how to determine feedback factor

absolutely covered! thanks all guys.

Quote DEDALUS:
I think there is no doubt that closed loop gain of voltage integrator is Acl = -1/(s*R*C). According to your derivation β=R/((1/jωC)+R). So why 1/β=1/(R/((1/jωC)+R)) = 1+1/(s*C*R) isn't equal to -1/(s*R*C)?

Hi DEDALUS:
The answer is simple. The approximation (for high loop gain) Acl=1/β is valid only for non-inverting circuits. For all inverting circuits it is Acl=-α/β.
The background is the feedback formula:

Acl=α*Ao/(1-β*Ao) with β negative and Ao=open loop gain.

For Ao very large this leads to the above expressions.

The factor α is the feedforward factor which is calculated in analogy to the feedback factor. It is the portion of the input signal arriving at the opamp input if the output is zero. For non-inv. applications α=1.
Regards
LvW

Hi LvW.

Thanks for explanation. If I understand correctly, for the case of integrator α = 1/(sC)/(1/(sC)+R), so Acl = -α/β = -1/(sCR).

Please, give some references to literature, where this feedback concept (with feedforward parameter) is described.

Thanks.

dedalus said:

Hi LvW.

Thanks for explanation. If I understand correctly, for the case of integrator α = 1/(sC)/(1/(sC)+R), so Acl = -α/β = -1/(sCR).
Please, give some references to literature, where this feedback concept (with feedforward parameter) is described.
Thanks.

Click to expand...

Yes, that's right.

Here is a selection of some references:
* All books on control theory, whereby the "feedforward factor" α very often is replaced by a box called "prefilter".
* Jerald Graeme: Optimizing opamp performance
*L.T. Bruton: RC-active circuits
*Sergio Franco: Design with opamps and analog integrated circuits
*All filter books: Because the quickest and simplest method to calculate transfer functions for inverting structures is to divide the process in three steps:
(a) find α and (b) find β and then (c) find the ratio of both.

Added after 4 minutes:

But I think the general formula for feedback is very well known and does not need some references:

Acl=Ao/(1-β*Ao).

And when there is another circuitry before the signal enters the input (as is the case for inverting applications), the above formula gets a corresponding factor α.

Hi LvW.

In your previous posts you've stated that Acl = A/(1+β*A) is valid not fol all cases, but only for non-inverting circuits. And in general Acl= α*A/(1+β*A), where Acl - closed-loop gain, A - open-loop gain, β - feedback factor, α - feedforward factor. I thought about this and I have counterarguments.

System that is described by the equation Acl= α*A/(1+β*A) is shown in the first attached figure. It can easily be shown that this system is equivalent to the system shown in the second figure. In both systems feedback network is present, but with different transfer functions. The key point here is in the definition of the feedback factor. It is defined as fraction of output signal that is fed back to the input. In the case of second system this is Sf2/So = β/α. And in the case of the first system this is (Sf1/So)/α (!), not Sf1/So (because input signal is signal applied to the input of prefilter α, not the signal at the input of summator, thus we must additionally divide by alpha). So in both cases feedback factor is equal as β/α (as expected, because systems are equivalent).

My conclusions:

1) We have used different definitions of feedback factor (I suppose you define feedback factor as Sf1/So, instead of (Sf1/So)/α). And indeed: I've derived β equal to -s*C*R, it's equal to ratio of β derived by you and α: -R/(1/sC+R) / 1/sC/(1/sC+R).

2) Acl = A/(1+f*A) can be applied for all cases, both inverting and noninverting (where f - feedback factor).

I want to add that in such books as "Analysis and design of analog integrated circuits" by Gray/Meyer, "Microelectronic circuits" by Sedra/Smith, "Design of analog CMOS integrated circuits" by Razavi, second representation of feedback system is used, but β/α is designate as β or f.

dedalus said:

Hi LvW.

