First Order HighPass Filter

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First Order High-Pass Filter
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• a resistor and a capacitor can be used together
  to create a simple high-pass filter

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• given Vi Vp sin(?t)
• find V0
• what do we know about V0?
• V0 will have the same frequency, ?, as the
  input, but will have a different amplitude and
  phase angle

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• i(t) Vi(t) / Ztotal
• Ztotal ZR ZC
• V0 i ZR

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• for a resistor, ZR R
• for a capacitor
• VC 1/(j?C) IC ZC 1/(j?C)

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• the amplitude of the output voltage, V0, will
  equal the amplitude of the input voltage, Vi,
  times the magnitude of the complex number

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• the break frequency, ?b, will be defined as
• and the magnitude of the output can be written as

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• the phase shift of Vo compared to Vi can be
  calculated as the angle of the complex number

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• to summarize,
• if Vi Vp sin(?t)
• the output voltage will have thesame frequency ?
• the amplitude of the output voltage will be
• the phase shift of the output will be

where
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• to summarize,
• if Vi Vp sin(?t)
• for large ?, Vo Vi, f 0
• for small ?, Vo 0, f 90º
• for ? ?b, Vo Vi, f 45º

we have a high-pass filter !
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• first order band-pass filter
• the left side represents a low-pass filter and
  the right side represents a high-pass filter
• there are two break frequencies

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• second order low-pass filter

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• i(t) Vi(t) / Ztotal
• Ztotal ZL ZR ZC
• V0 i ZC

• for a resistor, ZR R
• for a capacitor, ZC 1/(j?C)
• for an inductor, ZL j?L

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if L0 we get the same result as before
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• Higher-order low-pass filters
• Butterworth filters of higher order can be
  constructed, usually with op amps
• in general, for a Butterworth low-pass filter of
  order n,

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• higher-order filters have a much faster roll-off
  rate
• they will attenuate the high frequencies much
  better

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• Butterworth filters can be designed in Matlab
  (with the optional Signal Processing Toolkit)
• b,abutter(4,2pi100,'s')
• systf(b,a)
• bode(sys)
• grid

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