Some doubts in Oscillations in normal amplifiers and ring oscillators

[iVenky]

There are some questions that I have in oscillations. I will address oscillations that take place in amplifiers as normal oscillations and the one using inverters as ring oscillators.

In normal oscillations we need 360 phase shift or in other words both input and feedback signal should in phase with each other so that the resulting amplitude at the input will be high and the output will eventually oscillate. Here we basically assumed that there is no propagation delay. If the propagation delay is huge then the output will be oscillating but it will be distorted ( in the sense there will be no signal between periods of the sine wave). If there is propagation delay we will get oscillations even when there is 180 phase shift except the fact that every period will be out of phase with the previous one. Most important of all we need some initial ac signal for the oscillations to take place. We usually assume that small ac noise is always there in the circuits.

Now take ring oscillators. This basically needs propagation delay for oscillations to take place. This is not based on feedback concept at all ( I mean there is no mixing between input and feedback signal like in normal oscillations). Most important of all- Ring oscillators do not need any initial ac noise signal for oscillations to occur.These ideas prompted me to ask few questions-

Ring oscillations will occur only if there is odd number of inverters. Now what will happen if there is some ac noise signal initially in the ring oscillator? If there is ac noise signal even if we have even number of inverters it will oscillate, isn't it?
Another doubt- If we assume that there is no propagation delay and some ac noise is there initially, then odd number of inverters will not oscillate because barkhauson criteria is not satisfied and even number of inverters will oscillate because both feedback and input are in phase and hence barkhauson criteria is satisfied, isn't it?

Please clear my doubts

Thanks in advance.

 

[FvM]

I don't understand, why you are discussing propagation delay in distinction to phase shift. In my view, it's part of the total phase shift.

Ring oscillators aren't basically different from other oscillator types. An exact description is more difficult due to the multi stage non-linear behaviour, but an analysis in linear range, which can be assumed for the initial startup phase under some conditions, is similar to other oscillators.

The odd number of inverters point is rather trivial in my opinion. The even inverter count circuit simply has no stable DC operation point. A circuit with even number of inverters will fall into saturation after an initial oscillation period of unknown duration. You can see it as a FF with distinct metastability.

In my opinion, you misinterprete the AC noise point. All oscillators with stable DC operation point have an (only theoretical) chance to stay in an equilibrium state without oscillations. Practically they won't. Usually, an initial excitation is present, e.g a power-on transient. But if are able to suppress it completely, electronic noise is sufficient to stoke up oscillation in a linear circuit.

Ring oscillators don't behave different regarding noise than others.

A non-linear circuit with hysteresis may however initially show no oscillations and start after a sufficient start pulse. A similar effect exists in simulation of linear oscillators due to missing noise and numerical non-linearities.

 

[iVenky]

I don't understand, why you are discussing propagation delay in distinction to phase shift. In my view, it's part of the total phase shift.

Ring oscillators aren't basically different from other oscillator types. An exact description is more difficult due to the multi stage non-linear behaviour, but an analysis in linear range, which can be assumed for the initial startup phase under some conditions, is similar to other oscillators.

The odd number of inverters point is rather trivial in my opinion. The even inverter count circuit simply has no stable DC operation point. A circuit with even number of inverters will fall into saturation after an initial oscillation period of unknown duration. You can see it as a FF with distinct metastability.

In my opinion, you misinterprete the AC noise point. All oscillators with stable DC operation point have an (only theoretical) chance to stay in an equilibrium state without oscillations. Practically they won't. Usually, an initial excitation is present, e.g a power-on transient. But if are able to suppress it completely, electronic noise is sufficient to stoke up oscillation in a linear circuit.

Ring oscillators don't behave different regarding noise than others.

A non-linear circuit with hysteresis may however initially show no oscillations and start after a sufficient start pulse. A similar effect exists in simulation of linear oscillators due to missing noise and numerical non-linearities.

If you mean that barkhauson criteria is true for ring oscillators too then there is only 180 degree phase shift given by odd number of inverters. Do you mean that the delay gives you the additional phase shift? If so, what will happen if we have many inverters and more delay is added due to propagating (say a phase shift of 400 degrees) to 180 degree phase shift which again result in phase shift ( 180 + 400 = 580 = 220 degree) which doesn't satisfy Barkhauson criteria. So once again odd number of inverters will not oscillate.

Also Clear me this doubt- Is it possible to have oscillations with even number of inverters in ring oscillator?

 

[LvW]

iVenky, you shouldn`t overestimate the role of noise for the start-up behavior of harmonic oscillator circuits.
In contrast to some papers (and even textbooks) noise is not responsible to start oscillations.
These contributions assume that the noise spectrum contains a frequency component (the wanted oscillation frequency) for which the loop gain
is larger than unity. However, this assumption is inconsistent since at t=0 (start-up) the circuit is not yet in steady-state condition.
However, a loop gain of unity (as well as definition of the loop gain) is available for the steady-state only.
Measurements (and also simulations of the transient behaviour) have shown that real the start-up time (until the final amplitude is available)
is much shorter than the time that would be theoretically necessary under the assumption that noise is the cause of oscillations.
In fact, it is the instability caused by positive feedback that produces a "kick-off" immediately after power switch-on.
In this context, all parts with a "memory" like capacitors and inductors (without initial conditions at t=0) play a major role.

