• Bode plots play an important role in frequency domain analysis and design.
• Bode plots of a transfer function \(G(s)\) include two graphs:
  • Magnitude plot: \(20\log_{10}|G(j\omega)|\) (dB) in linear scale v.s. \(\omega\) in log scale.
  • Phase plot: \(\angle G(j\omega)\) (degree) in linear scale v.s. \(\omega\) in log-scale.

Consider the transfer function \(G(s) = \frac{s+1}{s^2+2s+3}\).

• Why choose \(20\log_{10}|G(j\omega)|\)?
  • We typical write a transfer function as \[G(s) = K\frac{(s+z_1)\dots(s+z_i)}{(s+p_1)\dots(s+p_n)}\]
  • The frequency response is \(G(j\omega) = G_1(j\omega)\dots G_n(j\omega)\). Hence,

\begin{align} 20\log|G(j\omega)|&=20\log|G_1(j\omega)|+\dots+20\log|G_n(j\omega)|\\ \angle G(j\omega)&=\angle G_1(j\omega)+\dots+\angle G_n(j\omega) \end{align}

• Why choose the frequency \(\omega\) in log-scale?
  • The magnitude plot can be easily approximated by straight lines
  • We can display a wider range of system behavior

As a result, Bode plots are plotted on semilog papers with the horizontal axis in log scale and the vertical axis in linear scale.

• A complex polynomial can always be factorized in to basic terms
  • \(K\)
  • \(s\)
  • \(\tau s + 1\)
  • \[\left(\frac{s}{\omega_n}\right)^2 + 2\zeta \left(\frac{s}{\omega_n}\right) + 1,\] where \(-1<\zeta < 1\). (Why?)
• To create the Bode plot of a complex rational transfer function, we only need to understand the Bode plots of simple polynomials

• Consider the following transfer function

\begin{align} G(s) = \frac{10(s+1)}{s(s^2+0.4s+4)} &= \frac{2.5\times (s+1)}{s \times \left[(0.5s)^2 + 2\times 0.1(0.5s)+1\right]}\\ &= \frac{G_1(s)G_2(s)}{G_3(s)G_4(s)}. \end{align}

• Therefore, we know that

\begin{align} \log|G(j\omega)| &=\log|G_1(j\omega)| + \log|G_2(j\omega)| -\log|G_3(j\omega)| -\log|G_4(j\omega)| \\ \angle G(j\omega) &=\angle G_1(j\omega) + \angle G_2(j\omega) -\angle G_3(j\omega) -\angle G_4(j\omega) \end{align}

• The magnitude is \(20\log|K|\).
• If \(K > 0\), then the phase is \(0^\circ\).
• If \(K < 0\), then the phase is \(-180^\circ\).

• The magnitude in dB is \[20\log|1/\omega| = -20 \log \omega\].
• The magnitude is a straight line passing \((1,\,0dB)\) with slope -20.
• The phase is \(-90^\circ\).

From what we know about \(s^{-1}\), we can prove that:

• The magnitude in dB is \(20N \log \omega\).
• The magnitude is a straight line passing \((0dB,\,1)\) with slope 20N.
• The phase is \(90N^\circ\).

• Here we will assume \(\tau > 0\). (\(\tau < 0\) will be discussed later)
• Observations:
  • The magnitude plot bends around \(10 = \tau^{-1}\).
  • The magnitude plot is a straight line when \(\omega \ll \tau^{-1}\) or \(\omega \gg \tau^{-1}\).
  • The phase is roughly \(0^\circ\) when \(\omega < 1 = 0.1/\tau\).
  • The phase is roughly \(-90^\circ\) when \(\omega > 100 = 10/\tau\).
  • The phase passes \(-45^\circ\) when \(\omega = 1/\tau\).

Asymptotes

• If \(\omega\tau\ll 1\), i.e., \(\omega\ll 1/\tau{}\), then \[G(j\omega{}) = \frac{1}{j\omega{\tau{}}+1} \approx 1,\,20\log\left|G(j\omega{})\right| \approx 0,\, \angle G(j\omega) \approx 0^\circ.\]
  • The magnitude plot is a horizontal line at 0dB.
  • The phase plot is a horizontal line at \(0^\circ\).
• If \(\omega\tau\gg 1\), i.e., \(\omega\gg 1/\tau{}\), then \[G(j\omega{}) = \frac{1}{j\omega{\tau{}}+1} \approx -j\omega^{-1}\tau^{-1},\, 20\log\left|\frac{1}{j\omega{\tau{}}+1}\right| \approx -20\log\omega{}-20\log\tau{},\, \angle G(j\omega) \approx -90^\circ.\]
  • The magnitude plot is a line passing \((1/\tau{},0)\) with a slope -20.
  • The phase plot is a horizontal line at \(-90^\circ\).
• \(1/\tau{}\) is called the corner frequency. It is where the two asymptotes of the magnitude plot intersect.
  • At the corner frequency, \(G(j/\tau{}) = 1/(1+j) = 0.5-0.5j = 0.707\angle -45^\circ\). The magnitude is \(-3dB\).

We can flip the plots of \(1/(\tau s+1)\) to get the plots of \(\tau s+1\)

Recall that

• \(\omega_n\) is called the natural frequency.
• \(\zeta\) is the damping ratio.

• If \(\omega \ll \omega_n\), then \(G(j\omega) \approx 1\)
  • The magnitude is 0dB.
  • The phase plot is \(0^\circ\).
• If \(\omega \gg \omega_n\), then \[G(j\omega) \approx 1/(j\omega/\omega_n)^2 = -\omega_n^2/\omega^2.\]
  • The magnitude is \(40\log \omega_n-40 \log\omega\). A straight line with slope -40 and passing \((\omega_n,0dB)\)
  • The phase is \(-180^\circ\).
• When \(\omega = \omega_n\) (corner frequency), \[G(j\omega) = \frac{1}{j^2+2\zeta j+1} = \frac{-1}{2\zeta}j.\]
  • The magnitude is \(-20\log(2\zeta)\).
  • the phase is \(90^\circ\).