This series of notes is devoted to parametric RLC-circuits, yet unexplored and underutilized. If the topic of parametric resonance and parametric amplifiers still captured the curriculum of radio engineering Universities, energy parametric RLC-circuits is unfairly bypassed. However, some special cases of such circuits, by reason of its not quite classical behavior, are of interest to seekers of free energy.

In the introduction we will introduce our readers to this really huge layer of knowledge, reveal some features of parametric resonance and parametric dependencies (PZ), will introduce the terminology which will be needed further. We will start with ordinary, familiar to all from childhood, swing. As we know, they can swing two ways.

1. Sitting on the swing (figure a). In this case, the person, swinging your legs back and forth creates the force \(\bar{F}\), directed parallel to the movement. The frequency of rocking of the feet corresponds to the frequency of the rocking of the swing: \(f_m = f_F\). There's nothing unusual here, everything is logical and understandable.

2. Standing on the swing (figure b). Here the person is squatting to the beat with a swing and in fact — change the height of their suspension. That is, this height is parametrically dependent on time. If you do not know about the physics and mathematics of this method, at first glance it seems that the swing should not be shaken, because the force exerted by the person directed along the suspension and perpendicular to the movement! This is the parametric resonance, the mathematics of which we will consider later, but for now only acquaint readers with the very principle. The frequency of rocking of the swing in this case is twice the frequency of rocking of the feet, as in one period of oscillation of the swing, the man crouches twice: \(f_m = 2f_F\). This is another important difference parametric resonance from the ordinary.

Electronic analog of the swing — oscillation circuit, consisting of capacitor, inductor and resistance. It is what we will consider next. The circuit can be parallel (Fig. a) or sequential (Fig. b), and in the simplest case, we can remove from the circuit or receptacle (Fig. c), or inductance (Fig. d), but the essence will not change. By the way, scheme c or d is called RL or RC circuit of the first order, i.e. consisting of only one active and one reactive element. When the reactive element two: capacity and inductance, such circuit will be of the second order (a or b). Here is traced the analogy with the name of differential equations in mathematics [1], as in the case of a single reactive element, the electrical circuit can be completely describe the diff. the equation of the first order, and in the case of two reactive elements — the equation of the second order.

We will introduce another classification of parametric circuits. If the parameter of the reactive element depends upon the original causes: the inductance from flowing through it from the current, and capacitance — voltage on it, then we call such chain of the first kind. The natural character of PZ is due to the natural characteristics of the materials studied radiokomponent. For example, the inductance of the coil depends on the permeability of the core, which depends on the magnetic field intensity, and therefore the current passing through it. The varicap and the variant change their capacitance depending on the applied thereto a reverse voltage. Mathematically these relationships are denoted as \(L = L(I_L)\), \(C = C(U_C)\).

If reactive elements are of a different character PZ, we will call such chains of the second kind. For example, when the inductance is independent of any voltage and capacity — from the current. Here is part of electro and magnetostriction parameter, as a function of time, depending on other parameters of the device: thrust power, shaft speed, etc. this subclass includes external PZ, where, for example, the inductance depends on the permeability of the core, which changes the external oscillator using the perpendicular (to the primary) magnetic field. Mathematically the definition of the parametric circuits of the second kind is given here.

Since all processes protektsia in time, and we can connect to their measurements in any time intervals, we introduce also such a division: full and partial cycle measurements. When the full cycle (FCC) energy in the reactive elements at the beginning and end of the measurement is missing. In castina cycle (PCC) energy in the reactive elements may be present either at the beginning or at the end of the measurements.

This classification is useful for quick analysis of the device. For example, if it uses PZ of the first kind with a full cycle, then the energy increase cannot be obtained. This includes, for example, flyback converters, oscillatory circuits with varicaps, etc. By the way, in universities, parametric amplifiers and circuits of the first kind are studied.

The generators of the second type, the receive energy gain is quite possible, even in the full cycle. In other words, in this case there is no need to create special external conditions to pre-pumping energy of the reactive elements, such as mechanical interruptions or periodic influence of the coil field of the permanent magnet. In the simplest case only two generators of sinusoidal waveform and of a material that changes its properties under the influence of one of them. More complex systems may use electro and magnetostrictive effects, but they need a strong electric or magnetic field, respectively.

The theoretical basis of parametric generators of the second type represented here, and payments can be made in a special calculator.

Next we will talk about the differences between RLC-circuits from mechanical swing. Most importantly — they have no reactive elements that can get ahead of the main process or be left behind. Therefore, the analogy of reactive energy with potential can be quite conditional.

All known active and reactive elements can be provided in two perpendicular coordinates (see figure). Along the axis indicated by the green color represented the positive and negative resistance \(R\). Positive resists the flow of current by converting the reactive energy of the electron in the active, forming, therefore, the resistance of the active power. It may be an ordinary resistor, filament, magnetic resistance, i.e., everything that ultimately is heated, cooled or lit. Negative resistance, on the contrary, nourishes the chain and in fact is the source current or voltage. In the middle of the axis at the point \(R=0\), are all superconductors.

In inductance and capacitance — electron converts its energy in them, she remains reactive. Therefore, these elements and formed in their current, voltage and power is called reactive. They are located on the blue axis. If we consider a circuit consisting of active and reactive elements, the current in the inductor will lag behind the current resistance at 90 degrees and the current through the capacitance, on the contrary — ahead by the same amount. The phase of the current through inductance and capacitance will be different by 180 degrees. All this is clearly visible in the figure.

All possible manifestations can be located either on one of the axes to be either the vector directed from the center and located at an angle to one of the coordinates. That is, any electronic component may be an energy source, resistance, capacity or inductance, or can be various combinations. In electronics a well-studied processes, the elements that can be formed in the right side of the figure, for example, the vectors \(a\) and \(b\). In the first case the circuit will have a capacitive character, and the second inductive. Also, obviously the behavior of the elements located on the y-axis. Right resistance, the left — the power source. But the region left of the figure are not well understood (for example, the vectors \(c\) and \(d\)), although for seekers of free energy that this region is of main interest. For example, under certain conditions, the reactive elements can become the source of power, because the vector of this element circuit is shifted to a negative resistance. This representation, in addition to its clarity, makes it easier to navigate in parametric circuits and their individual elements, which we describe later.