These are the curves for the coil when modeled as accurately as possible,
taking into account the non-uniform capacitance distribution and the
effects of mutual inductance and internal capacitance.
Simulated by tsim version 0.3c.
| Example 1: Typical coil without toroid | Example 4: With uniform external capacitance |
| Example 8: Typical coil with toroid | Example 7: With no external capacitance |
| Example 2: No internal capacitance | Example 9: Small external capacitance and topload |
| Example 3: No longitudinal coupling | Example 10: Inverted capacitance distribution |
| Summary: Notes on V/I profiles and distributed reactances | |
| Example 1: Typical coil without toroid |
These are the curves for the coil when modeled as accurately as possible,
taking into account the non-uniform capacitance distribution and the
effects of mutual inductance and internal capacitance.
Note the linear rise of V over the lower 70% of the coil, with a gentle fall of the slope above that. The current is a distorted cosinusoid, with a peak about 15% up the coil due to displacement currents in the internal capacitance. Note the slight concavity in the V profile below the current peak.
In this example Vt is around 454 volts with an input
current Ib of 0.0081 amps. Thus the transimpedance is
around 56K ohms.
| Example 2: With no internal capacitance |
The same coil as above, but with internal capacitance 'switched off'
to demonstrate the effect of Cint on the amplitude profiles.
The resonant frequency went up by 7.4% as a result.
The current peak at 15% height has disappeared. The V profile, if anything is slightly more linear without the internal capacitance.
The transimpedance has fallen slightly to around 51K ohms, but is still quite a bit higher than the cosine based prediction.
Note that the current peak in the lower part of the coil which has disappeared with the internal capacitance is due to the asymmetric effect of its displacement currents. Above the midpoint, displacement currents into the internal capacitance are always in phase with those going into the external capacitance, whereas below the midpoint the internal displacement currents are out of phase with those of the external capacitance.
The effect of this asymmetry on the current profile can be visualised
as a removal of current from the top half and a corresponding injection
of current into the lower half. Another way to regard this is as an
increase of the effective capacitance per unit length in the top half, and a
decrease in the lower half.
| Example 3: With no longitudinal coupling |
The coil, still with internal capacitance switched off as above but also
the mutual inductance profile has been replaced by an equivalent uniform self
inductance
per turn of Ldc/Nt. The resonant frequency fell by
1% on removal of the mutual inductive coupling, demonstrating that one of
the
effect of mutual inductance is to increase the velocity factor slightly, ie
the mutual inductance 'stiffens the spring'.
This coil is now operating
with no longitudinal coupling whatsoever.
The voltage rise is now noticeably less linear without the smoothing effect of the mutual inductance and is beginning to show the convex slope of a distorted sine curve. The current profile has hardly changed.
Transimpedance has fallen further to 48K ohms, a change of 3K from the
51K predicted with mutual inductance, suggesting that replacing mutual
inductance with a uniform equivalent self inductance might introduce
an error of around 5%.
| Example 4: With uniform external capacitance |
The model shown above without longitudinal coupling is now further
simplified by replacing the non-uniform external capacitance by a
uniform distribution.
Both the V and I profiles are now recognisably sinusoidal in shape, which is unsurprising as this simplified coil now qualifies as a standard uniform transmission line.
It can perhaps be judged from these charts, and the previous set of charts, that the two factors acting to linearise the voltage profile are the smoothing effect of mutual inductive longitudinal coupling, and the decreasing external capacitance with height. These two effects compensate for the current reduction occuring towards the top due to external capacitance.
The transimpedance has fallen further and is now exactly
the value calculated on the basis of a cosine current profile,
ie 4 * F * Ldc.
| Example 5: With exponential external capacitance |
Still with longitudinal coupling switched off, we now attempt to
exaggerate the effect of external capacitance by
introducing an artifical Cext profile based on an exponential
decay factor of 0.33 for each of 32 steps along the coil. This represents a
much steeper decay of Cext than is normally found,
with the distribution heavily biased towards the base.
The voltage profile is now significantly altered, achieving 90% of the final voltage in the first 20% of the coil. The current decays equally rapidly, and the proportionality between I(x) and dV(x)/dx is clear to see.
The transimpedance is a mere 22K ohms, demonstrating the poor utilisation of the coil in this extreme base-heavy external capacitance configuration.
Clearly, to prevent excessive stress near the base of the coil the
geometry should introduce as little external capacitance as possible
in this region.
| Example 6: With spot external capacitance |
We now return to the fully modeled coil with longitudinal coupling
restored.
This example shows the effect of a lumped external capacitance of 10pF applied at turn 625. This might be typical of an applied voltage probe.
