Harmonic Analysis of the Output Voltage Curve of a Machine-Valve Frequency Converter
Abstract Generated abstract
The paper analyzes the harmonic composition of the output voltage of a direct-link machine-valve frequency converter based on cyclic connection of synchronous generator windings to a load. Under assumptions of ideal valves, sinusoidal symmetrical generator voltages, active load, integer input-to-output frequency ratio, and instantaneous commutation, the authors derive Fourier coefficients for the segmented output-voltage curve and compute spectra for different frequency ratios and numbers of switched three-phase windings. The results identify the dominant higher harmonic orders, show that the fundamental amplitude increases with the number of generator windings but offers little benefit beyond six windings, and indicate that operation with switching frequency above generator frequency gives a more favorable spectrum than the opposite mode.
Full Text
HARMONIC ANALYSIS OF THE OUTPUT VOLTAGE CURVE OF A MACHINE–VALVE FREQUENCY CONVERTER
A. A. SKRIPILEV, B. E. TROFIMENKO, A. B. TSUKUBLIN
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
At the present time, in connection with the development of power semiconductor technology, frequency converters with a direct link have become widely used in autonomous electric-power supply sources. Their circuit arrangements may be quite diverse [1, 2, 3]. It is very important to determine the harmonic composition of the output voltage of such devices, which substantially affects the behavior of consumers and the elements of the filter.
Let us consider the harmonic composition of the output-voltage curve of a frequency converter with a direct link, whose operating principle is based on the cyclic connection of generator windings to the load at equal time intervals. The circuit of a three-phase–three-phase frequency converter is shown in Fig. 1. The form of the output-voltage curve will depend on the number of switched generator windings, on the ratio of the input and output frequencies, and on the nature of the load. Figure 2 gives the output-voltage curve for the case of an active load with six switched generator windings.
Fig. 1. Circuit of the frequency converter
Fig. 2. Converter output-voltage curve
The harmonic analysis of the output-voltage curve is carried out under the following assumptions:
1) the valves are ideal;
2) the ratio of the input frequency to the output frequency is an integer;
3) the load is active;
4) valve commutation occurs instantaneously; the valve commutation angle is \(\gamma=0\).
The last assumption does not introduce a large error into the calculation result, since the frequency converter includes a device for forced commutation of the valves.
We assume that the phase voltages of the generator are symmetrical and sinusoidal:
\[ U_A(t)^l = U_m \sin \left[\omega_{\mathrm{in}}\cdot t+\alpha-\frac{l-1}{m}\cdot 2\pi\right]; \]
\[ U_B(t) = U_m \sin \left[\omega_{\mathrm{in}}\cdot t+\alpha-\frac{l-1}{m}\cdot 2\pi-\frac{2\pi}{3}\right]; \tag{1} \]
\[ U_C(t) = U_m \sin \left[\omega_{\mathrm{in}}\cdot t+\alpha-\frac{l-1}{m}\cdot 2\pi-\frac{4\pi}{3}\right], \]
where \(l\) takes any integer value from 1 to \(m\);
\(U_m\) is the maximum phase value of the voltage of the synchronous generator; for simplicity we assume that \(U_m=1\);
\(m\) is the number of switchable three-phase windings of the synchronous generator.
For simplification of the analyses, we take the initial phase to be zero \((\alpha=0)\).
If the ratio of the input frequency to the output frequency is denoted by \(\nu=f_{\mathrm{in}}/f_{\mathrm{out}}\), then the equation of any \(n\)-th segment of the output voltage can be written in the form
\[ U_{\mathrm{out}(n)}=\sin \nu \left[\omega_{\mathrm{out}}\cdot t-\frac{2\pi(n-1)}{m\nu}\right]. \tag{2} \]
The equation of the output-voltage curve is written as
\[ U_{\mathrm{out}}=\sum_{n=1}^{N}\sin \nu \left[\omega_{\mathrm{out}}\cdot t-\frac{2\pi(n-1)}{m\nu}\right], \tag{3} \]
where \(N\) is the integer number of segments of the sinusoids of the primary phase voltage of the generator in the output-voltage curve, determined by the ratio of the output-voltage period to the duration of the connected state of each generator winding to the load.
