2021-09-20
Parametric resonance of the second kind in an RL circuit
Graphical research
Graphical research
In the first part of this work, we developed a mathematical model of parametric resonance (PR) for RL circuits of the second kind. In the second part, we examine the resulting formulas (1.10-1.14) using graphical methods that should clearly represent this unusual phenomenon. Also, it will become quite obvious to us why it is very easy to pass it by in real experiments.
To study this unusual PR by a graphical method, we will rewrite the formulas from the previous part in a relative (or specific) form, and as the limits of integration, we will choose only one period: \[\bar P_R(\alpha, Q) = \int \limits_0^1 {\sin(\omega t - \varphi)^2 \over 1 + Q(\alpha, t)^2}\, \partial t \qquad (2.1)\] \[\bar P_1(\alpha, Q) = \int \limits_0^1 {\sin(\omega t) \sin(\omega t - \varphi) \over \sqrt{1 + Q(\alpha, t)^2}}\, \partial t \qquad (2.2)\] It will be much more convenient to use the relative form of these integrals in further calculations, and to the absolute form, they are transformed as follows: \[P_R(\alpha, Q) = p_r\, \bar P_R(\alpha, Q), \quad p_r= {U_m^2 R \over R_r^2} \qquad (2.3)\] \[P_1(\alpha, Q) = p_1\, \bar P_1(\alpha, Q), \quad p_1= {U_m^2 \over R_r} \qquad (2.4)\] Here \(p_r\) and \(p_1\) are absolute powers without taking into account the influence of inductance, i.e. they represent the power values that would be in the circuit if the terminals of the L coil were closed. Also, remember that: \[\varphi = \arctan{Q}, \quad Q = {\omega L \over R_r} \qquad (2.5)\] \[Q(\alpha, t) = {\omega L(t) \over R_r} = Q\, [1 + m |\sin(\omega t n /2 + \alpha /2)|] \qquad (2.6)\] Let's designate the increase in efficiency as follows: \[K (\alpha, Q) = {P_R (\alpha, Q) \over P_1 (\alpha, Q) + P_2} \qquad (2.7) \] And for now, let's assume that \(P_2 = 0 \), i.e. no energy is consumed to change the inductance parameter. Later we will see how the picture changes when this power is not zero. In the meantime, let's build a graph using these formulas, and set the following initial parameters: \(n = 2, m = 0.2 \).
 Fig.2. The dependence of the increase in efficiency on the angle α and quality factor Q at: n=2, m=0.2, P2=0 |
This is the PR of the second kind for the RL-circuit in its pure form: taking into account its negative values and not taking into account the power consumption by the second generator (\(P_2 = 0\)). Negative values mean that as a result of the parametric process, more energy is returned to the source than this source spends. In this case, it is the active power of the first generator (\(P_1 \)). Moreover, after the maximum, it remains, albeit small, but still negative up to the highest values of Q.
Such presentation of the graph is not very convenient, and it would be the best option to take the module from the increase in efficiency. \[K (\alpha, Q) = \left| {P_R (\alpha, Q) \over P_1 (\alpha, Q) + P_2} \right| \qquad (2.8) \] and make this axis logarithmic. Now the same graph, more clearly, will look like this:
 Fig.3. Dependence of growth |efficiency| from the angle α and quality factor Q at: n=2, m=0.2, P2=0 |
Now let's look at this graph (Fig. 3) in more detail and compare its values with a calculator. The rather complex dependence of the efficiency maxima on the angle \(\alpha \) immediately catches the eye. In this case, the optimal angle is \(\alpha = - \pi / 2\), which is represented by the red curve in this graph. The lower the quality factor Q, the more - the current through the load, and hence the active power of the entire device. In addition, it is easier to make a coil with a lower Q factor than with a high one. Therefore, it is better to choose such angles at which Q will be less. Running a little ahead, in addition, from practice, we can say that after the maximum, the power consumption of the second generator begins to increase sharply, and the optimum operating point of the device is selected on the ascending part of the graph.
By the way, if for n=2 we make the angles \(\alpha\) positive, then the whole system will be subunit for any Q, so we do not consider these options.
Let's compare several values of this graph with a specialized calculator. For example, let's take the following values: \(Q = 17, n = 2, m = 0.2, \alpha = - \pi /2 \) (or "-90" degrees), check this point at calculator and on the red chart (point 1). Please note that the calculator takes a few seconds to calculate: wait for the task to complete. The calculator shows \(K_{\eta 2} = 2.17 \), which roughly corresponds to the graph shown.
The more periods are counted in the calculator, the more accurate the result will be, but the longer it will take to count. At the moment, it has a limitation on the number of periods, so the correct result can only be approached with some finite accuracy. For large m and Q, for acceptable accuracy, a large number of periods is required, 100 or more, which is not yet possible with the calculator.
Take one more point and accept the following values: \(Q=42, n=2, m=0.2, \alpha= -\pi/2\), check it in calculator and on the red chart (point 2). As we already know, after the maximum, the power P1 has negative values, which means that the efficiency tends to infinity, which is what the calculator shows.
Let's look at the graphs with a larger value of the modulation coefficient: m = 0.5 (Fig. 4).
