# Cascaded Op Amp circuit .. #2

#### LvW

Joined Jun 13, 2013
1,773
Hi Tesla, I did not mean that your simplification was not correct.
My doubts were poited to the 50k Resistor and the definition of the loop gain.
Anylyzing the loop gain of your circuit gives, of course, the same result as given in one of my earlier posts:

LG=(1+R4/R3)*R6/(R5+R6)*R1/(R1+R2)=....=+1.713

Because we are speaking about the outer (overall) loop, I have opened the loop for injection of a test signal at the non-inv. terminal of the most left opamp. All local loops remain closed. This approach is in accordance with system theory for multi-loop systems.

I will repeat that you modification (simplification) is correct - it is a clever idea!
And I also see your motivation to show what happens when we open also the local (negative) gain determining loop.
I know what you intend to show:
For defining the feedback factor this way, the faktor k is - as shown in your expression (post#38) - a sum of two factors with different sign (k+ and k-).

However, in this case it is really a problem to compute the overall loop gain (as your formula shows).
This is a classical problem for all high-gain loops because this definition does not allow the commonly used simplification (infinite open-loop gain).
Therefore, it is a classical approach to keep all local loops closed (unfortunately, this is not always possible).
However, it is not necessary to find the exact loop gain value. For our purpose it is sufficient to verify the condition for stability: (k+)< (k-)
But in the present case (values as in the original question) we have k+=0.57 and k-=0.5

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#### MrAl

Joined Jun 17, 2014
11,723
MrAl - sorry to say, but you are wrong. I really cannot understand your position - I do not "shift" anything.........
A simple visual inspection of the original circuit shows the following:
* There are two two main active blocks - two non-inverting amplifiers (both get their input signals at the non-inv. inputs)
* In between is a simple voltage divider which must be carefully dimensioned (in the given circuit, it is not) to keep the positive loop gain small enough.
* This series connection of non-inverting blocks is equipped with a resistive feedback loop.
* Question to you: Where do you see any "negative feedback in addition to the positive feedback" ?
Perhaps you can explain that?
Hello again,

The negative feedback is in all three circuits although the resistor enumeration is different
in all three circuits. These resistors in the three circuits are as follows:
R7 (10k) in the original circuit
R3 (10k) in the experimental circuit
R4 (10k) in Tesla23's circuit.

I said you were shifting the paradigm because originally you made it sound like you didnt
even have to analyze the circuit as long as you could see ANY positive feedback, then you
were saying that you had to analyze it to come up with the overall total feedback type.
I think you originally just spoke in more general terms though so dont worry about that
let's just try to agree on how these circuits work

I think you should be able to see Tesla23's point of view. In his original circuit that
is the same as the original circuit, the op amp U2 acts as a unity gain buffer. That is
because the sole function of that op amp section is to keep the inverting terminal at the
same potential as the junction of R5 and R6 which connects to the non inverting terminal
of U2. Now from what we know about op amps in the linear mode we know that the inverting
terminal is kept at the same voltage level as the non inverting terminal because if it is
not then the output changes in a direction to correct that. So it acts much like a servo.
Since the voltage levels are again independent of the currents through either of the resistors,
the U2 section acts just like a unity gain buffer except maybe for the speed of response.
So it's a kind of circuit 'trick' and of course it only works if the op amp is operating
in the linear mode. It does appear to be operating in the linear mode and it should unless
the current requirements at the inverting node become too large.
Another way to look at this is to look at a unity gain op amp with output connected directly
to the inverting input. We immediately recognize this as a unity gain buffer. Now break the
direct connection from output to input and replace it with a resistor of value say 1k. Note
now it still acts as a unity gain buffer as viewed from the non inverting terminal to output
unless any other connection to the inverting terminal is made and it sinks so much current
that the output of that op amp saturates trying to stay in the linear mode, but as long as
it stays in the linear mode it will act as a unity gain amplifier. That is how Tesla23 got
to the modified circuit with a more traditional unity gain buffer section.

I hope this explanation helps.

#### LvW

Joined Jun 13, 2013
1,773
Hello again,
I said you were shifting the paradigm because originally you made it sound like you didnt
even have to analyze the circuit as long as you could see ANY positive feedback, then you
were saying that you had to analyze it to come up with the overall total feedback type.
I think you originally just spoke in more general terms though so dont worry about that
let's just try to agree on how these circuits work
OK - when discussing DC stability of a system consisting of more than one active block and ome other circuitry, there is just one single loop which matters :
The "outer" overall feedback loop. In this context, it does not matter if there is any internal (local) feedback loop.

