I explained where he likely got that in post 4.

Hi,

Ok great. Looks like that site is all messed up

The solutions I had given for 'w' for the four and five stage versions show how much the expression changes as we go up in the RC stage count. I'm not even sure yet if there is a solution for 6 stages, didn't go that far yet.

 

I'm not even sure yet if there is a solution for 6 stages, didn't go that far yet.

If you like, you can find a solution for any number of stages - the only requirement is to have a phase shift of 180deg at one single frequency only.
But - the question is: Does it make sense to have more than 3 stages?

 

If you like, you can find a solution for any number of stages - the only requirement is to have a phase shift of 180deg at one single frequency only.
But - the question is: Does it make sense to have more than 3 stages?

4 identical stages only require a gain of 18.3878
Tapering the impedance level of each stage to be twice that of the preceeding one then needs a gain of 8.6178
It could make sense to reduce the gain requirement.

Edit; Tapering the impedance of the three stage version by a factor of two times for each subsequent stage reduces the gain requirement from 29 to 16.

 

If you like, you can find a solution for any number of stages - the only requirement is to have a phase shift of 180deg at one single frequency only.
But - the question is: Does it make sense to have more than 3 stages?

Yes that's an interesting question, and I have found also what the previous post was talking about. As the number stages increase, the required gain seems to decrease.
That's not a final analysis however, as there could be other issues that come up like noise or something.

On the slightly comical but still interesting side, since the required gain seems to decrease with each added stage, maybe if we add enough stages we will only need a gain of 1. Adding even more stages yet, maybe we don't even need that just connect one end of the string to the other with no op amp Free energy!


I mentioned the 6-stage question only because I had not calculated that yet and I do not like to say anything for certain about anything I had not calculated or tested, or both.

BTW, thanks for your interest in this topic.

 

4 identical stages only require a gain of 18.3878
Tapering the impedance level of each stage to be twice that of the preceeding one then needs a gain of 8.6178
It could make sense to reduce the gain requirement.

Edit; Tapering the impedance of the three stage version by a factor of two times for each subsequent stage reduces the gain requirement from 29 to 16.

Hi,

What do you mean by "tapering the impedance level"?

I don't think I've seen a gain less than 12 up to 10 stages, but I could be wrong I did the calculations rather quickly. I can test this later though. It's actually -12 but we all know it's always negative (or is it?).

Side note:
I think transmission line theory would tell us we might even see a phase shift of 360 degrees or more. I did not investigate this yet though.

[LATER]
I verified that for a 12 stage circuit the gain has to be -12.185 and it looks right.
This was when using 'w' around 7.75, but I did not check to see if there might be other solutions. The phase shift does come out to 180 degrees.

 

Hi,

What do you mean by "tapering the impedance level"?

For example, first stage R=10k, C=1 nF; next stage R=20k, C=.5 nF; next stage R=40k, C=.25nF; and so forth.

 

Side note:
I think transmission line theory would tell us we might even see a phase shift of 360 degrees or more. I did not investigate this yet though.

When we have a feedback loop containing an amplifier, the circuit will oscillate if - and only if - there is aone single frequency fo which can fulfill the oscillation criterion (loop gan magnitude larger tan one and loop phase shift 360 deg).
* In case of an inverting amplfier (phase shift topology), this condition requires that the rest of the loop causes a phase shift of 180 deg at this frequency fo.
* In case of a non-inverting amplifier (e.g. Wien oscillator) the passive feedback network must cause zero deg phase shift at f=fo

 

When we have a feedback loop containing an amplifier, the circuit will oscillate if - and only if - there is one single frequency of which can fulfill the oscillation criterion (loop gain magnitude larger than one and loop phase shift 360 deg).
* In case of an inverting amplifier (phase shift topology), this condition requires that the rest of the loop causes a phase shift of 180 deg at this frequency fo.
* In case of a non-inverting amplifier (e.g. Wien oscillator) the passive feedback network must cause zero deg phase shift at f=fo

Hi there,

I am not exactly sure why you are telling me this you did not really make your reasons that clear. I'd be happy to read more about what you have to say here.

As I think you know, the "loop gain larger than 1 and loop phase shift 360 degrees" is one condition necessary for oscillation, but this is often taken to also imply the "non sequitur" version where that means it will definitely oscillate which of course is not true.

The "360 degrees" criterion is also not quite complete. That's why I mentioned the brief theory on a transmission line. If we have enough stages we might look at the network as a degenerate case of a transmission line, which can have phase shifts of just about anything which of course includes 360, 720, 1080, etc., degrees, which would mean it could possibly oscillate and sustain those oscillations. I thought that was interesting because we usually do not consider phase shifts large than 360 degrees.
Since we often have that inverting amplifier which gives us that initial 180 degrees, that would mean that the network would have to have a phase shift of 180, 360+180, 720+180, etc., etc. With a non-inverting op amp though if we could get the entire phase shift of 360 degrees from some network, that would mean we just need some non-inverting gain.
Since I was solving for the angular frequency and the gain using an inverting op amp, I did not check this out yet but it sounds like it might be interesting even if just for the small academic value. Of course we usually do not do this because we are always interested in the lowest parts count, but it could be interesting just the same.

 

For example, first stage R=10k, C=1 nF; next stage R=20k, C=.5 nF; next stage R=40k, C=.25nF; and so forth.

Hi,

Oh I see what you mean now.
Yes, that provides a solution 'almost' like the buffered version because each successive stage loads the previous much less than in the usual version where all the R's are the same and all the C's are the same. That would result in a required gain less than what we calculate for the all-values-the-same versions. If I remember right, the four stage buffered version requires a gain of 4, so we would see something just over 4 for the tapered impedance transformed version as long as the successive stage impedances increased at a decent rate. A factor of 10 for each stage should provide some interesting results. This in theory would mean we could reach a frequency roughly 3 times that of the all-values-the-same versions using the same part number for the op amp. We'd also have to look at other effects too though such as noise entering the system.

