A summing amplifier is an op amp circuit that combines several inputs and produces an output that is the weighted sum of the inputs.
The summing amplifier is a variation of the inverting amplifier.
Applying KCL at node \(a\) gives
\(\displaystyle i = i_1 + i_2 + i_3 = \frac{v_1 - v_a}{R_1} + \frac{v_2 - v_a}{R_2} + \frac{v_3 - v_a}{R_3}\)
\(\displaystyle i = \frac{v_a - v_o}{R_f}\)
\(\displaystyle \frac{v_a - v_o}{R_f} = \frac{v_1 - v_a}{R_1} + \frac{v_2 - v_a}{R_2} + \frac{v_3 - v_a}{R_3}\)
But \(v_a = 0\) for an ideal op amp, since the non-inverting terminal is grounded. Hence,
\(\displaystyle \frac{- v_o}{R_f} = \frac{v_1}{R_1} + \frac{v_2}{R_2} + \frac{v_3}{R_3}\)
\(\boxed{v_o = -\left(\frac{R_f}{R_1}v_1 + \frac{R_f}{R_2}v_2 + \frac{R_f}{R_3}v_3\right)}\)