Research website of Vyacheslav Gorchilin
2021-09-17
Parametric resonance of the second kind in an RL circuit
Sinus modulation
what religion could not give, that is,
the knowledge of matter and the Cosmos"
Facets of Agni Yoga, vol. IV, 404
In this work, we will get acquainted with a relatively new phenomenon - parametric resonance (PR), which occurs in parametric RL chains of the second kind. Unlike nonparametric circuits, where inductance and capacitance are required to obtain resonance, one inductance will be enough here, the parameter of which is changed by one of the generators. The classical theory considers a parametric oscillator in which motion (or current) is investigated [1]. We will go the other way and propose energy ratios in the electric circuit, which will give us the optimal ratios of its constituent elements, needed by researchers to build free energy devices. We draw your attention to the fact that we will do all this using scientific calculation methods, which we will constantly refer to.
An attempt to obtain such an energy ratio of the I parametric RL circuit was undertaken by us here. But with all the simplicity of the iterative approach, it is rather difficult to obtain the final result of a steady-state process (for example, the current at the 1000th oscillation): this requires large computing power. In this case, an analytical study of the energetics of processes is completely impossible. Therefore, in this work we will apply an approximate method, which, in a certain range of values, still gives fairly accurate results. This method will allow obtaining energy relationships in a steady process and investigating it graphically.
It is known that PR, in its most general form, is an increase in the amplitude of oscillations as a result of parametric excitation. The phenomenon presented here differs from the classical one in that at resonance of the second kind, there is almost no change in the amplitude, on the other hand, the active power consumption of the master oscillator is noticeably reduced, and this leads to an increase in the efficiency of the circuit. We will see all this on the graphs in the second part of this work, where we examine this resonance by graphical methods and obtain the optimal ratios of elements for designing a real device. But first, we need to develop a mathematical model of this phenomenon, which we will continue to do in the first part.
To obtain the values of the classical nonparametric resonance, you need to know the frequency, inductance and capacitance of the circuit [2]. The PR in a parametric RL-chain of the second kind contains many more parameters: the value of the inductance, the value of its Q-factor and modulation coefficient, the ratio of the frequencies and phases of the master and modulating oscillator. This unusual resonance depends on all these parameters. This is reflected in the following scheme, according to which we will conduct further research (Fig. 1).
 Fig.1. Schematic diagram of a parametric RL circuit
In this diagram, G1 is a master oscillator, which is included in a series circuit consisting of a coil $$L (t)$$, the inductance of which is changed by the second generator G2, its active resistance $$r$$, and active load $$R$$. A current $$I (t)$$ flows in this circuit.
Now let's describe these elements in more detail. Let the master oscillator work according to the following law: $U (t) = U_m \sin (\omega t) \qquad (1.1)$ Here: $$U_m$$ is the amplitude value of the voltage G1, $$t$$ is the time, $$\omega = 2 \pi f$$, where $$f$$ is the frequency. The coil $$L (t)$$ is here a kind of current amplifier, and the amplification itself occurs under certain conditions, due to a change in its inductance according to the following law: $L(t) = L\, [1 + m |\sin(\omega t n /2 + \alpha /2)|], \quad n = 1,2,3,4... \qquad (1.2)$ Here: $$L$$ is the initial inductance of the coil (without modulation by the second generator), $$n$$ is the coefficient showing the ratio of the frequencies of the second and first generators: $$m$$ - modulation coefficient of the second generator, $$\alpha$$ - phase shift between the oscillations of the first and second generator.
It is necessary to draw your attention to the sine of the function, which is taken modulo in formula (1.2). This is done on purpose, to approximate this mathematics as closely as possible to real ferrite materials that will be used in the coil core. Since the magnetic fields from the first and second generators there are perpendicular to each other, the influence of the generators on each other will be only indirect - through a change in the magnetic permeability. This means that the sinusoidal modulation signal from the second oscillator will affect the first one equally in any half-cycle of the sinusoid.
I must say a few words about the current in this circuit. After solving the problem, it must be presented as follows: $I (t) = I_m \sin (\omega t - \varphi) \qquad (1.3)$ where $$I_m$$ is the amplitude value of the current, $$\varphi$$ is the phase shift between the oscillations of G1 and the current in the circuit. Also, to simplify the following calculations, we will combine two active resistances into one $R_r = R + r \qquad (1.4)$ and introduce the concept of the quality factor of the circuit [3]: $Q = {\omega L \over R_r} \qquad (1.5)$ We need the quality factor in the following formulas.
