James K. Roberge: 6.302 Lecture 11

JAMES ROBERGE: Hi. We've spent the last couple of sessions looking at compensating feedback systems. We started this discussion rather generally, without really considering the realities that the system's physical configuration oftentimes forces upon us. Last time we did an example where we considered some of the constraints that actually apply in a physical system.

The topology we were using for that demonstration, and in fact generally for our discussion to this point, has been what's known as series or cascade compensation. And for that, we have our general feedback system form, the kind we've been considering to this point, where we have a forward path, a, and a feedback path, f. Very frequently, the physical situation constrains the transfer functions of these two elements. We frequently consider the element shown a fixed element. f is very, very frequently constrained by the ideal closed loop gain for our configuration.

To implement the series compensation or cascade compensation that we've been considering up till now, we add one more-- or possibly more than one-- element somewhere in the loop. We add it in series or in cascade with the fixed elements. We could add it here, here, here, here. The effect of adding it in any of those places on the loop transmission is identical, of course, and so it has the same effect on stability, regardless of where we locate it. However, the actual closed-loop gain of the system varies as a function of where we locate the series compensating element.

Today I'd like to consider another way of compensating a feedback system, in particular feedback or minor loop compensation. The topology for feedback compensation is shown in this view graph. And this is a fundamentally different kind of topology from what we have with series compensation.

Here we have a two-loop system. We have, of course, the input and the output. And as always, we have a feedback element here. And this feedback element, labeled f1, is of course the element that links the output back to the summing point in the system. And so we choose f1 as a function of the desired or ideal closed loop gain of our feedback system. In particular, the ideal closed-loop gain of this system is simply 1/f1.

In addition to the overall loop which involves f1-- this is oftentimes called an outer or a major loop-- we have an inner or a minor loop here. And we show elements a2 and f2 associated with that minor loop. Very frequently in this sort of feedback topology, or system that's feedback-compensated, the elements a1 and a2 are fixed elements, and the element f1 is of course chosen as a function of the ideal closed loop gain. The way we then try to do compensation or try to effect compensation for this kind of a system very frequently boils down to the choice of the element used or the transfer function used for f2. So our hope is that we can select f2 in order to get acceptable performance from the overall system.

We can consider the transfer function Vout over Va. Va in the previous view graph was the voltage just out of the summing point in the major loop. And that transfer function is of course, as always, a1-- that's a cascaded element. And then we have a reduction of the minor loop. The minor loop has a forward path, a2, and a feedback path, f2. And so when we collapse the minor loop, we get a transfer function from the input to the output of the minor loop that's simply a2 over 1 minus the transmission of the minor loop, in particular 1 plus a2f2. Consequently, the transfer function from here to here, effectively the forward path of the major loop, is simply a1 times a2 over 1 plus a2f2.

Using that value for the forward path of our major loop system, we can write the overall output to input transfer function for the system, and in particular it's our forward path-- the expression here-- divided by 1 minus the loop transmission of the major loop. The negative of the loop transmission for the major loop is simply this expression times f1. Consequently, we get the input to output transfer function for the overall system as shown.

We could certainly adjust parameters in this system by choosing f2 to get a desired closed-loop transfer function, hopefully to accomplish stability objectives and so forth. And we could go through an exact mathematical development that would tell us the effect on overall performance of particular choices of the quantity f2. However, that's not usually what we do. The hope in minor loop compensation is that we're able to make the minor loop effectively honest, if you will, controlled by feedback at the major loop crossover frequency.

In other words, what we do is hope that the loop transmission of the minor loop, the a2f2 product, at the frequency, particularly, of crossover of the major loop-- we hope that that a2f2 product is large. If that's true, then the transfer function from here to here is about equal to 1/f2. If we didn't have everything else, if this were the only part of our feedback system, we'd of course argue that at any frequency where the loop transmission magnitude was large compared to 1, the input to output transfer function of that minor loop would simply be equal to 1/f2. And so our hope is that at the crossover of the major loop, that condition is satisfied, and the transfer function from here to here is about 1/f2.

