We make f 1/100. Let me do that. And again, I have to change the vertical scale on the oscilloscope.

We find out now though, the bandwidth is so deteriorated with the original value for the capacitor that we have to slow the signal generator. So let me do that, so that we reach final value.

All right, and here is the step response that corresponds to a 20 picofarad capacitor and a closed-loop gain of 100. The amplitude is down just a little bit. Let me increase that, so that we can easily measure the 10% to 90% rise time.

We notice the response is very much first-order. Here we're at 5 microseconds per division. 1, 2, just a little bit more than 3 divisions. So we have a 10% to 90% rise time of just about 15 microseconds. Again, since we're comparing, in this column, the response that we get with an equal value for c sub c, but progressively smaller values of f, we'd anticipate that this value would simply go as 1/f. And we see that that's what happens.

Again, we can get close to our original speed of response by removing the 20 picofarad capacitor, and replacing it with one that has been chosen. As I say, it's not quite 1/100 of the original value, but one that's been chosen to yield very close to the original dynamics. And we get that sort of a response much faster. Notice very nearly vertical rise on this kind of a time scale.

We can now speed up the generator once again. Look at the response in detail. And again, our 10% to 90% rise time. We're now back to a time scale of 200 nanoseconds per division. The 10% to 90% rise time is back to about 500 nanoseconds. 1, 2, 2 and 1/2 divisions at 200 nanoseconds per division. So again, we haven't quite gotten back to the original case. But we're roughly a factor of 30 faster than the case that resulted when we used the capacitor that was selected for unity gain stability, or for unity feedback. So we've made a very, very dramatic improvement in the bandwidth of the amplifier by simply choosing the compensating capacitor as a function of the closed-loop gain.

Finally, let me go back to the original value, the 20 picofarad compensating capacitor one last time. And now let's increase the closed-loop gain to 1,000.

We clearly are going very, very slowly. Let's slow down the generator. Things saturate. We've got to put in a smaller signal. We still haven't reached final value. All right.

Now we finally have things adjusted, so that we can see the rise time with the 20 picofarad capacitor at a closed-loop gain of 1,000. And if we make the rise time measurement this time, let's see. We're at 50 microseconds per division. So we have 1, 2-- again, just about 3 divisions for the rise time. So we have a rise time about 150 microseconds. Our amplifier, as we anticipated, has gotten a factor of 1,000 slower than the original case.

If we interpret this in bandwidth, in terms of bandwidth rather than rise time, here we had a bandwidth of very nearly 2 megahertz. Or actually, a little bit more. Remember, the product of rise time times bandwidth in hertz is about 0.35. So this corresponds to something like 2.3 megahertz of closed-loop bandwidth.

This corresponds to a factor of 1,000 smaller than that. Approximately 2 kilohertz. So here, at a closed-loop gain of 1,000, we're not even able to accomplish-- to get significantly through the audio band. The bandwidth in the amplifier is about 2 kilohertz.

If we had used a 741, which uses fixed compensation for all of these experiments, its numbers are such that we'd actually have about a kilohertz of bandwidth under these conditions. Well, let's see what we can do if we compensate the amplifier properly, instead of using the fixed compensation.

Let me remove the compensation entirely. When we do that, our approximation begins to break down a little bit. But we find out that even-- we can now speed up things.

Even with this large a value of f in the feedback path, or with a smaller value of f, this much attenuation in the feedback path, the amplifier is actually still under damped. We notice considerable overshoot to the step response with no compensation. We see there might be something like 30% overshoot, or 40% overshoot. A reasonably high frequency ring, ringing at about 300 kilohertz. We have 1 microsecond per division horizontally now. So let's see if we can compensate that.

If we really believed our numbers, we'd conclude that we needed 20 picofarads divided by 1,000. And that seems somewhat unrealistic. Actually, we need a capacitor a little bit larger than that. But let's get our capacitor this way.

I simply have two pieces of wire that I'll put into the compensating terminals. And we now have a very convenient method for adjusting capacitance at very small values. If we get these apart, we have the undercompensated case with a reasonable amount of overshoot. As we move the two wires together, we're able to get progressively better damping.

