Single op-amp IA with mismatched resistances

An instrumentation amplifier is a precision differential amplifier with high input impedance, high CMRR and an accurately set gain. It is used to amplify small differential signals from transducers in the presence of large common-mode noise.

Characteristics

• High and balanced input impedance
• High CMRR (more than 100 dB)
• Accurate gain set by one resistor, typically 1 to 1000
• Low offset, drift and noise; low output impedance

Single op-amp IA with mismatched resistors

            R1           R2
 V1 ----/\/\/----+----/\/\/----+
                 |             |
                (-)\           |
                    >----------+--> Vo
                (+)/
            R3   |
 V2 ----/\/\/----+
                 |
                 R4
                 |
                GND

Output voltage (superposition):

V1 alone (V2 = 0) is an inverting amplifier: −(R2/R1)V1. V2 alone is a divider followed by a non-inverting amplifier: V+ = V2·R4/(R3 + R4), so the output is (1 + R2/R1)·R4/(R3 + R4)·V2.

Vo = (1 + R2/R1)·R4/(R3 + R4)·V2 − (R2/R1)·V1

Differential and common-mode gains. Put V1 = Vc − Vd/2 and V2 = Vc + Vd/2, where Vd = V2 − V1 and Vc = (V1 + V2)/2:

Vo = Ad·Vd + Acm·Vc

Ad  = (R1R4 + R2R3 + 2R2R4) / [2R1(R3 + R4)]
Acm = (R1R4 − R2R3) / [R1(R3 + R4)]
CMRR = |Ad/Acm|

With matching R4/R3 = R2/R1: Acm = 0, Ad = R2/R1, and Vo = (R2/R1)(V2 − V1). Any mismatch makes Acm ≠ 0 and limits the CMRR.

Input impedance:

• At the V2 input: the current flows through R3 and R4 to ground, so Rin2 = R3 + R4.
• At the V1 input: the inverting terminal sits at V+ (virtual short), so I1 = (V1 − V+)/R1. With V2 = 0, Rin1 = R1.
• Differential input resistance (matched, R3 = R1): Rid = 2R1.
• Common-mode input resistance (matched, both inputs tied): Ricm = (R1 + R2)/2.

The input impedance is only a few kΩ and the two inputs are unequal. This is the main weakness of the one op-amp IA, and the reason for the three op-amp IA with buffered inputs.