On Some Mode Relations for a Low-Frequency Parametron Without DC Bias

A. I. Kutyavin

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

In the works of T. Hayashi and L. A. Bessonov [1, 2], the possibility is shown of obtaining parametric oscillations with the frequency of the principal power-supply source in the circuit shown in Fig. 1.

Fig. 1.

As follows from the circuit, the primary windings \(W_1\) are connected in series and aiding, while the tank windings \(W_2\) are connected in series and opposing. The secondary windings, together with the capacitance, form a resonant circuit tuned to the input frequency. A resistance \(R_H\), playing the role of the load, is included in the circuit.

The main difference between this parametron and the one widely used in computer-engineering and automation devices is the absence of a dc component in the magnetizing currents, which leads to excitation of the device at the fundamental frequency.

The simplicity of the circuit of a low-frequency inductive parametron without biasing possibly makes its use in automation and telemechanics promising.

In the present work an attempt is made to establish the basic relations between the effective values of the electrical quantities on the basis of an experimental investigation of a parametron having chokes with a transformation coefficient equal to unity.

The closer the transformation coefficient is to unity, the more noticeable is the difference in the function performed by each choke separately. As a rule, one of them may perform the function of a transformer, the other that of a double-supply choke. Such a distribution is accidental and depends on the phase of the supply voltage at the moment when oscillations are excited. The capacitance value also substantially affects the distribution of voltages between the chokes.

We investigated a parametron operating at a frequency of 50 cps. The basic core data were as follows: material É-42; lamination—Sh-32; stack—5.3 cm; area of active steel—14.4 cm²; length of the mean magnetic line—21.0 cm; number of turns \(W_1 = W_2 = 300\); total active resistance of the primary windings 11 ohms, of the secondary windings—14 ohms.

Let us construct a vector diagram for the case of excitation of the parametron with a capacitance of 7 μF from the experimental data given in Table 1, \(R_\mathrm{n}=0\) (Fig. 2).

Table 1

The voltage across the capacitance lags by an angle of \(90^\circ\) behind the secondary current and forms, with the remaining voltage vectors of the secondary circuit, a triangle which can be constructed by the method of intersections,

\[ \dot U_c = \dot U_{2\mathrm{t}} + \dot U_{2\mathrm{d}}, \tag{1} \]

where \(U_{2\mathrm{t}}, U_{2\mathrm{d}}\) are the voltages on the transformer and the double-supply choke, respectively.

We conventionally take the first choke to be the transformer, and the second to be the double-supply choke.

Taking into account that the active resistance of the secondary windings is small, we have \(E_{2\mathrm{t}} = U_{2\mathrm{t}}\).

The reactive component of the no-load current \(I_{\mathrm{ot}} \simeq I_{\mathrm{xt}}\) leads \(E_{2\mathrm{t}}\) by an angle of \(90^\circ\) and coincides in direction with the magnetic flux \(\Phi_0\). Having determined the active component of the current \(I_{\mathrm{at}}\), for example, according to I. I. Belopolsky [3], we find the direction of the no-load current of the transformer and the primary current of the parametron.

\[ \dot I_{\mathrm{xt}}=\dot I_{\mathrm{ot}}+\dot I_{\mathrm{at}}, \tag{2} \]

\[ \dot I_1=\dot I_{\mathrm{xt}}-\dot I_{2\mathrm{t}}. \tag{3} \]

Figure 2

The no-load current of the choke \(I_{\mathrm{xl}}\) is equal to the difference between the primary and secondary currents (Fig. 2):

\[ \dot I_{\mathrm{xl}}=\dot I_1-\dot I_{2\mathrm{d}}. \tag{4} \]

The applied supply voltage must be balanced by the geometric sum of the primary voltages on the choke and the transformer:

\[ \dot U_{\mathrm{p}}=\dot U_{1\mathrm{t}}+\dot U_{1\mathrm{d}}. \tag{5} \]

In a similar way, vector diagrams can be constructed for any operating mode of the parametron under consideration.

The amplitude of oscillations under parametric resonance is limited by the nonlinearity of the characteristics of the circuit elements; therefore three boundary-value problems can be distinguished:

1. Determination of the conditional volt-ampere characteristic (V–A characteristic) of a parametron with output at the fundamental frequency under excitation, with determination of the basic design relations.

