Volume 284　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1974

ANALYSIS OF THE OUTPUT VOLTAGE OF MODULATION AUTONOMOUS SOURCES OF STABILIZED FREQUENCY

V. A. DENISOV, Sh. S. ROIZ, N. F. ZHMAEV

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

In various autonomous power installations of moving objects, in order to obtain a sinusoidal voltage of a specified stabilized frequency when the rotational speed of the prime mover is variable, various sources are used, distinguished by great diversity both in principle of operation and in design implementation.

However, with the development of semiconductor technology, increasing attention is being paid to machine–valve systems that use the principle of amplitude modulation of an alternating high frequency by a specified low frequency. In this case, amplitude modulation is carried out both by a semiconductor converter and by an electric-machine generator [1, 2].

In the present work a comparative harmonic analysis is carried out of the output-voltage curve of a modulation machine–valve autonomous source under various laws of amplitude modulation. As the machine–valve system, the following system is used: a single-phase synchronous generator excited from the rotor side—a two-half-period thyristor reversible rectifier.

Fig. 1

Let us assume that amplitude modulation in the synchronous generator is performed according to a sinusoidal law; then the voltage curve at the rectifier output will have the form shown in Fig. 1. In the interval \(0—\pi\) the indicated curve can be expressed by the function

\[ \left. \begin{aligned} U&=(1-a+a\sin\theta)\sin k\theta &&\text{for } 0\leq\theta\leq\frac{\pi}{k},\\[6pt] U&=(1-a+a\sin\theta)\sin k\left(\theta-\frac{\pi}{k}\right) &&\text{for } \frac{\pi}{k}\leq\theta\leq\frac{2\pi}{k},\\[6pt] &\ldots \end{aligned} \right\} \]

\[ \begin{gathered} U=(1-a+a\sin\theta)\sin k\left(\theta-\frac{m\pi}{k}\right), \quad \text{for } \frac{m\pi}{k}\leqslant\theta\leqslant\frac{(m+1)\pi}{k} \\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \\ U=(1-a+a\sin\theta)\sin k\left[\theta-\frac{(k-1)\pi}{k}\right], \quad \text{for } \frac{\pi(k-1)}{k}\leqslant\theta\leqslant\pi \end{gathered} \tag{1} \]

Here

\[ k=\frac{f_1}{f_2} \]

is the multiplicity of the ratio of the carrier and modulating frequencies;

\(a\) is the depth of amplitude modulation;

\(2\pi\) is the period of the modulating frequency;

\(m\) is the current number of the half-sinusoid of the carrier frequency.

The expansion of the function in a Fourier series will contain only odd sine terms

\[ U=\sum_{n=1}^{\infty} b_{n(m)}\sin n\theta, \tag{2} \]

where \(n\) is the number of the odd (\(n=1, 3, 5, 7, 9, \ldots\)) harmonic in the output voltage,

\[ b_{n(m)}=\sum_{m=0}^{k-1}\frac{2}{\pi} \int_{\frac{m\pi}{k}}^{\frac{(m+1)\pi}{k}} \left[(1-a+a\sin\theta)\sin k\left(\theta-\frac{m\pi}{k}\right)\right]\sin n\theta\,d\theta \]

is the amplitude of the \(n\)-th harmonic of the sine series.

After several transformations and when the conditions \(k\ne n\), \(k+1\ne n\), and \(k-1\ne n\) are satisfied, we obtain

\[ \begin{aligned} b_{n(m)}={}&\frac{1}{\pi}\sum_{m=0}^{k-1}(1-a) \Biggl\{ -\frac{1}{k-n} \left[ \sin\frac{\pi}{k}(-nm)-\sin\frac{\pi}{k}(k-n-nm) \right] \\ &\quad +\frac{1}{k+n} \left[ \sin\frac{\pi}{k}(nm)-\sin\frac{\pi}{k}(k+n+nm) \right] \Biggr\} \\ &\quad +\frac{a}{2} \Biggl\{ \frac{1}{k+1-n} \left[ \cos\frac{\pi}{k}(m-nm)-\cos\frac{\pi}{k}(k+1-n+m-nm) \right] \\ &\quad +\frac{1}{k-1+n} \left[ \cos\frac{\pi}{k}(-m+nm)-\cos\frac{\pi}{k}(k-1+n-m+nm) \right] \\ &\quad -\frac{1}{k+1+n} \left[ \cos\frac{\pi}{k}(m+nm)-\cos\frac{\pi}{k}(k+1+n+m+nm) \right] \\ &\quad -\frac{1}{k-1-n} \left[ \cos\frac{\pi}{k}(-m-nm)-\cos\frac{\pi}{k}(k-1-n-m-nm) \right] \Biggr\}. \end{aligned} \tag{3} \]

