The materials presented in this work appeared due to the research in the field of ferromagnetic cores with a coil, the inductance of which varies depending on the current passing through it. Their results are published here, here and here, and allow shedding light on some of the achievements of free energy researchers who receive excess energy using RL or RLC circuits, containing a ferromagnetic core. In this paper, we propose the conditions under which such an excess becomes possible. As usual, in the first part of this note we will present the theory, and in the second - more practical developments.

As it turns out, inductive parametric systems, with COP greater than one, can be described by a classical theory using a non-classical approach to it, which we will demonstrate below. To do this, we will divide the transient process in the coil into two stages, in the first - we will consider the process of pumping current, and in the second - energy removal. The most important thing in such a process will be the different characteristics of the magnetic permeability during pumping and during removal. This approach is parametric chains of the second kind, the principle of construction, and the mathematics of which, we will try to describe from the very beginning, from the physical foundations. And for a detailed study of any topic, we will refer to earlier notes.

In this work, we will consider coils with a ferromagnetic core without a gap, which has no residual magnetization. There may be several options here. We can assume that the core has time to completely demagnetize from the end of one pulse to the beginning of the next, or else that the coercive force [1] of the ferromagnet is zero. In systems with magnetic memory, the expected effects may not work, because. they spend additional energy on the remagnetization of the core.

The first stage of the transient process is the saturation (pumping) of the L coil with the current I. This is achieved by connecting the coil to a circuit consisting of a power source U, active resistance R and a switch SW (Fig. 1a). The first step is to formulate an equation for this circuit, from where we can obtain the pump energy.

According to Faraday's law of electromagnetic induction [2], an electromotive force arises in a closed circuit [3]: \[ \mathcal{E} = \frac{\mathrm d \Phi}{\mathrm d t}, \quad \Phi = L\, I, \quad I=I(t) \tag{1.1}\] Where: \(\Phi\) is the magnetic flux, \(L\) is the inductance of the coil, \(I\) is the time-varying current in the circuit \(t\). Here we immediately take the EMF modulo, to compile the subsequent voltage balance in the circuit under study, after closing the SW key (Fig. 1a). The balance itself is compiled according to the Kirchhoff rule for stresses [4] \[ \mathcal{E} + R\, I = U \tag{1.2}\] and is completely obvious. In this circuit, \(R\) is an active resistance, which can consist of the resistance of the key channel, the active resistance of the coil wires and connecting wires.

Substituting expressions (1.1) into (1.2), we obtain the following equation: \[ \frac{\mathrm d}{\mathrm d t} \left[ L\, I \right] + R\, I = U \tag{1.3}\] So far, we have kept the inductance under the derivative sign because it changes depending on the current I passing through it. Such dependence can be described by a simple polynomial M(I), described in detail in this paper. If so, then such an inductance can be represented in this form \[ L = L_0\, M\!(I) \tag{1.4}\] where \(L_0\) is the initial inductance (no current). We substitute this expression into (1.3) and obtain the following differential equation: \[ L_0 \frac{\mathrm d}{\mathrm d t} \left[ M\!(I)\, I \right] + R\, I = U \tag{ 1.5}\] Given M(I), we can immediately say that such an equation cannot be solved analytically. Therefore, at the first stage, we will simplify it with the condition that we will not further take into account the resistance R. Let's see what happens to them, then to return it to the real chain.

In such a circuit, we assume that there are no losses, then the diff. the equation is much simpler: \[ L_0 \frac{\mathrm d}{\mathrm d t} \left[ M\!(I)\, I \right] = U, \quad R = 0 \tag {1.6}\] In this form, it can already be solved analytically, for which it is necessary to integrate the left and right parts of the expression over time: \[ L_0 \int \limits_0^t I\, \mathrm d [I\, M\!(I)] = \int \limits_0^t U\, I\, \mathrm d t \tag{1.7}\] Whence it immediately becomes obvious that the integrated right side of it is the energy expended by the power source on the transient process over the time interval \(0..t\): \[ \int \limits_0^t U\, I\, \mathrm d t = W \tag{1.8}\] In these equations, we mean that we choose the transition process time t from the condition for obtaining the maximum increase, which we do not yet know. But we know that at the moment of time t the current in the coil rises to the value I. We will substitute it into the integration boundaries in the following formulas.

