On Determining the Depth Coefficient of Inductance Modulation in a Parametric Oscillator

V. M. Vysotskaya

(Presented by the scientific seminar of the Electromechanical Faculty)

The electric circuit of an inductive-type parametric oscillator is an oscillatory circuit consisting of a variable inductance \(L\), a constant capacitance \(C\), and a load resistance \(R\) (Fig. 1).

Fig. 1.

When the inductive parameter of the oscillator rotates, under known conditions an alternating current arises and is maintained in its circuit, with a frequency half as large as the frequency of variation of the parameter \(L\).

Fig. 2.

Figure 2 shows the curve of harmonic variation of the inductance of the parametric oscillator.

For simplification one may make an assumption close to the truth, namely that the variable part of the generator inductance varies according to a cosine law. At the initial instant of time the inductance has a maximum, and the expression for it takes the form

\[ L=L_0+L_n \cos 2\omega t. \]

Here \(L_0\) is the mean value of the inductance,
\(L_n\) is the amplitude of its variable part.

As can be seen from Fig. 2,

\[ L_0=\frac{L_{\max}+L_{\min}}{2} \quad \text{and} \quad L_n=\frac{L_{\max}-L_{\min}}{2}. \]

Let us introduce \(m\), the depth of modulation of the parameter, i.e., the relative change of the parameter.

\[ m=\frac{L_n}{L_0}=\frac{L_{\max}-L_{\min}}{L_{\max}+L_{\min}}. \]

Thus, finally we obtain:

\[ L=L_0(1+m\cos 2\omega t). \]

To choose the parameters of the excitation circuit of a parametric generator and to determine the conditions for its stable operation, it is necessary to know the value of the modulation-depth coefficient \(m\).

The modulation-depth coefficient can be determined experimentally only after the parametric generator has been manufactured and put into operation. In this case the inductive reactances of the parametric generator \(X_{\max}\) and \(X_{\min}\) are measured, the inductances \(L_{\max}\) and \(L_{\min}\) are calculated, and then the modulation depth of the parametric generator is determined as

\[ m=\frac{L_{\max}-L_{\min}}{L_{\max}+L_{\min}}. \]

A method may be recommended for determining an approximate value of the modulation-depth coefficient during the design of a parametric generator for any profile shape of the stator and rotor core laminations.

For this it is necessary to construct the pattern of distribution of magnetic lines of force in the air gap for two cases:

1) when the rotor teeth are located opposite the stator teeth, i.e., for the greatest value of inductance (Fig. 3a);

2) when the rotor teeth are located opposite the stator slot, which corresponds to the smallest value of inductance (Fig. 4a).

Having constructed the curves of magnetic-induction distribution in the air gap for the cases indicated above, and replacing the area enclosed by these curves by the area of rectangles with a base equal to the calculated width of the stator pole arc \(b_i\), we find that the height of the rectangle in the first case (Fig. 3b) is equal to \(B_{\max}\), and in the second case (Fig. 4b) to \(B_{\min}\).

It is obvious that the following equality holds:

\[ \frac{L_{\max}-L_{\min}}{L_{\max}+L_{\min}} = \frac{B_{\max}-B_{\min}}{B_{\max}+B_{\min}}, \]

i.e.,

\[ m=\frac{B_{\max}-B_{\min}}{B_{\max}+B_{\min}}. \]

Experimental verification on a prototype parametric oscillator gave quite satisfactory results and showed that, using the indicated method, it is possible to determine the modulation-depth coefficient \(m\) of an inductive parametric oscillator for any profile of the inter-iron space.

Fig. 3.

Fig. 4.

LITERATURE

1. Papaleksiĭ, N. D. Parametric generation of alternating currents. Elektrichestvo, No. 11, 1938.

2. Zhezherin, R. P. On parametric generation. Elektrichestvo, No. 11, 1940.