Understanding Bode Plots and Op-Amp Analysis

Bode Plot Basics

Bode Plot: Illustrates how a signal's amplitude and phase change as it passes through a system.
Gain: Magnification of the signal, measured in dB (decibels).
- Gain of 6 dB corresponds to a quadrupling of signal (factor of 2²).
- Calculated as (10 \log_{10}\left(\frac{V_{out}}{V_{in}}\right)).
Phase: Shift of the sinusoid along the horizontal axis relative to its period, measured in degrees.
- 360° phase shift is equivalent to one period.

Vertical Axes: Two sets for phase and gain.
Logarithmic Axes: Both frequency and gain axes are logarithmic.
- Helps visualize a broad frequency range and gain values.
Gain Characteristics: Signals of various frequencies change in magnitude when passing through a system.
- Example: For an op-amp circuit at 100 kHz, the gain is nearly 20 dB (100x amplification).
- Gain diminishes at higher frequencies, reaching 0 dB at about 700 kHz (no gain/loss).
- At very high frequencies (e.g., 4 MHz): signals lose amplitude (-20 dB, 100-fold loss).
Phase Characteristics: Shows how the output signal phase shifts relative to the input signal.
- Example: Op-amp circuit has a 90° phase shift at 100 kHz.
- Phase approaches zero at 3 MHz (input and output signals are in phase).

Fourier Transform: Analyzes frequency content of a signal (e.g., peaks at certain frequencies).
Bode Plot: Illustrates how signals of different frequencies are modified by a system.
- Example: At 400 kHz, Bode plot shows amplification (observed change in frequency content of output signal).

Linear Time-Invariant (LTI) Systems: Bode plots valid for such systems.
- Linear: Effects on combined signals are equivalent to individual signals combined.
- Time-Invariant: System's effect on an input signal remains consistent over time.
Examples of non-LTI systems:
- Explicit time dependence in the system function.
- Nonlinear terms (e.g., cubic in signals).

Use Kirchhoff's Voltage Law to derive the Ordinary Differential Equation (ODE) for the circuit.
Example RC Circuit: 100 kΩ resistor and 1 μF capacitor.
- Apply a sinusoidal input and solve the ODE.
- Calculate gain and phase shift from the periodic solution.
- Ignore transient terms as they do not affect long-term behavior.
Transfer Function: Tool for analyzing LTI systems.
- Derived by taking the Laplace Transform of the ODE.
- Gain from the magnitude of the complex transfer function, phase from its angle.
- Plot over a range of frequencies to construct the Bode diagram.
Real-World Measurement: Probe a physical circuit and record input/output.
- Example: Use an Arduino to generate a chirp signal and record response.
- Compute the FFTs for input and output, then their ratio to get gain and phase.
- Compare experimental Bode plot to analytical plot.

Both Fourier transforms and Bode diagrams are useful tools for analyzing signal behavior in systems.
Bode plots specifically help visualize frequency response and are valid for LTI systems.
Practical methods and analytical techniques both yield consistent results in constructing Bode plots.