INVESTIGATION OF THE WAVEFORM OF THE OUTPUT VOLTAGE AND CURRENT IN THE CIRCUIT OF A FERROMAGNETIC FREQUENCY DIVIDER WITH HARD EXCITATION

V. L. Kosenko, A. I. Zaks

(Presented by the scientific seminar of the Department of Electrical Machines and General Electrical Engineering)

The quality of the energy transmitted from the converter to the load has a substantial influence on technical and economic performance, since transformation of a nonsinusoidal voltage causes large additional losses in the system as a whole. In this connection, it is important to solve the question of the optimal design variant of static frequency converters from the standpoint of obtaining a sinusoidal voltage at its output.

In the present article, an analysis is carried out of the waveform of the output voltage and current in the circuit of a ferromagnetic frequency divider by a factor of three.

The circuit of the frequency divider under investigation is shown in Fig. 1. It consists of series-connected nonlinear—NE—and linear—LE—elements, capacitors \(C\), and a special charging device CD, comprising a switch \(K\), a ballast resistance \(R_b\), and a low-power valve [1].

Fig. 1

When the divider is connected to the mains, in a series circuit containing a nonlinear inductance and a linear capacitance \(C\), owing to autoparametric resonance, powerful subharmonic oscillations are excited with a frequency three times lower than the frequency of the supply network. In practice these oscillations are nonsinusoidal, since at the output of the resonant system in this case there is inevitably penetration of the supply-network frequency and of the conversion harmonics from the output of the nonlinear element.

To estimate the waveform of subharmonic oscillations when the excitation process is periodic, one usually uses the harmonic coefficient [2]

\[ k_{\Gamma}=\frac{\sqrt{\sum U_n^2}}{U_1}, \]

where \(U_n\) is the amplitude of the \(n\)-th harmonic of the output voltage \((n>1)\), and \(U_1\) is the amplitude of the fundamental harmonic of the output voltage.

The amplitude value of the \(n\)-th harmonic of the output voltage is determined from the expressions:

\[ U_n=\sqrt{(U_n')^2+(U_n'')^2}; \]

\[ U_n'=\frac{2}{T}\int_0^T U(t)\cos\omega t\,dt; \]

\[ U_n''=\frac{2}{T}\int_0^T U(t)\sin\omega t\,dt. \]

The magnitudes of the cosine \(U_n'\) and sine \(U_n''\) components are determined graphically when the output-voltage curve \(U\) is expanded into a Fourier series [3].

The nonsinusoidal character of the subharmonic oscillations also depends on the position of the operating point characterizing the given operating mode of the divider within the region of existence of oscillations.

For a frequency divider by a factor of three, the boundaries of the region of existence of stable subharmonic oscillations were determined; they are shown in Fig. 2.

Fig. 2

As a result of the experimental investigation it was established that, when the inductance in the circuit of the nonlinear element is varied within the region of existence, the shape of the output-voltage curve also changes: from flattened—at small inductance values, near the right boundary of the region—to sinusoidal, which at large inductance values—near the left boundary of the region—is significantly distorted.

The shape of the output-voltage curve also depends on the capacitance of the capacitors connected in series in the circuit: at small values of \(C\), the shape of the curve approaches to a greater degree

to a sinusoidal one, then, as the capacitance of the capacitors increases, it changes to a flattened one, and at significant values of \(C\) the shape of the output-voltage curve approaches a rectangular one.

Consequently, within the region of existence it is possible to distinguish definite zones with oscillations containing different percentages of harmonic components. The dotted line here indicates the zone within which the shape of the output-voltage curve is closest to sinusoidal; the harmonic coefficient \(k_{\mathrm{g}}\) here varies from 12 to 18% (zone 1). The shape of the output voltage is distorted to a greater extent if the operating point of the divider is in zone 2, which in the figure is bounded by a dash-dotted line. Within this zone the harmonic coefficient varies from 18 to 40%.

Harmonic analysis of the voltage and current curves in zone 2 showed that, at induction values lying above zone 1, the third and fifth harmonics predominate in the output-voltage curve, while at induction values lying below zone 1 the influence of the third harmonic is mainly manifested; therefore the shape of the output-voltage curve here appears more flattened.

