In a previous EDTN column about op-amps, I pointed out that, contrary to the prevailing dc-oriented view of these components, the ac gain dominates the behavior of a classical op-amp -- identified by the three-letter acronym OPA -- in the majority of signal-processing applications. Some minor typos crept into this still-experimental e-column, during the launch into cyberspace. One of these was that the ac gain was shown as f_S/f₁, where f_S is the signal frequency and f₁ is the unity-gain crossover frequency. Obviously, this should have read f₁/f_S. From this starting point, we noted that a "100-MHz op-amp" will provide an ac gain of merely five at 20 MHz, which is a far cry from the "infinity" that is often reckoned as the ideal for an OPA. (Say after me, a hundred times: "Op-amps have very low open-loop ac gain".) Is this little detail important in practice? You betcha.

Low open-loop gain does a lot more harm than merely reducing the closed-loop gain magnitude: it very strongly influences distortion, a topic to be picked up in a later column, when we'll look at the consequences of the nonlinear behavior of the OPA. This topic is poorly treated in many of the textbooks about op-amps, which cling excessively to the idea of a simple, linear 'one-pole' model, having an idealized transfer function A(s) = A_O/(1+sT_O). These further columns will show how modern designs have evolved to address the matter of distortion, either by an appeal to high-speed technologies, or the use of novel amplifier architectures.

It was also pointed out that even if one had an OPA that was ideal in another one of the textbook ways, that is, having infinite bandwidth, it would be impossible to use it. I suggested that you should do a few little simulations to prove that to yourself. Did you try these? This column will play a few experimental games, using simulation based on some highly idealized op-amps. Much has been said about the 'dangers of simulation.' "If you really want to know what's what, you must build a breadboard," so say some sages.

To my mind, nothing could be further from the truth, or retrogressive. Modern integrated circuits rarely can be bread-boarded, for a variety of reasons which we need not discuss here; most savvy readers will be well aware of their limitations. By contrast, a few experiments using simulation can open one's eyes to numerous interesting subtleties of design which would absolutely go unnoticed in an attempt to 'build the real thing.' Clearly, the accuracy of the device models and numerical integrity are fundamental and crucial factors.

The purpose of computing is to gain insight, not generate numbers.

Few today would disagree about the potential of computers as levers for the mind. Like all power tools, however, they can be abused. For several years, I have been promoting the notion of Foundation Design. A circuit for study is first described in its simplest form, by using highly idealized device models; realism is then added systematically, layer by layer, using progressively more complete models.

The purpose of Foundation Design is to eliminate enigma & maximize insight.

I won't be able to fully explain or justify this philosophy of design in today's column, but hope to pick up that theme at a future time. In the few experiments that will be described here, we'll find that an extremely simple -- even somewhat naýve -- model for the OPA generates results that are, if not counterintuitive, at least unexpected. You are encouraged to repeat all these experiments in your own cyberlab; I'd be interested to hear of other insights you gain through their use.

The results may vary a little, depending on whether you're using a basic SPICE derivative, running with loose tolerances, or an advanced circuit simulator. For these experiments, done quickly in odd moments during a busy visit to MIT, I used my faithful Tecra 740CDT laptop, running a UNIX O/S (Solaris), and employing ADICE (Analog Devices Integrated Circuit Emulator), a powerful ally that has been the beneficiary of a massive amount of feedback from the design community, over a period of some twenty years. ADICE features highly accurate BJT and MOS models, super-models for passive elements (including monolithic inductors) and an extensive repertoire of interactive analysis modes, supported by a powerful macro language, going well beyond Berkeley SPICE.

So using this approach, let's now see how well some of my comments about the dangers lurking in an over-simplistic view of the OPA hold up to scrutiny. We can begin with some packaged results, obtained as follows. An ideal VCVS (that is, an 'e-element' in SPICE) having a flat gain of A_O, is used to represent the 'ideal' OPA. With it, I made a unity-gain inverting amplifier, having a resistor connecting the source to the inverting node of the OPA and another resistor from that node to the output; the non-inverting node is grounded. That's all there is to the topology.

I used resistors of 10 kW, a typical value in many less-demanding medium-frequency op-amp uses. But here's the critical bit: I modeled the distributed parasitics of the feedback resistor by breaking it up into N sections and placing N capacitors at every node along it, each connected to the perfect 'ground plane', that tantalizing node SPICE convention calls ý. (The very notion of such a 'Mecca,' a place of perfect rest, is seductive but dangerous; for the moment, however, we will conveniently pretend that there is such a zero-potential node).

