Introduction of an additional source of harmonic signal into the circuit of the electric power resonant amplifier

Annotation. Problem. The problems of the electric power industry, caused by the depletion of the natural resources of the planet and the need to replace them, initiate the development of new physical and technical solutions with the practical use of the known natural phenomena. Goal. The purpose of this work is to propose to introduce an additional source of a harmonic signal (voltage or current) into the circuit of a resonant amplifier of electric power, to obtain calculated analytical dependencies for numerical estimates of the characteristics of ongoing electromagnetic processes, which make it possible to give a fundamental justification for the real performance of the proposed circuit as a whole, taking into account the interaction of all its functional components. Methodology . This work, ultimately, involves the use of resonant phenomena in circuits with active-reactive elements and their theoretical analysis using the mathematical apparatus of the theory of electrical circuits. Result s. It is proposed to implement the resonant amplifier circuit in the form of four active-reactive closed circuits inductively coupled to each other. Moreover, in a practical embodiment, inductive couplings can be carried out using HF ferrites. The first circuit is the input circuit with the harmonic power source to be amplified. The second circuit generates amplified reactive power in the "voltage resonance" mode. The third circuit with an additional harmonic voltage source outputs reactive power from the second circuit in the "current resonance" mode. The fourth circuit, inductively coupled to the third circuit, contains the output load of the entire resonant amplifier. This is a resistor that simulates the release of active power. Originality. Physically, the introduction of an additional source in the third circuit is equivalent to the creation of a "negative active resistance", which makes it possible to create conditions for excitation of current resonance with the minimum possible distortion and, ultimately, to reduce the reverse effect on the amplifying processes in the second circuit (reactive power amplifier). The analysis and numerical evaluation of the characteristics of the proposed scheme of the active electric power resonant amplifier showed its fundamental viability. Practical value . As an example, calculations of currents and voltages in the circuit of an experimental model were made, which enabled to formulate recommendations for the selection of elements of a real active electric power amplifier with high efficiency for low-resistance output loads.


Introduction
As it follows from the information generalization represented in the different scientific publications the resonance can be a key to the energetic spike in the oscillatory systems of the different physical nature. For example, the historical facts of the bridge constructions mechanical destroying are well known to the world science but they are nonobvious in the sense of the their physical causality. There are many analogical questions to the resonant phenomena appearance in the heat processes, the electrical circuits and much other. Their analysis leads to the fundamental question formulation about a source the energy of which allows fulfilling a work what is impossible in traditional understanding of the physical processes cause-effect tie. There are different hypothesizes the essence of which consists in some universal substance existence which posses by the great energetic potential (for example, it can be the "dark matter", the "physical vacuum", the ether etc.). In dependence on the realization conditions this potential can become apparent in a sharp burst kind of the thermal energy, of the nuclear energy and finally of the electromagnetic energy [1][2][3]. Not stopping on the works dedicated to the fundamental questions of our Universe structure the undoubted interest of the world public to the known technical elaborations which are directed to solution of the modern power engineering problems should particularly extracted [4].

Analysis of publications
The efficiency of the electric resonance-rectifier circuit for the renewable energy conversion is analyzed in the work [5]. The scientific edition [6] is dedicated to a concise technical overview of energy technology: the sources of energy, energy systems and frontier conversion. Here are the advanced converters, catalysts, fuel cells, membranes, metal-hydrides, refrigerators and M.H.D. solar cells, finally. The articles [7,8] illuminate the theoretical investigations of the electromagnetic processes in Tesla transformer which was the first technically realized suggestion of the voltage resonant amplifier. The got results and the numerical estimates agree well with the qualitative conclusions of the Great Inventor. Appearance of the Patent [9] is conditioned by the practical interest to the power resonant amplify. The subject of the invention is related to the impact excitation systems in the electrical power engineering but It can find application in the uninterrupted power supply units, in the electromagnetic vibration transmitting apparatuses etc. Finally, the work [10] is dedicated to the experimental justification of the electrical power resonant amplifier workability. To the authors opinion the main result of the conducted investigations is the experimental fact when the output reactive electrical power exceeds more than ~33 times of the source input power.
The practical interest represents a scheme elaboration of the electrical energy resonant amplifier where (unlike the previous analogue!) the active electrical power is generated which can be used physically for the different works fulfilling.
The aim of the present investigation is the scheme suggestion of the active electrical power resonant amplifier, analysis and numerical estimates of the flowing electromagnetic processes characteristics for the principally justification of the suggested scheme real workability.

