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.

Substitute \(\zeta=1\) and \(\omega_n\) in the equation for the poles.

This double pole \(p\lt0\) is on the left real axis, as visualized in the \(s\)-plane

### Unit Step Response

Multiplying the Laplace transform of the unit step function, \(\Gamma(s)\), with the transfer polynominal gives the unit step response \(Y(s)\).

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]

Substitute \(K=p^2\) in the pole equations and use Heaviside’s cover up method to find the first two constants \(c_{0,1}\).

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]

The unit step response \(y(t)\) follows from the inverse Laplace transform of \(\eqref{eq:case2a_heaviside}\)

Substituting the constants \(\eqref{eq:case2a_constants1}\) and \(\eqref{eq:case2a_constants2}\) yields

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.

### Frequency Response

The frequency response \(y_{ss}(t)\) is defined as the steady state response to a sinusoidal input signal \(u(t)=\sin(\omega t)\,\gamma(t)\).

In Evaluating Transfer Functions, we have proven that

The transfer function \(H(s)\) for this RLC Filter is given by the transfer polynominal and the poles given by \(\eqref{eq:case2a_p}\)

Based on Euler’s formula, we can express \(H(s)\) in polar coordinates

This transfer function with the double poles at \(p\), evaluated for \(s=j\omega\) can be visualized with vectors from the poles to \(j\omega\).

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 \(\angle H(j\omega)\).

The output signal \(y_{ss}(t)\) for a sinusoidal input signal \(\sin(\omega t)\,\gamma(t)\) for \(\zeta=1\) follows as

The magnitude of the transfer function \(\eqref{eq:case2_frequencyresponse}\) expressed on a logarithmic scale

The magnitude of the frequency response has a relatively steep -40 dB/decade drop-off at \(\omega_n\) without signs of resonance.

The corresponding Nyquist plot