20210527
Longitudinal wave transmission lines
Authors: Gorchilin V.V., Znamenskiy A.V.
The title of this work does not correspond to the classical canons of theoretical electrodynamics, but it agrees well with the phenomena observed in practice.
It is from a practical point of view, on the basis of numerous experiences, that we will present here some systems for the transmission of electrical energy through a power transmission line (PTL),
the parameters of which will differ by orders of magnitude from the calculated ones.
For example, the diameter of power transmission line conductors can be an order of magnitude smaller than that required for power transmission by the classical method.
This means that the consumption of nonferrous metals for the transmission of the same power, with the same or even less losses,
accordingly, the cost of construction and installation work can be reduced significantly.
The performed technical and economic calculation on the example of a power transmission line with a length of 3 km and a capacity of 10 kW proves its financial feasibility.
The thing is that electrical energy can be transmitted using a longitudinal wave [1], in which charges move along the surface of the conductor, and which, by the way, does not exist in theory :)
In fact, and this has been shown by practice, when transferring energy in a wire, two waves always peacefully coexist: transverse and longitudinal.
If the first wave is well studied and theoretically substantiated, then the second one turns out to be a novelty even for experienced radio electronics engineers.
But for the effective transfer of energy, you just need to learn how to change and correctly find the relationship between them!
In this work, we will touch upon some aspects of this problem, show and compare the circuitry options for obtaining a longitudinal wave in a conductor,
we will derive some mathematical relationships and get an optimized device circuit for efficient transmission of electrical energy.
The block diagram of the generation and transmission of electrical energy using longitudinal waves is shown in Figure 1.
Generator G1 generates oscillations, which are brought to the required level of current and voltage in the reactive energy amplifier LC1.
Based on experimental data, we can say that the reactive power in LC1 should be several times higher than the active power that we get at the receiving end.
This fact, by the way, is one of the theoretical and practical problems of this direction.
Further, a part of this power enters the LL1 transmission line and is transmitted using longitudinal waves to the LC2 receiving circuit.
Its task is the reverse transformation of longitudinal waves into classical active power, which is then fed to the load Rn.
The reason for the appearance of a longitudinal wave is due to resonance processes occurring in the LC1LL1LC2 system and owes its origin to the reactive component.
If in industrial networks they are fighting with it in all sorts of ways, then here this component takes the most direct part in the transfer of energy.
It is interesting that the parameters of LC2, as practice has shown, with a sufficiently long transmission line, practically do not affect the resonant frequency.
Fig.1. Block diagram of the generation and transmission of electrical energy using longitudinal waves

With this method of transferring electrical energy, the active resistance of the power transmission line does not introduce losses into the transmitted power and therefore is not considered in this work.
Active losses are concentrated mainly in the reactive power amplifier LC1 and in the receiving circuit of LC2.
We will talk about reducing and optimizing losses in these circuit elements further.
The second wire, which is shown in the figure as a common point, in reality can be made in two versions:
in the form of a cable braid (Fig. 2a), or in the form of two earthing connections at the transmitting and receiving ends (Fig. 2b).
This point works as a counterweight or support, as was suggested in the Tesla transformer system [2], where they played the role of "zeroers".
Each of these options has its own advantages and disadvantages, but in this work we will consider the first option LL1: a cable with a center core and braid.
Due to the incomplete work on singlewire lines in the complex to be solved, as well as due to the upcoming R&D,
hereinafter, structural and basic electrical diagrams are presented in a generalized form without a number of technical subtleties,
without which and appropriate scientific and technical training, pure implementation will be difficult.
At the same time, in case of interest in financial support of the project, or R&D, the authors are ready to consider the appropriate proposals for interaction.
Transmitting and receiving circuitry
Figure 2 shows the basic block diagram of the transmittingreceiving circuit for the transmission of electrical energy by a longitudinal wave.
Here: G1 is a master oscillator, U1 is an amplifier, C1 is a resonant blocking capacitor, L1 is a resonant transmitting transformer, L2 is a broadband receiving transformer.
U1C1L1 form a reactive power amplifier (LC1 in Fig. 1). It is responsible for generating a longitudinal wave, which propagates along the LL1 conductor and enters the L2 receiving transformer.
