Difference between revisions of "RC Filters"

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===Reference Material===
 
===Reference Material===
 +
* Horowitz & Hill, ''The Art of Electronics, 2nd Ed.'', Ch. 1
 +
 
<latex>
 
<latex>
 
\documentclass[11pt]{article}
 
\documentclass[11pt]{article}
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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:
+
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}
 
\begin{align}
 
Z_R &= R \\
 
Z_R &= R \\
 
Z_C &= \frac1{j\omega C} \\
 
Z_C &= \frac1{j\omega C} \\
 
\end{align}
 
\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,
+
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,
 
\begin{equation}
 
\begin{equation}
 
1\ s = 1\Omega \cdot 1 F.
 
1\ s = 1\Omega \cdot 1 F.
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|R| &= \left|\frac1{j\omega C}\right| \\
 
|R| &= \left|\frac1{j\omega C}\right| \\
 
\omega_{-3dB} &= \frac1{RC} \\
 
\omega_{-3dB} &= \frac1{RC} \\
 +
\end{align}
  
 
\subsection*{Low-Pass Filter}
 
\subsection*{Low-Pass Filter}

Latest revision as of 10:11, 21 January 2014

Short Topical Videos[edit]

Reference Material[edit]

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

RC Filters

RC filters use 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 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 . 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:

Low-Pass Filter

Rc lowpass.png


A passive RC low-pass filter

High-Pass Filter

Rc hipass.png


A passive RC hi-pass filter