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Next: Peak frequency Up: bode Previous: Second order systems

The Bode plot of second order system

The transfer function of a second order system (e.g., RCL circuit with voltage across the capacitor C) as the output) is

\begin{displaymath}
H(s)=k_0 \frac{\omega_n^2}{s^2+2\zeta \omega_n s + \omega_n^2}
\end{displaymath}

where $k_0$ is an arbitrary gain factor. If we let $s=j\omega$, we get the frequency transfer function

\begin{displaymath}
H(\omega)=\frac{\omega_n^2}{-\omega^2+2\zeta \omega_n j \ome...
...-(\frac{\omega}{\omega_n})^2+j2\zeta(\frac{\omega}{\omega_n})}
\end{displaymath}

Specially, when $\omega=\omega_n$, we have

\begin{displaymath}H(\omega_n)=\frac{1}{j2\zeta} =\frac{-j}{2\zeta} \end{displaymath}

with magnitude

\begin{displaymath}\left\vert H(\omega_n) \right\vert = \frac{1}{2\zeta} \stackrel{\triangle}{=} Q \end{displaymath}

The Bode plot of this transfer function can be found as

\begin{displaymath}20 log_{10} \left\vert H(\omega) \right\vert =-10 log_{10}
[(...
...ac{\omega^2}{\omega_n^2})^2+(2\zeta\frac{\omega}{\omega_n}^2)]
\end{displaymath}

and

\begin{displaymath}
\angle H(\omega) = -tan^{-1} \frac{2\zeta \omega / \omega_n}{1-(\omega/\omega_n)^2}
\end{displaymath}

To obtain the asymptotic approximation of this function, consider the following three cases:

rcl_bode_plots.gif

rcl_c_bode_plot.gif


next up previous
Next: Peak frequency Up: bode Previous: Second order systems
Ruye Wang 2003-10-17