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Free energy in parametric RC and RL-circuits the first kind of first order
In this work we will try to attract the attention of seekers of free energy to parametric circuits. This unknown part of the electronics is still unpopular and even taboo subject. Meanwhile, under certain conditions, these chains can open up new possibilities for search of new sources of energy and increase The efficiency of the second kind.
Here we will consider the chain of the first order, i.e. involving RC or RL elements. By contrast to the classical approach would be that a capacity or inductance will behave in a nonlinear, or rather, parametrically depend on current or voltage. The aim of this note: using classical physics and mathematics to find the conditions under which such chains can give the energy gain, and also to formulas for calculating the free energy of these chains (for the first time!).
To do this in the beginning to define terminology; it could facilitate the perception of information and reduce the text. Under classifications here will be considered generators of the first kind and first order. In a partial cycle (PCC) energy in the reactive elements may be present, both at the beginning and end of the measurement, so we introduce two partial subkey cycle: PCCIE and PCCFE respectively. PCCIE is characterized by the fact that at the beginning of this cycle, the power source is deactivated, i.e., \(U(t)=0\).
RC или RL-цепочка с параметрическим реактивным сопротивлением
To simplify the formulas and reasoning will take the value of the resistance \(R\) is equal to one. Since is constant, to make it different from this value we can at any time. Then the General differential equation for chains of the first order will be [1]: \[Z(Y)\,\dot Y_t + Y = U(t), \quad Y=Y(t) \qquad (4.1)\] where: \(Z(Y)\) is the reactance of parametrically-dependent \(Y\). Resistance itself can be a capacitance or inductance depending on the form of a chain — RC, or RL, respectively. \(Y\) may be a voltage, if RC-circuit and the current, if it is a RL-circuit. Also, we need to remember how to mathematically convert the current to voltage (and Vice versa) in the reactive elements: \[\Phi(t) = Z(Y)\,\dot Y_t \qquad (4.2)\] where \(\Phi\) is the current in an RC circuit or a voltage in a RL-circuit. Now substituting (4.2) into (4.1) we get the characteristic equation: \[\Phi(t) + Y = U(t) \qquad (4.3)\]
For an RC circuit
Will domnain each member of \(\Phi\) and printeriem both parts: \[\int_0^T \Phi(t)^2\, dt + \int_0^T Y\,\Phi(t)\, dt = \int_0^T U(t)\,\Phi(t)\, dt \qquad (4.4)\] Enter the name of each member of this equation: \[W_R + W_F = W_E \qquad (4.5)\] If you look at the equation from left to right, its members are: active energy dissipation on resistance \(R\), the potential on the reactive element, and to the right of the equal sign is the energy that takes the power source for the whole process.
Let's separately consider \(W_F\). This is a very important element completely determines the free energy in such a system. Will do back substitution from formula (4.2) \[W_F = \int_0^T Y\,\Phi(t)\, dt = \int_0^T Z(Y)\,Y\,\dot Y_t\, dt \qquad (4.6)\] and after some transformations we find him General: \[W_F = \int_{Y(0)}^{Y(T)} Z(Y)\,Y\, dY \qquad (4.7)\] Note this formula under certain conditions it is in itself and contains potentially achievable free energy in a parametric reactance. One of the interesting features — the formula does not depend on time coordinate \(t\). Later we will return to it, but for now we will show that for RL-circuit proof is derived similarly.
For RL-circuit
Swap the first and second members of the characteristic equation (4.3), domnain every member of \(Y\), then printeriem both parts: \[\int_0^T Y^2\, dt + \int_0^T Y\,\Phi(t)\, dt = \int_0^T U(t)\,Y\, dt \qquad (4.8)\] As in the case of RC-circuit here we see all the same components: \[W_R + W_F = W_E\] and the potential energy of the reactive element \(W_F\) is exactly the same (see formula 4.7).
The proof for the FCC
If you understood the previous material, the FCC proof for appears very simple: it is enough to evaluate the energy \(W_F\). From the definition it is known that at the beginning and end of a full cycle, the energy in the reactive element is missing, this means that the limits of integration \(Y(0)\) and \(Y(T)\) in (4.7) equal to zero. Therefore, \(W_F\) is also zero, and the remaining two members of equation (4.5) are equal: \[W_R = W_E \qquad (4.9)\]

In chains of the first order, in the full cycle, it is impossible to gain energy even if the reactive element is parametric. It does not depend on the nature of parametric dependencies, nor from the selected interval time.