At the first stage of the transient process, we believe that the dependence of the inductance on the current is expressed through \(M_1\!(I)\), and the pump energy itself is denoted as follows - \(W_1\). Using the method of integration by parts of the left side of equation (1.7), we obtain its value: \[ W_1 = L_0\, M_1\!(I)\, I^2 - L_0 \int \limits_0^{I} I\, M_1\!(I)\, \mathrm d I \tag{1.9}\] In fact, this formula calculates how much energy the power source expended on the appearance of current I in the coil. Interestingly, there is no time in this equation, which should greatly simplify its application.

The next stage of our work will be the removal of energy from the coil, in which the current I is already present. We will do this according to the principle of a flyback converter [5], usingdiode D1 and load Rn (Fig. 1b). Here we also assume perfect lossless removal, with optimized load resistance. The main difference from the previous stage is that now the dependence of the coil inductance on the current will be different: \(M_2\!(I)\). How to achieve this from the same core, we will tell in the second part of this note.

Let's use the same principle and compose an equation for the current flow from (1.7). We know the initial current - I, the final current is zero, but the integration boundaries will need to be swapped: \[ W_2 = - L_0\, M_2\!(I)\, I^2 - L_0 \int \limits_{I}^0 I\, M_2\!(I)\, \mathrm d I \tag{1.10}\] That's right, compared to (1.9), the sign of energy is different, since we do not spend it, but receive it. But for further relations, it will be better to bring these energies to the same sign, implying that \(W_2\) is the energy received: \[ W_2 = L_0\, M_2\!(I)\, I^2 - L_0 \int \limits_0^{I} I\, M_2\!(I)\, \mathrm d I \tag{1.11}\] Obviously, this formula has the same form as (1.9), only with a different M(I). But for joining the two stages of the transient process, the coupling coefficient m is required. It differs little from unity, and smoothly connects the Stoletov curve at these two stages: \[ W_2 = m\, L_0 \left( M_2\!(I)\, I^2 - \int \limits_0^{I} I\, M_2\!(I)\, \mathrm d I \right) \\ m = { M_1\!(I) \over M_2\!(I) } \tag{1.12}\] Also, the coefficient m determines the magnetic memory of the core during the reverse current flow. The gain in efficiency of the second kind is thus found as the ratio of the two energies obtained in (1.9) and (1.12): \[ K_{\eta 2} = {W_2 \over W_1} \tag{1.13}\] At its core, formula (1.13) represents the potentially achievable free energy of a coil with a core in mathematical form. Then for an ideal chain, the final formula for the energy gain will look like this: \[ K_{\eta 2} = {1 - \frac{\large 1}{\large M_2\!(I) I^2} \int \limits_0^{I} I\, M_2\!(I)\, \mathrm d I \over 1 - \frac{\large 1}{\large M_1\!(I) I^2} \int \limits_0^{I} I\, M_1\!(I)\, \mathrm d I} \tag{1.14}\]

In our experiments, we use the parametric chain of the second kind, and different M(I) for pumping and removing energy from the coil. Therefore, we assume that K_η2 will be different from one, and for certain M(I) this parameter can be even greater than one, which will be discussed in more detail below. When operating in this mode, the device can be called a "current amplifier".

Until now, we have not taken into account the efficiency of our device, which consists mainly of losses in the wires, losses in radiation and losses in the switch. All this will definitely lower the achieved COP, which is mathematically taken into account as follows: \[ C\!O\!P = K_{\eta 2} \cdot \eta \tag{1.15}\] Here: \(\eta\) is the overall efficiency of the remaining elements of the device (except for the elements of the circuit under consideration), which is always less than one. This can be, for example, the efficiency of the power supply. We do not know it yet, and we give only a formula for its accounting. It gives an understanding that the increase in efficiency of the second kind can be completely leveled by the usual efficiency of the device, and COP can become less than unity. We must not forget about this and try to make the elements of the device as economical as possible in terms of energy losses.

