# Simple RC low-pass filter

Below is a LTSpice model for a simple RC low-pass filter.

\ \begin{align} Impedance \ of \ a \ capactor, X_C& = {1 \over j\omega C} \\ Impedance \ of \ a \ inductor, X_L& = {j \omega L} \\ \omega& = 2 \pi f_c \end{align}

Setup the input, V(in) as 1V amplitude with a frequency of 200Hz

$$\ {V(out) \over V(in)} = {{1 \over j \omega C }\over R+ {1 \over j \omega C}} = {1 \over {1+j \omega RC}}$$ $$\ \omega = 2 \pi f_c$$ $$\ f_c = {1 \over 2 \pi RC} = {1 \over 2 \pi * 1K * 1u} = 160Hz$$

## Why is the magnitude down 3dB when power is half ?

$$\ P_{dB} = {10 \ log ({P{out} \over P_{in}})}$$ $$\ P_{3dB} = {10 \ log ({{P_{out} \over 2}\over P_{in}})}$$ $$\ P_{3dB} = {10 \ log ({P_{out} \over P_{in}}) - 10 \ log2}$$ $$\ P_{3dB} = {10 \ log ({P_{out} \over P_{in}}) - 3dB}$$ $$\ P_{3dB} = {P_{dB} - 3dB}$$ In the bode plot, we can see the 3dB cutoff frequency on the cursor corresponding to a phase of 45$$^{\circ}$$

## Experiment: Effect of RC filter on square wave rise time

Tools:

Here is a snapshot of the 1KHz square wave scope cal signal on Ch1.

Now let's take this 1KHz square wave and pass it through the RC filter above. Here's our circuit on a breadboard:

When we pass this square wave through the above RC filter, it slows the rise/fall times as shown in the snapshot below on Ch2.

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