Purpose and Tasks
Electrical scheme, action principle.
The electrical equivalent scheme of the suggested resonant amplifier of the active electrical power consisting of the four resonant circuits is represented on Fig. 1. The first of them -1 with source of the harmonic voltage -E 1 is the amplifier input circuit. Its current and voltage transmit to the second serial circuit -2 with help of the coupling transformer «L 1T -L 2T ». Here the amplified reactive power from the output element (the capacitance -C 2 ) transmits to input (the capacitance -C 3 ) of the parallel resonant circuit -3. The latter one is inductively coupled with the serial circuit -4 the output element of which is modeled by resistor -R 4 . This is the load where the amplified active electrical power is liberated.
The particularity of the suggested scheme consists in what the parallel circuit -3 contains the additional source of the harmonic voltage -E 2 . Its appointment consists in the conditions creation for the "current resonance" regime in which a back influence on the serial circuit -2 is excluded.
A relation of the active power in the output element of the circuit -4 (load, resistor -R 4 ) to the power of the energy source in the input circuit -1 is the quantitative index of the electromagnetic energy conversion in suggested scheme of the resonant amplifier of the active electrical power.
Problem formulation. • The input serial circuit -1 contains the capacitance -C 1 , the inductance -L 1T (the primary winding of the coupling transformer between the circuits 1-2), the active resistance -R 1 and the source of the harmonic voltage E 1 (t) = E 1 ‧sin(ω ‧ t) (E 1 -is the amplitude, ω -is the angular frequency, t -is the time). • The amplifying serial circuit -2 contains the inductance -L 2T (this is the secondary winding of the coupling transformer between the circuits 1-2), the capacitance -C 2 (the output element), the inductance -L 2 and active resistance -R 2 (this can be the resistance of the winding inductances and coupling wires). • The parallel circuit -3 contains the capacitance -C 3 , the active resistance -R 3 (the resistance of the windings inductances and the coupling wires), the inductance -L 3 and the additional source of the harmonic voltage -E 2 (t) = E 2 ‧sin (ω ‧ t) (E 2 -is the amplitude). • The output serial circuit -4 contains the inductance -L 4 , the capacitance -C 4 and the resistor -R 4 which models the amplifier active load. • The frequencies of all resonant circuits are equal to each other resonant frequency.
The calculation dependencies for the workability theoretical justification of the suggested scheme are based on the physically "transparent" phenomenological statements and the strict mathematical approach with the electrical circuit theory methods usage [11].
We shall start from the "output" circuits of the amplifier resonant.
According to the equivalent scheme on Fig ; where k 34 ∈ [0, 1] -is the coefficient of the electromagnetic coupling level between the circuits -3 and 4; J 4 -is the current in the circuit -4 with the inductance -L 4 , with the capacitance -C 4 and the active resistance of the load -R 4 ; J 33 , J 3 -are the currents in branches of the circuit-3; J 33 -is the current in the branch with the inductance -L 3 ,with active resistance -R 3 and additional source of the harmonic voltage -E 2 ; J 3 -is the current in the branch with the capacitance -C 3 ; U C3 -is the voltage on the capacitance -C 3 .
The currents being excited can be found from the linear algebraic equations system (1) [9]. ; The current -J23 in the output from the capacitance -C 2 can be determined as sum the currents in the branches of the parallel circuit taking into account the resonance:  2  2  34 4  3  23  3  33  2  3  3  34  3  4 1 From (3) we receive that for J 23 = 0 the next condition has to be fulfilled: .
It should be marked that in practices, the necessary voltage of the additional source can be determined when the voltage amplitude variation till to obtain the zero current in the input to the circuit -3 from the capacitance -C 2 in the circuit -2.
With help of (2) and (4) we find the voltage and current in the load active resistance -R 4 : ; .
Let us return to analysis of the condition (4). If this condition is fulfilled the "current resonance" regime is excited in the parallel circuit -3. The current in the output from the capacitance -C 2 to the circuit -3 is equal to zero (J 23 = 0). The electromagnetic processes in the resonant circuits -2 and 1 are flowing independently on the processes in the circuits -3 and 4.
The last final conclusion allows analyzing excitation of the circuit -2 and 1 by the source of the harmonic voltage -E 1 without any coupling with the circuits -3 and 4. The state equations system has the view [11]: where J 12 -are the currents in the circuits -1 and 2, correspondingly; 12   ; .
The expressions for the currents being excited can be got from (7).
Should mark that parameter -Z in (8) can be interpreted as a module of the equivalent inductive resistance. It ties the source power voltage -E 1 with the resonant current -J 2 in the second circuit. And as it follows from the corresponding expression in (8) this tie has the strictly inductive character.
Let us rewrite the corresponding expression for J 2 separately for strict clearness in the further analysis of the flowing electromagnetic processes.
( ) ( ) ω Should mark that parameter -Z in (8) can be interpreted as a module of the equivalent inductive resistance. It ties the source power voltage -E 1 with the resonant current -J 2 in the second circuit. And as it follows from the corresponding expression in (8) this tie has the strictly inductive character.
Let us rewrite the corresponding expression for J 2 separately for strict clearness in the further analysis of the flowing electromagnetic processes. 1 2 ,

