RC network with voltage gain

[Enthalpy]

Hello you all!

It is known, but not universally, and intriguing anyway: an RC network can amplify a voltage. No power gain of course, here a low impedance drives the network and a high impedance observes the output.

The example network on top right here does that:

It is best understood as a three-cell lowpass RC, as on top left, where the output was moved to other terminals, between what was the usual output and what was the input. The reasoning is valid because the implicit low-impedance generator stays at the same place.

If, at some frequency, the network has some amplitude at the usual output with a phase between 90° and 270°, that is, it shows a component in phase opposition, then the voltage between the input and the usual output exceeds the input voltage. The component in phase quadrature adds even more voltage.

On the diagram, I compute the frequency for phase opposition (which becomes in-phase at the new output) and compute the gain there, but other frequencies bring more gain.

The high-pass shows the same behaviour. More than three cells would bring more gain. Higher R and C impedances near the usual output, too.

This was enough unexpected to me that I built both circuits at the diagram's bottom, and both oscillated immediately and quietly. As a result of the small loop gain, the waveform was a rather clean sine, at a frequency nearer to zero phase than to maximum gain.

Pretty useless I guess, but I find it puzzling.

Marc Schaefer, aka Enthalpy

[John Cuthber]

That's interesting.
Did you try making the first two circuits + checking if they really work?

[Enthalpy]

I breadboarded all the circuits.

I don't quite remember where I saw the claim of a voltage gain, maybe National Semiconductor's Linear Applications, or some source that is rarely wrong, unless I wouldn't have believed it. So I just built the RC circuit and measured. Once I had seen a voltage gain with a generator and oscilloscope, I built the two oscillators, and both ran smoothly.

I've made no use of this up to now, but it debugged a misconception ("RC circuit has no gain") of which I was quite sure.

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And just for fun... As I worked on active filters in the worldwide electronics research lab of a multinational company, I tried a state variable biquad cell where one op amp in a feedback loop was to provide a phase lag, so if all op amps in a package had the same Gain-Bandwidth Product, the phase lags would compensate, and this cell would be insensitive to the op amps' limited bandwidth. Nice idea. Found in a very reputable book about active filters, written by my professor, a known guy for filters.

Unfortunately, that magic biquad cell oscillated. I tried two more biquad circuits from that same book, and all oscillated. Putting some thoughts and equations on it, I found they had to be unstable.

So the well-known professor had put b*llocks in his reputable book, having computed a transfer function but not checked the stability, and without having breadboarded anything. I already knew such things happen.

Then came my bosses of the research lab, who were all unable to use a soldering iron nor an oscilloscope, and asked me to demonstrate that a Sallen-Key doesn't oscillate. Man, I tried to explain that it would have been known for long, but they insisted. Then - and I had already tested the oscillators of the first message here - I answered "the Sallen-Key is stable because the buffer op amp has a gain <1 and the RC network too, so nothing can oscillate" and that way I got of their stupid query.