Along these same lines, it is also a little interesting to consider values that are simply not the same, but not necessarily impedance transformed except by coincidence. I guess there are a lot of possibilities here that could be interesting and possibly practical.

 

Hi there,

I am not exactly sure why you are telling me this you did not really make your reasons that clear. I'd be happy to read more about what you have to say here.

OK - here are some words about the background of my response.
In post #25 you wrote "I think transmission line theory would tell us we might even see a phase shift of 360 degrees or more. I did not investigate this yet though."

This wording sounds a little unclear and misleading to me. A newcomer to the field of electronics might think that oscillation is also possible with a phase shift within the loop of e.g. 370 or 410 degrees ("...360 degrees or more".)
It was my intention only to make this point clear.

On the other hand, it is of course logical and need not necessarily be emphasized that in the example discussed here, a phase rotation of 360 degrees is identical to zero degrees or 720 degrees.

 

OK - here are some words about the background of my response.
In post #25 you wrote "I think transmission line theory would tell us we might even see a phase shift of 360 degrees or more. I did not investigate this yet though."

This wording sounds a little unclear and misleading to me. A newcomer to the field of electronics might think that oscillation is also possible with a phase shift within the loop of e.g. 370 or 410 degrees ("...360 degrees or more".)
It was my intention only to make this point clear.

On the other hand, it is of course logical and need not necessarily be emphasized that in the example discussed here, a phase rotation of 360 degrees is identical to zero degrees or 720 degrees.

Hi,

Ok, but I did offer a few examples too which where the 360, 720, etc., degrees.

One thing I should add though is that I am not sure if we could get a phase shift as high as 360 degrees with just the R's and C's as in the high pass filter network being used. Possibly only at infinite frequency. In that case, it would require an infinite gain which would be highly theoretical only.

 

Hi,

Ok, but I did offer a few examples too which where the 360, 720, etc., degrees.

One thing I should add though is that I am not sure if we could get a phase shift as high as 360 degrees with just the R's and C's as in the high pass filter network being used. Possibly only at infinite frequency. In that case, it would require an infinite gain which would be highly theoretical only.

The phaseshift oscillator we are here speaking of consists of an invereting amplifier and an R-C-lowpass or C-R highpass passive ladder network of at least 3rd order.
That means: The passive feedback path must be able to produce a 180deg phase shift at one single frequency only.
When we speak about 360deg (resp. 0 deg) we always are referring to the complete loop (including the amplifier) : LOOP GAIN (and not the passive filter network only)

 

Hi,
Ok, but I did offer a few examples too which where the 360, 720, etc., degrees.

No - this would not work.
We should ask what would happen for a loop gain which arrives at 0dB - for example - at 720 deg for a certain frequency.
The circuits loop gain would cross the 360 deg line for a lower frequency where the loop gain is much larger than 0 dB.

 

The phaseshift oscillator we are here speaking of consists of an invereting amplifier and an R-C-lowpass or C-R highpass passive ladder network of at least 3rd order.
That means: The passive feedback path must be able to produce a 180deg phase shift at one single frequency only.
When we speak about 360deg (resp. 0 deg) we always are referring to the complete loop (including the amplifier) : LOOP GAIN (and not the passive filter network only)

Hi,

It's nice to see you are interested in this topic.

Well actually when you say quote:
"When we speak about 360deg (resp. 0 deg) we always are referring to the complete loop"

what you are really saying here is:
"When I (and possibly others) speak about 360deg (resp. 0 deg) I am always referring to the complete loop"

I say that because I did not say that I was referring to the complete loop up to that point. I said I was referring to the network.

There is no doubt that in the simplest case we need a 360 degree total phase shift. It should not matter how we get there as long as we also have the right gain, and the circuit actually oscillates.

The problem that comes up with some networks is that they cannot produce a 360 degree phase shift without also reducing the gain to 0 (not 0db but actually no gain at all as in no output whatsoever for non zero input). That pushes them into a purely theoretical domain where as we see the phase approach 360 degrees we also have the gain approach infinity, which of course is just theoretical and not practical.
I will also note that this may not matter at all to someone building a physical oscillator because I do not think it would be possible using the kind of components we usually use for this.

We can look into this deeper but it's probably not worth the time since I can't see anyone trying to do this in real life. Real transmission lines are a bit different than purely RC networks.

Thanks for your input on this.

 

We can look into this deeper but it's probably not worth the time since I can't see anyone trying to do this in real life. Real transmission lines are a bit different than purely RC networks.

Thanks for your input on this.

Yes - I completely agree with you.

It's nice to see you are interested in this topic.

Yes that`s right.
To me, all oscillator topologies (with sinusoidal outputs) are very interesting systems because
* they must be as linear as possible (for a high-quality sinus), and at the same time
* they must contain a certain degree of non-linearity for a stable amplitude of the output signal

It is really a challenge to select a circuit (with a corresponding "good" dimensioning) to find the best trade-off between these two conflcting requierements.

 

Yes - I completely agree with you.



Yes that`s right.
To me, all oscillator topologies (with sinusoidal outputs) are very interesting systems because
* they must be as linear as possible (for a high-quality sinus), and at the same time
* they must contain a certain degree of non-linearity for a stable amplitude of the output signal

It is really a challenge to select a circuit (with a corresponding "good" dimensioning) to find the best trade-off between these two conflcting requierements.

Yeah that's an interesting point. Diodes and LDR's and light bulbs and the like enter the picture
I wonder if anybody tried to do that part with a microcontroller.