There, the energy transmitted by the longitudinal wave is taken off, followed by transformation into the classical total power, which is supplied to the load Rn.
Generally speaking, a resonant capacitance should also be located at the receiving end, which forms a receiving resonant circuit with L2, but in practice it turned out to be sufficient for the L2 transformer to be broadband.
Fig. 2. The basic block diagram of the receivingtransmitting circuit for the transmission of electrical energy by a longitudinal wave

The resonant frequency of the transmitting part, as it turned out in practice, can be calculated using the following formula:
\[\omega = {(1 + n) ^ {1/4} \over (L_ {1.1} C_1) ^ {1/2}} \qquad (1) \]
Where: \(n \) is the ratio of turns of the secondary and primary winding of the transformer L1, \(\omega \) is the angular frequency equal to \(\omega = 2 \pi f \).
Here \(f \) is the frequency of the master oscillator.
The main advantage of this circuitry is the galvanic isolation of the LC1 unit from the rest of the circuit: the power line and the receiver.
But such a denouement is not always required. For example, it is not needed to broadcast energy from solar panels, wind turbines and other similar devices.
In addition, using standard grounding of electrical networks and appropriate rectifier circuitry, it is also possible to avoid galvanic isolation.
This approach immediately gives a gain in optimizing active losses in L1 due to the absence of a secondary winding (Fig. 3).
But in this case, you can go further and get away from the two windings in the receiving coil L2.
However, in practice, it turned out that it is impossible to do the same as it was done with L1, here it was required to slightly change the approach, given that we are dealing with a longitudinal wave,
which must be converted into regular electricity (Fig. 4).
Coil L1 with this inclusion has its own characteristics.
For example, if it is wound on a ferrite ring, then even with sustained inductance parameters, a longitudinal wave will not form and the device will not work.
There are some nuances in the design of the transformer that allow the formation of a longitudinal wave due to the design of its core.
Contact the project lead for details.
The following schematic variant is also interesting, where C1 and L1 are interchanged (Fig. 5).
In this case, along the conductor LL1, in addition to the longitudinal wave, the constant component is also transmitted from the output of the amplifier U1, which in previous cases was cut off by the capacitance C1.
In classical circuitry, such an inclusion is called an Lshaped filter, and its elements can be calculated according to wellknown formulas.
This can be interesting for combined transmission of energy in different ways at the same time, which increases the utilization rate of transmission lines.
In addition, this method can be applied to existing industrial networks by adding several devices to the input and output of power lines.
Circuitry options shown in Figures 45 have lower active losses in comparison with the basic option due to the absence of interwinding losses during transformation.
But there are some conditions for their application.
The L2 coil, as it turned out in practice, should be one of the versions of the Tesla transformer [2,3], and one of its windings is enough.
Moreover, its optimal inductance is calculated using the following formula:
\[L_2 = {R_n \over \omega} \qquad (2) \]
where: \(R_n \)  load resistance Rn.
The optimal ratio of inductance and capacitance can be found like this:
\[\rho = \sqrt {L_1 \over C_1} \qquad (3) \]
Here: \(\rho \) is the known wave impedance of the transmission line.
The resonant frequency of the transmitting circuit is calculated by the formula (1), but taken with the following value: \(n = 0 \).
The rest of the parameters of the elements of the transmitting part are fairly well calculated by the formulas from [4].
The following note should be made about the L2 coil.
In it, in comparison with L1, large reactive energy does not circulate, which means that the diameter of the winding wire and its dimensions, in practice, can be several times smaller.
In the next part of this work, we will consider and compare the circuitry solutions of the amplifier U1, draw conclusions about the optimization of the entire device.
Materials used
 Koltovoy N.A. Book 5. Part 207. Longitudinal waves. [PDF]
 Wikipedia. Tesla_Transformer.
 Coil for electromagnets. US512340, 1893. Inventor: Nikola Tesla.
 Yuferev L.Yu., Roshchin O.A., Alexandrov D.V., Sokolov A.V. Investigations of the resonant power transmission system at increased frequency. [PDF]