ENERGY PERFORMANCE INDICATORS OF THYRISTOR SWITCHES

A. M. AZAROV, V. N. BAZAROV, B. P. SOUSTIN

In rectifier devices, in addition to voltage regulation, it is sometimes necessary to carry out instantaneous switching of a consumer from one network to another. In this case it is expedient to use thyristor switches (TS), in which switching from one network to another and regulation of the output voltage are accomplished by means of thyristors on the primary side of the transformer. An uncontrolled rectifier is installed on the secondary side. Below, comparative energy performance indicators of TS and controlled rectifiers (CR) with an equal number of controlled valves are considered.

Figure 1a shows the circuit of a single-phase TS. In analyzing this circuit, the valves are assumed ideal, commutation instantaneous, and the output—

Fig. 1.

current is completely smoothed, the leakage inductance and active resistances of the transformer windings are equal to zero, and the transformation ratio is equal to unity.

A distinctive feature of TC operation is that the primary current and the rectified voltage (Fig. 1b) are discontinuous regardless of the nature of the load. This follows from the operation of the TC, since the thyristor in the primary circuit is turned off at the instant when the line voltage is equal to zero. This circumstance—the discontinuity of the primary current—substantially improves the power factor of the device.

The rectified voltage is equal to:

\[ U_d=\frac{1}{\pi}\int_{\alpha}^{\pi}\sqrt{2}\,U_{2\phi}\sin\omega t\,d\omega t =\frac{U_{d0}}{2}(1+\cos\alpha), \]

where

\[ U_{d0}=\frac{2\sqrt{2}\,U_{2\phi}}{\pi} \]

is the rectified voltage at \(\alpha=0\).

In a lossless rectifier, with fully smoothed current, the active power is equal to the rectified power

\[ P=U_d I_d=\frac{U_{d0}I_d}{2}(1+\cos\alpha). \]

The consumed reactive power is determined as:

\[ Q=-\frac{1}{\pi}\int_{\alpha}^{\pi}\sqrt{2}\,U_{2\phi}I_d\cos\omega t\,d\omega t =\frac{U_{d0}I_d}{2}\sin\alpha. \]

The principal energy-performance indicator of a rectifier device is the power factor

\[ \chi=\nu\cos\varphi, \]

where \(\varphi\) is the phase-shift angle of the first harmonic of the primary current relative to the supply voltage;

\(\nu\) is the distortion factor, numerically equal to the ratio of the rms value of the first harmonic of the primary current to the rms value of the primary current,

\[ \frac{I_{1(1)}}{I_1}. \]

The amplitude and rms value of the first harmonic of the primary current are

\[ i_{1(1)m}=\frac{2}{\pi}\int_{\alpha/2}^{\pi-\alpha/2} I_d\sin\omega t\,d\omega t =\frac{4I_d}{\pi}\cos\frac{\alpha}{2} \]

\[ I_{1(1)}=\frac{i_{1(1)m}}{\sqrt{2}} =\frac{4I_d}{\sqrt{2}\pi}\cos\frac{\alpha}{2}. \]

The rms value of the current drawn from the network is expressed as follows:

\[ I_1=\sqrt{\frac{1}{\pi}\int_{\alpha}^{\pi} I_d^2\,d\omega t} =I_d\sqrt{\frac{\pi-\alpha}{\pi}}. \]

The displacement factor in this case is equal to:

\[ \cos\varphi=\frac{P}{\sqrt{P^2+Q_2^2}}=\cos\frac{\alpha}{2},\ \text{i.e.}\ \varphi=\frac{\alpha}{2}. \]

The distortion factor can be determined from the expression

\[ \nu=\frac{I_{1(1)}}{I_1} =\frac{4I_d\cos\frac{\alpha}{2}}{\sqrt{2}\pi I_d\sqrt{\frac{\pi-\alpha}{\pi}}} =\frac{2\sqrt{2}\cos\frac{\alpha}{2}}{\pi\sqrt{\frac{\pi-\alpha}{\pi}}}. \]

Then the power factor is determined as:

\[ \chi=\frac{2\sqrt{2}}{\pi\sqrt{\frac{\pi-\alpha}{\pi}}}\cos^2\frac{\alpha}{2}. \]

Let us introduce relative units

\[ U'_d=\frac{U_d}{U_{d0}}=\frac{1+\cos\alpha}{2};\quad P'=\frac{P}{U_{d0}I_d}=\frac{1+\cos\alpha}{2}; \]

\[ Q'=\frac{Q}{U_{d0}I_d}=\frac{\sin\alpha}{2};\quad \chi'=\frac{\chi}{\frac{2\sqrt{2}}{\pi}} =\frac{\cos^2\frac{\alpha}{2}}{\sqrt{\frac{\pi-\alpha}{\pi}}};\quad \nu'=\frac{\nu}{\frac{2\sqrt{2}}{\pi}} =\frac{\cos\frac{\alpha}{2}}{\sqrt{\frac{\pi-\alpha}{\pi}}}. \]

Table 1 gives expressions for the basic energy characteristics of the TK and UV, and comparative characteristics of the TK and UV were plotted from the data in the table (Fig. 2).

Table 1

It is evident from Fig. 2 that at a control angle \(\alpha=90^\circ\) the rectified voltage and active power of the UV are equal to zero, since \(\cos\varphi=0\) and

the rectifier draws only reactive energy from the network. The reactive power consumed by the TK reaches a maximum at \(\alpha=90^\circ\) and is half that for the UV at the same control angle.

The power factor of the TK increases at small control angles, which is due to improvement of the harmonic composition of the primary current.

