Filter + Amplifier Chain for a relay

Passive filters shape a signal's frequency content with resistors, capacitors, and inductors. An RC low-pass or high-pass has a cutoff at 1/(2πRC) where the response drops 3 dB; an RLC bandpass resonates at 1/(2π√(LC)) with sharpness set by its quality factor Q.

The op-amp is the fundamental building block of analog electronics. With negative feedback, its gain is set entirely by external resistors: an inverting amp gives −R2/R1, a non-inverting amp 1+R2/R1. An integrator uses a feedback capacitor to output the running integral of its input.

A Bode plot shows how a system responds across frequency: gain in decibels on top, phase shift below. This second-order low-pass passes low frequencies and rolls off at −40 dB/decade above its natural frequency. Low damping produces a resonant peak; high damping smooths it away.

To capture a signal faithfully you must sample above twice its highest frequency — the Nyquist rate. Sample too slowly and a high frequency masquerades as a lower one: aliasing. Push the signal frequency above half the sample rate and watch the reconstructed wave collapse to a false, slower tone.