What is the frequency of this phase shift oscillator?

[audioguru]

I tried a 741 opamp as an audio preamp in 1974. It was noisy and high audio frequencies were reduced and distorted. Its performance was so bad that I NEVER used a 741 again. An LM324 or LM358 is even worse.
Then Texas instruments developed the TL08x and TL07x opamps.

The TL07x opamps are low noise. Their minimum slew rate is 27 times the minimum slew rate of an expensive LM741A opamp. They have very low distortion. They have Jfet inputs so their inputs draw only a very small leakage current. AND they are inexpensive because they are used in Millions of audio products.

 

[Winterstone]

Megamox,

there is a severe error in your calculation as presented with your last reply:

You have used magnitudes for the gain function as well as the feedback factor - whereas both are complex functions.
Thus, you have completely neglected the phase characteristic which is one of the two oscillation conditions.

I think, there is no other way as described in my reply#11:

*Either you calculate with a 2nd order gain function and a first order lowpass in the feedback path (otherwise you cannot reach 180 deg phase shift at a finite frequency)
*Or you accept a small error due to the simplification as outlined in post#11. This approach gives a simple formula which was given also in post#11.

 

[Megamox]

I'll keep an eye out for the TL08x and TL07x series!

Strange, I was so sure the equations were correct but I'll go through them again. In regard to the phase shift components you mention, I think I assumed that the system would not oscillate without the necessary 360 deg phase shift around the closed loop and gave it no further thought than that. The equation seems to predict the simulated output correctly, so if it's wrong it's a bit worrying. For example I tested it again, attempting to make a 50Khz Sinewave Generator. The formula gives me R=100K and C=0.923nF. Simulating this circuit with those values gives me a 50.1Khz output, as attached, so again it looks spot on.... puzzling.

Megamox

 

[Winterstone]

Strange, I was so sure the equations were correct but I'll go through them again. In regard to the phase shift components you mention, I think I assumed that the system would not oscillate without the necessary 360 deg phase shift around the closed loop and gave it no further thought than that. The equation seems to predict the simulated output correctly, so if it's wrong it's a bit worrying. For example I tested it again, attempting to make a 50Khz Sinewave Generator. The formula gives me R=100K and C=0.923nF. Simulating this circuit with those values gives me a 50.1Khz output, as attached, so again it looks spot on.... puzzling.
Megamox

Did you check the correlation with "my formula"?

 

[Winterstone]

Hi megamox,

I have calculated the 3rd order equation with following results:

Opamp gain (two-pole model): A(s)=Ao/[1+s/w1)*(1+s/w2)]
Feedback (lowpass): H(s)=1/(1+sRC)

Using the oscillation criterion for the loop gain T(jwo)=A(jwo)*H(jwo)=1 gives a third-order expression, which can be simply solved setting Im(T(s=jwo)=0.
(Im =0 because the right side of the equation is a real number).
Important remark: Im=0 means to solve for R(T(jwo)=1

Result:

wo^2=Ao/[1/w1w2 + RC(1/w1 + 1/w2)]

This is the exact result for the angular oscillation frequency wo.
Following the two simplifications

w2>>w1 and RC>>1/w2 and setting Ao*w1=wT=GBW

we arrive exactly at the expression I have given (as an approximation) in my former posting #11: wo^2=GBW/RC=wT/RC

Note: The above simplifications are equivalent to the "verbal" assumptions/simplifications as mentioned in my former post#11

 

[Roff]

If I take Winterstone's equation and plug in some numbers, I get this:
GBW=10^7*2pi
RC=1e-2

Fo=√(2pi*10^9)/2pi
Fo=12.616kHz


If I take Megamox's equation,
Fo^2+Fo/(2piRC)-GBW/(2piRC)=0 and solve for Fo,
I get Fo=12.608kHz.

A sim on LTspice yields Fo=12.616kHz

Darned close. Not sure what the difference is, but IMHO it is insignificant.
I should point out that I did not have the 3rd pole necessary for oscillation, so I just had some serious peaking.

 

[Megamox]

Thanks Guys! I'm surprised I got as close as I did, taking as many shortcuts to avoid the math as I typically do! 0.08Khz difference... so excruciatingly close and yet so far . I really need to brush up on my complex analysis and transforms but as the working has been kindly laid out, it's made it a bit easier for me to get started

It's funny when I was looking into analysis of oscillator frequencies online, I don't think I ever saw GBW or Op amp factors in any of the formulas. Perhaps oscillators can be built whose frequencies are not a function of amplifier frequency response? Kind of like how amplifiers can be built that aren't a function of internal gain or transistor Beta.

Megamox

 

[audioguru]

There are a few sine-wave oscillator circuits that use RC networks:
Wien bridge, Phase-shift, Bubba, Quadrature and a few more.

 

[Roff]

Thanks Guys! I'm surprised I got as close as I did, taking as many shortcuts to avoid the math as I typically do! 0.08Khz difference... so excruciatingly close and yet so far . I really need to brush up on my complex analysis and transforms but as the working has been kindly laid out, it's made it a bit easier for me to get started

It's funny when I was looking into analysis of oscillator frequencies online, I don't think I ever saw GBW or Op amp factors in any of the formulas. Perhaps oscillators can be built whose frequencies are not a function of amplifier frequency response? Kind of like how amplifiers can be built that aren't a function of internal gain or transistor Beta.