Under the condition \(k=n\), \(k+1=n\), and \(k-1=n\), the coefficients are computed by the formula

\[ b_{n(m)}=\frac{1}{\pi}\sum_{m=0}^{k-1}(1-a) \left\{ \cos m\pi+ \frac{1}{k+n} \left[ \sin\frac{\pi}{k}(nm)-\sin\frac{\pi}{k}(k+n+nm) \right] \right\}+ \]

\[ +\frac{a}{2}\left\{ \frac{1}{k-1+n}\left[\cos\frac{\pi}{k}(-m-nm)-\cos\frac{\pi}{k}(k-1+n-m+nm)\right]- \right. \]
\[ \left. -\frac{1}{k+1+n}\left[\cos\frac{\pi}{k}(m+nm)-\cos\frac{\pi}{k}(k+1+n+m+nm)\right] \right\}. \tag{4} \]

In Fig. 2, the values of the coefficients of series (2) are given as a function of the frequency-ratio multiplicity for two cases of amplitude-modulation depth \((a=0\) and \(a=0.4)\), and in Fig. 3 (solid lines) the values of the coefficients are given as a function of amplitude-modulation depth for a frequency multiplicity equal to 8. The coefficients were calculated on a digital computer using formulas (3) and (4). For comparison, Fig. 3 (dashed lines) gives the values of the coefficients of the harmonic analysis of only the modulating (envelope) curve, without taking the carrier frequency into account. The construction was performed by formula (5)

Fig. 2

Fig. 3

\[ b_n=\frac{2}{\pi}\int_0^\pi (1-a+a\sin\theta)\sin n\theta\,d\theta, \]

\[ \left\{ \begin{array}{ll} (1-a)\dfrac{4}{n\pi}+a, & \text{for } n=1,\\[6pt] (1-a)\dfrac{4}{n\pi}, & \text{for } n\ne 1. \end{array} \right. \tag{5} \]

As is evident from the graphs in Figs. 2 and 3, as the frequency multiplicity increases above \(7\div 10\), the value of each coefficient of the harmonic series (2) practically does not change and is related to the corresponding coefficient of equation (5) by a constant quantity. The latter depends on the rectification circuit of the carrier frequency and, in the present case, may be taken equal to \(\dfrac{2}{\pi}\).

On the basis of the above, the further analysis of amplitude-modulated piecewise-smooth functions can be reduced to the analysis only of the modulating functions; for this it is necessary to choose a frequency ratio multiplicity no lower than \(7 \div 10\), and to introduce the rectification correction factor of the circuit into the resulting formula for determining the Fourier-series coefficients.

Figure 4 (a, b) shows the envelope curves of the output voltage of an autonomous source for the most simply implemented amplitude-modulation laws and the corresponding plots of the harmonic composition.

Fig. 4

The indicated curves and plots are constructed according to equations (6, 7, 8, 9):

Fig. 4, a

\[ \left. \begin{aligned} U&=1-a\cos 2\theta && \text{for } 0\leq \theta \leq \frac{\pi}{4},\\ U&=1 && \text{for } \frac{\pi}{4}\leq \theta \leq \frac{\pi}{2} \end{aligned} \right\}, \tag{6} \]

\[ b_n=\frac{8}{\pi^2 n}+\frac{8a}{\pi^2} \left[ \frac{\cos (n+2)45^\circ}{2(n+2)} + \frac{\cos (n-2)45^\circ}{2(n-2)} - \frac{n}{n^2-4} \right]; \tag{7} \]

Fig. 4, b

\[ U=1-\frac{a}{2}-\frac{a}{2}\cos 2\theta \quad \text{for } 0\leq \theta \leq \pi, \tag{8} \]

\[ b_n=\frac{4}{\pi^2} \left( \frac{2-a}{n} - \frac{an}{n^2-4} \right). \tag{9} \]

Of the amplitude-modulation laws considered, the law in Fig. 4b is more preferable, since in this case the best harmonic composition is achieved at a smaller modulation depth, which corresponds to better energy ratios.

Conclusions

1. In modulation autonomous sources, the ratio of the carrier and modulating frequencies must be no less than \(7 \div 10\), since at a smaller ratio a deterioration of the harmonic composition of the output voltage is observed.

2. The analysis of the output-voltage curve of an autonomous source may be carried out using the modulating function, provided that the ratio of the frequencies is greater than \(7 \div 10\) and that the rectification coefficient of the circuit is introduced into the formula for calculating the harmonic coefficients.

3. The most rational law of amplitude modulation in a synchronous generator should be considered to be the law obtained as a result of superposing, on the constant excitation flux, a sinusoidal flux varying at twice the frequency.

LITERATURE

1. M. P. Popov, I. I. Lukin. On the question of obtaining alternating current of constant frequency with a varying rotational speed of the generator drive. Izvestiya vuzov, “Energetika,” No. 1, 1970.

2. E. V. Glatko. Investigation of the modulation method for forming sinusoidal voltage in a shipboard shaft-generator installation. Author’s abstract of dissertation, L., 1972.

3. G. S. Zinoviev. Output voltage of directly coupled valve frequency converters. Izv. SO AN SSSR, No. 10, issue 3, 1965.