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 V_t is around 454 volts with an input current I_b of 0.0081 amps. Thus the transimpedance is around 56K ohms.

The same coil as above, but with internal capacitance 'switched off' to demonstrate the effect of C_int 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.

The coil, still with internal capacitance switched off as above but also the mutual inductance profile has been replaced by an equivalent uniform self inductance per turn of L_dc/N_t. 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%.

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 * L_dc.

Still with longitudinal coupling switched off, we now attempt to exaggerate the effect of external capacitance by introducing an artifical C_ext 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 C_ext 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.

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 C_ext 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.

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 C_ext.

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.

The coil is high off the ground so that C_ext 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.

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 C_ext 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 C_ext 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.

The following notes apply only to the 1/4 wave resonance.

Mutual Inductance

Mutual inductance acts to make the induced series emf at any point dependent on an average of the nearby current and therefore appears to 'smooth out' the voltage and current profiles. This longitudinal coupling acts to increase the velocity slightly and can be thought of as a 'stiffening' of the coil. This velocity increase is larger at higher frequencies and the effect is to reduce slightly the dispersion introduced by the internal capacitance.

External capacitance

This is the major determining factor of the current distribution, the rate of decrease of current -dI(x)/dx being proportional to the external capacitance C_ext at point x. End effects generally result in a bathtub shape to the distribution, as shown in this chart of the profile for Terry's coil over a ground plane without a toroid. The glitches at one-third and two-thirds height are artifacts of the tlap program which calculated the profile.

Internal capacitance

Displacement currents through this charge storage mechanism act with the external capacitance in the top half of the coil, and against it in the lower half. As a result, in the lower half of the coil, dI(x)/dx can become positive in regions where internal displacement current exceeds the external. This can cause a slight current peak near the bottom of the coil. When the external capacitance is reduced, the effect of C_int becomes more noticeable and the current peak moves up the coil towards the midpoint. Currents circulating through C_int can cause the transimpedance to exceed that of a coil with uniform current.

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.

Voltage profiles

These are always monotonically increasing, since to do otherwise would require a phase reversal of current coinciding with a voltage turning point, which is inconsistent with the quarter wave resonance. The shape of the V profile lies between a linear rise and a sine rise, and when internal capacitance effects are prominent, a slight concavity occurs in the lower half and a corresponding convexity in the upper half, thus forming a point of inflection coinciding with the current peak. In the absence of internal capacitance the dV(x)/dx has to be monotonically decreasing.

Current profiles

These lie somewhere between a uniform and a distorted cosine profile, determined mainly by the C_ext distribution and the amount of top loading. The current profile would be forced to be monotonically decreasing if it were not for the circulating action of the internal capacitance. There can be no zeroes of the current profile within the coil as this would require a phase reversal. A peak in the current (due to C_int) will cause a corresponding point of inflection in the voltage profile. Configurations involving minimum distributed external capacitance result in maximum ratio of V_t to stored energy for any given coil, and addition of top load improves the utilisation of the coil as described by its transimpedance.

Further details