Among the seekers of free energy this method has long been known, but its theoretical base is still not sufficiently covered. A small contribution to such a theory we will try to make in this post. Outset that here we use the approximate formulas for calculation of some quantities, for example, inductance and self-capacitance of the coil, but this is enough for qualitative and even quantitative understanding of the question.

The well-known formula [1] for finding the inductance of a single layer of the coil is: \[L = {D\,N^2 \over 0.45 + k} 10^{-6}, \quad k= \frac{H}{D} \qquad (1.1)\] where: \(D\) is a diameter of the coil (m), \(N\) is the number of turns, \(H\) is the height (length) of winding (m). This value is included in the classical Thomson formula for determining the resonant frequency of the circuit: \[f_{LC} = {1 \over 2 \pi \sqrt{L\,C}} \qquad (1.2)\] where: \(C\) is the sum of private and external capacitance of the coil. For convenience, we agree that the frequency will be considered in megahertz and the capacitance in picofarads. We give this formula to the form needed for further considerations: \[f_{LC} = {159 \sqrt{0.45+k} \over N\,D } \sqrt{D \over C} \qquad (1.3)\] From [2] the well-known formula for finding the semi and quarter-wave resonance for single layer coil: \[f_{SF} = \left({300\, \lambda \over \ell\,(1 + 0.9\,\lambda/k) }\right)^{0.8} \left(73\,h \over D^2\right)^{0.2} \qquad (1.4)\] where: \(\lambda\) — on the floor or a quarter-wavelength mode can take a value of 1/2 or 1/4, \(\ell\) is the length of the wire for winding coil (m), \(h\) — distance between coils (m). This formula further, we will also give a more convenient form.

Here we will describe the coil with a step of winding a little different from the diameter of the core wire, so the wire length with sufficient accuracy can be expressed as: \(\ell = \pi\,D\,N\) and a step winding as \(h = {H \over N}\). Substituting all this into formula (1.4) will present it in more readable form: \[f_{SF} = 98.4{k \over N\,D} \left({\Lambda \over \Lambda + k}\right)^{0.8}, \quad \Lambda= 0.9\,\lambda \qquad (1.5)\] To match the frequency of the first harmonic from the formula (1.3) and the frequency mode of the standing wave from the formula (1.5) is enough to equate them. After that and some transformations, we get the following relationship: \[{C \over D} = \left(\frac4\pi\right)^4 {0.45+k \over k^2} \left({\Lambda + k \over \Lambda}\right)^{1.6} \qquad (1.6)\] quantification of the described method requires knowledge of the wave impedance of the LC circuit, which is as follows: \(Z = \sqrt{L/C}\). Then, given (1.1), the previous formula is transformed thus: \[{Z \over N} = 10^3\left(\frac\pi4\right)^2 {k \over 0.45+k} \left({\Lambda \over \Lambda + k}\right)^{0.8} \qquad (1.7)\] the Result of its calculation has a dimension of Ohm/revolution, which can better reflect the pattern on the charts.

It would be interesting to find the speed of propagation of waves in the wire when combined frequencies. Considering the source [2], the relative velocity will be expressed as \[ \text{VC}(k) = \frac{V}{c} = 0.93 \left({\Lambda + k \over \Lambda}\right)^{0.2} \qquad (1.8)\] where: \(V\) — the speed of propagation of the wave, \(c\) is the speed of light.

For this we introduce the notation of some of the functions will transfer coil diameter \(D\) in centimeters and reflect on the chart previously obtained formula. Formula (1.6) we introduce the function \(\text{CD}(k) = 0.01\,\frac{C}{D}\), and for (1.7) — enter \(\text{ZN}(k) = \frac{Z}{N}\). As for the algorithm of calculation would require a comparison of \(\text{CD(k)}\) with its own capacity of the coil, we introduce the function (taken from source [3]): \[ \text{CDS}(k) = \frac{C_S}{D} = 0.1126\,k + 0.08 + {0.27 \over k^{0.5}} \qquad (1.9)\] here: \(C_S\) — self-capacitance of coils (pF). The diameter \(D\) in these formulas is reflected in centimeters.

