James K. Roberge: 6.302 Lecture 08

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JAMES K ROBERGE: Hi, today I'd like to continue the discussion that we started last time, looking at how we evaluate the stability of a feedback system based on frequency domain concepts. And then extend that development to one of the major issues of our subject. In particular, compensation. How do we modify a system in order to hopefully improve its performance?

As an illustration of how we might determine relative stability for a feedback system, let's consider a system where we've already performed the Nyquist test and found out that the system is, in fact, stable. We've chosen a two pole system for an illustration. And we know that any negative feedback system, which has two poles in its loop transmission with both poles in the left half of the s-plane will, in fact, be stable. In an absolute sense, we've looked at that situation before. So we know either by a Nyquist test, which we could do fairly easily for this system, or by any of the other experience that we've had that this system will, in fact, be stable in an absolute sense. And we'd like to look at the absolute stability of this system using the Nichols chart as our aid to evaluating absolute stability. I'm sorry, relative stability.

Well, in order to do that, we go ahead and focus on the corner of the Nichols chart that's of most interest to us. In particular, the part that lies near a magnitude of the loop transmission of 1, and an angle of the loop transmission of minus 180 degrees. So we focus on this corner of the Nichols chart.

And what we can do is plot our loop transmission. Or actually, the negative of the loop transmission, the af product, on the gain phase coordinates. These are the background coordinates for the Nichols chart.

And what I've shown here is precisely that construction for three different values of a0. We have a two pole system as you recall from the material on the blackboard, with a single pole on the negative real axis at s equals minus 1. Another pole at s equals minus 10. And so we were a two pole system. And what I've done is plot that transfer function for three different values of a0.

Here's the curve that results for an a0 of 8.5. Here's the curve that results for an a0 of 22, the middle curve. And here's the curve that results for an a0 of 50. These curves, of course, are effectively the left-hand part of the Nyquist construction.

If we had the entire gain phase plane running out to 0 degrees here and did our Nyquist construction, we'd have a curve that did something like so. Reached minus 180 degrees asymptotically, the contour would close this way and so forth. So what we have is simply the left-hand portion, or a piece of the left-hand portion of a Nyquist construction when we do a gain phase plot of the af product.

Omega is a parameter along these curves. Here we have, for example, on the a0 equals 8.5 curve, omega equals 2, omega equals 5, omega equals 10. Here on the a0 equals 22 curve, omega equals 2, omega equals 5, and so forth. So omega's a parameter along this curve in the gain phase plane.

And what we're able to do then, is determine the closed loop gain of our system by simply going ahead and looking at the magnitude of g over 1 plus g in these coordinates. And if we cared to the angle for various values of omega and a specific value of a0. And we can see how that works.

For example, if we look at the af product that corresponds to an a0 of 8.5, we find out that at sufficiently low frequencies, back near omega equals 0, this curve eventually levels out at 0 degrees and a magnitude of about-- a closed loop magnitude of about 0.9. We can get that either by analytic manipulations, or look at the complete chart. We'd find out that the low frequency magnitude for this system with a0 f0 equal to-- or a0 equal to 8.5. The low frequency closed loop magnitude would be about 0.9.

As frequency increases, here is omega equals 2, omega equals 5. We begin to climb up in magnitude. Here we touch a contour that's about a magnitude of 1. g over 1 plus g equals 1. So here is that contour. And we just about touch that somewhere on the order of omega equals 5. Somewhere in here. The curve gets a little bit busy, but somewhere in here at omega equals 5 we hit a closed loop magnitude of about 1. And then begin to drop off to lower closed loop magnitudes.

Here we're back down to 0.85, for example. And eventually, we get to a magnitude of about 0.7 at omega equals 10. And if we keep track of things beyond that, the closed loop magnitude drops off roughly as 1 over frequency squared. Since in this region for higher values of omega, the system's basically behaving in an open loop fashion. The magnitude of the loop transmission is small compared to 1, and the closed loop gain then is just about equal to the forward path gain.