If that's true, we're able to design using an approximation that says the negative of the loop transmission for the overall loop is simply equal to a1f1 divided by f2. The elements a1 and f1 are elements associated with the major loop. 1/f2 is the approximate input to output transmission of the minor loop. And again, our hope is, as I indicated, that the quantity a2f2, evaluated at omega c, where omega c is the crossover frequency of the major loop, is considerably larger than 1. That's the condition necessary to use this approximation for the transmission of the major loop as our design tool.

Why do we feel we might be able to accomplish this sort of a thing? Well, the thing that saves us generally, or the thing that allows us to make the minor loop have very high gain, or very high loop transmission magnitude, at the crossover frequency of the major loop really reflects the fact that there's relatively fewer elements in the minor loop then there are in the overall loop. We recognize of course that as we increase the complexity of the feedback system, increase the number of elements in the loop, because of the low-pass nature of those elements, it becomes progressively more difficult to stabilize the loop. We get more and more poles, really, associated with the energy storage. It becomes more difficult to stabilize the resulting loop.

Consequently, loops that contain fewer elements generally can be stabilized to have higher crossover frequencies than loops which include more elements. Here, the minor loop or the inner loop is a small fraction of the overall system, and so there's at least some hope that we're able to make this loop have a much higher crossover frequency than the overall loop. Consequently we would find out, if that were true, that the loop transmission magnitude of this loop would be large at the crossover frequency of the major loop.

Again, assuming that that condition's satisfied, we go ahead and we can express the transfer function Vout over Va. Again, this is effectively the overall forward path transmission of our major loop. This is an exact expression for it at frequencies where the magnitude of the minor loop transmission is much, much larger than one, and this is the case that we hope exists, particularly a crossover of the major loop so we can use this as our design tool. The effective forward path of our overall system is approximately a1/f2.

If we have the other extreme, if the magnitude of the minor loop transmission is small compared to 1, then the minor loop is effectively working on an open-loop basis. The forward path for our major loop system simply becomes a1a2. The transfer function, f2, doesn't really influence the results very much in that case.

This is a nice graphical interpretation of this. What we can do is plot the two limiting cases, plot the two approximations. In other words, we plot a1a2, and we also plot a1/f2. And we do that in Bode plot form. Here's an assumed form for the plot a1a2. I've assumed a two-pole kind of transmission for that. And then we also plot the quality a1/f2. That's the other limiting case. And I've shown that here. I've assumed in this plot that the quantity a1/f1 is single-pole in nature. And in fact, as I've drawn it, it reflects a pole at the origin. This would tell us that f1 is proportional to s, or at least one possibility is to have f1 proportional to s, and a1 be frequency-independent.

We'll see that that sort of transfer function is very frequently used for the quantity f2. I'm sorry, this should be f2. We find that the transfer function of the feedback element in the minor loop very frequently is a 0 at the origin. We'll see an example of that in the next lecture, when we do an example of this kind of compensation using an operational amplifier.

Another possibility that you may be familiar with is the use of a tachometer in a servo mechanism. And what one does with a tachometer is measure the output velocity of the system and feed that back in a way that forms a minor loop. Well, the tachometer, of course, introduces a differentiation between output position and its signal. So if we use a tachometer for minor loop feedback in a servo mechanism, when we model that sort of a system, we find that we form the minor loop where the value of the feedback transfer function in the minor loop is in fact proportional to s.

And in those sorts of cases, the quantity a1/f2 can very easily have this sort of form. Well, once we've plotted these two transfer functions, in particular the two limiting approximations-- a1/f2, which is the approximation that holds when the loop transmission of the minor loop is large compared to 1, and the a1a2 product, which is the approximation that holds when the transmission of the minor loop is small compared to 1-- we can very easily determine which of these approximations holds at any given frequency.

