Exam help! A loop-gain greater than one

[Mr.MEB]

OK then it's a bit more clear for me (i'm a bit slow).
I don't know exactly, maybe some analogy with mechnaic system coud help:
imagine a ball that stand on the top of an hill. At the beginning the system is in equilibrum because the top of the hill is flat. But when you push (v.small noise) a little bit (a certain threshold) it will go unstable (and maybe oscillate).
It's all that I can tell.
Good luck with your exam, if you don't find any other explanations you can try these ones. how you gonna sell depends on your marketing skills.

 

[jasmin_123]

Thank you, Mr.MEB for trying. The question is tougher than it may seem. I have already done many simulations and asked all the professors around. They have suggested many mathematical proofs and none intuitive. Too bad, but I will not give up on my 'marketing skills.' I still have two weeks.

 

[zorro]

Hi jasmin_123,

An oscillator oscillates with a waveform such that, if you open the loop at some point, the output is exactly the same as the input.

Intuitively: imagine an amplifier excited by a waveform such that the output is the same as the input. Now remove the external excitation replacing it by the output of the amplifier. The waweform if self-maintained.

Now, let’s consider sinusoidal oscillations. Let’s call G the absolute value of AOL*B and “phase” its phase.

The above concept means that opening the loop, at some frequency G has to be 1 and the phase has to be 0 (or some multiple of 2*pi radians).
Another essential condition (sometimes overlooked) is that the slope of phase vs. frequency has to be negative in order to maintain stable the oscillation.

Intuitively: suppose that, once the oscillator started, some external cause tends to introduce a phase lag in the signal output. This is equivalent to a decrease in the instantaneous oscillation frequency. The amplifier then has to correct for that lag; for this reason the phase slope has to be negative, otherwise (positive slope) the frequency of oscillation tends to diverge, the signal is not self maintained and the oscillation stops.
[I recall that the slope of the phase, -d(phi)/d(freq), is a figure of merit of the topology of an oscillator because it is directly related to frequency stability].

If these conditions are met, that frequency is the frequency of oscillation.

The gain G has to be >1 at the frequency at which phase is N*2*pi (with negative slope) in order to start oscillation.
You can have G>1 at other frequencies, but the circuit does not oscillate if at those frequencies the phase conditions are not met.

Best regards

Z

 

[The Joy of Electronics]

Hi... U guys really got me thnking!

As far as I know loop gain greater than one always causes instability. Because amplifiers are Minimum Phase Systems.

Plz tell me the truth if Im wrong.

 

[jasmin_123]

Hi, zorro, thanks for trying, but I cannot accept your explanations
because they are not correct. Attached is some nice example I have found
on the Net.
---
The absolute value of AOL*B does NOT "have to be >1
at the frequency at which phase is N*2*pi (with negative slope) in
order to start oscillation."

 

[jasmin_123]

Hi... U guys really got me thnking!

As far as I know loop gain greater than one always causes instability. Because amplifiers are Minimum Phase Systems.

Plz tell me the truth if Im wrong.

Hi, The Joy of Electronics, this is a common misconception (see the attachment).

 

[laktronics]

Hi Jasmin,
I do not understand the 'catch' in your question, but at least intuitively speaking, one should think that a linear system with loop gain >1 and phase n2Π should necessarily be unstable. In an ideal case of a zero noise system, just like a balanced ceter of gravity toy, it may appear to be stable but with the slightest external excitation, the system would become unstable and the amplitude would continue to grow since the signal gets amplified every time it passes through the loop, as the loop gain is >1.
By the way, do you have an electronic circuit that justifies your question and if so, can we have a look at that?
With regards,
Laktronics.

 

[jasmin_123]

Hi, Laktronics,

>...a linear system with loop gain >1 and phase n2Π should necessarily be unstable.
>... since the signal gets amplified every time it passes through the loop, as the loop
> gain is >1.

This is absolutely wrong.

>By the way, do you have an electronic circuit that justifies your question and if so,
>can we have a look at that?

One of the simple circuits in my replies of 10 Feb 2007 is stable(!) for a loop gain >1
(at zero phase) and the other is unstable(!) for a loop gain <1 (at zero phase).
Download them!

Enjoy

 

[zorro]

Hi jasmin_123,

Hi, zorro, thanks for trying, but I cannot accept your explanations
because they are not correct. Attached is some nice example I have found
on the Net.

Please explain where is the incorrectness of my explanations.

Have you checked whether the inner loop in 11.pdf (that has the topology of a phase shift net oscillator with U1) is stable or not?
If it is unstable, the results of the two circuits you posted are consistent.
My explanation is for analyse if closing a loop causes instability in a stable system. That was the explanaiton you requested, ok?

Could somebody give an INTUITIVE(!) explanation why a loop-gain greater
than one does not necessarily causes instability?

Regards

Z

 

[jasmin_123]

The gain G has to be >1 at the frequency at which phase is N*2*pi (with negative slope) in order to start oscillation.

The gain G does NOT have to be >1 at the frequency at which phase is N*2*pi (with negative slope) in order to start oscillation. In the attached example, G=0.1, but the oscillations do start!

