In the third part of this work, we found the optimal phase shift between the oscillations of two generators: G1 and G2. In this part, we will develop over-unity criteria for parametric RL-circuits with parametric resonance (PR) of the second kind, and also, the methodology for calculating such circuits. Here, in all calculations, we assume that the optimal phase shift is used.

From Figure 5 we already know that the power on the load does not depend on the phase shift \(\alpha \), and therefore formula (2.1) can be written as follows: \[\bar P_R(Q) \approx \frac{1}{Q^2} \int \limits_0^1 {\cos(\omega t - \varphi)^2 \over (1 + m |\sin(\omega t n /2 + \alpha /2)|)^2}\, \partial t \approx {1 \over 2 Q^2 (1 + 0.6 m)^2} \qquad (4.1)\] where on the right is the expression found by the method proposed in the previous part. From the same part, we can take the formula (3.12) found there for the power of the master generator G1: \[\bar P_1 (\alpha, Q) = A (Q) - B (\alpha, Q) \qquad (4.2) \] where for n=2 \[A (Q) = \frac {1} {2 Q ^ 2} \left [1 - {2m \over \pi} + 0.5 m ^ 2 \right], \quad B ( \alpha, Q) = \frac {m (1 + 1 / Q)} {3 \pi Q} [1 - 1.15 m + 1.4 m ^ 3] \qquad (4.3) \] a for n=1 \[A (Q) = \frac {1} {2 Q ^ 2} \left [1 - {2m \over \pi} + 0.5 m ^ 2 \right], \quad B ( \alpha, Q) = \frac {m} {15 \pi Q} [1 - 0.05 m] \qquad (4.4) \] Recall that \(A (Q) \) and \(B (\alpha, Q) \) here are solved for the optimal phase shift, which we already found a little earlier .

And the last expression, for the beginning of work, which we will borrow from the first part, will be formula (1.14), which is designed to obtain an increase in efficiency: \[K _ {\eta 2} = {\bar P_R \over \bar P_1 + \bar P_2} \qquad (4.5) \] Here we rewrote it for relative (specific) capacities.

Now it remains for us to add everything together to obtain the criteria that will be needed to construct over-unity parametric RL-chains of the second kind. For this, we substitute expressions (4.1), (4.2), (4.3) into formula (4.5): \[K _ {\eta 2} = {d \over a - 2b (Q + 1) + 2 Q ^ 2 \bar P_2} \qquad (4.6) \] where for n=2 \[a = 1 - {2m \over \pi} + 0.5 m ^ 2, \quad b = {m (1 - 1.15 m + 1.4 m ^ 3) \over 3 \pi}, \quad d = {1 \over (1 + 0.6 m) ^ 2} \] a for n=1 \[a = 1 - {2m \over \pi} + 0.5 m ^ 2, \quad b = {m (1 - 0.05 m) \over 15 \pi}, \quad d = { 1 \over (1 + 0.6 m) ^ 2} \] We will build on it in the following calculations. Recall that here \(\bar P_2 \) is the power consumption by the modulating generator G2, which is still unknown to us.

In such problems, a certain dilemma arises - what to do with negative power values? But such can appear if P₂ is small enough or B is large enough. At the same time, the real circuit can go into gear, and if there is no special protection there, some of its elements may burn out. Since a mathematical model is being deduced here, we will approach this from the point of view of the boundary condition: \[B (\alpha, Q) - A (Q) \lt \bar P_2 \qquad (4.7) \] which follows from (4.2). In other words, the criteria found below will work analytically only up to this boundary. If they go beyond it, then for a real circuit this will mean that additional protections will be required in the form of some restrictions (current or voltage).

We have already done the most difficult thing and, thanks to the three previous parts of this work, we have obtained formula (4.6). And from it we can easily find the extremum of the function and derive the optimal quality factor Q: \[Q ^ {*} = {b \over 2 \bar P_2} \qquad (4.8) \] If in this formula we substitute the values of m and P₂ the same as in Figures 9-10, and mark the power maxima on them, then we will see that they coincide. Thus, we have confirmed the graphical method - analytical.

If you display the obtained optimal quality factor graphically, you get Figure 16. It shows the dependence of the optimal Q-factor on the power of the modulating generator P₂, for different values of the modulation factor: \(Q^{*}(m, P_2) \).

Fig.16. For n=2. Dependence of the optimal quality factor Q on the power of the modulating generator P2, at different values of the modulation factor m

An increase in efficiency at optimal Q can be obtained by substituting (4.8) into (4.6): \[K _ {\eta 2} ^ {*} = {d \over a - 2b - 0.5 b ^ 2 / \bar P_2} \qquad (4.9) \]

Graphically, this result can be displayed in Figure 17, where the dependence of the efficiency gain on the power of the modulating generator P ₂ , for different values of the modulation coefficient: \(K _ {\eta 2} ^ {*} (m, P_2) \). It is understood that the figure of merit is optimal here, found by formula (4.8).

Fig.17. For n=2. Dependence of the efficiency gain on the power of the modulating generator P2, at different values of the modulation coefficient m, and at optimal Q

These graphs may be needed for preliminary design of a parametric RL circuit of the second kind, when creating a real device. Also, the following graph is useful for this (Fig. 18), where the dependence of the power at the load (P_R), on the power of the modulating generator P ₂ , at different values of the modulation factor: \(P_ {R} ^ {*} ( m, P_2) \).

