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Capacitor soakage, dielectric absorption, fractional derivative capacitor

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Published on May 15, 2013

Watch how the voltage bounces back after the capacitor has been shorted. It reaches 1.8% of the original voltage here. An ideal capacitor should have stayed at 0 Volts.

The capacitor has a value of 220 uF/63 Volts, measured to 212 uF and with ESR (equivalent series resistance) of 0.18 ohms. It is connected to 10.0 Volts for about a minute, then discharged through a 3.9 ohms resistor for 6 seconds.

This effect is also called dielectric relaxation or battery action. It has been known for as long as one has had capacitors. It is a memory phenomenon which usually is approximated by a multi-stage ladder network of R's and C's.

Alternatively, it is also a demonstration of a real-life device that can be described by a fractional derivative. The impedance is 1/[(j omega)^a C], where a is slightly less than 1. This expression involves both a reactance, as expected for a capacitor, and a frequency dependent resistance for the losses. In practice a has been found to be from 0.9776 to 0.9999 for low value capacitors and even smaller for a high value electrolytic capacitor such as here (Westerlund and Ekstam. "Capacitor theory," IEEE Trans. Dielectrics and Electrical Insulation, 1994)

It is even more elegant to give the voltage current relationship i = C d^a u/dt^a. This is a fractional derivative of order a, which unlike a first order derivative, involves a memory process. (i = current, u = voltage, a = 1 is an ideal capacitor ).

The charge/discharge cycle here follows Bob Pease "Understand capacitor soakage to optimize analog systems" from 1982, http://www.datasheetarchive.com/files...

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