So we can plot the closed loop response of the system for that case. And we've done that here. We start out, this would be about 0.9. The magnitude peaks up just a little bit, reaches a magnitude of about 1 as we saw from the Nichols construction. Drops off to 0.707, or 0.7. And we said that that occurs at a frequency of about 10. So on our omega axis, we'd find out for this particular system, we had a magnitude of 0.7 at a frequency of 10. So we're able to determine a good bit from this kind of construction. We are able to determine, for example, the so-called 1/2 power frequency of the closed loop system. We're also able to get a very good indication of relative stability.

Here we have very little peaking for a0 f0 equals 8.5, and we conclude that's a fairly stable system. That's a very, very well-behaved system, in fact.

Let's look at what happens for an a0 of 22. This construction on the Nichols chart. Here again, if we went through the numbers, we'd find out that at very low frequencies, the closed loop magnitude was actually about 0.95. It'd be 22/23, which is about 0.95.

As we go to higher frequencies, we begin to get some peaking. Here, at omega equals 5, we're on a contour magnitude of g over 1 plus g of 1.05. So we're beginning to peak up a little bit.

At omega equals 10 here on our construction, we have a magnitude of 1.3.

And for a0 equals 22, we reach a maximum magnitude, our closest proximity in a Nichols chart sense to the minus 1 point. We reach a maximum magnitude of 1.4 at omega is a little bit above 10. Here's omega equals 10. Here's omega equals 20. Here's the highest point we reach on the Nichols chart, a magnitude of 1.4. And so that's the maximum closed loop magnitude we get.

We've dropped off to a magnitude of 0.7 by approximately omega equals 20. So we're able to draw the closed loop response for that system using the information that we've gained from the Nichols chart.

Here we start out at about 0.95. We peak up to a maximum value of about 1.4 for this particular value of a0 f0, or a0. This is the curve that corresponds to a0 equals 22. And the 0.707 frequency when we're down to about 1 over the square root of 2 of the low frequency gain, occurs in the system at about 20 radians per second. This point is about 20 radians per second.

As we go to progressively larger values for a0, the system again, becomes progressively less stable. The relative stability of the system drops. And that corresponds very nicely to what we've seen, for example, in a root locus development where we saw that as we increase a0 in a two pole system, the damping ratio of the complex conjugate, the dominant pole pair, the only complex conjugate pole pair for that two pole system, gets progressively smaller. And so we'd expect this behavior that the system becomes progressively less stable for larger values of a0 in the two pole system. And our Nichols construction confirms that.

Let's look for one final value of a0. In particular, an a0 of 50. Here's the Nichols chart construction that corresponds to that situation. Again, we would start out actually at a value of g over 1 plus g of about 0.98. For omega equals 0, we'd have a value of 50/51, about 0.98.

Here at omega equals 2, we'd be up to 0.99.

At omega equals 5, we'd be up to 1.02. We intersect that curve on the Nichols chart.

At omega equals 10, we're up to 1.2.

At omega equals 20, we hit our peak response, the p closed loop response. And the magnitude is just about 2. So we hit a maximum resonant peak with a magnitude of 2 at omega equals about 20. And then we're down to 0.7 by somewhere below omega equals 50.

Here's omega equals 50. Here's omega equals 20. Here's the 0.7 curve value for g over 1 plus g. So we hit that at omega equals 40 or 45, something like that.

And again, we can draw that construction or determine the corresponding closed loop gain for that value of a0. In particular, an a0 of 50. We'd find that the peak magnitude is about 2 as we've seen from the Nichols chart. And that the 0.707 frequency is about 40 or 45 radians per seconds, somewhere in here. So we can quite quickly, with the aid of the Nichols chart, determine the closed loop transfer function of the system. And of course, get a very graphic indication of the deteriorating stability as we go to larger and larger values for a0

Well, that sort of a construction is fairly easily done. Although, in reality, we normally don't even have to do that much work in order to get a good indication of relative stability via frequency domain techniques. And in particular, it happens that the closed loop response of the system is very nicely characterized by several parameters relating to the loop transmission or the af product. And I'd like to define those, and then see how we use them.

Here we have what might be a sort of typical af product Bode plot. We're running, of course, on a common frequency scale here. So omega goes in this direction. These are the usual Bode plot coordinates. A logarithmic axis for frequency, a logarithmic axis for magnitude. Here's a magnitude of the af product equal to 1. We're evaluating af of j omega. And we'd have logarithmic spacing here, 10, 100, 1,000, or whatever.