---------- Post added at 09:28 ---------- Previous post was at 09:26 ----------

If you mean that barkhauson criteria is true for ring oscillators too then there is only 180 degree phase shift given by odd number of inverters.

iVenky, the Barkhausen condition applies to linear circuits only.

 

[iVenky]

iVenky, you shouldn`t overestimate the role of noise for the start-up behavior of harmonic oscillator circuits.
In contrast to some papers (and even textbooks) noise is not responsible to start oscillations.
These contributions assume that the noise spectrum contains a frequency component (the wanted oscillation frequency) for which the loop gain
is larger than unity. However, this assumption is inconsistent since at t=0 (start-up) the circuit is not yet in steady-state condition.
However, a loop gain of unity (as well as definition of the loop gain) is available for the steady-state only.
Measurements (and also simulations of the transient behaviour) have shown that real the start-up time (until the final amplitude is available)
is much shorter than the time that would be theoretically necessary under the assumption that noise is the cause of oscillations.
In fact, it is the instability caused by positive feedback that produces a "kick-off" immediately after power switch-on.
In this context, all parts with a "memory" like capacitors and inductors (without initial conditions at t=0) play a major role.

---------- Post added at 09:28 ---------- Previous post was at 09:26 ----------



iVenky, the Barkhausen condition applies to linear circuits only.

I just couldn't get you. Theoretically transient state dies out only at infinity. What this has to do with noise and positive feedback? Don't we have loop gain greater than unity at t=0?

 

[LvW]

I just couldn't get you. Theoretically transient state dies out only at infinity. What this has to do with noise and positive feedback? Don't we have loop gain greater than unity at t=0?

No, at t=0 capacitors are empty and the circuit does not represent a linear transfer function at all. It has not yet reached its steady-state.
You are right with your argument "infinity". But this applies in theory to each amplifier and each frequency-dependent circuit. As you know, in practice we consider the transient period to be concluded after a finite time period.

 

[FvM]

In my opinion, all has been said about the oscillation start-up, initial inbalance and noise topic. I don't feel a need to contribute to it further.

If you mean that barkhauson criteria is true for ring oscillators too then there is only 180 degree phase shift given by odd number of inverters. Do you mean that the delay gives you the additional phase shift? If so, what will happen if we have many inverters and more delay is added due to propagating (say a phase shift of 400 degrees) to 180 degree phase shift which again result in phase shift ( 180 + 400 = 580 = 220 degree) which doesn't satisfy Barkhauson criteria. So once again odd number of inverters will not oscillate.

Also Clear me this doubt- Is it possible to have oscillations with even number of inverters in ring oscillator?

A key point has been already mentioned by LvW, the Barkhausen criterion is only valid for linear oscillators. Strictly spoken, the operation of most oscillators is involving a certain degree of non-linearity required for amplitude stabilization. But ring oscillators are highly non-linear and the Barkhausen criterion can be at best applied as a rough estimation.

The total loop phase is comprised of the basic feedback polarity (usually 180°) and the phase of the circuit poles and zeros. Propagation delay is a term in non-linear circuit description, but you can roughly relate it to the group delay of a linear circuit.

Because the loop phase of a circuit with delay is frequency dependent, it's rather unlikely, that a ring oscillator with some gain does not fulfill the n*360 degree phase criterion for a certain frequency. The question is, if stable oscillations are possible.

As already mentioned, an exact description of non-linear oscillations is theoretically difficult. I think, it's a better idea to get a clear understanding of linear oscillators before proceeding to nonlinear ones.

I'm under the impression, that I gave a plausible explanation why a ring oscillator with even inverter count doesn't oscillate. I also guessed about the possibilty, that it can oscillate for a certain time. I fear, you'll have difficulties to prove that it doesn't oscillate at all. But because stationary states without oscillations exist as well, the oscillations are surely not stable.

 

[iVenky]

I fear, you'll have difficulties to prove that it doesn't oscillate at all.

I would be really happy if someone could prove it.

Thanks anyway.

 

[FvM]

I would be really happy if someone could prove it.

What's the problem behind your question?

As a first point, did you understand that an even number inverter chain can work as latch? If so, the problem reduces to the question if you can make it oscillate despite of it's latching behaviour. My hypothesis is, that an oscillator needs a convergent DC operation point besides all oscillation criteria. Presumed you have a (theoretically) stationary solution for the even number of inverters ring oscillator. What happens, if a infitisemal small DC inbalance is created? It will cause an exponentially growing duty cycle inbalance and immediately end up in a latching state.

The question "will it oscillate" is similar to "will a globe stay on top of needle tip"?