High-pass filter

If a capacitor and a resistor are connected in series to an AC source they have the same current. The maximum potential difference across the resistor is proportional to the resistance, while the maximum potential difference across the capacitor is proportional to the capacitive reactance.

The capacitive reactance of the capacitor is inversely proportional to frequency. At high frequencies XC is small, so the maximum potential difference across the capacitor is also small. Thus, most of the potential difference is across the resistor. At low frequencies the opposite is true. If we take the potential difference across the resistor to be the output voltage, the output voltage is high at high frequencies and low at low frequencies. This is a high-pass filter - high-frequency signals pass through virtually unchanged, but low-frequency signals are attenuated.

If the input voltage is ΔVin the output voltage is:
ΔVout = ΔVin
 R Z
= ΔVin
 R (XC2 + R2)1/2

Low-pass filter

We can use the same circuit as a low-pass filter if we take the potential difference across the capacitor to be the output voltage. At high frequencies most of the potential difference is across the resistor, and the potential difference across the capacitor is small. The opposite happens at low frequencies. Thus, low frequency signals get through the filter with large amplitudes, while high-frequency signals get through with very small amplitudes.

If the input voltage is ΔVin the output voltage is:
ΔVout = ΔVin
 XC Z
= ΔVin
 XC (XC2 + R2)1/2