QUALITATIVE ANALYSIS OF PARAMETRIC OSCILLATIONS IN ODD-ORDER INSTABILITY REGIONS

R. A. Vainshtein, N. V. Kolomiets

(Presented by the scientific seminar of the Department of Electric Power Stations)

It is known that in nonlinear systems with a periodically varying energy-storage parameter, a wide variety of oscillatory processes may occur. One of the most important properties of such systems is that, under certain conditions, oscillations of various frequencies arise in them at an unchanged frequency of the exciting force. Some regimes, for example frequency division by two and frequency repetition, have been studied rather well [1—3] and have found broad practical application. Regimes in which oscillations of higher frequencies arise have been studied less, although they too may have useful applications.

In the present article, oscillations at odd harmonics are considered in the circuit of an inductive parametron without premagnetization by direct current, shown in Fig. 1. The inductive parametron consists

Fig. 1. Schematic circuit of an inductive parametron.

of two identical transformers on closed ferromagnetic cores (A and B). The windings with number of turns \(w_1\), called excitation windings, and the windings with number of turns \(w_2\)—the tank windings—are connected so that there is no electromagnetic coupling between them. When the excitation winding is supplied with a sinusoidal voltage of frequency \(\omega\), parametric oscillations with frequencies that are multiples of \(\omega\) may arise in the oscillatory circuit formed by the windings \(w_2\) and the capacitance \(C\).

Parametric oscillations with a frequency equal to \(\omega\) are considered in [2, 4], etc. In [2], theoretical and experimental investigations of these oscillations were carried out in sufficient detail. On paramet—

... in parametric oscillations with frequency greater than \(\omega\), only some experimental data are known. Below, certain relations are obtained for the analysis of parametric oscillations with frequencies \(n\omega\), where \(n\) is an odd integer.

We specify the induction in cores A and B in the steady state in the form

\[ \begin{aligned} b_{\mathrm A}&=B_1\sin\omega t+B_n\sin(n\omega t+\psi);\\ b_{\mathrm Б}&=B_1\sin(\omega t+\varphi)-B_n\sin(n\omega t+\psi); \end{aligned} \tag{1} \]

where \(B_1\) is the induction amplitude of the first harmonic;
\(B_n\) is the induction amplitude of the \(n\)-th harmonic.

To compose the differential equation of the oscillatory circuit, we write the system of equations for the circuit of Fig. 1:

\[ \begin{aligned} i_2+i_c+i_g&=0;\\ i_1w_1+i_2w_2&=h_{\mathrm A}l_{\mathrm{cp}};\\ i_1w_1-i_2w_2&=h_{\mathrm Б}l_{\mathrm{cp}};\\ i_c&=C\frac{dU_2}{dt};\\ i_g&=gU_2;\\ U_2&=qw_2\frac{d(b_{\mathrm A}-b_{\mathrm Б})}{dt}; \end{aligned} \tag{2} \]

where \(l_{\mathrm{cp}}\) is the length of the mean magnetic line of the core;
\(q\) is the cross-sectional area of the core;
\(h_{\mathrm A}\) and \(h_{\mathrm Б}\) are the magnetic-field strengths in the cores. The remaining notation is clear from the figure.

We approximate the magnetization curve by the expression

\[ h=\alpha \operatorname{sh}\beta b, \tag{3} \]

where \(\alpha\) and \(\beta\) are approximation coefficients.

Substituting for \(h_{\mathrm A}\) and \(h_{\mathrm Б}\), respectively, \(\alpha\operatorname{sh}\beta b_{\mathrm A}\) and \(\alpha\operatorname{sh}\beta b_{\mathrm Б}\), and using (2), we obtain the equation

\[ \frac{d^2(b_{\mathrm A}-b_{\mathrm Б})}{dt^2} +g\frac{1}{C}\frac{d(b_{\mathrm A}-b_{\mathrm Б})}{dt} +\frac{l_{\mathrm{cp}}\alpha}{2qw_2^2C} \left(\operatorname{sh}\beta b_{\mathrm A}-\operatorname{sh}\beta b_{\mathrm Б}\right)=0. \tag{4} \]

In equation (4) we substitute (1) and expand the hyperbolic sine of a periodic argument into a Fourier series, whose coefficients, as is known, are Bessel functions of a purely imaginary argument.

