Operational Amplifiers
Learning Objectives
- State the ideal op-amp assumptions and explain why they simplify circuit analysis
- Derive the gain equations for the inverting and non-inverting amplifier configurations
- Apply the "virtual short" concept to analyze op-amp circuits quickly
- Identify common op-amp circuits: inverting, non-inverting, voltage follower, differential amplifier, comparator
- Explain the difference between open-loop and closed-loop gain, and the role of negative feedback
- Recognize real-world op-amp limitations: finite gain, offset voltage, bandwidth, slew rate
Quick Answer
An operational amplifier ("op-amp") is a high-gain differential voltage amplifier — with two inputs (inverting "−" and non-inverting "+") and one output — that engineers wrap in a feedback network to build precise, predictable circuits. Rather than using the op-amp's raw (and unreliable) open-loop gain of 100,000 or more, negative feedback trades away excess gain for stability, linearity, and a gain set entirely by external resistors. Op-amps are the workhorse building block of analog electronics: they amplify sensor signals, build active filters, sum and subtract voltages, compare signals, and generate waveforms — all using the same handful of ideal-op-amp rules.
The Ideal Op-Amp: Two Golden Rules
Real op-amps are complicated internally (dozens of transistors), but for 90% of circuit analysis, you only need two idealized assumptions, valid whenever there's negative feedback:
- No current flows into either input terminal. (Ideal input impedance is infinite.)
- The voltage difference between the two inputs is zero. (Ideal open-loop gain is infinite, so feedback drives V+ = V−.) This is called the virtual short — the two inputs behave as if shorted together for voltage purposes, even though no actual current flows between them.
These two rules alone let you solve almost every basic op-amp circuit with simple algebra — no differential equations needed.
An ideal op-amp additionally has: infinite bandwidth, zero output impedance, and zero input offset voltage. Real devices fall short of all of these (more on that below), but the ideal model is remarkably accurate for everyday designs.
The Inverting Amplifier
In the inverting configuration, the input signal drives the inverting terminal through resistor Rin, and a feedback resistor Rf connects the output back to that same inverting terminal. The non-inverting terminal is grounded.
Because the op-amp forces V+ = V− (virtual short) and V+ = 0 V (grounded), the inverting input sits at 0 V too — this point is called a virtual ground. Since no current enters the op-amp input, all the current flowing in through Rin must flow out through Rf (Kirchhoff's current law):
(Vin − 0)/Rin = (0 − Vout)/Rf
Solving for the gain:
Vout / Vin = − Rf / Rin
Worked example: If Rin = 10 kΩ and Rf = 100 kΩ, the gain is −100 kΩ/10 kΩ = −10. A 0.2 V input produces a −2 V output — amplified by 10 and inverted (flipped in polarity).
The Non-Inverting Amplifier
Here the input signal drives the non-inverting terminal directly, and the feedback network (Rf and a ground resistor R1) connects from the output to the inverting terminal, forming a voltage divider.
Since V− = V+ = Vin (virtual short, and Vin is applied directly to the + terminal), and the divider sets V− = Vout × R1/(R1 + Rf):
Vout / Vin = 1 + Rf / R1
Notice the gain is always ≥ 1 and always positive (no inversion) — a key practical difference from the inverting configuration. Setting Rf = 0 and R1 = ∞ (i.e., no feedback resistor, direct output-to-input wire) gives the special case of the voltage follower (unity-gain buffer), with gain = 1. Its purpose isn't amplification — it's impedance isolation: it presents a very high input impedance to a delicate sensor while driving a low-impedance load without loading down the source.
Other Key Configurations
- Differential amplifier: Amplifies the difference between two input voltages while rejecting signals common to both (common-mode rejection) — essential for measuring small signals riding on large noisy backgrounds, like ECG signals or thermocouple outputs.
- Summing amplifier: An inverting-amplifier variant with multiple input resistors, producing an output proportional to the weighted sum of several input voltages — used in audio mixers.
- Comparator: An op-amp used open-loop (no feedback) to compare two voltages; the output slams to one supply rail or the other depending on which input is higher. Used in zero-crossing detectors and simple ADCs.
- Integrator/Differentiator: Replace a resistor with a capacitor in the inverting configuration to produce an output proportional to the integral or derivative of the input — building blocks for oscillators and analog computers.