Automobile transport, Vol. 51, 2022
The necessary condition of the extremum existence for the function -Z = Z(ωM 12 ) is being written in the view [12]: ω ω (10) As it follows from the expression (10), the equivalent resistance module reaches the minimum under ( ) And the corresponding resistance minimum will equal to - In the terms of the parameters of the circuits -2 and 1 the realization condition of the minimal value of the equivalent resistance -Z has the view: The estimate of the electromagnetic coupling coefficient which provides the secondary current maximum -J 2max follows from the expression (11).
Physically, the found minimum of the equivalent resistance ting the secondary current with the voltage -E 1 of the power source and determining the maximum power amplifying can be explained by the minimally possible return of the energy from the secondary circuit -2 to the primary circuit -1. At that all this process is being provided by the electromagnetic coupling level between the circuits correspondingly to the formula (12).
To calculate the integral coefficient of the active electrical power conversion in the suggested scheme of the resonant amplifier the formulas should be written for the current in the circuit -1 and the voltage in the "output element" -L 2 of the circuit -2.
Taking into account that 3 According to (13) and (14) the power amplitudes in the circuit -1, in the circuit -2 and in the circuit -4 (the amplifier output of the active electrical power) will be determined by the following dependencies.
The introduced power -2 E P (of the additional source) normalized on the power -2m P which is introduced from the circuit -2 (under the resonant conditions) is being determined by the expression: where 4m P − is the output power in the load active resistance in the terms of the amplifier parameters.
As it follows from the formula (18) the active power in the load which is determined with taking into account power of the additional source can be considered as some efficiency conditional characteristic which permits evaluating the minimally possible value of the output power in the scheme of the considered amplifier.

Analysis, numerical estimates
From physical consideration it is obvious that for the amplifier efficiency maximum a contribution of the power additional source in exciting the "current resonance" in the parallel circuit has to be minimal. As it follows from dependencies (4), (17) and (18), for this it is necessary quite high Q-factor -Q 3 >> 1 and quite weak electromagnetic coupling with the serial circuit in aggregate with quite small Q-factor, so that -2 34 4 1 k Q ⋅ << . Simultaneously, the dependence for the power conversion coefficient -(16), (17) and (18) demands increasing parameter -2 34 4 k Q ⋅ . The efficiency illustrations of the experimental model of the active power resonant amplifier are represented on Fig. 2, Fig. 3. The following initial data were accepted for calculation: ω = 2‧π‧25000 Гц, L 1T = L 2T = L 3 = 14,8 мкГн, L 2 = 169 мкГн, R 2 = 0,35 Ом, R 1 = R 3 = 0,1 Ом, k34 = 0,1.
Practically straight-proportional tie of the conversion integral coefficient and the power of the additional source follows from the calculations results on Fig. 2. Really, a substitution of (17) into the expression for K 1-4 from (16) leads to the relationship: Physically, this tie supposes the growth possibility of the conversion coefficient but with the simultaneous growth of the power of the additional source.
Finally, last comment. The curves on Fig. 3 illustrate the output power falling down in the dependence on the load resistance with and without taking into account the power of the additional source.