Shows the math of a critically-damped RLC low pass filter. Visualizes the poles in the Laplace domain. Visualizes the step and frequency response. Part of the article RLC Low-pass Filter.\(\)
Coinciting Real Poles (critically-damped case)
In the critically-damped case, the two poles from the transfer polynominal coincite on the negative real axis.
Split up this complicated fraction into forms that are in the Laplace Transform table. According to Heaviside, this can be expressed as partial fractions. Note the factor \(\frac{c_2}{s-p}\). [swarthmore, MIT-cu]
Given \(c_0\) and \(c_1\), constant \(c_2\) can be found by substituting any numerical value (other than \(0\) or \(p\)) in equation \(\eqref{eq:case2a_heaviside}\). In this case, we substitute \(s=-p\) [MIT-ex4]
As shown in the graph below, this unit step response is a relatively fast rising exponential curve, demonstrating the shortest possible rise time without overshoot.
Step response for critically-damped case
Frequency Response
The frequency response \(y_{ss}(t)\) is defined as the steady state response to a sinusoidal input signal \(u(t)=\sin(\omega t)\,\gamma(t)\).
This transfer function with the double poles at \(p\), evaluated for \(s=j\omega\) can be visualized with vectors from the poles to \(j\omega\).
Transfer function evaluated at \(s=j\omega\) for critically-damped case
The square of the length of the vector corresponds to \(|(H(j\omega)|\), and minus twice the angles with the real axis corresponds to phase shift \(\angle H(j\omega)\).
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