I'm trying to derive the ripple factor for a full wave rectifier with a filter.
This parameter is defined as the ratio of the root mean square (rms) value of the ripple voltage (Vr_rms) to the absolute value of the DC component of the output voltage:
γ = Vr_rms / Vdc
I use the following plot with an assumption of |AB| = T/2.
First I want to write an expression for ripple rms voltage Vr_rms.
The ripple amplitude as seen in the plot is Vr. This can be approximated(using Taylor series expansion's first term for exponential decay) and given for full wave rectifier with a filter as:
Vr = |XY| = t/τ * Vm, and in this case t = T/2 and τ = R*C
Vr = Vm / (2*CRf) and rms value of a triangle wave is given as Vp/sqrt(3) so:
Vr_rms = Vm / (2*sqrt(3)xCRf)
Now I want to find the average DC voltage Vdc. This can be approximated as:
Vdc = Vm – (Vr/2)
This yields the ripple factor:
γ = Vr_rms / Vdc
γ = [Vm / (2*sqrt(3)xCRf) ] / [Vm – (Vr/2)]
γ = [1 / (2*sqrt(3)xCRf) ] / [1 – (Vr/(2*Vm))] and since Vr/Vm = 1/(2*CRf)
γ = [1 / (2*sqrt(3)xCRf) ] / [1 – (1/(4*CRf))]
I'm stuck at this point I cannot progress and the texts give this factor as:
How is this derived? I cannot reach this..
Best Answer
The RMS value of a triangle wave is its peak value/sqrt(3). In your case the peak value of the triangle wave is Vr/2, not Vr which is the peak-to-peak value. This correction will produce a factor of 4 in your final result instead of 2. If you then make the approximation that RC>>T, then CRf>>1. With that approximation, the denominator in your final equation can be approximated as 1 since 1/(4CRf)<<1. The result is that your final equation reduces to the desired result.