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!

 

[rezeew]

I would like to point out that the approximation you have used is only valid for small values of T/R_LC. In this case, T/R_LC is much less than 1, and therefore the difference between e^{-T/R_LC} and 1-\frac{T}{R_LC} is very small. This is why the approximation is still valid and can be used in this derivation.

However, if T/R_LC is not small, then the difference between e^{-T/R_LC} and 1-\frac{T}{R_LC} becomes significant and the approximation may no longer be accurate. In this case, a more precise calculation using the actual values of T/R_LC would be necessary.

It is important to note that approximations are often used in scientific calculations for simplification and ease of understanding, but they should always be used with caution and their limitations should be acknowledged. In this case, the approximation is valid for the given context, but may not be accurate in all situations.