Consider the summing amplifier in Figure 9.14 (a). Assume the op-amp has a finite open-loop differential gain \(A_{o d}\). Using the principle of superposition, show that the output voltage is given by

\[v_{O}=\frac{-1}{1+\frac{\left(1+R_{F} / R_{P}\right)}{A_{o d}}}\left[\frac{R_{F}}{R_{1}} v_{I 1}+\frac{R_{F}}{R_{2}} v_{I 2}+\frac{R_{F}}{R_{3}} v_{I 3}\right]\]

where \(R_{P}=R_{1}\left\|R_{2}\right\| R_{3}\). Demonstrate how the expression will change if more or fewer inputs are included.

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RF 0 R v=0 V12 R V13 R3 (a) Virtual ground RF Un ww R VII- R R 2 = (0. www R3 V13 0- = ww 3=0- (b) vo Figure 9.14 (a) Summing op-amp amplifier circuit and (b) currents and voltages in the summing amplifier