............................
System that is described by the equation Acl= α*A/(1+β*A) is shown in the first attached figure. It can easily be shown that this system is equivalent to the system shown in the second figure. In both systems feedback network is present, but with different transfer functions. The key point here is in the definition of the feedback factor. It is defined as fraction of output signal that is fed back to the input. In the case of second system this is Sf2/So = β/α. And in the case of the first system this is (Sf1/So)/α (!), not Sf1/So (because input signal is signal applied to the input of prefilter α, not the signal at the input of summator, thus we must additionally divide by alpha). So in both cases feedback factor is equal as β/α (as expected, because systems are equivalent).

Click to expand...

Hi dedalus !
I'm sorry, but I cannot agree. Instead, I'm afraid you are wrong.
Let me explain:
* I agree, that both systems are equivalent - as far as the input-output relation is concerned!
* However, both systems have different feedback factors. This is no surprise, because the open loop gain (forward path) in both cases is different. Thus, when the input-output relation is constant, there must be another feedback factor!
* In my former posting I have explained that the calculation of the feedback factor of course (!)) is and must be based on the superposition theorem.
This implies for the calculation of the feedback factor, that you have to take into consideration the input at the opamp (!) - not the signal input (when both are different!)
*This can be verified by setting Vin=0 in Fig. 1 (according to the superposition theorem). Then, the forward box α has no influence on feedback.
*Summary: Both forward open loop gains are different; both feedback factors are (and must be!) different; but both loop gains and both closed loop transfer functions are identical.
*Is there anywhere a logical break?

Regards
LvW

Apart from the question, if it gives a correct feedback factor, the "feedback2" scheme is rather the realization of a calculus
trick than a description of a real system. I understand, that you try to describe the discussed -α/β gain inverting amplifier structure
with the "α-less" scheme known from literature. In my opinion, it won't get us anywhere.

In contrast, the "feedback1" scheme describes a real system, it's parameters can be directly related to circuit elements, it's
actually instructive.

FvM said:

...............
...............
In contrast, the "feedback1" scheme describes a real system, it's parameters can be directly related to circuit elements, it's
actually instructive.

Click to expand...

Yes, completely agreed.
It is instructive because this block diagram allows identification of the feedback factor (β) by simple inspection. The forward factor (prefilter) α is outside the loop and, therefore, does not influence the feedback at all.

Hi LvW! Hi FvM!

LvW said:

*Is there anywhere a logical break?

Click to expand...

No, there is no any logical break. But in my arguments there is no logical break too, and I want to illustrate this.

I will find feedback parameters of integrator (third attached figure) using both representations of feedback system and than compare the results. I shown these structures in the attached figures again (with minor corrections).

First of all, I want to define what I mean under the "feedback parameters". These parameters are: close-loop gain, Acl = Sout/Sin; loop gain, Al = product of transfer functions in the feedback loop, in the first case Al = A1*β, and in the second case Al = A2*f; feedback factor, ff = transfer function of the feedback network, in the first case ff = β, in the second case ff = f; and open-loop gain, Aol = tranfer function of the amplifier in the forward path, in the first case Aol = A1, in the second case Aol = A2.

If we apply first representation of the feedback system to the integrator we will get next results:

α = -1/(sC)/(R+1/(sC))
β = R/(R+1/(sC))
A1 = A

Aol = A
ff = R/(R+1/(sC))
Al = A*R/(R+1/(sC))
Acl = (-1/(sC)/(R+1/(sC)))*A/(1+A*R/(R+1/(sC))) = -A/(1+sCR*(1+A))
lim Acl (A->inf) = -1/(sCR) = α/β.

If we apply second representation of the feedback system we will get next results:

f = -sCR
A2 = -A/(sC)/(R+1/(sC))

Aol = -A/(sC)/(R+1/(sC))
ff = -sCR
Al = A*R/(R+1/(sC))
Acl = -A/(sC)/(R+1/(sC))/(1+A*R/(R+1/(sC))) = -A/(1+sCR*(1+A))
lim Acl (A->inf) = -1/(sCR) = 1/f.

Resume:

1) Loop gain and close-loop gain are the same. Why? Because they have the same definition regardless of the representation of the feedback system.
2) Feedback factor and open-loop gain are different. Why? Because their definitions (used in this derivation) depend on the representation of the feedback system.