\[ N=\frac{T_{\mathrm{out}}}{\tau}. \tag{4} \]
The duration of the connected state of the generator windings to the load depends only on the number of switched three-phase generator windings and on the ratio of the input and output frequencies. If the period of the switching frequency of the three-phase windings is denoted by \(T_{\text{sw}}\), then
\[ \tau=\frac{T_{\text{sw}}}{m}. \tag{5} \]
The period of the generator-winding switching frequency is determined as follows, depending on the relation between \(f_{\text{in}}\) and \(f_{\text{sw}}\). If
\[ f_{\text{sw}}<f_{\text{in}},\quad \text{then}\quad T_{\text{sw}}=\frac{1}{f_{\text{in}}-f_{\text{out}}}, \tag{6} \]
if
\[ f_{\text{sw}}>f_{\text{in}},\quad \text{then}\quad T_{\text{sw}}=\frac{1}{f_{\text{in}}+f_{\text{out}}}. \]
Substituting the value of \(T_{\text{sw}}\) into expression (5), after transformations we find, for
\[ f_{\text{sw}}<f_{\text{in}}\qquad \tau=\frac{\nu}{m}\cdot\frac{1}{f_{\text{in}}(\nu-1)}, \]
for
\[ f_{\text{sw}}>f_{\text{in}}\qquad \tau=\frac{\nu}{m}\cdot\frac{1}{f_{\text{in}}(\nu+1)}. \tag{7} \]
Hence the number of sinusoidal segments of the primary phase voltage of the generator in the output-voltage curve is determined by:
\[ N=m(\nu\mp 1). \tag{8} \]
Here the plus sign corresponds to the operating mode of the frequency converter with \(f_{\text{sw}}>f_{\text{in}}\), and the minus sign to \(f_{\text{sw}}<f_{\text{in}}\).
On the other hand, the duration of the connected state of the generator windings to the load will be equal to
\[ \tau=\frac{2\pi}{N}=\frac{2\pi}{m(\nu\mp 1)}. \tag{9} \]
For a quantitative estimate of the amplitude of each harmonic, we expand the output-voltage curve (3) into a Fourier series. The coefficients of the Fourier series are written as:
\[ a_k=\frac{1}{\pi}\int_{t_{n-1}}^{t_n}\sum_{n=1}^{N} \sin \nu\left[\omega_{\text{out}}\cdot t-\frac{2\pi(n-1)}{m\nu}\right]\cdot \cos k\omega_{\text{out}}\cdot t\,d(\omega_{\text{out}}\cdot t). \tag{10} \]
\[ b_k=\frac{1}{\pi}\int_{t_{n-1}}^{t_n}\sum_{n=1}^{N} \sin \nu\left[\omega_{\text{out}}\cdot t-\frac{2\pi(n-1)}{m\nu}\right]\cdot \sin k\omega_{\text{out}}\cdot t\,d(\omega_{\text{out}}\cdot t), \]
Solving equations (10), taking into account (7), (8), (9), we obtain
\[ a_k=\frac{1}{2\pi}\left\{ -\frac{1}{\nu+k}\sum_{n=1}^{N} \left[ \cos\frac{2\pi\left[n(k\mp 1)+(\nu\mp 1)\right]}{N} -\cos\frac{2\pi(n-1)(k\mp 1)}{N} \right] -\frac{1}{\nu-k}\sum_{n=1}^{N} \left[ \cos\frac{2\pi\left[n(k\mp 1)-(\nu\mp 1)\right]}{N} -\right. \]
\[ -\cos \frac{2\pi(n-1)(k\mp 1)}{N}\Biggr]\Biggr\}; \tag{11} \]
\[ b_k=\frac{1}{2\pi}\left\{ -\frac{1}{\nu+k}\sum_{n=1}^{N}\left[ \sin \frac{2\pi[n(k\pm 1)+(\nu\mp 1)]}{N} -\sin \frac{2\pi(n-1)(k\pm 1)}{N}\right]\right. \]
\[ \left. -\frac{1}{\nu-k}\sum_{n=1}^{N}\left[ \sin \frac{2\pi[n(k\mp 1)-(\nu\mp 1)]}{N} -\sin \frac{2\pi(n-1)(k\mp 1)}{N}\right]\right\}. \]
The upper signs refer to the converter operating mode with \(f_{\mathrm{per}}<f_{\mathrm{in}}\), the lower signs to \(f_{\mathrm{per}}>f_{\mathrm{in}}\).
