Difference between revisions of "RC Filters"

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\section*{RC Filters}
 
\section*{RC Filters}
  
 +
RC filters uses 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 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 $RC$.  In a great triumph of SI units,
 
\begin{equation}
 
\begin{equation}
\omega_{-3dB}=\frac1{RC}
+
1\ s = 1\Omega \cdot 1 F.
 
\end{equation}
 
\end{equation}
 +
 +
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} \\
  
 
\subsection*{Low-Pass Filter}
 
\subsection*{Low-Pass Filter}

Revision as of 15:32, 30 August 2012

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RC Filters

RC filters uses resistors () and capacitors () 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:

where . 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 . In a great triumph of SI units,

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:

Failed to parse (unknown function "\begin{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}\,\!}