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

Vo/Vi

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

Frequency Response

 * ω = 0 .  $$V_o = V_i$$
 * ω = $$\frac{R}{L}$$ .  $$V_o = V_i/2$$
 * ω = 00 .  $$V_o = 0$$


 * [[Image:Bode_Low-Pass.PNG|200px]]

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

Low-pass Filter, RC Configuration

 * [[Image:1st_Order_Lowpass_Filter_RC.svg|200px]]

Vo/Vi

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

Frequency Response

 * {| class="wikitable"

! $$\omega = 0$$ !! $$\omega = \frac{1}{RC}$$ !! $$\omega = \infty$$
 * $$V_o = V_i$$ || $$V_o = V_i/2$$ || $$V_o = 0$$
 * }
 * }

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