And similarly, on the same common frequency scale, we plot the angle associated with the af product. Possibly going from 0 degrees to minus 180 degrees, and then beyond. So here we have the usual Bode plot presentation for an af transfer function. And let's define a couple of quantities.

One of these is the crossover frequency, omega c. And that's simply the frequency where the magnitude of the loop transmission goes to 1.

I'd also then like to define the phase margin of the system. And to determine the phase margin, assuming we have a stable system-- incidentally for any of this, we really have to go back and check via a Nyquist test whether the system is stable or not. But we've very rapidly learn to recognize familiar patterns. And if we have sort of a well-behaved, maybe typical common kind of a system where the magnitude is sort of a monotonic decreasing function of frequency, the angle starts out possibly close to 0 degrees. Or in the case where our loop transmission includes an integration, which is a fairly common case in mechanical systems, the angle of the af product would start at minus 90 degrees. And then again, drops off and goes through minus 180 degrees only once. It doesn't wiggle back and forth around the minus 180 degree line.

If we have this sort of a well-behaved kind of a system, then in fact, the Nyquist test simplifies greatly. And we can convince ourselves that providing the angle has not yet reached minus 180 degrees when the magnitude has gotten down to 1, the system will be stable. And so here we've shown that situation. The magnitude of the af product is 1. The angle is somewhat more positive than minus 180 degrees. And for this sort of a simple well-behaved system, why that guarantees stability if we conducted the Nyquist test.

Once we've convinced ourselves the system is stable, we can look at the crossover frequency. In other words, look at the frequency where the magnitude of the af product is 1. And this distance is the phase margin. In other words, the phase margin represents the difference between the angle of the af product at the crossover frequency and minus 180 degrees.

In some sense, it's a measure of the amount of additional negative phase shift we could add to the loop transmission before the system became unstable. If we could somehow reach in and add negative phase shift to the system, if we added this much of negative phase shift to the system, why the angle of af would be minus 180 degrees at the frequency where its magnitude is 1. And that would give us a minus 1 or a pole on the imaginary axis, a closed loop pole on the imaginary axis. The system would be unstable.

The final quantity that we'd like to define is a quantity called the gain margin. And in particular, the gain margin is the amount of additional gain we could provide to the system assuming we kept the angle fixed before we incurred instability. So in this picture, it's the amount we could push up the af curve before the system became unstable.

And to do that, we could look at the point where the frequency where the angle went through minus 180 degrees. If we pushed the magnitude curve up by this much, we'd have a magnitude of 1 when the angle of the af product was minus 180 degrees. That again, would be the borderline case between stability and instability. And so this distance reflects the gain margin of the system.

We can look at those same qualities in a gain phase plot if we care to. We frequently do use gain phase plots for Nichols constructions and so forth. Incidentally, drawing a gain phase plot is a little bit more cumbersome than drawing a Bode plot. Mainly because we can make asymptotic approximations. And particularly, to the magnitude portion of a Bode plot, get a very, very good approximation as a series of straight line segments where we just sort of round in a little bit where two straight line segments intersect.

Drawing a gain phase plot is a little bit more difficult. Possibly the easiest way to do develop a gain phase plot is to simply draw the Bode plot first, and then transfer values of magnitude and angle that correspond to a given frequency. So here we have such a gain phase plot. Let's assume that increasing omega runs along this direction, which would be the case corresponding to the Bode plot the we've shown. As omega increases, the magnitude goes down. The angle goes more negative. The curve goes in this direction. The gain phase curve goes in this direction.

In these coordinates, we get the crossover frequency omega c as by looking at the parameter omega along the curve. So here is the frequency omega c as omega runs along the curve. The distance between minus 180 degrees and the actual angle of af at omega c is the phase margin of the system. So that's this distance. And the gain margin is effectively this distance. In other words, we could push up the magnitude curve or the gain phase curve by this amount before we incurred instability. And so this then, is the gain margin, or that ratio. The ratio of this gain into 1 is the gain margin of the system. So again, we can define the quantities phase margin, gain margin, and crossover frequency in either a Bode plot presentation or a gain phase presentation.

