LOW-FREQUENCY INDUCTIVE PARAMETRON WITHOUT BIASING BY DIRECT CURRENT*

R. A. VAINSTEIN, A. V. SHMOILOV

Recommended by the scientific seminar of the Department of Electric Stations

A high-frequency parametron with biasing of the excitation circuit by direct current, as a storage element of digital computers, has been used for 15 years already [1]. Here a low-frequency parametron without biasing is described, which is proposed for use in protection against ground faults in networks with a compensated neutral and in networks with an isolated neutral. Fig. 1 shows the basic circuit of such a parametron.

Fig. 1.

* The work was carried out under the supervision of Professor I. D. Kuteĭavin.

The parametron consists of two cores T1 and T2, made of electrical steel, on each of which there are two windings: \(W_{\mathrm{в}}\) (excitation windings) and \(W_{\mathrm{к}}\) (circuit windings).

The windings \(W_{\mathrm{в}}\) are connected to each other in series aiding, while the windings \(W_{\mathrm{к}}\) are connected in series opposition. The windings \(W_{\mathrm{к}}\), together with the capacitor \(C\), form an oscillatory circuit whose resonant frequency is close to \(50\ \text{cps}\). The load resistance \(Z_{\mathrm{н}}\) is connected to the terminals of the windings \(W_{\mathrm{к}}\). Owing to the indicated connection of the windings, when alternating current is passed through the windings \(W_{\mathrm{в}}\), the e.m.f. of mutual induction at the terminals of the windings \(W_{\mathrm{к}}\) is equal to 0.

When current passes through the windings \(W_{\mathrm{в}}\) during one period, the instantaneous magnetic permeability of the cores T1 and T2 changes in accordance with the curve of dynamic magnetic permeability.

Fig. 2.

Figure 2 shows the curve of the dynamic inductance \(L_{(i)}\) of the circuit windings as a function of current and the curve of current variation during one period. Since \(L_{(i)}\) is an even function, and \(i_{(t)}\) is \(2\pi\)-periodic in time, the function \(L_{(t)}\) (see Fig. 2) is a \(\pi\)-periodic function of time, i.e., the frequency of variation of the inductance is equal to twice the frequency of the excitation current.

As is known, for a certain value of the excitation current, and consequently for a certain degree of modulation of the inductance of the circuit windings, parametric oscillations arise in the oscillatory circuit at a frequency equal to half the frequency of variation of the inductance.

Fig. 3.

To formulate the differential equation of an oscillatory circuit with periodically varying inductance, we adopt the equivalent circuit shown in Fig. 3.

In this circuit \(L_{(t)}\) is the periodically varying inductance of the circuit windings.

\(C\) is the capacitance of the capacitor,

\(g\) is the active conductance of the circuit.

It replaces the load conductance \(Z_{\mathrm{n}}\) and takes into account the losses in the circuit.

By Kirchhoff’s first law,

\[ i_l+i_c+i_g=0. \tag{1} \]

By Kirchhoff’s second law,

\[ u_l=u_c=u_g=u=\frac{d\psi}{dt}, \]

where \(\psi\) is the flux linkage.

Expressing in equation (2) the currents \(i_l\), \(i_c\), and \(i_g\) through the flux linkage and differentiating it with respect to time, we obtain:

\[ \frac{d^2\psi}{dt^2}+\frac{g}{C}\cdot\frac{d\psi}{dt}+\frac{\psi}{L_{(t)}\cdot C}=0. \tag{2} \]

Equation (3), by means of the substitution \(\psi=u\cdot e^{-\frac{q}{2C}t}\), is reduced to the equation:

\[ \frac{d^2u}{dt^2}+\left(\frac{1}{L_{(t)}\cdot C}-\frac{g^2}{4C^2}\right)\cdot u=0. \tag{3} \]

The inductance \(L_{(t)}\) in equation (3) may be specified in the form of a Fourier series, where the angular frequency of the first harmonic is equal to \(2\omega\), where \(\omega\) is the angular frequency of the excitation current.

