Operational Amplifiers

An operational amplifier — universally called an op-amp — is a high-gain electronic amplifier built on a single integrated-circuit chip. It has two inputs (labelled + and −) and one output, and its job is to amplify the difference in voltage between those two inputs by an enormous factor. The name 'operational' comes from the 1940s, when these devices were first used to perform mathematical operations such as addition, subtraction, and integration in analogue computers. In practice, a real op-amp is modelled using a set of idealised properties that make circuit analysis straightforward: infinite open-loop gain, infinite input impedance, zero output impedance, and infinite bandwidth. Although no real chip perfectly meets these criteria, modern op-amps (such as the ubiquitous LM741 or TL071) come close enough that the ideal model gives accurate, useful predictions for most school-level circuits. By connecting a handful of resistors around an op-amp, engineers create two fundamental building-block circuits: the inverting amplifier and the non-inverting amplifier. These appear everywhere — in audio equipment, medical sensors, radio receivers, and scientific instruments. Understanding how they work is the gateway to understanding all analogue signal processing.

Key takeaways
  • The Four Ideal Properties
  • Negative Feedback — the Key Idea
  • The Inverting Amplifier
  • The Non-Inverting Amplifier
  • Saturation — When the Ideal Breaks Down

The Four Ideal Properties

An ideal op-amp obeys four rules. (1) Infinite open-loop voltage gain A: the output equals A × (V₊ − V₋), and A → ∞. (2) Infinite input impedance: no current ever flows into either input terminal. (3) Zero output impedance: the output can drive any load without its voltage drooping. (4) Infinite bandwidth: it amplifies every frequency equally. These idealisations let us apply two golden rules when negative feedback is present: the output will do whatever is necessary to make V₊ = V₋ (the inputs are at the same potential), and no current flows into the inputs.

Negative Feedback — the Key Idea

Connecting the output back to the inverting (−) input creates negative feedback. Think of it like cruise control in a car: if the car speeds up beyond the set speed, the system reduces throttle to correct it. Similarly, if the op-amp output rises, some of that rise is fed back to the − input, which pushes the output back down. This self-correcting loop forces V₊ ≈ V₋ and makes the gain stable and predictable — determined entirely by external resistors, not by the huge and variable open-loop gain of the chip.

The Inverting Amplifier

In the inverting configuration, the input signal Vᵢₙ is connected through a resistor R₁ to the − input, and a feedback resistor Rƒ runs from the output back to the − input. The + input is connected to ground (0 V). By the golden rules, V₋ = V₊ = 0 V — this point is called a virtual earth. Because no current enters the op-amp, all the current through R₁ must flow through Rƒ. Applying Ohm's law: Vᵢₙ/R₁ = −Vₒᵤₜ/Rƒ, giving a closed-loop gain G = −Rƒ/R₁. The negative sign means the output is 180° out of phase with the input — it is inverted.

The Non-Inverting Amplifier

In the non-inverting configuration, the signal is applied directly to the + input. A voltage divider made of Rƒ and R₁ feeds the output back to the − input. The golden rules give V₋ = Vᵢₙ (since V₋ = V₊). The divider equation is V₋ = Vₒᵤₜ × R₁/(R₁ + Rƒ). Setting these equal and solving yields G = 1 + Rƒ/R₁. This gain is always greater than or equal to 1, and the output is in phase with the input — there is no inversion. The special case where Rƒ = 0 and R₁ = ∞ gives G = 1; this is called a voltage follower or buffer, which copies the voltage while providing current.

Saturation — When the Ideal Breaks Down

A real op-amp is powered by supply rails (e.g. ±15 V). The output cannot exceed these rails; if the calculated output would be larger, the amplifier saturates and clips the signal. This is why gain must be chosen carefully relative to the supply voltage. For example, with a ±12 V supply and a gain of −10, an input exceeding 1.2 V in magnitude will saturate the output.

The Key Equation

G=−RfR1(inverting)G=1+RfR1(non-inverting)G = -\frac{R_f}{R_1} \quad \text{(inverting)} \qquad G = 1 + \frac{R_f}{R_1} \quad \text{(non-inverting)}
  • G — Closed-loop voltage gain (dimensionless): Ratio of output voltage to input voltage when negative feedback is applied.
  • R_f — Feedback resistor (ohm (Ω)): Resistor connecting the output back to the inverting (−) input.
  • R_1 — Input resistor (ohm (Ω)): Resistor through which the input signal is applied to the inverting input.

Worked Example

Problem. An inverting amplifier is built with R₁ = 10 kΩ and Rƒ = 47 kΩ. The input signal is a 0.2 V (peak) sine wave. Calculate (a) the closed-loop gain, (b) the peak output voltage, and (c) state whether this output fits within ±15 V supply rails.

G=−4.7,Vout,peak=−0.94 VG = -4.7, \quad V_{out,peak} = -0.94\text{ V}

Answer. Gain = −4.7; peak output = −0.94 V (inverted); safely within ±15 V rails.

Figure
Diagram
Inverting Op-Amp AmplifierA→∞−+GNDR₁VᵢₙVₒᵤₜRƒvirtual earthClosed-loop gain: G = −Rƒ / R₁ (output inverted, in-phase currents at virtual earth)
Origins · 1 minRobert John Widlar and how this was discovered
RJ
Robert John Widlar
1937–1991 · American

Widlar was a pioneering analogue integrated-circuit designer at Fairchild Semiconductor and National Semiconductor. He designed the μA702 and μA709 — some of the first practical monolithic op-amps — and later the LM101 and LM741, the latter of which became the most widely used op-amp in history and is still in production today. His innovations in bias current reduction and internal compensation made op-amps reliable, cheap, and accessible to every engineer. Widlar was famously eccentric; he once brought a sheep to the National Semiconductor campus and let it graze on the lawn in protest of excessive grass-mowing noise. Despite his quirks, colleagues regarded his circuit intuition as almost supernatural.

  • Voltage Dividers — The non-inverting amplifier's feedback network is a voltage divider; understanding this circuit is essential for deriving the gain formula.
  • Negative Feedback in Amplifiers — Negative feedback is the core principle that makes op-amp gain stable and predictable; both circuits rely entirely on it.
  • Ohm's Law and Resistance — All gain calculations use Ohm's law to relate currents and voltages across R₁ and Rƒ at the virtual earth node.
  • Comparators — An op-amp used without feedback acts as a comparator — the open-loop mode that contrasts sharply with the linear feedback configurations studied here.