e-choke with capacitor

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Use enough robust FET.
 

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E-Choke may, like E-Cigarette, be rather a trade name than description of an electrical equivalent.

An important characteristic of a choke is that it can store (magnetic) energy. Try to disconnect a real choke with good current running in it and you will see hazard. This small fellow, the E-Choke, cannot store much energy and cannot be used, for instance, in a Buck converter.

De facto, it is a ripple filter based on what I have heard being called a capacitance multiplier. The similarity, judged from euro21's diagram, with Juma's Easy Peasy Capacitance Multiplier is striking.

This means, the E-choke is not used as the choke in a PI-filter.
The E-Choke, as other ripple filters designed as source followers, should be rather fast in step response seen from the output. Then, a large output capacitor is not needed. A smaller low ESR capacitor should be sufficient.

Could it be that it is slow in reacting to input voltage variations, not output changes?

I believe that gerittube is right in that charging a large capacitance at the output during startup can cause some nasty surge currents.
 
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