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Negative Feedback and Opamps

We will now analyse the opamp in a negative feedback circuit, and derive a virtual model for the opamp which is valid only when operating in negative feedback, Figure 10. The input is v_in=v₊. The negative feedback connection is the one from the opamp output to the input through resistor R₂.

  

Figure 10: Opamp in negative feedback circuit.

Let's combine the model and negative feedback circuit, Figure 11.

  

Figure 11: Opamp in negative feedback circuit with opamp model.

KCL at node (-) gives, using the model (6) and the circuit,

$$\frac{\overline{v}_{in}-v_-}{R_{in}} + \frac{0-v_-}{R_1} + \frac{v_{out}-v_-}{R_2} = 0$$

and solving gives

$$v_- = \frac{R_1}{R_1+R_2}v_{out} + (\frac{R_1 R_2}{R_1+R_2})\frac{\overline{v}_{in}-v_-}{R_{in}} .$$

Next, KVL gives

$$v_{out} = A(\overline{v}_{in}-v_-) - i_{out} R_{out} .$$

We will now derive approximations. Since $R_{in} \approx \infty$, we get $1/R_{in} \approx 0$, and hence

$$v_- \approx \frac{R_1}{R_1+R_2} v_{out}$$

(so K=R₁/(R₁+R₂)). Since $R_{out} \approx 0$, we get also

$$v_{out} \approx A(\overline{v}_{in}-v_-)$$

Combining, we get

$$v_{out} \approx A\overline{v}_{in}-A \frac{R_1}{R_1+R_2}v_{out}$$

and solving gives

$$v_{out} \approx \frac{A}{1+ A\frac{R_1}{R_1+R_2}} \overline{v}_{in}$$

Now since $A \approx \infty$, we get, approximately,

$$v_{out} = (1+\frac{R_2}{R_1})\overline{v}_{in} .$$ (8)

This is the main formula for this negative feedback circuit, since it gives the gain as

$$A'= (1+\frac{R_2}{R_1})$$ (9)

Thus the negative feedback circuit consisting of the opamp and two resistors is an amplifier with gain A'.

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