Physics · Electronics · A2 Level

The Ideal Operational Amplifier

A single small chip that amplifies the tiny difference between two input voltages by a colossal factor is the workhorse behind an enormous range of practical circuits, from audio amplifiers to precision instrumentation. Before any of those circuits can make sense, you need a clear picture of what the device does in its simplest, unmodified form.

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Terminals and connections

The operational amplifier, almost always abbreviated to op-amp, is a high-gain electronic amplifier built as a single integrated circuit, with two inputs and one output. It is the fundamental building block for every op-amp circuit covered in the rest of this topic, so getting a clear picture of its basic behaviour now pays off throughout everything that follows.

An op-amp has five terminals that matter for this course, falling into two distinct groups:

Power supply connections. Two terminals, typically labelled $+V_S$ and $-V_S$ (or a positive supply and a ground/negative supply), connect the chip to the external power source it needs simply to operate. These supply rails set the absolute limits within which the output voltage can vary, but they do not themselves carry any signal information.
Signal connections. Three terminals carry the actual information the op-amp is processing: the non-inverting input ($V_+$), the inverting input ($V_-$), and the single output ($V_{out}$).

It is worth being explicit that the "+" and "−" labels on the two inputs do not mean one is a positive terminal and the other negative in the way a battery's terminals are — both inputs can sit at any voltage within the supply range. The labels instead describe how each input affects the sign of the output: a rise in $V_+$ pushes the output in the positive direction, while a rise in $V_-$ pushes it in the negative direction.

Operational amplifier triangle symbol with labelled terminals — The standard op-amp symbol: a triangle with the non-inverting input ($V_+$) and inverting input ($V_-$) on the flat side, the output ($V_{out}$) at the point, and the two supply-rail connections shown separately.

The idealised model: infinite gain, infinite input resistance

Throughout this module, unless stated otherwise, the op-amp is treated using a simplified, idealised model that makes circuit analysis far more tractable. This idealisation is flagged explicitly here because a later lesson in this topic, "Real Operational Amplifiers", returns to it and describes the ways an actual physical device falls short of the ideal.

The idealised model makes two key assumptions:

Infinite open-loop gain. The open-loop gain, $A_{OL}$, is the factor by which the op-amp amplifies the difference between its two inputs when there is no feedback connection from the output back to either input — "open-loop" simply means the loop from output back to input is open, i.e. disconnected. In the ideal model, $A_{OL}$ is taken to be infinite: the output responds to an arbitrarily small difference between $V_+$ and $V_-$.
Infinite input resistance. The two input terminals are treated as drawing no current at all, however they are connected. No real current ever flows into either input.

With these assumptions, the open-loop behaviour of the op-amp is summarised by a single defining relationship:

$$V_{out} = A_{OL}(V_+ - V_-)$$

This says the output voltage is the open-loop gain multiplied by the difference between the non-inverting and inverting input voltages. Feedback circuits — where part of the output is fed back into an input to control the overall gain precisely — are the subject of the next topic; for now, every circuit in this lesson is open-loop, meaning this equation applies directly with no feedback path present.

Worked exampleWhy the open-loop output can't sit in between

An op-amp is powered from supply rails of $+9\ \text{V}$ and $-9\ \text{V}$, and has a (large but finite, realistic) open-loop gain of $A_{OL} = 2\times10^{5}$.

Real open-loop gains are typically in the range $10^4$–$10^6$ — enormous, even though not literally infinite.

Suppose $V_+ - V_- = 1\ \text{mV} = 1\times10^{-3}\ \text{V}$, a very small but non-zero difference.

This is well within the kind of tiny input difference that might occur from noise or a genuine small signal.

$V_{out} = A_{OL}(V_+ - V_-) = (2\times10^{5})(1\times10^{-3}) = 200\ \text{V}$

But the supply is only $\pm9\ \text{V}$ — the op-amp physically cannot produce an output of 200 V.

The output instead saturates: it is driven to (and stuck at) the positive supply rail, close to $+9\ \text{V}$, since that's the largest voltage the device can physically deliver.

The output cannot exceed the supply voltage that powers the chip, however large $A_{OL}(V_+-V_-)$ works out to be on paper.

Only an input difference smaller than about $9/(2\times10^{5}) = 45\ \mu\text{V}$ would keep the (finite-gain) output from hitting a rail. In the ideal, infinite-gain model, that threshold shrinks to exactly zero.

So in the open-loop configuration, the output essentially only ever takes one of two values — close to $+V_S$ or close to $-V_S$ — never a smooth in-between value, for any input difference that isn't vanishingly small.

Shown ✓

Use as a comparator

This all-or-nothing behaviour is not a flaw to be avoided — used deliberately, it makes the open-loop op-amp directly useful as a comparator: a circuit that compares two voltages and reports, via a clear high or low output, which one is larger.

Connect one voltage to $V_+$ and the other to $V_-$, with no feedback path. Whenever $V_+ > V_-$, the output swings to (and saturates at) very close to the positive supply rail; whenever $V_+ < V_-$, it swings to very close to the negative supply rail. The output switches sharply between these two states as soon as the relative sizes of $V_+$ and $V_-$ flip, giving an unambiguous digital-like signal indicating which input is currently the larger. This comparator behaviour requires no feedback whatsoever — it is a direct, open-loop use of the device exactly as described by $V_{out} = A_{OL}(V_+-V_-)$.

Open-loop comparator vs. closed-loop amplifier

Keep the comparator use of this lesson clearly separate from the feedback-controlled amplifier circuits coming up in the next topic. Used open-loop, as here, the op-amp's enormous gain makes it act as a switch — comparing two voltages and driving the output to one of two extremes. Once a feedback path is added from the output back to an input (closed-loop operation), the same device instead becomes a precise, controllable amplifier capable of producing a smooth, proportional output. Same chip, same defining equation — two very different jobs depending on whether feedback is present.

Key takeaways

An op-amp has power supply connections ($+V_S$, $-V_S$), which don't carry signal, and signal connections: non-inverting input $V_+$, inverting input $V_-$, and a single output.
The idealised model used throughout this module assumes infinite open-loop gain $A_{OL}$ and infinite input resistance (no current drawn at either input).
The open-loop defining relationship is $V_{out} = A_{OL}(V_+ - V_-)$, valid when there is no feedback path from output to input.
"Open-loop" means no feedback connection is present; feedback and closed-loop operation are covered in the next topic.
Because $A_{OL}$ is so large (infinite in the ideal model), even a tiny input difference drives the output to one of its two extreme values — it saturates at close to $+V_S$ or $-V_S$.
This all-or-nothing open-loop output makes the op-amp directly useful as a comparator, switching sharply to indicate which of $V_+$ or $V_-$ is larger.
The idealisation is refined later, in the "Real Operational Amplifiers" lesson, which considers how actual devices depart from these assumptions.

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Q Factor and Bandwidth

Inverting Amplifier