General conclusions:

1) In my derivation and in your ,LvW, derivation of feedback factor different definitions of feedback factor were used:
According to definition used by me, feedback factor is fraction of output signal fed back to the input of the whole system.
According to definition used by you, LvW, feedback factor is fraction of output signal fed back to the input of the basic amplifier.
That's why we got different results.

2) I absolutely agree that second representation can result (in the case of inverting application) in "unnatural" description of the feedback amplifier: this representation can lead to the fact that open-loop gain is not equal to the gain of the basic amplifier. But this doesn't mean that this representation is incorrect and cannot be used for all feedback configurations.

P.S.

FvM, in one of your previous post in this topic you've said that your literature is Gray/Meyer, Razavi, and you've asked me, whether I used definition of Aol, Acl, ff from these sources. Yes, I'm. But why don't you use them?

But why don't you use them?

Click to expand...

I don't want to use the structure for a circuit, where it's not approriate. The output of the substractor in your feedback2 scheme is
not the signal at the amplifier input in case of the inverting amplifier. The description doesn't help me to analyze this type of circuits.

Hi DEDALUS,

first of all, I like to say that this discussion for my opinion is not superfluous at all, as it can lead to a better insight in feedback theory as well as to a better understanding of the technical meaning of defined parameters.
Having this in mind, I comment your last posting as follows:

1) Loop gain and close-loop gain are the same. Why? Because they have the same definition regardless of the representation of the feedback system.

Yes, without any doubt.

2) Feedback factor and open-loop gain are different. Why? Because their definitions (used in this derivation) depend on the representation of the feedback system.

No. Not their DEFINITION depends on the feedback topology. Instead, the RESULT (the expression) applying this general definition depends on the representation. According to your own words, the feedback factor in circuit No. 2 is ff=f. Thus, you have used the classical and single definition for the feedback factor.

General conclusions:
1) In my derivation and in your ,LvW, derivation of feedback factor different definitions of feedback factor were used:
According to definition used by me, feedback factor is fraction of output signal fed back to the input of the whole system.

What do you mean with "whole system"? Signal input? If yes, your "new" definition makes no sense, because nothing can be fed back to the input of the whole system, since the voltage source has zero input resistance and no feedback signal can be developped.

2) I absolutely agree that second representation can result (in the case of inverting application) in "unnatural" description of the feedback amplifier: this representation can lead to the fact that open-loop gain is not equal to the gain of the basic amplifier. But this doesn't mean that this representation is incorrect and cannot be used for all feedback configurations.

As far as I remember, nobody has claimed, that the 2nd representation is "incorrect". But, it has no practical meaning and, more than that, cannot be used to identify the correct feedback factor for the circuit under discussion. And this was the original question to the beginning of this discussion!

But, on the other hand, in practical control sytems there are some cases in which such manipulation of block representation (transfer of some specific properties from the forward to the feedback path) may have some advantages - however, of course without any violation or modification of any definition!
LvW

FvM said:

The output of the substractor in your feedback2 scheme is not the signal at the amplifier input in case of the inverting amplifier.

Click to expand...

Yes, this is actually my second conclusion.

FvM said:

The description doesn't help me to analyze this type of circuits.

Click to expand...

This is your personal preference. It does help me to analyze feedback circuits (and this type too).

Have you any doubts that the second representation can be used for analyzing all feedback amplifiers (regardless inverting or non-inverting), and the resulting loop gain and close-loop gain will be the same as in the case of using first representation?

And also, do you agree with my first conclusion, that both answers (my and LvW's) are correct with corresponding definitions of feedback factor?

Added after 58 minutes:

LvW said:

What do you mean with "whole system"? Signal input? If yes, your "new" definition makes no sense, because nothing can be fed back to the input of the whole system, since the voltage source has zero input resistance and no feedback signal can be developped.

Click to expand...

