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Ripple Voltage Derivation (Full-Wave Rectifier)

  1. Jan 4, 2013 #1

    JJBladester

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    Gold Member

    1. The problem statement, all variables and given/known data

    Derive the ripple voltage of a full-wave rectifier with a capacitor-input filter.

    2. Relevant equations

    ripple_voltage.jpg

    Where [itex]V_{r(pp)}[/itex] is the peak-to-peak ripple voltage and [itex]V_{DC}[/itex] is the dc (average) value of the filter's output voltage.

    And [itex]V_{p(rect)}[/itex] is the unfiltered peak rectified voltage.

    3. The attempt at a solution

    [tex]v_{C}=V_{p(rect)}e^{-t/R_LC}[/tex]

    [itex]t_{dis}\approx T[/itex] when [itex]v_C[/itex] reaches its minimum value.

    [tex]v_{C(min)}=V_{p(rect)}e^{-T/R_LC}[/tex]

    Since [itex]RC> > T[/itex], [itex]T/R_LC[/itex] becomes much less than 1 and [itex]e^{-T/R_LC}[/itex] approaches 1 and can be expressed as

    [tex]e^{-T/R_LC}\approx 1-\frac{T}{R_LC}[/tex]


    Therefore,

    [tex]v_{C(min)}=V_{p(rect)}\left ( 1-\frac{T}{R_LC} \right )[/tex]

    [tex]V_{r(pp)}=V_{p(rect)}-V_{C(min)}=V_{p(rect)}-V_{p(rect)}+\frac{V_{p(rect)}T}{R_LC}=\frac{V_{p(rect)}T}{R_LC}=\left ( \frac{1}{fR_LC} \right )V_{p(rect)}[/tex]

    My issue is with the approximation that I bolded above. If [itex]e^0[/itex] approaches 1, then how does the expression [itex]e^{-T/R_LC}[/itex] approach [itex]1-\frac{T}{R_LC}[/itex]?
     
  2. jcsd
  3. Jan 4, 2013 #2

    berkeman

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    Staff: Mentor

    Those are the first 2 terms of the series expansion for e^x

    BTW, in your initial problem statement, that should be "with a capacitor-output filter", not "input" filter, right?

    Also, are you given as part of the problem statement that T << RC? That's certainly not always the case for FWRs with output filter caps. If you want to minimize ripple, that is a requirement though.
     
  4. Jan 4, 2013 #3

    JJBladester

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    Gold Member

    Berkeman,

    Thanks for clarifying about the series expansion of e^x.

    The text does say "For a full-wave rectifier with a capacitor-input filter..." I took "input" to mean that the capacitor takes the full-wave rectified input waveform and transforms it into a ripple waveform.

    The T << RC approximation is simply given as "which is usually the case..." In the chapter I'm studying, it is an introduction to diodes/rectifier circuits and the goal is to get DC waveform that is as close to a horizontal line (constant voltage) as possible.
     
  5. Jan 4, 2013 #4

    berkeman

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    Staff: Mentor

    Got it, thanks for the clarifications. And yeah, being able to assume T << RC simplifies the math a lot! :smile:
     
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