# Full-wave Rectifier with Smoothing Capacitor

Discussion in 'Homework Help' started by xxxyyyba, Jan 9, 2015.

1. ### xxxyyyba Thread Starter Member

Aug 7, 2012
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2
Hi!
I'm studying Full-wave rectifier. How would one mathematicaly descibe effect of smoothing capacitor on output voltage waveform?

2. ### Papabravo Expert

Feb 24, 2006
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A mathematical description would say that the derivative of voltage with respect to time is substantially reduced. While the input goes to zero, then negative, and finally crosses zero and begins rising it intersects the output and brings it back up to the peak.

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3. ### MikeML AAC Fanatic!

Oct 2, 2009
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The rectifier charges the filter capacitor to the peak value of the input sine wave minus any IR drop from the transformer and the forward drop of the rectifier(s).

The energy stored in the capacitor has to supply the load current until the next peak of the full-wave rectified waveform.

What is the relationship between voltage in the capacitor and the load current? How does time effect that relationship?

Study these three circuits, and explain the three load currents.

Last edited: Jan 9, 2015
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4. ### studiot AAC Fanatic!

Nov 9, 2007
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It changes the waveshape from a full wave rectified sinewave, that drops to zero every half cycle, to a sawtooth that never drops below a certain value.

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5. ### WBahn Moderator

Mar 31, 2012
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A mathematical description of the would involve a part of the period that is a linear ramp (assuming a constant load current) and a sinusoidal ramp. The transition point would be a function of the filtering capacitor and the load current (as would the slope of the linear ramp). The mathematical model could further take into account the effect of the diodes and the source impedance.

6. ### MikeML AAC Fanatic!

Oct 2, 2009
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Look at post #3 again.

Oct 2, 2009
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8. ### MrAl Distinguished Member

Jun 17, 2014
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Hi,

This partly depends on how detailed you want to get about how the capacitor affects the output. We always use a sort of approximation unless every single part of the circuit is known exactly. For example, if you know the series inductance and resistance (as well as the cap value and the load resistance) we can describe this circuit very well, but if you only know the resistance we can only approximate to some degree. Most people take a very approximate view anyway and deal with any differences once they get to the real life circuit breadboard.

For one example if we had resistance in series and a load resistor, we could describe the action of the cap as a filter:
Vout=Vin*RL/(s*C*RL*Rs+RL+Rs)

This is a description of the filter in the frequency domain. Vin here is the full wave rectified sine wave which would have to be expressed in the frequency domain also.

I am not sure what kind of solutions you are used to seeing so it is hard to guess what would be best to show you. If you provide some background information about your studies and/or lab work so far it would help me choose the best solution to show you.

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9. ### xxxyyyba Thread Starter Member

Aug 7, 2012
249
2
Thanks for replies!
Is it possible to find analytical expression for output voltage? It would be necessary to express input voltage firstly, right? I tried to find on web mathematical approach to this problem but I didn't get any result.

Oct 2, 2009
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11. ### xxxyyyba Thread Starter Member

Aug 7, 2012
249
2
Here is Full-wave rectifier circuit.

Here are input AC and output DC voltage waveforms:

If we put capacitor parallel to our load resistor, output DC voltage waveform would be like this:

I want to represent rectified waveform mathematically (in terms of Fourier series if it is possible) and then to analyse circuit with capacitor placed in parallel to load resistor in order to find analytical expression for output voltage (red color). I hope you understand what I mean.

Last edited: Jan 10, 2015

Jul 18, 2013
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The degree (%) of ripple will depend on actual current.
zero Ia = zero ripple.
Max.

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13. ### MrChips Moderator

Oct 2, 2009
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How do you represent the rectified waveform mathematically?
You cannot do that in one formula. You have to do it piece-wise, from one phase angle to the next. As I said before, you have to determine when the rectifier diodes begin conducting and end conducting.

14. ### WBahn Moderator

Mar 31, 2012
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As has been stated several times, you model each part of the waveform separately and determine the transition points. If you want to find the Fourier series for this, then use the definition of the Fourier series on the resulting equations.

15. ### studiot AAC Fanatic!

Nov 9, 2007
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Why is this in the homework section?

Is this homework or a project and if so what was the actual question?

16. ### xxxyyyba Thread Starter Member

Aug 7, 2012
249
2
It's not homework question but this section looked most appropriate to me.

17. ### studiot AAC Fanatic!

Nov 9, 2007
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Then you should note the following.

1) Folks have already posted more than the hoemwork rules allow. You should ask a moderator to move this to a more appropriate section, and read the rules.

2) The actual red voltage waveform you show is an idealisation. The correct wavform depends upon the ciurrent, which depends upon the load. The current waveform is far from a copy of the voltage weaveform, but comes in discrete large short pulses (infinite in the idealisation). The analysis of this is quite complicated.

Mar 6, 2009
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19. ### MrAl Distinguished Member

Jun 17, 2014
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Hi there,

I think i understand what you want to do here. It is interesting to do this but it may not relate to the real life circuit as well as we would like to see because the diodes conduct in a very strange way.

To do it the way you want to do it, first note that you would be taking the absolute value of the sine wave to mimic the full wave rectification:
Vs=Vpk*sin(w*t)
Vx=abs(Vpk*sin(w*t))=Vpk*abs(sin(w*t))

Vx is now the full wave rectified sine.

Next, you would find the Fourier series for abs(sin(w*t)) and multiply by Vpk.
Once you have that you would push all the components through the filter one by one, but to make this work you have to assume some series resistance.
The series resistance together with the capacitor and load resistor form the filter:
Vout=Vin*RL/(s*C*RL*Rs+RL+Rs)

where Rs might be very low like 0.1 ohms.

You must compute the effect of the filter for every significant harmonic using that filter equation, and that provides you with an amplitude and phase angle for each harmonic. You can then reconstruct the time wave using those results knowing the time equation for each harmonic is A*sin(n*w*t+ph).

Try this yourself first and see what you can come up with. You should get a DC output with some ripple.

What this does not model is the relatively high impedance of the rectifier bridge when the cap has a higher voltage than the input sine wave. To do that you would have to solve for the start and end of the conduction time and only integrate from the start to end when computing the Fourier series. If you are not interested in too much accuracy this is easy to estimate.

Last edited: Jan 11, 2015
20. ### xxxyyyba Thread Starter Member

Aug 7, 2012
249
2
Hi MrAL,
I found Fourier series for full wave rectified sine. I got $v_x(t)=\frac{2V_p_k}{\pi}+2V_p_k\sum_{n=2}^{\infty}\frac{(-1)^{n+1}-1}{\pi(n-1)(n+1)}\cos(nw_0t)$.
Here is plot of series for period T=0.02s, Vpk=12V:

And now I should analyse circuit below, right?

Last edited: Jan 11, 2015