Now the value of these quantities is that they very, very accurately predict important features of the closed loop response of the system. If we assume we have a topology with a frequency independent feedback path, the one we've been looking at in the Nichols construction, we can then get a very good indication of important parameters of the system by looking simply at these quantities. In general, the speed of response of the system scales directly with crossover frequency. Assuming we can maintain equivalent values of gain margin and phase margin for two systems that have different crossover frequencies, any measure of the dynamics of those systems-- the half power frequency, the rise time in response to a step, any of those sorts of measures of speed of response will scale directly with crossover frequency. A higher crossover frequency, of course, corresponding to a wider bandwidth system, a faster rise time system.

Gain margin is kind of an override kind of a thing. Not much happens to deteriorate system performance until we get down to gain margins that become sort of too small. And what's too small?

Well, if you have a gain margin of 3 or more possibly, or maybe a little bit more than that, why it turns out that the gain margin has relatively little influence on the relative stability of the system. It does give you a measure of how much margin you have. How tolerant or intolerant you are to changes in various system parameters. How sensitive you are to changes in certain quantities that affect the loop transmission magnitude.

For example, if we had a gain margin of 3, then increasing the loop transmission magnitude by a factor of 3 would cause the system to become unstable.

But if we have a gain margin of 3 and are sure we'll maintain it, it has relatively little influence on the stability, the relative stability, of the system.

The big indicator of relative stability is the phase margin. And in fact, there is a very good relationship between phase margin and a quantity that we might define as m sub p. m sub p is a frequency domain concept. And in fact, the m sub p is the ratio of the maximum resonant peak that we get for a system in our standard form with unity feedback. The maximum peak we'd get divided by the low frequency magnitude of the closed loop gain.

In the case that we worked out on the Nichols chart a little bit earlier, for all three of our systems we had very nearly a magnitude of the closed loop gain of 1 at sufficiently low frequencies. And so the peak values here very nearly reflect m sub p. For the case where a0 f0 was 22, we'd have an mp of about 1.4. For a case where the a0 f0 product is 50, we have about 2 for the maximum magnitude of the resonant peak.

And we get a very, very good indication of relative stability by focusing simply on mp. We can design our systems when we go to do it really using that as our principal indicator of relative stability.

There's a very good empirical relationship between mp and the phase margin. And that's simply that m sub p is about equal to 1 over sine of the phase margin. That's not a mathematically exact relationship. But just a relationship that empirically exists for most systems of interest. And we can at least offer a plausibility argument for why that relationship might hold, or might be a good approximation as follows.

Here we've shown a much simplified Nyquist chart and eliminated a lot of the additional contours that confuse the issue. And we've focused once again on the magnitude equals 1, angle equals minus 180 degree region of the Nichols chart. And what I've assumed is that there's a straight line relationship in this vicinity at least, between the magnitude and angle of the af product. So here are several possible af products with different slopes in the gain phase plane.

And what I'll assume is that-- or we'll do the construction such that all of these systems, all of these af products have phase margins of 45 degrees. That says that they all go through the point magnitude of af equals 1, angle of af equals minus 135 degrees. This distance then is 45 degrees. We have a phase margin correspondingly of 45 degrees. So all of the curves are constructed to go through this point, magnitude equals 1, angle equals minus 135 degrees.

And we simply have different slopes. Approximating different kinds of af products all with the same phase margin. And we can now begin to look at the mp, the maximum value of the closed loop responses that we get for those various curves.

And if, for example, we look at this curve Here we're up to a closed loop magnitude of 1.2. Here we're up to a closed loop magnitude of 1.3. And we get to our mp, our maximum value for the closed loop magnitude, somewhere in here. Somewhere between the 1.4 and the 1.3 curve. Maybe 1.35.

We have a little greater slope like so for our assumed af product. Again, with a phase margin of 45 degrees. Why, let's see. Here we have a closed loop response magnitude of 1.2. Here we're up to 1.3. And that's the maximum mp for this particular curve. mp is 1.3. We have a little greater slope and chase things through. We find out that the mp that results is about 1.4.

However, if we have an even flatter curve, we can get to mps higher than 1.4. For example, if we had a curve like this, we'd get up to about 1.7 for an mp. But those result in progressively smaller values of gain margin.

