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Operational Amplifiers

Negative Feedback

Overview

Negative feedback is a crucial concept in operational amplifiers that enhances stability and control.

Important Equations

  • Closed-loop Transfer Function (Gain):

Negative feedback loop

\[ H_{closed}(j\omega) = \frac{A(j\omega)}{1 + A(j\omega)B(j\omega)} \]
  • Where \( A(j\omega) \) is the open-loop gain

  • \( B(j\omega) \) is the feedback factor

  • The Non-Inverting Amplifier examples:

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Stability Considerations

Overview

Understanding stability is essential to ensure that negative feedback systems do not lead to oscillations or performance degradation.

Stability Analysis

  • Oscillations: Occur if system poles are near or on the imaginary axis
  • Nyquist Stability Criterion: Determines stability based on the system's loop gain
  • Routh-Hurwitz Criterion: Used to analyze stability in the time domain

Gain and Phase Margins

  • Gain Margin (G.M.): Distance (in dB) from 0 dB to magnitude at frequency \( f_{\pm180} \)
  • Phase Margin (P.M.): Distance (in degrees) from \( \pm180^\circ \) to the frequency where gain crosses 0 dB

  • Loop Gain Analysis:

    \[ Loop Gain = A(j\omega)B(j\omega) \]

Stability Guidelines

  • A system is stable if G.M. ≥ 10 dB and P.M. ≥ 45°
  • If the loop gain crosses \( -1 \) at a critical frequency, the system may oscillate

Example: Transfer Function for Active Inverting Low-Pass Filter (LPF)

This example demonstrates how to compute the transfer function and the cut off frequnecy of a first-order inverting LPF using op-amp feedback.

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Example solved here

Example: Stability via Gain and Phase Margin

This example applies gain and phase margin concepts to assess system stability.

Example solved here