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

The Bode plot of second order systems

We first consider the transfer function of a 2nd order system (e.g., RCL circuit with voltage across the capacitor C as the output):

\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(j\omega)=\frac{\omega_n^2}{-\omega^2+2\zeta \omega_n j \om...
...-(\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(j\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(j\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:

Next consider a second type of 2nd order system (e.g., RCL circuit with voltage across the resistor R as the output):

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

As the denominator of the frequency response function is already considered above, all we need to do now is to find the log-magnitude of the numerator:

\begin{displaymath}20 log_{10} \left\vert j2\zeta \omega/\omega_n \right\vert=20 log_{10} ( 2\zeta \omega/\omega_n ) \end{displaymath}

The is a straight line with a slope of 20 dB/decade. In particular, when $\omega=\omega_n$, it is $20 log_{10} (2\zata)$, which cancels the denominator $-20 log_{10} (2\zata)$, in other words, at $\omega=\omega_n$, $H(j\omega)=1$ and $20 \log_{10} \left\vert H(j\omega)\right\vert=0$. The phase of the numerator is

\begin{displaymath}\angle (j2\zeta \omega/\omega_n)=\pi/2 \end{displaymath}

Finally consider a second type of 2nd order system (e.g., RCL circuit with voltage across the inductor L as the output):

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

the log-magnitude of the numerator:

\begin{displaymath}20 log_{10} \left\vert (-(\omega/\omega_n)^2 \right\vert=40 log_{10} ( \omega/\omega_n ) \end{displaymath}

The is a straight line with a slope of 40 dB/decade. In particular, when $\omega=\omega_n$, it is $20 log_{10} (1)=0$. The phase of the numerator is

\begin{displaymath}\angle (-(\omega/\omega_n)^2)=\pi \end{displaymath}

rcl_bode_plots.gif

rcl_c_bode_plot.gif


next up previous
Next: Peak frequency Up: active_filter Previous: Second order systems
Ruye Wang 2008-11-24