# What is the derivation for the error due to opamp open loop gain in this case?

For an ideal opamp the unity gain buffer has a voltage gain of exactly 1; so Vin = Vout:

But if we neglect all non-idealities except the DC open loop gain Aol, how can we mathematically/analytically demonstrate the effect of the Aol to the error between the input and the output voltage? I have read once that if Aol=infinity then Vin = Vout but if Aol=10000 then Vin is not exactly equal to Vout. But I cannot derive myself why neither I can find a source in internet for derivation of this.

• I am really surprised - You did not "find a source in internet for derivation of this"? If I had time and motivation enough - I could give you at least 100 sources....how many minutes did you spend for searching? – LvW Mar 18 at 11:10
• I guess I was searching with wrong keywords. I should have googled "negative feedback" probably. – pnatk Mar 18 at 11:39

But I cannot derive myself why neither I can find a source in internet for derivation of this.

Consider this: -

At the point marked with the red X we can say: -

$$(V_{IN}-V_{OUT})\cdot A_{OL} = V_{OUT}$$

$$A_{OL}\cdot V_{IN} = V_{OUT}\cdot(1 + A_{OL})$$

$$\dfrac{V_{OUT}}{V_{IN}} = \dfrac{A_{OL}}{1+A_{OL}}$$

Can you take it from here and plug some numbers in?

• Yes this was exactly I was looking for. – pnatk Mar 18 at 10:35