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!