lecture_EE1C1_lec5_opamps_2025-10-07

📌 Overview

This lecture provides a comprehensive introduction to Operational Amplifiers (Op-Amps), essential components in modern electronics. We will explore their fundamental principles, ideal characteristics, and practical applications in various circuit configurations.

Key Topics Covered:

• Introduction to Op-Amps: Understanding their role as active elements and their primary functions.
• Ideal vs. Non-ideal Op-Amps: Exploring the parameters of an ideal op-amp and how real-world devices differ.
• Feedback Principles: Analyzing the concepts of negative and positive feedback and their impact on stability.
• Common Op-Amp Circuits:
  • Inverting Amplifier: A circuit that amplifies and inverts the input signal.
  • Non-inverting Amplifier: A circuit that amplifies the input signal without phase inversion.
  • Summing and Difference Amplifiers: Circuits that perform mathematical operations on input signals.
  • Cascaded Op-Amps: Combining multiple op-amp stages to achieve higher gain and specific filtering characteristics.

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🎯Learning Objectives

By the end of this lecture, you will be able to analyze and design basic op-amp circuits, apply the “golden rules” for ideal op-amps, and understand their practical applications in signal processing and amplification.

• Understand the basic operation and purpose of an operational amplifier.
• Distinguish between ideal and non-ideal op-amp characteristics.
• Apply the concept of negative feedback to stabilize op-amp circuits.
• Analyze and design fundamental op-amp circuits, including inverting, non-inverting, summing, and difference amplifiers.
• Recognize the advantages of cascaded op-amp configurations.

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💡Key Concepts & Definitions

• Operational Amplifier (Op-Amp): A high-gain electronic voltage amplifier with a differential input and, usually, a single-ended output. Op-amps are among the most widely used electronic devices today, used in a vast array of consumer, industrial, and scientific devices.
• Active Element: An electronic component that can supply energy to a circuit, such as a battery, a generator, or an op-amp.
• Passive Element: A component that consumes or stores energy, such as a resistor, capacitor, or inductor.
• Open-Loop Gain (A): The gain of the op-amp without any external feedback from the output to the input. In an ideal op-amp, this value is infinite.
• Negative Feedback: A configuration where a portion of the output signal is fed back to the inverting input, which helps to stabilize the amplifier’s gain and improve its performance.
• Positive Feedback: A configuration where a portion of the output signal is fed back to the non-inverting input, which can lead to oscillation and is used in circuits like oscillators.
• Voltage Saturation: The state where the op-amp’s output voltage is limited to its maximum or minimum supply voltage levels.
• “Golden Rules” for Ideal Op-Amps:
  1. The input currents to both the inverting and non-inverting terminals are zero.
  2. The voltage difference between the inverting and non-inverting terminals is zero (with negative feedback).

➗ Formulas

• Inverting Amplifier: $V_{\text{out}}=-V_{\text{in}}\frac{R_f}{R_1}$
• Non-inverting Amplifier: $V_{\text{out}}=V_{\text{in}}\left(1+\frac{R_f}{R_1}\right)$
• Summing Amplifier: $V_{\text{out}}=-R_f\left(\frac{V_1}{R_1}+\frac{V_2}{R_2}+\dots \right)$
• Difference Amplifier: $V_{\text{out}}=\frac{R_f}{R_1}(V_2-V_1)$

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✍️ Notes

Introduction to Op-Amps

• What is an Op-Amp?

  • An operational amplifier is a versatile electronic device that serves as a cornerstone of analog circuit design. It is an active element, meaning it requires a power source to function and can inject power into a circuit.
  • The name “operational amplifier” comes from its original use in analog computers, where it performed mathematical operations such as addition, subtraction, integration, and differentiation.

• Key Functions:

  • Voltage Amplification: The primary purpose of an op-amp is to act as a voltage-controlled voltage source, amplifying the difference between its two input terminals.
  • Signal Processing: In combination with passive components like resistors and capacitors, op-amps can be configured to perform a wide range of signal-processing tasks.