With "input of the whole system" I mean input terminal (of course feedback voltage cannot shunt the input source). E.g. in the case of integrator the feedback signal (in the second representation) is voltage across the resistor, which is connected to the input terminal of the integrator (in the first representation it's the voltage across the resistor too, but with the input shorted to ground, so actually feedback signal is voltage across inputs of the op amp (basic amplifier) with input shorted to ground).

LvW said:

No. Not their DEFINITION depends on the feedback topology. Instead, the RESULT (the expression) applying this general definition depends on the representation. According to your own words, the feedback factor in circuit No. 2 is ff=f. Thus, you have used the classical and single definition for the feedback factor.

...

But, it has no practical meaning and, more than that, cannot be used to identify the correct feedback factor for the circuit under discussion.

Click to expand...

Yes, I agree. I've expressed incorrectly. I had to use "expression" instead of "definition".

Why do you think, that expression from the first representation is correct and from the second - is incorrect? If both expressions are derived from the same definition, using different, but equivalent, representations of the system, why one of them is correct and other incorrect? Isn't this a logical break?

I think that the term "correct feedback factor" has no sense. The question "What is correct expression for feedback factor?" without specifying representation of the system is similar to the question "What is the voltage at the node?", without specifying reference node. Actually feedback factor is intermediate parameter, used for defining loop gain, parameter of primary interest. To find loop gain we multiply feedback factor by open-loop gain, which also has no "correct expression". Expression for open-loop gain is also depends on representation of the system. But their product (loop gain) doesn't depend (similar to the subtracting of two voltages). So the term correct "correct loop gain" does make sense, but the terms "correct feedback factor" and "correct open-loop gain" don't.

dedalus said:

.............
.............
Why do you think, that expression from the first representation is correct and from the second - is incorrect? If both expressions are derived from the same definition, using different, but equivalent, representations of the system, why one of them is correct and other incorrect? Isn't this a logical break?

I think that the term "correct feedback factor" has no sense. The question "What is correct expression for feedback factor?" without specifying representation of the system is similar to the question "What is the voltage at the node?", without specifying reference node. Actually feedback factor is intermediate parameter, used for defining loop gain, parameter of primary interest. To find loop gain we multiply feedback factor by open-loop gain, which also has no "correct expression". Expression for open-loop gain is also depends on representation of the system. But their product (loop gain) doesn't depend (similar to the subtracting of two voltages). So the term correct "correct loop gain" does make sense, but the terms "correct feedback factor" and "correct open-loop gain" don't.

Click to expand...

OK, DEDALUS, you are right. I have to admit, that a definition cannot be "correct".
A definition only can make sense or not.
Insofar, I correct my self and replace the wording "correct feedback factor" by "commonly used feedback factor".
And, more than that, I do not find any error in your block diagram modification with associated calculation. Of course, you can do that - and I have mentioned in my last posting, that in some cases this procedure is applied (in control system design).
However, as far as your preferred definition is concerned, I refuse to call the term α/β "feedback factor" as you propose. Try to find any other name for it.
At first, otherwise it would lead to misunderstandings and at second, it is simply not true that this is a factor which determines the part of the output signal which is fed back to the input.
Example 1: inverting amplifier: α/β=R1/R2
Example 2: integrator: α/β=1/sRC

In both cases, this ratio of two impedances can be larger than 1 (as you have claimed already at the beginning of this thread). But in this case, it cannot be a portion of the output signal and does not meet the common definition of "feedback factor".
In summary, you may define this ratio as you want - and use it - but do not call it "feedback factor". OK?
Regards
LvW

LvW said:

Insofar, I correct my self and replace the wording "correct feedback factor" by "commonly used feedback factor".

...

However, as far as your preferred definition is concerned, I refuse to call the term α/β "feedback factor" as you propose. Try to find any other name for it.
At first, otherwise it would lead to misunderstandings and at second, it is simply not true that this is a factor which determines the part of the output signal which is fed back to the input.
Example 1: inverting amplifier: α/β=R1/R2
Example 2: integrator: α/β=1/sRC

In both cases, this ratio of two impedances can be larger than 1 (as you have claimed already at the beginning of this thread). But in this case, it cannot be a portion of the output signal and does not meet the common definition of "feedback factor".
In summary, you may define this ratio as you want - and use it - but do not call it "feedback factor". OK?

