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Path: utzoo!utgpu!jarvis.csri.toronto.edu!rutgers!cs.utexas.edu!mit-eddie!mit-amt!mit-caf!ankleand
From: ankleand@mit-caf.MIT.EDU (Andrew Karanicolas)
Newsgroups: sci.electronics,sci.physics
Subject: Re: HV Cap Fun!
Keywords: capacitor,energy,paradox
Message-ID: <2449@mit-caf.MIT.EDU>
Date: 1 Jun 89 02:41:29 GMT
References: <4924@m2c.M2C.ORG> <3806@mit-amt> <20772@quacky.mips.COM>
Reply-To: ankleand@mit-caf.UUCP (Andrew Karanicolas)
Organization: Microsystems Technology Laboratories, MIT
Lines: 69

In article <20772@quacky.mips.COM> vaso@mips.COM (Vaso Bovan) writes:
>A Paradox of Capacitor Energy Storage
>
>I've heard several competing answers to this paradox. None is entirely
>satisfactory:
>
>Consider an ideal 2uF (for computational ease) capacitor charged by a 10 volt 
>source. Eventually, the energy stored is (1/2)*CV^2=100 joules.
>
>Consider the capacitor to be isolated from the voltage source, and then
>directly shorted across an identical (ideal) capacitor. Eventually, the
>voltage across each capacitor will be 5V. Now, there are two equally
>charged capacitors, each storing (1/2)*CV^2= 25 joules, for a total of
>of 50 joules.  What happened to the other 50 joules ?


Interestingly enough, this problem came around on sci.electronics about
this time last year; this capacitor problem is a classic EE problem.

Basically, the difficulty in determining what 'happened' to the other
50 Joules arises from an implication in the problem statement.  The
implication is as follows:


	suddenly connecting the capacitors together is tantamount
	to saying that the capacitor voltage in the circuit changes
	in a step-function manner, u(t)

	since the capacitor current is related to the capacitor 
	voltage by i(t)=Cdv(t)/dt, a step change in the capacitor
	voltage results in the capacitor current changing in a
	delta-function manner, delta(t)

	using energy conservation, one will then implicitly attempt
	the evaluation of the following, undefined, integral:

		/ +inf
	       |
	       | u(t) * delta(t) dt
	       |
	      / -inf

	the fact that this integral is not defined is why there
	is a 'paradox'.  

	The point is that the *details* of how the capacitors are
	connected together determines what the final energy of the
	system will be.  Modelling a series resistance shows that 
	half of the original energy is dissipated in the resistor.
	More elaborate models of electromagnetic radiation can also
	be considered.  A switch that has time varying resistance can
	also be worked into the problem.  Clearly, intricate models for
	the switching mechanism can be devised.

	However, there is a fundamental explanation for the 'missing'
	energy in the mathematics in the case of the 'paradoxical'
	problem where things occur suddenly.

	One source to see a more complete treatment of this problem:

		"Circuits, Signals and Systems"
		Siebert, William McC
		McGraw Hill 1986


Andy Karanicolas
MIT Microsystems Laboratory
ankleand@caf.mit.edu
ankleand@charon.mit.edu