Effective Capacitance and ESR of different caps in Parallel

[jstefanop]

Im trying to figure out the effective capacitance and ESR of a cap bank for a buck converter so I can optimize its compensation network.

I have 3x 50uF ceramics with 2.2 milliOhm ESR @250khz, and 1x 420 uF poly cap with 7mOhm ESR (these values are derated).


What would be the equivalent capacitance and ESR of this bank? I referenced this for my calculations : https://www.power-mag.com/pdf/feature_pdf/1387888355_Murata_Feature_Layout_1.pdf

and got around 220uF capacitance with 1.65mOhm ESR. The ESR sounds about right but I'm not sure about the effective capacitance value. According to the equations in the article the individual parallel capacantce of the poly cap Cpk= 20.6uF which sounds really low.

I was trying to find another reference to a way to figure out these values so i could cross check my numbers but i couldn't find anything. Any insight would be appreciated!

 

[albbg]

First of all we have to convert the RC series of each capacitor into an equivalent RC parallel (valid only at a given frequency).

The impedance of the series circuit is Zs = Rs + jXs, where the reactance Xs=-1/(w*Cs)
The impedance of the parallel circuit is Zp = Rs*jXs/(Rs+jXs) the using a little math:

Zp = Rp*Xp^2/(Rp^2+Xp^2) * jXp*Rp^2/(Rp^2+Xp^2)

we want Zp = Zs. Then we also have imag(Zp)/real(Zp) = imag(Zs)/real(Zs). Thus:

Xs/Rs = Rp/Xp ==> Rp = Xp*Xs/Rs

Substituting this into the imaginary part of Zp, after simplifying:

Xs = Xp*Xs^2/(Xs^2+Rs^2)

Since Xs = -1/(w*Cs) and Xp = -1/(w*Cp)

Cp = Cs/[1 + (w*Cs*Rs)^2]

for each of the three 50 uF, 2.2 mohm I obtained an equivalent parallel of 48.6 uF, 76 mohm
for the 420 uF, 7 mohm I obtained an equivalent parallel of 18.8 uF, 7.3 mohm

so the total parallel equivalent is 164.4 uF, 5.7 mohm

Then I've to convert this back to its series equivalent, simply using:

Rs = Rp*Xp^2/(Rp^2+Xp^2), and Xs = Xp*Rp^2/(Rp^2+Xp^2)

I've obtained 240.8 uF, 1.8 mohm that is valid at a frequency of 250 kHz

I hope it' correct, however it's in line with your calculation

Notice that the raw (wrong) estimate obtained summing the original capacitances and paralleling the original ESR gives 570 uF, 0.66 mohm