Now reflect these formulas in the form of graphs for a quarter-wavelength mode, when \(\lambda = 1/4\). The left graph captures the full range of possible values of \(k\), and the right focuses only on necessary to us in the future:

As an example to compare, here are the same graphs, but for \(\lambda = 1/2\):

From the graphs good shows one interesting pattern: when the values of \(k\) greater than seven, the combination of the waves to search is useless, because the self capacitance of the coil becomes greater, the aggregate (the blue graphics are above orange). And if to consider, what it is necessary to add the capacity of the earth, the real maximum value of this coefficient will be much less. For \(\lambda = 1/2\) the maximum value of this coefficient becomes very small: 0.2 and below.

But if the scheme binding to complement another coil, connected in series with the external capacitance, the orange graph can shift up, and this means that the maximum value of \(k\) will be greater. This leads to two schemes: with a small \(k\), but without the additional coil and with a large \(k\), but with another coil.

Go straight to the diagram (1.2) where the additional coil L2 here performs two functions: shifts the graph of the function \(\text{CD}(k)\) up and running by the matching transformer to the load Rn. Soglasovat load is carried out by inserting Rn into a part of the turns of the coil, but the chart shift is not so obvious, so a little at this stop.

Self-resonant frequency of coil L1 without external circuit is as follows: \[\omega^2 = \frac{1}{C_s L_1} \quad \omega = 2\pi f \qquad (1.10)\] where: \(f\) — resonance frequency, \(C_s\) — self-capacitance of the coil L1. The resonant frequency of the scheme (1.2) is found using the classical theory of coupled circuits: \[\omega^2 = \frac{1}{L_1 C_s} \sqrt{1 \pm K} \qquad (1.11)\] where \(K\) is a coupling coefficient between circuits formed by the coils L1 and L2 together with their own tanks. To delve into this method until we are, we can do a separate article, and the last formula only want to show qualitatively achievable result — increase of the resonance frequency due to the additional coil L2. But in the diagram (1.1) let us consider more detail.

The methodology will be carried out on the scheme (1.1) in which there is no additional inductance, and hence the value of \(k\) you need to choose small enough. We will proceed from the fact that you need to find a point \(k\), where the graph of the function \(\text{CD}(k)\) over \(\text{CDS}(k)\) in two or three times. So you need to do to have a variable capacitance C1 was a reserve for the variation. Choose a point \(k=0.1\), while \(\text{CD}(0.1)=2.6\), and \(\text{CDS}(0.1)=0.95\). Also, if you look at the graph with \(\lambda = 1/2\) you can see that at this point he's approaching schedule with \(\lambda = 1/4\). Thus we can partially seize and the half-wave mode.

Now define the wave impedance, it must match the value of the load Rn. First, according to the schedule we find the value of the wave function in the chosen point \(\text{ZN}(0.1)=85\). This means that we can now calculate the number of turns depending on the load value is based 85 Ohms on the coil. For example, our load will be equal to 1kω, then the number of turns in the coil L1 must be equal to 1000/85 = 11.8.

Next, choose the diameter of the coil L1 and the value of C1. For example, let the diameter is equal to 25cm. Then, knowing \(\text{CD}(0.1)=2.6\), find the average value of the variable capacitance: C1 = 2.6*25 = 65пФ. But it is necessary to subtract the self capacitance of the coil; looking ahead to say that she will be about 30 pF. Therefore, C1 will be equal to 35pf, so the total range of adjustment of this capacitor may be, for example, from 15 to 65пФ.

Now let's check the results in the calculator. He calculates the formulas (1.1-1.9) more precisely, so with his help we will need to adjust the results to more real and also to find the diameter of the wire. Substitute into the calculator the previously obtained data and look at the result. On the horizontal axis of this calculator, we have observed all possible values of \(k\), but we should be interested only in point of 0.1 (the height of the winding coil is less than its diameter 10 times). As you can see, the point of intersection of the graphs shifted slightly to the left of \(k=0.1\). By selecting the "external tank" and the diameter of the wire to achieve an exact match, such as here. Then the final parameters of our device are the following:

• The resonant frequency of the oscillator, MHz: 2.56
• The length of the wire (coil L1), m: 9.34
• The core diameter of wires, mm: 1.9
• Coil diameter, mm: 250
• The number of turns in coil: 11.8
• Height (length) of winding, mm: 25
• Variable capacitor C1, pF: 15/65

If the device will work with the ground, you will need to make the same selection of the thickness of the wire and the capacitance of the capacitor C1, but taking into account the capacity of the earth. Another point is that the resistance of the actual load may be greater than calculated and should be selected experimentally. Also, you may need to find a more suitable method of energy extraction.