# High Pass Filter – Simulations, Calculators, and Theory

High pass filters (HPF) are circuits that allow higher frequencies to pass through them while blocking lower frequencies. This can be useful in may applications like filtering out the lower bass frequencies of a music signal to a tweeter speaker (speaker designed to produce high frequencies). Additionally these filters can remove unwanted DC signals from a circuit by providing AC coupling. An example of this can be found in an oscilloscope’s AC coupling option. This is effectively a high pass filter tuned to block very low frequencies, mainly 0 Hz which is DC. Another example of this would be if you where generating a sin wave using an oscillating circuit that only produced values in the positive range but you wanted to remove the DC offset and send the AC signal, oscillating around 0, to a speaker.

## RC High Pass Filter

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

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

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

## RC High Pass Filter Simulation (Video)

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

Vin (optional)

## Theory (Approximation)

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

At high frequencies

Using the capacitive reactance equation we can see that as \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 HPF equation turns into V_{out} \approx V_{in} thus passing all high frequencies

V_{out} = V_{in} {R_1 \over R_1 + {1 \over(2 \pi {f} C_1)\uparrow}\downarrow}\uparrow
V_{out} \approx V_{in} {R_1 \over R_1 + 0} \approx V_{in}\cancel{R_1 \over R_1} \approx V_{in}
\boxed{V_{out} \approx V_{in}}

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} \approx \infty when this happens the RC high pass filter equation turns into V_{out} \approx 0 thus blocking all low frequencies including DC

V_{out} = V_{in} {R_1 \over R_1 + {1 \over(2 \pi f C_1)\downarrow}\uparrow}\downarrow
V_{out} \approx  V_{in} {R_1 \over R_1 + \infty } \approx  V_{in}0 \approx 0
\boxed{V_{out} \approx 0}

## 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

V_{out} = V_{in}{R_1 \over{R_1 - {j \over{w C_1}}}} = V_{in}{1 \over{1 - {j \over{R_1 w C_1}}}}

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{(1 - {1 \over{R_1^2w^2 C_1^2}})^{1 \over {2}}}}}

## Usages

### AC Coupling

Oscilloscopes have options to remove the DC offset of a signal by turning on AC Coupling. This is just a high pass filter because the lowest frequency is 0Hz and that is DC. So a high pass filter with a very low cutoff frequency can be used to remove the DC component thus providing AC coupling.

### DC Blocking

Another way you can look at this is that capacitors block DC and allow AC to travel across them. The higher the frequency the less reactance (think resistance to AC) the capacitor provides.

As the frequency goes up the reactance of the capacitor goes down. This means low frequencies will experience heavy reactance where as higher frequencies will see little to no reactance

\downarrow X_{C_1} = {1 \over(2\pi (f \uparrow) C_1)}\downarrow

Low Pass Filters