[Phys-L] charging a capacitor reversibly (or nearly so)

[John Denker <jsd@av8n.com>]

On 08/06/2014 07:36 AM, Bob Sciamanda wrote:

> Perhaps this is a correctly simulated instance of "Mandated Energy
> Dissipation".   EG., when a battery charges a capacitor, 1/2 of the
> battery-produced energy must be dissipated.

More-or-less everybody assumes that, but it's not necessarily
true!

A whole lot of very smart people have gotten that wrong
over the years.

If we are clever, we can dissipate a whole lot less than 
half of the energy.  Proof by construction:

A battery, by definition, is a collection of /cells/.
Assume all N cells are in series.  Assume we have access 
to the individual cells.  Let the cell voltage be u (small
u) while the battery voltage is V (big V) where V = N u.

Charge the capacitor step by step.  Start by hooking it 
up to just one cell.  This moves a charge (C u) across an
average voltage drop of u/2, so it dissipates energy on 
the order of (C u^2/2).  This is very small, since u is 
small.  At the next step, hook the capacitor to two cells 
in series.  This moves an additional charge (C u) across
an average voltage drop of u/2.  Again the dissipation
per step is on the order of (C u^2/2).  After N steps
the capacitor has the full charge (N C u) i.e. (C V),
and the total dissipation is (N C u^2/2) i.e. (C V u/2)
which is N times smaller than the "conventional" but
unclever result (C V^2/2).

If you don't have access to the individual cells, you
can achieve the same result using an inductor and a
transistor to build a buck/boost DC-to-DC converter.

This has a number of real-world applications.  For
starters, real-world CFL and LED light bulbs include 
high-efficiency DC-to-DC converters.
  http://www.pavouk.org/hw/lamp/en_index.html
This is a huge market.

Also note that modern computers, memories, and logic 
chips are based on using switches to charge up capacitors.
Cutting down on energy requirements (including cooling 
requirements) is always desirable, so there is interest 
in "reversible computing" aka "adiabatic computing".

This provides yet another example showing the value of
/interdisciplinary/ work.  If you're going to invent
adiabatic computing, you need to know a fair bit about
the application area (chip design), and also a fair bit 
of fundamental physics.