The most logical way is to plot the loop gain (magnitude and phase) and to check the stability criterion.
Because your loop has some nodes with very low output resistances (e.g. output A3 which is a follower) it is not a problem to open the loop without considering the load at the opening.
To keep the operating point (dc feedback) I suggest to connect the load at the A3 output through a huge inductance (100 H) and to inject a test signal of 1 volt (ac simulation) via a large capacitor (100 F) behind the inductance. Don`t forget to ground the "normal" signal input.
The ouput of A3 gives you with very good accuracy the loop gain response.

1. After running stability analysis, I found that the system is unstable.
Even though ALL the inner loops are stable, the outer feedback loop is oscillating. Is there a way to find out what(component) is making the system unstable?

2. When I increase the resistance of the resistor(by a factor of 100) connecting the output of A3 to the input of A4, the system is stable and does not oscillate. I am looking for a better solution.  Any suggestions?

shivamm wrote on Jun 7^th, 2012, 2:34pm:


At first, it is really no surprise that the loop is unstable - because of the various opamps and RC elements connected in a loop.
Nevertheless, if the magnitude ofthe loop gain is small enough it can be stable - as demonstrated by you (increased resistor value between A3 and A4). You ask for a "better solution",
Why do you think this is a bad solution?
It is quite normal in such a case either to reduce the loop gain (that`s what you did) or to increase the loop phase (for example by incorporating a PDT1/lead-lag controller).
But in any case - as I have suggested - check the loop gain in order to see what must be done to arive at the wanted phase margin.
And show us the simulation results!
Don`t forget to include an inductor at the output of A3. Otherwise your dc operating point is lost! I cannot understand why the former reply (wave) argues against it.

Lex wrote on Jun 8^th, 2012, 2:28am:


I don`t expect any bias problem, because the non-inv. A3 input is connected to another opamp output.
However, this is one of the reasons the loop must remain closed for dc.
Thus, opening the loop is allowed only for ac signals which makes the inductor necessary!

Hi Kevin,

Of course, I agree to the contents of your reply - however, for the circuit under discussion i don`t expect that non-linear effects could produce some unexpected results that canot be explained based on an ac analysis.
But - as mentioned - in principle you are right in requiring a transient analysis.
As a good example for your doubts: There are some circuits that fulfill Barkhausen`s oscillation criterion (ac analysis with unity loop gain) without being able to oscillate.
However, that is no surprise knowing that Barkhausen`s condition is a necessary one only.
B.

PS: by the way - are you familiar with the application of describing functions for oscillator design?

Kevin - thank you for providing the link for your tutorials.
I had a short look only - and I am very happy to see that you consider the BJT as voltage controlled (in contrast to many books, which seem to know only the relation Ic=beta*Ib)
However, I am not to happy to read: "If the slope of the loop gain is -20db/dec when the loop gain finally falls to 0db, the system will be stable, and unstable if the slope is greater than -20db.

And what about 25 0r 30 dB? I do not refer to the asymptotic lines but to the real response.

More than that: Do you consider the Barkhausen criterion to be false?
(Quote: This is sometimes called the Barkhausen Criteria, and is false.)

I think, there is a common agreement that Barkhausens condition for oscillation is a necessary one only!
For my opinion, Barkhausen himself did mention this restriction in his book.

In the near future I will further go through your tutorial. (Are you interested in further comments?).

Regards
B.

Kevin Aylward wrote on Jul 14^th, 2013, 9:15am:


Yes ?  I do not. When I observe that the inverse feedback factor in the Bode plot meets the second pole (approx.) - that means between the 20dB/dec roll-off and the 40 dB/dec roll-off - I know that the phase margin is approx. 45 deg.
The straight lines in the diagram are a great help to draw the magnitude response - however, for evaluation of stability issues I strongly recommend to use - as good as possible - the real curve in between.

Kevin Aylward wrote on Jul 14^th, 2013, 9:15am:

Yes - that´s what I was saying.

B.

PS: Regarding "feel free to comment": My question was if you are interested in some comments.

Example: If there is a pole in the right half plane of LG +1, it can be shown that the impulse response of any transfer function described by LG+1, will result in an ever increasing sinusoidal response. In practice this increase will limit when the amplifier saturates at its supply rails. This results in steady state oscillations. This implies that the amplifier is analyzed such that it is reasonable linear for small signals, but can limit at large signals in a way such that this limiting does not change the small signal phase response at the limiting point.

I suppose, you speak about a complex pole pair. More than that, such a condition will not ALWAYS result - as mentioned - in steady-state oscillations. There are cases where latch-up occurs.