We discussed this paper over a decade ago and it was online at the time but, today, I had to use the Wayback Machine
to find it. Making it a thread here so that it is not lost forever. I believe that there was also a .pdf of it. Most of the
theory also applies to power amps.
There are a few Figures that were not captured.
A few very old links where it was discussed:
This one is almost 20 years old:
https://www.diyaudio.com/community/...rning-cordell-otala-and-gilbert-papers.48873/
Fairly certain that it was referenced here also, not saying that this is a good thread but here it is:
https://www.diyaudio.com/community/threads/whats-your-reasoning-and-not-whats-your-belief.41770/
Are Op Amps Really Linear?
By
Barrie Gilbert,
ADI Fellow; Manager, NW Labs, Beaverton, OR
Everyone knows that op amps are the most linear building blocks in the analog repertoire. If you want nonlinear behavior, you had better look to multipliers or other arcania. But we recognize that every real amplifier has a
bit of nonlinearity, like it or lump it, so maybe we ought to take a look at just how good is that standard op amp, the voltage-mode amplifier identified as an 'OPA' in previous columns, when used at frequencies above a few Hertz.
It wasn't so many years ago, in the heyday of the uA741, that a guy called Otala discovered this early 1-MHz OPA performed a lot worse than might expected at audio frequencies, so he wrote a paper in the Journal of the Audio Engineering Society in which he coined a brand new term -- transient intermodulation distortion, or 'TIM' -- to describe in audio lingo what most users of op amps had already discovered, namely, the little matter of a
limited slew-rate. It was a phenomenon peculiar to IC op amps. If you had grown up with vacuum tubes, you had plenty of reason to raise your eyebrows, since, unlike bipolar transistors (with their extremely high gm and a correspondingly small range over which linearity prevails in the familiar differential-pair gain cell), tube amplifiers rarely, if ever, got themselves into this pathological state of affairs.
Predictably, Otala's 'discovery' only added welcome fuel to the fire of those who were quite convinced that these new-fangled transistor amplifiers were genetically incapable of pleasing the audiophile ear, a notion which, like all myths, persists unabated, in spite of the fact that the root cause of TIM is now completely understood, and can easily be avoided in transistor amplifiers expressly designed for the most demanding audio applications. Nevertheless, Mister Otala was on to something, and although there are many splendid op amps out there, not all deliver quite what the textbooks promise. Accordingly, for a few minutes, we will examine a major source of distortion in op amps. Writing this in Kailua-Kona, Hawaii, with the thunder of the surf just feet away from this lanai, as the palms and hibiscus breathe the balmy post-dawn air, I have ample time on my hands.
The dominant mantra intoned over the application of op amps goes something like this: Never mind what's inside that cute little triangle; the function of the overall linear block -- invariably just a simple amplifier or filter stage -- is determined (almost entirely) by the passive components that are added to provide feedback. The triangle thing is merely the power-house that, come rain or shine, makes everything work out right in the end, by a kind of op-amp magic. The parenthetical inclusion is a concession to those who know a bit better, and it's the focus of this piece.
Reduced to essentials, most voltage-mode op amps, OPAs, are based on a topology like that shown in Fig. 1. To develop the theory, our device is here connected as a simple amplifier with a closed-loop gain of G, determined by the ratio (R1+R2)/R2, which can alternatively be expressed in terms of the feedback fraction b = 1/G. Because the dominant source of nonlinearity is in the
input cell, the distortion will be lowest in the voltage-follower mode.
In the interests of clarity and analytical simplicity, we will assume here that the
output is a
perfect sinusoid, having an amplitude E and angular frequency w, that is, E
sinwt, and work
backwards from this output to deduce what VIN
would need to be to generate that output. This may seem an odd approach, and it's certainly not essential to do it this way. However, many analyses involving the exponential behavior of transistors lead to transcendental solutions when pursued in the forward direction, and that's true in this case. A reverse-direction analysis quickly generates the key insights, and points towards the required modifications to effect a solution, at the price of a slight but not serious loss of rigor.
