What is the output expression for an inverting summing amplifier with multiple inputs V1, V2, … using input resistors R1, R2, … and a feedback resistor Rf?

What is the output expression for an inverting summing amplifier with multiple inputs V1, V2, … using input resistors R1, R2, … and a feedback resistor Rf?

• A. Vout = -Rf (V1/R1 + V2/R2 + …)
• B. Vout = - (V1/R1 + V2/R2 + …) / Rf
• C. Vout = - V1/R1 - V2/R2 - …
• D. Vout = -Rf (V1/R1 + V2/R2 + …)/?

Answer: (A)

The inverting summing amplifier uses negative feedback to hold the inverting input at a virtual ground (approximately 0 V when the noninverting input is grounded). The currents from each input flow through their respective input resistors into that summing node. By Kirchhoff’s current law, the sum of those input currents must be exactly the current through the feedback resistor. With the node at ~0 V, each input current is Vi/Ri, and the current through the feedback path is Vout/Rf. Setting up the currents gives V1/R1 + V2/R2 + … + Vout/Rf = 0, which leads to Vout = -Rf (V1/R1 + V2/R2 + …). This shows how the output scales with the sum of the input contributions and why the sign is negative. The other forms either misplace the Rf factor (dividing by Rf instead of multiplying) or omit the feedback’s role, giving the wrong relationship.

The inverting summing amplifier uses negative feedback to hold the inverting input at a virtual ground (approximately 0 V when the noninverting input is grounded). The currents from each input flow through their respective input resistors into that summing node. By Kirchhoff’s current law, the sum of those input currents must be exactly the current through the feedback resistor. With the node at ~0 V, each input current is Vi/Ri, and the current through the feedback path is Vout/Rf. Setting up the currents gives V1/R1 + V2/R2 + … + Vout/Rf = 0, which leads to Vout = -Rf (V1/R1 + V2/R2 + …). This shows how the output scales with the sum of the input contributions and why the sign is negative. The other forms either misplace the Rf factor (dividing by Rf instead of multiplying) or omit the feedback’s role, giving the wrong relationship.