To obtain some qualitative relations from the solution of equation (3), we shall adopt:

1) a linear dependence of the amplitudes of the terms of the Fourier series on the current in the circuit;

2) we shall neglect all terms of the series except the first harmonic component and the constant term, taking the phase of the first term to be equal to 0, i.e.

\[ L_{(t)}=L_0+\Delta L\sin 2\omega t =L_0\left(1+\frac{\Delta L}{L_0}\sin 2\omega t\right) =L_0(1+m\sin 2\omega t), \]

where

\(L_0\) is the constant term of the series (see Fig. 2),

\(\Delta L\) is the amplitude of the first term of the series,

\(m\) is the relative degree of modulation of the inductance, or the modulation coefficient;

3) for convenience in reducing equation (3) to canonical form, we adopt the law of variation of the inductance

\[ L_{(t)}=\frac{L_0}{1+m\sin 2\omega t}, \tag{4} \]

which will not lead to a fundamental change in the solution of equation (3). Substituting (4) into (3) and denoting

\[ \left. \begin{gathered} \frac{q}{2C}=\delta,\quad \frac{1}{L_0C}=\omega_0^2,\\[4pt] \omega_0^2-\delta^2=\omega_{\mathrm{sv}}^2,\quad \frac{\omega_{\mathrm{sv}}^2}{\omega^2}=\beta,\quad \frac{\omega_0^2 m}{\omega^2}=\gamma,\quad \omega t=x, \end{gathered} \right\} \tag{5} \]

we obtain:

\[ \frac{d^2u}{dx^2}+(\beta+\gamma\sin 2x)\cdot u=0, \tag{6} \]

where

\(\delta\) is the damping of the circuit,

\(\omega_0\) is the resonant frequency of the circuit in the absence of modulation,

\(\omega_{\mathrm{sv}}\) is the natural frequency of the circuit in the absence of modulation.

The remaining coefficients have no designation.

Since \(\omega_{\mathrm{sv}}=\omega\), then \(\beta=1\).

Then equation (6) becomes

\[ \frac{d^2u}{dx^2}+(1+\gamma\sin 2x)=0. \tag{7} \]

Expression (7) is the Mathieu equation.

The solution of this equation, according to [2], is

\[ u=Ae^{\mu x}\cdot R_1(x)+Be^{-\mu x}\cdot R_2(x), \tag{8} \]

where \(R_1(x)\) and \(R_2(x)\) are \(\pi\)- or \(2\pi\)-periodic functions,

\(\mu\) is an exponent depending on the coefficient \(\gamma\).

From the theory of Mathieu equations [2] it is known that the functions \(R_1(x)\) and \(R_2(x)\) contain series of periodic functions whose arguments are equal to \(\kappa x\), where \(\kappa=1,2,3\ldots\).

According to the method of slowly varying amplitudes, the periodic functions \(R_1(x)\) and \(R_2(x)\) are series of sinusoidal or cosinusoidal functions of \(\kappa x\).

For small \(m\) in the circuit there exist conditions close to resonance at the frequency \(\omega\). Therefore it may be assumed that the higher harmonics are, for the most part, suppressed.

In connection with the above, the solution of the Mathieu equation in the present case may be sought in the form:

\[ u=Ae^{\mu x}\cdot \sin (x+\varphi_1)+Be^{-\mu x}\cdot \sin (x+\varphi_2). \tag{9} \]

The functions \(\varphi_1\) and \(\varphi_2\) in expression (9) are slowly varying in time [2]. In the first approximation they may be regarded as constant.

Substituting the solutions (9) successively into equation (7) and neglecting terms of the second order of smallness and terms with arguments \(3x\), we obtain 4 equations for determining \(\mu\), \(\varphi_1\), and \(\varphi_2\).

\[ \left(\frac{\gamma}{2}-2\mu\right)\sin\varphi_1=0,\qquad \left(\frac{\gamma}{2}-2\mu\right)\cos\varphi_1=0,\qquad \left(\frac{\gamma}{2}+2\mu\right)\sin\varphi_2=0, \]

\[ \left(\frac{\gamma}{2}-2\mu\right)\cos\varphi_2=0. \tag{10} \]

