# Making Low-Pass and High-Pass Filters with RC Circuits

With simple RC circuits, you can build first-order RC low-pass (LPF) and high-pass filters (HPF). These simple circuits can give you a foundational understanding of how filters work so you can build more-complex filters.

## First-order RC low-pass filter (LPF)

Here's an RC series circuit — a circuit with a resistor and capacitor connected in series. You can get a low-pass filter by forming a transfer function as the ratio of the capacitor voltage *V*_{C}*(s)* to the voltage source *V*_{S}*(s)*.

You start with the voltage divider equation:

The transfer function *T(s)* equals *V*_{C}*(s)/V*_{S}*(s)*. With some algebra (including multiplying the numerator and denominator by *s/R*), you get a transfer function that looks like a low-pass filter:

You have a pole or corner (cutoff) frequency at *s = –1/(RC)*, and you have a DC gain of 1 at *s* = 0. The frequency response starts at *s* = 0 with a flat gain of 0 dB. When it hits *1/(RC)*, the frequency response rolls off with a slope of –20 dB/decade.

For circuits with only passive devices, you never get a gain greater than 1.

## First-order RC high-pass filter (HPF)

To form a high-pass filter, you can use the same resistor and capacitor connected in series shown earlier, but this time, you measure the resistor voltage *V*_{R}*(s)*. You start with the voltage divider equation for the voltage across the resistor *V*_{R}*(s)*:

With some algebraic manipulation (including multiplying the numerator and denominator by *s/R*), you can find the transfer function *T(s) = V*_{R}*(s)/VS(s)* of a high-pass filter:

You have a zero at *s = 0* and a pole at *s = –1/(RC* ). You start off the frequency response with a zero with a positive slope of 20 dB/decade, and then the response flattens out starting at *1/(RC)*. You have a constant gain of 1 at high frequencies (or at infinity) starting at the pole frequency.