For example, this distance. Here is the magnitude 1 point and here's our curve going through minus 180 degrees. That distance happens to correspond to a gain margin of 2.2 for the system that has an mp of 1.5. Larger values of mp require correspondingly smaller values of gain margin.

As I indicated before, if we have enough gain margin, where enough is possibly 3. Then in fact, this approximation is a very, very good one.

We can do this same sort of construction, the same kind of construction that I've just done, choosing different values for phase margin. We might try it for 30 degrees and for 60 degrees and so on. And when we do that and sort of empirically fit the resulting data points, we find that this approximation is a good one.

It happens that the ease of doing this sort of thing is so-- it's such a simple construction to do where we can quickly and rapidly draw Bode plot. We can go ahead and from the Bode plot determine, first of all, is the system stable? We can translate a Bode plot into Nyquist coordinates and determine rapidly if the system's stable. We're then able to go ahead and from the Bode plot, evaluate the crossover frequency very rapidly. We're able to determine the gain margin and see if it's quote, enough. And then we're able to find the phase margin of our system. And those constructions are very rapid. And give us remarkably good results when we ask about the performance of a feedback system.

Once we know the phase margin and the crossover frequency, we can make excellent predictions about the closed loop response of the system. And because it's so easy to do, I think almost all of the numerical work that one does with feedback systems really is based on these kinds of frequency domain concepts. We may use root locus as an aide to quickly seeing what's going on in a system, and what might happen if we change to 0 location, or what might happen if we change the location of a pole, and that sort of thing. We can use root locus concepts to explore those issues very quickly.

But when we want to get numerical results, when we'd like to go ahead and determine the correct value for certain system parameters, we almost invariably do that using Bode plot frequency domain kinds of techniques.

I'd now like to introduce the idea of compensation. Compensation simply indicates that we're interested in modifying a feedback system. Hopefully in order to improve its performance.

When we put together a feedback system, why as we might imagine, what usually happens is we find out that we can't get the desired values of desensitivity, or the desired values of speed of response without paying a stability penalty that we're unwilling to pay. And so the question is, what can we do in order to modify the system to improve its performance? that process of modification is known as compensation.

And what we do is look for certain figures of merit or indicators of system performance that guide us in this design process. Well, what might be good indicators of system performance? In particular, the relative stability of a feedback system. It's very hard to make sweeping general statements, in that so much depends on the particular application. For example, if you're designing an aircraft landing system, the thing you want to avoid is any undershoot. And so here we have a very, very conservative kind of requirement where we have to make sure that the aircraft doesn't undershoot. That's the bad condition in an aircraft landing system.

Furthermore, the degree to which we're willing to approach instability, if you will, the proximity we're willing to come to a system that's unstable, depends in part on how sure we are of the parameters. And the fact, will they remain fixed? Or is a system that has a minor stability problem today. Will it get worse as time goes on because of some unfortunate change in parameters? So we really have to have a good feeling for the actual details of the system when we go through this process.

But as sort of general guidelines we find, of course, that as we compromise stability, as we go to a relatively less stable system, the advantages we gain in exchange for that are generally a higher crossover frequency. If we have a sort of typical af plot like this where the magnitude drops down monotonically. The angle, again, is a function at least in the vicinity of the crossover frequency that's a monotonic decreasing function. Now we find out that as we might, for example, increase the af product, crossover moves out. The system gets faster. Because we've increased the af product, we get grader desensitivity. But unfortunately, the price we pay is one of reduced phase margin. Consequently, reduced stability. So there's almost always that trade-off where higher values of crossover frequency and greater amounts of desensitivity invariably lead to smaller phase margins. Consequently, less relative stability.

But if we look at some of the indicators of relative stability, we find out that for many systems of interest anyway, we achieve satisfactory compromises between speed of response and relative stability.

If we look either for overshoots in response to a step that are possibly 10% to 50% of the amplitude of the step. This is a time domain measure and we've earlier looked at step responses for systems. If we define the peak overshoot as the percentage overshoot from a final value for the step, typically systems are designed for peak overshoot. So possibly 10% to 50%. We might if we look at the corresponding frequency domain evaluations of stability, mp, the amount of resonant peak that we have in the closed loop frequency response, again people off times look for mps or design for mps somewhere between 1 and 2, or maybe a little bit greater than 1. 1 may be an overly conservative value of mp. We may have compromised speed of response too much for stability in that case.

