Research website of Vyacheslav Gorchilin
2022-08-19
The Tesla transformer is like a pump of charges from the Earth. Part 2
Earlier we obtained the dependence of the optimal solitary capacity on the form factor and diameter of the TT coil. Based on this, it is now possible to obtain the remaining parameters of the TT, which we will do in this part of the work.
Optimal increase for TT
Formula (1.4) from the previous part of this work shows what part of the energy we can pump with a TT coil in one period: $K_{\eta} = {C_L \over C_S} \tag{2.1}$ From there we take the values ​​of the self-capacitance of the coil $$C_L$$ (1.12) and the optimal solitary capacitance $$C_S$$ (1.16): $K_{\eta} = {a \over 4 \varepsilon / (9\,b\,m) - a - 0.0885\, k^{0.75}} \tag{2.2}$ This increase is correctly calculated under the condition that the optimal solitary capacity (1.16) has values ​​greater than zero. The latter means that not any $$k$$ form factor is suitable. For example, if this denominator approaches zero, then the increase in efficiency will tend to infinity. This, in turn, means the zero value of the solitary capacitance, which in reality cannot be, because. it also includes the capacitance of the key (arrester), connecting wires, etc. In addition, the small form factor leads to unstable operation of the TT and the rest of the circuit.
The overall efficiency consists of the efficiency of the charge of a solitary capacitance $$C_S$$, which is completely consumed by the load, and the gain factor: $\eta = \eta_C + K_{\eta} \tag{2.3}$ The efficiency of charging any capacity from a power source is about 50%, so we can assume that $$\eta_C \approx 0.5$$. Then the overall efficiency of the entire device will be as follows: $\eta \approx 0.5 + K_{\eta} \tag{2.4}$ It should be noted that the charge of a solitary tank can be performed with a better efficiency, for example 70%, in this case $$\eta_C$$ should be increased to 0.7. Also, we must not forget that this coefficient, in the general case, should include the efficiency of driving circuits, which we do not take into account yet, because we do not know this data.
 Fig.2. TT coil efficiency (η) versus form factor (k), with ε=1. Red graph: m=1, blue graph: m=1.2
In this work, instead of efficiency, it seems to be more correct to use the energy efficiency coefficient COP.
Output Power
Knowing the overall efficiency of the device, we can estimate its power. It can be calculated through the energy supplied to a solitary container in one period (formula 2). Multiplying it by the frequency, we get the consumed (input) power: $P_{in} = 2 f_r E_s = f_r C_S U_S^2 \tag{2.5}$ Only in this case we double this energy, because 50% is spent on the charge of the tank, and 100% of the total energy. In this formula, $$f_r$$ is the resonant frequency of the TT, which is inverse to the period from formula (1.8) $f_r = 1/T = {1 \over 2\pi \sqrt{L C}} \tag{2.6}$ or, it can be obtained from expression (1.7): $f_r = {c \over 4\pi D N \sqrt{\varepsilon}} \tag{2.7}$
Then the output power at the load will be found in the standard way: by multiplying the input power by the total efficiency, which is found from (2.5): $P_{out} = \eta P_{in} \tag{2.8}$
Calculation example
Take example of our coil, where its parameters are already known: diameter, number of turns, and all the capacities necessary for the calculation. The only parameter we don't know is m, which we'll take as 1.2. The latter means that with such a solitary capacitance, we will have to tune our coil to the resonant frequency by introducing a magnetic core into it.
Using formula (2.7), we find its resonant frequency - 400 kHz. The main thing for us now is to decide on the charge voltage of a solitary capacitance. Let's start with $$U_S = 7500\, V$$, and using formula (2.8) we find the power supplied to the coil: $P_{in} = f_r C_S U_S^2 = 133\, W$ Here $$C_S = 5.9\cdot 10^{-12} F$$, which is determined from the form factor $$k=3$$ according to graph 1, provided that ε=1 and m=1.2.
From formulas (2.2-2.4), or from graph 2 (blue curve), we find the overall efficiency of the device: $\eta_0 = 1.317$ Recall that for now we consider the efficiency without taking into account losses in the master oscillators and the device circuit. Using formula (2.8), we find the output power: $P_{out} = \eta_0 P_{in} = 175\, W$ From here it is possible to calculate the increase received from the earth pump, with the given HP: $P_{+} = P_{out} - P_{in} = 42\, W$ In practice, with a coil power of 133 W and a quarter-wave distribution, in which the current will be concentrated in the lower part of the TT, significant heating can be obtained. This may require additional cooling, and hence additional power costs. In addition, if a spark gap is used in the circuit, then it will not be able to operate at such a frequency.
The problem with the arrester can be solved if it is beaten, for example, in every 20th oscillation, and the charging voltage of the solitary capacitance $$U_S$$ is raised to 33`000 Volts. In this case, the power picture will remain approximately the same, and the arrester will operate at a frequency of 20 kHz, which is quite acceptable.
Here it is assumed that the length of the pulse supplied to the isolated capacitance using a key (spark gap) will be less than the half-cycle of oscillations of the TT coil. Therefore, at high operating frequencies, you should take care of a high-quality key or a quick spark break in the spark gap.
Conclusions
The efficiency of the TT coil does not depend on its diameter and the number of winding turns, but depends on the size of the form factor, the relative permittivity of the frame and the tuning core. For some form factors, the operation of the TT in charge pump mode is not possible. This follows from formula (2.2).
With a small form factor value (one or less), we get the best efficiency gain ratios, but also a relatively large instability in the operation of the entire device. Moving in this direction, we can come to a flat Tesla coil [1], which will be discussed in one of the subsequent works.
The second option is to increase the inductance of the TT coil by introducing a tuning magnetic core into it. To do this, when calculating it, the adjustment factor m is more than one. Usually, it should not exceed the value in 1.4. Tuning the coil, therefore, comes down to adjusting the magnetic core of its calculated resonant frequency. This will allow, almost without changing the design of the TT, to radically change the power ratio: input and output, in favor of the latter.
It is possible to further increase the efficiency by increasing the relative permittivity ε (more than one). In this case, the self-capacity of the coil will increase. In practice, this may mean, for example, impregnating the turns of the coil with varnish.
In the above calculations, we consider only one oscillation period in the coil under study. But if the key (spark gap) supplies a potential to a solitary container once in several periods, then all oscillations, except for the first one, will be free, and their number is determined by the quality factor of the entire TT system. No energy is spent on creating free oscillations, but nevertheless, the charge pump continues to work. Therefore, the real efficiency (COP) may be higher. However, the efficiency of the circuitry part of the device is also not taken into account, since is unknown. In reality, all this can lead to an adjustment in the resulting energy increase.
All calculations are made on the assumption that high-quality grounding will be used in practice, corresponding to the size and power of the TT. Connection to a heating battery, or to water pipes, is not considered grounding.
Based on the results of this work, a specialized calculator has been developed.

Materials used
1. N.Tesla. Coil for electro-magnets. No. 512,340. Patent Jen. 9.1894. [PDF]