Re: Help with this equation!

To: tesla-at-poodle.pupman-dot-com
Subject: Re: Help with this equation!
From: Tesla List <tesla-at-stic-dot-net>
Date: Fri, 30 May 1997 09:25:06 -0500
Approved: tesla-at-stic-dot-net
Subject:      Re: Help with this equation!
       Date:  Fri, 30 May 1997 10:45:29 +1000
       From:  Ian Holland <ian-at-digideas-dot-com.au>
Organization: Digital Ideas Pty Ltd
         To:  Tesla List <tesla-at-pupman-dot-com>
 References:  1


Tom Heiber wrote:

> This is an equation I got from Bert Pool's file.
> =======================================================================
>     EQUATION 10: CAPACITANCE OF A TOROID
>                                                     ___________________
>                                                    /    2
>          C =(1+ (0.2781 - d2/d1)) x  2.8  x      /  2 pi  (d1-d2)(d2/2)
>                                                /   -------------------
>                                             \/      4 pi
> 
>      C = capacitance in picofarads (+- 5% )
>     d1 = outside diameter of toroid in inches
>     d2 = diameter of cross section (cord) of toroid in inches
> 
>     Equation courtesy of Bert Pool
> 
> =======================================================================
> 
> CAN ANYONE help me solve for D1 as in D1=........
> I have been trying to do this without any results (I suck at math)

Tom,

Hi from downunder!

My initial thought, given the nature of the equation, was that you could
only solve for d1 &/or d2 by an iterative approach, however I have found
an approach which may suit your needs:

Taking "Bert's #10" equation and cleaning it up we have,

C = 1.4 (1.2781 - d2/d1) . sqrt( PI (d1-d2) d2 )

If we substitute D for d2 and A for the aspect ration of the toroid,
(i.e. A = d1/d2), we can put the equation in a form where only one
diameter term exists and it is then easy to solve for D:

C = 1.4( 1.2781 - D/(AD) ) . sqrt( PI (AD-D)D )
  = 1.4( 1.2781 - 1/A ) . D . sqrt( PI(A-1) )

Hence D = C / ( 1.4( 1.2781 - 1/A) . sqrt( PI(A-1) ) )

Just select a value for the aspect ratio that appeals to you, say 3, the
capactiance you want and out pops D.  For mathematical reasons A must be
greater than 1 (or else solution is not real) and should be greater than
2 for practical reasons (or the toroid intersects itself.

Hope this helps.

What is the origin of this equation?  While it is relatively
straightforward problem to derive the eqn for the capacitance of an
isolated sphere (one simple integral), my maths is far too rusty to do
so for a more complicated shape like a toroid.  Is there a derivation of
the equation from Bert's list anywhere?  (I would be surprised if it
came from a textbook in its current form given the number of terms that
can be cancelled directly or eliminated by simple rearrangement).

Best regards from one of the many generally silent but avid followers of
the contents of this list,

Safe and sucessful coiling to all.

Ian.
-- 
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Ian Holland                               E-mail:   Ian-at-DigIdeas-dot-com.au
Carnegie, Victoria AUSTRALIA              Callsign: VK3YQN
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