Ripple Voltage Derivation (Full-Wave Rectifier)

[JJBladester]

Homework Statement

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

Homework Equations

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

And V_{p(rect)} is the unfiltered peak rectified voltage.

The Attempt at a Solution

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

t_{dis}\approx T when v_C reaches its minimum value.

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

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

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

Therefore,

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

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)}

My issue is with the approximation that I bolded above. If e^0 approaches 1, then how does the expression e^{-T/R_LC} approach 1-\frac{T}{R_LC}?

 

[berkeman]

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.

 

[JJBladester]

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.

 

[berkeman]

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