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.