Harmonic Composition of the Current of Single-Bridge Thyristor Reactive-Power Sources as a Function of Commutation Duration

A. I. ZAITSEV, V. N. MISHIN, D. F. ZENKOV

(Presented by the scientific-technical seminar of the Research Institute of AEM)

One of the principal shortcomings of valve-type controlled reactive-power sources (RPS) based on AC-to-DC converters with artificial commutation is the considerable content of higher harmonics in the current curve [1], [2]. To improve the shape of the current curve, it has been proposed to use 12-phase converters [1], which complicates the power section of the RPS. The harmonic composition of the RPS current can also be improved by increasing the duration of current transfer from phase to phase, i.e., by increasing the angle

\[ 0 < \gamma < \frac{\pi}{3}. \]

The minimum value of the nonlinear-distortion coefficient, corresponding to \(\gamma=\gamma_{\mathrm{opt}}\), is equal to \(K_{\mathrm{n.d\,min}}=3.8\%\).

Figures 2 and 3 show the plot of the equation \(K_{\mathrm{n.d}}(\gamma)\) and the plot of the function

\[ \frac{I_{(k)}}{I_{(1)}}(\gamma)=I_{k*}(\gamma) \]

for the 5th, 7th, 11th, and 13th harmonics.

Depending on the magnitude of the commutation angle, two operating modes of the converter of the RPS can be distinguished:

1) \(\gamma < \dfrac{\pi}{3}\)—during the commutation period, three valves of the converter conduct current;

2) \(\gamma \geq \dfrac{\pi}{3}\)—over a certain commutation interval, four valves conduct current, two in each group of the bridge.

Expanding the current curve in a Fourier series, we obtain an expression for determining the rms value of the current harmonics

\[ I_{(k)}=\frac{\sqrt{2}I_d}{\pi\gamma k^2}\sin k\frac{\pi}{3}\sin k\frac{\gamma}{2}, \]

where \(I_d\) is the magnitude of the current flowing on the dc side of the RPS converter;

\(\gamma\) is the magnitude of the commutation angle;

\(k=6n\pm1\) is the harmonic number \((n=0,1,2,3\ldots)\).

The rms value of the current generated by the RPS is

\[ I_{\mathrm{d}}=I_d\sqrt{\frac{2}{3}-\frac{\gamma}{3\pi}}, \qquad 0<\gamma<\frac{\pi}{3}, \tag{2} \]

\[ I_{\mathrm{d}}=I_d\sqrt{1-\frac{2\gamma}{3\pi}-\frac{\pi}{9\gamma}+\frac{\pi^2}{81\gamma^2}}, \qquad \frac{\pi}{3}<\gamma<\frac{2\pi}{3}, \tag{3} \]

and the nonlinear-distortion coefficient of the RPS current is determined as

\[ K_{\mathrm{n.i}}= \frac{\sqrt{I_{\mathrm{d}}^{\,2}-I_{(1)}^{\,2}}}{I_{(1)}} = \sqrt{\frac{2\pi(2\pi-\gamma)\gamma^2}{144\sin^2\dfrac{\gamma}{2}}-1}, \qquad 0<\gamma<\frac{\pi}{3}, \tag{4} \]

\[ K_{\mathrm{n.i}}= \sqrt{ \frac{ \pi^2\gamma^2\left(1-\dfrac{2\gamma}{3\pi}-\dfrac{\pi}{9\gamma}+\dfrac{\pi^2}{81\gamma^2}\right) }{ 24\sin^2\dfrac{\gamma}{2} } -1 }, \qquad \frac{\pi}{3}<\gamma<\frac{2\pi}{3}, \tag{5} \]

where \(I_{(1)}\) is the rms value of the first harmonic of the RPS current.

Analysis of equations (4) and (5) shows that, in the range of commutation angles \(0<\gamma<\dfrac{\pi}{3}\), as the commutation angle increases the nonlinear-distortion coefficient of the RPS current decreases monotonically; whereas in the range \(\dfrac{\pi}{3}<\gamma<\dfrac{2\pi}{3}\), the function \(K_{\mathrm{n.i}}(\gamma)\) has two extrema, one of which is the minimum of the nonlinear-distortion coefficient. The optimum value of the commutation angle at which the content of higher harmonics in the RPS current curve is minimal is determined by the equation

\[ \tg\frac{\gamma_{\mathrm{opt}}}{2} = \frac{ 9\pi\gamma_{\mathrm{opt}}^{2}-6\gamma_{\mathrm{opt}}^{3}-\pi^{2}\gamma_{\mathrm{opt}}+\dfrac{\pi^{3}}{9} }{ 18\pi\gamma_{\mathrm{opt}}-18\gamma_{\mathrm{opt}}^{2}-\pi^{2} }, \tag{6} \]

whence, solving by numerical methods, we obtain \(\gamma_{\mathrm{opt}}=69.75^\circ\), and the mini-

small value of the nonlinear-distortion coefficient corresponding to \(\gamma=\gamma_{\mathrm{opt}}\) is \(K_{\mathrm{n.i.min}}=3.8\%\).

Figures 2 and 3 show the graph of the equation \(K_{\mathrm{n.i}}(\gamma)\) and the graphs of the functions

\[ \frac{I_{(k)}}{I_{(1)}}(\gamma)=I_{k*}(\gamma) \]

for the 5th, 7th, 11th, and 13th harmonics.

Fig. 2

It follows from (4) and (5) and Figs. 2 and 3 that increasing the commutation angle is most effective at small values of the latter. At commutation angles greater than \(\pi/3\), an increase in the commutation duration changes the content of higher harmonics in the current of the RPM relatively less. The appropriate value of the commutation angle should be determined by technical-and-economic calculations, taking into account the damage caused to the genera-

Fig. 3

... of higher harmonics into the network. As a first approximation, one may use GOST 183-66, which determines the value of the coefficient of nonlinear distortion of the voltage of alternating-current generators, taking for the RPS the same degree of current distortion. Then, calculating the RPS for \(K_{\text{н.и}}<0.05\), we obtain \(\gamma>\dfrac{\pi}{3}\). Thus, the content of higher harmonics in the RPS current can be reduced by increasing the commutation angle to sufficiently small values when using a single-bridge converter, i.e., without complicating the power section.

REFERENCES

1. B. Ya. Gumanovskii, V. I. Pleskov, S. I. Voropaev, A. A. Poskrobko. Thyristor reactive-power source. Elektrichestvo, 1970, No. 11.

2. A. I. Zaitsev, V. N. Mishin, D. F. Zenkov, L. V. Aleksandrova. On the calculation of thyristor reactive-power sources (RPS). Reports of the VII Scientific-Technical Conference, Tomsk, 1971.