Re: Solving the DC coil mystery

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Bert Hickman by way of Terry Fritz <twftesla-at-qwest-dot-net>" <bert.hickman-at-aquila-dot-net>

Steve and all,

The proof follows. Assume a constant voltage source (V) and an initially
uncharged capacitor being charged through resistor R by closing the switch
at time T=0 and then leaving it closed forever (since it takes forever to
fully charge the cap). 

          Close at 
           T = 0                C
           -----      R        | |
    <------o   o----/\/\/\-----| |----
    +                          | |    |
                    ----->    + Vc-   |
                      I               |
    V                                 |
                                      |
    -                                 |
    <---------------------------------

By solving the circuit equations it can be shown that the charging current
is a maximum at time T=0, declining exponentially towards 0 as the
capacitor charges:
     
        I(t) = V/R(e-(t/RC))

The final energy residing in the capacitor after it is fully charged is:

        Ecap = 0.5*C*(V^2)

The energy dissipated by current flowing through the resistor during the
charging process is:

        Er = R*(I^2)

Substituting I(t) from above:  

        Er = R*(V^2/R^2)*e-(2t/RC) = ((V^2)/R)*e-(2t/RC)

and integrating from time T=0 to infinity:

        Er = (RC/2)*(V^2)/R) = 0.5*C*(V^2)

This tells us that the total energy dissipated in the resistor is the same
as the total energy that's eventually stored in the capacitor, giving us a
50% charging efficiency. And, since R cancels out, the result is
independent of the value of R! (Not at all intuitive!)

-- Bert --
-- 
Bert Hickman
Stoneridge Engineering
Email:    bert.hickman-at-aquila-dot-net
Web Site: http://www.teslamania-dot-com

S & J Young wrote:
> 
> Hi Rick & others,
> 
> Yes, you are right.  See below.
> 
<SNIP>
> Bert Hickman published the key to my mystery a few weeks ago.  I quote:
> 
> (Bert or others, please elaborate on the "It can be shown ...".  It has
> taken me a while to decide this is true, and I would like to see the
> calculations / reasoning that prove it.)
> 
> "It can be shown that if a capacitor is charged through a resistor, and if
> the charge time is sufficient to virtually fully charge the cap (Tcharge >
> 3RC), then the series resistor will dissipate the SAME amount of energy as
> the energy that ends up being stored in the capacitor. Knowing this
> simplifies the problem a bit.
> 
<SNIP>
> --Steve