From: Paul
Date: Mon, 16 Oct 2000 07:04:55 +0100
Subject: Re: [TSSP] Surprising secondary voltage profiles
Terry,
Yes, the measured at predicted curves are stunningly different,
something we really must resolve.
Terry wrote:
> I used a tiny antenna placed very near the coil and measured
> the RMS voltage...
How was the RMS extracted from the RF probe signal?
> I retuned the coil for
> maximum signal to sort of compensate for detuning by the antenna.
The closeness of the probe, and the retuning, might upset the
measurements, if the probe presents a capacitance
approaching that of the topload + the self cap of the part of the coil
above the probe, you may have been making the probe position the
effective electrical 'end' of the coil, ie the part of the coil
above the probe is starved of current and there would be little or
no V rise above the probe.
eg
V |
| . . . . .
| .
| .
| .
| .
|.
------------------------- X
^ probe
Did you have to retune more than a few percent?
You could check for this with a simultaneous V probe on the toroid.
The toroid volts should remain constant-ish as the moving probe works
its way up the coil.
Referring to Malcolm's Ruler, I'm not at all convinced that
the mechanical analogy can be applied. The differential equations
which apply to the beam displacement are very different to those
which apply to the electrical potential. It's a long time
since I did mechanics theory, but I managed to look up some
formulae describing the resonating beam.
Apparently, the displacement V(x,t) of position x, time t, is
governed by
EI d^4 V(x,t) / dx^4 = - lambda d^2 V(x,t)/dt^2
where the d are partial differentials. E is the uniform
elastic modulus, I is the uniform inertia, and lamda is
the uniform mass density per unit length.
The x and t dependency of displacement V(x,t) can be separated,
to get, for the x dependency, ie the displacement profile,
d^4 V(x)/dx^4 = k^4 V(x)
where k is a constant for a particular mode of vibration.
The solution to this 4th order equation, with the boundary conditions
as for Malcolm's Ruler, is an evil looking thing,
V(x) = C * ( cos(kx) - cosh(kx) +
(sin(kx) - sinh(kx)((-cos(kL) - cos(kL)))/(sin(kL)-sinh(kL)))
and the plotted solution looks exactly like the curve of the ruler.
An approximation to V(x) is that which occurs when the beam is
bent by application of a force at the top:
V(x) = 0.5 Vmax ( 3(x/L)^2 - (x/L)^3) for 0 <= x <= length L
None of this is very reminiscent of the much simpler differential
equations governing a transmission line in the absence of
longitudinal coupling. The sine profiles of the uniform transmission
line are distored in the x direction - stretched and squeezed when
the line made is non-uniform, but there is no point of inflection
allowed in the absence of longitudinal coupling. (I think I can
prove that.)
The only hope of rescuing Malcolm's Ruler is, as you say, the
possibility that the longitudinal coupling, ie internal C and
mutual inductance, comes to the rescue and enables the V profile
to go concave.
Let's look at what it would take to do this. In my last email
I said that
dV(x) = w L I(x) dx
and so for V to be concave, I(x) needs to be increasing.
The mutual inductance modifies this so that the I(x) is replaced
by a weighted average of the I in the region around x - the
weighting being the mutual inductance profile. If that region,
ie the 'span' of the mutual coupling were to extend from the
top half to the bottom half of the coil, the large currents lower
down might induce a significant extra dV/dx in regions higher up.
The problem is however, if you now slide that coupling region a
little way up the coil in order to examine the dV/dx at a point
higher up, all the contributing currents going into the weighting
region are reduced a little from what they were lower down.
Thus, even if a substantial portion of dV(x) is due to mutual
inductance from higher current regions lower down, the dV(x)
will still reduce with height, ie the V slope will be convex.
Thus, on the face of it, a uniform mutual inductance profile
cannot make V concave. On the contrary, mutual inductance
I think will act to flatten the V profile towards linear, by
making dV/dx depend on a local weighted average of the currents
around x, rather than just the spot current at x - a kind of
smoothing process.
So that leaves the hope that the displacement currents of the
internal capacitance can come to the rescue. The problem here is
that the direction of the displacements currents is the wrong way
around to do this, it acts to remove current just when you
want it to insert current. Consider any two points A and B on
the coil, A higher than B, and therefore at a higher voltage.
Consider a quarter cycle during which the voltage is rising
from zero towards a peak. Current is leaving both A and B to
charge up the external capacitance. But since A is at a
higher voltage than B, current will also leave A towards B as
the internal capacitance between A and B is charged up, thus
taking more current from the high point A than would be the
case without internal capacitance.
The two points could have been chosen anywhere, so the general
flow of internal capacitance displacement current is from high
to low as the voltage is rising.
Thus internal capacitance acts to divert current away from the
coil higher up, and insert it lower down, thus reducing series
current with height, and tending to make the V profile more
convex.
I believe this 'descending' internal C displacement current is
the cause of the current peak which occurs just above the bottom
of the coil, which *is* associated with a small concavity in the V
profile near the bottom, ie the effect you wanted is there, but
at the wrong end of the coil.
My understanding of this is summarised in pn1310 page 4, eq 5.5
which gives differential equations applying to the V and I profiles
of a tesla coil.
The results from tsim are based on a model which takes into account
the mutual inductance and internal C. According to this model, they
have an effect on dispersion, but so far all the coils looked at
the V profiles have been an almost linear rise over say the lower
60% of the coil, with a gentle levelling off above that. The current
is always a very distorted cosine, with no tendency to rise in the top
half of the coil - it falls quite rapidly in this region.
The possibility remains that I've dropped a minus sign in the program
(or in my head!) which might reverse the effects described above, so
I'll try to prove this one way or another without reference to tsim.
> One thing that does trouble me is that E-Tesla5 uses the curves I
> measured and gives very good accuracy for loaded and unloaded coils.
> I tried your graphs back then as secondary voltage profiles and the
> program showed significant errors. Not just simple scaling but it
> appeared the voltage profiles were not working...
Concerning E-Tesla5, I have a gut feeling that you should be
weighting with V^2 rather than V but I'm unable to explain myself
at the moment, but its something to do with weighting to get
an equivalent stored energy, rather than a 'mean' capacitance.
I'll ponder this further.
Cheers,
--
Paul Nicholson,
Manchester, UK.
--
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.