FOUR-STABLE TWO-FREQUENCY PARAMETRON

R. A. Vainshtein, A. V. Shmotilov

(Presented by the scientific seminar of the Departments of Electric Power Stations and Electric Networks and Systems)

Bistable elements are widely used for performing logical operations in computing technology, communications, telemechanics, etc., owing to their high reliability and simplicity of design. The use of bistable elements for carrying out complex or large-scale logical operations requires circuits with a large number of such elements. The use of multistable devices for this purpose makes it possible to reduce the number of elements, while the reliability and simplicity of multistable elements should not be inferior to the corresponding qualities of bistable elements. In storage elements [6, 7], increasing the number of equilibrium states does not lead to complication of the circuit. However, their field of application is quite limited, since the equilibrium states are not stable. On the other hand, the creation of a stable multistable element leads to complication of its design.

At present, much attention is being given to multistable parametrons, for example [4, 5]. In them, stable states are characterized by different phases of oscillation. For a definite phase of oscillation of the pump frequency there correspond \(n\) possible stable phases of the \(n\)-th subharmonic, shifted relative to one another by

\[ \frac{2\pi}{n}. \]

However, for large values of \(n\), the excitation of oscillations encounters considerable difficulties; therefore, in practice, excitation of oscillations only at the second subharmonic is used.

This article describes a device that is called a four-stable two-frequency parametron. This parametron has 4 stable states, characterized not by different phases of subharmonic oscillations, but by two opposite phases of oscillations of the first parametric resonance (oscillations at half the frequency of parameter modulation) and by two opposite phases of oscillations of the second parametric resonance (oscillations at the frequency of parameter modulation). Excitation of oscillations of both the first and the second parametric resonances is produced by varying the magnitude of the factor that modulates the parameter. The reliability and simplicity of this device are not inferior to the reliability and simplicity of the bistable parametron, which has found wide application in technology. The circuit of the device under consideration is shown in Fig. 1.

It consists of two identical ferromagnetic cores, on which two windings each are wound: \(W_n\), the pump windings, and \(W_k\), the circuit windings. The windings \(W_n\) are connected in series opposition. An alternating current \(i\) is passed through them. The windings \(W_k\) are connected in series and, together with a capacitor \(C\) connected to their terminals, form an oscillatory circuit. Owing to this connection of the windings, when current is passed through the windings \(W_n\) the emf of transformation in the oscillatory circuit is equal to 0. The action of the current in the windings \(W_n\) on the oscillatory circuit consists in the fact that this current saturates the cores and thereby produces a periodic modulation of the inductance of the oscillatory circuit. When the amplitude of the current in the windings \(W_n\) is varied (within a certain range), ferroresonant oscillations arise in the oscillatory circuit [1].

Fig. 1

The electromagnetic processes in the oscillatory circuit formed by the periodically varying inductance \(L(t)\) of the windings \(W_k\), the capacitor \(C\), the active resistance of the wires \(W_k\), and the losses in the ferromagnetic material are satisfactorily described by Mathieu’s nonlinear equation. The unstable solution of this equation at the first subharmonic frequency of the inductance modulation describes the phenomenon of the first parametric resonance [2]. It is known [2] that, in a parametric system with a nonlinear element, oscillations can be obtained at the frequency of parameter modulation (the second parametric resonance) and oscillations of higher orders.

However, to obtain oscillations of the required order it is necessary each time to tune the system (to create the corresponding resonance conditions and introduce the necessary nonlinearity).

