Wire length,resonance, and Q (fwd)

From:  Jim Lux [SMTP:jimlux-at-earthlink-dot-net]
Sent:  Monday, May 25, 1998 10:39 AM
To:  Tesla List
Subject:  Re: Wire length,resonance, and Q (fwd)

>  > Going forward;
> > This thread is always interesting when it comes up from time to time.
The big
> > question which I keep asking myself is, "What are the optimum
parameters?" as
> > John hinted towards. I wonder if there is really is an optimum. It
appears that
> > each coils optimum setting is unique to the geometry, size, power
levels, ground
> > plane, and a host of other variables.
> > 
> They are pretty conflicting: A high secondary reactance to yield a 
> high voltage, a low secondary reactance to yield a high current. 
> Using a large top C removes the burden of the low reactance 
> requirement from the resonator itself. However, Vout can only reach 
> what Es and top C will allow.

Of course, "optimum" implies that there is some attribute (or combination
of attributes) that one wants to maximize (or minimize). Just what might
that be? Given most comments, I suspect that what people want out of their
coils is long sparks.  

This is very interesting, because long sparks don't take particularly high
voltages, once you get above a certain point (around a MV). After that it
is more dependent on the stored energy and the "shape" of the Efield.

So, you want the output voltage to be around a megavolt (whether you
consider it as a open ended transmission line, a parallel LC, or whatever).
You also want enough stored energy (typically in the Ctop) to support the
growth of the leader. You want a low resonant frequency so the leader has
time to grow before the polarity reverses.

Bazelyan and Raizer point out that the continually increasing capacity of a
long spark channel as it extends is quite significant. The channel is very
low resistance, so you have to pump ever more charge into it to keep the
voltage up at the end, which is necessary for it to keep growing. The
changing circuit parameters means that approximating the system as a lumped
LC with a damped ringing waveform probably isn't particularly accurate.

BTW, is it possible that both the 1/4 and lumped approximations are correct
simultaneously? Perhaps the "optimum" number of turns and dimensions is
when the propagation time up the coil is 1/4 period and when it is a
parallel resonant circuit. I will surmise that the actual length of the
wire probably isn't important, since in a close wound coil, the propagation
speed is more determined by the characteristic impedance, which is a
function of the distributed L and C. Commercial "delay line" coax has a
helical center conductor for this reason: it gets the L/meter really high,
so the Z0 is really high.

My gut feel is that the dimensions of a typical coil are so tiny compared
to a wavelength, that the lumped approximation should be more accurate.
This is good, because it means we can do the calculations of leader growth,
etc, using a fairly straightforward set of differential equations, which
can be numerically integrated.