The Operational Amplifier

What you will learn

• The ideal op amp: with negative feedback and in its linear region, no current enters either input ($i_p = i_n = 0$) and the two input voltages are equal ($v_p = v_n$).
• Saturation: the output can never go beyond the supply voltages. Outside that range the two rules no longer hold.
• The inverting amplifier: $v_o = -\tfrac{R_f}{R_s} v_s$.
• The summing amplifier: several inputs, each scaled by $R_f$ over its own input resistor, then added and inverted.
• The noninverting amplifier: $v_o = \left(1 + \tfrac{R_f}{R_s}\right) v_g$, and with $R_f = 0$ the voltage follower, which copies a voltage without loading it.
• The difference amplifier: $v_o = \tfrac{R_b}{R_a}(v_b - v_a)$ when the resistor ratios on the two inputs match.
• Cascades: the output of one stage drives the next, and each stage must stay inside its supplies.

Later in this course

In your program

• Electrical and Electronics: amplifiers, filters, voltage regulators and the control loops of motor drives are built around op amps.
• Computer, Communication and Telecom: every analog signal that reaches a processor, from a microphone, an antenna or a sensor, passes through op-amp stages that scale and shift it for the analog-to-digital converter.
• Biomedical: an ECG amplifier is a difference amplifier: it amplifies the tiny difference between two electrodes and rejects what both pick up.
• Mechanical and Industrial: strain gauges, thermocouples and pressure sensors give millivolts; op amps turn them into volts a controller can read.
• Surveying: the sensors inside levels, total stations and GNSS receivers feed op-amp stages before their signals are digitised.

As an engineer

• Two rules, then node equations. Check for negative feedback, set $v_n = v_p$ and $i_n = i_p = 0$, and write KCL at the inverting input. Every circuit in this chapter yields to that recipe.
• Always check the supplies. An answer outside the supply range is not an answer: the op amp saturates, and the output sits at a supply voltage.
• The inputs draw no current; the output can. The output current comes from the supplies, which is why an op amp can drive a load that would collapse a divider.

In real life

• Monitoring a solar battery bank. A divider scales 60 V down to 5 V for a display, but the display’s own resistance pulls the reading down. A voltage follower between them fixes it (S7).
• From a sensor to a microcontroller. A temperature sensor gives 10 mV per °C; an op amp scales and shifts that signal so that the microcontroller’s 0–5 V input range is used in full (S15, and Nilsson’s Practical Perspective for this chapter).

How to study this chapter, and where AI fits

• AI tools are allowed on the practice sheet. Use them to check your work, not to replace it.
• In a 2025 study of Gemini 2.5 Pro on undergraduate circuit problems, misread source polarities caused about a third of its wrong answers, and misread current directions nearly as many (arXiv 2512.10159). In op-amp circuits these slips show up as a feedback current sent the wrong way, which flips the sign of the gain, or as $v_n$ set to 0 when the noninverting input is not grounded. S11 shows a direction slip, and S12 a formula used where it does not apply: find them before you trust any answer.
• Check every answer yourself: KCL at the inverting input must hold with your $v_o$, and $v_o$ must lie between the supplies.
• The quiz is device-free, and every question is a twin of a practice-sheet question. If you can do the sheet on your own, you will do well on the quiz.