And if we use as our indicator of relative stability the damping ratio associated with the dominant closed loop pole pair, again people are off times generally happy with systems that have damping ratio somewhere between 0.3 and 0.7 for the dominant pole pair.

What control do we have over these parameters? What kind of changes can we make to a system in order to modify these parameters, and at the same time change crossover frequency and so forth? Well, of course, again, we're very much dependent on the precise details of the system. And I'm sure that as we go along, we'll see many examples of this. We'll have a number of demonstrations later in the course and we'll see how-- or the constraints that affect us when we look at actual physical systems.

But let's consider one possible modification, a very simple one. In particular, changing the af or a0 f0 product. We've already looked at that in the earlier example today when we did the Nichols chart construction. We found out in a very direct way how changes, in that case, in a0 affected mp for the system.

What we assumed in doing that construction was that we were able to change a0, change the magnitude of the loop transmission, without making any change in the angle associated with the af product. And that is realistic in some cases, but not always.

For example, if we have a servomechanism, a mechanical feedback system, where the frequency is associated, the critical frequency is associated with many of the elements in the loop are quite low corresponding to mechanical frequencies. A few Hertz in many cases. We're generally able to change the loop transmission magnitude at will electronically. If the signal exists in electrical form at some point, we can put in an amplifier that has a bandwidth so far in excess of the mechanical bandwidths that we don't modify, we don't add any additional poles. We don't change the phase characteristics of the loop transmission. We're able to simply modify the magnitude characteristics. And we can either increase the magnitude or decrease the magnitude with relatively little trouble.

However, if we, for example, consider an operational amplifier. And in particular, a very fast operational amplifier. Maybe an amplifier that has 100 megahertz unity gain frequency, and such kinds of operational amplifiers exist. Then if we want to increase the loop transmission magnitude, basically we have to add another stage. And that may be very, very difficult to do without dramatically changing the nature of the phase curve. So in that case, we have relatively less freedom on the manipulations we're able to do with the a0 f0 product. Because changing the a0 f0 product affects something else.

Let's, however, look at one case involving an amplifier, an operational amplifier, where we are able to modify the a0 f0 product. In particular, we're able to lower it, at least, without modifying the angle associated with the af product.

And let's look at the configuration that I've shown. In particular, we have a non-inverting operational amplifier configuration. The input applied to the non-inverting input terminal of the operational amplifier. And then, feedback from the output through a resistive attenuator to the inverting input terminal of the amplifier. And we've looked at this case earlier. And we found out that providing we can ignore loading that the input of the amplifier applies to the feedback network, and loading that the feedback network applies to the output of the amplifier, we get a very nice physical correspondence between this topology and the corresponding block diagram. An input, we subtract from that input a fraction of the output voltage. In particular, a fraction that's dependent on the ratio R1 over R1 plus R2. Simply the attenuation ratio of this network.

The difference between those two voltages, the signal right here, is the differential signal applied to the input of the operational amplifier. That gets scaled by an amount a of s to generate the output voltage. So we get a very, very nice correspondence between the physical embodiment and the block diagram.

Now, suppose we build up one of these, and what we find is that the system is not acceptably stable. We suspect that the loop transmission magnitude is too large. Either the system's absolutely unstable or we have too large an mp. Or the damping ratio of the dominant pole pair is too small. Or somehow we've gotten an indication that the system is not acceptably stable. How can we lower the loop transmission magnitude?

We might believe that we have this kind of a picture associated with the loop transmission. Again, sort of very frequently occurring well-behaved one where the magnitude drops off with increasing frequency. The angle drops off. And in that situation, we increase phase margin as we lower crossover frequency.

If we can attenuate the af product without changing the angle characteristics, we get increasing values of phase margin as we push this curve down. The difficulty is that there's an interdependence between the ideal closed loop gain, which is simply the reciprocal of the feedback transfer function-- in particular, R1 plus R2 over R1. That's the ideal closed loop gain-- and the loop transmission. The negative of the loop transmission for this system is simply a of s times R1 over R1 plus R2.

And it seems that we have a problem because we have to choose the ratio R1 over R1 plus R2 on the basis of the ideal closed loop gain. But once we make that choice, we've pinned the value of the loop transmission.

