# RC Filters

## RC Filters

RC filters uses resistors (${\displaystyle R}$) and capacitors (${\displaystyle C}$) 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:

{\displaystyle {\begin{aligned}Z_{R}&=R\\Z_{C}&={\frac {1}{j\omega C}}\\\end{aligned}}\,\!}

where ${\displaystyle j\equiv {\sqrt {-1}}}$. Regardless of whether you are construction 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 ${\displaystyle RC}$. In a great triumph of SI units,

${\displaystyle 1\ s=1\Omega \cdot 1F.\,\!}$

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:

figure}"): {\displaystyle \begin{align} |Z_ R| & = |Z_ C| \\ |R| & = \left|\frac1{j\omega C}\right| \\ \omega _{-3dB} & = \frac1{RC} \\ \begin{figure} \includegraphics[width=2in]{rc_ lowpass.png} \caption{A passive RC low-pass filter} \end{figure} \begin{figure} \includegraphics[width=2in]{rc_ hipass.png} \caption{A passive RC hi-pass filter} \end{figure} \end{align}\,\!