Re. Reactive to resistive ballast ratio of sqrt(3):1 ? (fwd)

---------- Forwarded message ----------
Date: Fri, 15 May 98 14:33:09 EDT
From: Gary Lau  15-May-1998 1423 <lau-at-hdecad.ENET.dec-dot-com>
To: tesla-at-pupman-dot-com
Subject: Re. Reactive to resistive ballast ratio of sqrt(3):1 ?

Wow!  It's been a long time since I've tried my hand at differential
equations.  I'll just have to believe you!

My experience has only been with NST-based coils, small potatoes by some
standards, so I have no experience with inductive and resistive ballast.
But this got me to thinking, could NST-based coils also benefit from this
analysis?  Most NST coilers use some sort of R/C/L network between the
main gap and the NST for protecting the NST.  Analysis of these networks
has focused solely on their protective merit.  Could these networks also
be affecting coil performance?  Certainly a much harder problem to model,
given the lack of a good model for current-limited transformers.

With more questions than answers,
Gary Lau
Waltham, MA USA

>Date: Wed, 13 May 98 22:41:34 EDT
>From: Jim Monte <JDM95003-at-UCONNVM.UCONN.EDU>
>To: tesla-at-pupman-dot-com
>Subject: Reactive to resistive ballast ratio of sqrt(3):1 ?
>I have been thinking about the advantages of including resistive ballast
>in the charging circuit, and I stumbled on something a little intersting.
>I found that an arc is most stable with the reactive and resistive
>components in the ratio of sqrt(3):1.  How does this compare with
>values that people find to work well in practice?
>Below is the justification for my stability claim:
>Consider the circuit below representing the charging portion of a
>primary when the gap is firing.
>                      - Ia ->
>      +--- Z = R + jX -------+
>      |                      |
>      V                      Ra
>      |                      |
>      +----------------------+
>V is the secondary voltage of the HV transformer, Z is the ballast
>impedance (transformed via the turns ratio of the transformer if on
>the primary side, but this leaves the X/R ratio unchanged anyhow), and
>Ra is a crude approximation of the spark gap as a resistance which will
>later approach zero.
>I defined stability as the partial derivative of the magnitude of
>the arc current with respect to the arc resistance --
>d|Ia|/dRa, found its minimum (actually I found the minumum of its square,
>d|Ia|**2/dRa, which was a bit easier to do and has the same minimum
>location), and then differentiated again to show that the extremum
>was in fact a minumum.
>Current magnitude squared is
>|Ia|**2 = V**2/((R+Ra)**2+Z**2)
>Defining stability as the first partial with respect to Ra,
>d|Ia|**2/dRa = -2*V**2*(R+Ra)/((R+Ra)**2+Z**2)**2
>Taking the second partial,
>d2|Ia|**2/dRa2 = 2*V**2*(3*(R+Ra)**2-X**2)/((R+Ra)**2+X**2)**3
>  Setting to zero for min, X**2 = 3*(R+Ra)**2.
>  Letting Ra -> 0, X = sqrt(3)*R
>Taking the third partial,
>d3|Ia|**2/dRa3 = 12*V**2*(R+Ra)*(X**2-2*(R+Ra)**2)/((R+Ra)**2+X**2)**4
>Evaluating at X**2 = 3*(R+Ra)**2,
>d3|Ia|**2/dRa3 = 12*V**2*(R+Ra)**3/((R+Ra)**2+X**2)**4  > 0,
>so this is a minimum.
>Comments?  Does this seem to be a reasonable approach?
>Jim Monte