Re: Wire Length (fwd)

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Original poster: Gerry Reynolds <greynolds@xxxxxxxxxx>



---------- Forwarded message ----------
Date: Thu, 21 Dec 2006 13:28:34 GMT
From: Paul Nicholson <paul@xxxxxxxxxxxxxxxxxxx>
To: tesla@xxxxxxxxxx
Subject: Re: Wire Length (fwd)

Bart wrote:

> 1.8125 = Radius 1
> 24.4375 = Radius 2
> 50 = Height 1
> 50 = Height 2
> 91 = Turns
> 18 = Wire Awg

I ran this planar spiral through simulation software:-

Mode frequencies - unloaded
---------------------------
         Center ground             Rim ground
f1         388.6 kHz                 464.2 kHz
f3        1164   kHz                1236   kHz 
f5        1904   kHz                1972   kHz

Low frequency (bulk) L and C
----------------------------
Ldc                  4.205 mH      
Cdc                  51.68 pF      

Equivalent L and C
------------------
         Center ground             Rim ground
Lee @ f1     5.62 mH                 9.99 mH
Les @ f1     4.82 mH                 6.18 mH

Cee @ f1    40.5 pF                 30.7 pF
Ces @ f1    34.8 pF                 19.0 pF

V/I distributions
-----------------
(In these plots, the position given is the percentage turns from
 the cold end)

 Center ground:-
   http://www.abelian.demon.co.uk/tmp/baps-cg.f1.gif
   http://www.abelian.demon.co.uk/tmp/baps-cg.f3.gif
   http://www.abelian.demon.co.uk/tmp/baps-cg.f5.gif

 Rim ground:-
   http://www.abelian.demon.co.uk/tmp/baps-rg.f1.gif
   http://www.abelian.demon.co.uk/tmp/baps-rg.f3.gif
   http://www.abelian.demon.co.uk/tmp/baps-rg.f5.gif

The f5 plots are looking a bit crumpled in places - limited
spatial resolution of internal capacitance and small number
of turns is the cause.   

In both configurations the highest voltage gradients per turn
are occuring in the outer half of the winding.

Bart, if you've settled on a primary design, I can try to
predict the mode amplitudes and the time domain response.

The effective series inductance at f1 (quarter wave resonance)
in the rim-grounded configuration is almost 50% higher than the
DC inductance. Look at the current distribution to see why: the
current reaches a max at about 15% of the way into the coil,
at which point the current is about twice the rim current.
The induced voltage due to this extra current (brought about
by internal C coupling between outer and inner parts of the
winding) adds to the total voltage induced across the coil for
a given rim current, ie the effective inductance is pushed up.

If Bart can inject a measured current at f1 into the rim
and simultaneously measure the center voltage, he should be
able to confirm this effective inductance.

Even more dramatic is the increase in the energy storage 
effective inductance (Lee) which is over twice the DC
inductance!  

(The need for separate effective inductances to represent
the induced voltage (Les) and stored energy (Lee) is one of
the many subtleties arising from a non-uniform current
distribution in a coil).

Lee can be measured by a simultaneous measurement of input
impedance (as seen looking into the rim) and Q factor.
Then calculate

  Lee (measured) = (Q * Zin)/(2 * pi * Fres)

This is a tricky pair of measurements to do accurately, but
the sheer size of Lee compared to Ldc might enable Bart to
demonstrate this point with the instruments he has available.

See

  http://www.abelian.demon.co.uk/tssp/pn2511.html

section 7 for definitions of these effective reactances.

Let's compare the DC inductance with the (Wheeler, 1928)
approximate formula posted by Shaun,

           (NR)^2
 L =  -----------------
           8R + 11W

Here, W = 24.4375 - 1.8125 = 22.625 inches
      R = (24.4375 + 1.8125)/2 = 13.125 inches

which gives L = 4.03 mH, an error of -4% compared with our
more accurate calculation of 4.205 mH, so not bad at all.

--
Paul Nicholson
--