Here's a critically damped case. No overshoot. The closed-loop poles appear to all be on the real axis. And let's see if we can get back to pretty close to the original situation. Something like that. So in this way, by adjusting the capacitance, we're able to very well control the dynamics, the transient response.

Also, we get back to a very respectable value for rise time. Here we're at 500 nanoseconds per division. If we measure the 10% to 90% rise time, we find out it's just about two divisions on that time scale from here to here. So we get about a 1 microsecond rise time. Roughly a factor of 150:1 improvement in rise time and correspondingly, bandwidth.

When we compensate with this small value of capacitor, we still have something like a 350 kilohertz closed-loop bandwidth at a gain of 1,000.

We might question the practicality of all of this when, in fact, in order to compensate, we have to do something like this and wave two wands at each other. But in reality, we find out we can do that fairly well.

If we back off just a little bit from sort of the optimum situation and conclude that we're going to overcompensate a little bit, maybe only get 100 or 150 kilohertz of bandwidth instead of trying to stretch things out and get 350 kilohertz, we can do that quite repeatedly.

Furthermore, there are techniques which certainly have been used. For example, you can lay out a printed circuit board and peel back foil or something to make the adjustment to tailor particular circuit values to a specific operational amplifier. But as I say, if you're a little bit concerned about that, why you can certainly get most of the benefits, be a little bit more conservative by simply over compensating a little bit, and ending up with still far, far better dynamics than those that result if you use a fixed compensating element.

Well, single pole compensation is a very good general purpose kind of compensation. It's the one that's normally used when fixed compensation amplifiers are sold. As I say, it can be certainly designed so that the amplifier is stable in a variety of feedback connections. One of its problems is simply that it's overly conservative in many applications, as we've seen.

Another possibility is to consider the following. Let's suppose that this is the open-loop transfer function of the amplifier without compensation when we consider the effects of loading by the compensating network. And in general, why the compensating network doesn't-- the loading of the compensating network usually doesn't change that curve much. So let's suppose somehow we know that this is the uncompensated open-loop transfer function for the amplifier.

If we use single pole compensation, and possibly adjust it for unity gain at a megahertz or something like that, why this is the approximating function. Our approximation tells us that at least over from here to this frequency, and possibly beyond, we'll have basically a single pole roll off associated with the open-loop transfer function of the amplifier.

We might wonder though, we're throwing away a good bit of amplifier capability. This area here represents a region where the amplifier has considerably more open-loop gain potential than we're actually using. We've had to squeeze down the open-loop gain over a very wide range of frequencies in order to make the amplifier stable.

Well, what would happen if we used a somewhat different kind of roll-off, a somewhat more heroic roll-off if you will? Let's keep a single pole roll-off in the vicinity of crossover. Let's assume this is a plot of the af product. So it includes f. Let's keep a single pole roll-off in the vicinity of crossover. But let's roll-off as 1 over s squared at lower frequencies. Maybe something like so.

Our approximation now becomes 1 over s squared, or some constant over s squared. And then a single pole roll-off. The actual open-loop transfer function of the amplifier, again, would be the lower of these two curves at all frequencies. So it might look something like that.

But the important part is that we've picked up a good bit of area in here. We've increased desensitivity over a very wide range of frequencies by using two-pole compensation as opposed to one-pole compensation.

In order to implement two-pole compensation, we want to make the ratio g sub m over 2 Y sub sub c of s have the general form K tau s plus 1 over s squared. The K over s squared would give us the 1 over s squared roll-off at low frequencies. We then put in a zero, which causes our loop-transmission to go as 1 over s in the vicinity of crossover. Hopefully getting enough phase margin for acceptably stable performance.

In this case, we really have to use a two port network in order to achieve our desired short circuit transfer admittance. We want to get a short circuit transfer admittance of the form some constant times s squared over tau s plus 1. We can't do that with a single element, as we did in the case of one-pole compensation, where we wanted a short circuit transfer admittance just proportional to s.

We can, however, get the desired form for the short circuit transfer admittance out of a T network as shown.

Here we have two capacitors, a single resistor. If we measure the ratio of I sub n to V sub n, we find out it has a desired form. Some constant times s squared over another constant times s plus 1. And by appropriately choosing C1, C2, and R, we can adjust the constant. So this kind of a T network will, in fact, give us the sort of two-pole roll-off that we'd like to have if we use that kind of a network for compensation.

