Instrumentation amplifier and its gain

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

 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.