From: Paul
Date: Fri, 01 Sep 2000 21:48:44 +0100
Subject: Re: [TSSP] Mystery of the missing loss
This might be a good time to summarise the results
of this weeks experiments by Terry, stimulated by the
inability of the computer model to predict Q factors
(and therefore input impedance).
The loss in the model is almost entirely based on winding
resistance which is predicted using Medhurst's empirical
tables. Unsurprisingly this under-estimates the loss, but
the under-estimate is quite a large amount, eg
Q Measured Model
f1 84 147 With toroid.
f1 57 221 No toroid.
Not only are the predictions wildly in error, the trend
in Q on fitting the toroid is opposite, a major qualitative
difference.
I can see only four plausible explanations:
1/ The software implementation is faulty and fails to
properly represent the physical model we are using.
2/ The Medhurst factor only applies to coils with an
essentialy uniform current distribution. His measurements
were made using a very large external capacitance to
resonate with the coils, which would force a uniform current
distribution. In the case of a tesla secondary, the
current is far from uniform. However the model takes
this into account quite accurately, since it divides
the Medhurst resistance equaly among all the turns and
then computes the loss based on the current at each turn.
Nevertheless the possibility remains that the effective
series resistance is fundamentaly higher in tesla
secondaries than that predicted by Medhurst.
3/ The return current path of the external capacitance
(of both coil and toroid) will contribute some I^2R
loss. This has been assumed negligible by requiring a
ground plane of at least a coil height in radius so that
the loss resistance experienced by displacement currents
landing on the ground plane should be less than 1 ohm.
The effect of external flux impinging on the wider
surroundings is an unknown quantity and may thus be
a significant factor.
4/ The continuous ground plane sheet cutting across the
B field near the base of the coil is bound to contain
circulating eddy currents and is therefore an energy
absorber. This extra loss is also an unknown quantity.
I'll now look at each of these four in more detail:
1/ Coding errors.
I found one this week, an incorrectly applied formula
for skin effect, which was over-estimated. Fixing this
has actualy made things worse. There is always the chance
of more errors but the code territory in which they may
lie is getting smaller. We predict the correct resonances
over a wide range directly from the geometry with no fiddle
factors, so there cannot be much wrong with the network
equation setup and elimination steps. The Q factors and
input Z are arrived at by different means but are
nevertheless commensurate with one another.
2/ Winding loss.
Perhaps the effective AC resistance is being underestimated.
The model predicts 175 ohms effective and Terry at one
time arrived at around 390 ohms. The W factor was put in the
code to see what the coil would look if the series resistance
was higher, so lets apply it... W = 390/175 = 2.23, gives
Q Measured Model
f1 84 66 With toroid.
f1 57 100 No toroid.
which remains qualitatively wrong.
3/ External E-field loss.
The resonating capacitance of Terry's coil is made up
of the following component contributions:
Coil Coil
internal + external + toroid
3.16pF + 8.49pF + 25.42pF = 37.07pF With toroid.
3.29pF + 12.04pF + 0.00pF = 15.33pF No toroid.
In what must have been a quite delicate experiment,
Terry replaced the toroid with 14.7pF of stray
capacitance and 10.7pF of vacuum variable, thereby
replacing some 30% of the potentialy lossy external
capacitance with a reasonably high Q lumped component.
This reduced the Q to 87% of the toroided value, whereas
if external field loss was significant, we might have
expected the Q to improve.
In the model, if we set the external capacitance loss
factor Dp to correspond with a couple of hundred ohms Rct,
we get
Q Measured Model
f1 84 57 With toroid.
f1 57 44 No toroid.
and again the predicted figures remain stubbornly the
wrong way around.
4/ Eddy current loss in external B field.
Placing the coil on top of the foil ground plane reduced
the inductance from its nominal value of 75.4mH to 74.1mH,
a strong indication that inductive coupling into shorted
loops is occurring. This offers a potential source of
significant loss.
