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Operational Amplifiers

Summary

This chapter provides comprehensive coverage of operational amplifiers (op-amps), the versatile building blocks of analog electronics. Students will learn the ideal op-amp model and how negative feedback creates stable, predictable circuits. The chapter covers fundamental configurations including inverting and non-inverting amplifiers, voltage followers, summing amplifiers, and integrator/differentiator circuits. Practical limitations like bandwidth, slew rate, and input offset are also addressed. Mastering op-amps enables students to design sophisticated audio processing circuits.

Concepts Covered

This chapter covers the following 30 concepts from the learning graph:

  1. Operational Amplifier
  2. Ideal Op-Amp
  3. Op-Amp Symbol
  4. Inverting Input
  5. Non-Inverting Input
  6. Op-Amp Output
  7. Open-Loop Gain
  8. Closed-Loop Gain
  9. Negative Feedback
  10. Positive Feedback
  11. Virtual Short
  12. Virtual Ground
  13. Inverting Amplifier
  14. Non-Inverting Amplifier
  15. Voltage Follower
  16. Buffer Amplifier
  17. Summing Amplifier
  18. Difference Amplifier
  19. Instrumentation Amplifier
  20. Integrator Circuit
  21. Differentiator Circuit
  22. Op-Amp Bandwidth
  23. Gain-Bandwidth Product
  24. Slew Rate
  25. Input Offset Voltage
  26. Input Bias Current
  27. Common Mode Rejection
  28. CMRR
  29. Op-Amp Saturation
  30. Rail-to-Rail Op-Amp

Prerequisites

This chapter builds on concepts from:

title: Operational Amplifiers description: Master the versatile op-amp - the building block of analog electronics generated_by: chapter-content-generator v0.03 date: 2026-01-30

Introduction: The Swiss Army Knife of Electronics

If passive components are the nouns of circuit language, operational amplifiers are the verbs. They do things: amplify, buffer, sum, subtract, integrate, differentiate, compare, and more. A single IC costing less than a dollar can perform tasks that would require dozens of discrete transistors.

The operational amplifier (op-amp) earned its name from early analog computers where it performed mathematical "operations" like addition and integration. Today, op-amps are everywhere: in your phone's microphone preamplifier, your laptop's audio output, every sensor interface, and countless industrial control systems.

The beauty of op-amps lies in how negative feedback tames their enormous gain into precise, predictable behavior. With just two rules—and a handful of resistors—you can design amplifiers with exactly the gain you need, every time. This chapter teaches you those rules and how to apply them.

The Ideal Op-Amp Model

The ideal op-amp is a simplified model that makes analysis straightforward. Real op-amps approach this ideal closely enough that the model works remarkably well for most designs.

Ideal op-amp characteristics:

Property Ideal Value Why It Matters
Open-loop gain (A) ∞ Any input difference creates huge output
Input impedance ∞ No current flows into inputs
Output impedance 0 Can drive any load without voltage drop
Bandwidth ∞ Works at all frequencies
CMRR ∞ Rejects common-mode perfectly
Slew rate ∞ Output changes instantly

The Op-Amp Symbol

The standard op-amp symbol is a triangle with five terminals:

  • Inverting input (−): Marked with minus sign
  • Non-inverting input (+): Marked with plus sign
  • Output: At the triangle apex
  • Power supplies (V+ and V−): Often omitted from schematics

The output voltage is proportional to the difference between the inputs: [V_{out} = A(V_+ - V_-)]

Where A is the open-loop gain (typically 100,000 to 1,000,000 for real op-amps).

The Power Supply Convention

Op-amp schematics often don't show power connections, but they're always there! Typical supplies are ±15V, ±12V, ±5V (dual supply) or +5V, +3.3V (single supply). The output can only swing between these rails.

Open-Loop vs. Closed-Loop Gain

Open-Loop Gain

Open-loop gain (A or A_OL) is the op-amp's intrinsic gain with no feedback—typically 100,000 or more (100 dB).