Remember a series connection of two transistor stages with overall feedback.
One of the stages may have an internal emitter feedback (RE) which, of course, is considered during calculation of that single stage - but nobody would say "in the circuit, we have two loops.
More than that, it is even not possible to combine these two loops into a common expression.

As I have already mentioned - it was a misunderstanding between us. I spoke about the overall loop and you had also the local loops in mind. Thats all.

Regarding Teslas circuit - I have placed my comments already. No further comments necessary from my side.

#### Tesla23

Joined May 10, 2009
543
Hi Tesla, I did not mean that your simplification was not correct.
My doubts were poited to the 50k Resistor and the definition of the loop gain.
Anylyzing the loop gain of your circuit gives, of course, the same result as given in one of my earlier posts:

LG=(1+R4/R3)*R6/(R5+R6)*R1/(R1+R2)=....=+1.713

Because we are speaking about the outer (overall) loop, I have opened the loop for injection of a test signal at the non-inv. terminal of the most left opamp. All local loops remain closed. This approach is in accordance with system theory for multi-loop systems.
I look for tricks otherwise I can't be bothered spending much time on problems that seem to have been contrived to extend (annoy?) students. The positive feedback path is a bit obscured in this circuit and I thought it helped to reveal it. Sorry if I missed your earlier post where you showed the relative sizes of the positive and negative feedback paths - I only did a quick skim of the earlier posts.

#### MrAl

Joined Jun 17, 2014
11,723
OK - when discussing DC stability of a system consisting of more than one active block and ome other circuitry, there is just one single loop which matters :
The "outer" overall feedback loop. In this context, it does not matter if there is any internal (local) feedback loop.

Remember a series connection of two transistor stages with overall feedback.
One of the stages may have an internal emitter feedback (RE) which, of course, is considered during calculation of that single stage - but nobody would say "in the circuit, we have two loops.
More than that, it is even not possible to combine these two loops into a common expression.

As I have already mentioned - it was a misunderstanding between us. I spoke about the overall loop and you had also the local loops in mind. Thats all.

Regarding Teslas circuit - I have placed my comments already. No further comments necessary from my side.
Hello again and thanks for the reply.

Well i am saying that there is a positive feedback, a negative feedback, and an overall feedback.
I believe you are limiting yourself too much here by insisting there is only one way to look at a circuit with feedback. Sure, in theory the overall response is dependent on the overall feedback, but to be succinct we may have to talk about two different feedbacks in a CIRCUIT while we may be most interested in the overall net feedback. It seems like you are saying that we can never mention any negative feedback if the overall net feedback is positive, but that cant be true in a circuit and also cant be true in a block diagram.

But i think you are limiting yourself here too much. For example, if i were to say "increase the negative feedback a little" you make it seem like you would not be able to perform this simple function because you dont believe in any negative feedback if the net feedback is positive. You mean you have never seen an exercise like:
"Label the negative feedback and the positive feedback and compute the overall net feedback."
Please clarify this if i misunderstand maybe we can hash this out a bit more.

#### Jony130

Joined Feb 17, 2009
5,539
Did you try to run a simulation of this configuration?
Yes, I did

Do you have any papers on this subject ( DC stability )?

#### LvW

Joined Jun 13, 2013
1,773
I believe you are limiting yourself too much here by insisting there is only one way to look at a circuit with feedback.
...............
But i think you are limiting yourself here too much.
...............
Please clarify this if i misunderstand maybe we can hash this out a bit more.
Well, my answer can be very short:
* When I see a circuit (containing more than one single active block) with feedback - in particular, when somebody has problems to understand the principle behaviour - at first, I check if the fundamental precondition for proper operating is fulfillerd: DC bias point within the linear region of the transfer function.

* For fulfilling this precondition, the overall feedback for DC must be negative.
In this case, an uneven number of sign inversions within the loop is required.

* It was a very simple task to verify that the circuit under discussion consists of two non-inverting amplifiers with overall feedback . In this context, it does not matter at all if the circuit contains some local (negative) feedback loops for realizing the fixed gain properties of the two amplifiers.

* Result: No sign inversion within the main loop for DC - hence, no negative feedback. A pretty simple consequence !
That was the contents of my first post . In another post I have given the value for the DC loop gain.

* Yes, I have limited myself to the DC loop gain (and its sign) only. Why? Because this was my only concern - nothing else.
Perhaps one could discuss - in addition - some applications and/or other properties of the circuit.
Therefore, I must admit that I really do not know what remains to be "clarified" .

