What is the transfer function of a difference amplifier with resistors R1, R2, R3, R4 and the usual matching condition?

What is the transfer function of a difference amplifier with resistors R1, R2, R3, R4 and the usual matching condition?

• A. Vout = (R2/R1) (V2 − V1)
• B. Vout = (R1/R2) (V2 − V1)
• C. Vout = (R2/R1) (V1 − V2)
• D. Vout = (R4/R3) (V2 − V1)

Answer: (A)

Difference amplifiers subtract two input voltages with a gain set by resistor ratios. In the standard four-resistor configuration, the inverting input gets V1 through R1 and feedback from the output through R2; the non-inverting input is fed from V2 via R3 and to ground via R4. With an ideal op-amp, the two input voltages are equal (virtual short), so V+ = V−. The voltage at the non-inverting input is the divider V+ = V2 · [R4/(R3+R4)]. Using the inverting-node current relation (currents leaving the node sum to zero) and substituting V− = V+, you get Vout = - (R2/R1) V1 + [(R1+R2)/R1] · V+. Substituting V+ and simplifying shows that, to make the output depend symmetrically on V2 and V1, the resistor ratios must match: R2/R1 = R4/R3. When this matching condition holds, the V2 term becomes (R2/R1) V2, giving Vout = (R2/R1) (V2 − V1). So the transfer function is Vout = (R2/R1) (V2 − V1).

Difference amplifiers subtract two input voltages with a gain set by resistor ratios. In the standard four-resistor configuration, the inverting input gets V1 through R1 and feedback from the output through R2; the non-inverting input is fed from V2 via R3 and to ground via R4. With an ideal op-amp, the two input voltages are equal (virtual short), so V+ = V−. The voltage at the non-inverting input is the divider V+ = V2 · [R4/(R3+R4)]. Using the inverting-node current relation (currents leaving the node sum to zero) and substituting V− = V+, you get Vout = - (R2/R1) V1 + [(R1+R2)/R1] · V+. Substituting V+ and simplifying shows that, to make the output depend symmetrically on V2 and V1, the resistor ratios must match: R2/R1 = R4/R3. When this matching condition holds, the V2 term becomes (R2/R1) V2, giving Vout = (R2/R1) (V2 − V1). So the transfer function is Vout = (R2/R1) (V2 − V1).