If the operating point of the divider is in zone 3 at induction values lying beyond the limits of zone 2, the shape of the output-voltage curve is strongly distorted because of the presence of the third, fifth, and seventh harmonics. The harmonic coefficient in this region varies from 40% and higher.

Figure 3 gives oscillograms taken for various operating points of the divider within the region of existence of the oscillations shown in Fig. 2.

The appearance of the output-voltage and current curves for operating point \(A\) of the divider, located in zone 1, is shown in Fig. 3a. The shape of the output-voltage curve is distorted here to a lesser degree. Harmonic analysis showed that the harmonic coefficient here is 15%, the third harmonic \(k_3 = 8\%\), the fifth \(k_5 = 8\%\), and the seventh \(k_7 = 7\%\). The largest harmonic by amplitude in the current curve in the divider circuit is the third, which is 10%; the fifth and seventh are 29% and 8%, respectively.

How the shape of the output voltage changes if the divider operates at the same induction as at point \(A\), but at a larger value of the capacitance of the capacitors (in this case operating point \(B\)), is clearly seen from the oscillogram presented in Fig. 3b. The harmonic composition of the output-voltage curve here is as follows: \(k_{\mathrm{g}} = 26\%\), \(k_3 = 8\%\), \(k_5 = 13\%\), \(k_7 = 2.5\%\). For the current curve we have \(k_{\mathrm{g}} = 40\%\), \(k_3 = 18\%\), \(k_5 = 8.8\%\), \(k_7 = 8\%\).

If, at this same value of the capacitance of the capacitors in the divider circuit, the induction in the nonlinear element is increased to \(1.15\ \text{T}\) (operating point \(C\), lying in zone 2), then the output-voltage curve will be distorted to a greater extent (Fig. 3c). For the voltage curve here \(k_{\mathrm{g}} = 40\%\), \(k_3 = 12\%\), \(k_5 = 20\%\), \(k_7 = 2.8\%\); for the current curve in the divider circuit we have \(k_{\mathrm{g}} = 93\%\), \(k_3 = 29.7\%\), \(k_5 = 46\%\), \(k_7 = 8\%\). From the analysis presented it is seen that the fifth harmonic has the predominant value.

Figure 3d presents an oscillogram of the output voltage and current in the circuit for operating point \(D\), i.e., when the divider operates at larger values of the induction and of the capacitance of the capacitors in the circuit. Harmonic analysis of these curves showed that, for the voltage curve, \(k_{\mathrm{g}} = 35\%\), \(k_3 = 25\%\), \(k_5 = 9\%\), \(k_7 = 2.6\%\); for the current curve, \(k_{\mathrm{g}} = 70\%\), \(k_3 = 50\%\), \(k_5 = 35\%\), \(k_7 = 20\%\). Consequently, here the third harmonic predominantly prevails. It should also be noted that, at significant capacitances of the capacitors, i.e., where, with increasing in-

a)

b)

c)

d)

Fig. 3

induction from the minimum value to the maximum, the operating point does not pass through zone 1, and the waveform of the output-voltage curve changes only slightly. The waveform of the output-voltage curve here approaches a rectangular one and, as the induction increases, changes somewhat, mainly owing to an increase in the third harmonic.

Thus, the analysis carried out of the waveform of the output voltage and the current in the divider makes it possible, when designing it, to determine the operating point within the region of existence of oscillations, proceeding from the requirements imposed on the waveform of the output voltage.

Conclusions

1. In the experimental investigation of the divider it was established that, within the region of existence of oscillations, the waveform of the output voltage and of the current in the divider circuit varies depending on the induction of the nonlinear element and on the capacitance of the capacitors \(C\).

1. As a result of the harmonic analysis performed on the voltage and current curves at various points of the region, zones with a definite percentage content of harmonics were established; on this basis it is possible to choose the operating point of the divider with a satisfactory waveform of the output-voltage curve.

References

1. A. A. Artem′ev. A ferromagnetic frequency divider by a factor of three. Transactions of the A. A. Zhdanov GPI (“Static Converters in Automation Devices and Elements of Electrical Equipment”), vol. XXIV, issue 7, Gorky, 1968.

2. D. M. Bamdas, S. V. Shapiro, L. N. Davydova. Ferromagnetic Frequency Dividers. Energiya Publishing House, 1967.

3. L. A. Bessonov. Theoretical Foundations of Electrical Engineering. Moscow, Vysshaya Shkola, 1967.