The sharp fall in current at the point of application is prominent,
starving the upper portion of the coil of current, and causing a
slight convex knee in the voltage profile as dV(x)/dx is suddenly
reduced. Above and below the capacitance impulse the current
profile has the same shape as the unaffected coil.
| Example 7: No external capacitance |
This simulation is of the normal coil with longitudinal coupling
but with virtualy no external capacitance, as might be obtained
by a coil operating far removed from other objects (*).
The linear-V weighted internal capacitance remains 1.98pF.
The reader will recognise this as a half wave resonance and this is the lowest mode available to the coil in the absence of an external current path. This configuration is of interest because it presents the minimum possible capacitance to the resonator, and thus maximises the ratio of end-end voltage to stored energy for a given coil.
When referenced to the potential at one end, the half wave voltage profile is a distorted sine, appearing concave in the lower half and convex in the upper half.
The current profile is particularly interesting, as it is enforced by the largely symmetric internal capacitance distribution. There is no longer any external capacitance displacement current to break the symmetry. This mode represents the 'groundstate' resonance of a free coil.
(*) A small amount of Cext must remain to provide a
return path for the simulator to energise the coil through the base.
These profile charts refer to operation just below the resonant frequency,
since tsim is not configured to explore the even-wave modes
properly.
The quantisation is caused by limited output file precision in tsim.
| Example 8: Typical coil with toroid |
The same fully modeled coil as in example 1, but this time with a toroid applied, contributing
around 25pF of topload. With this geometry the linear-V weighted equivalent
internal and external capacitances are now 1.91pF and 8.49pF. As usual the
toroid, by its shielding effect on the coil, has stolen some of the coil's
Cext.
The non-zero current at the top of the coil is apparent and its value of 4.5mA is consistent with the top voltage of 278 volts into a 25pF capacitance.
The transimpedance is 43K ohms, much higher than the 29K predicted by a cosine current profile, and almost equal to the uniform current value of 45K ohms.
The voltage profile now remains more linear over the whole length
of the coil, and the input impedance is significantly lower than that
of the unloaded coil.
| Example 9: With small external capacitance and small top load |
The coil is high off the ground so that Cext is about a quarter
of its normal value. A small top loading capacitance of about 5pF is
applied to the top of the coil.
The result is a largely uniform current distribution all along the coil, and a fairly linear rise in V.
This external capacitance configuration can also be brought about by inverting the resonator, ie treating the top of the coil as the grounded end through which the drive is applied, and the coil base is then the open circuit hot end.
Transimpedance is very high under these conditions, 122K ohms, since maximum use is being made of the inductance. In fact the transimpedance prediction based on uniform current is only 95K ohms, and the extra impedance appears to be due to circulating currents in the internal capacitance, as hinted at by the hump in the current profile. The current peak at 15% height which was starting to appear in example 1 has enlarged and moved almost to the half way point on the coil. Thus we seem to get more than our money's worth out of the coil in this configuration.
Note that due to this current hump, there is a very slight concavity in the lower half of the V profile, and a corresponding convexity in the upper half.
This configuration is worthy of further study as it highlights the extra
peculiarities of the transmission line in the presence of longitudinal
coupling.
We would expect a normal
(non-inverted) Tesla coil fitted with a large topload to approach these
profiles in the absence of a nearby ground plane. The action of the
internal capacitance is probably responsible for the
good performance reported in these cases.
| Example 10: Inverted capacitance distribution |
This time the fully modeled coil, without a toroid, is subjected to
an artificial
external capacitance distribution generated by an exponential
decay factor of 0.2 for each of 32 steps, but starting from the top end,
in other words, the maximum Cext is at the top.
The current profile is almost uniform along 80% of the coil, falling rapidly beyond this point.
This configuration is similar to the previous example except that in this case the external capacitance is quite high. It is heavily biased towards the top and we are just beginning to see the formation of the midpoint current hump.
The transimpedance is 58K, so clearly the ability of the coil to raise a top voltage is not upset by this inversion of the normal Cext distribution. While this situation may therefore be advantageous to a CW coil, a capacitor discharge system will suffer from the reduced ratio of topvolts to stored energy.
The base input impedance is about the same as that of the coil
with toroid.
| Summary |
The following notes apply only to the 1/4 wave resonance.
The chart shows the distribution of internal capacitance between a point in the middle of the coil and the rest of the coil. The flat top is due to limited spatial resolution in the tlap laplace solver program from which these distributions were obtained.
Internal capacitance reduces the velocity and the reduction is greater
at higher frequencies, causing dispersion, and this is the main cause of the
non-harmonic relationship of the overtones to the fundamental resonance.