The system of Fig. 1 has favorable advantages in this respect. Modulation of the inductance is carried out by magnetizing ferromagnetic cores having a nonlinear dependence of magnetic permeability on the magnetizing current \(i_n\). The curve of the variation of inductance in time \(L(t)\) (Fig. 2) contains an infinite series of harmonics with frequencies that are multiples of the frequency of inductance modulation. Depending on the magnitude of the magnetizing current, either the first or the second harmonic component will predominate in the curve \(L(t)\). Accordingly, in the system of Fig. 1 the first necessary condition will be created for excitation of, respectively, the first or the second parametric resonance. The second necessary condition is the creation of the corresponding resonance conditions in the oscillatory circuit, as follows from the expressions

\[ \omega L_{\mathrm{cp}} = \frac{1}{\omega C}, \tag{1} \]

\[ 2\omega L_{\mathrm{cp}} = \frac{1}{2\omega C}, \tag{2} \]

where \(2\omega\) is the frequency of the first harmonic of the inductance modulation;

\(L_{\mathrm{av}}\) is the average inductance over the magnetization period at the first parametric resonance;

\(L'_{\mathrm{av}}\) is the average inductance over the magnetization period at the second parametric resonance;

\(C\) is the capacitance of the oscillatory circuit.

From (1) and (2) we obtain

\[ L_{\mathrm{av}}=\frac{1}{4}L'_{\mathrm{av}}. \tag{3} \]

The first and second conditions for obtaining oscillations of the first and second parametric resonances are entirely compatible in the system of Fig. 1, given an appropriate choice of its parameters. If the parameters are chosen so that the phenomenon of the first parametric resonance occurs at a slight saturation of the cores, when the curve \(L_{(t)}\) is only slightly distorted (that is, its first harmonic component predominates) and the average component \(L_{\mathrm{av}}\) satisfies expression (1), then with a further increase in the magnetizing current the curve \(L_{(t)}\) begins to be distorted in the direction of a decrease in the average component according to the dynamic inductance curve \(L_{(i)}\) and an increase in the even harmonic components, as shown in Fig. 2. When \(L_{(t)}\) is distorted to the degree at which the corresponding second harmonic component has been formed in the curve and the average component reaches the value \(L'_{\mathrm{av}}\), oscillations of the second parametric resonance arise in the oscillatory circuit, that is, oscillations at the frequency of parameter modulation.

Thus, with an appropriate choice of parameters in the system of Fig. 1, it is possible to obtain oscillations of the first and second parametric resonances by passing the corresponding magnetizing current through winding \(W_{\mathrm{m}}\).

It is known [3] that the phase of oscillations of a parametron operating in the regime of the first parametric resonance can be controlled by applying to the oscillatory circuit a small signal that predetermines the phase of the parametric oscillations.

Fig. 2

Since the phenomenon of the second parametric resonance is physically no different from the phenomenon of the first parametric resonance, the phase of the oscillations of the second resonance can likewise be controlled by applying the corresponding signal to the oscillatory circuit.

Since in the system of Fig. 1 one can obtain oscillations of both the first and the second parametric resonances, depending on the current \(i_n\), it is possible to control the phase of the oscillations of both resonances by applying simultaneously to the oscillatory circuit two memorized signals: at the half-frequency and at the frequency of the inductance modulation. However, in this case a difficulty arises: will the signals simultaneously applied to the circuit influence one another in controlling the phases of the oscillations of the parametric resonances?

In order to answer this question, let us carry out a qualitative analysis of this influence. Let two signals be applied to the oscillatory circuit,

\[ i_c = i_{c1} = i_{c2}, \]

where

\[ i_{c1} = I_{m1}\sin \omega t \]

is the memorized signal;

\[ i_{c2} = I_{m2}\sin(2\omega t+\varphi) \]

is a parasitic signal, which can affect the phase of the memorized signal;

\[ \omega \]

is the half-frequency of the inductance modulation;

\[ \varphi \]

is the initial phase.

We represent the parasitic signal as consisting of two orthogonal components,

\[ i_{c2}=I_{s2}\sin 2\omega t+I_{c02}\cos 2\omega t, \]

where

\[ I_{s2}=I_{m2}\cos \varphi,\qquad I_{c02}=I_{m2}\sin \varphi . \]

For the memorized signal, the conditions for parametric input of energy into the oscillatory circuit must be satisfied. That is, the resonance conditions in the oscillatory circuit at the frequency \(\omega\) must be satisfied, and, if the change in inductance is represented by a rectangular law [3], the decrease in inductance must coincide with the passage of the current \(i_{c1}\) through a maximum. Figure 3 shows the variation of inductance and of the signal currents.

