Frequency Response and Bode Plots

Frequency Response

• It is the steady-state response of a system to a sinusoidal input signal.
• Used for designing analog filters and feedback control systems.
• It also helps in characterizing communication channels.

RC Circuit Example

• Relationship between input $v_{in}(t)$ and output $v_{out}(t)$ is described by: $RC \frac{dv_{out}(t)}{dt} + v_{out}(t) = v_{in}(t)$
• Transfer function is given by: $H(s) = \frac{V_{out}(s)}{V_{in}(s)} = \frac{1}{RCs + 1}$
• With $s = j\omega$, the transfer function becomes: $H(j\omega) = \frac{1}{1 + j\omega RC}$
• Magnitude of the frequency response: $|H(j\omega)| = \frac{1}{\sqrt{1 + (\omega RC)^2}}$
• Phase of the frequency response: $\angle H(j\omega) = -\arctan(\omega RC)$

Bode Plot

• It is a graph showing magnitude and phase of frequency response versus frequency.
• Magnitude is in decibels (dB): $20\log_{10}|H(j\omega)|$
• Phase is in degrees: $\angle H(j\omega)$
• Frequency is on a logarithmic scale.

Bode Plot of RC Circuit Example

• Magnitude in dB: $20\log_{10}|H(j\omega)| = -10\log_{10}(1 + (\omega RC)^2)$
• Phase in degrees: $\angle H(j\omega) = -\arctan(\omega RC)$
• Cutoff frequency: $\omega_c = \frac{1}{RC}$
• Magnitude Approximation:
  • Approximately 0 when $\omega \ll \omega_c$.
  • Approximately $-20\log_{10}(\omega RC)$ when $\omega \gg \omega_c$.
• Phase Approximation:
  • Approximately 0 when $\omega \ll \omega_c$.
  • Approximately -90 degrees when $\omega \gg \omega_c$.

General Transfer Function

• General form: $H(s) = \frac{K(s + z_1)(s + z_2)\cdots(s + z_m)}{(s + p_1)(s + p_2)\cdots(s + p_n)}$ where $z_i$ are zeros and $p_i$ are poles.
• Magnitude in dB: $20\log_{10}|H(j\omega)| = 20\log_{10}|K| + \sum_{i=1}^m 20\log_{10}|j\omega + z_i| - \sum_{i=1}^n 20\log_{10}|j\omega + p_i|$
• Phase in degrees: $\angle H(j\omega) = \angle K + \sum_{i=1}^m \angle(j\omega + z_i) - \sum_{i=1}^n \angle(j\omega + p_i)$

First-Order Factors

• Zero at $s = -z$: $H(s) = s + z$
  • Magnitude:
    • Approximately $20\log_{10}|z|$ when $\omega \ll |z|$.
    • Approximately $20\log_{10}|\omega|$ when $\omega \gg |z|$.
  • Phase:
    • Approximately 0 when $\omega \ll |z|$.
    • Approximately 90 degrees when $\omega \gg |z|$.
• Pole at $s = -p$: $H(s) = \frac{1}{s + p}$
  • Magnitude:
    • Approximately $-20\log_{10}|p|$ when $\omega \ll |p|$.
    • Approximately $-20\log_{10}|\omega|$ when $\omega \gg |p|$.
  • Phase:
    • Approximately 0 when $\omega \ll |p|$.
    • Approximately -90 degrees when $\omega \gg |p|$.

Second-Order Factors

• Complex conjugate poles at $s = -\zeta\omega_n \pm j\omega_n\sqrt{1 - \zeta^2}$: $H(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$
  • Magnitude: $20\log_{10}|H(j\omega)| = 20\log_{10}\left|\frac{\omega_n^2}{(j\omega)^2 + 2\zeta\omega_n (j\omega) + \omega_n^2}\right|$
  • Phase: $\angle H(j\omega) = \angle\left(\frac{\omega_n^2}{(j\omega)^2 + 2\zeta\omega_n (j\omega) + \omega_n^2}\right)$

Summary Points

• Frequency response analyzes steady-state response to sine inputs.
• Bode plots graph magnitude and phase.
• Useful for analyzing system stability and performance.