2. Limitation of the amplitude of parametric oscillations by the maximum.

3. Limitation of the oscillation amplitude by the minimum.

The greatest oscillation amplitude occurs at the minimum pump voltage, and the minimum occurs at the “critical” load,

an increase of which leads to a disruption of the parametric oscillations.

1. Some Relations under Excitation

Let us consider the operation of a parametron under excitation, with no load and with a very small active resistance of the windings of the oscillatory circuit.

To clarify the boundary conditions, the V–A characteristics were constructed for one choke (curve 1, Fig. 3) and for two chokes together (curve 2). Curve 4 was found experimentally when various capacitances were included in the oscillatory circuit and characterizes the magnitude of the voltage on the capacitor at the moment of excitation, as well as the conditional V–A characteristic of the inductive

Fig. 3

part of the parametron with output at the fundamental frequency. It does not start from the origin, since at small capacitances in the circuit there is resonance with the doubled frequency of the power-supply source (zone I). The V–A characteristic of the minimum capacitance at which parametric resonance at the fundamental frequency is already possible is tangent to the V–A characteristic of one choke. The inclusion of larger capacitances leads to saturation of the transformer (\(B_t > 1\) tesla); in the circuit there is a slow increase of voltage across the capacitance, and the oscillations are unstable (zone III).

In zones II and III, the conditional V–A characteristic of the inductance of the parametron with output at the fundamental frequency is represented with sufficient accuracy by the one-and-a-half V–A characteristic of one choke (curve 5).

\[ \dot U_{Ly}=1{,}5U_L=f(I_0). \tag{6} \]

As follows from Table 1, the supply voltage at the instant of excitation is, to a first approximation, equal to the voltage across the capacitance

\[ U_{\mathrm{p.v}} = U_{\mathrm{c.v}}. \tag{7} \]

Table 1 gives experimental and some calculated data for the parametron at the instant of excitation. The basic calculated relationships can be found rather simply when the conditional V–A characteristic of the parametron inductance is known. We shall regard as given the V–A characteristic of one choke of the parametron and the basic parameters of the cores, including the winding data. With a transformation ratio equal to unity, the V–A characteristic of the conditional inductance of the parametron is determined from expression (6). The point of intersection of the V–A characteristic of the capacitance (straight line 3) and the V–A characteristic of the conditional inductance (curve 5) gives us the value of the voltage across the capacitance and of the supply voltage at the instant of excitation. The calculated values of the supply voltages are given in the third row of the table.

In an inductive parametron, as in voltage ferroresonance, the actual value of the current of the oscillatory circuit at excitation is greater than the current determined by the formula

\[ I_2 = \frac{U_{\mathrm{c.v}}}{X_c}. \tag{8} \]

To find the actual value of the secondary current, we carry out constructions as in voltage ferroresonance [4].

The segment \(oa\) on the ordinate axis, enclosed between the origin and the tangent to the V–A characteristic of the conditional inductance of the parametron, is the voltage required for ferroresonance at the given capacitance.

Fig. 4.

The tangent is drawn parallel to the V–A characteristic of the capacitance, the tangent of whose angle of inclination to the ordinate axis is numerically equal to \(\frac{1}{\omega C}\). Taking into account the voltage drop across the active resistances of the windings of the oscillatory circuit (straight line 6, Fig. 3), we find the calculated value of the secondary current

parametron (segment \(aa_1\)). The calculated and experimental secondary currents—\(I_{2r}, I_2\)—are given in Table 1.

The distribution of the secondary voltages over the windings \(U_{2r}\) and \(U_{2d}\) makes it possible to determine the magnetizing currents \(I_{0r}\) and \(I_{0d}\) from the current–voltage characteristic of a single choke; or, conversely, the voltage can be found from the magnetizing currents.

The vector diagram at the instant of excitation of the parametron can be constructed from the already known data and taking into account the ratios \(I_{0r}/I_2\) and \(I_{0d}/I_2\).

As the calculations showed, when the capacitance is varied from 3 to 24 \(\mu\text{F}\), the ratio \(I_{0d}/I_{0r}\) varies between 1.2 and 1.8.

The vector diagram (Fig. 4) was constructed without allowance for magnetic leakage for a parametron with a capacitance of 7 \(\mu\text{F}\), and with the assumption that \(I_{0r}=I_2\), \(I_{0d}=1.5I_2\). In the construction, the voltage drops across the active resistances of the windings were taken into account.