Fig. 2.

The angle \(\alpha_m\), at which the distortion factor assumes its maximum value, is found from the condition

\[ \frac{\partial \nu'}{\partial \alpha}=0; \]

after transformation we obtain

\[ \pi\left(1-\frac{\alpha}{\pi}\right)\tg\frac{\alpha}{2}=1, \]

whence

\[ \alpha_m=46^\circ. \]

The boundary of the range of improvement of \(\nu'\) due to regulation is determined from the equality

\[ \nu'=\frac{\cos\frac{\alpha}{2}}{\sqrt{\frac{\alpha-\pi}{\pi}}}=1, \]

whence \(\alpha_{\mathrm{gr}}=90^\circ\).

Similarly, for \(\chi'\) one has

\[ \alpha_m=20^\circ \]
\[ \alpha_{\mathrm{gr}}=40^\circ. \]

In analyzing the operation of the three-phase TK circuit (Fig. 3, a), we shall adopt the previous assumptions and compare the characteristics of the TK and of a three-phase bridge UV.

At control angles \(\alpha\leqslant60^\circ\), the operation of the TK does not differ in any way from the operation of the UV.

The rectified voltage for \(\alpha>60^\circ\) is

\[ d_\alpha=\frac{3\sqrt{6}}{\pi}\int_{\alpha}^{\frac{2\pi}{3}} U_{2\phi}\sin\left(\omega t+\frac{\pi}{3}\right)\,d\omega t =U_{d0}\left[1+\cos\left(\frac{\pi}{3}+\alpha\right)\right], \]

Fig. 3

\[ U_{d0}=\frac{3\sqrt{6}}{\pi}U_{2\phi} \]
— rectified voltage at \(\alpha=0\).

The active and reactive powers consumed are determined by

\[ P=P_d=U_d I_d =U_{d0}I_d\left[1+\cos\left(\frac{\pi}{3}+\alpha\right)\right], \]

\[ Q=-\frac{3\sqrt{6}}{\pi} \int_{\alpha}^{\frac{2}{3}\pi} U_{2\phi}I_d\cos\left(\omega t+\frac{\pi}{3}\right)\,d\omega t = U_{d0}I_d\sin\left(\frac{\pi}{3}+\alpha\right). \]

The amplitude and rms value of the first harmonic of the primary current

\[ i_{1(1)m} = \frac{1}{\pi} \int_{\frac{\alpha}{2}}^{\frac{2}{3}\pi-\frac{\alpha}{2}} I_d\,d\omega t + \frac{1}{\pi} \int_{\frac{\pi}{3}+\frac{\alpha}{2}}^{\pi-\frac{\alpha}{2}} I_d\,d\omega t \]

\[ I_1=\sqrt{\frac{1}{\pi}\int_{\alpha/2}^{2\pi/3-\alpha/2} I_d^2\,d\omega t+\int_{\pi/3+\alpha/2}^{\pi-\alpha/2} I_d\,d\omega t} = I_d \sqrt{\frac{4}{3}-\frac{2\alpha}{\pi}}. \]

Displacement factor

\[ \cos\varphi=\frac{P}{\sqrt{P^2+Q^2}}=\cos\left(\frac{\pi}{6}+\frac{\alpha}{2}\right), \]

i.e.,

\[ \varphi=\frac{\pi}{6}+\frac{\alpha}{2}. \]

The distortion factor is found from the expression

\[ \nu=\frac{\sqrt{P^2+Q^2}}{3U_1 I_1} =\frac{2\sqrt{6}}{\pi}\cdot \frac{\cos\left(\frac{\pi}{6}+\frac{\alpha}{2}\right)} {\sqrt{\frac{4}{3}-\frac{2\alpha}{\pi}}}. \]

Then the power factor is

\[ \chi=\nu\cos\varphi= \frac{2\sqrt{6}}{\pi}\cdot \frac{\cos^2\left(\frac{\pi}{6}+\frac{\alpha}{2}\right)} {\sqrt{\frac{4}{3}-\frac{2\alpha}{\pi}}}. \]

Introduce relative units

\[ U'_d=\frac{U_d}{U_{d0}};\quad P'=\frac{P}{U_{d0}I_d};\quad Q'=\frac{Q}{U_{d0}I_d};\quad \chi'=\frac{\chi}{\frac{3}{\pi}};\quad \nu'=\frac{\nu}{\frac{3}{\pi}}. \]

Table 2

Using the data of Table 2, comparative energy characteristics of three-phase circuits of thyristor switches and rectifier devices are plotted (Fig. 4).

On the basis of the foregoing, the following conclusions may be drawn:

Fig. 4

1. The use of thyristor switches makes it possible to improve all energy characteristics \((\nu, \chi, \cos \varphi, P, Q)\) at lower power consumption.

2. The loading of the network by reactive currents is reduced; in this case reactive energy is not returned to the network, but is used to maintain the current in the load.

3. The harmonic composition of the consumed current is improved, which is especially important when operating from a generator of comparable power.

4. In a single-phase thyristor switch, when stabilization of the output voltage is necessary, it is advisable to have a control angle \(\alpha = 20^\circ\), which corresponds to operation of the rectifier device with the maximum power factor.

5. The disadvantages include the need to have additional uncontrolled valves.

REFERENCES

1. V. S. Bogryi. Analysis of three-phase current rectification circuits with controlled silicon valves on the primary side of the transformer. Proceedings of the Mordovian Research Institute of Economics, issue II, 1964.

2. N. Kh. Sitnik. Power Semiconductor Engineering. “Energiya,” 1968.