Soon it will be 120 years since the launch of the famous Wardencliff tower [1], which allowed Nikola Tesla to transmit electrical power over a fairly long distance, but until now, physicists do not have a more or less worthwhile model and explanation of how such a transfer became possible. There are hypotheses about the excitation of the ether, through which this phenomenon becomes possible, but all kinds of experiments set up during this time have not unequivocally proved its existence. There is an explanation for such transmission using classical transverse TEM waves [2], as well as an interesting explanation of transport using Zenneck surface shear waves [3]. But Tesla himself stated that his waves are of a non-Hertz type and are longitudinal, about which he argued with Hertz and even, as a result of the experiments set at that time, proved to him that he was right. Let's try to understand this difficult matter from the point of view of classical electrodynamics, derive the formulas allowing to get closer to understanding these processes, and we will make calculations of real structures with their help.

As it turned out, nothing is impossible here, and the phenomenon of electric power transfer by longitudinal waves can be explained by the physics known today. True, at first it will be necessary to move away from wave processes altogether in order to explain the process of energy connection with the help of displacement currents. And since these currents are directly related to longitudinal waves, at the end of this work we will again return to the declared wave model. We will not be limited here to only one Tesla tower, and we will generalize this phenomenon to any designs of wireless transmission of electrical energy with a longitudinal wave type.

Charges can be delivered to the transmitting ionospheric capacitor in different ways, for example, by a laser beam. Moreover, this can be done both from the Earth and from its satellite - a solar power plant located in orbit and rotating synchronously with the planet. By the way, energy transport along a laser beam is now being studied in scientific laboratories and is a very promising way of directed energy transfer over relatively short distances, just what is needed here.

Interestingly, the receiving capacitor may not necessarily be located in the ionosphere, which forms a certain capacitance C with the Earth. It can also be placed at a low altitude (C3), then the energy transfer will be carried out through the capacitance C₀ formed between the ionosphere and this capacitor. In this case, the receiver power, with the same capacitor area, will be less, but this area can be proportionally increased.

The advantage of this transmission method is approximately the same distance between the ionosphere and the receiving capacitor C3, which means - approximately constant capacity of communication C₀ in any point of the planet. This will make it possible to calculate and produce receivers of electrical energy, regardless of their geographic location.

This method of transmitting electrical power through the atmosphere depends on the distance between transmitter and receiver, where the receiving and transmitting capacitors are located at a relatively small distance from the planet's surface, at an ionospheric height (Fig. 2a). Apparently, it was this option that was used by Tesla in his towers, of which, in fact, there were several, built in different periods of his work [4]. We will explore this method further in this work, but since the model should turn out to be generalized, the formulas obtained here will be true for the first option as well.

Thus, the first and second methods involve the transfer of energy through the capacitance \(C_0\), and hence through the bias current. We will fix this fact and for now we will not use wave processes, which will certainly appear at distances of the order of a quarter of a wavelength from the frequency of the master oscillator. Later, we will return to this issue.

The design of the transmitting and receiving transformers - coils L1 and L2 - we deliberately do not consider here, because they can have different principles of high voltage formation, for example [5]. And also, we will make some assumptions and tolerances to simplify our model, which will allow us to calculate such devices and understand the principle of their operation.

It must be said right away that in our model, the transmitting and receiving capacitors (\(C_{S1}, C_{S2}\)) have the shape of a ball, but can be recalculated into an equivalent design, for example, a torroid. The height of their location above the ground is 10 or more of their diameters. In our model, we will consider one transmitting and one receiving capacitor, which include the capacitance of the ball relative to the ground and the solitary capacitance of the coils corresponding to them - L1 and L2 (Fig. 2b): \[C_1 = C_{S1} + C_{L1} + C_{G1}, \quad C_2 = C_{S2} + C_{L2} + C_{G2} \tag{1}\] These balls are at a distance \(d\) from each other and there is an electrical interaction between them, formed by the bond capacitance \(C_0\), which is detailed in here. The distance between the balls is quite large, respectively, their generalized capacity is much greater than the bond capacity: \[C_1 \gg C_0, \quad C_2 \gg C_0 \tag{2}\] We will make the next logical assumption if we assume that the circuits L1C1 and L2C2 are in resonance. After all, we know that the Tesla transformer is configured in this way. Mathematically, this means that the reactances of inductance and capacitance are equal at a known frequency: \[\mathbf{i} \omega L_1 - {\mathbf{i} \over \omega C_1} = \mathbf{i} \omega L_2 - {\mathbf{i} \over \omega C_2} = 0 \tag{3}\] where: \(\omega = 2\pi f\) is the circular frequency, and \(f\) is the frequency of the master oscillator G1.

