Two rules do almost all the work

An op amp is a high-gain differential amplifier in a small package, typically a DIP-8 with pin 7 as the positive supply and pin 4 as the negative supply. It amplifies the difference vd = v2 - v1 between its inputs.

For an ideal op amp with negative feedback, two consequences carry the whole analysis:

• No current enters either input terminal, because the input resistance is taken as infinite.
• The two input voltages are equal, v1 = v2, the virtual short. It follows from infinite open-loop gain: a finite output needs a vanishing difference at the input.

Apply KCL at the inverting node with those two rules and every standard configuration falls out in a line or two.

The standard configurations

• Inverting: input through R1 to the inverting terminal, feedback Rf. Gain Av = -Rf/R1. It amplifies and flips the sign.
• Non-inverting: input to the plus terminal. Gain Av = 1 + Rf/R1, always at least one, never inverting.
• Voltage follower: the non-inverting case with Rf = 0, so vo = vi. Gain of one, used as a buffer so one stage stops loading the next.
• Summing: several inputs each through their own resistor into the inverting node, vo = -(Rf/R1·v1 + Rf/R2·v2 + …).
• Difference: amplifies v2 - v1. When R1 = R2 and R3 = R4 it becomes a plain subtractor, vo = v2 - v1.
• Cascade: stages in series, and the overall gain is the product of the stage gains.

Watch for

• Feedback must go to the inverting terminal. Positive feedback saturates the amplifier instead of amplifying.
• The output cannot exceed the supply rails, so a gain that looks fine on paper can clip in practice.
• In a difference amplifier the resistor ratios must match, or common-mode signal leaks into the output.