Transfer function

Filters can remove low and/or high frequencies from an electronic signal. My article on RC Low-pass Filter introduced a first order low-pass filter. The second order filter introduced here improves the unit step response and the the roll-off slope for the frequency response. As we will learn, even this passive filters may exhibit resonance near the natural frequency.\(\)

\(u(t)\) Instead of \(\Delta v(t)\), we use the European symbol for voltage difference: \(u\). The letter ‘u’ stands for “Potentialunterschied”.

We examine the properties of a filter consisting of a series circuit of an inductor \(L\), resistor \(R\) and capacitor \(C\). The output is taken across the capacitor as shown in the schematic below. We show the transfer function and derive the step and frequency response.

own work
RLC circuit

This article describes a low-pass filter, but the same principles apply to high and band pass filters and can even be extended to to resonators. For example, taking the voltage over the inductor results in a high-pass filter, while taking the voltage over the resistor makes a band-pass filter.

Prerequisite reading includes Laplace Transforms, Impedance and Transfer Functions.

Transfer Function

In the RLC circuit, shown above, the current is the input voltage divided by the sum of the impedance of the inductor \(Z_L\), the impedance of the resistor \(Z_R=R\) and that of the capacitor \(Z_C\). The output is the voltage over the capacitor and equals the current through the system multiplied with the capacitor impedance.

$$ \begin{align} Y(s) &= I(s)Z_C=\frac{U(s)}{Z_L+Z_R+Z_C} Z_C \Rightarrow \nonumber \\ H(s) &= \frac{Y(s)}{U(s)}=\frac{Z_C}{Z_L+Z_R+Z_C} =\frac{\frac{1}{sC}}{sL+R+\frac{1}{sC}} = \frac{1}{s^2LC+sRC+1} \label{eq:voltagedivider} \end{align} $$

The denominator of \(\eqref{eq:voltagedivider}\) is a second-order polynomial. The roots to this polynomial are called the system’s poles.

$$ \shaded{ H(s) = K\frac{1}{(s-p_1)(s-p_2)} } \label{eq:transferpolynomial} $$
where
$$ \begin{align*} K &= \frac{1}{LC} \\ p_{1,2} &= -\frac{R}{2L} \pm \sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}} \end{align*} $$

Note that the factor \(K\) can be expressed as the product \(p_1\cdot p_2\) by applying the identity \((a+b)(a-b)=a^2-b^2\)

$$ K = p_1\cdot p_2=\frac{1}{LC} \label{eq:p1p2} $$

Here is where is gets interesting:

The poles in \(\eqref{eq:transferpolynomial}\) may be real or complex conjugates.

To highlight this, we parameterize the poles in terms of the natural frequency \(\omega_n\) and damping ratio \(\zeta\).

$$ \begin{align} p_{1,2} &= -\frac{R}{2L} \pm \sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}}\nonumber\\ &= \frac{1}{\sqrt{LC}} \left( -\sqrt{LC}\frac{R}{2L} \pm \sqrt{LC}\sqrt{\left(\frac{R}{2L}\right)^2-\frac{1}{LC}} \right) \nonumber \\ &= \underbrace{\frac{1}{\sqrt{LC}}}_{\omega_n} \left( -\underbrace{\frac{R}{2}\sqrt\frac{C}{L}}_{\zeta} \pm \sqrt{\underbrace{\left(\frac{R}{2}\sqrt\frac{C}{L}\right)^2}_{\zeta^2}-1} \right) \\ \end{align} $$

Assign the natural frequency \(\omega_n=\sqrt{\frac{1}{LC}}\) and the damping ratio \(\zeta=\frac{R}{2}\sqrt{\frac{C}{L}}\). These parameter choices will become evident as we examine complex conjugate poles.

$$ \shaded{ H(s) = K\frac{1}{(s-p_1)(s-p_2)} } \label{eq:transferpoles} $$
where
$$ \begin{align*} K &= \frac{1}{LC} \\ p_{1,2} &= \omega_n\left(-\zeta \pm \sqrt{\zeta^2-1}\right) \\ \zeta &= \frac{R}{2}\sqrt{\frac{C}{L}} \\ \omega_n &= \sqrt{\frac{1}{LC}} \end{align*} $$

This dampening coefficient \(\zeta\) determines the behavior of the system. Given that the value of \(\frac{1}{LC} >0\) and \(\frac{R}{L}\geq0\), we can identify three classes of pole locations.

Effect of dampening coefficient \(\zeta\) on system behavior
ζ Pole location Referred to as
\(\zeta>1\) different locations on the negative real axis overdamped \(R>2\sqrt\frac{L}{C}\)
\(\zeta=1\) coincite on the negative real axis critically damped \(R=2\sqrt\frac{L}{C}\)
\(\zeta<1\) complex conjugate poles in the left half of the s-plane underdamped \(R<2\sqrt\frac{L}{C}\)

In the overdamped case, \(\zeta>1\), the following relation between \(R\), \(L\) and \(C\) must be true

$$ \begin{align} \zeta&>1\nonumber\\ \implies \frac{R}{2}\sqrt{\frac{C}{L}}&>1 \nonumber \\ \implies R&>2\sqrt\frac{L}{C} \end{align} $$

Following sections determine the step and frequency response for each of these classes.
  • Two different real poles
  • Coinciting real poles
  • Complex poles