In the model we are developing, there is no internal resistance of the generator G1 and losses for the buildup of the transmission transformer L1, because they are unknown. But when calculating the efficiency of the entire installation, they can be introduced at any time. The buildup of L1 itself is also not considered, because its options can be any, but instead the generator is connected in series in a circuit with this transformer, schematically representing the most general approach to this problem. The same can be said about the receiving coil L2: the load R is connected in series with it, and in the case of voltage transformation, it is simply recalculated in proportion to the ratio of the turn of the primary and secondary windings. Then the equivalent circuit of our model will look like in Figure 2b, which is a classic coupled contours [6].

A feature of such circuits is a fully reactive load on the generator G1 with the active load R disconnected. Thus, the transmitter can consume a minimum of energy (for losses) when the receiver is turned off, and start consuming energy only when it is connected. This is a key difference from classical shear wave transmitters, where the antenna is an active load for the generator and consumes active power regardless of the number of connected receivers.

It would be possible to start the calculation from scratch, and write down all the currents and voltages, and only then derive the values we need from the resulting system of equations, but we'd better use the ready-made solution from [6] and immediately write out the currents in the circuit, according to Figure 2b: \[I_1 = {Z_2 U_G \over Z_1 Z_2 - Z_0^2}, \quad I_2 = {Z_0 U_G \over Z_1 Z_2 - Z_0^2} \tag{4}\] Here: \(Z_1, Z_2, Z_0\) is the complex resistance in the circuit, \(U_G\) is the voltage of the generator G1. First, we write down all the complex resistances in the most accurate form, as it was done in [6], and then we simplify these expressions according to our assumptions: \[Z_1 = \mathbf{i} \omega L_1 - {\mathbf{i} \over \omega C_I}, \quad C_I = C_1 + {C_0 C_2 \over C_0 + C_2} \tag{5}\] \[Z_2 = R+ \mathbf{i} \omega L_2 - {\mathbf{i} \over \omega C_{II}}, \quad C_{II} = C_2 + {C_0 C_1 \over C_0 + C_1} \tag{6}\] \[Z_0 = {- \mathbf{i} \over \omega (C_1 + C_2 + C_1 C_2 / C_0)} \tag{7}\] Let us now recall our assumptions and simplify these formulas. Following inequality (2) and tolerance (3) we get: \[Z_1 \approx {\mathbf{i} C_0 \over \omega C_1^2} \tag{8}\] \[Z_2 \approx R+ {\mathbf{i} C_0 \over \omega C_2^2} \tag{9}\] \[Z_0 \approx {- \mathbf{i} C_0 \over \omega C_1 C_2} \tag{10}\] From here we can find the currents in the circuit: \[I_1 \approx {U_G \over R} \left[ {C_1^2 \over C_2^2} - {\mathbf{i} \omega C_1^2 R \over C_0} \right] \tag{11}\] \[I_2 \approx {U_G \over R} {C_1 \over C_2} \tag{12}\]

Taking only the real part of the first current (11), we find the active power consumed by the generator for the entire process: \[P_G = I_{R1} U_G = {U_G^2 \over R} {C_1^2 \over C_2^2} \tag{13}\] And compare it with the active power given by the circuit to the load R: \[P_R = I_2^2 R = {U_G^2 \over R} {C_1^2 \over C_2^2} \tag{14}\] As we can see, these powers are equal, which means that the principle stated earlier is observed: the active power of the generator is consumed only if the active load is turned on (excluding losses, of course).

