Back in around 1963, we had a whole bunch of controversies at Philbrick Researches. We were getting a bunch of new solid-state operational amplifiers, some with high voltage gain, and others with moderate gain. A lot of debates and arguments erupted. One guy even argued that an op amp with low input impedance was better than high impedance. I didn't quite agree with him on that.

Also, most of these op amps had a collector-loaded output with high output impedance, quite different from the low output impedances of the cathode-followers of our older vacuum-tube amplifiers. Even though these op amps didn't have terribly high voltage gain at full load (such as A_{V} = 50,000 at R_{L} = 5 kΩ), the gain would rise by a factor of 10, if you only had to drive a light load. So, a user could achieve better precision with light loads—a nice feature.

(A while ago, there was an engineer who liked to argue that an ordinary op amp with a high dc gain, such as 2 million, didn't have any better useful gain because you could only use this gain at frequencies below 1 Hz. He refused to hear my point that if you asked this amplifier to amplify a square wave at 100 Hz, the op amp with high gain would settle (in much less time than 0.1 ms) to a 10-times better accuracy than an op amp with a mediocre gain, such as 200,000. He was wrong, and he didn't want to hear about it.)

One Philbrick engineer ran a full analysis with equations. Using a big flock of them, he showed that an op amp with high output impedance didn't suffer appreciably in ordinary applications. They're the same equations as the ones I worked out again last week, below.

It's been well known for years, even before 1960, that the gain of an op amp in a typical case is:

−V_{OUT }= Ideal Relationship × Error Factor

where the Error Factor is:

\[1/(1 + 1/A_{V }× ß\]

For example: the ideal relationship for a simple inverting amplifier is:

−V_{OUT} = V_{IN }× R_{F}/R_{IN}

and the Error Factor is:

\{1/\[1 + (R_{F} + R_{IN})/A_{V }× R_{IN}\]\}

Further, that Philbrick engineer defined the Relative Error, R.E., as:

R.E. = V_{OUT} (1 − Error Factor)/ V_{OUT }× Error Factor

But, this all assumes that the voltage gain, A_{V}, is largely invariant of output load. If the amplifier's output resistance, R_{O}, isn't low, then these assumptions aren't valid. These engineers used a few pages to show this *(see Equation Listing)*.

I checked out the equations. It looked like fun! If I chose the right values, I could make this equation blow up! I went out in the lab and built up a unity-gain inverter. I picked a couple of *appropriately selected* resistors, and plugged in a couple of *appropriately selected* op amps. These were experimental op amps with A_{V} = 50,000 specified at a light load of 100 kΩ—but a much lower A_{V} when R_{L} was 5 kΩ. I ran some measurements, then went over to the design area and declared, "This equation blows up, and the Relative Error is—infinity!"

The engineers said, "No, that can't be." But I declared that I had measured these op amps as unity-gain inverters, and the Relative Error was infinity! That's much higher than 1000! In fact, using these new experimental amplifiers, the signal gain of the unity-gain inverter was zero!

The engineers couldn't believe this, and they spilled out of their offices to see what I was doing wrong. I showed them that V_{IN} was a certain amplitude, several millivolts peak-to-peak (p-p), and on the same scale, the output was 0.0 mV p-p. They were horrified (and very skeptical). But I showed them that our standard P65As didn't have this kind of problem. Their gain was pretty good. What was going wrong? They asked me to show them the schematic of what I was doing. Fine *(see the figure).*

So I showed them the math, that when the R_{F} is very low, the Error Factor can fall rapidly toward zero. Then, too, the Relative Error could be outrageously bad (large) just because the signal gain became about zero, and the output signal was thus—about zero. I pointed out that maybe the "Relative Error" wasn't a very good or useful parameter if it behaved so badly!

I showed them that the Error Factor becomes 0.0 when R_{F} = 1/g_{m}, where "g_{m}" is the transconductance of the op amp, or ΔI_{OUT} per ΔV_{IN}. Effectively, the voltage gain of the op amp is:

A_{V} = g_{m }× R_{L}, (or to be more precise: × R_{L} parallel to R_{O} parallel to R_{F})

In this case, the g_{m} of the little experimental op amp was barely 1.11 mho. So, the equations bear out the results that you get when you wire it up. When R_{F} = 0.9 Ω, the signal gain will be zero, no matter the ratio of R_{IN} to R_{F}. The Error Factor will be zero, so the output will be zero, and if the Relative Error is defined as (Error in V_{OUT})/(V_{OUT}), then the Relative Error will be—infinity!

Even though my example was aimed to provide ridiculous results, it indicated that an amplifier with low g_{m} is likely to give relatively poor gain accuracy in ordinary applications. The op amps with poor g_{m} and poor voltage gain, even at loads as heavy as 5 kΩ, were largely discredited. From then on, a minimum gain of 20,000 was tolerated only for the cheapest op amps, and all new general-purpose designs had to do 50,000 minimum, with a 5-kΩ load. So the g_{m} was at least as high as 10 mhos.

(A way to look at this example is that op amps with low R_{O} can provide better gain accuracy with heavy loads. They can be said to have better g_{m}. Most modern high-performance op amps have emitter-follower outputs and, thus, lower R_{O} and higher voltage gain with any reasonable load. Although many rail-to-rail op amps don't have follower outputs—they get their precision in other ways—not all available op amps have very good gain accuracy with heavy loads.)

In the future (in several months), I'll talk more about the transconductance (g_{m}) of op amps. It's not always the best way to analyze a circuit, but it can provide some very good insights!

All for now. / Comments invited!

RAP / Robert A. Pease / Engineer*[email protected]*—or:

Mail Stop D2597A

National Semiconductor

P.O. Box 58090

Santa Clara, CA 95052-8090

Note: I really wanted to print this column on the *theoretical* aspects of finite gain first, but it was more important to first print last month's *practical* column about finite gain *(April 30, p. 95)*. Also, concerning last month's column, Bob Milne suggested that an air-conditioner's normal thermostat wouldn't work very well if the *outside* temperature varied while you were only trying to hold a reasonably stable *indoor* temperature. Of course, he's right, but it wouldn't be too hard to swap in a suitable thermostat to control the indoor temperature. /RAP