Incidentally, you can leave out the capacitor at the output of the VCVS: being an ideal E-generator, it won't make any difference to the response, in any modality. To be realistic, these capacitances should be quite small. If the total parasitic is 1 pF (it will be considerably lower in a practical PC board layout) we'll need to add ten little 100 fF caps when N = 10. The particular values of R and C don't affect the basic problem: they only determine the time-scaling. If you think the experiment makes more sense using a total capacitance of only, say, 0.1 pF, go ahead. Note that at this point, we're choosing to leave out the parasitics of the input resistor, though they'd be about the same in practice, and we can add them later.

We can start by using a transient analysis, to provide a more obvious indication of whether the amplifier is stable or otherwise. I used a voltage source providing a short pulse, running from zero to 1 V and back to zero, with a rise-time of 10 ps, lasting a few nanoseconds: I started with a 1-ns pulse. Yes, that's pretty zippy, but remember, we're exploring the consequences of using an OPA that is supposed not to have any 'frequency limitations.' Now, we really don't need to ask whether this feedback system will be stable should this super-idealized infinite-bandwidth OPA also have infinite gain: we ought to be instinctively aware that it would scream like a banshee. (If that's not obvious to you, consider a career in logic design.) The question can be couched more precisely: What is the minimum value of the open-loop gain A_O, let's call it A_CRIT, at which the complete circuit becomes unstable? That is, the excitation pulse causes the system to enter a region of behavior in which there is an exponentially-varying oscillation. More strictly specified, A_CRIT is that gain for which an oscillation is established by the pulse, but which neither increases nor decreases over time. Thus, A_CRIT represents a knife-edge balance between a decay in the oscillations caused by the power losses in the resistors and an increase in their amplitude caused by the signal power provided by the OPA.

So, what would you guess A_CRIT is going to be? You would be correct in believing that the number of sections, N, into which the feedback resistor is split would have a bearing on the question. You might envisage, as I did, a table in which we list N in the left hand column and A_CRIT on the right. The experimental results (which are not significantly less rigorous than by using a direct appeal to mathematics, by the way, since we're just letting SPICE intrinsically do the math for us, through the equations involving C and R) surprised me when I first did this, many years ago.

The chief thing to notice here is how very low the gain can be if this system is to be stable. Although we may argue about the realism of the modeling, it seems to be saying that if you really could find an OPA having infinite bandwidth, which the textbooks blithely assume to be the 'best case scenario,' it had better not have a gain of more than about fifteen in this mundane and commonplace application, which directly contradicts the other 'ideal' of 'infinite gain'! Stated in the most critical terms: infinite bandwidth and infinite gain are not ideals for an operational amplifier in any practical sense; they are, rather, extreme limit cases that serve solely to simplify the mathematics in textbook op-amp studies and thereby help to promote the underlying notions of an "operational amplifier" in the clearest possible terms. This simple experiment points out the inadvisability of actually introducing infinite bandwidth devices, even as a temporary model.

Now, while a transient analysis is fine for looking at the general lay of the land, it doesn't afford the surgical precision that is available to us in this simulation study. In the first place, it takes considerable empirical juggling to steer A_CRIT into the ballpark. Secondly, the amplitude of the envelope does some interesting things as the width of the test pulse is varied. This doesn't alter the criteria, but it leads to a final amplitude of oscillation which can be lower than that during the pulse period. Indeed, if the pulse width is adjusted to be equal to the discovered value of f_CRIT, the oscillation amplitude after the pulse will drop to zero! (The second edge, having the same amplitude but opposite sign, exactly cancels the response caused by the first edge.) Thirdly, we'd like to invoke some macros that allow us to put the simulator into an automatic mode, and search for these special values, using very high values of N, without manual intervention. Finally, we'd like to improve the numerical precision of the simulation, looking for subtle trends.

Accordingly, I replaced the pulse generator with a simple ac source, and swept frequency while adjusting the open-loop gain, A_O, of the VCVS still representing the 'infinite-bandwidth' OPA. This of course leads to a magnitude response that (provided A_O is in the right general range) will show a peak at some frequency. Here, my automatic algorithm was allowed to chase down A_CRIT, by searching for that value of A_O which generates the highest ac gain at resonance. It uses a kind of successive approximation, in which the frequency range also converges on f_CRIT, leading to very accurate results, in a few seconds, for each value of N.

There is a wonderful, almost fractal-like amount of detail near the resonance, and you need to be careful that your numerical techniques are accurate enough to fully explore this space. My voltage gains at resonance were between about 150 dB and 200 dB (10 billion), sometimes a little higher. The most definitive indication that you're hovering on the exact value of A_CRIT is that the phase transition at f_CRIT will suddenly flip in direction, within the span of a Hertz or two. Table II shows the values I obtained from this experiment.

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