Click to expand...

Hi LvW.

1) I suppose α/β is typo and you've meaned β/α. α/β is closed-loop gain (with ideal basic amplifier), not feedback factor in any case.

2) I absolutely disagree that β/α cannot be called "feedback factor". I want to emphasize that this is not my definition of feedback factor as you've claimed. I follow terminology accepted in next books:

Sedra/Smith "Microelectronic Circuits" (6th ed, page 805).
Razavi "Design of Analog CMOS Integrated circuits" (page 247).
Johns/Martin "Analog Integrated Circuit Design" (page 233)
Baker/Li/Boyce "CMOS. Circuit Design, Layout and Simulation" (1st ed, page 526)

(I must admit that in Gray/Meyer/Hurst/Lewis "Analysis and Design of Analog Integrated Circuits", which I prefer from all, the term f = β/α isn't called "feedback factor", but "network transfer function". Though this name cannot avoid misunderstandings, because β can be called "network transfer function" too)

3) I want to remind you, that this forum is called "Analog IC design and layout", not "Control theory", not "Filter design". Thus I think, your attempt to apply terminology accepted in Control theory is absolutely incorrect here. As a consequence your statement that β is "commonly used feedback factor" (instead of β/α) is incorrect here too (please refer to the listed books).

Hi DEDALUS,

1) I suppose α/β is typo and you've meaned β/α. α/β is closed-loop gain (with ideal basic amplifier), not feedback factor in any case.

Yes, sorry. It's really a typo.

2) I absolutely disagree that β/α cannot be called "feedback factor". I want to emphasize that this is not my definition of feedback factor as you've claimed. I follow terminology accepted in next books:

Sedra/Smith "Microelectronic Circuits" (6th ed, page 805).
Razavi "Design of Analog CMOS Integrated circuits" (page 247).
Johns/Martin "Analog Integrated Circuit Design" (page 233)
Baker/Li/Boyce "CMOS. Circuit Design, Layout and Simulation" (1st ed, page 526)

OK, if it is so, I have the following question:
How is this factor (beta/alpha) defined in words? (Like beta, which results simply from the superposition theorem as the portion which is fed back to the forward amplifier for Vin=0).


3) I want to remind you, that this forum is called "Analog IC design and layout", not "Control theory", not "Filter design". Thus I think, your attempt to apply terminology accepted in Control theory is absolutely incorrect here. As a consequence your statement that β is "commonly used feedback factor" (instead of β/α) is incorrect here too (please refer to the listed books).

"Absolutely incorrect"? Really?
OK, since I am not directly involved in analog IC design&layout I must believe you.
However, I would be really surprised, when the terminology and the definitions of some describing parameters would depend on whether I discuss a lumped element circuitry or an IC design.
But may be, I am wrong. In this case, I would have learned something - especially, when you answer my question as formulated above under 2).

LvW

LvW said:

OK, if it is so, I have the following question:
How is this factor (beta/alpha) defined in words? (Like beta, which results simply from the superposition theorem as the portion which is fed back to the forward amplifier for Vin=0).

Click to expand...

Hi LvW.

In any of these books there is no explicit definition of this factor "in words". I quote the autors:

Sedra/Smith said:

The output xo is fed to the load as well as to a feedback network, which produces a sample of the output. This sample xf is related to xo by the feedback factor β,
xf = β*xo.

Click to expand...

(In Razavi's book H(s) is used instead of A, and G(s) is used instead of β in the block diagram)

Razavi said:

... where H(s) and G(s) are called the feedforward and the feedback networks, respectively. ... we replace G(s) by a frequency-independent quanity β and call it the "feedback factor".

Click to expand...

Johns/Martin said:

The feedback term, β, represents the feedback factor, ...

Click to expand...

Baker/Li/Boyce said:

The feedback factor, β, is defined as
β = xf/xo

Click to expand...