The effect on the amplifier transfer function, again, assuming an uncompensated transfer function including loading that's sort of three-pole. Here's the 1 over s squared 1 over s approximation. And again, we get the lower of those two curves at all frequencies.

Here we might argue that between about this frequency and this one, the approximation, the 1 over s squared and then a zero kind of approximation, very accurately predicts the behavior of the amplifier. So the hope is that we have the crossover in our major loop would occur somewhere in this region. We could use the approximation to predict behavior.

If we look at the angle, we see a pattern that's very much like that which results with lag compensation. The compensated angle dips down close to minus 180 degrees possibly, if this range of frequencies is large. Comes back up in the vicinity of crossover. Crossover in the major-loop is assumed to occur in here somewhere. Comes back up, so that we get adequate phase margin at crossover. And then, eventually, trails off. This is the kind of a pattern that we very frequently get with lag compensation. So the 1 over s squared sort of minor-loop compensation, or the kind of minor-loop compensation that forces a 1 over s squared roll-off over a wide range of frequencies, is the parallel to lag compensation in the series compensated case.

Again, I'd like to look at the effects of using, or the results of using 1 over s squared compensation. Here we have another part in our system, another portion, where we have the amplifier that allows us to look at 1 over s squared compensation. We use a somewhat different configuration here. Actually, we use the amplifier connected as a unity gain inverter. And the reason for that is what we're able to very easily look at the error signal. We connect the amplifier as a unity gain inverter. And let me, if I can, look at a view graph that shows that topology.

Here's the configuration. We simply have an equal valued input and feedback resistor. We have the T network that does compensation. In this case, we chose elements in the T, so that unity game frequency in the amplifier is about 1 and 1/2 megahertz. The 2:1 attenuation provided by the feedback network forces overall crossover at about 1 megahertz, or something like that. The zero comes in at about 100 kilohertz. So this combination of values, as I say, gives us the 1 over s squared roll-off. Breaks back to 1 over s in the vicinity of crossover. And the advantage of the inverter topology as opposed to the non-inverting configuration we used earlier, is that the signal at this point is a direct measure of the error in the system. We can look at this point with an oscilloscope. And ideally, this voltage should be zero. The output should ideally be the negative of the input. Assuming no loading by the amplifier, this point should reflect error directly.

If the output we're exactly the negative of the input, this point would be at zero. To the extent that it's, not we get a direct measure of error. And so this configuration allows us to readily look at the error of the system. And that's the reason that we use it in preference to the non-inverting configuration.

So as I say, we have that amplifier here. We have a buffer amplifier that allows us to look at the error signal and present that signal on an oscilloscope when we wish to. And we're able to very quickly change from 1/s to 1 over s squared compensation, maintaining crossover frequency by simply lifting the resistor in the middle leg of the T.

We're first going to look at the small signal response of the amplifier. And let's look first at what happens with 1/s compensation.

Here we have removed the resistor in the T network. Simply, switched it out, open circuited the resistor. And we get the rise time that we show. Let me see if I can make things go just a little bit faster here. We get a rise time, in this case, of a couple of hundred nanoseconds, 300 nanoseconds, something like that. Reflecting roughly a megahertz crossover. Reasonably well damped response.

If we switch to 1 over s squared compensation, the zero is back. I think actually, if we go through the numbers for this network, the zero is back about a factor of 5 or 6 or 7 from crossover. We notice very little change as I go from 1 over s to 1 over s squared compensation. Very little change in the relative stability. The phase margin of the two systems is very nearly comparable. The zero is located well enough below crossover. The equivalent to locating the zero of a lag network far enough below crossover, so that the stability is comparable.

There is one effect that I'll point out and we'll talk about later. Notice that when we use 1/s compensation, the response goes up to final value, pretty much settles there. And the speed of the response is compatible with the 1 megahertz crossover frequency.

If I use 1 over s squared compensation, the basic response is very much the same. That relative stability is very much the same. But we notice a long tail. We can see that. Let me up the amplitude just a little bit and look at the final part of the response.

