9.12 : Passive Filters

Passive filters are utilized to shape the frequency spectrum of signals across a diverse array of applications. These filters, using only passive elements like resistors (R), inductors (L), and capacitors (C), are capable of selectively allowing or blocking certain frequency ranges without the need for external power sources.

Low-Pass Filters

Low-pass filters are designed to transmit signals with frequencies lower than the cutoff frequency, ω_c, and attenuate those above it. The cutoff frequency is defined by:

They are commonly employed to route audio signals to woofers or subwoofers, ensuring that only bass frequencies reach these devices. The typical transfer function H(s) for an RC low-pass filter is given by:

In this equation, V_o is the output voltage, V_i is the input voltage, ω is the angular frequency, R is the resistance, and C is the capacitance.

High-Pass Filters

In contrast, high-pass filters allow frequencies above ω_c to pass while blocking those below. These filters are vital in directing high-frequency components to tweeters in audio systems. An RC high-pass filter has the following transfer function:

Band-Pass Filters

Band-pass filters are engineered to pass a band of frequencies, making them ideal for channels dedicated to mid-range audio frequencies.

Band-stop Filters

These are also known as notch filters; band-stop filters reject a specific band of frequencies to avoid interference, which is particularly useful in complex electronic systems where multiple signals overlap.

The design of passive filters is a delicate balance of selecting the right component values to achieve the desired cutoff frequencies and attenuation levels.