The 101 of Analog Signal Filtering

Summary

Signal filters are ubiquitous in electronics; on this blog alone, they cropped up in articles on digital-to-analog converters, radio receivers, audio amplifiers, and probably more. In principle, the relative simplicity of these circuits should make the underlying theory easy to self-study; in practice, most introductory texts rapidly devolve into a foul mix of EE jargon and calculus. On Wikipedia, unsuspecting visitors are usually greeted by some variant of this:“We wish to determine the transfer function H(s) where s = σ + jω (from Laplace transform). Because |H(s)|² = H(s)H(s)¯ and, as a general property of Laplace transforms at s = jω, H(−jω) = H(jω)¯, if we select H(s) such that:\(H(s)H(-s) = \frac {{G_0}^2}{1+\left (\frac{-s^2}{\omega_c^2}\right)^n}\)The n poles of this expression occur on a circle of radius ωc at equally-spaced points, and symmetric around the negative real axis. For stability, the transfer function, H(s), is therefore chosen such that it contains only the poles in the negative real half-plane of s. The k-th pole is specified by …”There are some aspects of signal processing that require solving novel integrals or navigating the complex S-plane — but in today’s article, let’s try a gentler approach. The following write-up still assumes familiarity with electronic concepts such as capacitance and reactance, so if you need a refresher, start here and here.Have a look at the following circuit:Let’s assume that the capacitor is discharged and that the circuit’s input terminal gets hooked up to a standard, benchtop voltage supply. Further, let’s say there’s no load connected to the output leg. Chances are, you already know that the current through the resistor (blue) will initially shoot up to I = V/R as the capacitor starts charging, but then gradually decay to zero. Meanwhile, the voltage across the capacitor’s terminals (yellow) will increase from 0 all the way to what’s present on the input leg — in my example, 48 volts:A 100 µF capacitor charge...