Consider a frequency, for example, between here and here, where the a1a2 product exceeds the a1/f2 product. In other words, this condition. If that's true and we simply divide both sides by a1/f2, we find that over this entire frequency range, the a2f2 product is large compared to 1, or at least greater than 1. If we consider the entire frequency range, it's certainly large, much greater than 1, possibly over this frequency range.

And if we look above, we find out that at frequencies where that condition is satisfied, in particular where a2f2 is much, much larger than 1, an approximation to the forward path of our system is simply a1/f2. So that says that in this frequency range, we should use this line as an approximation to the forward path transfer function of our overall system.

If we reverse the inequality-- in other words, if we're working at frequencies either below this frequency or above this one, where a1a2 is less than a1/f2-- the inequality simply reverses, and we find out that here and here, in those frequency ranges, we should use a1a2 as our approximation to the forward path transfer function of the system.

Well, a very simple way to do this is to simply use the lower of these two curves at all frequencies. That's a consequence of simply checking the inequalities. And so we conclude, at least where the much greater than and much less than conditions were satisfied-- for example, in this frequency range, in this frequency range, and in this frequency range-- we could simply select the lower of the two curves as being a very good approximation for the forward path transfer function of our overall system.

We have to worry a little bit about what happens where the magnitude the magnitudes are about equal-- in other words, in this frequency range and in this frequency range-- and if we did that carefully, we find out for this sort of situation where the slopes of the two curves differ by one unit on logarithmic coordinates at the intersections, we get basically the usual sort of single-pole matching between the two curves.

So the only place we have to take any particular care is in the immediate vicinity, where the magnitudes are equal. Other than that, we can just go ahead and approximate the forward path transfer function of our system, as simply the lowest of the two curves. We'll very rapidly relax this and make an even cruder approximation, and really not worry too much about exactly what goes on in this vicinity and this vicinity, but simply go blindly ahead and take the lower of the two curves at all frequencies. We find out that very frequently gives us completely satisfactory information for use as a design tool.

I mentioned that one common kind of a system that uses this sort of minor loop compensation is that of a servo mechanism where tachometric feedback is used. The tachometer provides a 0 at the origin. At least if we consider the input to the tachometer to be a mechanical position, then it provides a differentiation of 0 at the origin in its transfer function.

There's another case where minor loop feedback is frequently used. And that's in the design of most modern operational amplifiers. This can be done either internal to the amplifier-- there are a number of amplifiers available where there's an internal minor loop compensating element. The 741, which is familiar to most of you, is probably the best example or the most widely-used example of that kind of an amplifier. There are also several operational amplifiers available where provision is included for the user to complete compensating the amplifier, really by forming a minor loop. And an amplifier like that, we might be able to represent this way. Here we have the symbol for the operational amplifier, anyway.

And a very common way of designing these amplifiers is to basically use a two-stage design. There's some sort of an input differential stage that takes the difference between the two voltages applied to the input terminals of the operational amplifier. It generally provides an output current proportional to that difference. And then there's a second stage that provides very high voltage gain, that uses the output of the first stage and amplifies it by a large amount.

And by using appropriate design techniques in these two-stage amplifiers, or amplifiers that have really only two stages of voltage gain, it's possible to get the kinds of open-loop gains at low frequencies that one normally associates with operational amplifiers, somewhere on the order of 100,000 to 1 million, implying a voltage gain-- if it's equally distributed between the two stages-- of something like 1,000 per stage.

They way one then forms a minor loop is to connect the compensating network from the output of the second stage back to the input of the second stage. So a minor loop is formed. In general, we could form that minor loop using a two-port compensating network. And we'll see how that works.

But we form the minor loop around, simply, the second stage of the operational amplifier. And consequently, if we compare this topology with our earlier topology for a minor loop-compensated system, this might represent the first stage of the operational amplifier. This would represent the second stage, with the two-port network providing the function f2, be feedback around the second stage, which forms the minor loop. And then of course when we go to connect the operational amplifier in an overall feedback system, in some overall feedback connection, we add an external path, f1, to complete the major loop. Once again, here's the way we implement the previous topology for an operational amplifier.