Figure 18. For n=2. Dependence of the power at the load (PR), on the power of the modulating generator P2, at different values of the modulation factor m, at optimal Q

In this case, the formula by which the graph is drawn is derived from expressions (4.1) and (4.8): \[P_R ^ {*} = {2 d {\bar P_2} ^ 2 \over b ^ 2} \qquad (4.10) \] Recall that all powers are relative here. To convert them into absolute values, you can use the formulas (2.3-2.4) or apply the technique that will be presented below.

Exactly the same calculation is applicable for the coefficient showing the ratio of modulating and master frequency, equal to one. Only as \(a, b, d \) you need to use the values described in expression (4.6) for n=1. Then the optimal figure of merit will be found according to the formula (4.8), the optimal increase in efficiency - according to the formula (4.9), and the optimal power at the load - according to the formula (4.10). The graphs drawn from them will certainly differ from n = 2 (not shown here).

The calculation will be carried out according to the scheme of a parametric RL-circuit of the second kind, presented in Figure 1. Previously, we measured the following parameters:

• amplitude value of the master oscillator voltage G1: \(U_m = 10 \, V \)
• coil inductance without changing the parameter: \(L = 1 \, mH \)
• resistance of the coil wire: \(r = 0.01 \, Ohm \)
• load resistance: \(R = 0.5 \, Ohm \)
• modulation factor of the coil inductance: \(m = 0.2 \). For charts 16-18 this is the green curve
• the active power of the G2 generator to create this modulation: \(P_2 = 0.04 \, W \)
• ratio of frequencies G2 to G1: \(n = 2 \)

Methodology. First, we find \(p_1 \) by formula (2.4): \[p_1 = {U_m ^ 2 \over R + r} = 196 \qquad (4.11) \] and then - the relative (or specific) power of the generator G2: \[\bar P_2 = {P_2 \over p_1} = 2 \cdot 10 ^ {- 4} \qquad (4.12) \] Further, using formula (4.8), we find the optimal quality factor for this power. The same can be done using the graph (Fig. 16, green curve), from which, for \(\bar P_2 = 2 \cdot 10 ^ {- 4} \), we determine \(Q = 41 \). By the way, this is the key parameter from which we can find the frequency of the master oscillator G1 (2.5): \[f = {Q (R + r) \over 2 \pi L} = 3328 \, Hz \qquad (4.13) \] Applying formula (4.9) or the graph (Fig. 17, green curve), we can find the increase in the efficiency of the circuit. For \(\bar P_2 = 2 \cdot 10 ^ {- 4} \), we define \(K _ {\eta 2} = 4.5 \). By the way, we can find the same value in Figure 9 , on the red curve, at the maximum of the function. It is also interesting that if the power of the G2 generator is less than only one and a half times, then the circuit can go into spacing and it will be necessary to introduce restrictions on the increase in current or voltage in it. This is clearly seen from the same graph (Fig. 17), when the green curve begins to rush sharply upward. But for our tasks, such a regime is just a positive factor.

It remains to find the relative power in the load according to the formula (4.10) or according to graph 18 (green curve). For \(\bar P_2 = 2 \cdot 10 ^ {- 4} \), we define \(\bar P_R = 2.3 \cdot 10 ^ {- 4} \). And now it needs to be converted into absolute power at the load, which can be found by the formula (2.3): \[P_R = \bar P_R {U_m ^ 2 R \over (R + r) ^ 2} = 44 \, mW \qquad (4.14) \] Do not forget that the phase between oscillations of generators G2 and G1 is optimal and is calculated in of the previous part of this work. Check this result, and see the shape and values of the current (red graph), it is possible in calculator. Note that if the modulation of the inductance is removed altogether (turn off the generator G2), then the power in the load is about 30% will grow, and the efficiency of the circuit will decrease by 7 times, which can sometimes confuse the researcher.

How to increase the output power? It depends on the square of the oscillation amplitude of the master oscillator. If it is increased from ten, for example, to one hundred volts, then the power at the load will increase 100 times and become equal to 4.4 watts. But in this case, in a real circuit, it is likely that the inductance control will also become more complicated, and the power of G2 will increase in proportion to the square of the voltage.

This work shows the possibility of obtaining parametric resonance in an RL circuit of the second kind, in which the inductance parameter changes with sinusoidal oscillations. Since the modulation coefficient m is positive, it is obvious that that in real ferromagnets the increasing section of the Stoletov curve should be used. The coefficient itself is considered in the range from zero to 0.5: \(m \le 0.5 \), which gives an acceptable accuracy for coincidence with real results.

At its core, the end result of this work is reflected in Figure 17, which presents the over-unity criteria: with known power of the modulating generator P₂ and modulation factor m, all values to the left of the corresponding curve, are superunit. Of course, this assumes that the phase angle and quality factor are optimal. At the same time, the optimal phase shift can be found from Figures 12-15, and the optimal figure of merit - according to the formula (4.8) or - according to the graph 16.

Several versions of this technique have been developed, complementing each other. However, there are some approximations in them that had to be done to get a qualitative and quantitative picture. Also, there are some clarifications, which we will discuss below.

The method does not take into account the current, large values of which can, for example, greatly heat the coil. The diameter of the coil wire depends on this value, and hence its design. Also, here the efficiency of losses in the circuit is not taken into account, which can consist of heating the coil, its internal capacity, heating the output part of the generators. However, these losses, in principle, can be summed up with P₂.

The active resistance of the coil is actually the resistance of the wire at a certain frequency. Due to the skin effect, the same wire can have different resistances at different frequencies. In this work, we consider \(r \) as the resistance of a wire at an operating frequency that we initially do not know. A check is required: how much this resistance changes at the frequency found as a result of the calculation. If more than 20%, then a recalculation should be made taking into account the new data.

Refinement of this methodology will be made as real results are obtained or alternative approaches to PR of the second kind are obtained.