Terry made a comparative set of measurements to test
the effect of splitting the ground plane into two halves,
the idea being to break the largest eddy current paths.
Splitting the ground plane raised the inductance from 74.1mH
to 75.1mH, almost restoring its nominal value. The Q factors
were:
Ground plane
Continuous Split
84 71 With toroid.
57 49 No toroid.
The increase in inductance confirms the split was effective
in reducing the total coupling coefficient to the ground plane,
but both the Q factors actually went down, by the same proportion,
to 85% of the original. This behaviour is at first sight
surprising, but on closer examination, might begin to look
plausible. The effect of a shorted loop on the secondary depends
on the ratio of mutual inductance to resistance of the loop in
question. A loop of negligible resistance but high mutual
inductance with the coil, will pull the secondary inductance to
a lower value without contributing much to the loss. A loop of
smaller mutual inductance, but with a higher resistance will
cause less shift of the secondary inductance, but may offer a
proportionaly higher loss.
In a continuous sheet placed across the B field, current loops
of all sizes and all possible centers will be induced. However,
the current in any small current loop tends to be cancelled by
an opposite and almost equal current in a neighbouring loop, so
on average the net current flow is cancelled in all loops except
those centered on the coil/field axis, so the result is a circular
current sheet flowing around the axis.
Breaking the ground plane in half causes the eddy currents to
form into two separate not-so-circular current sheets in each
half, centers offset from the axis. Mutual inductance is roughly
proportional to the loop area, and loss resistance proportional
to the loop circumference, so if the mean radius of the current
sheet was more than halved by the split, the inductive pull will
go down by more than 1/4 which is about what we see. Furthermore
the loss per loop will go down, but probably by less than half,
since the nice circular current loops in the uncut ground plane
are replaced by rather squashed ones in the split ground plane
and thus a longer perimeter. Bearing in mind that there are now
two of these current sheets, each with more than half the loss of
the original, it is I think possible for the total loss to go up
slightly when the ground plane is split.
So much for a handwaving description, its jolly hard to actualy
put numbers to this effect. I've been trying to sum the losses
due to the circular current sheet in the uncut ground foil,
treating it as concentric current loops and integrating. Initial
impression is that a hundred or so ohms can be reflected into the
secondary loss resistance from this, but I'm drowning in Legendre
polynomials and nowhere near certain of the figure.
Whatever the eddy current loss is, it will increase in proportion
to f cubed. I can at least test the effect on the model, in a
crude sort of way, by adding a contribution to the winding
resistance which is proportional to f-cubed.
We get an encouraging result when using a loss due to eddy
currents of 136 ohms at the lower resonance,
and 136 times the frequency ratio cubed at the higher
resonance:
Q Measured Model
f1 84 84 With toroid.
f1 57 58 No toroid.
Further weight is added to this theory by Terry's report that the
input impedances at f3 and f5 are too high to reliably measure,
which would be expected if the resonator contains a significant
loss proportional to f cubed.
I'll be bold and make a prediction, based on the above
modification to winding resistance, of the Q factors at f3.
Perhaps, if sufficient ground plane survives in Terry's lab,
he might be able to check these out:
Q Measured Model
f3 ?? 14 With toroid.
f3 ?? 11 No toroid.
The Q factors at f5 are about 4.
Congratulations to anyone making it this far through this
posting! I hope that others will agree that eddy currents
in the ground plane provide the best explanation of the
missing loss. If this is the case, it may well be present
in a number of tesla coil installations, for example the
Thor system exhibits a measured inductance over 5% smaller
than the Nagaoka value, and the measured Q of 222 compares
with a prediction of 350. Thor operates above a sheet metal
floor within metal walls.
I believe further work should be done to investigate eddy
current losses in ground planes as I am sure they are
generally under appreciated. I'm very interested to
hear what others on the list have to say on this subject,
especially if they can interpret Terry's measurements any
other way.
Regards all,
--
Paul Nicholson,
Manchester, UK.
--
Maintainer Paul Nicholson, paul@abelian.demon.co.uk.