I'd like to look at a very simplified representation of how such an operational amplifier works, so that we can begin to look at the performance of this kind of an amplifier. I've taken a few liberties here with the construction of actual amplifiers. There are some minor differences between the topology I'll present, which is very schematic in nature, and the actual topologies used. But they don't change any significant answers, and so we'll go ahead and look at this somewhat simplified representation, bearing in mind that it does give us the correct answer for the transfer function of the operational amplifier.

We might have a differential input stage. I've shown it constructed out of PNP transistors, and that's the way it's done in amplifiers like the 741. Or at least the topological function of the input stage is the same as if it were constructed simply of PNP transistors. And that differential pair of transistors is generally biased with a current source. I've assumed for purposes of calculating the transfer function of the amplifier that we apply an input to only one side of the differential pair, ground the other side. So here's our input signal, Vi.

And again, in a little distortion of reality but one that does lead us to the correct result, I assume that the first stage is loaded with a current source, in fact a fixed-magnitude current source. And what we'd generally do would be to select this current source to have a value 1/2 of the current used to bias the entire first stage, the rationale for that of course being that if these transistors are well-matched, when we had Vi equal to 0, the current applied to the emitter of the differential pair would divide equally between the two members of the pair. Consequently, the quiescent current with 0 volts applied to the differential pair-- the quiescent collector part of this transistor would be 1/2 of the bias current.

The second stage of the amplifier is modeled even more diagrammatically than the first stage. Here we at least show a couple of transistors. The second stage, we simply assume has some input admittance. While I've shown it as a resistor or a conductance, in general it's a frequency-dependent element. Let's call that Yi. And the voltage across the input of the second stage, the incremental voltage, is a little bit of a problem with grounds, but at least the incremental component of the voltage across the input to the second stage we'll call Va.

And the output voltage of the second stage, which we assume is identically the output voltage of the operational amplifier-- usually there's some kind of buffering, an emitter follower, or something like that between the second stage itself and the output to make the second stage less sensitive to loads applied to the amplifier.

Anyway, we assume that we can model the second stage as a dependent voltage source. And we'll call the gain from the input to the second stage to the output of the second stage simply a sub i. So we model the second stage as an input admittance, Y sub i, and a dependent generator whose magnitude is ai, times the voltage applied to the input of the second stage. We then form a minor loop by connecting a compensating network from the output to a two-port network. And the second port of the two-port network is connected back to the input of the second stage.

We'll find it convenient to represent the two-port in terms of its short-circuit transfer admittance. At least as I've shown things, where we apply the voltage from a voltage source to one side of the two-port, we don't have to worry about parameters of the two-port at this side of the network. They can't really affect operation if this model is valid, since the input parameters on the two-port can't affect the voltage of the source. However, we do have to worry about modeling the two-port at its output side. And we'll do that in an admittance form, where we show the two-port as a Y sub a output admittance at this port and a short-circuit transfer admittance, a dependent current source which is dependent on the voltage applied to the side of the two-port.

So we can model our two-port at this side, this admittance, this dependent current source. Of course, to evaluate the quantity Y sub c, we'd perform an experiment where we'd incrementally short this port of the two-port, and would then measure the relationship between that short-circuit current that we observed and the voltage, V sub n. We find that Y sub c was simply the ratio of the short-circuit current to the voltage applied to the part of the network.

Let me point out that my convention here is a little bit at variance with standard two-port convention. And I've chosen to do that simply to save some minus signs in the development. The usual convention defines currents in a two-port being into the network. The way I've defined my short-circuit transfer admittance is considering the current out of the network. And my reason for doing that is just that it saves some minus signs in the development, and hopefully some confusion. I've always found that I work a little bit better with fewer minus signs. So that's the reason for violating standard conventions in this development.