A
bipolar implementation is shown, since many monolithic OPAs use this technology. A basic differential-pair Q1, Q2, also cast in pnp form, and having a near-perfect current-source IT in its 'tail', senses the difference between the applied input VIN and some fraction of the output, while being insensitive to common-mode levels. In the case of a voltage follower, of course, the distinction between the signal and the common-mode voltage is somewhat fuzzy; the
real value of the high common-mode rejection ratio (CMRR) afforded by OPAs is much more apparent when the amplifier is connected in a high-gain mode, and a small input signal is accompanied by an interfering common-mode signal.
Formally, this input cell is a
nonlinear transconductance, whose output currents IC1 and IC2 are applied to the npn current-mirror Q3, Q4, which generates the difference IC1 - IC2. This current is then
integrated by the 'HF compensation' capacitor, CC, in the main voltage-gain stage provided by the common-emitter stage Q5, which is forced to operate at a constant collector current of IT. The resulting output voltage is buffered by what is here shown as an ideal voltage-controlled voltage-source (VCVS) but which in most cases will be the familiar Class-AB complementary emitter-follower that provides the high current-gain to drive the load, RL.
In a real OPA, some of the open-loop distortion will arise in this VCVS stage, but in order to keep our sights focused on the main distortion mechanism, we can ignore that here. Notice that CC is connected to the
final output node, not to the collector of Q5, as is often the case. This minor modification means that back-and-forth flow of the HF displacement current in CC is not supported by Q5 but, rather, by the output stage. Consequently, there is no variation in IC5 (it's held steady at IT) and thus
the VBE of Q5 is likewise constant with output voltage.
All this groundwork may seem very tedious, but it is with a view to getting at the
root cause of distortion. Numerous such detailed considerations are crucial to the design of an op amp capable of ultra-linear HF performance, and each needs to be eliminated independently. We're only considering the first of many, here.
Now we're ready to start the sums. Unfortunately, there is no painless way to avoid mathematics, but what we
can do is use simple, even rudimentary, models for the transistors, in the spirit of Foundation Design. This approach to the quest for insight was mentioned in the
"Spicing Up The Op Amp", and it's time to be a little more specific about this notion. It works very well for the bipolar junction transistor (BJT), which will probably remain a major technology -- certainly in the high-performance arena -- for at least the next decade, and beyond, in spite of the impressive advances in analog CMOS design, only made more difficult by the almost total emphasis on digital applications in the on-going development of sub-micron technologies. (Some of the reasons for holding to this view were stated in
"Why Bipolar?".)
The
Level-0 model for the BJT is simply a voltage-controlled current-source (VCCS) having an exact exponential relationship between its collector current, IC, and its base-emitter voltage, VBE; this is the
heart of the BJT:
IC = IS
exp (VBE/VT) (1)
where IS is the saturation current (and, only slightly whimsically, can be regarded as the BJT's
soul, since it mediates so much of the device's personality) having a value of some 3.6E-18 amps for a VBE of 800 mV at IC = 100 mA and temperature of 27ıC. It's amazing to me, after having stared at this equation for the better part of my life, how very profound it is, traceable to fundamental aspects of carrier statistics in semiconductor materials. It is the well-spring of BJT magic. It takes but a moment to find that the transconductance dIC/dVBE of a single device is:
gm = IC/VT (2)
where VT is, of course, the thermal voltage kT/q, 26 mV at 30ıC. This remains as true for a modern complex SiGe heterojunction transistor as it was for the primitive junction-alloy devices that came along quite shortly after Bardeen, Brattain and Shockley went whooping up and down the halls of Bell Labs jubilantly shouting
Eureka! It's fair to say that (1) and (2) are the most remarkable of all equations in modern electronics.
The
Level-0 model also conveniently omits such pesky set-backs as the finite base current and Early voltage of a BJT, its ohmic resistances and parasitic capacitances, base transit time and other effects. Accordingly, we set BF = BR = VAF = VAR = 1E6, and most other parameters to zero. Crazy? Not really: This is just what first-order textbook analyses do, without drawing attention to the fact. It is nonetheless surprising just how much of the reality of an IC's behavior emerges from the application of this simple
translinear model to circuit analysis. For example, the minimum permissible supply voltage will usually be correctly modeled; the shot noise will be right on the money; most of the temperature behavior; and, for the present purposes, an important distortion mechanism in OPAs will be quite accurately predicted.