```Thus spake jwisnia@aol.com (JWISNIA) in sci.electronics.misc:

>Today I was trying to show someone that the total energy of the I^2*R
>losses when resistively charging a capacitor from a dc source is exactly
>equal to the energy stored in the capacitor after it has charged for an
[snip]
>I'd appreciate someone helping me out here, my calculus is rusty.....and
>never was very good anyway!

/            +  Vr   -
___/  ___________/\/\/\/\____
_|_       -----> I     R       |
/ + \                        ___|___
|     | V                  C  _______
\_-_/                           |
|_____________________________|
<---- I

When the switch is closed (time t=0), the capacitor acts as a short
circuit. Current is V/R. Once the cap charges up, current flow is 0.
The charging process is exponential, so instantaneous current is:

V      -t/RC
I=--- * e
R

(This means I'm waving my hand, instead of actually doing the
integration!)

Instantaneous voltage across resistor is:

-t/RC
Vr= V * e

Instantaneous power dissipated in resistor is voltage * current:

V * V     -2t/RC
Pr= ----- * e
R

Integrate this to get total energy dissipated in resistor:

oo /
| V * V    -2t/RC
Er= | ----- * e        dt
|   R
0 /
_              _
|                |
V * V * R * C   |    -oo       0 |
= ------------- * |  e      -  e   |
R * (-2)      |                |
|_              _|

_         _
V * V * C    |           |
= ---------- * | 0   -   1 |
(-2)      |_         _|

V * V * C
= ---------
2

The capacitor charges to voltage V, so the energy it stores is also:

V * V * C
Ec = ---------
2

>BTW, we got into this while discussing "charge pump" voltage multiplying
>power supplies.  I'd read an article in a recent mag which claimed they

Here is one charge pump circuit. It works best if the square-wave
oscillator has a low output impedance, and if Schottky diodes are
used:

VCC
O
|
_|_
\ /
__V__)
|  | + (  |       /
VCC             |  |      |   |\ |
O         .----+  +------+---| >|------+--------->  2 * VCC
|         |    |  |          |/ |      |
|         |    |  | +           /    + | +
______|_____    |                         ___|___
|            |   |                         _______
| square-    |   |                            |
| wave       +---+                         ___|___
| oscillator |   |                          _____
|            |   |                           ___
|____________|   | + |  |             /       .
|        |   |  |            | /|
|        '---+  +-----+------|< |------+---------> -2 * VCC
|            |  |     |      | \|      |
___|___       + |  |     |      /         |
_____                  _|_            ___|___
___                   \ /            _______
.                   __V__)           + | +
(  |             ___|___
|              _____
___|___            ___
_____              .
___
.
(I love ASCII art!) More capacitor-diode stages can be added to get
higher output voltages. I don't know how efficient it is.

Hope this helps.

>Thanks guys...
>Jeff Wisnia  W1BSV
>Winchester, MA.
>                                  "Common sense isn't very common..."

--
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```

Date: Fri, 23 Aug 1996 05:22:49 GMT

Original Subject: Re: Help my rusty math please