Fig. 3

From examination of the curves we conclude: the parametric input of energy for the component \(i_s\) occurs as it passes through 0. Consequently, this component does not affect the phase locking of \(i_{c1}\). For the component \(i_{c02}\), the input and output of energy coincide

with the maximum value \(i_{c02}\). The total energy input over the period is equal to 0, and therefore \(i_{c02}\) likewise does not affect the phase locking of \(i_{c1}\).

Thus, a signal of frequency \(2\omega\), for any initial phase \(\varphi\), cannot predetermine an incorrect storage of the phase of a signal of frequency \(\omega\).

Now let the signal being stored vary according to the law

\[ i_{c2}=I_{m2}\sin 2\omega t, \]

and the spurious signal according to the law

\[ i_{c1}=I_{m1}\sin(\omega t+\varphi_1). \]

We transform

\[ i_{c1}=I_{s1}\sin\omega t+I_{c01}\cos\omega t, \]

where

\[ I_{s1}=I_{m1}\cos\varphi_1, \]

\[ I_{c01}=I_{m1}\sin\varphi_1, \]

\(\varphi_1\) is the initial phase.

In this case the conditions of parametric energy input for the signal of frequency \(2\omega\) must be satisfied. For consideration, let us replace the second harmonic component of the inductance \(L(t)\) by a periodic curve varying according to a rectangular law. Figure 4 shows the inductance and the signals as functions of time. From consideration of Fig. 4

Fig. 4

we note: the energy input for the spurious components of the signal \(i_{s1}\) and \(i_{c01}\) somewhat predominates over the energy extraction (the input is carried out twice per half-period of the frequency \(\omega\), at instantaneous signal values equal to \(\frac{\sqrt{2}}{2}\) of the maximum values of the corresponding signals, whereas the extraction occurs once per half-period at the maximum signal values). However, taking into account the damping of the circuit, the resonance tuning of the oscillatory circuit to the frequency \(2\omega\), and the commensurability of the signals of frequencies \(\omega\) and \(2\omega\), one may assert that no incorrect storage of the phase of the signal of frequency \(2\omega\) will occur due to the signal \(i_{1c}\).

Thus, the mutual influence of signals at any phases between them does not lead to incorrect storage of the phase of the oscilla-

… of both the first and the second parametric resonances. That is, when the signals to be stored are applied simultaneously to the oscillatory circuit, the signal at half the frequency of the inductance modulation controls the oscillations of the first parametric resonance, while the signal at the inductance-modulation frequency controls the oscillations of the second parametric resonance. Consequently, the system of Fig. 1 makes it possible to implement a four-stable two-frequency parametron, in which two phases of the oscillations of the first parametric resonance, differing by \(180^\circ\), and two phases of the oscillations of the second parametric resonance, differing by \(180^\circ\), are stored by applying to the oscillatory circuit two small signals carrying the necessary information.

At the Department of Electrical Stations of the Tomsk Polytechnic Institute, tests were carried out on devices according to the circuit of Fig. 1. The tests were performed for devices whose cores were made of electrical steel, at a current frequency in the \(W_n\) winding circuit of 50 Hz.

All tested specimens made it possible to select the necessary optimal capacitance so as to obtain voltages across the capacitance at the first and second parametric resonances of equal amplitude.

Fig. 5

A deviation of the capacitance downward from the optimum leads to a decrease in the amplitude and power of the oscillations of the first resonance. An increase in the capacitance relative to the optimum leads to: 1) an increase in the power consumed by the circuit of the \(W_n\) windings, 2) an increase in the amplitude and power of the oscillations of the first resonance, 3) a decrease in the amplitude and power of the oscillations of the second resonance.

Experiments were also carried out on controlling the phase of the oscillations of the first and second resonances. For this purpose, additional signal windings \(W_{c1}\) and \(W_{c2}\) were wound (Fig. 5). The terminals of the windings \(W_1\) are connected to a source of regulated current at a frequency of 50 Hz. A small signal at a frequency of 50 Hz, whose phase can be varied, is passed through the windings \(W_{c1}\). A small signal at a frequency of 100 Hz, whose phase can be varied, is likewise passed through the windings \(W_{c2}\).