Calculations of the coefficients \(a_k\) and \(b_k\) according to (11) were carried out on a computer for various values of the ratios of the input and output frequencies and of the number of switched three-phase windings of the generator.
The calculations showed that the amplitude of the first harmonic of the output voltage, for all ratios of the input and output frequencies when the number of switched three-phase windings is six, remains constant, equal to \(0.954\,U_m\). The largest percentage in the output-voltage curve is contributed by harmonics whose orders are determined by the following relation:
\[ k=Nq\mp 1, \tag{12} \]
where \(q\) is any positive integer; moreover, the amplitudes of the higher harmonics decrease as \(q\) increases.
In the operating mode with a switching frequency greater than the frequency of the synchronous generator, harmonics of higher orders appear in the output-voltage curve in comparison with the same mode when the switching frequency is less than the frequency of the synchronous generator. Figure 3 gives a diagram illustrating the dependence of the amplitudes
Fig. 3. Dependence of the amplitudes of the higher harmonics on the harmonic number for \(\nu=4\) and \(m=6\) with \(f_{\mathrm{per}}<f_{\mathrm{in}}\) (solid line) and \(f_{\mathrm{per}}>f_{\mathrm{in}}\) (dashed line)
of the harmonics on the harmonic number \(k\) for the ratio of the input and output frequencies \(\nu=4\) and the number of switched three-phase windings \(m=6\), for \(f_{\mathrm{per}}<f_{\mathrm{in}}\) (solid line) and for \(f_{\mathrm{per}}>f_{\mathrm{in}}\) (dashed line). The amplitude of the first harmonic is not shown in the figure.
Figure 4 shows the spectral composition of the output-voltage curves for various ratios of the input and output frequencies with
Fig. 4. Spectral composition of the output voltage of the frequency converter for \(f_{\mathrm{per}}>f_{\mathrm{in}}\) at various \(\nu\)
\(f_{\mathrm{per}}>f_{\mathrm{in}}\) and the number of independent switched generator windings equal to six. The amplitudes of the higher harmonics, as the multiplicity of the input and output frequencies increases, while remaining constant in magnitude, are shifted toward higher orders. This regularity is characteristic for any number of independent generator windings at any ratio of the switching frequency to the generator frequency.
With a decrease in the number of switched three-phase generator windings, the amplitude of the first harmonic of the output voltage decreases. At \(m=3\), the amplitude of the fundamental harmonic of the output voltage decreases by 14% in comparison with the amplitude value of the fundamental voltage harmonic at \(m=6\).
With a decrease in the number of three-phase generator windings, the percentage of higher harmonics increases; moreover, the amplitudes of the higher harmonics, increasing, shift toward lower orders; the amplitude of the higher harmonic having the greatest value in the spectrum of the output voltage at \(m=3\) amounts to 43% of the amplitude value of the fundamental harmonic.
The dependence of the amplitude of the first harmonic on the number of switched three-phase windings of the synchronous generator is shown in Fig. 5.
Fig. 5. Dependence of the amplitude of the first harmonic on the number of independent three-phase generator windings.
The constant component of the voltage is absent in the output-voltage curve for any integer ratio of the input and output frequencies for different numbers of three-phase windings.
Thus, as a result of the study of the harmonic composition of the output-voltage curve of a frequency converter with direct coupling, it has been established:
- Among the higher harmonics, the greatest values are possessed by harmonics with orders determined by the expression
\[ k = mq(\nu \mp 1) \mp 1 . \]
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The amplitude of the fundamental harmonic increases as the number of generator windings increases, but increasing the number of windings beyond six for the purpose of improving the harmonic composition of the output voltage is inexpedient (Fig. 5).
-
In the operating mode of the converter with a switching frequency greater than the generator frequency, the spectral composition of the output voltage is better than in the operating mode in which the switching frequency is less than the generator frequency.
References
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I. Ya. Bernshtein. Thyristor Frequency Converters without a DC Link. Energiya Publishing House, 1968.
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N. Z. Mastaev, G. S. Myshkin. Criteria for evaluating methods of frequency conversion for converters with direct coupling. Elektrotekhnika, 1972, No. 1.
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R. D. Iesse. W. I. Spaven. Constant–Frequency A–C Power Using variable Speed Generation, “Application and Industry”, 1960, N 46.