Closed loop gain of inverting stage

[houly]

Hello all,
I need help to understand an equation which permit to know the closed loop gain Avc for an inverting stage (between the first pole frequency and the second pole frequency) with considering Av (open loop gain), Rf (feedback resistor), R1 (input resistor) :

Avc=-Av*Rf/R1/√((Av)²+(1+Rf/R1)²)

I don't see how they could deduce this ...

 

[erikl]

For the frequency range between the first pole frequency and the second pole frequency you get a -20dB/decade decay, whereas your above equation doesn't depend on frequency, but produces a fixed Avc value for fixed values of Av, Rf & R1 : Avc = -Rf/R1 for Av » Rf/R1 . So it can't be correct -- at least for the considered frequency range.

 

[houly]

Hello Erikl and thanks for your help, well, I finally scan the book where I have found this expression. I don't know how they can write it

**broken link removed**

**broken link removed**

 

[LvW]

Avc=-Av*Rf/R1/√((Av)²+(1+Rf/R1)²)

Houly, where does this formula comes from?
Other than erikl, I recognize that Av is the open-loop gain that, of course, is frequency dependent.
However, I doubt that the formula is correct.
If Av is frequency-dependent (that means: A complexe function) why you express the denominator as a magnitude (not the numerator)? This would give a closed-loop gain with a pole frequency that is identical to the lower pole of the open-loop gain - and this, obviously, is wrong.

 

[houly]

LvW, this is my question, I don't know where does this expression comes from ! I can't explain how they deduce it (see the2.18 and 2.19)

 

[LvW]

I have written the above before reading the scanned text from houly, nevertheless, I cannot agree (as explained).
It would be interesting to learn about the source of the text (reference).

 

[houly]

oh ok... sorry.
I agree with you and Erikl and I'm suprised to find it in a old book :
"Handbook of operational amplifier circuit design" Stout and Kaufman 1976 McGraw-Hill ISBN 0-07-061797-X

 

[LvW]

You will find the correct formula in each reliable textbook and in many internet contributions .