2021-09-26

Parametric resonance of the second kind in an RL circuit

Optimal phase shift

Optimal phase shift

In part two, we graphically examined the mathematical model developed in part one of this work and received the first results in the form of graphs, from which some conditions follow for creating a parametric resonance in an RL circuit of the second kind. The disadvantage of graphical research is the two-dimensional representation of the process, although in reality, much more simultaneously changing parameters are involved in it. Therefore, in this part we will take a slightly different approach and first find the optimal angles between the oscillations of the modulating and master oscillators. They will be very important when designing real free energy devices. And we will apply the formulas obtained here in the next section to obtain superunit criteria.

From the previous part, we already know that the Q factors with which we will deal exceed the number eight (\(Q \ge 8 \)), and if so, then we can simplify the specific power integrals, while maintaining an acceptable accuracy of calculations. Also, for the convenience of presenting the results in further calculations, we replace the sine functions with cosine functions in formulas (2.1) and (2.2), after which we obtain:

\[\bar P_1 (\alpha, Q) \approx \int \limits_0^1 {\cos (\omega t) \cos (\omega t - \varphi) \over Q (\alpha, t)} \, \partial t = \frac {1} {Q} \int \limits_0 ^ 1 {\cos (\omega t) \cos (\omega t - \varphi) \over 1 + m C_s} \, \partial t \qquad (3.1) \] \[C_s = | \cos (n \omega t / 2 + \alpha / 2) | \] The angle \(\alpha \) is divided by two here because of the doubled, in fact, the frequency value obtained after taking the modulus from the cosine. Consider the integrand without the dividend and expand it in the Maclaurin series, up to the third term [1]: \[{1 \over 1 + m C_s} \approx 1 - m C_s + (m C_s)^2 \qquad (3.2)\] At the same time, the modulation coefficient should be less than one, and for acceptable accuracy in our further calculations it will be enough for \(m \le 0.5 \). This approach will give us the opportunity to get away from the fraction, which means that further it will be possible to work with the usual product of cosines. But for this it is necessary to do one more step and expand the cosine modulus (\(C_s \)) in a Fourier series [2-5]: \[C_s \approx \frac {4} {\pi} \left ({1 \over 2} + {\cos (n \omega t + \alpha) \over 3} - {\cos (2n \omega t + 2 \alpha) \over 15} \right) \qquad (3.3) \] Here we will also take the first three terms and this will be quite enough, as we can see further. The accuracy of this representation is shown in Figure 11, where f1 (t) is the cosine without modulus, f2 (t) is the modulus of the cosine, and f3 (t) is an approximate expansion of f2 in a Fourier series up to the third term.

Fig.11. For n = 2: cosine function - f1 (t) [green graph], cosine modulus function - f2 (t) [blue graph], approximate expansion of f2 in the Fourier series up to the third term - f3 (t) [red graph] |

To obtain the products of cosines in the integral, the antiderivatives of which can be easily obtained, it is necessary to take one more step and expand the product in the divisor of the integrand (3.1): \[\cos (\omega t) \cos (\omega t - \varphi) = \frac12 [\cos (2 \omega t - \varphi) + \cos (\varphi)] \qquad (3.4) \] Then the final integral will be like this: \[\bar P_1 (\alpha, Q) \approx \frac {1} {2Q} \int \limits_0 ^ 1 [\cos (\varphi) + \cos (2 \omega t - \varphi)] \, [1 - m C_s + (m C_s) ^ 2] \, \partial t \qquad (3.5) \] where \(C_s \) is found by formula (3.3). Despite the apparent complexity of this integral, in terms of the large number of terms in its integrand, most of them will be equal to zero.

Optimal phase shift at n=2

As in the second part of this work, here we first find a solution for the ratio of the frequencies of the modulating and master oscillators n = 2, and only then - for n = 1. For this, we will leave in the integrand (3.5) only those terms that, after integration, will differ from zero. And this is possible only for \(\cos (\varphi) \) multiplied by constant terms, or for the product of cosines, in which the variable argument is \(2 \omega t \): \[\bar P_1(\alpha, Q) \approx \frac{\cos(\varphi)}{2Q} \left[1 - {2m \over \pi} + 0.5 m^2 \right] - \frac{2 m}{3 Q \pi} \int \limits_0^1 \cos(2\omega t - \varphi) \cos(2\omega t + \alpha) [1 - 1.15 m + 1.4 m^3] \, \partial t \qquad (3.6)\] Taking this integral, and making some transformations, we get: \[\bar P_1 (\alpha, Q) = A (Q) - B (\alpha, Q) \qquad (3.7) \] where \[A(Q) = \frac{\cos(\varphi)}{2Q} \left[1 - {2m \over \pi} + 0.5 m^2 \right], \quad B(\alpha, Q) = \frac{m \cos(\varphi + \alpha)}{3 \pi Q} [1 - 1.15 m + 1.4 m^3]\] In this formula, A is responsible for the power \(\bar P_1 \) without taking into account the parametric and almost does not depend on m, and B - for the parametric addition to it.