Neglecting terms containing Bessel functions above the \(n\)-th order, and after separating in equation (4) the cosine and sine components, we obtain

for the first harmonic

\[ \left[-jI_1(j\beta B_n)\right]I_{n-1}(j\beta B_1) \cos(n-1)\frac{\varphi}{2} \cos\left(\psi-\frac{n\varphi}{2}\right) =-kB_1\sin\frac{\varphi}{2}g, \tag{5} \]

\[ \left[-jI_1(j\beta B_n)\right]I_{n-1}(j\beta B_1) \cos(n-1)\frac{\varphi}{2} \sin\left(\psi-\frac{n\varphi}{2}\right)- \]

\[ -\left[-j I_1(j\beta B_1)\right] I_0(j\beta B_n)\sin\frac{\varphi}{2} = -kB_1\sin\frac{\varphi}{2}\omega C; \tag{6} \]

for the \(n\)-th harmonic

\[ j I_n(j\beta B_1)\left[I_0(j\beta B_n)+I_2(j\beta B_n)\right]\sin\frac{n\varphi}{2}\cos\left(\psi-\frac{n\varphi}{2}\right) = -nkB_n g, \tag{7} \]

\[ j I_n(j\beta B_1)\left[I_0(j\beta B_n)-I_2(j\beta B_n)\right]\sin\frac{n\varphi}{2}\sin\left(\psi-\frac{n\varphi}{2}\right) + \]

\[ +I_0(j\beta B_1)\left[-j I_1(j\beta B_n)\right] = nkB_n n\omega C; \tag{8} \]

where

\[ K=\frac{\omega w_2^2 q}{a l_{\mathrm{cp}}}; \]

\(I\) is the Bessel function.

We consider the operation of the inductive parametron in the no-load mode; active losses are neglected, i.e., \(g=0\). In this case, from equations (5) and (7) it follows that

\[ \psi-\frac{n\varphi}{2}=\frac{\pi}{2}. \tag{9} \]

To determine the amplitude of the \(n\)-th harmonic, we transform expression (8), using the well-known relation between Bessel functions

\[ I_{\nu-1}(x)-I_{\nu+1}(x)=\frac{2\nu}{x}I_\nu(x). \]

Then we expand the Bessel function of the first order in a power series and retain the first term, containing the argument to the first power. After these transformations, taking (9) into account, we obtain

\[ B_n= \frac{2j I_n(j\beta B_1)} {2n^2 k\omega C-\beta I_0(j\beta B_1)} \sin\frac{n\varphi}{2}. \tag{10} \]

The coefficient preceding \(\sin(n\varphi/2)\) represents the maximum possible value of the amplitude of the \(n\)-th harmonic.

As will be shown below, expression (10) well reflects the character of the dependence of the amplitude \(B_n\) on the angle \(\varphi\). However, calculation of the absolute values of the amplitudes from this expression gives values that differ greatly from the actual ones because of the assumptions made. Therefore, to determine the maximum amplitude of the \(n\)-th harmonic in the first approximation, we shall assume that the current \(i_1\) in the excitation circuit at the boundary of the onset of oscillations is sinusoidal.

We take the induction in the core to be

\[ b=B_1\sin\omega t+B_n\sin n\omega t. \tag{11} \]

Since the cores are under identical conditions, because oscillations in the circuit do not yet exist, the inductions of the first and \(n\)-th harmonics in both cores are equal in magnitude and coincide in phase. As soon as oscillations arise, a phase shift will occur between the first harmonics of the inductions in the cores by an angle \(\varphi\), the limiting value of which, as follows from (10), is equal to \(\pi/n\). The inductions of the \(n\)-th har-

harmonics at \(\varphi=-\dfrac{\pi}{n}\) will be shifted in phase relative to each other by the angle \(\pi\). The magnitude of the voltage of the \(n\)-th harmonic in the oscillatory circuit will then assume its maximum possible value, which will be determined by twice the value of the amplitude of the induction of the \(n\)-th harmonic at the boundary of excitation of parametric oscillations.

To determine \(B_{n\max}\) in (3), substitute (11). After simple transformations and expansion of the hyperbolic sine in a Fourier series, from the equation \(H_n=0\) we obtain an expression for the maximum value of the amplitude of the \(n\)-th harmonic

\[ B_{n\max}=\frac{2j I_n(j\beta B_1)}{\beta I_0(j\beta B_1)} . \tag{12} \]

Substituting expression (12) into (10), we finally obtain

\[ B_n=\frac{2j I_n(j\beta B_1)}{\beta I_0(j\beta B_1)}\sin\frac{n\varphi}{2}. \tag{13} \]

The possibility of determining \(B_{n\max}\) by such an approximate method is explained by the following factors:

1) the amplitudes of the inductions of the first harmonic are equal and, in practice, remain constant in magnitude at the boundary of excitation and throughout the entire region of existence of parametric oscillations;

2) the amplitude of the induction of the \(n\)-th harmonic is much smaller than the amplitude of the induction of the first harmonic.

