20220818
Tesla transformer as a pump of charges from the Earth
"Try to assemble and correctly configure the Tesla transformer. You will immediately understand…"
Tariel Kapanadze
Tariel Kapanadze
Tesla transformer (hereinafter referred to as TT) is one of the most mysterious phenomena of the end of the century before last, and even the beginning of this century [1].
In appearance, this is an ordinary coil in which wave processes occur, which is why it is sometimes even confused with a quarterwave resonator.
However, the resonant frequency of a TT is calculated as if this coil had no wave processes at all  according to the formulas of a lumped oscillatory circuit.
A TT with an inductor does not obey the gear ratios of a conventional transformer, but requires taking into account wave processes.
With such contradictions, this unusual transformer resembles waveparticle duality in quantum mechanics.
But if you try to combine these processes, then the efficiency of its work increases many times.
Another property of the TT is that it can work as a pump of charges from the Earth,
of which our planet is enough to provide electricity to small households.
In general, such a pump can work on any other planet where the mantle is electrically conductive.
This task is devoted to this work, in which we will consider the condition for an increase in energy from the condition for the influx of additional charges,
we will find the optimal ratios in the TT coil with maximum efficiency, and give an example of calculating its design.
There are good studies where the search for the optimal sizes of TTs is carried out, at which the distributed and concentrated resonances are combined,
which gives good results in terms of energy, not yet explained from the classical point of view.
Also, there are experimental data on the operation of the TT as a charge pump.
However, there are almost no works devoted to the mathematics of these processes.
But, as you know, it is the mathematical model that often allows the researcher to "look beyond the horizon"; and see the whole picture from a more general perspective.
In this paper, we will not consider the construction of generators, since they can have different circuit solutions.
A much more important task is to consider the principle of pumping external energy with the help of TT.
Therefore, we will begin our work with the principle of charge capture from the medium, to which many famous researchers of free energy attached great importance.
Efficiency Gain
At one of D. Smith's conferences [2], the following principle of energy capture from the environment is presented, which is very well described by the author himself (Fig. 1a):
Capacitor plate [A] with a certain voltage applied to it,
causes duplication on the capacitor plate [E] of energy,
obtained from the environment (ground).
Thanks to the input diode [C] and the output diode [B], the energy present at [E]
flows through the transformer to ground.
Useful energy is obtained from the transformer.
When calculating our pump, we will do about the same, but as "diodes" we use the properties of the TT coil, which we will discuss below.
Now we are modernizing the scheme for our tasks and calculate the possible increase.
Fig.1. Ways to implement a pump of charges from the Earth

Figure 1b shows a block diagram of a device with two transformers spaced \(\ell\),
where the transmitting TT_{r} generates an electric potential at the receiving TT_{t}, which attracts charges from the ground.
This is the "classic" an approach to the problem, in which the transmitting TT must form powerful electrostatic charges on its solitary capacitance \(C_S\).
There are other ways to implement this principle.
We will work with a simplified circuit (Fig. 1c), where the electric potential \(U_S\) is supplied through a switch or spark gap SG,
and instantly charges a solitary container \(C_S\), the charge of which attracts additional charges from the ground.
The whole trick is that the charge from the ground will be drawn into the transformer coil \(L \) until
until the potentials on the solitary capacitance \(C_S\), and the selfcapacitance of the coil \(C_L\), do not align.
The potential source must be galvanically isolated from the Earth.
In such a simple way, we connect to the "huge reservoir of energy" that N. Tesla spoke about.
Consider the operation of the circuit (Fig. 1c) for one period.
As mentioned earlier, we must achieve equality of potential:
\[U_S = U_L \tag{1.1}\]
At the same time, the potential energy of a solitary container for the entire period is:
\[E_S = {C_S U_S^2 \over 2} \tag{1.2}\]
Then the potential energy in the selfcapacitance of the coil will be:
\[E_L = {C_L U_L^2 \over 2} = {C_L U_S^2 \over 2} \tag{1.3}\]
Consequently, in one period we acquire additional energy, the relative value of which we will call the increase in efficiency:
\[K_{\eta} = {E_L \over E_S} = {C_L \over C_S} \tag{1.4}\]
This increase, we believe, is spent further on the active load.
The load itself, and various ways of transferring the received energy to the load, are not considered here.
In fact, \(C_S\) also includes the capacitance of the elements connected to it, for example, a spark gap (its electrode and interelectrode gap).
The general meaning of this whole undertaking is to collect as many charges as possible from the earth, with the help of a relatively small capacitance that attracts them.
The principle of the pump turns out to be very simple  a smaller charge should attract a larger charge to itself, due to the potential difference.
But for smaller losses of such a pump, and further processing of the charge into electric current, the TT must have some properties, presented below.
Condition for the correct operation of the TT
For the fullfledged operation of the TT, it is necessary to combine two resonant frequencies: in a distributed and concentrated mode, which is called wave resonance in the environment of free energy seekers.
This condition is met
if during one oscillation wave
manages to walk along the conductor from one end of the coil to the other:
\[\lambda = {c \over \sqrt{\varepsilon}} T \tag{1.5}\]
Here: \(\lambda\)  wavelength;
\(c\)  speed of light;
\(T\)  period (time) of one oscillation.
The relative permittivity \(\varepsilon\) makes a small contribution to the wave propagation velocity,
has a value of about unity, and in a singlelayer coil wound on a frame, it is considered as follows [4]:
\[\varepsilon = 1 + {4 t\, (\varepsilon_t  1) \over D} \tag{1.6}\]
In this expression, \(t\) is the thickness of the coil frame dielectric, \(\varepsilon_t\) is its relative permittivity.