Well, the way out of this problem, or this interdependence, is to add one more component. Suppose we put in a resistor between the inputs R. If we do that, we can remodel the system as follows.

We have our input applied to the non-inverting input of the amplifier. That corresponds directly. We have the resistor R between the non-inverting and the inverting input terminals. Let's define the voltage between those two terminals as V sub e.

And then I'd like to change my representation of the feedback network a little bit to make a Thevenin-equivalent for it. And in order to do that, we get a Thevenin-equivalent generator that's simply a scaled function of the output voltage. In particular, R1 over R1 plus R2 times V out. Let's define that voltage as Va.

In parallel, or with an output resistance, that's the parallel combination of the R1 R2 network. So all I've done is make a Thevenin-equivalent for this point, looking back into the network at this point. When we do that, we see V out scaled by the R1 over R1 plus R2 ratio and the Thevenin-equivalent output resistance of that generator is R1 in parallel with R2. So we've remodeled that way.

Once we do that, we can get our block diagram quite directly. What we find out is that this voltage Ve, which is the voltage we're sort of trying to generate at this point in the block diagram, we can consider the superposition of two parts. We get the contribution that exists when this generator is 0, but Vi is present. That's simply Vi times R over R plus R prime. So that's this path.

Plus, we get the contribution that's present when this generator is active, but this one is 0. That comes in with a minus sign. A positive voltage on this generator gives us a negative contribution to Ve. And the amount is simply Va times, again, the attenuation ratio R over R plus R prime. So this voltage in our block diagram would be Va. We would scale Va by an amount R over R plus R prime and subtract it from this voltage. The result would be Ve.

Similarly, Va is simply the output voltage scaled by the ratio R1 over R1 plus R2. That's how we define Va over here. So we get that corresponding block diagram.

What we're then able to do is do a little bit of block diagram manipulation. And if you remember, in this particular situation where we have the same factor, R over R plus R prime, on both inputs to a summing point, we're able to slide that quantity passed the summing point. When we do that, we get rid of these two blocks and we slide that attenuation ratio passed the summing point. So we gain a block in here. And the block that we gain is simply the attenuation ratio R over R plus R prime.

Now what we see is we have broken the interdependence between ideal closed loop gain and loop transmission magnitude. Because the ideal closed loop gain for this system, assuming large enough loop transmission magnitude, is simply the reciprocal of the feedback path. In particular, R1 plus R2 divided by R1. So we haven't changed the ideal closed loop gain.

However, the loop transmission now, for this system, picks up an additional term. Let's see. We have a of s. We have R1 over R1 plus R2. But we have an additional attenuation factor, R over R plus R prime. And so what we're able to do is modify the loop transmission magnitude, or at least lower it, without making corresponding changes in the ideal closed loop gain. And so if we had a system where we found that the system was inadequately stable, this sort of a system, for whatever ideal closed loop gain we wanted, we'd be able to improve the stability by adding this one resistor. This is actually fairly frequently done.

What we find out is that there are certain operational amplifiers sold, for example, that are not guaranteed to be unity gain stable. In other words, if we wrap feedback from the output of the amplifier back to its input, just put a wire from the output back to the inverting input, such amplifiers are not guaranteed to be stable.

If they are stable under those conditions, we speak of them as being unity gain stable. What we really mean is that they're stable in a loop that has an f equal to 1.

Well, suppose we'd like to take such an amplifier and run it at a closed loop non-inverting gain of 2, for example. But the manufacturer only guarantees stability if f is less than 1/5. In other words, if we attenuate the signals fed back by a factor of more than 5.

We can use this sort of approach in order to attenuate the signal fed back to the amplifier. Lower the loop transmission magnitude so that the amplifier configuration becomes stable. As I say, there are commercially available amplifiers that are intended to run that way. You gain certain advantages in the design of the amplifier if you design it other than for stability with direct feedback. And when that happens, if you want to run the amplifier at closed loop gains, this sort of an approach allows you to maintain stability.

The modification of the loop transmission magnitude assumed to be done without changing the angle associated with the af product, is only one of the ways we can modify a system in order to compensate it. In order to hopefully improve its performance.

Next time we'll look at several other options for improving the performance of feedback systems or compensating such systems. Thank you.