This enormous gain means that even a tiny input difference (microvolts) drives the output to the supply rails. Open-loop operation is essentially useless for linear amplification but perfect for comparators.

Closed-Loop Gain

Closed-loop gain (A_CL) is the overall gain when feedback is applied. It depends on the feedback network, not the op-amp's open-loop gain.

\[A_{CL} = \frac{A_{OL}}{1 + A_{OL}\beta}\]

Where β is the feedback fraction. When \(A_{OL}\beta >> 1\):

\[A_{CL} \approx \frac{1}{\beta}\]

The closed-loop gain depends only on the feedback network—not on the op-amp's gain! This is the magic of negative feedback.

Negative Feedback: The Taming Force

Negative feedback connects a portion of the output back to the inverting input. It's the key to stable, predictable op-amp circuits.

How it works:

  1. Any increase in output feeds back to inverting input
  2. This reduces the input difference (V+ − V−)
  3. Which reduces the output
  4. System settles at a stable equilibrium

Benefits of negative feedback:

  • Stable, predictable gain
  • Reduced distortion
  • Increased bandwidth
  • Reduced output impedance
  • Reduced sensitivity to component variations

Positive Feedback

Positive feedback connects output to the non-inverting input. Instead of stabilizing, it drives the output harder in the same direction.

Uses:

  • Oscillators (intentional instability)
  • Comparators with hysteresis (Schmitt triggers)
  • Not used for linear amplification!
Feedback Type Connection Effect Use
Negative Output → V− Stabilizing Amplifiers
Positive Output → V+ Destabilizing Oscillators, comparators

The Golden Rules: Virtual Short and Virtual Ground

For ideal op-amps with negative feedback, two simple rules solve almost every circuit:

Rule 1: Virtual Short

Virtual short: The voltage difference between the inputs is essentially zero. [V_+ \approx V_-]

Why? With infinite gain, even a tiny difference would rail the output. Negative feedback forces the inputs to be equal.

Rule 2: No Input Current

No input current: The inputs draw essentially zero current. [I_+ \approx I_- \approx 0]

Why? The input impedance is infinite, so no current flows into the op-amp inputs.

Virtual Ground

Virtual ground is a special case of virtual short when the non-inverting input is grounded: [V_+ = 0 \Rightarrow V_- = 0]

The inverting input sits at 0V even though it's not directly connected to ground—it's virtually grounded through the feedback mechanism.

Diagram: Op-Amp Golden Rules Visualizer

Op-Amp Golden Rules Visualizer Type: microsim Bloom Level: Understand (L2) Bloom Verb: explain Learning Objective: Students will explain how the golden rules (virtual short, no input current) emerge from negative feedback by observing voltage and current values in an inverting amplifier. Visual elements: - Inverting amplifier circuit with labeled nodes - Voltage values displayed at each node (V+, V−, Vout) - Current arrows showing direction and magnitude (zero into inputs) - "Virtual ground" label when V− ≈ 0 - Input difference (V+ − V−) shown approaching zero Interactive controls: - Slider: Input voltage Vin (-5V to +5V) - Slider: Rf/Ri ratio (1 to 100) - Toggle: Show ideal vs real op-amp (with finite gain) - Display: V+, V−, |V+ - V−|, input currents Step-through mode: - Step 1: "Apply input voltage" - Step 2: "Feedback forces V− toward V+" - Step 3: "With V+ = 0, V− becomes virtual ground" - Step 4: "Output settles at -Vin × Rf/Ri" Annotations: - "V+ = V− (virtual short)" - "I+ = I− = 0 (no input current)" - "Virtual ground when V+ is grounded" Default parameters: - Vin = 1V - Ri = 10kΩ, Rf = 100kΩ - Gain = -10 Canvas layout: - Circuit diagram: 450 × 350 pixels - Control area: 100 pixels below Implementation: p5.js

Fundamental Op-Amp Configurations

Inverting Amplifier

The inverting amplifier is the most common op-amp configuration. Input connects through Ri to the inverting input; feedback resistor Rf connects from output to inverting input.

