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Re: Reactive to resistive ballast ratio of sqrt(3):1 ? (fwd)
---------- Forwarded message ----------
Date: Fri, 15 May 98 18:34:48 EDT
From: Jim Monte <JDM95003-at-UCONNVM.UCONN.EDU>
To: tesla-at-pupman-dot-com
Subject: Re: Reactive to resistive ballast ratio of sqrt(3):1 ?
Maybe I'd better clarify what I did since I did not solve a differential
equation. While the spark gap is firing, the most significant load that
the HV transformer sees is the gap, which is nearly a short circuit.
However, while the gap is firing there will hopefully be current flowing
through this same gap from the Tesla coil's primary. That is, the gap
is common to both the HV transformer and the Tesla coil's primary. The
instantaneous current in the gap will be changing greatly due to the
coil's resonance, and as a result, the dynamic impedance of the gap will
also be changing. It would seem reasonable that stability would be
enhanced if the HV transformer saw a more constant load. What I noticed
in my original post is that the current drawn from the HV transformer
while the gap fires is most independent of the resistance of the gap
when the ballast has a X/R ratio of sqrt(3). My question in that post
is whether any of this is meaningful in practice.
There is an error that propagated through the last 2 lines of the
original post. I will include the correction below. This does not
affect the ratio of sqrt(3) or the fact that the ratio is a minimum.
Jim Monte
>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!
<snip>
>>
>>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
d3|Ia|**2/dRa3 = 24*V**2*(R+Ra)*(X**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,
d3|Ia|**2/dRa3 = (3/16)*V**2/(R+Ra)**5 > 0,
>