METHOD FOR STUDYING THE FREQUENCY STABILITY OF GENERATORS

A. A. Kuz’min and V. Ya. Sul’yan

(Presented by the scientific seminar of the Radio Engineering Faculty)

A very timely problem is the study of the frequency of generators. The present work considers one of the possible methods for measuring frequency instability with the aid of an oscillograph.

The essence of the measurement is as follows. If a voltage is applied to the horizontal-deflection plates of the oscillograph,

\[ u_1 = u_{m_1}\sin(\omega_2 t+\varphi_0), \]

and to the vertical-deflection plates—a voltage with a frequency \(n\) times greater,

\[ u_2 = u_{m_2}\sin n\omega_2 t, \]

then on the screen of the cathode-ray tube we observe an interference pattern (Figs. 1, 2). For \(\varphi_0=0,\ \dfrac{\pi}{n}\) (which can always be ensured with the aid of a phase shifter), the forward and reverse motion of the beam coincide—

Fig. 1.

are obtained. If the voltage \(u_1\) is fed to the oscilloscope through a four-terminal network with a steep phase characteristic, for example through an oscillatory circuit with high \(Q\), then a change in the generator frequency by \(\Delta f_g\) will cause an increment in the phase shift between the voltages \(u_1\) and \(u_2\) equal to

Fig. 2.

\[ \Delta \varphi = \operatorname{arctg} 2 \frac{\Delta f_g}{f_g} Q_k , \tag{1} \]

where \(Q_k\) is the \(Q\) of the circuit. This increment can be measured from the segments of the figures of the interference pattern (Fig. 1) and determined from the formula [1]

\[ \sin \Delta \varphi = \frac{aa'}{bb'} \cdot \sin \frac{\pi}{n}. \tag{2} \]

Equating expressions (1) and (2), we obtain

\[ \xi = \frac{1}{2Q_k}\cdot \frac{a \sin \frac{\pi}{n}} {\sqrt{1-a^2\sin^2\frac{\pi}{n}}}, \tag{3} \]

where \(a=\dfrac{aa'}{bb'}\) and \(\xi=\dfrac{\Delta f_g}{f_g}\) is the relative change in frequency. For small \(\Delta \varphi\), expression (3) takes the form

\[ \xi = \frac{a\pi}{2Q_k n}. \]

The sensitivity of the proposed method is sufficiently high and is limited by the thickness of the beam. For example, for \(Q_k=100\), \(n=10\), \(bb'=100\) mm

and \(aa'=2\ \text{mm}\) (tube EO37, \(\varnothing\ 125\ \text{mm}\)) the relative change in frequency is \(\xi=3.14\cdot 10^{-5}\).

Thus, at a generator frequency of \(100\ \text{kHz}\), the absolute frequency change that can be measured is \(3.14\ \text{Hz}\).

Experimental verification of this method was carried out according to the block diagram shown in Fig. 3. The voltage from the generator under study 1 (GSS-6) was fed to the vertical-deflection plates of the oscilloscope through the auxiliary phase shifter 4. Through the frequency divider 2 (PS-10000) and the high-\(Q\) circuit 3, the voltage was fed to the horizontal-deflection plates.

Fig. 3.

To set \(\varphi_0=0\), switch \(P_1\) is placed in position 1. When the line voltage was changed by 30%, the frequency \(f_2=600\ \text{kHz}\) changed by \(75\ \text{Hz}\).

The proposed method is distinguished by the simplicity of the equipment used and has sufficient accuracy. With the aid of this method it is possible to investigate the frequency instability of generators as a function of various factors, as well as its drift over time.

REFERENCES

1. K. M. Shulzhenko, E. N. Silov, and V. Ya. Supyan. Reports of the Tomsk City Scientific and Technical Conference on the topic “Automation of Production Processes.”