Real-World Limitations
| Ideal Assumption | Real Op-Amp Behavior | Practical Consequence |
|---|---|---|
| Infinite open-loop gain | Finite (typically 10⁵–10⁶) | Closed-loop gain slightly less than the ideal formula predicts at very high gains |
| Zero input offset voltage | A few µV to a few mV mismatch | Small DC error at the output, matters in precision circuits |
| Infinite bandwidth | Gain-bandwidth product is fixed (e.g., 1 MHz for a 741) | Gain drops off at high frequency — a gain-of-100 amp might only work cleanly to 10 kHz |
| Zero output resistance | A few tens of ohms | Output droops slightly under heavy load |
| Infinite slew rate | Finite (e.g., 0.5 V/µs for a 741) | Large fast signals get distorted into a ramp instead of a sharp edge |
Worked example — gain-bandwidth tradeoff: A popular general-purpose op-amp (like the LM741) has a gain-bandwidth product (GBW) of about 1 MHz. If you configure it for a gain of 100 (non-inverting), the usable bandwidth shrinks to roughly GBW/gain = 1 MHz/100 = 10 kHz. This is why high-gain, high-bandwidth applications (like RF amplifiers) need specialized op-amps or multiple cascaded lower-gain stages instead of one huge-gain stage.
Why It Matters
Op-amps let engineers build precise analog functions — amplification, filtering, summing, comparing — using cheap resistors to set the exact behavior, rather than relying on the messy, temperature-dependent, unit-to-unit-varying characteristics of the op-amp's internal transistors. This is the entire point of negative feedback: it trades away "free" open-loop gain for predictability. Nearly every piece of analog signal-conditioning circuitry — from a microphone preamp to a sensor interface to an active filter — is built from one or more op-amp stages.
Common Mistakes
Misconception 1: "The virtual short means current can flow between the op-amp's + and − input terminals." Why it's wrong: The "virtual short" refers only to the voltage being equal at both inputs (V+ = V−); it does not mean the inputs are electrically connected. No actual current flows into or between the input pins of an ideal op-amp. Correct understanding: Virtual short = equal voltages, but the inputs remain open — infinite input impedance means zero input current, by rule 1.
Misconception 2: "A comparator and a non-inverting amplifier are basically the same circuit with different resistor values." Why it's wrong: A comparator is used open-loop (no feedback path), letting the op-amp's astronomically high open-loop gain saturate the output to one supply rail based on which input is larger. A non-inverting amplifier always has a feedback resistor network that sets a specific, finite, linear gain. Correct understanding: Feedback determines linear vs. saturated behavior — without feedback, an op-amp behaves like a binary comparator, not a proportional amplifier.
Misconception 3: "Since the inverting amplifier's gain formula only involves Rf and Rin, the actual resistor values used don't matter as long as the ratio is right." Why it's wrong: The ratio sets the ideal gain, but real resistor values also affect noise, input impedance, output loading, and bias current errors. Extremely small resistors draw excessive current and waste power; extremely large resistors pick up more noise and interact badly with the op-amp's input bias current. Correct understanding: Designers pick a practical resistor range (often kΩ to low hundreds of kΩ) that hits the target ratio while keeping noise, power, and bias-current errors in check.
Comparison and Connections
| Feature | Inverting Amplifier | Non-Inverting Amplifier | Voltage Follower |
|---|---|---|---|
| Output polarity | Inverted (negative gain) | Same as input (positive gain) | Same as input |
| Gain formula | −Rf/Rin | 1 + Rf/R1 | 1 (fixed) |
| Input impedance | ≈ Rin (moderate) | Very high (ideal: infinite) | Very high |
| Typical use | General amplification, summing, filtering | Amplifying high-impedance sources | Buffering/isolating a sensor |
| Feature | Comparator | Linear Amplifier |
|---|---|---|
| Feedback | None (open-loop) | Negative feedback present |
| Output behavior | Saturates at supply rail (binary) | Proportional to input (analog) |
| Typical use | Threshold detection, zero-crossing | Signal amplification/conditioning |
Practice Questions
Recall 1: State the two golden rules of ideal op-amp analysis. Answer guidance: (1) No current flows into either input; (2) the voltage between the two inputs is zero (virtual short) when negative feedback is present.
Recall 2: Write the gain formula for both the inverting and non-inverting amplifier configurations. Answer guidance: Inverting: Vout/Vin = −Rf/Rin. Non-inverting: Vout/Vin = 1 + Rf/R1.
Understanding 1: Explain why the non-inverting amplifier's gain can never be less than 1, while the inverting amplifier's gain can be any magnitude, including fractional values. Answer guidance: Non-inverting gain = 1 + Rf/R1; since Rf/R1 ≥ 0, gain ≥ 1 always. Inverting gain = −Rf/Rin can be made less than 1 in magnitude simply by choosing Rf < Rin.