The system (16) has the following solutions:

\[ \mu=\frac{\gamma}{2};\qquad \left. \begin{aligned} &\varphi_1=\frac{\pi}{2},\quad \varphi_2=0\ \text{or}\\ &\varphi_1=-\frac{\pi}{2},\quad \varphi_2=\pi; \end{aligned} \right\} \tag{11} \]

\[ \mu=-\frac{\gamma}{2};\qquad \left. \begin{aligned} &\varphi_1=0,\quad \varphi_2=\frac{\pi}{2}\ \text{or}\\ &\varphi_1=\pi,\quad \varphi_2=-\frac{\pi}{2}. \end{aligned} \right\} \tag{12} \]

To the solutions (17) and (18) there correspond the following solutions of the original differential equation (3)

\[ \psi=A\cdot e^{\left(\frac{\gamma}{2}\omega-\delta\right)t}\cdot \cos\omega t + B\cdot e^{-\left(\frac{\gamma}{2}\omega+\delta\right)t}\cdot \sin\omega t, \]

\[ \psi=-Ae^{\left(\frac{\gamma}{2}\omega-\delta\right)t}\cos\omega t - Be^{\left(\frac{\gamma}{2}\omega+\delta\right)t}\cdot \sin\omega t, \tag{11'} \]

\[ \psi=Ae^{-\left(\frac{\gamma}{2}\omega+\delta\right)t}\cdot \sin\omega t + Be^{\left(\frac{\gamma}{2}\omega-\delta\right)t}\cdot \cos\omega t, \]

\[ \psi=-Ae^{-\left(\frac{\gamma}{2}\omega+\delta\right)t}\cdot \sin\omega t - Be^{\left(\frac{\gamma}{2}\omega-\delta\right)t}\cdot \cos\omega t. \tag{12'} \]

In the solutions (11′) and (12′), the values of the constants \(A\) and \(B\) depend on random initial conditions. At the moment when the parametron is switched on, disordered fluctuation currents flow in the oscillatory circuit, creating a random initial flux linkage. Since the initial flux linkage is not determined, the constants \(A\) and \(B\) also remain undetermined.

The nature of the dependences (11′) and (12′) indicates that, for certain constants, they essentially reduce to one of them.

Analyzing the solutions (11′) and (12′), it should be noted:

1. The cosine components of the initial flux linkage increase, while the sine components decay.

2. The increase may occur in two opposite phases, depending on the sign of the cosine component of the random initial flux linkage.

3. Increase is possible under the condition

\[ \frac{\gamma}{2} - \omega \ge \delta \quad \text{or} \quad m \ge g L_0 \omega . \]

Since the phase of the oscillations is determined by the sign of the cosine component of the fluctuation flux linkage, in order to predetermine the phase of the parametron oscillations it is necessary to feed into the circuit an initial signal whose flux linkage exceeds in magnitude the largest possible cosine component of the random initial flux linkage, and whose phase coincides with one of the opposite phase values of the cosine component.

This property of the parametron under consideration can be used to fix one of the directions of the zero-sequence current in a single-phase ground fault in networks with compensated neutral and in networks with isolated neutral. The distribution of the total zero-sequence current in networks with isolated neutral and of the active component of the zero-sequence current in networks with compensated neutral during a ground fault makes it possible to determine unambiguously the damaged line. These currents can be used to create the initial signal in the oscillatory circuit of the parametron.

The increase of the cosine components in expressions (11′) and (12′) proceeds to infinity. This conclusion is valid for the linear differential equation (2), whose solution we have considered. In reality, the inductance entering into equation (2) is a nonlinear function of the excitation current and of the current in the oscillatory circuit, and, consequently, equation (2) is nonlinear.

Experiments carried out with the system shown in Fig. 1 indicate the establishment of the amplitude of the oscillations. The establishment of a definite oscillation amplitude can be explained, in particular, as follows. As the oscillations increase, the modulation depth decreases, which is due to nonlinearity. The increasing oscillations increase the losses in the circuit with steel.

REFERENCES

1. Parametrons. Collection of articles. Translated from Japanese and English. IL, Moscow, 1962.

2. A. Ango. Mathematics for Electrical and Radio Engineers. “Nauka,” Moscow, 1964.