Operational Amplifier: The Op-Amp Building Block

An amplifier with almost infinite gain

An operational amplifier is a differential amplifier with enormous open-loop voltage gain — on the order of one hundred thousand to a million — two inputs (inverting, marked minus, and non-inverting, marked plus) and one output. Left alone, without any external components, that gain is so large that it is essentially useless as a linear device: a fraction of a millivolt of difference between the two inputs drives the output to one supply rail or the other almost instantly. The op-amp only becomes tame, and genuinely useful, once you close a feedback path from the output back to the inverting input.

The two golden rules

With negative feedback in place, and assuming the op-amp is not saturated, two idealized rules describe the circuit almost exactly, and they are the entire toolkit for hand-analysing op-amp circuits:

Rule 1 — no current flows into either input
        (the inputs present essentially infinite impedance)

Rule 2 — the op-amp's output does whatever is needed
        to force the two inputs to the same voltage

Rule 2 only holds because of negative feedback: the output voltage is fed back to the inverting input through some external network, and because open-loop gain is so enormous, only a vanishingly small input difference is needed to produce any realistic output. Treat that difference as exactly zero and you have Rule 2. Rule 1 comes from the input stage being built from transistors whose gate or base draws negligible current at the input node.

The inverting amplifier and virtual ground

Ground the non-inverting input and feed the signal through a resistor Rin into the inverting input, with a feedback resistor Rf from the output back to that same node. By Rule 2 the inverting input sits at 0 volts — a virtual ground, held there by feedback rather than by a direct wire. By Rule 1, all the current arriving through Rin has nowhere to go but through Rf, since none of it can enter the op-amp. That single observation gives the whole transfer function:

Iin  = Vin / Rin          // current into the virtual-ground node
Iin  = -Vout / Rf         // the same current, leaving through Rf
Vout = -(Rf / Rin) * Vin  // inverting gain, set purely by resistor ratio

Notice that the gain depends only on the ratio of two resistors — not on the op-amp's own open-loop gain, its temperature drift, or manufacturing spread — as long as the open-loop gain is large enough for the golden rules to hold. That insensitivity to the amplifying device's own imperfections is the entire point of negative feedback, and it is why op-amp circuits are so predictable and so widely used.

Non-inverting and summing variants

Feed the signal into the non-inverting input instead, and put the same Rin/Rf feedback divider from output to ground on the inverting input. Now Rule 2 says the inverting node must sit at Vin, and the same current argument gives Vout = (1 + Rf/Rin) · Vin — positive gain, always at least 1, and very high input impedance since the signal sees the op-amp's input directly rather than through a resistor.

Add more input resistors converging on the same virtual-ground node of an inverting amplifier and Rule 1's current-summing argument extends directly: the output becomes a weighted sum of every input, each scaled by -Rf divided by its own input resistor. This summing amplifier is the building block of analogue mixers and of classic digital-to-analogue converters, where each input resistor is weighted in a binary sequence.

Where the ideal model breaks down

Real op-amps draw a small input bias current, have a finite gain-bandwidth product that limits how fast the output can respond, and cannot swing all the way to the supply rails. At high frequency the loop gain drops and the golden rules become progressively less accurate, which is why op-amp circuits are always specified with an operating frequency range in mind. None of this changes the reasoning above — it only bounds where it applies.

Frequently asked questions

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