 Fig.4. Dependence of growth |efficiency| from the angle α and quality factor Q at: n=2, m=0.5, P2=0 |
As we can see, all the maxima have shifted to the left edge of the graph, which means that with a larger value of the modulation coefficient m, a lower Q-factor of the Q coil can be used. It is undesirable to use even larger values of m, because the accuracy of our calculations will decrease, since the current shape will start strongly differ from sinusoidal.
And now let's estimate the relative power in the active load PR and the generator power G1 (P1), and look at their dependence on the phase shift and the quality factor (Fig. 5-6).
 Fig.5. Dependence of PR on the angle α and quality factor Q at: n=2, m=0.5, P2=0 |  Fig.6. Dependence of P1 on the angle α and quality factor Q at: n=2, m=0.5, P2=0 |
It is easy to guess that the power PR does not depend on the phase shift, but strongly depends on the quality factor, and the power P 1 depends both on the phase and on the quality factor, and where the efficiency graph (Fig. 4) is the maximum, there is a transition to negative values on the power graph (Fig. 6). Please also note that the working section begins with powers that are an order of magnitude or two less from the nominal values. Thus, in a real device, you do not need to chase the maximum output power, but you should select its optimal value, at which the power consumption will be minimal, and the increase in efficiency will be maximum.
It remains to look at the same dependences, but with a different frequency ratio: n=1. Here the effect is observed only with a positive phase shift (Fig. 7-8).
 Fig.7. Dependence of growth |efficiency| from the angle α and quality factor Q at: n=1, m=0.2, P2=0 |  Fig.8. Dependence of growth |efficiency| from the angle α and quality factor Q at: n=1, m=0.5, P2=0 |
With such a frequency ratio coefficient, a strong dependence on the phase shift is observed and only a small range of it turns out to be working. Also, in order to reach the working area, a relatively high Q-factor of the coil is needed here, several times higher than with n=2 (compare Figures 4 and 8). Based on all this, such a frequency ratio is not recommended for use.
But what if we take n more than two? As it turned out, with such a ratio of frequencies, there are no superunit values \(K_{\eta 2}\) in any part of the graph. This point is another difference from the classical PR, where resonance is possible at \(n> 2 \). You can verify this if you try to get \(K_{\eta 2} > 1\) in calculator. In any case, this pattern works as long as the current shape does not differ much from sinusoidal. If we take rectangular pulses as a modulating signal, then it becomes possible to obtain over-unity values, since the higher harmonics of these impulses will work here. We will consider this option in one of the following works.
Now let's see what the change in efficiency will be with a non-zero power consumption of the second generator, and consider two examples: \(P_2 = 2 \cdot 10^{-4} \) and \(P_2 = 5 \cdot 10^{-4} \). Recall that all powers here are relative.
 Fig.9. The dependence of the increase in efficiency on the angle α and quality factor Q at: n=2, m=0.2, P2= 2*10-4 |  Fig.10. The dependence of the increase in efficiency on the angle α and quality factor Q at: n=2, m=0.2, P2=5*10-4 |
Figures 9 and 10 clearly show that the increase in efficiency is very dependent on the power consumption of the second generator: with this power of 2*10-4 the maximum gain reaches 4.5, but with a power of 5*10-4 the maximum gain will be less than 1.3. It is clear that in the latter case, it is impossible to catch the desired effect in a real device, since we do not take into account the efficiency of all circuit elements, which neutralize the increase. On the other hand, if P2 is relatively small, then when it falls into the required Q-factor, we may well detect it. It is interesting that if this power turns out to be less than 2*10-4, then at a certain Q, the whole circuit can run wild, and its elements can fail. Also, you should pay attention to the shift angle between the voltage G1 and the current in the circuit: its best value, for n=2, remains equal to "-90" degrees, as noted earlier.
Conclusions
In this work, we got acquainted with the parametric resonance of the second kind, which arises in the parametric RL circuit. It differs from the classical PR in that there is almost no change in the amplitude at resonance, on the other hand, the active power consumption of the master oscillator is noticeably reduced, which leads to a sharp increase in the efficiency of the circuit. The second difference is that resonance of the second kind is possible when the ratio of the frequencies of the modulating and master oscillators is 1:1 and 2:1. If anything, this applies to a sinusoidal modulating signal.
The optimal shift angle between the oscillations of the modulating and master oscillator are:
- at a frequency ratio of 1:1, the optimal shift is 45 degrees;
- at a frequency ratio of 2:1, the optimal shift is "-90" degrees.
1:1 frequency ratio is not recommended for use in real circuits, since at the same time, in order to achieve the effect, several times higher quality factor or several times lower power consumption by the modulating generator is required, which may not be a feasible requirement.
The necessary effect, at which an acceptable increase in efficiency is achieved, is obtained in a certain range of the quality factor of the circuit and a shift angle of "-90" degrees, which can be seen from graph 9. In real circuits, this effect can be obtained by preliminary calculation of the Q-factor range and subsequent adjustment of the master oscillator frequency, which also changes the Q-factor according to formula (2.3). In this case, the phase shift between the oscillations of the modulating and master oscillator should be stable and be "-90" degrees, and the frequency ratio between these generators should be 2:1. When setting up the device, you do not need to chase the maximum output power, but you should select its optimal value, at which the power consumption will be minimal, and the increase in efficiency will become the maximum.
In the next section, we will improve the graphical method and find analytical values for the optimal angles.