Now we come to the most interesting - the calculation of real power transmission systems. But for this we need to start not from the generator voltage - it is unknown to us and can be, in principle, any, depending on the circuitry of the exciting stage. But, from the initial conditions, we know the voltage \(U\), which is the potential on the radiating solitary capacitor of the Tesla transformer (see Fig. 2b). Let's find it mathematically: \[U = U_G + I_1 \mathbf{i} \omega L_1 \tag{15}\] Making known transformations, we find the stress ratio: \[ {|U| \over U_G} = {C_1 \over C_0} \sqrt{1 + \left({C_0 \over \omega C_2^2 R} \right)^2} \tag{16}\] Moreover, \(U\) is taken modulo here, i.e. looking for the effective value of the voltage. We already know that active power, without taking into account losses, is completely transferred from the generator to the load. It remains to find the power in the load, taking into account the voltage obtained in (16): \[ P_R = {|U|^2 \over R} {C_0^2 \over C_2^2} {g^2 \over 1 + g^2}, \quad g = {\omega C_2^2 R \over C_0} \tag{17}\] The formula turned out to be useful, but it can be optimized even more if you find the optimal value of the load R. In radio engineering, this is called load matching. So, the optimum is found from the well-known mathematical technique of equating the derivative with respect to R to zero: \[ R^{*} = {C_0 \over \omega C_2^2} \tag{18}\] Now it is clearly seen that the optimal load value \(R^{*}\) depends on the communication capacity, and hence on the distance between the receiver and the transmitter. The power in the load, at its optimal (coordinated) value, will be as follows: \[ P_R^{*} = {\omega C_0 |U|^2 \over 2} \tag{19}\] And the last thing to do is to substitute expressions for finding the connection capacity into formula (19). It can be taken from here, formula (23): \[ P_R^{*} = {\omega \pi r^3 \varepsilon_0 \over 2 d^2} |U|^2 \tag{20}\] Recall that here: \(r\) is the radius of the ball, \(d\) is the distance between the transmitting and receiving ball, \(\varepsilon_0\) is the absolute dielectric constant. Now we can proceed to the calculation of the known working Tesla installation.

We believe that the load in the receiver is maximally consistent with the circuit. Then formula (20) can be used for calculation. We know that the radius of the ball on this tower was 10.5 meters, the master oscillator frequency was 150 kHz, and the distance between transmitter and receiver was 42,000 meters. At the same time, Tesla lit incandescent bulbs with a total power of 10 kW. The only thing that is not known for sure is the voltage on the transmitting ball; figures are given from various sources: from 12 to 100 million volts. The latter value is obtained if we take into account the streamers emanating from the tower, the length of which reached 40 meters [7]. But for the calculation, we will take some average value - 35 MV. Substituting these data into formula (20) we get the value of the power at the load - 10.1 kW, which is quite consistent with the declared one! Of course, losses are not taken into account here, but the ionospheric effect is also not taken into account, which, although partially, could well enhance signal transmission.

In this work, we have considered two methods of wireless transmission of electrical energy using longitudinal waves. We have shown how such power can be transmitted by displacement currents, which are formed in any capacitor, in our case - in the coupling capacitance between the receiver and the transmitter, and made calculations on this basis. Continuing the reasoning in this vein, we can say that the displacement currents, in turn, are carried by longitudinal waves, which closes the original problem. However, we did not consider the wave process itself, because this would greatly complicate the story, and the power calculation itself would not change much from this. The nodes and antinodes of longitudinal waves are approximately the same as in transverse waves. whence it is clear that the energy receiver must be located in wave antinodes.

When making assumptions and assumptions, the following possible effects were not taken into account:

• transmission amplification due to the ionospheric effect described in the first method of power transmission through longitudinal waves;
• increasing the receiving power due to the suction of electrons from the ground;
• losses in the transmission stages and in the atmosphere.

It is possible that some of these effects could cancel each other out.

In the proposed mathematical model, in coils L1 and L2, there are no transformer windings, which, as a rule, have a small number of turns. But the recalculation of the voltage of the generator G1 and the load resistance R can be done at any time by simply changing their values in proportion to the ratio of the turns of the primary and secondary windings.

All the formulas obtained from this work can be used to calculate transfer systems.delivery of electrical energy by longitudinal waves through the atmosphere of the planet. For example, simple ratios can be obtained from them to calculate the size of the transfer ball for a certain transmitted power, optimal load, etc.