Instrumentation Amplifier Gain
G = (1 + 2R1/Rg) × (R3/R2)
Quick answer
This calculator solves G = (1 + 2R₁/Rg) × (R₃/R₂) for Gain. With R1 (input stage) 10 kΩ, Gain Resistor 0.47 kΩ, R3 (diff stage) 22 kΩ, R2 (diff stage) 47 kΩ, the result is 20.39. It solves in 2 directions, so you can enter the values you have and get the one you need.
Calculator
Formula
Description
The three-op-amp instrumentation amplifier (INA) is the standard topology for precision differential measurement. The first stage (two op-amps) provides adjustable gain set by a single resistor Rg, with the formula G₁ = 1 + 2R₁/Rg. The second stage (difference amplifier) provides fixed gain G₂ = R₃/R₂, typically set to unity. The total gain is G₁ × G₂. The key advantage is that the gain is set by a single resistor (Rg) while maintaining excellent CMRR, high input impedance, and low drift. Dedicated INA ICs (INA128, AD620, AD8421) integrate all three op-amps with laser-trimmed resistors.
Variables
- G — Total voltage gain (V/V)
- R₁ — Input stage resistors (Ω, both equal)
- Rg — Gain-setting resistor (Ω)
- R₃ — Difference stage feedback resistor (Ω)
- R₂ — Difference stage input resistor (Ω)
Practical Notes
For INA ICs with R₃/R₂ = 1 (unity second stage), G = 1 + 2R₁/Rg. Common IC gain formulas: INA128/INA129: G = 1 + 50kΩ/Rg, AD620: G = 1 + 49.4kΩ/Rg. CMRR improves with higher gain and better resistor matching. For Rg = ∞ (open), G = 1 (unity gain). For maximum CMRR, the R₃/R₂ ratio must be precisely matched between the two sides of the difference amplifier — even 0.1% mismatch limits CMRR to about 66 dB.
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| G | Gain | — | — |
| R1 | R1 (input stage) | kΩ | mΩ, Ω, kΩ, MΩ |
| Rg | Gain Resistor | Ω | mΩ, Ω, kΩ, MΩ |
| R3 | R3 (diff stage) | kΩ | mΩ, Ω, kΩ, MΩ |
| R2 | R2 (diff stage) | kΩ | mΩ, Ω, kΩ, MΩ |
Step by step
- 1. What you know: R1 (input stage) 10 kΩ, Gain Resistor 0.47 kΩ, R3 (diff stage) 22 kΩ, R2 (diff stage) 47 kΩ
- 2. The relation: G = (1 + 2R₁/Rg) × (R₃/R₂)
- 3. Substitute: R1 = 10 kΩ, Rg = 0.47 kΩ, R3 = 22 kΩ, R2 = 47 kΩ
- 4. Result: Gain = 20.39
Worked examples
Minimum range
Given: R1 (input stage) 100 Ω, Gain Resistor 1 Ω, R3 (diff stage) 0.22 kΩ, R2 (diff stage) 0.47 kΩ
Result: Gain = 94.09
Low range
Given: R1 (input stage) 1 kΩ, Gain Resistor 10 Ω, R3 (diff stage) 2.2 kΩ, R2 (diff stage) 4.7 kΩ
Result: Gain = 94.09
Typical
Given: R1 (input stage) 10 kΩ, Gain Resistor 0.47 kΩ, R3 (diff stage) 22 kΩ, R2 (diff stage) 47 kΩ
Result: Gain = 20.39
High range
Given: R1 (input stage) 100 kΩ, Gain Resistor 4.7 kΩ, R3 (diff stage) 0.22 MΩ, R2 (diff stage) 0.47 MΩ
Result: Gain = 20.39
Maximum range
Given: R1 (input stage) 1 MΩ, Gain Resistor 47 kΩ, R3 (diff stage) 2.2 MΩ, R2 (diff stage) 4.7 MΩ
Result: Gain = 20.39
Solving for each variable
Solve for Gain
R1 (input stage) 100 Ω, Gain Resistor 1 Ω, R3 (diff stage) 0.22 kΩ, R2 (diff stage) 0.47 kΩ → 94.09
Solve for Gain Resistor
Gain 25, R1 (input stage) 100 kΩ, R3 (diff stage) 0.22 MΩ, R2 (diff stage) 0.47 MΩ → 3.816 kΩ
How Gain changes with R1 (input stage)
| R1 (input stage) (input) | Gain (result) |
|---|---|
| 1 kΩ | 2.46 |
| 2.5 kΩ | 5.448 |
| 5 kΩ | 10.43 |
| 7.5 kΩ | 15.41 |
| 10 kΩ | 20.39 |
| 15 kΩ | 30.35 |
| 20 kΩ | 40.31 |
| 40 kΩ | 80.14 |
| 70 kΩ | 139.9 |
| 100 kΩ | 199.7 |
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<p><a href="https://vbatpower.com/formulas/opamp-instrumentation/">Instrumentation Amplifier Gain</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Instrumentation Amplifier Gain?
Use G = (1 + 2R₁/Rg) × (R₃/R₂). With R1 (input stage) 10 kΩ, Gain Resistor 0.47 kΩ, R3 (diff stage) 22 kΩ, R2 (diff stage) 47 kΩ, Gain works out to 20.39.
What is the formula for Instrumentation Amplifier Gain?
The formula is G = (1 + 2R₁/Rg) × (R₃/R₂). The three-op-amp instrumentation amplifier (INA) is the standard topology for precision differential measurement.
How do you find Gain Resistor when you know the other values?
Rearrange for Gain Resistor: with Gain 25, R1 (input stage) 100 kΩ, R3 (diff stage) 0.22 MΩ, R2 (diff stage) 0.47 MΩ, Gain Resistor is 3.816 kΩ.
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