# frequency of oscillator

Discussion in 'Homework Help' started by jun1119, Jun 15, 2012.

1. ### jun1119 Thread Starter New Member

Jun 15, 2012
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Hi all

I would like to know how can this op amp oscillator be analysed to determine its output frequency?

From simulation, the frequency is 645Hz

Thanks
jun

2. ### Ron H AAC Fanatic!

Apr 14, 2005
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Post the .asc file so we can simulate it without having to recreate the schematic.

3. ### WBahn Moderator

Mar 31, 2012
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I'm not too sure how to analyze the circuit. My first impression was to simply treat the two branches as voltage dividers and assume that the output voltage was running at some unknown frequency. Then apply the ideal assumption that the opamp is keeping the votlage at the two input pins equal. However, when I solve for the frequency I get a purely imaginary frequency of j281kHz.

I'm very much interested to see the proper way to analyze it, too (oscillators are not my forte...yet).

4. ### Ron H AAC Fanatic!

Apr 14, 2005
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I let XC1=XC2+XL1 and I got a real frequency of 7.17kHz.
However, it is not a sinusoidal oscillator. It puts out a semblance of a square wave when simulated. Furthermore, results get weird when you change the GBW of the op amp.
I used the universal opamp2 in LTspice. AVol was constant at 1e8. Slew rate was constant at 1e8. I varied only GBW.

 GBW Freq 1e6 285Hz 1e7 625Hz 1e8 521Hz

Go figure.

5. ### WBahn Moderator

Mar 31, 2012
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I did that, too, (except using impedances) and got the same results I got setting the input voltages equal.

$
\frac{1}{j \omega C_1} \ = \ \frac{1}{j \omega C_2} \ + \ j \omega L_1
\
j \omega C_2 \left( \frac{1}{j \omega C_1} \right) \ = \ j \omega C_2 \left( \frac{1}{j \omega C_2} \ + \ j \omega L_1 \right)
\
\frac{C_2}{C_1} \ = \ 1 - \omega^2 L_1 C_2
\
\omega^2 L_1 C_2 \ = \ 1 - \frac{C_2}{C_1}
\
\omega \ = \frac{1}{sqrt{L_1 C_2}} \sqrt{1 - \frac{C_2}{C_1} \ }
\
\omega \ = \frac{1}{sqrt{10mH 100nF}} \sqrt{1 - \frac{100nF}{33nF} \ }
\
\omega \ = \frac{1}{sqrt{(1000p)s^2}} \sqrt{1 - 3.03\ }
\
\omega \ = \frac{1}{sqrt{(1000p)s^2}} \sqrt{1 - 3.03\ }
\
\omega = 31,623\frac{r}{s} \ j \sqrt{2.03}
\
\omega = j45,056\frac{r}{s}
\
f = j45,056\frac{r}{s} \left( \frac{1Hz}{2\pi r/s} \right)
\
f = j7.17kHz

$

I did make a major goof because when I converted from rad/s to Hz, I multiplied by 2pi instead of dividing. Because I did that directly on the calculator without writing anything down, I robbed myself of the ability to check units, which would have caught the mistake.

So the frequency is still, as far as I can see, imaginary. Plus, it is an order of magnitude off of what the simulations show.

If the output is not a sine-wave, then a fundamental assumption of the above analysis is wrong. My next thought would be to redo it with one or two harmonics added in and see if that moves the results closer.

That imaginary frequency bothers me. Just a thought, but I wonder if that is preventing it from running closer to a sine wave at the nominal frequency. Try swapping the C1 and C2 in the simulation and see what effect that has. The calculated frequency then becomes 4.12kHz.

6. ### Ron H AAC Fanatic!

Apr 14, 2005
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I reviewed my calcs and found I had slipped a sign, so my results agree with yours.
I tried swapping the caps. I added an initial condition of I(L1)=1nA, and got a sinusoidal oscillation of 7.09kHz that decayed slowly with an envelope time constant of about 0.5 sec.
I changed the circuit as below, to add a DC input reference (GND), and to make the net feedback slightly positive. This results in a stable oscillator,in simulation, with fairly low harmonic distortion, even with no level control circuitry, as is usually required. Not sure how this works. Maybe I'll try it with a "real" op amp model.

EDIT: I tried it with an LT1022A. I had to make some changes to get it to sustain oscillations, but with R3=1Meg and R4=2Meg, it looks good.

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• ###### weird oscillator.asc
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Last edited: Jun 17, 2012
7. ### t_n_k AAC Fanatic!

Mar 6, 2009
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Staying with the OP's original circuit, one could undertake the "traditional" stability analysis in which the positive feedback path is "broken" and the open loop transfer function [w.r.t the feedback signal] is obtained.

A bode plot of the the open loop TF would then provide some understanding of the closed loop stability - based on gain / phase margins. Of course this might not predict the large scale limit cycle oscillations which are also likely.