The triangle of primary voltages was constructed by the method of intersections. In Table 1, for the capacitance of 7 \(\mu\text{F}\), the complete experimental and calculated data obtained from the vector diagram are given. The latter are written in the denominators of the fractions. In rows 8 and 9 of the table for 7 \(\mu\text{F}\), not the magnetizing currents but the no-load currents are entered.

2. Some relations at the minimum pumping voltage

When the supply voltage is reduced, the voltage across the capacitor increases, but it is limited along the ordinate axis by curve 2 (Fig. 3). At capacitances whose current–voltage characteristics are almost tangent to the current–voltage characteristic of a single choke (curve 1), there is one more limitation of the oscillations with respect to the maximum. The reason is that the voltage across the capacitance under parametric resonance with output at the doubled frequency—and, consequently, the current–voltage characteristic of the inductive part of the parametron—is expressed by curve 2. It turns out to be, as it were, “occupied” by parametric resonance at the doubled frequency. This limitation is shown in Fig. 3 by a dashed line.

Table 2

Regardless of the operating mode of the parametron, the expression

\[ \dot U_c < \dot U_c + \dot U_{Rн} = \dot U_{2r} + \dot U_{2d}, \tag{1′} \]

is valid, where \(U_{Rн}\) is the voltage drop across the load.

On the basis of the experimental data, it may be said that the maximum voltage on the capacitor is determined by the point of intersection of the capacitor current-voltage characteristic and curve 2, projected onto the ordinate axis; the secondary current is determined by the projection, onto the abscissa axis, of the segment bounded by this point.

The experimental and calculated data for determining the limitations on the capacitor voltage by the maximum and minimum are given in Table 2. In the numerator of the fraction the experimental data are given, and in the denominator, the calculated data.

Reducing the pumping voltage to the pull-out point leads to a considerable decrease in the voltage on the transformer and to slight saturation of the choke. The minimum voltage on the transformer at which oscillations are still possible is determined by the point of intersection of the capacitor current-voltage characteristic and that of one choke (segment \(bb\)):

\[ U_{2T\min}=bb . \tag{9} \]

The voltage on the double-supply choke, in the first approximation, is equal to the voltage on it for the case of excitation,

\[ U_{2D}=U_{2D,\mathrm{exc}} . \tag{10} \]

The minimum pumping voltage can be determined with sufficient accuracy in zone II as the segment between the origin and the point of intersection of the tangent to the current-voltage characteristics of both chokes (curve 2) (Table 3).

Table 3

3. Some relations at a “critical” load

The minimum voltage on the capacitor at which oscillations are still possible depends on the magnitude of the load, as well as on the active resistances of the secondary windings, and cannot be less than the voltage on the transformer at the moment of excitation:

\[ U_{C\min}\geq U_{2T,\mathrm{exc}} . \tag{11} \]

It should be noted that condition (11) is valid when the circuit is supplied with a voltage equal to the excitation voltage or exceeding its maximum by a factor of 1.2. In this case the pull-out and excitation of parametric oscillations are clearly visible. At a higher pumping voltage, significant load oscillations lead to a smooth change in the output voltage.

The magnitude of the secondary current is bounded by the perpendicular to the abscissa axis passing through the point on the capacitor current-voltage characteristic \(U_{C\min}\). Table 2 gives the experimental and calculated values of the voltages and currents for this case.

As a guide, the “critical” load resistance may be taken as equal to \((0.2 \div 0.3)X_c\) for series connection with the capacitance, and \((2.3 \div 4)X_c\) for parallel connection. A larger resistance for a series load corresponds to a smaller one for a parallel load.

Conclusions

1. In an inductive parametron with an output at the fundamental frequency, one of the chokes operates in transformer mode, the other in the mode of a double-power-supply choke, which is not observed in the same parametron with an output at the doubled frequency.

2. The paper proposes the use of a conditional V–A characteristic of the inductive part, which makes further calculations sufficiently simple (expression 6).

3. The basic relations between voltages and currents for the parametron in limiting regimes can be determined with sufficient accuracy by a graphical method using vector diagrams.

4. The paper presents certain data reflecting the physical processes of an operating parametron.

Literature

1. Tihiro Hayashi. Forced Oscillations in Nonlinear Systems. IL, 1957.
2. L. A. Bessonov. Nonlinear Electrical Circuits. “Vysshaya shkola,” 1964.
3. I. I. Belopolsky. Calculation of Low-Power Transformers and Chokes. GEI, 1963.
4. L. A. Bessonov. Theoretical Foundations of Electrical Engineering. “Vysshaya shkola,” 1964.