# Re: Racing Spark Prediction

```Original poster: Paul Nicholson <paul@xxxxxxxxxxxxxxxxxxx>

Dimitry wrote:

> M[x][y] ... i mistakenly confused "x" with secondary and "y"
> with primary...

Thanks Dimitry, you saved me a post to point that out.  You
obviously understand the thing now.   For the sake of anyone
else digging into GeoTC, the 2-D array M represents the
distributed self inductance of the secondary - each element
M[a][b] represents the mutual inductance between the sections
of the coil labelled 'a' and 'b'.   In the lumped model, self
inductance is just a single number, in a distributed model it's
a density function of two position variables.  In the half-way
land occupied by our numerical models, the self inductance is
represented by the 2-D array in the computer, which in turn is
representing a square, symmetric matrix.

Very roughly, the coil modelling goes as follows:  Define
arrays V and I to represent the initially unknown voltage
and current profiles - these are what we want to calculate.
Then, set up an equation that would calculate V if we happened
to know I, something like

V = [M] I

where [M] is a square matrix full of inductance coefficients, and
V,I are column vectors.  I put [M] in square brackets to remind us
that it's a square matrix and the other things are arrays.

Then make up another equation - one which would generate I given V,
say,
I = [C] V

where C is another square matrix - the capacitance matrix in
which each element C[a][b] gives the capacitance between sections
'a' and 'b'.

Now the key is that any voltage and current profile that the coil
can support must satisfy both of these equations at the same time,
so we solve them as a pair of simultaneous equations.   Easy - we
just substitute to get rid of either V or I, say

V = [M] [C] V

We know both [M] and [C] because we calculate them quite accurately
from the given geometry of the coil, and we multiply them together
to give a single square matrix [A] which then represents the entire
reactance of the coil.  So, you have V = [A] V for any voltage
profile that will fit on the coil.  Find an array of numbers, V,
that when fed through the square matrix [A], comes out the same.
If you find one, you can call it an eigenvector of [A] and it
represents one of the resonant modes of the coil.

Ok, there are a few minor details glossed over, such as the fact
that I've left out a few differential coefficients, but the general
idea comes across, I hope.

GeoTC does things slightly differently to save work, which is one
reason it only computes the lowest resonant mode, but the principle
is pretty much the same.

Dimitry wrote:
> to plot primary induced voltage profile we need to calculate
> Mp(x) instead of M(x,y), so probably at first we need to modify
> "function XX_compute_Mdc( c1, c2)" to return a vector instead of
> single number value?

Right.  And the 'lumped' mutual inductance is then just the straight
sum of all the elements.

Gerry wrote:
> Im wondering if there is a third component to the voltage
> profile...

> For a 1000 turn coil probably 1000 little traveling waves each
> starting from a different turn of the coil if you get my meaning
> here.

That's fine.  These are all contained within the calculated self-
resonance of the coil, and your description in term of travelling
waves initiated on a turn by turn basis is a valid way to look
at the thing.   After all, our models start by setting up the
equations for a single turn or section, whatever, of the coil
using a general label say 'x' to indicate some arbitrary place
on the coil, and then we say - right, apply those equations to
every section of the coil, simultaneously, so we end up with a
whole stack of equations, identical but for a different value
of 'x' in each.  That's how we build up, row by row, the matrix
system mentioned above.

> The current flowing by the traveling waves creates a magnetic
> field that propagates at the speed of light to all other turns
> in a direct path instead of following the wire path.

Yes, and ditto the voltage and electric field.  We can forget
the speed of light, seeing as the coils are so small compared
with a wavelength, you can just say that the coupling is
instantaneous which makes the arithmetic a lot easier and only
loses a miniscule bit of accuracy.

> Does this paradigm seem reasonable???

Yes, although it doesn't add "a third component to the voltage
profile".  Superposition works here - model the response as if
each turn is excited (somehow) in isolation and add it all up
to get the familiar 'total' voltage.

Each section of the coil can be thought of as being a simple
harmonic oscillator in its own right, each coupled to every
other by the E and H mechanisms.  The combined motion of the
them all is what we see as the V/I distributions.  This is
much like a vibrating beam or string, but those are simpler
because each 'oscillator' is coupled only to its immediate
neighbours.  In the resonating coil, you have long range
coupling between oscillators which makes the behaviour much
richer - dispersive propagation, complex characteristic
impedance, etc.

--
Paul Nicholson
Manchester, UK.
--

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