Shows the math of a overdamped RLC low pass filter. Visualizes the poles in the Laplace domain. Calculates the step and frequency response. Part of the article RLC Low-pass Filter.\(\)
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Two Different Real Poles (overdamped case)
The two poles from the transfer polynominal are on separate locations on the negative real axis.
Note that \(p_1\lt p_2\lt0\) and \(|p_1|>|p_2|\), as visualized in the \(s\)-plane
\(s\)-plane for overdamped case
Unit Step Response
The unit step response shows how the system reacts to the input going from \(0\) to \(1\) volt at time \(t=0\). This input is called the Unit Step Function, here represented by \(u(t)=\gamma(t)\). The unit step response gives an impression of the system behavior in the time domain.
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 that we need to set up a partial fraction for each descending power of the denominator. [swarthmore, MIT-cu]
The unit step response \(y(t)\) follows from the inverse Laplace transform of \(\eqref{eq:case1a_heaviside}\) and substituting the constants \(\eqref{eq:case1a_constants}\)
As shown in the graph, the unit step response is a relatively slow decaying exponential curve (with \(p_1\lt p_2\lt 0\)). The figure was generated using the source code listed in the appendix.
Unit step response for over dampened case
Frequency Response
The frequency response \(y_{ss}(t)\) is defined as the steady state response to a sinusoidal input signal \(u(t)=\sin(\omega t)\,\gamma(t)\). It describes how well the filter can distinguish between different frequencies.
This transfer function with poles at \(p_1\) and \(p_2\), evaluated for \(s=j\omega\) can be visualized with vectors from the poles to \(j\omega\).
Transfer function evaluated at \(s=j\omega\) for over dampened case
The product of the length of the vector corresponds to \(|(H(j\omega)|\), and minus the sum of the angles with the real axis corresponds to phase shift \(\angle H(j\omega)\).
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