As the previous posts indicate the problem arises of characterizing the real op-amp behavior vs an ideal op-amp model.

8. ### Ron H AAC Fanatic!

Apr 14, 2005
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Don't let us hinder you from taking on this analysis.
Just don't do it for me, because I hate to say it, but I don't care.

Mar 31, 2012
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10. ### Ron H AAC Fanatic!

Apr 14, 2005
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I have seen that app note before. It's a very good paper. You won't find any LC oscillators there, but the general theory might be applicable (but not to the original circuit).
IIRC, Ron Mancini was an app engineer for Burr-Brown before they sold to TI. I remember him coming to see me where I worked. I think it was regarding wideband current-feedback amplifiers, which we were using as TV video amplifiers. Mancini is (or was) a bright guy. Not sure where he is now.

11. ### t_n_k AAC Fanatic!

Mar 6, 2009
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For the OP's original schematic [but using an ideal op amp instead] breaking the positive feedback branch would yield the open loop Laplace Transfer function ...

$G_o(s)=\frac{$$LC_2s^2+RC_2s+1$$}{$$RC_1s+1$$$$LC_2s^2+1$$}$

which after substitution of the actual values gives a gain margin of "infinity" and a phase margin of 51°.

So in theory the arrangement would not oscillate at any particular sinusoidal frequency. Whilst not conclusive, this makes me wary of the earlier analysis giving an oscillation frequency of j7.17kHz. I also have a conceptual difficulty with an imaginary solution for a notionally real condition for a physical system.

The oscillations observed in the simulation by the OP are presumably therefore limit cycle excursions with the op-amp switching between its ± output limits. If the op-amp were ideal I doubt one could justifiably conjecture the existence of oscillations even in that mode.

12. ### WBahn Moderator

Mar 31, 2012
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I think we are in agreement here. The analysis resulted in an imaginary sinusoidal oscillation frequency, which I conjectured probably meant that the system could not oscillate sinusoidally. As soon as that is said, then the meaning of the 7.17kHz becomes very murky. Someone that understands this stuff well could probably extract useful information from it, but I certainly can't. The simulated output showing that it is a quasi-square wave output supports that and let me to wonder if redoing the analysis under the assumption that you would have a few harmonically related components might yield a real value for the fundamental frequency. I don't know.

This is very much in keeping with the discussion in the TI App Note I linked previously. That was a good read. Of course, there's only so much that can be put into a five page paper, but I think that overall he chose well what he put in there.

13. ### t_n_k AAC Fanatic!

Mar 6, 2009
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Thanks WBahn,

I've realised my analysis using phase and gain margin based on the inbuilt functions in my Scilab application would be "looking" for the 0dB / 180° [negative feedback mode] transition. With +ve feedback one would look for 0° phase shift at 0dB. The attached Bode plot is for the OP's circuit open loop response showing a highly unstable [0dB / 0°] point at about 5kHz - visible in the attachment. This may give some insight as to why small scale oscillations might fail to occur in practice. A study of the well-known Wien bridge circuit is informative as the latter's open loop response transition is well defined & stable at the critical frequency. The second attachment shows a typical Wien bridge open loop Bode plot with circuit parameters adjusted such that oscillation just initiates. Chalk and cheese.

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• ###### Wein Bridge open loop Bode plot example.jpg
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Last edited: Jun 18, 2012
14. ### vk6zgo Active Member

Jul 21, 2012
677
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This sounds pretty close to the 5.033kHz at which the LC series network in the feedback to the non inverting input is resonant.
At resonance,the feedback would have 0 degrees phase shift,as you say,hence oscillation.
The other input is presumably negative feedback,to prevent oscillations increasing to clipping point.
It would be interesting to build a real oscillator like this to see what happens!

Last edited: Jul 21, 2012
15. ### ramancini8 Active Member

Jul 18, 2012
462
130
Hi guys. I am Ron Mancini, and I am on this site as ramancini8 which is my email name. I did write the referenced paper, and thanks for the generous comments. That paper does not deal with inductors because I seldom used them.

16. ### JoeJester AAC Fanatic!

Apr 26, 2005
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Your works are referenced alot Ron. It's good to see your visiting what I consider is the best electronics forum.

17. ### ramancini8 Active Member

Jul 18, 2012
462
130
I am Ron Mancini, and I joined this group as ramancini8 earlier this month. I did write the mentioned article, but I don't think it will help much here because I tended to stay away from textbook circuits. If I was going to do the analysis the open loop gain A*R2/(Xc2+R2)* Xc1/(Xc1 + R1) and A is complex. Thanks for the great comments about my work

18. ### ramancini8 Active Member

Jul 18, 2012
462
130
Sorry for the second post but I forgot about a second page; got to give us old timers some lattitude.