We notice a fairly prolonged tail. The time constant associated with that tail is somewhere on the order of a microsecond. We're at 500 nanoseconds per division. And from here to here, the tail may shrink to 1 minus 1 over e of final value. So we have a time constant of roughly a microsecond. We'll have more to say about that in a little while. But that's the only really noticeable difference that we see in the small signal input to output step responses of the system.

Now let's look at errors for certain kinds of inputs, or for one input in particular. What I'd like to do is drive the system with a triangle wave, drive our amplifier with a triangle wave. And let's put in a much larger one then we now have. Let's begin to go up.

In fact, this amplifier has an output dynamic range of about plus or minus 10 volts. And let's drive very close to that. Here we are at 5 volts per division. Let me run the input amplitude up to the point where we're getting an output amplitude of about 20 volts peak to peak, plus or minus 10 volts. So let's move that up in there. So that's the input signal and the output signal really, to the amplifier. To a good degree of approximation, the output signal-- which is the one we're actually looking at-- is simply the negative of the input signal. So there is our triangle. We're running now at about 5 kilohertz. We're at 50 microseconds per division. The period of this wave form is about 4 divisions, or 200 microseconds. We're running at about 5 kilohertz.

Let's first look at the error signal when we have 1/s compensation. We have the error signal on the second channel. And let me put that on simultaneously.

We need a much larger-- look at a much smaller signals here, a much higher scale factor. And what we see is an error signal. Let's get a zero line for that right there.

What we see is an error signal that's very, very nearly a square wave. Let's see, we're at 10 millivolts per division, and so it goes somewhere on the order of plus 15 millivolts and minus 15 millivolts. Is that reasonable?

Well, let's see. We said that when we use single pole compensation, the open-loop transfer function of the operational amplifier over a wide range of frequencies is proportional to 1/s. That's what we mean by having a single pole roll-off. The amplifier looks sort of like in an integrator from its input terminals, it's differential input terminals, to its output.

Or, alternatively, the output voltage of the amplifier is the integral of the voltage applied at its input, at the very input terminals in the amplifier, not the input of the overall feedback loop.

Fine, if we have a triangle wave at the output and that's the integral of the input-- the input, of course, is the derivative with some scale factor of that output signal. We have a triangle wave at the output, we have a square wave at the input. So this is perfectly reasonable. This is the sort of response we'd anticipate if we got a triangle wave or force a triangle wave at the output, and have an amplifier whose open-loop transfer function rolls off as 1/s.

Now let's see what happens when we switch to 1 over s squared compensation. So to do that, I will simply throw this switch. And notice that the error most of the time gets much smaller. There are a series of really, what about to impulses that we can't see very well. Let's see if we can brighten them up. There's a series of impulses here, here, here, here, here. And the rest of the time, the error is very, very nearly 0. To within our ability to measure it, this line is 0. Does that make sense?

Well, here we have changed the amplifier so that over a wide range of frequencies, its open-loop transfer function falls off as 1 over s squared. In other words, over a wide range of frequencies, the output is the second integral of the error signal. Or correspondingly, the error signal is the second derivative of the output signal. We have a series of ramps at the output, a triangle wave at the output. Why, we'd anticipate that the input signal would be the second derivative of the triangle wave, or a series of impulses. Of course, they're not perfect impulses. They don't go to infinity and all that, reflecting the fact that we don't have a 1 over s squared transfer function at all frequencies. But we get a series of small impulses.

And the important point is that most of the time, the error is much, much lower for 1 over s squared compensation than it is for 1/s compensation. That simply reflects the fact that the open-loop transfer function with 1 over s squared compensation is much, much larger at many, many frequencies. Recall that the comparison between the two was as shown here.

We have the orange curve showing the 1 over s squared compensation, the yellow curve showing the single pole compensation. And over all of these frequencies, we have considerably higher open-loop gain from the two-pole compensated amplifier. We'd anticipate improved desensitivity for any signal where the majority of the signal frequency components lay in this region, where most of the signal energy was concentrated in this region. The triangle wave that we've chosen happens to be one like that, where the 5 kilohertz fundamental and the appropriate harmonics lie in this region, or most of them lie in this region.

We get greatly improved desensitivity, a much smaller error signal in the system. In fact, we haven't particularly cheated by picking a particularly fortunate test signal.

