Transfer function

Shows the math of a first order RC low-pass filter. Visualizes the poles in the Laplace domain. Calculates and visualizes the step and frequency response.

Filters can remove low and/or high frequencies from an electronic signal, to suppress unwanted frequencies such as background noise. This article shows the math and visualizes the system response of such filters.\(\)

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

One of the simplest forms of passive filters consists of a resistor and capacitor in series. The output is the voltage over the capacitor \(y(t)\) as shown in the schematic below.

own work
Schematic RC filter

This type of filter is called an Infinite-Impulse Response (IIR) filter, because if you give it an impulse input, the output takes an infinite time to go down to exactly zero.

Even though this article shows a low pass filter, the same principles apply to a high pass filter where the output is taken over the resistor.

We will derive the transfer function for this filter and determine the step and frequency response functions. Required prior reading includes Laplace Transforms, Impedance and Transfer Functions.

In this article will will use Laplace Transforms. The alternate method of solving the linear differential equation is shown in Appendix B for reference.

Transfer Function

TransferFunction
Transfer function

In the RC circuit, shown above, the current is the input voltage divided by the sum of 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. The transfer function follows as the quotient of the output and input signals.

$$ H(s) = \frac{Y(s)}{U(s)} = \frac{Z_C}{Z_C+Z_R} = \frac{1/sC}{1/sC+R} = \frac{1}{sRC+1} \label{eq:voltagedivider} $$

The denominator of \(\eqref{eq:voltagedivider}\) is a first-order polynomial. The root of this polynomial is called the system’s pole

$$ \shaded{ H(s) = K\frac{1}{s-p},\quad K = \frac{1}{RC},\quad p = -\frac{1}{RC} } \label{eq:transferpolynomial} $$

This first order system \(\eqref{eq:transferpolynomial}\) has no zeros and one stable pole \(p\) on the left real axis \(p\lt 0\) as visualized in the \(s\)-plane.

own work
\(s\)-plane

Moving on from this “unit step response of RC low-pass filter”, continue reading about the Unit Step Response.