We won't be able to calculate the COP directly yet, because we do not know the efficiency of the device η, but we can calculate the increase in efficiency of the second kind using the formula (1.13-1.14). Recall that the dependence M(I) can be represented as follows \[ M(I) = {1 + k_{12} I^2 \over 1 + k_{22} I^2 + k_{23} I^3} + {1 \over \mu_i} \tag{1.16}\] where: μ_i is the initial relative magnetic permeability of the coil core, k₁₂ k₂₂ k₂₃ -- coefficients of the Stoletov curve (details). In the example on the following two graphs, created by the formula (1.14), coefficients at M(I) taken from a real coil with a core that has μ _i=2000.

On graphs 2 and 3, the increase in efficiency is clearly visible even with small differences in the coefficients. These graphs show the optimal value of current I, at which the maximum effect is achieved. Recall that this current is reached in the coil at the end of the first step of the transient. If the difference between M(I) at different stages of the transition process is greater, then the increase in efficiency may increase.

Graphs (Fig. 2.3) describe the ideal circuit, and therefore - potentially achievable results, without taking into account losses. It's time to talk about the restrictions in formulas (1.9-1.14). Active resistance in the circuit (Fig. 1a) plays a large role in the entire transient process, which is devoted to a separate work. From here it becomes obvious that this resistance will also affect the COP calculated by us.

Next, we present a technique for calculating a real parametric RL-chain of the second kind. In such a circuit, the active resistance R (Fig. 1a) must be taken into account at the first stage of the transient process (pumping). In this case, you will have to solve equation (1.5) with numerical methods \[ L_0 \frac{\mathrm d}{\mathrm d t} \left[ M_1\!(I)\, I \right] + R\, I = U \tag{ 1.17}\] and then, to obtain the energy costs, the found current values I(t) from (1.17) will need to be added to the following integral: \[ W_1 = U \int \limits_0^{t_1} I(t)\, \mathrm d t \tag{1.18}\] Where: \(t_1\) is the end time of the first stage of the transient process (pumping time).

If the coil is sufficiently high-quality, and its active resistance is very small, then it can be ignored at the second stage of the transient process, and the removal energy formula will be the same as in (1.12). At the same time, we assume that the energy is harvested in an ideal way, with an optimal load. Then the expression for the increase in efficiency and COP will be the same as in (1.12-1.13).

If the coil is medium-quality, and / or its reverse circuits have a relatively high resistance, then the diff. the equation for the removal of energy will look like \[ L_0 \frac{\mathrm d}{\mathrm d t} \left[ M_2\!(I)\, I \right] + (R_n + R_L)\, I = 0 \tag{1.19}\] where: \(R_L\) is the active resistance of the coil and the reverse circuit. Its solution will be similar to formula (1.12), but with a small addition: \[ W_2 = {m\, L_0\, R_n \over R_n + R_L} \left( M_2\!(I_1)\, I_1^2 - \int \limits_0^{I_1} I\, M_2\!(I)\, \mathrm d I \right) \\ m = { M_1\!(I_1) \over M_2\!(I_1) } \tag{1.20}\] In this case, the end time of the first stage should be related to the current in the coil pumped into it during this time: \[ I_1 = I(t_1) \tag{1.21}\] This time itself should be chosen based on the maximum increase, which will be visible after obtaining the energy ratios in (1.13), from where the increase in efficiency will be visible. The definition of growth COP will be the same as in formula (1.15). We will analyze examples of real circuits in the second part of this note.

According to the author, all electrical processes in nature operate on the basis of electronic synthesis, detailed here and here. Every time we turn on a light bulb or a heater, we start this process. But in its usual form, it is not very effective, because. the electrons race around in circles thousands of times, using almost no internal energy. This problem concerns the entire electric power industry, and it cannot be solved at once. But it is necessary to take steps in this direction now, what this site is dedicated to, and, specifically, this note :)

In the second part of this work, we will consider the well-known practical developments of some researchers of free energy, from the standpoint of the theory presented here. And also, we will give methods and examples of obtaining various M(I) at different stages of the transient process, in order to achieve effective COP values (in development).