The resistor-capacitor (RC) low-pass filter circuit diagram is given below. The voltage across the resistor is given by Ohm's law and is V_{R} = I x R. The voltage across the capacitor, V_{out} lags the current by 90^{0} and is given by V_{out} = I x X_{C}, where X_{C} is the capacitive reactance.

Looking at the diagram, Pythagoras's Theorem can be used to obtain the expression V_{in} = (V_{R}^{2} + V_{out}^{2}) ^ 0.5.

The impedance (ie combination of resistance and reactance) looking from the supply into the circuit is given by Z = (R^{2} + X_{C}^{2}) ^ 0.5.

The current in the circuit, I, is equal to V_{in} / Z.

RC Low-Pass Filter Excel Spreadsheet

MathML formula | Formula Image | Straight text formula |
---|---|---|

${f}_{c}=\frac{1}{2\pi RC}$ | f_{c} = 1 / 2πRC |

Where

f_{c} = cut off frequency, Hertz

= that frequency at which output power has fallen by half

= that frequency at which output voltage has fallen by 1/(2^0.5) = 0.707

R = resistance in Ohms

C = capacitance in Farads

The cut off frequency is defined as that frequency at which the output power has fallen by 50%, ie 1/(2^0.5) = 0.707 of the voltage.

I intend to build the circuit below to test my understanding of RC low-pass filters. The 1 Ohm resistor makes little difference to the operation of the circuit, but can be used as a convenient way to measure current (a high input impedance voltmeter connected across the 1 ohm resistor will give a reading in volts that corresponds 1:1 with the current in amps).