# RC Filters

### Reference Material

• Horowitz & Hill, The Art of Electronics, 2nd Ed., Ch. 1

## RC Filters

RC filters use resistors (R) and capacitors (C) to make circuits that have frequency-dependent responses to input waveforms. They are passive circuits that operate on the same principle as the voltage divider, but make use of the imaginary, frequency-dependent impedances of capacitors. Recall that the impedances of resistors and capacitors are given by:

\begin{align} Z_ R & = R \\ Z_ C & = \frac1{j\omega C} \\ \end{align}\,\!

where $j\equiv \sqrt {-1}$. Regardless of whether you are constructing a low-pass or high-pass filter, RC filters have a characteristic frequency at which their frequency response evolves most rapidly. This timescale is given by the product RC. In a great triumph of SI units,

$1\ s = 1\Omega \cdot 1 F. \,\!$

The frequency response of filters is typically given by the cutoff frequency at which a signal is attenuated by 3dB. For RC filters (both low-pass and high-pass), this happens when the magnitude of the impedance of the resistor and the capacitor are equal:

\begin{align} |Z_ R| & = |Z_ C| \\ |R| & = \left|\frac1{j\omega C}\right| \\ \omega _{-3dB} & = \frac1{RC} \\ \end{align}\,\!

### Low-Pass Filter

A passive RC low-pass filter

### High-Pass Filter

A passive RC hi-pass filter