This is intuitively clear: even if the additional energy appears in the growing cycle, it kompensiruet to falling, or Vice versa. A special case of the FCC with the capacitive circuit is discussed in more detail here.
The increment of energy in PCCIE
As in the case of PCCIE power is missing and \(U(t)=0\) in equation (4.5) the energy \(W_E\) is zero and \[W_R = -W_F \qquad (4.10)\] and accumulated to the beginning of this cycle, the energy in the reactive element will be expressed, obviously, so: \[W_0 = \frac{Z_0\, Y_0^2}{2} \qquad (4.11)\] where \(Z_0\) and \(Y_0\) is capacitance-inductively and voltage-current at the initial moment of the cycle. Thus, the ratio of the increment of energy will be, as the ratio \(W_R\) — energy scattering on the resistance to \(W_0\) is the initial energy: \[K_{\eta 2} = \frac{W_R}{W_0} = -\frac{W_F}{W_0} \qquad (4.12)\] Returning to the formula (4.7) we first need to determine the limits of integration. \(Y(0)\) — it will be \(Y_0\) and \(Y(T)\), by definition, PCCIE, is zero. Substituting the obtained data, we obtain the final formula for the free energy in PCCIE: \[K_{\eta 2} = { 2 \over Z_0\, Y_0^2} \int_{0}^{Y_0} Z(Y)\,Y\, dY \qquad (4.13)\] the formula immediately shows that if a parametric dependency is missing and \(Z(Y)=Z_0\), then any increment of power and \(K_{\eta 2}=1\).
Above was the General approach to the problem. For the particular case of PCCIE inductive circuit see separate work with calculations and specialized calculator.
Conditions for PCCFE
PCCFE from the definition it follows that equation (4.5) remains unchanged, but the lower boundary of the integral (4.7) becomes equal to zero \(Z(0)=0\). We rewrite this equation as \[W_R = W_E - W_F \qquad (4.14)\] it is Obvious that to obtain the energy surplus energy is allocated to resistance should be more than consumed by the power source, and this means that the summand is \(W_F\) must be less than zero: \[W_F \lt 0 \quad \Rightarrow \quad \int_{0}^{Y(T)} Z(Y)\,Y\, dY \lt 0 \qquad (4.15)\] This question may seem rather complex to display in the real world. But in fact the physical meaning of all this — the possible constant component, arising in variable fields. For example, if the high voltage and high frequency, the DC component can manifest in the form of electrostatic deposited on the surrounding device.
The meaning of the negative \(W_F\) from the point of view of electronics is the availability of the required plot of CVC with negative differential resistance. You can also offer an exotic option when shift between voltage and current reaches 180 degrees (with increasing voltage on the inductor, the current through it decreases).
Mathematical sense is to find the optimal curve or the coefficients of an exponential series, which we next consider. Let the parametric dependency is described by a power series of the three members of the: \[Z(Y) = 1 + k_1 |Y| + k_2 Y^2 \qquad (4.16)\] Such a number, for example, can describe the change of permeability of ferrite depending on the intensity of the magnetic field (chart). Take from it the integral and compare with the zero according to the formula (4.15): \[\frac12 Y_T^2 + \frac{k_1}{3}Y_T^2 |Y_T| + \frac{k_2}{4}Y_T^4 \lt 0, \quad Y_T=Y(T) \qquad (4.17)\] Thus, the task of finding the conditions of free energy for PCCFE, in such parametric dependence is reduced to finding the range of the coefficients in the inequality, with known \(Y_T\), or Vice versa — the search for the optimal \(Y_T\) with known coefficients: \[1 + \frac{2\,k_1}{3}|Y_T| + \frac{k_2}{2}Y_T^2 \lt 0 \qquad (4.18)\] in More detail, this example is parsed here.
How to find the change in efficiency in this case? It is enough to compare the energy dissipation by the resistance with the spent power source: \[K_{\eta 2} = {W_R \over W_E} = {W_E - W_F \over W_E} = 1 - {W_F \over W_E} \qquad (4.19)\] from Here we immediately see that if we keep the condition (4.15), then \(K_{\eta 2}\) is greater than one. A more complete formula for efficiency at PCCFE will be like this: \[K_{\eta 2} = 1 - {1 \over W_E} \int_{0}^{Y(T)} Z(Y)\,Y\, dY \qquad (4.20)\]
In this work we proved that it is impossible to gain energy in parametric circuits of the first order in the full cycle (FCC) because the energy dissipated in the resistance is always equal to the energy expended by the power formula (4.9). But if the cycle is incomplete, the receiving gain becomes achievable task. If the reactive element comprises potential energy in the beginning of the cycle (PCCIE), the increase can be found by the formula (4.13). If in the reactive element of the energy remains at the end of the cycle, then the conditions for receiving allowances, we can find according to the formula (4.15), and the increment of efficiency by (4.20).
You need to understand that in this note a mathematically strictly proved potentially achievable values of the increment of energy, part of which, in the real reactance, can be spent inefficiently, for example, on heating. However, on the basis of evidence about the energy increment in the fractional cycles, one can obtain special cases for engineering calculations, which, in turn, will allow you to build a real device with high efficiency.

© Vyacheslav Gorchilin, 2017
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