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Re: High Voltage Output

Original poster: Jim Lux <jimlux-at-earthlink-dot-net> 

At 11:37 AM 6/1/2004 -0600, you wrote:
>Original poster: "Chris Fanjoy" <zappyman-at-hotmail-dot-com>
>   As someone relatively new to this hobby (still building my first TC) I 
> haven't fully grasped all the principles involved. One thing that has me 
> puzzled is this: what determines the high voltage output of a Tesla coil? 
> If not the step-up ratio of the coil itself, then how about:
>-Firing rate of spark gap
>-Size of tank capacitor
>-Operating frequency of coil
>-All of the above?
>   Just curious, as this may be something to consider as I design my own TC.
Overall, it can get complex, but here goes with the simple explanation..

The TC transfers energy from the primary to the secondary.  The energy 
available to transfer is determined by the primary voltage and the primary 
capacitor = 1/2 *Cp * Vp^2

That is the absolute maximum energy that can be transferred to the 
secondary capacitor (the top load + parasitic C from the secondary 
winding).  In reality, there are losses..

If you know the secondary capacitance, then you can calculate the voltage 
from the same equation:  = 1/2 * Cs * Vs^2

Hence the usual equation you'll see:  Vsec = Vpri * sqrt( Cpri/Csec)
Since the primary and secondary are part of an LC resonant circuit, and 
resonate at the same frequency, we know that Lpri*Cpri = Lsec * Csec...
That is: Lsec/Lpri = Cpri/Csec
We can substitute that in and get another familiar equation:

Vsec = Vpri * sqrt( Lsec/Lpri)

In reality, the max voltage is limited by two effects:

1) Loss... A fair amount of the energy in the primary capacitor is lost in 
the spark gap (hence the interest in low loss switching with solid state 
devices) and in such mundane things as the resistance of the wires in 
primary and secondary.

2) As soon as a spark starts to form on the secondary top load, the voltage 
tends not to rise any more, and the energy flowing into the secondary goes 
to making the spark bigger (both in terms of heating up the air, and in 
terms of charging the increased capacitance of the secondary.. the spark 
itself has some significant capacitance to ground).  The breakout voltage 
is primarily determined by the radius of curvature of the topload and the 
surface roughness.  At the highest, it's probably 30 kV/cm or 70 kV/inch of 
radius, so a 4" diameter dryer vent pipe toroid isn't going to get a whole 
lot higher than say, 280 kV, and probably a whole lot less.

You'll note that turns ratio doesn't enter into this anywhere!

If what you're interested in is big sparks, that's determined more by the 
energy available to grow the spark, and since sparks have a limited life 
time, you need high average power too.  I have seen several meter long 
sparks drawn from only 60 kV (but there was several hundred amps in the 
arc).  Likewise, my Van deGraaff, which on a good day might get to 300 kV, 
can only make sparks 30-40 cm long, and they're pretty thin and wispy.

A good rule of thumb for spark gap tesla coils is spark length (in inches) 
= 1.7 * Sqrt(average input power in watts)