Efficiency of a Modulation Converter for Autonomous Stable-Frequency Sources

V. A. DENISOV, Sh. S. ROIZ

(Presented by the scientific seminar of the departments of electrical machines and apparatus and general electrical engineering)

The development of autonomous stable-frequency sources with preliminary amplitude modulation of the output voltage in an electromechanical generator is impossible without evaluating the efficiency of the modulation converter, which is the principal power part of the source. A distinctive feature in determining the efficiency of such converters is that the depth of preliminary modulation may be different.

Let us consider the modulation converter of an autonomous source shown in Fig. 1. The operating principle of the converter is as follows. When the generator excitation winding is supplied with direct and alternating sinusoidal currents, a modulated voltage is produced in the output winding \(W_2\). The depth of amplitude modulation \((a)\) is determined by the ratio of the currents in winding \(W_1\). The final modulation of the generator output voltage is carried out by means of a rectifier, which, in step with the alternating current of winding \(W_1\), forms positive and negative half-waves of voltage at the output. Figure 2a shows the nature of the variation of the induction and voltage at the output of the synchronous generator, and Fig. 2b shows the nature of the variation of the voltage at the rectifier output. To simplify the graphical construction, the sinusoidal form of the carrier frequency is replaced by a triangular one.

Fig. 1. Modulation converter of an autonomous stable-frequency source: SG — single-phase synchronous generator; \(B\) — semiconductor controlled two-half-period rectifier; \(\Phi\) — filter

In accordance with Fig. 1, the efficiency of the modulation converter can be expressed as the product of the efficiencies of the elements included in it:

\[ \eta = \eta_{\mathrm{sg}} \eta_{\mathrm{B}} \eta_{\Phi}. \tag{1} \]

In the general case, the efficiency of the synchronous generator can be determined by the formula

\[ \eta_{\mathrm{st}}=\frac{1}{\dfrac{P_{\mathrm{mx}}}{P_2}+\dfrac{\Sigma P}{P_2}+1}, \tag{2} \]

where

\[ P_{\mathrm{mx}} \]
— mechanical losses in the generator;

\[ \Sigma p \]
— losses in the copper and steel of the generator;

\[ P_2 \]
— power at the generator output.

In the absence of modulation in the generator, i.e., for \(a=0\), we have

\[ \Sigma P_{(a=0)}=P_{\mathrm{M}1}+P_{\mathrm{M}2}+P_{\mathrm{C}2}. \tag{3} \]

or

\[ \Sigma p_{(a=0)}=I_{\mathrm{в}}^{2}r_1+\frac{U_m^2}{2r_2}+\frac{B_m^2}{2}\varepsilon, \tag{3a} \]

where

\[ p_{\mathrm{M}1}=I_{\mathrm{в}}^{2}r_1 \]
— losses in the rotor copper;

\[ p_{\mathrm{M}2}=\frac{U_m^2}{2r_2} \]
— losses in the stator copper;

\[ p_{\mathrm{C}2}=\frac{B_m^2}{2}\varepsilon \]
— losses in the stator steel;

Fig. 2. Output functions of the modulation converter:
a — change in the induction and output voltage of the synchronous generator;
b — change in the rectifier output voltage

\[ \varepsilon \]
— coefficient accounting for the specific losses, the remagnetization frequency, the weight, and the grade of the stator steel. With modulation present, i.e., for \(a\ne0\), we obtain

\[ \Sigma p_{(a\ne0)}= I_{\mathrm{в}}^{2}r_1\left[(1-0.5a)^2+\frac{a^2}{8}\right] +\frac{U_m^2}{r_2}F^2 +B_m^2\varepsilon F^2 +I_{\mathrm{в}}^{2}\frac{a^2}{8}\xi . \tag{4} \]

Here

\[ F= \sqrt{ \frac{1}{\pi}\int_{0}^{\pi} \left[(1-0.5a-0.5a\cos 2\omega t)\sin k\omega t\right]^2\,d\omega t } = \frac{(1-0.5a)^2}{2}+\frac{a^2}{16} \]

— effective value of the output function (Fig. 2a);

\[ \xi \]
— coefficient accounting for the specific losses, the grade of steel,

frequency of remagnetization, the winding parameters, and the rotor geometry. Comparing expressions (3a) and (4), we find

\[ \sum p_{(a\ne 0)}=2F^{2}\sum p_{(a=0)}+I_{v}^{2}\frac{a^{2}}{8}\,\xi . \tag{5} \]

Taking into account that the generator output power under modulation is also related to the effective value of the output function,

\[ P_{2(a\ne 0)}=2F^{2}P_{2(a=0)}, \tag{6} \]

and substituting (5) and (6) into (2), we obtain the final expression for the efficiency of the synchronous generator

\[ \eta_{\mathrm{sg}}= \frac{1}{ \dfrac{p_{\mathrm{mx}}}{2F^{2}P_{2(a=0)}}+ \dfrac{\sum p}{P_{2(a=0)}}+ \dfrac{I_{v}^{2}a^{2}\xi}{16F^{2}P_{2(a=0)}} }. \tag{7} \]

Using equation (7), for a model prototype made on the basis of the PTO-700 production converter, the efficiency of the synchronous generator was calculated as the depth of preliminary modulation was varied (curve 1, Fig. 3). Analysis of the results obtained shows that, as the modulation depth increases, the efficiency of the synchronous generator decreases because of the relative increase in the share of mechanical losses and the appearance of losses in the rotor steel. However, this decrease is insignificant, since the remagnetization frequency of the rotor steel is much lower than the remagnetization frequency of the stator steel \((k=f_{1}/f_{2}=7\div 10)\).

Fig. 3. Dependence of the efficiency of the modulation converter and its elements on the depth of preliminary modulation: 1 — synchronous generator; 2 — filter; 3 — modulation converter

In view of the fact that the losses in the rectifier are small, we take \(\eta_{v}=1\).

The efficiency of the filter is found through the distortion factor of the voltage curve at the rectifier output (Fig. 2b) [1]. For this it is necessary to carry out a harmonic analysis of the indicated curve and find the relative values of the higher harmonics

\[ \eta_{\phi}= \frac{1}{ \displaystyle\sum_{n=1}^{\infty} \left[ \frac{b_{(2n-1)}}{b_{1}} \right]^{2} }. \tag{8} \]

Here

\[ \frac{b_{(2n-1)}}{b_{1}} = \frac{3}{2}\cdot \frac{(2-a)\left[(2n-1)^{2}-4\right]-a(2n-1)^{2}} {(2n-1)\left[(2n-1)^{2}-4\right](3-a)} - \]

are the relative values of the higher harmonics, and \(n\) is the ordinal number of the harmonic.

Figure 3 (curve 2) shows the dependence of the filter efficiency on the depth of preliminary modulation, constructed on the basis of (9).

In the graph, the maximum efficiency of the filter corresponds to the absence of the 3rd harmonic in the harmonic spectrum of the output voltage.

Substituting (7) and (8) into (2), one can obtain the values of the efficiency of the modulation converter as the depth of the preliminary modulation is varied (Fig. 3, curve 3).

Conclusions

1. The efficiency of a modulation converter in autonomous stable-frequency sources with preliminary amplitude modulation in the synchronous generator tends toward the nominal value of the efficiency of the synchronous generator.

2. In a modulation converter, the efficiency of the synchronous generator decreases only slightly as the depth of the preliminary amplitude modulation is increased.

REFERENCES

1. V. M. Griko. On the question of the efficiency of modulation frequency converters. Izvestiya vuzov, “Energetika,” No. 4, 1967.