Gain: [A_V = -\frac{R_f}{R_i}]

The negative sign indicates phase inversion (180°).

Analysis using golden rules:

  1. V− = V+ = 0 (virtual ground)
  2. Current through Ri: \(I_i = \frac{V_{in}}{R_i}\)
  3. This current must flow through Rf (no current into op-amp)
  4. \(V_{out} = 0 - I_i R_f = -V_{in}\frac{R_f}{R_i}\)

Input impedance: \(Z_{in} = R_i\) (looking into the circuit)

Non-Inverting Amplifier

The non-inverting amplifier has input at V+ and a voltage divider feedback network.

Gain: [A_V = 1 + \frac{R_f}{R_i}]

Always positive (non-inverting) and always ≥ 1.

Analysis:

  1. V− = V+ = Vin (virtual short)
  2. V− comes from voltage divider of Vout
  3. \(V_- = V_{out} \cdot \frac{R_i}{R_i + R_f}\)
  4. Therefore: \(V_{out} = V_{in}(1 + \frac{R_f}{R_i})\)

Input impedance: Very high (approaches ideal ∞)

Voltage Follower (Buffer)

The voltage follower (or buffer amplifier) is a special case of non-inverting amplifier with Rf = 0 and Ri = ∞.

Gain: [A_V = 1]

Output equals input: \(V_{out} = V_{in}\)

Why use it?

  • Provides very high input impedance (doesn't load the source)
  • Provides very low output impedance (can drive heavy loads)
  • Isolates stages from each other

Applications:

  • Buffer between high-impedance source and low-impedance load
  • Sample-and-hold circuits
  • Active probes for oscilloscopes
Configuration Gain Phase Z_in
Inverting −Rf/Ri 180° Ri
Non-inverting 1 + Rf/Ri 0° Very high
Voltage follower 1 0° Very high

Diagram: Basic Op-Amp Configurations

Basic Op-Amp Configurations Type: microsim Bloom Level: Apply (L3) Bloom Verb: calculate Learning Objective: Students will calculate gain for inverting, non-inverting, and follower configurations by selecting resistor values. Visual elements: - Three circuit diagrams side by side (inverting, non-inverting, follower) - Input and output waveforms for selected configuration - Gain calculation shown step-by-step - Phase relationship indicated Interactive controls: - Radio buttons: Select configuration - Slider: Ri (1kΩ to 100kΩ) - Slider: Rf (1kΩ to 1MΩ) - Slider: Input voltage (sinusoidal amplitude) - Display: Calculated gain, input/output relationship For each configuration: - Circuit schematic with component values - Gain formula with calculation - Output waveform with correct amplitude and phase Default parameters: - Inverting: Ri = 10kΩ, Rf = 100kΩ → Gain = -10 - Non-inverting: Ri = 10kΩ, Rf = 90kΩ → Gain = +10 - Follower: Gain = 1 Canvas layout: - Circuit/waveform display: 600 × 400 pixels - Control area: 100 pixels below Implementation: p5.js

Arithmetic Circuits

Summing Amplifier

The summing amplifier adds multiple input signals. Each input has its own input resistor connected to the virtual ground at V−.

For equal input resistors (R): [V_{out} = -\frac{R_f}{R}(V_1 + V_2 + V_3 + ...)]

For different input resistors: [V_{out} = -R_f\left(\frac{V_1}{R_1} + \frac{V_2}{R_2} + \frac{V_3}{R_3}\right)]

Applications:

  • Audio mixing (combining multiple channels)
  • Weighted averaging
  • Digital-to-analog conversion

Difference Amplifier

The difference amplifier subtracts one input from another.