Now,

**the proof for the optimal angle**between the oscillation of the master oscillator voltage (G1) and the oscillation of the modulating oscillator (G2) looks very simple. It is clear that for greater efficiency, the power of the generator G1 (\(P_1 \)) should be as low as possible, and this is achievable when \(B (\alpha, Q) \) will be as large as possible. This will happen if \(\cos (\varphi + \alpha) \) is as large as possible. \[\cos (\varphi + \alpha) = \cos (\varphi) \cos (\alpha) - \sin (\varphi) \sin (\alpha) \approx {\cos (\alpha) \over Q} - \sin (\alpha) \qquad (3.8) \] Let's remind that: \[\varphi = \arctan (Q), \quad \cos (\varphi) \approx {1 \over Q}, \quad \sin (\varphi) \approx 1\] Let's find the optimal angle by searching for the extremum of the function: \[\alpha = - \arctan (Q) + k \pi \qquad (3.9) \] where: \(k\) are integers. Let's choose \(k = 0\) from the calculation of the smallest angle and negative arctangent, and plot the dependence of the optimal angle on Q on the graph. As we can see, for n=2, the optimal angle is close to "-90" degrees, which is what we had to prove. This confirms our conclusions from the graphical method presented in the second part of this work and the corresponding Figures 2-4. The calculator demonstrates this quite clearly. Compare how the efficiency changes for the same circuit parameters: for angle -90 degrees, and for angle -84 degrees. Based on this, we derive A and B from formula (3.7) for n = 2, but taking into account the optimal phase shift: \[A(Q) = \frac{1}{2 Q^2} \left[1 - {2m \over \pi} + 0.5 m^2 \right], \quad B(\alpha, Q) = \frac{m (1 + 1 / Q)}{3 \pi Q} [1 - 1.15 m + 1.4 m^3] \qquad (3.10)\] We will need these formulas to find the criteria for superunity in the fourth part of this work.

Optimal phase shift at n=1

In this case, we will do the same as for n=2, but first we define the Fourier series expansion for this coefficient: \[C_s \approx \frac {4} {\pi} \left ({1 \over 2} + {\cos (\omega t + \alpha) \over 3} - {\cos (2 \omega t + 2 \alpha) \over 15} \right) \qquad (3.11) \] after which we will use formulas (3.4-3.5). Then the general expression for n=1 will look like this: \[\bar P_1 (\alpha, Q) = A (Q) - B (\alpha, Q) \qquad (3.12) \] where \[A(Q) = \frac{\cos(\varphi)}{2Q} \left[1 - {2m \over \pi} + 0.5 m^2 \right], \quad B(\alpha, Q) = - \frac{m \cos(\varphi + 2\alpha)}{15 \pi Q} [1 - 0.05 m]\] It is immediately striking that this power is several times less than the same, but for n=2. The result can be compared with formula (3.7). With this, we confirmed the conclusion drawn by the graphical method in the previous part of this work and proved that such a frequency ratio is ineffective. Nevertheless, we find the optimal phase shift for n=1, which is derived in the same way as in the previous case: \[\alpha = - \frac12 (\arctan (Q) + k \pi) \qquad (3.13) \] Let's choose \(k = -1 \) from the calculation of the smallest angle and positive arctangent, and plot the dependence of the optimal angle on Q on the graph. As we can see, the optimal phase shift at n=1 is close to 45 degrees, and more accurately it can be determined from Figures 14 and 15. It should be noted that for n=1 the quality factor Q is very large and it is possible to determine this angle with sufficient accuracy as a constant equal to 45 degrees. In the calculator, it might look like so.

Based on this, we derive A and B from formula (3.12) for n=1, but taking into account the optimal phase shift: \[A(Q) = \frac{1}{2 Q^2} \left[1 - {2m \over \pi} + 0.5 m^2 \right], \quad B(\alpha, Q) = \frac{m}{15 \pi Q} \left[ 1 - 0.05 m \right] \qquad (3.14)\]

Is an increase possible for n>2

To answer this question, let us turn to formula (3.5) and try to get \(B (\alpha, Q) \), for example, for n=3. Then formula (3.3) will be written as follows: \[C_s \approx \frac {4} {\pi} \left ({1 \over 2} + {\cos (3 \omega t + \alpha) \over 3} - {\cos (6 \omega t + 2 \alpha) \over 15} \right) \qquad (3.15) \] For nonzero values of the antiderivative integral (3.6), we need the values of the cosine with a variable argument equal to \(2 \omega t \). As you can see, there are no such values here, as well as for other values of n>2. It means that: \[B (\alpha, Q) = 0, \quad n \gt 2 \qquad (3.16) \] In this case, there is no parametric component, and therefore the efficiency will always be the classic subunit. This rule can also be checked in calculator. The rule applies without exception, but the shape of the current must be close to sinusoidal. Non-sinusoidal oscillations contain multiple harmonics, which we do not take into account here.

In the next section, based on the formulas obtained here, we will find the criteria for the over-unity of the parametric RL circuit at resonance of the second kind and develop a method for preliminary calculation of the device based on it.

__The materials used__

- Wikipedia. Taylor and Maclaurin Series.
- Wikipedia. Fourier Series.
- Habr. Harmonic oscillations. [IMG]
- Bessonov L.A. Theoretical foundations of electrical engineering. Volume 1. Electrical circuits. Some properties of periodic curves with symmetry. [IMG]
- Tables DPVA.ru. Fourier series expansions of the main periodic functions (periodic pulses / signals). [IMG]