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Re: Bogus proof?

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> Subscriber: jd231825-at-engr.colostate.edu Tue Feb 11 23:07:41 1997
> Date: Tue, 11 Feb 1997 16:11:13 -0700
> From: Jeff Detweiler <jd231825-at-engr.colostate.edu>
> To: tesla-at-pupman-dot-com
> Subject: Bogus proof?
> 
> Hi all,
> 
> I was doing some reading up on transmission line theory and I don't
> understand what the 1/4 wavelength principle of the secondary has to do
> with resonance. Consider this proof:
> 
> Velocity of a wave travelling down a transmission line is:
> 
> v = l/sqrt(LC)         l = length of transmission line
>                        C = capacitance of the length "l" transmission
line
>                        L = inductance " " " "
> 
> And we know the resonant frequency of a secondary coil is:
> 
> f = 1/(2*pi*sqrt(LC))        eq. 2
> 

Here's the problem. This formula is only applicable to a discrete
LC circuit where all the physical dimensions of L and C are much
smaller than a wavelength (how long is the wire?). In that case
there is exactly one resonant frequency. A tesla coil with no 
top terminal does not closely meet this condition and therefore
any use of this formula is burrying many details. If you drive an
unloaded tesla coil with a signal generator, you will find many
resononant frequencies not just one. Therefore you've got a 
true transmission line. 

If, on the other hand, you put a large torriod capacitor on top 
of the coil you overwhelm the effects due to the distributed 
capacitance and therfore you'll measure one strong
resonance and the many, but very much attenuated,
extra resonances. Now your f=1/2piLC formula will work
assuming you can compute the C.

All you can say in general is that f depends on L and C
whether distributed or not. Though for some simple structures
you can actually compute the relevant L's and C's.

-Ed Harris
> also, since
> 
> v = f*lamda    lambda = wavelength
>                f = frequency
> 
> then:
> 
> l/sqrt(LC) = f*lamda         eq. 3
> 
> substituting resonant eq. 2 into eq. 3 for "f":
> 
> l/sqrt(LC) = lambda/(2*pi*sqrt(LC))
> 
> cancelling terms and solving for "l" the length of the transmission line:
> 
> l = lambda/(2*pi)
> 
> Thus at resonance, the actual physical length of the wire should be 1/2pi
> of the wavelength, and not 1/4. So where is this proof bogus? What
exactly
> does the 1/4 wavelength frequency have to do with resonance? I thought
> resonance is only a function of the L and C of the coil. I hope Fr.
McGahee
> will include this in the Guide.
> 
> Thanks,
> Jeff Detweiler