RE: Need Formula for length of spiral

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

Original poster: "Pete Komen by way of Terry Fritz <twftesla-at-qwest-dot-net>" <pkomen-at-zianet-dot-com>

Sorry Jim, but the r^2 doesn't make sense to me.  I suggested that the
length of the wire is PI * Average diameter * number of turns.  Take the
case of three turns: the center turn length is PI * Diam., the outer turn is
PI * (diam + delta); the inner turn is PI * (diam - delta).  Add them all
together and the + and - PI * delta terms cancel leaving 3 * Diam * PI.

R^2 gives units of area (a unit of length squared).  For a close wound coil
the area of the windings divided by the width of the wire would give a close
approximation of the length but not r^2.  Note that the width is arbitrary.

You said, "You know that the sum of all areas of the rings is the area of
the circle,
which is proportional to r^2, so therefore, the sum of the circumferences
must also be proportional to r^2."

I agree with the first part, but the conclusion is wrong.  Consider your
painting with three wide rings of constant width.  The total length of the
rings is proportional to r^2/width (of the ring).  Since the width term is
not fixed your conclusion is invalid.  In any circle of radius R, any
arbitrary number of rings could fit, thus making the total length (or
average circumference) of the rings arbitrarily different from R^2.

Regards,

Peter Komen

-----Original Message-----
From: Tesla list [mailto:tesla-at-pupman-dot-com]
Sent: Thursday, February 14, 2002 11:37 AM
To: tesla-at-pupman-dot-com
Subject: Re: Need Formula for length of spiral

Original poster: "Jim Lux by way of Terry Fritz <twftesla-at-qwest-dot-net>"
<jimlux-at-earthlink-dot-net>

A further aspect which occurred when driving to work this morning..

Consider the length of each turn (spiral or not, but concentric circles, in
any case).  They're going to be some form of 1+2+3+4+5+.... which is a
N^2+N kind of thing...

Another analogy (due to Leibniz, and the Chinese, much earlier). Imagine
you are painting in a circle with concentric rings of constant width.  The
area (or circumference, since it is constant width) of a given ring is
proportional to its radius.

You know that the sum of all areas of the rings is the area of the circle,
which is proportional to r^2, so therefore, the sum of the circumferences
must also be proportional to r^2.

A spiral winding typically doesn't start at the center of the circle, so
you're really working on an annulus (a circle with a circular hole in the
middle), but the idea of dependence on r^2 still holds.

And, just to beat this into the ground somewhat more, I considered the case
of the spiral where r = k * theta.  There are other spiral forms, which may
have different relations (because the assumption of equally spaced rings,
as above, doesn't hold). (examples: r = r0*k^theta (equiangular/log
spirals))

For those more analytically inclined..

consider a small length of the winding ds.  ds = r*dtheta where r =
r0+k*theta, so, the total length is

integral[theta start, theta end] of (r0+k*theta)*dtheta



Tesla list wrote:
>
> Original poster: "Jim Lux by way of Terry Fritz <twftesla-at-qwest-dot-net>"
<jimlux-at-earthlink-dot-net>
>
> since the length is essentially integrating a linearly varying function
> (radius) there will need to be some squared term in the equation. Granted,
> for small enough ranges of parameters, a linear approximation will
probably
> work.
>
> ----- Original Message -----
> From: "Tesla list" <tesla-at-pupman-dot-com>
> To: <tesla-at-pupman-dot-com>
> Sent: Wednesday, February 13, 2002 8:37 PM
> Subject: Re: Need Formula for length of spiral
>
> > Original poster: "Steve Stuart by way of Terry Fritz
<twftesla-at-qwest-dot-net>"
> <sstuart-at-glasscity-dot-net>
> >
> > Try:
> >        L = (Do - Di) / 2 * 1.6 * pi * T
> >
> > Where:
> >        L  = conductor length
> >        Do = outside diameter
> >        Di = inside diameter
> >        T  = number of turns
> >
> > It will give you a pretty close approximation
> >
> > 73 de Steve
> > ·¸¸·´¯`·¸¸·´¯`·¸¸·´¯`·¸¸·´¯`·¸¸·´¯`·¸¸·
> > w8an-at-w8an-dot-net
> > http://www.w8an-dot-net
> >
> > Tesla list wrote:
> > >
> > > Original poster: "John Tomacic by way of Terry Fritz
> > <twftesla-at-qwest-dot-net>" <tesla_ownz_u-at-hotmail-dot-com>
> > >
> > > Hi everyone,
> > >
> > > Does anyone have a formula that I can use to calculate the length of
> wire
> > > required in a flat spiral coil? I have the formula for inductance,
> however,
> > > I really need the wire length.
> > >
> > > Thanks,
> > >
> > > John
> > > SST coiling in Ottawa.
> > >
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