operational amplifier question

[emperror123]

i have a question on this analog electronics

i would like to ask, if we would like to get expression on Vout/ Vin as the attachment show

which they ask V2/Vs in terms of k and Ri

which formula can be form for that

i only get a formula

which is V2/Vs = - R2/R1, but i cant get on what they request


Circuit Diagram

 

[Jony130]

You can use nodal analysis to find voltage on node A (Vi)
And if you know VA then V2 = -K * VA
So we can write
[LATEX]\frac{ Vs - V_A}{R1} = \frac{V_A}{Ri} + \frac{V_A - ( -K * V_A) } { R2}[/LATEX]

 

[emperror123]

thxx you so much

what the value to put in if K and ri is infinity

 

[MrAl]

Hello,


After you analyze the circuit you get something like this:
V2=-(k*Ri*Vs*R2)/(R1*R2+Ri*R2+k*Ri*R1+Ri*R1)

and to find out what happens when k gets infinite you can take the limit as k approaches plus infinity:
limit[k -->+inf](-(k*Ri*Vs*R2)/(R1*R2+Ri*R2+k*Ri*R1+Ri*R1))

and we get:
V2=-Vs*R2/R1

and note that now the response no longer depends on Ri as it did when k was finite. This is one of the main points of an op amp where we use a large gain to counter the effects of something else to get better accuracy overall. In practice k is not infinite but can be very very high like 100,000 or 1 million.

 

[emperror123]

Ri is infinity?

 

[MrAl]

Hi,

Ri can be infinity, but it doesnt have to be. That's the whole point of the large gain. The only value it can not be is zero.

 

[Jony130]

For Ri =∞

[LATEX]\frac{Vout}{Vin}=-\frac{K*R2}{R1 + K*R1 + R2}=\frac{K}{1+( \frac{R1}{R2}*(1+K))}[/LATEX]

 

[emperror123]

if k is infinity

then v2/Vs = -(Ri*R2)/(R1*R2+Ri*R2+2*Ri*R1)

if Ri is infinity


V2/Vs = - (K*R2)/(R1+K*R1+R2)

if k and Ri is infinity

then
V2/Vs = -R2/R1

 

[Jony130]

No, the only think you need to get this result :
V2/Vs = -R2/R1 is K =∞
Ri doesnt have to be infinity.

 

[emperror123]

ok now i understand wat u have been said, thxx all
problem solved

 

[emperror123]

how do i analyze the effect of overall gain?

 

[Jony130]

Hmm, don't understand

K is open loop gain

And close loop gain (overall gain) is equal

[LATEX]\frac{Vout}{Vin}=-\frac{K*R2}{R1 + K*R1 + R2}=\frac{K}{1+( \frac{R1}{R2}*(1+K))}[/LATEX]

For Ri =∞

 

[emperror123]

for example
when i do Ri = 1k and k = 100
i get the gain about -4.505

when i do Ri = infinity and k = 10
i get about -0.01

then like when both is infinity
it get about -5

so it ask to analyze the effect of the above gain value
how do i explain on the above 3