Understanding 2: Why is negative feedback necessary to make the ideal op-amp assumptions (like virtual short) valid? Answer guidance: Without feedback, the op-amp's enormous open-loop gain immediately saturates the output to a supply rail for any tiny voltage difference at the inputs. Negative feedback continuously adjusts the output to keep the input difference near zero, which is exactly what makes V+ ≈ V− true.
Application 1: Design an inverting amplifier with a gain of −5 using a 20 kΩ input resistor. What feedback resistor is needed? Answer guidance: Rf = |gain| × Rin = 5 × 20 kΩ = 100 kΩ.
Application 2: A sensor produces a very weak signal from a high-impedance source. Which op-amp configuration would you choose to interface with it, and why? Answer guidance: A non-inverting amplifier or voltage follower, because its very high input impedance avoids loading down (and distorting) the high-impedance sensor signal.
Analysis 1: A student builds a non-inverting amplifier with Rf = 90 kΩ and R1 = 10 kΩ expecting a gain of 10, but at 50 kHz the measured gain has dropped to about 2. Explain the likely cause using the gain-bandwidth product concept. Answer guidance: The op-amp's gain-bandwidth product limits usable bandwidth at high gain (bandwidth ≈ GBW/gain). If GBW is around 1 MHz, gain-of-10 bandwidth is ~100 kHz, so gain should still hold at 50 kHz unless GBW is lower or the op-amp is being pushed near its slew-rate limit — the student should check the op-amp's datasheet GBW and slew rate specification against the actual test frequency and signal amplitude.
Analysis 2: Compare and contrast why an inverting summing amplifier is useful for audio mixing versus simply connecting multiple audio sources' outputs together with a single resistor each and no op-amp. Answer guidance: Directly tying signal sources together through resistors causes each source to load and interact with the others, distorting signals and creating crosstalk. A summing op-amp's virtual ground isolates each input resistor so each source only "sees" its own resistor to a fixed 0 V node, cleanly combining signals without inter-source loading.
FAQ
Q1: What does "op-amp" stand for and why is it called "operational"? A: Operational amplifier — the name comes from their original use in analog computers to perform mathematical "operations" like addition, subtraction, integration, and differentiation on voltages.
Q2: Why do op-amps need two power supply rails (like +15 V and −15 V)? A: To allow the output to swing both positive and negative around 0 V, since many analog signals (like audio) are naturally centered on zero. Some op-amps are designed for single-supply operation with the output referenced to a mid-supply voltage instead.
Q3: What is "input offset voltage" and why does it matter? A: It's the small internal voltage mismatch (a few µV to mV) between the ideal and real behavior of an op-amp's input stage, caused by manufacturing imperfections. It creates a small, unwanted DC output error, which matters most in high-precision or high-gain circuits.
Q4: Can an op-amp amplify a DC signal? A: Yes — unlike AC-coupled amplifier stages, a properly biased op-amp circuit amplifies signals all the way down to DC (0 Hz), which is one of its major advantages over some transistor amplifier stages.
Q5: What happens if you don't use any feedback resistor at all? A: The op-amp runs open-loop with essentially infinite gain, so any tiny voltage difference between its inputs (even input offset voltage or noise) drives the output to slam against one supply rail — this is exactly how a comparator works.
Quick Revision
- Ideal op-amp rules: zero input current, zero voltage difference between inputs (virtual short) under feedback.
- Inverting amp gain: −Rf/Rin; inverting input sits at virtual ground (0 V if + input is grounded).
- Non-inverting amp gain: 1 + Rf/R1; always ≥ 1, no inversion.
- Voltage follower: gain = 1, used for buffering/impedance isolation, not amplification.
- Comparator = op-amp with no feedback; output saturates to a supply rail.
- Differential amplifier amplifies the difference between two inputs, rejecting common-mode noise.
- Real op-amps have finite gain, nonzero offset voltage, limited bandwidth, and finite slew rate.
- Gain-bandwidth product (GBW) is roughly constant: higher gain configurations have lower usable bandwidth.
- Negative feedback trades excess open-loop gain for stability, linearity, and predictable gain.
- Summing amplifiers add multiple weighted inputs via a shared virtual-ground node.
Related Topics
Prerequisites: Analog Signal Fundamentals, basic circuit laws (Ohm's Law, Kirchhoff's Current Law)
Related: Filters and Amplifiers, Feedback Systems, Signal Conditioning
Next: Filters and Amplifiers, Oscillators