Megamox

No self-respecting oscillator is significantly dependent on the GBW of its op amp.

 

[Winterstone]

It's funny when I was looking into analysis of oscillator frequencies online, I don't think I ever saw GBW or Op amp factors in any of the formulas. Perhaps oscillators can be built whose frequencies are not a function of amplifier frequency response? Kind of like how amplifiers can be built that aren't a function of internal gain or transistor Beta.

Megamox

Megamox, the answer is easy and simple: Nothing in the electronic world is IDEAL.
In the present case, we speak about "linear" oscillators (I suppose, you know why I have used quotation marks).
It is common practice in opamp-based filter and oscillator design to assume IDEAL amplifier properties (neglecting input and output impedances, frequency-dependent gain).
Note, this is in contrast to your example, which intentionally exploits the frequency-dependent gain of the opamp.

Thus, there is always a SYSTEMATIC error due to this simplification. However, if we use a suitable opamp that justifies these assumptions (in particular: GBW large enough) these systematic errors will be lower than all other errors and uncertainties like deviations of parts values from the calculated values (tolerances, aging, parasitics).
That is the reason, opamp parameters do not appear in filter/oscillator formulas.

By the way, in some cases (high pole frequencies) it might be necessary to include a simple opamp integrator model in the filter/oscillator design process. Some authors have created a set of corresponding formulas (method of "prewarping" or "predistortion" of transfer functions). In another method, the circuit is designed conventionally and in a second step one selected circuit branch is modified in order to correct the influence of non-ideal opamp parameters (using a simulation program in conjunction with a realistic opamp model).

 

[Winterstone]

Thanks Guys! I'm surprised I got as close as I did, taking as many shortcuts to avoid the math as I typically do! 0.08Khz difference... so excruciatingly close and yet so far . I really need to brush up on my complex analysis and transforms but as the working has been kindly laid out, it's made it a bit easier for me to get started
Megamox

Indeed, the difference is rather small. However, as you have indicated to "brush-up" on your complex analysis perhaps you are interested at which step in your calculation you have introduced an error.

Here is my explanation:
At first, without mentioning you have assumed that the phase of the product (gain*feedback)=loop gain will be real and unity (equivalent to my setting Im=0, R=1 in my calculation).
That`s what I also have done in my first approximation (post#11). This is a simplified but reasonable approach as long as the feedback lowpass as well as the opamp work far enough above their (first) pole frequency.
That means, you have calculated with magnitudes only (like X=1/wC) . This works as long as you have to deal with multiplications or divisions only.
However, your derivation causes one single expression that is a sum, which combines a real and an imaginary part (1+R/X=1-jwRC).
At this point, you were not consistent because you have neglected the „j“, since the correct magnitude is SQRT[1+(wRC)^2].
Thus, this is the only difference between your formula and the correct one.
The error introduced, however, is pretty small because
SQRT[1+(wRC)^2]/(1+wRC)=0.9987 >>>> error of 0.13%.

That is the reason, Roff has found that the frequency according to your formula is slightly lower than using the correct formula.

With regards
W.

 

[Megamox]

Thanks! Sincerely appreciate it

Megamox

 

[Bob Scott]

If I take Winterstone's equation and plug in some numbers, I get this:
GBW=10^7*2pi
RC=1e-2

Fo=√(2pi*10^9)/2pi
Fo=12.616kHz


If I take Megamox's equation,
Fo^2+Fo/(2piRC)-GBW/(2piRC)=0 and solve for Fo,
I get Fo=12.608kHz.

A sim on LTspice yields Fo=12.616kHz

Darned close. Not sure what the difference is, but IMHO it is insignificant.
I should point out that I did not have the 3rd pole necessary for oscillation, so I just had some serious peaking.

I think it might be that the phase shift has to be just a fraction of a degree greater than 270 because the gain at the oscillation frequency is finite.

 

[Winterstone]

I think it might be that the phase shift has to be just a fraction of a degree greater than 270 because the gain at the oscillation frequency is finite.

Hi Bob, could you please elaborate the above statement (phase shift of which function? Why 270 deg?).

 

[Bob Scott]

Hi Bob, could you please elaborate the above statement (phase shift of which function? Why 270 deg?).

Hi Winterstone.

You don't need a full 360 degrees phase shift for positive feedback to cause oscillation. You just need >270. If there is 271 degrees shift, cosine(271) = 0.0175, that is a small positive feedback vector. It will oscillate if amplification factor is over 1/ cos(271) = 57.2X, or 36dB gain.

57.2 * .0175 > 1

With your high boost filter AKA "oscillator circuit", you get 180 degrees shift from feeding the output of the RC filter through the negative input of the op-amp, and you get almost 90 degrees from the RC circuit itself. So far, the total phase shift is slightly less than 270 degrees. More comes from the op-amps natural open loop gain slope of 6Db/octave at the point that this falling gain slope intersects with the circuits rising frequency response plot.