Moreover, oscillations can decay for a G >1 at a zero phase.
---
The question as stated in my message of Feb 03:

The loop gain AOL*B is real, positive, and greater than one, and this does not NECESSARILY mean that the oscillator is unstable. It can be stable. Why?
---
An INTUTIVE explanation is needed, without Nyquist, root locus, and so on.

 

[zeeshanzia84]

Well, as far as I understand, there is no such thing as a loop-gain alone. LOOP GAIN is a function of frequency.

When I tell you that you have a positive feedback system and that the loop-gain is greater than one AT A frequency where the phase becomes 180 degrees or more shifted. That system will behave as a negative fb for THAT range of frequencies. And ofcourse a negative fb system cannot be unstable.

Thus, even though the +ve fb-system has a loop-gain greater than one (but only at frequencies where phase is 180 or more)....it is not unstable!!!

HOPE THIS HELPS

 

[zorro]

Hi jasmin_123,

I ask again: is the inner loop (the block formed by U1, the three Rs and the three Cs) stable? (Sorry, I do not have the time to ckeck that stability). Its topology is that of a classical oscillator, but its stability depends of its loop gain.
Ut is needed to know that in order to proceed.
Regards

Z

 

[jasmin_123]

Thus, even though the +ve fb-system has a loop-gain greater than one (but only at frequencies where phase is 180 or more)....it is not unstable!!!

Hi, zeeshanzia84,

We are talking about a +ve fb-system that has a loop-gain greater than one (at a frequency where phase is 360)... is it unstable???

Added after 22 minutes:

I ask again: is the inner loop (the block formed by U1, the three Rs and the three Cs) stable?

The block formed by U1, the three Rs and the three Cs is unstable.

---

Well, guys, I felt that we started to make circles and asked the tutor for a hint (at the expense of the 10 bonus points, too bad).
---
The hint was as follows:

"The loop-gain greater than one, at a frequency where the phase is zero, provides no more information about oscillator stability than the color of the oscillator capacitors does!"

Does it say anything to you???

 

[zorro]

I ask again: is the inner loop (the block formed by U1, the three Rs and the three Cs) stable?

The block formed by U1, the three Rs and the three Cs is unstable.

Good.

It can be seen that if the open loop system is unstable it is normal that there are oscillations even if the outer loop has loop gain < 1:
This inner loop has a gain (from the left input of the adder to OUT) of finite value at any (sinusoidal) frequency.
Now imagine that U2 has gain 0 (zero). The open loop gain is 0. Still the circuit oscillates.
But these oscillations are not responsibility of the outer loop! They are originated in the open loop system.

Regards

Z

 

[jasmin_123]

Zorro,

I understand what you have written above, but what is your explanation of the below:

Why an oscillator with a loop gain G >1 at zero phase can be stable?
This means that the oscillations will decay.

A primitive logic implies that if G>1 then any signal will grow wile traveling aroun the loop. This means that the input of the amplifier will grow and so will do its output. Hence, an oscillator with a G>1 at zero phase can never be stable! But it can be!

 

[zorro]

Dear Jazmine,

It is agreed that an amplifier with transfer function exactly 1 at some frequency would maintain oscillations at that frequency, at any amplitude (all linear, all ideal). It is a point of equilibrium in which the output is the same as the input and the loop can be closed staying in that condition.

If G is real and G>1 for some frequency, the output is no more the same as the input, and that is not an equilibrium point. Not necessarily the ouput has to grow (although it does in many circuits) when the loop is closed.

The output waveform at which the amplifier would “oscillate” is a function at which the output would be exactly the same as the input. In our ideal case, those functions (natural modes) are complex exponentials and linear combinations of them.These natural modes correspond to the poles of the closed loop circuit.
The fact that G>1 (G real) at some sinusoidal waveform does not assure that there is a natural mode with growing amplitude.
The classical examples of conditional stability are the best illustration of such cases.

Is this clear?
Regards

Z

 

[jasmin_123]

If G is real and G>1 for some frequency, the output is no more the same as the input, and that is not an equilibrium point. Not necessarily the ouput has to grow (although it does in many circuits) when the loop is closed.

Do you know why?

 

[zorro]

If G is real and G>1 for some frequency, the output is no more the same as the input, and that is not an equilibrium point. Not necessarily the ouput has to grow (although it does in many circuits) when the loop is closed.

Do you know why?

Sorry, I don't understand your question. What are you referring to?

 

[jasmin_123]

Hi, zorro,

Why not necessarily the output has to grow? If signal at some frequency is increasing while traveling over the loop and remains in phase (G>1 at zero phase implies just this), then the output does necessarily have to grow.
---
The output always have to grow for a G>1, P(G)=0. Why for the same G>1, P(G)=0 it does grow in some cases and does not grow in the other cases? Why?

Jasmine

 

[rabel126]

it depand ipon application line in sinosidal oscillators loop gain is 1 but for multivibrators the loop gain is greater than one somethime we use we use +ve feedback bcz it of application and some time it is -ve