Electronics Handbook/Circuits/Low-pass Filter

Low-pass Filter

A low-pass filter has a stable voltage at low frequencies and increasing voltage at high frequencies.

Low-pass Filter , LR Configuration

Picture

V_o/V_i

$\frac{V_o}{V_i}=\frac{Z_R}{Z_R+Z_L}=\frac{R}{R+j\omega L}=\frac{V_o}{V_i}=\frac{1}{1+j\omega T}$

$T=\frac{L}{R}$

Frequency Response

ω = 0 . $V_o=V_i$

ω = $\frac{R}{L}$ . $V_o=V_i/2$

ω = 00 . $V_o=0$

LR has a stable voltage at low frequencies and increasing voltage at high frequencies.

Low-pass Filter , RC Configuration

V_o/V_i

$\frac{V_o}{V_i}=\frac{Z_C}{Z_R+Z_C}=\frac{\frac{1}{j\omega C}}{R+\frac{1}{j\omega C}}=\frac{1}{1+j\omega CR}=\frac{1}{1+j\omega T}$ với

$T=CR$

Frequency Response

$\omega =0$	$\omega =\frac{1}{RC}$	$\omega =\mathrm{\infty }$
$V_o=V_i$	$V_o=V_i/2$	$V_o=0$

RC has a stable voltage at low frequencies and increasing voltage at high frequencies.

Summary

1. Low-pass filters have a stable voltage at low frequencies and increasing voltage at high frequencies.
2. Low-pass filters can be constructed from RC or LR.
3. Low-pass filters have a general formula:

$\frac{V_o}{V_i}=\frac{1}{1+j\omega T}$ . T = RC = $\frac{L}{R}$

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