Basic configuration (equal R): [V_{out} = \frac{R_f}{R_i}(V_2 - V_1)]

The output is proportional to the difference between the two inputs.

Limitation: Input impedances are different and relatively low.

Instrumentation Amplifier

The instrumentation amplifier is a precision difference amplifier with:

  • Very high input impedance on both inputs
  • Excellent CMRR (common-mode rejection)
  • Gain set by a single resistor

Structure: Three op-amps (two input buffers + difference amp)

Applications:

  • Sensor interfaces (strain gauges, thermocouples)
  • Medical instrumentation (ECG, EEG)
  • Precision measurement

Integrator and Differentiator

Integrator Circuit

The integrator replaces Rf with a capacitor, producing output proportional to the integral of the input.

\[V_{out}(t) = -\frac{1}{RC}\int V_{in}(t) \, dt\]

In frequency domain: [H(j\omega) = -\frac{1}{j\omega RC}]

Gain increases at low frequencies—acts as a low-pass filter with -20 dB/decade slope.

Applications:

  • Analog computing
  • Active low-pass filters
  • Waveform generation (triangle wave from square wave)

Differentiator Circuit

The differentiator replaces Ri with a capacitor, producing output proportional to the rate of change of input.

\[V_{out}(t) = -RC\frac{dV_{in}}{dt}\]

In frequency domain: [H(j\omega) = -j\omega RC]

Gain increases at high frequencies—acts as a high-pass filter, but amplifies noise!

Practical issue: High-frequency noise is amplified. Usually add a small series resistor for stability.

Bandwidth and Frequency Response

Op-Amp Bandwidth

Real op-amps have finite bandwidth. The open-loop gain decreases with frequency at -20 dB/decade above a low corner frequency (often just a few Hz).

Unity-gain frequency (f_T or GBW): The frequency where open-loop gain drops to 1 (0 dB).

Typical values: 1 MHz to 100 MHz for general-purpose op-amps.

Gain-Bandwidth Product

Gain-bandwidth product (GBW) is approximately constant for a given op-amp:

\[GBW = A_{CL} \times BW\]

Implication: Higher gain means lower bandwidth!

Example: Op-amp with GBW = 10 MHz - At gain = 10: BW = 1 MHz - At gain = 100: BW = 100 kHz - At gain = 1000: BW = 10 kHz

Slew Rate

Slew rate is the maximum rate of change of the output voltage:

\[SR = \left|\frac{dV_{out}}{dt}\right|_{max} \text{ in V/μs}\]

It limits the output at high frequencies even if gain-bandwidth allows.

Maximum frequency for full output swing: [f_{max} = \frac{SR}{2\pi V_{peak}}]

Example: SR = 1 V/μs, desired 10V peak output: [f_{max} = \frac{10^6}{2\pi \times 10} = 15.9 \text{ kHz}]

Above this frequency, output cannot follow the full swing.

Parameter Effect Typical Values
GBW Gain × Bandwidth trade-off 1-100 MHz
Slew rate Large signal speed limit 0.5-100 V/μs
f_T Unity-gain frequency ≈ GBW

Practical Op-Amp Limitations

Input Offset Voltage

Input offset voltage (V_OS) is the small DC voltage that must be applied between the inputs to make the output exactly zero.

Typical values: 1-10 mV (general purpose), < 100 μV (precision)

Effect: DC error at output = V_OS × (1 + Rf/Ri)

Mitigation: Use precision op-amps, add trimming circuit, or AC-couple input.

Input Bias Current

Input bias current (I_B) is the small DC current that flows into the inputs.

Typical values: 10 nA - 10 μA (BJT input), < 1 pA (JFET/CMOS input)

Effect: Voltage drop across source resistance appears as offset.

Mitigation: Use FET-input op-amps, or match source impedances.

