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
No self-respecting oscillator is significantly dependent on the GBW of its op amp.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
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
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
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.
Hi Bob, could you please elaborate the above statement (phase shift of which function? Why 270 deg?).
Bob, please can you give a reference or a justification of this statement, which seems to be a novel oscillation criterion.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
Yes, you are right I took a close look at the plot and when gain drop to 0 the phase shift is 0.2°.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.
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