# Why don't oscillators amplitude saturate or converge to zero ?

#### MathematicianKing

Joined Nov 10, 2017
34
Hello my dear and lovely friends

One question I have had for a while now is about oscillators. I have not yet studied oscillators in depth, but I have designed a few simple phase shift ones, which I can understand but not quite.

As the title says, my question is the following...

As I understand, most oscillators work by amplifying a signal, then phase shifting the output so that the output phase matches the input phase, and then iterating this indefinitely. However, there seems to be something weird about this. If the amplifier amplifies the input too much, the signal will expand to much and saturate to something close to a square wave it seems to me. And if the amplifier amplifies too little, the signal will die out. So there is only one magical Av, so that the output is amplified just right so that the amplitude stays constant. Now the question is, mathematically it's easy to see that there is such a single amplifying factor to make things work, but in reality, it seems impossible to keep this constant amplitude, firstly because of noise, and then because transistors will fluctuate their parameters over time.

So how one earth do we manage to keep these amplitudes at a desired value ? In reality it's impossible to keep the Av factor constant all the time.

I know that after the amplifier gives its output, some loss is due inside the phase shifter, and so the amplifier has to amplify it just right so that the losses in the phase shifter are cared for, and the input to the amplifier is just right so that the output will be the same as before.

What happens here? How can the amplitude stay constant? It seems there is also a question about the frequency of oscillation. How can the frequency itself remain constant?

I have thought about this, and my guess as to why the frequency is constant is because the phase shifter is also a low pass filter, so that any high frequencies die out because their coefficients are too small, and only the lowest frequency will be high enough to be amplified just right ?

So it seems that the amplitude is related to the frequency, because if the low pass filter filters out all high frequencies, and the very lowest frequencies remain, then we can adjust the gain of the amp, so that the losses inside the filter will be higher than the coefficients of the higher frequencies that come out of the amp, and only the lowest frequency has a high enough coefficient so that it survives after passing through the phase shifter/filter. I am not sure of this.

How can the amplitude/gain of the amplifier stay constant and not either go to zero, or keep growing until it hits the +ve rail?

There's another way of looking at the phase shifter which explains how it works, but doesn't explain why the loss in the filter matches the gain of the amp: When the amp's output is going high, it is charging the capacitors inside the passive low pass filter, which increases the voltage at the amp's base, which in turn then decreases the output voltage, and this starts discharging the caps, which decreases the voltage at the base, and the cycle repeats. But still it doesn't make clear why the gain of the amplifier is just right so it doesn't either blow up or go to zero...

Could anyone explain to me how the amplitude can be just right ?

#### AlbertHall

Joined Jun 4, 2014
12,324
"Why don't oscillators amplitude saturate or converge to zero ?"

The short answer is that they do unless measures are taken to prevent it.

If the gain is set just a little higher than required to sustain oscillation then the amplitude will increase until clipping begins. The effect of the onset of clipping is to effectively reduce the loop gain. The amplitude will stabilise at the amplitude where the loop gain is unity. This means that the output will be distorted to some extent. Oscillators designed to be low distortion will include some circuitry to reduce the gain as the amplitude increases e.g. a wien bridge oscillator may include a filament lamp whose resistance increases with increased current in the feedback.
http://web.ewu.edu/groups/technology/Claudio/ee331/Lectures/L8.pdf

#### OBW0549

Joined Mar 2, 2015
3,566
How can the amplitude/gain of the amplifier stay constant and not either go to zero, or keep growing until it hits the +ve rail?
To keep the oscillations from decaying to zero, the amplifier is designed to have more than the minimum gain needed to sustain oscillation; that's the easy part.

To keep the oscillations from growing in magnitude until the peaks of the sine wave bang into the supply rails and cause distortion, one common approach is to feed the amplifier's output into a circuit which senses the oscillation amplitude and then use the output of that circuit to adjust the amplifier's gain through some sort of control mechanism such as a JFET acting as a voltage-controlled resistor. If the oscillations grow larger than the desired level, the amplifier's gain is reduced slightly; if the oscillations fall below the desired level, the gain is increased. The whole thing operates continuously, somewhat similar to the AGC (Automatic Gain Control) in a radio receiver's IF section.

The diagram below shows one example of this technique, taken from Fig. 3 in this TI app note, Sine Wave Generation Techniques:

#### ErnieM

Joined Apr 24, 2011
8,373
To make an oscillator you set up an amplifier and connect output back to input, with a gain greater than one so the signal grows and grows...

So say you have an amplifier with a gain of 2. Put in 2 volts, get out 4. Put in 5 volts, get out 10. Put in 10 volts, get out 12. Same with 15 volts in, it's limited at 12 volts.

So... somewhere between 5 and 10 volts the gain changes to ZERO, since a change in the input no longer makes a change in the output.

That is what gain limiting does for you, it will adjust the gain to exactly one to maintain the oscillations. It will always introduce some distortion, though this can be minimized.

