Research website of Vyacheslav Gorchilin
All articles/Parametric Circuits
Energy parametric circuits of the second kind. Introduction
Considering the power of parametric circuits (PMTS) of the first kind we have come to the conclusion that to increase the efficiency of the second kind (\(\eta_2\)) to them either in principle impossible, either — it seems quite a complex task, which somehow smoothly brings us to the PMTS of the second kind in which the change in \(\eta_2\) is osnovoobrazuyuschey. But for a more systemic understanding of the processes we begin his story not with electronics, but ... quantum mechanics!

If quantum mechanics hadn't shocked you, then you don't understand! Niels Bohr

Remember known from school a thought experiment with schrodinger's Cat [1], and two possible versions of events: the cat is alive and cat is dead? Since the work of this famous physicist, science has leaped forward and now knows quite a lot about parallel dimensions or parallel worlds [2]. As it turned out, there is nothing unusual, and the "parallelization" can happen at singular points, the so-called Bifurcation Points [3]. But really, all the more interesting...
Распараллеливание миров

In fact, it's not like really! French proverb

All possible States of matter (challenges) exist simultaneously in a certain community, called Reality. This term also can be called unmanifested Reality. This is evidenced by for example the motion of a photon in the optical waveguide where the phase appears only when it reaches the photodetector, and before that both possibilities exist simultaneously. Other experiments with photons allow to observe oscillations of the probability of such capabilities [4].
But the Reality itself we are not very interesting, yet it does not manifest itself in one of the Realities in the form of certain sets of States or possibilities. In one of those Realities we live and call it our world! The set of such States give rise to certain laws of interaction of matter that we observe in it Laws of Physics. From this it becomes immediately clear that the different Realities of physical laws may differ. Note this important paragraph because it further we will need.

Everything is a number! Pythagoras (one Dedicated)

And what unites all of these Realities? You will be surprised — this is math, its laws are the same everywhere! While we are at school and told that this science is applied, but in fact the opposite is true: mathematics is the Queen of all Sciences, but physics, chemistry and other Sciences is just a different manifestation. Here's a simple example: the harmonic oscillator without any loss [5]. He describes a mathematical formula that is the same for all his forms: \[ {\ddot {x}}+\omega _{0}^{2}x=0 \] In mechanics — it can be a pendulum or the system load-spring is in electronics — oscillating circuit, in chemistry, the fluctuations of electrons, atoms, and molecules, etc. But what does all this have to do with the stated theme, you ask? Very direct.
A little math

Mathematics is our link with reality!

