[[File:media/image1.emf|139x122px]]The circuit shown at right has an AC generator of peak EMF E<sub>max</sub> and frequency '''' connected in series with a resistor ''R'' and a capacitor ''C''. Let’s analyze the behavior of this circuit at extreme values of the frequency ''''.

''<span class="underline">Important note</span>: Since there is no inductor present, you may think that we’re back to the material of “RC Circuits” that we studied a month ago. But not so! Those “RC circuits” were connected to '''DC batteries''' of constant voltage! This one is being driven by an '''AC generator''', which makes it behave in a totally different way. This is an RLC circuit like any other … the fact that L = 0 just simplifies things a little bit.''

[[File:media/image2.emf|114x98px]](a) Draw a phasor diagram for this circuit, showing the relative phases of the voltages ''V<sub>R</sub>'', ''V<sub>C</sub>'', and E.

(b) Now draw phasor diagrams for the cases when the driving frequency '''' is <span class="underline">extremely small</span>, and when it is <span class="underline">extremely large</span>. In both cases, please answer the following questions:

* How does the peak voltage ''V<sub>R</sub>''<sub>,max</sub> across the resistor compare with the generator E<sub>max</sub>?<br />
(Are they about the same? Or is one very-much smaller than the other?)
* How does the peak voltage ''V<sub>C</sub>''<sub>,max</sub> across the capacitor compare with the generator E<sub>max</sub>?<br />
(Are they about the same? Or is one very-much smaller than the other?)

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A common application of the circuit we’ve been considering is in '''audio amplifiers''', which process music signals containing many different frequencies. In this application, the generator EMF is produced by something like a CD player, or an old-fashioned record player. This '''input signal''' ''V''<sub>in</sub> is sent through the resistor and capacitor, and an '''output signal''' ''V''<sub>out</sub> is produced by connecting an output device (like a speaker) in parallel across either ''R'' or ''C''. Here are the two configurations:

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== Circuit A ==

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== Circuit B ==

Suppose our music input comes from an old record-player, and is subject to hiss-and-pop scratch noise. These distortions of the music are very short-lived, and so are of very high frequency. Thus, we would like our amplifier to <span class="underline">remove</span> signals of very <span class="underline">high frequency</span>. This is called a '''low-pass filter''' (or “scratch filter”) → when the input signal is of low frequency, the input signal ''V''<sub>in</sub> is passed through to ''V''<sub>out</sub> without much change … but high-frequency input signals are blocked.

(c) Based on your phasor diagrams, which of the circuits shown would provide a '''low-pass filter'''?

The loud speaker in Circuit B records the voltage across the capacitor. In the high ω limit , V<sub>C,max</sub> approaches zero and hence the high frequency components are blocked. Circuit B is then the low pass filter.

(d) Would the ''other'' one provide a '''high-pass filter''' (also called a “rumble filter”) which deletes very-low frequency bass noise?

The loud speaker in Circuit A records the voltage across the resistor. In the low ω limit, V<sub>R,max</sub> → 0 and hence the low frequency components are blocked. Circuit A is thus a high pass filter.

(e) Low-pass and high-pass filters can also be constructed using inductors rather than capacitors. Use your expertise with phasor diagrams to determine which of these two circuits would provide a '''low-pass filter''' and which would provide a '''high-pass filter'''.

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Circuit A

Circuit B

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