#### neonstrobe

Joined May 15, 2009
190
Although FETs can be linear-ish, they are not linear. That means that the distortion won't be particularly low in the simple FET circuit. But to follow on with the answers already posted, the tricky part of designing an oscillator is that amplitude control problem. Quite often it involves more complex circuitry than the oscillator. Many experimenters used to use a negative temperature coefficient resistor (thermistor type R53) that was specifically designed as an automatic gain control element. It had a time constant of about 14s and would allow reasonable control of the amplitude but often accompanied by "bounce" - the amplitude increases and decreases at a low frequency until the control element eventually stabilizes. Filament lamps were used in Hewlett and Packard's original oscillator, and they can be used again, but also have bounce. The simplest option I recommend is to use a filament lamp, but only as a fine adjustment in the feedback loop with a second feedback network giving a gain of about 3.1-3.2, say, for a Wien bridge circuit, enough anyways to make sure it kixstarts. The lamp then has less effect on the overall feedback and the "bounce" is significantly damped.

#### MathematicianKing

Joined Nov 10, 2017
34
To keep the oscillations from decaying to zero, the amplifier is designed to have more than the minimum gain needed to sustain oscillation; that's the easy part.

To keep the oscillations from growing in magnitude until the peaks of the sine wave bang into the supply rails and cause distortion, one common approach is to feed the amplifier's output into a circuit which senses the oscillation amplitude and then use the output of that circuit to adjust the amplifier's gain through some sort of control mechanism such as a JFET acting as a voltage-controlled resistor. If the oscillations grow larger than the desired level, the amplifier's gain is reduced slightly; if the oscillations fall below the desired level, the gain is increased. The whole thing operates continuously, somewhat similar to the AGC (Automatic Gain Control) in a radio receiver's IF section.

The diagram below shows one example of this technique, taken from Fig. 3 in this TI app note, Sine Wave Generation Techniques:

View attachment 139915

That is great.

Is it possible to keep the oscillations steady without such mechanism ? Is there a "simpler" way ? I think I've seen many oscillators that do not use such feedback mechanisms and still keep it steady, and this is what I don't get.

What I want to know is if in general, is it really impossible to keep the amplitude steady and undistorted without using feedback mechanisms ?

#### MathematicianKing

Joined Nov 10, 2017
34
To make an oscillator you set up an amplifier and connect output back to input, with a gain greater than one so the signal grows and grows...

So say you have an amplifier with a gain of 2. Put in 2 volts, get out 4. Put in 5 volts, get out 10. Put in 10 volts, get out 12. Same with 15 volts in, it's limited at 12 volts.

So... somewhere between 5 and 10 volts the gain changes to ZERO, since a change in the input no longer makes a change in the output.

That is what gain limiting does for you, it will adjust the gain to exactly one to maintain the oscillations. It will always introduce some distortion, though this can be minimized.

I think I understand now! Thank you. That was nice and easy. So you just let it saturate, but adjust the circuit to minimize this clipping. Alright...

#### MathematicianKing

Joined Nov 10, 2017
34
Although FETs can be linear-ish, they are not linear. That means that the distortion won't be particularly low in the simple FET circuit. But to follow on with the answers already posted, the tricky part of designing an oscillator is that amplitude control problem. Quite often it involves more complex circuitry than the oscillator. Many experimenters used to use a negative temperature coefficient resistor (thermistor type R53) that was specifically designed as an automatic gain control element. It had a time constant of about 14s and would allow reasonable control of the amplitude but often accompanied by "bounce" - the amplitude increases and decreases at a low frequency until the control element eventually stabilizes. Filament lamps were used in Hewlett and Packard's original oscillator, and they can be used again, but also have bounce. The simplest option I recommend is to use a filament lamp, but only as a fine adjustment in the feedback loop with a second feedback network giving a gain of about 3.1-3.2, say, for a Wien bridge circuit, enough anyways to make sure it kixstarts. The lamp then has less effect on the overall feedback and the "bounce" is significantly damped.

Alright, so It's really a complex subject! I was suspecting it to be. I seen some simple oscillators where the output is steady but very distorted. Anyhow you can always filter the output from even a square wave, so doesn't seem to be a problem.

#### OBW0549

Joined Mar 2, 2015
3,566
What I want to know is if in general, is it really impossible to keep the amplitude steady and undistorted without using feedback mechanisms ?
Unfortunately, yes: if you want the absolute minimum distortion, a gain-control feedback mechanism is the only game in town. If you need amplitude control but can tolerate some modest amount of distortion, some sort of "soft clipping" mechanism may suffice.

Bear in mind that some level of distortion, however low, is inevitable because the amplifier itself can never be perfectly linear.

#### crutschow

Joined Mar 14, 2008
33,962
Anyhow you can always filter the output from even a square wave, so doesn't seem to be a problem.
It depends upon how much distortion you can tolerate.
It takes a lot of filtering to turn a square-wave into a low distortion sine-wave.