Next, we will speak the language of mathematics, and how it is implemented in our reality: through the first, second or even third magnetic field using a conventional spin or charge, using centered or distributed — profile case of this reality, which a little later. In the meantime, address by idealized mathematical models of the inductance, the resistance and the voltage source, we find the conditions for the increase or change, \(\eta_2\), and also derive the definition for parametric circuits of the second kind. Параметрическая RL-цепь первого порядка These findings may surprise many of the classics.
The supply voltage U is changed according to the law \(U=U(t)\); the resistance R is constant, but in the future it can be done menyushina in time, and make the sign of the integral; the inductance L is changed under the action of passing through it the current is \(L=L_0\,M(I)\), where \(M(I)\) — relative inductance (at zero current it is equal to one).
We rewrite the formula (4.8), which focused on the PMTS of the first kind, directly to our RL-circuit: \[ R\int_0^T I^2\, dt + \int_0^T L(I)\,I\,\dot I\, dt = \int_0^T U(t)\,I\, dt, \quad I=I(t) \qquad (1.1) \] Here you need to remember this equation represents the equation of energy balance for the circuit elements, which simplistically can be written as: \[ W_R + W_L = W_U \qquad (1.2) \] Also recall, as was the solution of the integral \(E_L\) for the PMTS of the first kind: \[ W_L = \int_0^T L(I)\,I\,\dot I\, dt = L_0\int_0^T M(I)\,I\,\dot I\, dt = L_0\int_{I(0)}^{I(T)} M(I)\,I\, dI \qquad (1.3) \] Since rassmatrivali full period when \(I(0)=I(T)\), then the whole integral is equal to zero: \(W_L=0\). Physically, this means that how much energy the inductor will get as much and give, and it does not depend on properties of the coil or its core. It followed evidence that the full period of \(W_R = W_U \), i.e. is consumed by the power source energy is completely dissipated in the active resistance that is observed in ordinary practice and in the PMTS of the first kind. What you can change?
PMTS of the second kind
What if the time period \(0 .. T\) to be split into two sections: \(0 .. T_1\) and \(T_1 .. T\), and each of them to establish its dependence on \(M(I)\)? Let's see what we can do in this case: \[ W_L = L_0\int_{I(0)}^{I(T_1)} M_1(I)\,I\, dI + L_0\int_{I(T_1)}^{I(T)} M_2(I)\,I\, dI \qquad (1.4) \] Since in the full period of \(I(0)=I(T)\), the equation can be rewritten as: \[ W_L = L_0\int_{I(0)}^{I(T_1)} M_1(I)\,I\, dI - L_0\int_{I(0)}^{I(T_1)} M_2(I)\,I\, dI = L_0\int_{I(0)}^{I(T_1)} \left[ M_1(I) - M_2(I) \right]\,I\, dI \qquad (1.5) \] In this formula, and shown the main difference between the PMTS of the first kind from the second. In the PMTS of the first kind of dependence of magnetic permeability versus current is always the same: \(M_1(I) = M_2(I) = M(I)\), and therefore: \(W_L=0\), and the PMTS of the second kind, they are different: \(M_1(I) \neq M_2(I)\) and therefore: \(W_L \neq 0\). At different ratios of relative inductance this could mean either an increase in \(\eta_2\), or its reduction, but in reality, additional heating resistance, or Vice versa, its cooling! But the implementation of this mathematics of our Reality can make their adjustments in the form of additional fields and radiation, including yet unknown to science.
Based on (1.2) and (1.5) we can immediately find the rate of change of efficiency of the second kind: \[ K_{\eta 2} = {W_R \over W_U} = 1 - {W_L \over W_U} = 1 + \frac{L_0}{W_U} \int_{I(0)}^{I(T_1)} \left[ M_2(I) - M_1(I) \right]\,I\, dI \qquad (1.6) \]
Also, the second-order parametric resembles in its asymmetry the second-order phase transitions for matter [6].
Return to our reality
Formula (1.5) opens up incredible, in fact, opportunities in the energy sector, which may be manifested in our world in different guises. What are the limitations? About them we don't know yet, but something already known. If we are talking about the application of this mathematics in electronics and generally, in the electricity sector, the main limitation will be the internal energy of the electron. The second limitation is the saturation of the ferromagnetic material, which usually leads to restriction of the maximum magnetic flux. The third, and the main obstacle is the classic use of conductive and magnetic materials that will not allow us to reach the PMTS of the second kind, since the current through the inductor in forward and reverse direction flows according to the same laws, and therefore \(M_1(I) = M_2(I) = M(I)\).
But with all this there is a solution — the use of completely different principles of current flow in different directions, which is conventionally shown on the left. It depicts a conductor through which current in the forward direction (blue), due to the skin effect, flows over its surface, and back through the section (red) [6]. If the material of the conductor — ferromagnet, under certain conditions it can be achieved with different dependencies when \(M_1(I) \neq M_2(I)\). This example was suggested to the author of a brilliant researcher and engineer Dmitry and is not the only solution. Options for the practical implementation of this mathematics is much more.
In this introduction, we deliberately considered the simplest RL circuit in order to disclose the principle of the change of the efficiency in the PMTS of the second kind. Later, in a more General model, we calculate the chain of the second order, where will be present the parametric capacity that will allow researchers to find more practical solutions in the real plane.
Further development of this topic led to the development of the theory of parametric generators of the second kind (read here).
The materials used
  1. Wikipedia. Schrodinger's Cat.
  2. The universe splits with every quantum experiment.
  3. Wikipedia. Point Bifurcation.
  4. Physicists for the first time observed true interference of three photons.
  5. Wikipedia. Harmonic Oscillator.
  6. Wikipedia. Phase transition of the second kind.
  7. Wikipedia. Skin Effect.