To determine the amplitude of the first harmonic for each value of \(n\), let us consider the stability of the zero solution of equation (4). We write the differential equation in variations. The induction in the cores before the onset of oscillations will be taken as

\[ b=B_m\sin\omega t . \tag{14} \]

Let a differentially small current \(\Delta i\) appear in the oscillatory circuit; then the inductions in cores A and B will be equal to

\[ b_A=b+\Delta b; \]

\[ b_B=b-\Delta b; \tag{15} \]

where \(\Delta b\) are small variations of the induction relative to regime (14). The differential equation of the oscillatory circuit, neglecting losses and using expressions (3) and (15), takes the form

\[ \frac{d^2\Delta b}{d\tau^2}+\frac{\alpha\beta l_{\mathrm{cp}}}{2q\omega_2^2\omega^2 C}\,ch\,\beta b\,\Delta b=0, \tag{16} \]

where \(\tau=\omega t\).

After substituting (14) into (16) and expanding the function \(ch\,\beta B_m\sin\omega t\) in a Fourier series, we obtain Hill’s equation

\[ \frac{d^2\Delta b}{d\tau^2}+\left(\Omega_0+2\sum_{k=1}^{\infty}\Omega_k\cos 2k\omega t\right)\Delta b=0, \tag{17} \]

where \(\Omega_0\) and \(2\Omega_k\) are the constant component and the \(k\)-th harmonic of the trigonometric polynomial in equation (17).

Next we need to know only the expression for

\[ \Omega_0=\frac{\alpha \beta l_{\mathrm{av}}}{2q\omega_{2\omega}^{2}C} I_0(j\beta B_1), \tag{18} \]

since we restrict ourselves to the zeroth approximation of the stability condition

\[ \Omega_0=n^2. \tag{19} \]

Substituting (18) into (19), we have

\[ \frac{\alpha \beta l_{\mathrm{av}}}{2q\omega_{2\omega}^{2}C} I_0(j\beta B_1)=n^2. \tag{20} \]

From (20) one can determine graphically the amplitude of the first harmonic at the boundary of the onset of oscillations with frequency \(n\omega\).

Fig. 2. Vector diagram of voltages for the first harmonic.

Fig. 3. Calculated (a) and experimental (b) dependences of the voltages on the oscillatory circuit on the input voltage: —— voltage on the oscillatory circuit with frequency \(\omega\); — — — the same with frequency \(n\omega\).

The relation between the input voltage \(U_1\) and the voltages on the oscillatory circuit \(U_2\) with frequencies \(\omega\) and \(n\omega\) is found, using (13), (20), and also on the basis of the vector diagram of voltages for the first harmonic shown in Fig. 2, by varying \(\varphi\)

\[ U_1=2q\omega w_1B_1\cos\varphi/2; \tag{21} \]

\[ U_{2(1)}=2q\omega w_2B_1\sin\varphi/2; \tag{22} \]

\[ U_{2(n)}=2qn\omega w_2B_n \sin \frac{n\varphi}{2}. \tag{23} \]

Figures 3a and b present calculated and experimental data for a circuit with the following parameters: \(q_{\mathrm{st}}=5\cdot 10^{-4}\ M^{2}\);

\[ l_{\mathrm{av}}=0.306\ \mathrm{m};\quad w_{1}=570;\quad w_{2}=725;\quad C=0.5\ \mu\mathrm{F};\quad \beta=4.8;\quad \alpha=2.06. \]

The calculations and experiments were carried out for parametric oscillations with frequencies \(3\omega\), \(5\omega\), \(7\omega\).

Conclusions

1. The relations obtained give a picture qualitatively similar to that of the actual processes.

2. The analysis carried out made it possible to reveal the most essential aspect of the oscillations under consideration, namely that the maximum possible phase shift between the first harmonics of the inductions in the cores is limited and is related to the order of the oscillations as \(\dfrac{\pi}{n}\).

References

1. A. M. Bamdas, S. V. Shapiro, L. N. Davydova. Ferromagnetic frequency dividers. “Energiya,” 1967.

2. L. A. Bessonov. Nonlinear electrical circuits. “Vysshaya shkola,” 1964.

3. I. N. Lisitskaya, L. A. Sinitskii, Yu. M. Shumkov. Analysis of electrical circuits with magnetic and semiconductor elements. Kiev, “Naukova dumka,” 1969.

4. T. Hayashi. Nonlinear oscillations in physical systems. “Mir,” 1968.