Ideal vs. Non-ideal Op-Amps

To simplify circuit analysis, we often use an ideal op-amp model. The key characteristics of an ideal op-amp are:

1. Infinite Open-Loop Gain ($A\to \infty$): The gain is so large that even a tiny voltage difference at the inputs can drive the output to its maximum voltage.
2. Infinite Input Resistance ($R_i\to \infty$): The op-amp draws no current from the input source, meaning it does not load the circuit connected to it.
3. Zero Output Resistance ($R_o=0$): The output voltage is independent of the load connected to it.

In reality, non-ideal op-amps have very large but finite gain, high but finite input resistance, and low but non-zero output resistance. For most applications, the ideal model provides a sufficiently accurate approximation.

Feedback Principles

• Negative Feedback:

  • This is the most common configuration for op-amp circuits. A portion of the output signal is connected back to the inverting input (usually through a resistor).
  • Benefits:
    • Stabilizes Gain: The closed-loop gain becomes dependent on the external feedback components, not the op-amp’s massive open-loop gain.
    • Reduces Distortion: Improves the linearity of the amplifier.
    • Increases Bandwidth: Allows the amplifier to work effectively over a wider range of frequencies.

• Positive Feedback:

  • The output is connected back to the non-inverting input. This configuration is inherently unstable and is typically used in oscillators and Schmitt triggers, which are not covered in this course.

The “Golden Rules” for Ideal Op-Amp Analysis

When analyzing an op-amp circuit with negative feedback, we can use two simplifying assumptions, often called the “golden rules”:

1. Zero Input Current: No current flows into the inverting ($i_{-}$) or non-inverting ($i_{+}$) terminals.
   • $i_{-}=i_{+}=0$
2. Virtual Short Circuit: The voltage at the non-inverting terminal ($v_{+}$) is equal to the voltage at the inverting terminal ($v_{-}$).
   • $v_{+}=v_{-}$

Important Note: These rules only apply when the op-amp is operating in its linear region (i.e., not in saturation) and with negative feedback.

Common Op-Amp Circuits

1. Inverting Amplifier

• Topology: The input signal is applied to the inverting terminal through a resistor $R_1$, and the non-inverting terminal is grounded. A feedback resistor $R_f$ connects the output to the inverting terminal.
• Operation: The output voltage is an amplified and inverted version of the input. The gain is determined by the ratio of the feedback resistor to the input resistor.
• Example: If $R_f=10\text{ k}\Omega$ and $R_1=2\text{ k}\Omega$, the gain is $-5$. A $1\text{V}$ input will produce a $-5\text{V}$ output.

2. Non-inverting Amplifier

• Topology: The input signal is applied directly to the non-inverting terminal. The inverting terminal is connected to ground through $R_1$ and to the output through $R_f$.
• Operation: The output voltage is an amplified, non-inverted version of the input. The gain is always greater than or equal to 1.
• Voltage Follower: A special case where $R_f=0$ and $R_1\to \infty$. The gain is exactly 1 ($V_{\text{out}}=V_{\text{in}}$). This circuit is used as a buffer to isolate a high-impedance source from a low-impedance load.

3. Summing Amplifier

• Topology: An extension of the inverting amplifier with multiple input signals, each connected through its own resistor to the inverting terminal.
• Operation: The output is a weighted sum of the input voltages, with each weight determined by the ratio of the feedback resistor to the corresponding input resistor.

4. Difference Amplifier

• Topology: A combination of inverting and non-inverting configurations that amplifies the difference between two input signals.
• Operation: When the resistor ratios are balanced ($R_1=R_2$ and $R_3=R_4$), the output is directly proportional to the difference between the two input voltages ($V_2-V_1$).

5. Cascaded Op-Amps

• Concept: Multiple op-amp circuits can be connected in series (cascaded) to achieve higher overall gain or to combine different functions.
• Total Gain: The total gain of a cascaded system is the product of the individual stage gains ($A_{\text{total}}=A_1\times A_2\times \dots$).

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🔗 Resources

• Presentation:

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❓ Post lecture

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📖 Homework

• Complete the exercises in SGH4 related to op-amp circuits.
• Review the examples of inverting, non-inverting, and summing amplifiers in the textbook.
• Prepare for the next lecture, where we will introduce new circuit elements: capacitors and inductors.