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#### LvW

Joined Jun 13, 2013
1,773
Do you have any papers on this subject ( DC stability )?
Hi Jony, yes - there are some papers dealing exclusively with DC feedback and DC stability for similar circuits.
Let me try to find corresponding links or references.
LvW

#### MrAl

Joined Jun 17, 2014
11,723
Well, my answer can be very short:
* When I see a circuit (containing more than one single active block) with feedback - in particular, when somebody has problems to understand the principle behaviour - at first, I check if the fundamental precondition for proper operating is fulfillerd: DC bias point within the linear region of the transfer function.

* For fulfilling this precondition, the overall feedback for DC must be negative.
In this case, an uneven number of sign inversions within the loop is required.

* It was a very simple task to verify that the circuit under discussion consists of two non-inverting amplifiers with overall feedback . In this context, it does not matter at all if the circuit contains some local (negative) feedback loops for realizing the fixed gain properties of the two amplifiers.

* Result: No sign inversion within the main loop for DC - hence, no negative feedback. A pretty simple consequence !
That was the contents of my first post . In another post I have given the value for the DC loop gain.

* Yes, I have limited myself to the DC loop gain (and its sign) only. Why? Because this was my only concern - nothing else.
Perhaps one could discuss - in addition - some applications and/or other properties of the circuit.
Therefore, I must admit that I really do not know what remains to be "clarified" .
Yes it is reasonable to want to understand the overall working of the circuit but it is also reasonable to understand the inner workings too. The bottom line in any discipline is does a particular concept do anything for us in understanding a concept or something else useful. Intracircuit analysis is certainly userful and we do it all the time such as when combining parallel resistors. The simplification of the sub circuit helps understand the overall circuit easier. If we see intracircuit positive feedback along with negative feedback it becomes harder to tell what the overall feedback operation is, but when we see just negative feedback alone it usually means the overall circuit has negative feedback. It's also useful when we want to point out different things about the inner workings of the circuit.

But since you stated that the circuit "has positive feedback" long before any analysis was done, i took it to mean that ANY inner circuit positive feedback to you meant the entire circuit overall operation had to act as if it had positive feedback. Perhaps you just spoke too soon.

But also since you are concerned only with the overall circuit feedback type i will try to remember never to mention any intracircuit feedback when discussing these circuits with you

You know what else is funny though, some electronic engineers with 30 plus years experience dont realize that adding positive feedback can increase input impedance as well as gain and sometimes bandwidth.

Ok so good luck with your future circuits...

#### LvW

Joined Jun 13, 2013
1,773
But since you stated that the circuit "has positive feedback" long before any analysis was done, i took it to mean that ANY inner circuit positive feedback to you meant the entire circuit overall operation had to act as if it had positive feedback. Perhaps you just spoke too soon.
......"long before any analysis was done" ....

How do you know?
Don't you think I thought a little before I wrote my answer? Without "any analysis" ?
Perhaps you "spoke too soon"?
It was also not a big mathematical problem to find the loop gain for this single-loop arrangement .
Was something wrong in my analysis?

PS: I am involved in feedback systems (control systems, filters, oscillators) since more than 40 years.....and I know what I am speaking of.

#### MrAl

Joined Jun 17, 2014
11,723
......"long before any analysis was done" ....

How do you know?
Don't you think I thought a little before I wrote my answer? Without "any analysis" ?
Perhaps you "spoke too soon"?
It was also not a big mathematical problem to find the loop gain for this single-loop arrangement .
Was something wrong in my analysis?

PS: I am involved in feedback systems (control systems, filters, oscillators) since more than 40 years.....and I know what I am speaking of.
Ok, good luck to you friend

#### LvW

Joined Jun 13, 2013
1,773
But since you stated that the circuit "has positive feedback" long before any analysis was done, i took it to mean that ANY inner circuit positive feedback to you meant the entire circuit overall operation had to act as if it had positive feedback. Perhaps you just spoke too soon.
I must admit that I am always a bit surprised and disappointed when in a technical-objective discussion suddenly personally colored statements are made. My fault?

#### LvW

Joined Jun 13, 2013
1,773
Hi Jony, yes - there are some papers dealing exclusively with DC feedback and DC stability for similar circuits.
Let me try to find corresponding links or references.
LvW
Hi Jony, in case you are still interested here are some references for papers dealing with DC stability:

* Green M., Wilson, A: "How to identify unstable DC operating point.". IEEE Trans. CAS, Vol. 39, No 10, pp. 820-832

* Biolek D., Biolek, Z.: "Potential stability of DC operating points in Circuits with controlled sources",
ICECS 96, Rhodos, Greece, Vol. 1, pp.101-104

* Biolek D., Biolek, Z.: "Stability of DC operating points and corresponding dynamical systems", MS`97 Melbourne, pp236-240.