If we used a sinusoid anywhere from this frequency to this frequency, and the numbers here are something like 10 hertz and 100 or 150 kilohertz. Any signal in that region or any single whose dominant spectral components lay in that region, would get a considerably smaller error signal with 1 over s squared compensation than with 1 over s compensation.

Well, why don't we always use 1 over s squared compensation? One of the problems is it's nowhere near as general purpose. The 1 over s squared compensation really has to be very carefully tailored to the particular application. We have to cross over in the region where the open-loop transfer function is dropping off as 1/s. If we cross over at lower frequencies, we run into a stability problem because of limited phase margin.

In the case of single pole compensation, why we're able to, if we lower crossover frequency, the system becomes slower. But stability isn't compromised. However, that's not the case with 1 over s squared. So we have to be a little bit more careful in our choice. We have to know the application somewhat better.

Another consideration is what happens if we somehow force another pole in the loop. Suppose we capacitively load the amplifier, which has the effect of forcing another pole.

If we do that, why we find out that with 1 over s squared compensation, stability, or instability, is a very real possibility. And because we can force crossover back to relatively lower frequencies where the amplifier even without the additional load has limited phase margin, we then add the additional pole from the load capacitor, and instability is a very real possibility. So there are those kinds of disadvantages.

But again, if it's carefully chosen, if we know the application, we know what loads to anticipate, we know the capacitors aren't included, it gives us very dramatically improved desensitivity as we've seen from the demonstration.

There's one final effect associated with either the two-pole compensation or the same sort of thing happens with lag compensation. Recall the sort of prolonged settling transient we saw when we applied small steps to the system. We can explain that as follows.

Here's a root locus diagram. Here we have assumed two low frequency poles. They're not actually coincident as I've shown them, but this models the two-pole roll-off that starts at low frequencies. We have a zero, which flattens out the curve to a 1/s roll-off in the vicinity of crossover. And then I have assumed some additional high frequency poles associated with the amplifier.

We've seen this kind of pattern before. We get a root locus picture that, depending on the relative spacing of the poles and zero, either circles around, comes back, one branch coming back to the zero, another one going out here. Meeting an incoming branch, and then finally breaking off like so. Or, again, for a different arrangement of the zero and poles, does something like this. We looked at this case in detail when we first introduced the idea of root locus.

In any case, we'd normally operate this amplifier with its dominant closed-loop poles out here somewhere. And under those conditions, the transient responses corresponding to the two amplifiers look pretty much the same because they're dominated by-- or the two possibilities look pretty much the same. They're dominated by this closed-loop pole pair.

However, down in this region, there's a pole zero doublet. There' a zero associated with our compensation, if you will. The zero appears in the forward path, the tau s plus 1 expression in the forward path.

This branch, for the types of a0 f0 that we normally use, has moved in quite close to the zero. So we have a low frequency zero. And at just a slightly higher frequency, a pole. They very nearly cancel. They're separated by a few percent. The basic transient response is that associated with the complex conjugate pole pair.

But what we do is look, if you will, at the complex conjugate pole pair through a filter that has the following form. Here's the closely spaced doublet, epsilon's a small number. As the situation was shown in the view graph, we have a zero at a somewhat lower frequency than the pole. So we can visualize the system as being one that gives us the step response corresponding to the two poles, the dominant pole pair. But we view that step response through a filter that has this transfer function.

Well, the step response of this transfer function is of the following form. This is basically the uncompensated, or the overcompensated scope probe.

If you adjust the transient response of a scope probe, what you're really doing is trying to get a pole and a zero doublet to cancel each other. So this is the case where we haven't perfectly compensated the scope probe. This is the high pass version of the thing where the zero is located at a slightly lower frequency, or a slightly longer time constant than the pole.

The step response of that sort of system is as follows. If we put in a unit step, we'd get-- in this case, with a zero at a lower frequency, we get an output that overshoots final value by an amount epsilon. Decays with a time constant corresponding to the pole location.

In our particular case, the pole is located at a considerably lower frequency than the dominant response associated with the dominant pole pair. And so what we see is the high frequency response, the overshoot if you will, as the dominant response. But there's a long tail that reflects the much longer time constant associated with the pole of the pole zero doublet. And that sort of thing happens, as I say, with lag compensation, as well as with two-pole compensation when we do feedback compensation.