The wavelength will be found as the product of the length of one turn \(\pi D\), by the total number of turns \(N\):
\[\lambda / 4 = \pi D N \tag{1.7}\]
Here \(D\) is the diameter of the TT coil.
The wavelength here is divisible by 4, since we consider
that quarter wave standing wave mode.
At the beginning of this work, we already mentioned the dualism of TT, and now it's time to face it in practice.
If earlier we considered the coil by its wave properties,
then the resonant TT frequency, and hence the period of one oscillation, we find according to the classical Thomson formula [3] for concentrated resonance:
\[T = 2\pi \sqrt{L C} \tag{1.8}\]
Now we need to combine these completely different properties of TT: concentrated and distributed.
Mathematically, we can do this by equating expressions (1.5) and (1.8).
From here, having made simple transformations, we will obtain a formula under which the condition for the correct operation of the TT is satisfied:
\[(2 D N)^2 = {c^2 \over \varepsilon} L C \tag{1.9}\]
Note. All formulas in this work are presented in the SI system of units
Coil calculation
\(C\) from (1.71.8) actually contains several capacitances affecting the resonant frequency of the TT:
\[C = C_L + C_G + C_S \tag{1.10}\]
\(C_L\) is the selfcapacitance of the TT coil, \(C_G\) is the ground capacitance, \(C_S\) is the solitary capacitance (Figure 1c).
Let's introduce a form factor that will show the ratio of the winding height \(H\) to the diameter of the coil:
\[k = {H \over D} \tag{1.11}\]
This parameter is very easy to operate in further reasoning.
For example, the selfcapacitance of the coil is found by the following formula [5], where the form factor is decisive:
\[ C_L = a D \cdot 10^{10} , \quad a = 0.1126 \left(0.7174 + k + {0.933 \over \sqrt{k}}+ {0.106 \over k } \right) \tag{1.12}\]
The inductance of the coil is determined by a fairly accurate Welsby formula [6]:
\[ L = bDN^2 m \cdot 10^{6}, \quad b= {1 \over 0.45 + k  0.005/k} \tag{1.13}\]
Here we introduce one more additional parameter  \(m\), which is equal to one if the coil works without a magnetic core.
If we introduce such a core to adjust the frequency, then this parameter becomes slightly more than one.
Also, in this paper, we present the experimentally obtained dependency (from the authors) to find the ground capacitance
\[ C_G = \varepsilon_0 H^{0.75 }D^{0.25} = \varepsilon_0 D\, k^{0.75} \tag{1.14} \]
which we will use in what follows.
Here \(\varepsilon_0\) is the absolute permittivity equal to \(8.85\cdot 10^{12}\) F/m [8].
This formula assumes that the earthing of the TT is appropriate for its dimensions.
Substituting these expressions into (1.9), we obtain:
\[ 4 D \varepsilon = 9 \left( a D \cdot 10^{10} + \varepsilon_0 D\, k^{0.75} + C_S \right) \cdot 10^{ 10} \tag{1.15}\]
where we find the optimal solitary capacity:
\[ C_S = D \left( {4 \varepsilon \over 9\, b\, m}  a  0.0885\, k^{0.75} \right) 10^{10} \tag{1.16}\]
As we can see, this capacity depends only on the coil diameter and form factor.
We managed to remove the winding turns for the time being, which should simplify further calculations.
Let's take a closer look at the resulting dependency:
Fig.2. Dependence of the optimal solitary capacitance in picofarads on the form factor k, at D=1 dm and ε=1.
Red graph: m=1, blue graph: m=1.2

Graph 2 shows that when m=1 and the form factor value is less than 0.4, the value of the solitary capacitance becomes less than zero,
which indicates the absence of optimal values beyond this interval.
When we adjust the coil by introducing a small (relative to its size) magnetic core (m=1.2) into it,
then the allowable range of form factor values narrows slightly, and can start from 0.5.
At the same time, the optimal values of the solitary capacitance become slightly less, which, based on formula (1.4), gives a theoretically greater effect from the operation of the TT.
If we take a special case, when the coil winding height is equal to its diameter, then the optimal solitary capacitance will be as follows:
\[ C_S= 24.3\cdot 10^{12}\,D , \quad k=1, \quad \varepsilon=1, \quad m=1 \tag{1.17}\]
that is, 2.43 pF for every decimeter of coil diameter.
But with one of the popular form factors, with \(k= 3\), this ratio will become equal to 8.47 pF*dm.
In these calculations (1.17) it is assumed that the coil is wound on a frame with a relative permittivity equal to unity,
and the adjustment of the inductance of the coil by the magnetic core is not performed.
Knowing the optimal solitary capacitance, we can calculate the rest of the TT parameters: efficiency increase, power and geometric dimensions.
This will be discussed in the second part of this work.
Materials used
 Wikipedia. Tesla Transformer.
 DONALD L. SMITH, FEBRUARY 14. 2004
 Wikipedia. Thomson's Formula
 Alan Payne. SELFRESONANCE IN COILS, 2014.
 D.W. Knight. The selfresonance and selfcapacitance of solenoid coils, July 2013. [PDF]
 WELSBY V.G. : The Theory and Design of Inductance Coils, Second edition, 1960, Macdonald, London.
 Wheeler H.A. Numerical Methods for Inductance Calculation. Empirical Formulae  Wheeler's Continuous Inductance Formula
 Wikipedia. Dielectric permittivity.