# Low Pass Filter – Simulations, Calculators, and Theory

Low pass filters (LPF) are circuits that allow lower frequencies to pass through while blocking higher frequencies. This can be useful in may applications like filtering out the higher frequencies music signal to a bass speaker (speaker designed to produce low frequencies). Additionally these filters can remove unwanted noise from a circuit.

## RC Low Pass Filter

The simplest LPF circuit is the RC low pass filter. This LPF is just a resistor in series with a capacitor where the output is taken in between the two components.

The basic idea for how this filter works is at low frequencies the capacitor C_1 acts like a open and therefor making the output equal to the input. Conversely, at high frequencies the capacitor C_1 acts like an short so that the entire input signal is shorted to ground and making the output equal zero.

Below is a video of us demonstrating how to do a KiCad simulation of an RC LPF circuit.

## RC Low Pass Filter Simulation (Video)

This is a video of a KiCad low pass filter simulation showing the effects of adjusting the resistor and capacitor in a RC low pass filter.

Vin (optional)

## Theory (Approximation)

For the RC LPF approximation equation just use X_{C_1} for the capacitor and treat it like a resistor. Below is the equation for a voltage divider

V_{out} = V_{in}{X_{C_1} \over{X_{C_1} + R_1}}
\boxed{V_{out} = V_{in}{{1 \over(2 \pi f C_1)} \over{{1 \over(2 \pi f C_1)} + R_1}}}

At high frequencies

Using the capacitive reactance equation we can see that as frequency goes up \uparrow f \Rightarrow {1 \over(2\pi (f \uparrow ) C_1)}\downarrow \Rightarrow X_{C_1} \downarrow and at a high enough frequency X_{C_1} \approx 0 . When this happens the RC LPF equation turns into V_{out} \approx V_{in} {0\over 0 + R_1} \Rightarrow V_{out} \approx 0 thus blocks all higher frequencies

V_{out} = V_{in} {\downarrow X_{C_1} \over \downarrow X_{C_1} + R_1} \approx V_{in}{0 \over 0 + R_1} \approx 0
\boxed{V_{out} \approx 0}

At low frequencies

Conversely, as frequency goes down \downarrow f \Rightarrow {1 \over(2\pi (f \downarrow ) C_1)}\uparrow \Rightarrow X_{C_1} \uparrow and at a low enough frequency X_{C_1} \gg R_1 so we can ignore R_1 because X_{C_1} is the dominant factor. When this happens the RC LPF equation turns into V_{out} \approx V_{in} \cancel{X_{c_1}\over X_{c_1}} \Rightarrow V_{out} \approx V_{in} thus passes all lower frequencies

V_{out} = V_{in} {\uparrow X_{C_1} \over \uparrow X_{C_1} + R_1} \approx V_{in}\cancel{X_{C_1} \over X_{C_1}} \approx V_{in}
\boxed{V_{out} \approx V_{in}}

## Theory (Complex solution)

The complex solution uses X_{C_1} = {j \over {w C _1 }} instead of X_ {C_1} = {1 \over {w C _1}} where w = 2 \pi f

Here is the calculation

V_{out} = V_{in}{{j \over{w C_1}}\over{{j \over{w C_1}} + R_1}} = V_{in}{j \over{C_1 R_1 w + j}}

taking the absolute value of a complex number will give us its magnitude. Meaning if we take the absolute value we will get the amplitude and not worry about the phase.

Here is the calculation

\boxed{V_{out} = V_{in}{1 \over{\sqrt{{R_1^2w^2 C_1^2}+1}}}}

## Usages

### Filter out high frequency noise

High frequency noise can be filtered out with a low pass filter allowing for a cleaner output wave form.

### Filter out high frequency signals

In the cause of audio, the higher frequency audio can be filtered out leaving on the bass frequencies that can then be passed to a subwoofer

High Pass Filter