---
title: "Non-inverting op-amp voltage gain"
subject: "Fundamental of Electrical and Electronics Engineering"
chapter: 6
key: "semester_1/fundamental_of_electrical_and_electronics_engineering:non-inverting-amplifier-gain"
source: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/non-inverting-amplifier-gain
site: BCT Engineering
---

# Non-inverting op-amp voltage gain

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

- Asked in: 2081 Baishakh (new course), 4 marks
- Video: [Non-inverting op-amp circuit](https://www.youtube.com/watch?v=_Ut-nQ535iE) — Khan Academy
- 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/non-inverting-amplifier-gain
- Whole chapter: https://www.bctengineer.com/md/semester_1/fundamental_of_electrical_and_electronics_engineering/chapter-6

## Question

Derive voltage gain of closed loop non-inverting op-amp configuration.

## Short answer

Virtual short: V− = V+ = V_i. Current in R_f and R_1: (V_o − V_i)/R_f = V_i/R_1. Thus V_o = V_i (1 + R_f/R_1). Closed‑loop voltage gain A_CL = 1 + R_f/R_1.

## Answer

In a **non-inverting amplifier** the input signal is applied to the non-inverting (+) terminal, and part of the output is fed back to the inverting (−) terminal through the divider Rf and R1 (negative feedback). The output is in phase with the input.

#### Circuit

```text
              Rf
        +---/\/\/\---+
        |            |
  R1    |   |\       |
 +-/\/\-+---|-\      |
 |      V−  |  >-----+---o Vo
GND   Vi o--|+/
        V+  |/
```

#### Assumptions (ideal op-amp)

1. Infinite open-loop gain, so V+ = V− (virtual short).
2. Infinite input resistance, so no current enters either input terminal.

#### Derivation

Because of the virtual short:

```text
V− = V+ = Vi
```

No current enters the (−) terminal, so the same current I flows through Rf and R1. KCL at the inverting node:

```text
(Vo − V−)/Rf = (V− − 0)/R1
(Vo − Vi)/Rf = Vi/R1
Vo − Vi      = Vi (Rf/R1)
Vo           = Vi (1 + Rf/R1)
```

Equivalently, Rf and R1 form a voltage divider: V− = Vo × R1/(R1 + Rf) = Vi.

**Closed-loop voltage gain:**

```text
A_CL = Vo/Vi = 1 + Rf/R1
```

#### Points to note

- The gain is always ≥ 1 and positive (no phase inversion).
- Gain depends only on the external resistors, not on the op-amp's open-loop gain.
- Input resistance is very high (ideally infinite), so it does not load the source.
- **Example:** Rf = 9 kΩ, R1 = 1 kΩ gives A_CL = 1 + 9 = 10; Vi = 0.5 V gives Vo = 5 V.
- Special case: Rf = 0 or R1 = ∞ gives A_CL = 1, the **voltage follower** (buffer).

With finite open-loop gain A: A_CL = A/(1 + Aβ), β = R1/(R1 + Rf), which tends to 1/β = 1 + Rf/R1 when Aβ ≫ 1.

## Check yourself

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

1. In the ideal non-inverting amplifier derivation, why is V- equal to Vi?
   A. Because the op-amp has infinite open-loop gain, so V+ = V- (virtual short) and V+ = Vi
   B. Because the feedback resistor Rf is zero
   C. Because no current flows through R1
   D. Because the output is grounded
   Answer: A. Infinite open-loop gain forces the two inputs to the same voltage. Since the input signal is on V+, V- = Vi. (Key point: Virtual short)

2. Which equation comes from KCL at the inverting node?
   A. (Vo - Vi)/Rf = Vi/R1
   B. (Vo + Vi)/Rf = Vi/R1
   C. Vo/Rf = Vi/R1
   D. (Vo - Vi)/R1 = Vi/Rf
   Answer: A. No current enters the (-) terminal, so the current through Rf, (Vo - V-)/Rf, equals the current through R1 to ground, V-/R1. (Key point: KCL at inverting node)

3. What is the closed-loop voltage gain of the non-inverting amplifier?
   A. -Rf/R1
   B. Rf/R1
   C. 1 + Rf/R1
   D. 1 - Rf/R1
   Answer: C. Solving the KCL equation gives Vo = Vi(1 + Rf/R1), so the gain is positive and at least 1. (Key point: Closed-loop gain)

4. Which condition turns the non-inverting amplifier into a voltage follower with gain 1?
   A. Rf = R1
   B. Rf = 0 or R1 = infinity
   C. Rf = infinity
   D. R1 = 0
   Answer: B. With Rf/R1 = 0 the gain 1 + Rf/R1 becomes 1; this is the buffer. (Key point: Voltage follower)

## Related in Fundamental of Electrical and Electronics Engineering

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

## Related in other subjects

- [Voltage gain of non-inverting op-amp](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-noninverting-gain) — Basic Electronics Engineering, chapter 4, asked 2×
- [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×
- [Ideal op-amp properties; inverting gain](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Aopamp-props-inverting-gain) — Basic Electronics Engineering, chapter 4, asked 3×
- [Non-inverting: zero at -4, pole at -8, HF gain 2](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Anoninverting-zero-4-pole-8) — Filter Design, chapter 6, asked 4×