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Why Medhurst derived fr is approxiamlty correct and the transmission line equation for the resoance frequency of a 1/4 wave helical resonator or Tesla coil.

Hi all,

The following is an updated copy of a previous post that was lost.

I have completed most of my analysis and  I have reached my goal of trying
to understand what initially appeared to me as violations of circuit laws. I
have now even solved the boundary problem for the current and voltage
profiles in a Tesla coil.

I started of with the view that the lumped tuned circuit equation using
Medhurst C could not possibly produce the correct frequency. This was due to
my belief and apparently that of others  that Medhurst C was the true self C
of the coil. It is actually the value you must subtract from a much
larger(>100x)  tuning capacitor to calculate the measured resonance
frequency of a coil with one end grounded and the coil isolated. i.e. a
correction factor for true self C and the internal turn to turn capacitance.
Where as true self C is the capacitance you would measure with a low
frequency (<< fr) LC bridge with the coil isolated.  It is the sum of the
distributed capacitance of each turn to ground and is an intrinsic property
of the coil.  It is only a function of its geometry and surrounding.  It is
independent on the mode of oscillation of the coil. For a close wound coil
its value is equal to the capacitance of a hollow cylinder with the same
dimensions as the windings.

If you differentiate a formula for Med C it confirms that the true self C is
approximately constant along the coil with a small increase at the ends.
Incidentally it also confirms its for an isolated coil.  Then with simple
circuit analysis referring a constant distributed C to one end shows that
true self C of the coil is 3 times Med C for very short coils and greater
than 10x the 1/4 wave resonant frequency,  and  2 times Med. C for  long
coils and at least 10 times the 1/4 wave resonate frequency.  So a quick bit
of lumped approximation shows that the lumped equation using Medhurst C
could produce approximately the correct answer.  Which has been confirmed by
measurements on a variety coils.

A more accurate analysis using my (not a claim of originality) transmission
line equation (given below) gives  Med. C as being  2.46 or (PI**2)/4 at the
1/4 resonate frequency for long coils. I did not realise the significance of
this when I first  posted the above comments.  It simply means that Med. C
can only be proved to be valid an theoretical grounds for frequencies at
least 10x the 1/4 wave resonance of the coil or putting it an other way when
the propagation time of the coil is insignificant.  Which is precisely what
Medhurst was using it for.

Please note that I do not necessarily suggest that Med. C when used in a
practical calculation produces the wrong measured 1/4 wave resonance
frequency. As you can see from the above for an average coil  the ratio
between Med. C and true self C must be between 2 and 3 compared to the 2.46
factor which has no end effects, then add the error reduction effect of
the squareroot and you have a reasonable  fr from Med. C. There are also
other effects which could conspire to correct it. This readily accounts
measurements on coils that are compatible with the lumped tuned circuit
equation using Med C.


When I first checked the transmission line equation I erroneously used Med C
which produces the wrong answer not surprising considering the above.

Now if you use the true self C per unit length  and L/length in a
transmission line equation for fr you do get a reasonable answer.   But I
now know that the standard transmission line equations are not supposed to
apply because of the mutual coupling between turns or you should be using
the inductance per unit length which is zero.  So why the correct answer
with L/length???

I now know from my analysis that the correct parameter that must be used in
the transmission line equation is the mutual inductance per unit length
which is a function of frequency.  But for coils greater than 3 diameters
long this has the same numerical value as the total inductance dived by the
length of the coil (for wavelengths greater the length of the coil).  Hence
the equation works.  Which means that if you measure the true L and C of
your coil the 1/4 wave resonance is the reciprocal of the four times
squareroot of the product of the L and C measured in H and F respectively.
Strictly this only true for long  isolated coils (no end effects) close
wound and at least ten turns per diameter length . In fact there is
justification for believing that the end effects are small. It also does not
include the effects of internal C or top load.

The same reason that makes the velocity equation valid also makes the wave
impedance equation valid again accurate only for long coils (no end
effects).  I don't know what the end effects will have on the impedance.

Revelation is a wonderful feeling but I feel a little sorry that there are
no new circuit laws for Tesla coils or at least I know of none.

I have also completed my analysis of the simple addition law for top C and
Med. C and true self C which I will post in due time.

This is not attempt  to resurrect the wave/lumped debate.

It is just the conclusions of my application of circuit laws to Tesla Coils
on which subject I am happy to receive or invite comment.

Regards Bob (Robert Alwyn Jones  a circuitlawophile)