Closed Loop OPAMP Inverting Amplifier Formula Derivation and Example

Conditions:

1. Voltage at non inverting input (+) = voltage at inverting input (-).
2. There is no current flow between positive and negative input.

There is no current flow between positive and negative. So,

$ I_i = I_f $

It is $ v_i - 0 $, because voltage at inverting and non inverting inputs are same. Since the positive terminal is connected to the ground, both positive and negative pins have $ 0 volts $. So on the left side of $ R_i $ voltage is $ v_i $ and on right side it is $ 0 $. The voltage difference across the resisitor $ R_i $ is $ \Delta v = v_i - 0 $. Using the formula $ V = IR $,

$ I_i = \frac{v_i - 0}{R_i} $

Similarly,

$ I_f = \frac{0 - v_o}{R_f} $

Finally the formula for the output voltage,

\begin{align} I_i & = I_f \\ \\ \frac{v_i - 0}{R_i} & = \frac{0 - v_o}{R_f} \\ \\ \frac{v_i}{R_i} & = \frac{- v_o}{R_f} \\ \\ v_o & = - \frac{R_f}{R_i} v_i \end{align}

An example circuit with solution:

Here, I have shown the answer in the picture. Given, $ v_i = 2 V, R_i = 2 k \mathrm{\Omega}, R_f = 10 k \mathrm{\Omega} $ task is to find out $ v_o $. Using the formula above we get,

\begin{align} v_o & = - \frac{R_f}{R_i} v_i \\ \\ & = - \frac{10 k \mathrm{\Omega}}{2 k \mathrm{\Omega}} 2 V \\ \\ & = -10V \end{align} So, the output is opposite sign of the input voltage for inverting amplifier.