Capacitor Voltage Decay at Constant Power - John Dunn, Consultant, Ambertec, P.E., P.C.

The discharge curve of a capacitor's voltage versus time for a resistive load is an exponential decay curve of Tau = RC and for a constant current shunt, is a linear downward slope of dV/dt = i/C . However, the discharge curve for a constant power load is quite different from either of those.

An example of constant power decay, taken from real life, is shown here. The question at hand was for a 470 µF capacitor shunted by a constant power load of 714 watts, how long would it take the capacitor voltage to decay to 200 volts from various starting voltages.

In the sketch below, each value of time, "Tnnn", is the time value at 200 volts from having started at "nnn" volts. Thus, starting at 400 volts, the decay of capacitor voltage to 200 volts takes T400 = 39.5 mSec.

 

The calculation can be made using the recursive differential equation method from which the graphical plots shown here were made or it can be done using the derived equation on the right.

Comments

[Birocchi]

Without integration, it can be seen also in the following way:

Initial energy in the C: Es=1/2CVs^2
Final energy in the C: Ef=1/2CVf^2
Energy supplied by the C: E=Es-Ef=1/2C(Vs^2-Vf^2)
Energy to the load: E=PdeltaT

so:

PdeltaT=1/2C(Vs^2-Vf^2)

[George Storm]

Constant power loads can cause other, even more "interesting" behaviours. These are mostly a result of their negative slope conductance (Gslope = - P/V^2).
One example is the potential (pun originally accidental) of regulated power supplies to oscillate when their principal loading is by a constant-power load.

BTW, for this case I find the analytic approach more revealing...

[John Dunn]

Hi, George.

Indeed so! Please see:

http://licn.typepad.com/my_weblog/2010/11/switchmode-dynamic-input-impedance-john-dunn-consultant-ambertec-pe-pc.html#more

[Carl Schwab]

Your notes are also useful when studying the use of supercaps for electrostatic storage in place of electrochemical storage for powering vehicles. Of course the time scales are quite different. But what comes out of this is that the supercap pack only has to discharge to 1/3 peak voltage to deliver all but 1/9 of the stored energy.

The advantage is that the supercap pack can be recharged much more quickly than a rechargeable battery. Also the supercap is more efficient than a battery. Nano-engineering is providing the path to both better rechargeable batteries an MUCH better supercaps. Thanks for the blog.