The phase of the oscillations in the tank circuit of the parametron under test is monitored by means of an electronic oscilloscope synchronized by a voltage at the mains frequency of 50 Hz.

Experimental Results

1. The phase of the oscillations of the first parametric resonance is controlled both when a 50 Hz signal with the required phase is applied to the winding \(W_{c1}\), and when a 50 Hz signal with the required phase is applied simultaneously to the winding \(W_{c1}\) and a 100 Hz signal with various phases to the winding \(W_{c2}\).

2. Control of the phase of the oscillations of the second parametric resonance is effected both when a 100 Hz signal with the required phase is applied to the winding \(W_{c2}\), and when a 100 Hz signal with the same phase is applied simultaneously to the winding \(W_{c2}\) and a 50 Hz signal with various phases to the winding \(W_{c1}\). The signals required for controlling the phase of the oscillations of both the first and the second resonances are small in comparison with the magnitudes of the excited oscillations.

The ratio of the 50 Hz signal to the amplitude of the oscillations of the first resonance, measured at the terminals of the capacitor \(C\), lies within the limits

\[ \frac{1}{35} \div \frac{1}{45} \]

depending on the magnitude of the unbalance in the oscillatory circuit. The more identically the cores are made, the smaller this ratio.

The ratio of the 100 Hz signal to the amplitude of the oscillations of the second resonance, measured at the terminals of the capacitor \(C\), in the experiments was approximately

\[ \frac{1}{50}, \]

and in this case the unbalance in the oscillatory circuit has practically no effect on the change of the above-mentioned ratio.

Conclusions

Periodic oscillations in the system of Fig. 1, when the amplitude of the alternating current in the windings \(W_n\) is varied, are considered with the aid of the theory of parametric systems.

In accordance with this theory, the phenomenon of periodic oscillations in this system at half the frequency of the inductance modulation is explained as the phenomenon of the first parametric resonance, and the phenomenon of oscillations at the frequency of the inductance modulation—as the phenomenon of the second parametric resonance.

On the basis of these conclusions, the possibility is shown of controlling the phase of the oscillations of the first parametric resonance in the system of Fig. 1. For the first time, the possibility is shown of controlling the phase of the oscillations of the second parametric resonance.

On the basis of the theoretical conclusions and experiments, it is proposed to use the system of Fig. 1 as a parametric system for obtaining the first and second parametric resonances without changing the capacitance of the oscillatory circuit. The possibility of independent control of the phases of the oscillations of the first and second-

second parametric resonances when two signals to be stored are simultaneously applied to the oscillatory circuit.

As a result of all the foregoing, it is proposed to use the system of Fig. 1 as a four-stable two-frequency parametron storing 4 signals: 2 based on the phenomenon of storing the phases of oscillations of the first parametric resonance, differing by \(180^\circ\), and 2 based on the phenomenon of storing the phases of oscillations of the second parametric resonance, differing by \(180^\circ\).

The use of a four-stable two-frequency parametron in logic circuits of computer technology, communications, telemechanics, etc., will make it possible to reduce the number of logic elements.

The reliability and simplicity of the device considered are no worse than the reliability and simplicity of two-stable elements.

LITERATURE

1. L. A. Bessonov. Nonlinear Electrical Circuits. “Vysshaya shkola,” 1964.
2. I. D. Papaleksi. Collected Works. Publishing House of the Academy of Sciences of the USSR, 1948.
3. Parametrons. Collection of articles. Translated from Japanese and English, IL, 1962.
4. R. L. Wigington. A new principle in computing technology. Cybernetics Collection, issue 2, 1961.
5. S. A. Akhmanov, A. S. Roshal. Izv. vuzov. “Radiotekhnika,” No. 2, 1961.
6. Frimen. Elektrotekhnika, No. 24, 1962.
7. R. A. Lipman. A i T, No. 10, 1963.