The effect of the first stage on incremental signals is simply to inject or remove a current from this node that's proportional to an applied input voltage. If the input transistors are perfectly matched, under quiescent conditions, the collector current of this transistor is identically equal to the current associated with this current source. If we then apply an incremental input voltage about the quiescent operating point of 0 volts for the input voltage, that voltage effectively divides equally between the two input transistors.

Consequently, we get an incremental change in the collector current of this transistor that's 1/2 of the input voltage-- the amount of voltage that's applied to the base-to-emitter junction of this transistor being 1/2 of the input voltage, because of the symmetry of the circuit-- times the transconductance of this transistor. We assume that the transconductance of the two input transistors are equal. So the effect of the first stage is to simply cause an incremental current at this node and the constant proportionality between that incremental current and the input voltage is simply gm divided by 2.

We're then able to model the overall system as follows. We can replace the differential pair by a dependent current source whose magnitude is gmVi divided by 2. So that dependent current source reflects the transmission from the input voltage to this current. We combine these two admittances to get an admittance that's Yi plus Ya. The voltage across that admittance is Va, and the output voltage as it was above is simply ai times that voltage, V sub a.

We complete the minor loop, or model the effect of the two-port network, as far as its feedback effect, simply as a dependent generator whose magnitude is Y sub c times, now, Vo, recognizing that Vo and Vm are identical. And so we represent the dependent generator associated with the two-port in this way. We've already taken care of the output admittance of the two-port, since we've lumped it into this admittance.

We can then use this representation to determine what becomes the open-loop transfer function of the operational amplifier. We of course have considered the operational amplifier from an input applied to the differential input stage to the output of the amplifier. So this is what we commonly consider the open-loop transfer function of the operational amplifier. We can most easily determine that quantity by first writing a node equation at this node. And that's the equation shown here.

We have a current coming out of the node that's gmVi over 2. If we can move that over to the left, we get a minus gmVi over 2 on the left side of the equation. There's a current going out of the node in question through the combined admittance, Yi plus Ya. The voltage across that admittance is Va, so we get a current through that combined admittance that's simply Va times Yi plus Ya. And we get a current into the node that's Yc times Vo. That represents the short circuit transfer admittance of our compensating two-port.

The second equation reflects the fact that Vout is simply minus ai times Va, the relationship for the dependent generator in the second stage. Rearranging terms, we get 0 equals aiVa plus Vo. And when we solve this set of equations for the input-to-output transfer function, we obtained gm over 2 times ai over Yi plus Ya over 1 plus aiYc over Yi plus Ya.

There's a very nice physical significance associated with the terms in that final equation. In particular, let's consider the numerator. Here we have gm over 2 times ai over Yi plus Ya. What's that? Well, consider what would happen if we disabled the feedback network. If we made Yc equal to 0, possibly by disconnecting the input port of the compensating network from Vout and simply shorting it.

That would of course disable this generator, since Vm would than be 0. However, we would keep the loading associated with the two-port network. We lump that admittance, Y sub a, with Y sub i, so all we've done really is make Yc, or the magnitude of this generator, equal to 0. And then calculate the gain that results in the absence of minor loop feedback. Disabling this generator effectively makes f in the minor loop 0.

Well, the gain would simply be gm over 2. That gets us to a current, divided by Yi plus Ya. That gets us the voltage at this point. We multiply that voltage by a sub i to get the overall output voltage. So the physical significance of the quantity gm over 2 times ai over Yi plus Ya is that it's the open-loop voltage gain of the operational amplifier with the minor loop disabled, but considering the loading that the output side of the two-port compensating network applies to the second stage. We see that because the quantity Y sub a appears in the expression.