Hi DEDALUS,

thank you for the excerpts from the 4 textbooks as contained in your last posting.
When I compare the corresponding equations and the associated figure (Feedback.jpg) with the two feedback representations as given by you as Feedback1.jpg (april 14th and 15th) I am really happy to see that general rules, equations and the terminology of electronics and control theory apply also to the domain of "Analog IC design" (surprise, surprise,..).

[Quote DEDALUS (April 15th): I want to remind you, that this forum is called "Analog IC design and layout", not "Control theory", not "Filter design". Thus I think, your attempt to apply terminology accepted in Control theory is absolutely incorrect here. As a consequence your statement that β is "commonly used feedback factor" (instead of β/α) is incorrect here too )]

Regards
LvW

In addition. I tried to understand, what you want to show with your reference to Grey/Hurst et al:

(I must admit that in Gray/Meyer/Hurst/Lewis "Analysis and Design of Analog Integrated Circuits", which I prefer from all, the term f = β/α isn't called "feedback factor", but "network transfer function". Though this name cannot avoid misunderstandings, because β can be called "network transfer function" too)

Click to expand...

I assume, you refer to paragraph 8.1 Ideal Feedback Equation. It discusses the feedback network transfer function, designated
with the letter f, obviously identical to feedback factor β in the above presented schemes. But there isn't anything like β/α, neither at
this place nor anywhere in chapter 8 Feedback or 9 Frequency Response and Stability of Feedback Amplifiers.

Interestingly, the inverting amplifier is discussed in paragraph 8.9, but without referring to the said transfer function, because the
representation is targeting to port impedances. So we can't conclude, how the authors would deal with factor α in their ideal feedback
configuration.

Hi.

LvW said:

thank you for the excerpts from the 4 textbooks as contained in your last posting.
When I compare the corresponding equations and the associated figure (Feedback.jpg) with the two feedback representations as given by you as Feedback1.jpg (april 14th and 15th) I am really happy to see that general rules, equations and the terminology of electronics and control theory apply also to the domain of "Analog IC design" (surprise, surprise,..).

Click to expand...

Yes, it's incorrect to say that the terminology of control theory cannot be applied to the "analog ic design" domain, because actually the term ("feedback factor") is the same, BUT the things that this term represents are different. By the way, I have not said, that general rules and equations cannot be applied in the "analog ic design" domain.
I wanted to say, that there is no "commonly used feedback factor" as well as "correct definition of feedback factor". Every domain can have it's own "commonly used" parameters.

FvM said:

I assume, you refer to paragraph 8.1 Ideal Feedback Equation. It discusses the feedback network transfer function, designated
with the letter f, obviously identical to feedback factor β in the above presented schemes. But there isn't anything like β/α, neither at
this place nor anywhere in chapter 8 Feedback or 9 Frequency Response and Stability of Feedback Amplifiers.

Click to expand...

Actually f in the Gray/Meyer's book is β/α (I've shown this in my post, 13 April). I think you was misleaded with double usage of β symbol.

dedalus said:

Yes, it's incorrect to say that the terminology of control theory cannot be applied to the "analog ic design" domain, because actually the term ("feedback factor") is the same, BUT the things that this term represents are different. By the way, I have not said, that general rules and equations cannot be applied in the "analog ic design" domain.
I wanted to say, that there is no "commonly used feedback factor" as well as "correct definition of feedback factor". Every domain can have it's own "commonly used" parameters.

Click to expand...

Hi DEDALUS,

to clarify things and to come to a common understanding, I have one final request to you:
Can you please define the term "feedback factor" for an electronic circuit according to your understanding ? That also means: How can this parameter be measured/calculated/simulated?
Because I am interested in a general definition applicable to the domain you are speaking of, please don't use any circuit example and no block diagram representation.
(Here I repeat the definition I rely upon: The feedback factor is defined as a portion of the output signal of an amplifier which is fed back to the input of this amplifier. For measurement, calculation or simulation of this portion the input signal which is applied in normal operation has to be set to zero).
Thank you
LvW