Common-Mode Rejection Ratio (CMRR)

Common-mode rejection ratio measures how well the op-amp rejects signals that appear on both inputs simultaneously.

\[CMRR = 20\log_{10}\left(\frac{A_{differential}}{A_{common}}\right) \text{ dB}\]

Typical values: 80-120 dB

Why it matters: Real-world signals often have common-mode interference (60 Hz hum, ground noise). High CMRR rejects this interference while amplifying the desired differential signal.

Op-Amp Saturation

Saturation occurs when the output reaches its maximum or minimum voltage, determined by the power supply rails.

For ±15V supplies, typical output swing: ±13V to ±14V (1-2V from rails)

Rail-to-rail op-amps can swing output very close to the supply rails—important for low-voltage, single-supply designs.

Self-Check Questions

1. Design an inverting amplifier with gain of -20. If Ri = 10kΩ, what value of Rf is needed?

Using the inverting amplifier gain formula: [A_V = -\frac{R_f}{R_i}]

\[-20 = -\frac{R_f}{10k\Omega}\]
\[R_f = 20 \times 10k\Omega = 200k\Omega\]

Use Rf = 200 kΩ (or nearest standard value 220 kΩ for gain of -22).

2. An op-amp has GBW = 5 MHz. What is the bandwidth when configured for gain of 50?

Using the gain-bandwidth product relationship: [GBW = A_{CL} \times BW]

\[BW = \frac{GBW}{A_{CL}} = \frac{5 \text{ MHz}}{50} = 100 \text{ kHz}\]

At gain of 50, the amplifier's bandwidth is limited to 100 kHz.

3. Why does a voltage follower have gain of exactly 1, even though it's a non-inverting configuration?

The voltage follower is a non-inverting amplifier with Rf = 0 and Ri = ∞ (open circuit).

Non-inverting gain formula: [A_V = 1 + \frac{R_f}{R_i}]

With Rf = 0: [A_V = 1 + \frac{0}{R_i} = 1 + 0 = 1]

The output connects directly to V−, so the virtual short (V+ = V−) means Vout = V+ = Vin. The gain is unity regardless of any resistor values.

4. An integrator has R = 10kΩ and C = 100nF. What is the output after 1ms if Vin = constant 1V?

For an integrator with constant input: [V_{out}(t) = -\frac{1}{RC}\int_0^t V_{in} \, dt = -\frac{V_{in} \cdot t}{RC}]

\[RC = 10 \times 10^3 \times 100 \times 10^{-9} = 1 \text{ ms}\]
\[V_{out}(1\text{ ms}) = -\frac{1\text{V} \times 1\text{ ms}}{1 \text{ ms}} = -1\text{ V}\]

After 1 ms, the output is -1V. The integrator ramps linearly when the input is constant.

Summary

Operational amplifiers are the foundation of analog circuit design:

  1. Ideal op-amp model simplifies analysis - Infinite gain, infinite input impedance, zero output impedance

  2. Negative feedback creates stability - Output adjusts to minimize input difference

  3. Two golden rules solve most circuits - Virtual short (V+ = V−) and no input current

  4. Inverting amplifier gain = -Rf/Ri - Simple, predictable, phase-inverting

  5. Non-inverting amplifier gain = 1 + Rf/Ri - Always ≥ 1, non-inverting

  6. Voltage follower is the ultimate buffer - Unity gain, very high Zin, very low Zout

  7. Summing and difference amplifiers do math - Add, subtract, weighted average

  8. Integrators and differentiators process signals - Mathematical operations become circuit operations

  9. GBW limits high-frequency, high-gain operation - Trade gain for bandwidth

  10. Real op-amps have imperfections - Offset, bias current, limited CMRR, saturation

With op-amps in your toolkit, you can design active filters, precision amplifiers, signal conditioners, and countless other circuits. The ideal op-amp model and the two golden rules give you the analytical power to tackle any configuration. Real-world limitations exist, but understanding them lets you choose the right op-amp for the job and design around the constraints. Op-amps truly are the Swiss Army knife of electronics—master them, and you've gained a superpower.