Notes on Stability of Closed Loop Systems

Introduction

• Topic of discussion: Stability of a closed loop system.
• Relation to feedback amplifiers and frequency compensation.
• Importance of stability and consequences of instability.

Review of Previous Concepts

• Frequency response of CS amplifier discussed in the last class.
• Miller's Theorem used for pole analysis of systems.
• Importance of feedback amplifiers in the context of CS amplifiers.

Stability of Closed Loop Systems

• Stability pertains specifically to closed loop systems.
• Open loop systems do not have stability issues.
• Need for stability arises from the requirements of control and predictability in system behavior.
• Consequences of instability include oscillations and unpredictable outputs.

Feedback in Amplifiers

• Amplifiers often have very high open loop gain.
• Feedback is needed to control gain and reduce variations caused by temperature and other factors.
• Example: CS amplifier with gain variations of approximately 20% when designed for a gain of 10.
• Solution: Implement feedback to stabilize and accurately set gain.

Basic Feedback System Structure

• Components:
  • Open loop amplifier (gain A).
  • Feedback network to sense output.
  • Input signal (Vin) and output signal (Vout).
• Feedback can be positive or negative. Negative feedback subtracts the feedback signal from the input, stabilizing the system.

Advantages of Negative Feedback

• Reduces noise and sensitivity to variations in component values.
• Improves stability.
• Changes input/output impedance favorably.
• Increase in bandwidth: Bandwidth increases by a factor of (1 + Aβ).
• Controlled gain even if the open loop gain is poorly defined.

Understanding Feedback Factor (β)

• Feedback Factor (β): Proportion of output fed back to input.

• Closed loop gain with feedback can be expressed as:

  [ A_f = \frac{A}{1 + A\beta} ]

Conditions for Stability in Feedback Systems

• Instability Condition:
  • If the loop gain magnitude equals 1 and the phase equals -180 degrees, instability occurs.
• Magnitude and Phase Analysis:
  • Loop gain expressed as a function of frequency (A(jω)β).
  • If both conditions occur at a specific frequency, the system will oscillate.

Stability Analysis Techniques

• Root Locus: Analyzes the poles of the system in the s-plane to determine stability.
  • Poles on the left half-plane indicate stability.
• Bode Plot: Magnitude and phase plots used to analyze stability.
  • Check if the phase crosses -180 degrees when the magnitude is 1.

Example: Analyzing Single-Pole and Two-Pole Systems

• Single-Pole Systems: Always inherently stable if the poles remain in the left half of the s-plane.
• Two-Pole Systems: Stability can be analyzed through root locus and Bode plots.
  • Poles can potentially enter the right half-plane as feedback (β) increases, indicating instability.

Summary of Key Points

• Understanding stability in feedback systems is crucial for designing reliable circuits.
• Negative feedback provides various advantages, including stability, noise reduction, and controlled gain.
• Analyzing stability can be done through root locus and Bode plot techniques.
• Instability in feedback systems can lead to oscillations and uncontrolled outputs, requiring careful design consideration.

Conclusion

• Feedback systems enhance amplifier performance through improved gain control and stability.
• Continuous evaluation of stability is necessary during system design to prevent oscillations and ensure predictable behavior.