Non-inverting op-amp voltage gain

In a non-inverting amplifier the input signal is applied to the non-inverting (+) terminal, and part of the output is fed back to the inverting (−) terminal through the divider Rf and R1 (negative feedback). The output is in phase with the input.

Circuit

              Rf
        +---/\/\/\---+
        |            |
  R1    |   |\       |
 +-/\/\-+---|-\      |
 |      V−  |  >-----+---o Vo
GND   Vi o--|+/
        V+  |/

Assumptions (ideal op-amp)

1. Infinite open-loop gain, so V+ = V− (virtual short).
2. Infinite input resistance, so no current enters either input terminal.

Derivation

Because of the virtual short:

V− = V+ = Vi

No current enters the (−) terminal, so the same current I flows through Rf and R1. KCL at the inverting node:

(Vo − V−)/Rf = (V− − 0)/R1
(Vo − Vi)/Rf = Vi/R1
Vo − Vi      = Vi (Rf/R1)
Vo           = Vi (1 + Rf/R1)

Equivalently, Rf and R1 form a voltage divider: V− = Vo × R1/(R1 + Rf) = Vi.

Closed-loop voltage gain:

A_CL = Vo/Vi = 1 + Rf/R1

Points to note

• The gain is always ≥ 1 and positive (no phase inversion).
• Gain depends only on the external resistors, not on the op-amp's open-loop gain.
• Input resistance is very high (ideally infinite), so it does not load the source.
• Example: Rf = 9 kΩ, R1 = 1 kΩ gives A_CL = 1 + 9 = 10; Vi = 0.5 V gives Vo = 5 V.
• Special case: Rf = 0 or R1 = ∞ gives A_CL = 1, the voltage follower (buffer).

With finite open-loop gain A: A_CL = A/(1 + Aβ), β = R1/(R1 + Rf), which tends to 1/β = 1 + Rf/R1 when Aβ ≫ 1.