For the op amp below

- Open-loop gain is \$A=2\times10^5\$
- Input resistance is \$R_i=2\,M\Omega\$
- Output resistance is \$R_o=50\,\Omega\$

I am asked to calculate the close-loop gain \$Vo/Vs\$ and find \$i_o\$ when \$V_S=1\$ .

(Schematic diagram above from "Fundamentals of Electric Circuits" by Charles Alexander and Matthew Sadiku.)

Then I've redrawn the circuit and defined the currents as shown in the figure.

^{simulate this circuit – Schematic created using CircuitLab}

Then according to those currents above I've applied KCL and also I've got these equations below.

\$I = I_1+I_2\qquad I_2=I_3+I_4\$

$$I_4=\frac{V_1}{5\,k\Omega}$$

$$I_3=\frac{V_1-V_o}{40\,k\Omega}$$

$$I_3+I_1=\frac{V_o}{20\,k\Omega}$$

$$A \times V_d=2\times10^5\times 2\,M\Omega\times I$$

$$I=\frac{V_S-V_1}{2\,M\Omega}$$

Then I've also written some equalities according to KVL in the closed loops but I was never able to find a relation between \$V_S\$ and \$V_o\$ or just ended up crosschecking the same equations. It all got tangled up.

- How would you solve this?

## Best Answer

Assuming this is a problem from a book and we can assume ideal parameters to the opamp where no specifications are otherwise given, then:

^{simulate this circuit – Schematic created using CircuitLab}Assuming the equivalent circuit at the top for the opamp, the nodal equations are:

$$\begin{align*} \frac{V_X}{R_1}+\frac{V_X}{R_2}+\frac{V_X}{R_{IN}}&=\frac{V_O}{R_1}+\frac{V_S}{R_{IN}}\\\\ \frac{V_O}{R_1}+\frac{V_O}{R_3}+\frac{V_O}{R_{OUT}}&=\frac{V_X}{R_1}+\frac{\left(V_S-V_X\right)\cdot A_\text{OL}}{R_{OUT}}\\\\\therefore\\\\V_X&\approx 0.999955\cdot V_S\\\\V_O&\approx 8.999593\cdot V_S \end{align*}$$

The script I used in sympy (worth getting) is:

Just nodal is enough here. The main thing is to figure out the opamp model from the specs you were given.