---
title: "Summing amplifier using op-amp"
subject: "Fundamental of Electrical and Electronics Engineering"
chapter: 6
key: "semester_1/fundamental_of_electrical_and_electronics_engineering:summing-amplifier-op-amp"
source: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/summing-amplifier-op-amp
site: BCT Engineering
---

# Summing amplifier using op-amp

Fundamental of Electrical and Electronics Engineering · Chapter 6: Operational Amplifier and Oscillator

- Asked in: 2081 Chaitra (new course), 4 marks
- Video: [Summing Amplifiers - Op Amp Circuits](https://www.youtube.com/watch?v=p4FQyOwjhLA) — The Organic Chemistry Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/summing-amplifier-op-amp
- Whole chapter: https://www.bctengineer.com/md/semester_1/fundamental_of_electrical_and_electronics_engineering/chapter-6

## Question

Derive the expression of summing amplifier using OP amp.

## Short answer

The inverting summing amplifier uses virtual ground at the op‑amp input. Input currents sum, flowing through Rf: Vo = –Rf (V1/R1 + V2/R2 + V3/R3). Equal resistors give Vo = –(V1+V2+V3).

## Answer

A **summing amplifier** (adder) is an op-amp circuit whose output is proportional to the (weighted) sum of several input voltages. The common form is the **inverting summing amplifier**.

#### Circuit

```text
 V1 o--[R1]--+
             |        Rf
 V2 o--[R2]--+-----/\/\/\-----+
             |                |
 V3 o--[R3]--+   |\           |
             +---|-\          |
              V− |  >---------+--o Vo
           GND --|+/
                 |/
```

#### Assumptions (ideal op-amp)

1. V+ = 0 (grounded), and because of the virtual short V− = V+ = 0 (virtual ground).
2. No current enters the op-amp input.

#### Derivation

Input currents:

```text
I1 = (V1 − 0)/R1 = V1/R1
I2 = V2/R2
I3 = V3/R3
```

No current enters the (−) terminal, so by KCL all these currents flow through Rf:

```text
If = I1 + I2 + I3
```

The output voltage, with the inverting node at 0 V:

```text
0 − Vo = If × Rf
Vo     = −Rf (V1/R1 + V2/R2 + V3/R3)
```

**Special cases**

- **Equal resistors** R1 = R2 = R3 = Rf = R: **Vo = −(V1 + V2 + V3)** (inverting adder).
- R1 = R2 = R3 = R, Rf = R/3: **Vo = −(V1 + V2 + V3)/3** (averaging amplifier).
- Unequal resistors give a **scaling (weighted) summer**, Vo = −(a1V1 + a2V2 + a3V3) with ai = Rf/Ri.

**Example:** V1 = 1 V, V2 = 2 V, V3 = −0.5 V with all resistors 10 kΩ: Vo = −(1 + 2 − 0.5) = −2.5 V.

Because the inverting node is a virtual ground, the inputs do not interact with each other. Uses: audio mixers, digital-to-analog converters (weighted resistors), and adding a DC offset to a signal. A second inverting amplifier of gain −1 can be added to get a non-inverted sum.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why is the inverting node at 0 V in the summing amplifier?
   A. V+ is grounded and the virtual short makes V- = V+ = 0
   B. Rf is shorted to ground
   C. The inputs sum to zero
   D. The output is zero
   Answer: A. V+ = 0 and the virtual short forces V- to 0 V, a virtual ground. (Key point: Virtual ground)

2. What does KCL give at the inverting node?
   A. If = I1 + I2 + I3, as no current enters the op-amp
   B. If = I1 - I2 + I3
   C. If = 0
   D. If = (I1 + I2 + I3)/3
   Answer: A. All input currents Vi/Ri flow through Rf because the input draws no current. (Key point: Current summation)

3. What is the output voltage?
   A. Vo = Rf(V1/R1 + V2/R2 + V3/R3)
   B. Vo = -Rf(V1/R1 + V2/R2 + V3/R3)
   C. Vo = -(V1 + V2 + V3)/Rf
   D. Vo = (1 + Rf/R1)(V1+V2+V3)
   Answer: B. 0 - Vo = If Rf, so Vo = -Rf(V1/R1 + V2/R2 + V3/R3). (Key point: Output expression)

4. With R1 = R2 = R3 = R, what Rf makes the output -(V1+V2+V3)/3?
   A. R
   B. 3R
   C. R/3
   D. R/9
   Answer: C. Vo = -(Rf/R)(V1+V2+V3); a factor 1/3 needs Rf = R/3. (Key point: Averaging case)

## Related in Fundamental of Electrical and Electronics Engineering

- [Non-inverting op-amp voltage gain](https://www.bctengineer.com/question.md?key=semester_1%2Ffundamental_of_electrical_and_electronics_engineering%3Anon-inverting-amplifier-gain) — Fundamental of Electrical and Electronics Engineering, chapter 6

## Related in other subjects

- [Op-amp properties and summing amplifier](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-props-summer-v1-2v2-3v3) — Basic Electronics Engineering, chapter 4
- [Find coefficients of op-amp summing circuit](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Asumming-amplifier-coefficients) — Advanced Electronics, chapter 1
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- [Ideal op-amp properties; non-inverting gain](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-props-noninverting-gain) — Basic Electronics Engineering, chapter 4, asked 3×