Teaching guide for the module

RC and RLC filters: Teacher's Guide

Teaching guide for the Filters simulator: explain in class the frequency response of low-pass, high-pass, band-pass and series RLC filters. Bode plot in amplitude and phase, cutoff frequency, quality factor, decibels. Designed for electronics teachers.

Module: Filters · Three types: RC · Bandpass · RLC Series · Bode plot + oscilloscope

Physical phenomenon

A filter is a circuit that passes signals in certain frequency ranges and attenuates others. It exploits the frequency-dependent impedance of reactive components (inductors and capacitors) introduced in the AC module.

The main types covered by the module:

RC low-pass: passes low frequencies, attenuates high ones. The cutoff frequency is $f_{c} = \frac{1}{2 π R C}$ , defined as the frequency at which the output amplitude drops to $1/ 2$ of the passband value (corresponding to $- 3 dB$ ).
RC high-pass: symmetric behaviour: passes high frequencies, attenuates low ones. Same $f_{c}$ formula.
Band-pass: combination that passes a frequency range around a central frequency.
Series RLC: sharper response, with a clean peak at resonance frequency $f_{0} = \frac{1}{2 π L C}$ . The quality factor $Q$ measures peak narrowness.

The Bode plot is the standard representation of frequency response: frequency on the x-axis (logarithmic), amplitude in decibels ( $20 lo g_{10} ∣ H (f) ∣$ ) and phase in degrees on the y-axis. This visualisation linearises behaviours that would be exponential in linear space, and highlights the slope of attenuations, typically $- 20 dB/decade$ per reactive component of the filter.

Key concepts

Cutoff frequency $f_{c}$ : the boundary between passband and stopband, defined at $- 3 dB$ .
Decibel (dB): logarithmic amplitude scale: $- 3 dB$ corresponds to $1/ 2$ , $- 20 dB$ to $1/10$ , $- 40 dB$ to $1/100$ .
Slope (roll-off): $- 20 dB/decade$ for first-order RC filters, $- 40 dB/decade$ for series RLC.
Frequency-dependent phase shift: the filter modifies the signal's phase too, not just its amplitude. Central for audio and control applications.
Passband: range of frequencies transmitted without significant attenuation.
Quality factor $Q$ : selectivity measure of an RLC filter: higher means narrower passband.

How to use it in the classroom

Opening: the RC low-pass filter. Load the RC configuration. Show the Bode plot: amplitude is flat at low frequencies and then falls with $- 20 dB/decade$ slope. Visually identify the $- 3 dB$ point and link it to the value $f_{c} = 1/ (2 π R C)$ computed mentally. Vary $R$ or $C$ and observe how $f_{c}$ shifts.

Development: time domain. Press the SCOPE tab in the visualisation panel. Show the input sinusoid (blue) and the output one (gold). At low frequency the two practically overlap; as the frequency increases, the output amplitude shrinks and a growing phase shift appears. This is exactly what the Bode plot shows in logarithmic language, here seen in the time domain.

Deeper exploration: RLC resonance. Switch to the series RLC configuration. The Bode plot now shows a resonance peak at $f_{0}$ instead of a simple cutoff frequency. Vary $R$ : at the same $L$ and $C$ , a small resistance produces a tall narrow peak (high selectivity), a large resistance produces a low broad peak (low selectivity). This is the quality-factor concept directly visualised.

Closing: applying the concept. Ask students to suggest where a low-pass would be useful (audio noise, antialiasing before A/D conversion), where a high-pass (DC offset removal), where a band-pass (radio station tuning).

Real-world examples

Audio crossovers. In multi-way speakers, low-pass and high-pass filters separate the frequencies sent to the woofer (low) and the tweeter (high).
Antialiasing in A/D converters. A low-pass eliminates components above the Nyquist frequency before sampling, otherwise aliasing artefacts appear that cannot be undone.
Mains EMI filters. RC and RLC filters remove radiofrequency disturbances from the power lines of sensitive equipment.
Equalisers. Combinations of parametric filters control individual frequency bands in professional audio.
Analogue radio receivers. A band-pass filter tuned to the desired station's frequency selects the signal while rejecting other stations.

Classroom discussion questions

What does it mean that at $f_{c}$ the amplitude drops to $- 3 dB$ ? How much of the signal remains in numerical ratio?
If I double $C$ in an RC low-pass, where does the cutoff frequency shift?
Why does a series RLC filter have double the slope ( $- 40 dB/decade$ ) of an RC?
An audio recording has annoying high-frequency noise. Which filter would you apply and why?
In an RLC band-pass with low $R$ , the peak is tall and narrow. What practical consequence in "tuning" a radio station?

Related modules

AC Behaviour (R, L, C): the conceptual prerequisite: frequency-dependent impedance is what makes filtering possible.
Capacitor Charge & Discharge: the time constant $τ = R C$ of the transient is tightly linked to the cutoff frequency $f_{c} = 1/ (2 π R C)$ of the filter.