Parametric RL Net Equation
Let us compose the equation of the electric circuit (Fig. 1) using the Kirchhoff method, and calculate the voltage and current values by the method of complex amplitudes [4]: $U = I \, R_r + I \, \mathbb {i} \omega L \qquad (1.6)$ where $$\mathbb {i}$$ is a complex unit, and the value of current and voltage is reflected here in complex form. From here we derive the current: $I = {U \over R_r + \mathbb {i} \omega L} = {U \over R_r} {1 - \mathbb {i} Q \over 1 + Q ^ 2} \qquad (1.7)$ Take the modulus of the current and display its amplitude value: $I_m = {U_m \over R_r} {1 \over \sqrt {1 + Q ^ 2}} \qquad (1.8)$ Now you can bring all this to the initial current and voltage, which already depend on time: $I (t) = {U_m \over R_r} {\sin (\omega t - \varphi) \over \sqrt {1 + Q ^ 2}} \qquad (1.9)$ where $\varphi = \arctan {\omega L \over R_r} \qquad (1.10)$ This angle determines the phase shift between the voltage of the master oscillator and the current in the circuit [4, p/p 4]. For clarity, this can be seen in the following graph, where voltage is displayed in green and current is displayed in red. Also, this angle separates the active and reactive power in any system, and if, for example, it is zero, then this power is active, if it is 90 degrees, then it is reactive. And yet, the value of the cosine of this angle is indicated in the passport of any induction motor. Here we will need it for the same purposes.
Next, we will proceed in a not entirely standard way and immediately determine that this method is suitable for small values of the modulation coefficient $$m$$, when the current shape does not differ much from sinusoidal (example). Here we need to remember formula (1.2). The fact is that, in fact, the Q factor in the parametric circuit changes along with the inductance within a small range: $Q(t) = {\omega L(t) \over R_r} = Q\, [1 + m |\sin(\omega t n /2 + \alpha /2)|] \qquad (1.11)$ where $$Q$$ we take from formula (1.5).
Power balance
Based on the previously presented formulas, it will be easy to find the power balance. To do this, we first find the average power dissipated at the load R (Fig. 1): $P_R = {R \over T} \int \limits_0 ^ TI (t) ^ 2 \, \partial t = {U_m ^ 2 R \over R_r ^ 2 \, T} \int \limits_0 ^ T {\sin (\omega t - \varphi) ^ 2 \over 1 + Q (t) ^ 2} \, \partial t \qquad (1.12)$ where $$Q (t)$$ we take from formula (1.11), and $$\varphi$$ - from formula (1.10). In this case, $$T$$ is an integer number of oscillations of the generator. Next, we get the average active power that the generator G1 consumes to power the RL circuit: $P_1 = {1 \over T} \int \limits_0 ^ TU (t) \, I (t) \, \partial t = {U_m ^ 2 \over R_r \, T } \int \limits_0 ^ T {\sin (\omega t) \sin (\omega t - \varphi) \over \sqrt {1 + Q (t) ^ 2}} \, \partial t \qquad (1.13)$ We only have unknown active power $$P_2$$, consumed by the generator G2 to parametrically change the inductance. We can learn it from a real circuit, or we can assume the value of this power. It goes without saying that in a real device it should be as small as possible. Then the power balance, which is a change in efficiency of the second kind, will be as follows: $K _ {\eta 2} = {P_R \over P_1 + P_2} \qquad (1.14)$ By the way, if the inductance is constant (nonparametric) and energy is not spent on changing it, then this efficiency turns into a classic subunit efficiency: $K _ {\eta 2} = \eta = {R \over R_r}, \quad m = 0, \quad P_2 = 0 \qquad (1.15)$
In principle, formulas (1.10–1.14) are already ready to present a new phenomenon and describe the second kind of PR for the RL chain. But they cannot be considered in an analytical form, so we will use graphical methods, which we will present in the second part of this work. We will see on the graphs the general picture of the PR of the second kind and find the optimal ratios of the circuit elements for the maximum manifestation of this effect in real circuits.

The materials used
1. Wikipedia. Parametric oscillator.
2. Wikipedia. Oscillating circuit.
3. Wikipedia. Inductor. Goodness.
4. Chapter 6. Lecture 10. The method of complex amplitudes. [PDF]