Voltage gain of non-inverting op-amp

A non-inverting amplifier uses an op-amp with negative feedback to give an output in phase with the input and a gain set by two resistors.

Circuit and derivation

              |\
 Vin o--------|+ \
              |   >------+------o Vo
         +----|- /       |
         |    |/        [Rf]
         |               |
         +---------------+
         |
        [R1]
         |
        GND

R1 is connected from the inverting input to ground and Rf from the output to the inverting input (negative feedback). The input Vin is applied directly to the non-inverting terminal.

Derivation (ideal op-amp)

• No current enters the op-amp inputs.
• Virtual short: V− = V+ = Vin.
• The voltage at the inverting node is set by the divider R1, Rf: V− = Vo × R1/(R1 + Rf).

Equating V− = Vin:

Vin = Vo R1/(R1 + Rf)

Av = Vo/Vin = 1 + Rf/R1

The gain is positive (the output is in phase with the input) and always at least 1. For Rf = 0 or R1 = ∞ (open), the circuit becomes a unity gain buffer (voltage follower), Av = 1.

Example: R1 = 1 kΩ, Rf = 9 kΩ gives Av = 1 + 9 = 10. The input resistance is very high (equal to the op-amp input resistance) and the output resistance is very small.