BTW, if you wanted to stabilize the original schematic to use it as intended as a high boost filter, you can just add a resistor in series with the capacitor. Choose a value that levels off the rising slope of the circuit's response curve before it intersects with the op-amp's falling open loop response line.

Edit: Right now, your filter creates a rising 6dB/octave closed loop reponse vs frequency curve that intersects the falling 6dB/octave open loop response line of the op-amp at 4,883 Hz. That is the mean* between the 15.9 Hz corner frequency of the filter at unity gain and the GBP of the op-amp, "typical value" on the TI datasheet is 1.5 MHz at unity gain. That mean frequency is pretty close to your observed 5KHz.

* mean: SQRT(15.9 *1,500,000) = 4,883

 

[Winterstone]

Hi Bob, may I place some comments to your contribution? I kindly ask you to respond to my first comment below.

Hi Winterstone.
You don't need a full 360 degrees phase shift for positive feedback to cause oscillation. You just need >270. If there is 271 degrees shift, cosine(271) = 0.0175, that is a small positive feedback vector. It will oscillate if amplification factor is over 1/ cos(271) = 57.2X, or 36dB gain.
57.2 * .0175 > 1

Bob, please can you give a reference or a justification of this statement, which seems to be a novel oscillation criterion.

With your high boost filter AKA "oscillator circuit", you get 180 degrees shift from feeding the output of the RC filter through the negative input of the op-amp, and you get almost 90 degrees from the RC circuit itself. So far, the total phase shift is slightly less than 270 degrees. More comes from the op-amps natural open loop gain slope of 6Db/octave at the point that this falling gain slope intersects with the circuits rising frequency response plot.

Yes, and this "more comes" is in fact another 90 deg phase shift resulting in a total shift of 360 deg as outlined already in post#11

BTW, if you wanted to stabilize the original schematic to use it as intended as a high boost filter, you can just add a resistor in series with the capacitor. Choose a value that levels off the rising slope of the circuit's response curve before it intersects with the op-amp's falling open loop response line.

Yes, that is the well known classical method to stabilize this circuit if it is intended to be used as a differentiating unit (within a limited frequency range).

Edit: Right now, your filter creates a rising 6dB/octave closed loop reponse vs frequency curve that intersects the falling 6dB/octave open loop response line of the op-amp at 4,883 Hz. That is the mean* between the 15.9 Hz corner frequency of the filter at unity gain and the GBP of the op-amp, "typical value" on the TI datasheet is 1.5 MHz at unity gain. That mean frequency is pretty close to your observed 5KHz.

* mean: SQRT(15.9 *1,500,000) = 4,883

This approximative expression was derived already in post#11.

 

[Jony130]

Very interesting thread and I want to ask a questions.
What about osculation amplitude? I use a stimulation to find open loop Bode plot for this two circuit
**broken link removed**
And the bode plots

As you can see both circuit should oscillate. But in simulation the second circuit (R1 = 220K C1 = 100nF) show damp oscillations as if there was not enough gain in the loop.
Can you explain why is so?

 

[Winterstone]

Hi Joni,

to understand why only one of the circuits does oscillate you should have a very detailed look on the phase response in the vicinity of the (expected) oscillation frequency.
Is the phase really at zero deg. or somewhat above/below?

Supplement: I suppose, in the 2nd case the phase has not yet reached 0 deg at the gain crossing frequency (0 dB).
That means, if the phase reaches 0 deg, the gain has dropped already slightly below 0 dB.

 

[Jony130]

Supplement: I suppose, in the 2nd case the phase has not yet reached 0 deg at the gain crossing frequency (0 dB).
That means, if the phase reaches 0 deg, the gain has dropped already slightly below 0 dB.

Yes, you are right I took a close look at the plot and when gain drop to 0 the phase shift is 0.2°.

I also build this circuit in breadboard and I could not make it to oscillate. And I don't know why.
Circuit start osculation when I connect capacitor directly to the output terminal.

**broken link removed**

and the signal at output terminal with C1 = 1uF

 

[Winterstone]

Hi Joni,

At first, the circuit under discussion is NOT intended to use as an oscillator - because sometimes it oscillates and sometimes not.
Both of your loop gain simulations are a good example - and you see that the condition of oscillation depends on the used RC time constant, in conjunction with the open-loop characteristic of the opamp.
Thus, no surprise that it "fails" sometime. I suppose for lowering the RC time constant the circuit will (perhaps) oscillate.

As to your second simulation: This is a complete different case. You have loaded the opamp with a capacitor - thereby creating an additional output lowpass (in conjunction with the finite Rout of the opamp).
Thus, you create additional phase shift - and when the opamp now is used as a unity gain amplifier (as in your case), which is the most oscillation-sensible configuration, the whole circuit oscillates.
But the question if it oscillates or not strongly depends on the used opamp and its second pole.
But watch the frequency: It will NOT be determined by any external RC time constant but it is simply the transit frequency - under the assumption that the slewr rate is large enouigh to allow such oscillations.
In your case, it seems that the slew rate is a determining factor.