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Poles and Zeros

From the quadratic formula, the two poles are located at

$\displaystyle s =
-\eta \pm \sqrt{\eta^2 - \omega_0^2}
\;\isdef \;
-\frac{1}{2RC} \pm \sqrt{\left(\frac{1}{2RC}\right)^2 - \frac{1}{LC}}
$

and there is a zero at $ s=0$ and another at $ s=\infty$. If the damping $ R$ is sufficienly small so that $ \eta^2 < \omega_0^2$, then the poles form a complex-conjugate pair:

$\displaystyle s = -\eta \pm j\sqrt{\omega_0^2 - \eta^2}
$

Since $ \eta = 1/(2RC) > 0$, the poles are always in the left-half plane, and hence the analog RLC filter is always stable. When the damping is zero, the poles go to the $ j\omega$ axis:

$\displaystyle s = \pm j\omega_0
$


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