---
title: "Instrumentation amplifier and its gain"
subject: "Instrumentation I"
chapter: 4
key: "semester_4/instrumentation_i:instrumentation-amplifier-gain"
source: https://www.bctengineer.com/q/semester_4/instrumentation_i/instrumentation-amplifier-gain
site: BCT Engineering
---

# Instrumentation amplifier and its gain

Instrumentation I · Chapter 4: Electrical Signal Processing and transmission

- Asked in: year not given, 6 marks
- Video: [Instrumentation Amplifier using OpAmp (Basics, Circuit & Output Equation Derivation) Explained](https://www.youtube.com/watch?v=-LuqQec4YRk) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/instrumentation_i/instrumentation-amplifier-gain
- Whole chapter: https://www.bctengineer.com/md/semester_4/instrumentation_i/chapter-4

## Question

What is an Instrumentation amplifier? Derive the expression for its gain.

## Short answer

An instrumentation amplifier is a high-precision differential amplifier with high input impedance, high CMRR, and gain set by a single resistor. The three-op-amp circuit yields gain = (R3/R2)(1 + 2R1/Rg) by amplifying the differential input in the first stage and converting to single-ended in the second.

## Answer

An **instrumentation amplifier (IA)** is a high-precision differential amplifier that amplifies the small difference between two inputs and rejects the common-mode signal. It has very high input impedance, a high and stable gain set by one resistor, very high CMRR, and low offset and drift. It is used with transducers such as strain-gauge bridges, thermocouples and ECG electrodes.

#### Three op-amp circuit
```text
 V1 ---------(+)\
                 A1 >----+------ Vo1 --[R2]--+----[R3]----+
       +-----(-)/        |                   |            |
       |                 |                   +---(-)\     |
       +------[R1]-------+                          A3 >--+--- Vo
       |                                     +---(+)/
      [Rg]                                   |
       |                                     |
       +------[R1]-------+                   |
       |                 |                   |
       +-----(-)\        |                   |
                 A2 >----+------ Vo2 --[R2]--+
 V2 ---------(+)/                            |
                                            [R3]
                                             |
                                            gnd
```
(Input stage: A1 and A2, each with R1 in its feedback path and Rg between their inverting inputs. Output stage: A3 as a difference amplifier with R2 input resistors and R3 feedback and ground resistors.)

#### Gain derivation
**Input stage.** By the virtual short, the voltage at the inverting input of A1 is V1, and that of A2 is V2. The current through Rg is

I = (V1 − V2)/Rg

The same current flows through both R1 resistors (the op-amp inputs draw no current), so

Vo1 − Vo2 = I(R1 + Rg + R1) = (V1 − V2)(1 + 2R1/Rg)

**Output stage.** A3 is a difference amplifier with equal resistor ratios (R2 in each input path, R3 for feedback and to ground):

Vo = (R3/R2)(Vo2 − Vo1)

Combining these, the overall result is

**Vo = (R3/R2) (1 + 2R1/Rg) (V2 − V1)**

(the sign depends on how the inputs are labelled; the magnitude is the gain).

- **Gain = (R3/R2)(1 + 2R1/Rg)**; it is varied by changing Rg alone.
- The input stage amplifies only the differential signal; the common-mode signal passes with unity gain (it produces no drop across Rg), so the CMRR is increased by the gain of the first stage.
- Example: R1 = 10 kΩ, Rg = 1 kΩ, R2 = R3: gain = 1 + 20 = 21.

## Check yourself

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

1. What does an instrumentation amplifier do?
   A. Amplifies the sum of two inputs
   B. Amplifies the small difference between inputs and rejects the common-mode signal
   C. Integrates its input
   D. Has low input impedance
   Answer: B. It is a high-precision differential amplifier with very high input impedance and CMRR. (Key point: IA features)

2. The gain magnitude of the three op-amp IA is:
   A. (R3/R2)(1 + R1/Rg)
   B. (R2/R3)(2R1/Rg)
   C. (R3/R2)(1 + 2R1/Rg)
   D. 1 + 2R1/R3
   Answer: C. Input stage gives (1 + 2R1/Rg) and the difference stage gives R3/R2. (Key point: Gain expression)

3. With R1 = 10 kΩ, Rg = 1 kΩ and R2 = R3, the gain is:
   A. 11
   B. 20
   C. 41
   D. 21
   Answer: D. Gain = 1 + 2·10/1 = 21. (Key point: Gain setting)

## Related in Instrumentation I

- [Piezoelectric transducer sensitivity and output voltage](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Apiezoelectric-transducer-output-voltage) — Instrumentation I, chapter 3
- [Single-phase electrodynamometer wattmeter](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Aelectrodynamometer-wattmeter) — Instrumentation I, chapter 7
- [Gauge factor of resistance strain gauge](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Astrain-gauge-factor) — Instrumentation I, chapter 3

## Related in other subjects

- [Common collector amplifier Ri, Ro and gain](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Acc-amplifier-derivation) — Electronics Devices and Circuit, chapter 2, asked 4×
- [Short note: Instrumentation amplifier](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Ashort-note-instrumentation-amplifier) — Advanced Electronics, chapter 4
- [Active load doubles differential amplifier gain](https://www.bctengineer.com/question.md?key=semester_4%2Fadvanced_electronics%3Aactive-load-gain-twice-passive) — Advanced Electronics, chapter 1, asked 2×
- [CE amplifier input resistance and overall gain](https://www.bctengineer.com/question.md?key=semester_3%2Felectronics_devices_and_circuit%3Ace-input-resistance-gain-680k) — Electronics Devices and Circuit, chapter 2