The denominator has a very familiar-looking feedback form, 1 plus something. It'd be very reassuring if that something were minus the loop transmission. So let's consider the quantity a sub i times Yc over Yi plus Ya. Well, as we suspected, that's precisely the negative of the loop transmission. And in particular, it's the transmission around the minor loop. Let's consider the minor loop. If we forget about a driving input to the overall system here, around the minor loop we get a product, Y sub c divided by this admittance, to get us a voltage here. So we get a Yc over Yi plus Ya. We multiply that by a sub i, and if we keep track of things, there's a minus sign associated with that loop transmission. So the quantity Yc over Yi plus Ya, all multiplied by ai, is the negative of the transmission around the minor loop.

So we have the numerator of this expression being the gain the amplifier would have if the minor loop were disabled but its loading were considered. The denominator of this expression is 1 minus the loop transmission of the minor loop. And again, there's the familiar limiting cases. This goes right back to what we did earlier in our general discussion of feedback compensation. Let's consider the quantity Vout over Vi, the open-loop transfer function of the amplifier, for those two limiting cases.

Well, if the magnitude of the transmission around the minor loop, the aiYc over Yi plus Ya product, is much larger than 1-- in other words, if this term is big compared to the one in the denominator-- we can ignore the 1. The ai's cancel out. The Yi plus Ya's cancel out. And under those conditions, the open-loop transfer function of the amplifier is simply gm divided by 2 times Y sub c.

In the other limiting case, where the magnitude of the transmission around the minor loop is small compared to 1, we find out that the input-to-output transfer function for the amplifier, its open-loop transfer function, is simply that that would result with the minor loop disabled. That's the effect of having the transmission around the minor loop small. And in that case, we get gm over 2 times ai over Yi plus Ya.

Exactly as we did in our general discussion of minor loop compensation, a very convenient way to look at this and a very nice way to do design is to draw these two quantities. We draw g sub m over 2Y y sub c, which is the approximation that we would use when the transmission around the minor loop has a magnitude large compared to 1.

And we similarly draw the open-loop gain of the amplifier in the absence of minor loop feedback, but considering the loading associated with the two-port feedback network. That quantity is of course g sub m, ai over 2Yi plus Ya. We draw those two quantities on our usual Bode plot coordinates. And we look to see which curve is greater in magnitude.

If in fact, for example, this quantity-- the open-loop transfer function of the amplifier and the absence of minor loop feedback-- is greater than this quantity, the implication, which we can get by simply multiplying both sides of this equation by the reciprocal of this term, is that the magnitude of the transmission around the minor loop is large compared to 1. If that's true, our earlier approximation tells us that we use this as our estimate of the open-loop transfer function of the amplifier. It's exactly the sort of thing that we did over here.

In this case, this curve would be gm over 2Y sub c. So we can show what would happen for our operational amplifier. This curve would be g sub m over 2Y sub c. And this curve would be not this quantity, but rather g sub m over 2 times ai over Yi plus Ya. So we plot those two quantities.

Once again, as in our general discussion of minor loop feedback, we'd estimate the overall forward path transfer function of our system in this case, the open-loop transfer function of the operational amplifier, as simply the lower of those two curves at all frequencies.

We'll find out that this sort of estimate works very well for at least one available operational amplifier. In fact, most modern operational amplifiers, the ones that are currently being designed, use this sort of two-stage topology. Several of the available ones allow the user to select the compensating element. We'll find out that that gives us a very, very powerful handle on the performance of the operational amplifier. There are a number of popular operational amplifiers which do use minor loop compensation.

A good example of that is the 741. And it's a fine amplifier in certain applications, but it really doesn't have anywhere near the flexibility of those amplifiers that we're able to compensate ourselves. And in fact, I feel that one of the reasons for the popularity of the 741 is simply that an awful lot of people who are designing with operational amplifiers haven't made the effort to learn how to compensate those amplifiers themselves.

Hopefully this sort of development will help. In fact, because of the ease of applying this approximation and, as we'll see next time, because of the very good results that we get using the approximation, why we're able to very readily determine a quote optimum compensation for many, many feedback configurations using these sorts of operational amplifiers. So I'd like to look into that next time and do an actual example using a commercially available operational amplifier, determining the kinds of compensation we should use as a function of application.

Thank you.