Compensation of Load Reactive Energy in a Three-Phase Bridge Inverter with Pole Commutation in Sinusoidal PWM Mode

M. A. Zhitkov, V. A. Dobruskin

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

An inverter with pole commutation [1], like an inverter with individual commutation [2], under certain conditions is capable of compensating the reactive energy of the load internally.

The main indicator of the compensating action of the inverter is the unidirectionality of the current consumed by it from the power source. Under identical conditions, the range of variation of \(\cos \varphi_{\text{н}}\) corresponding to full compensation of the reactive power of the load is smaller for an inverter with pole commutation than for an inverter with individual commutation. This is explained by the specificity of the control law, under which, in a half-period of the output frequency, \((n-1)\) control signals are formed (\(n\) is the modulation ratio).

Taking into account the additional shift between current and voltage in the load phases, one may conclude that operation of an inverter with pole commutation in the PWM mode is possible with a load whose power factor is not less than \(\cos \varphi_{\text{н}}=0.5\), since otherwise the average value of the current consumed by the inverter becomes negative.

When the inverter is supplied from a source with unidirectional conductivity (a rectifier), it is necessary to compensate the energy arriving from the inverter at the dc buses by installing a capacitor in parallel with the source. To choose its value, it is necessary to analyze the variation of the instantaneous values of the current consumed by the inverter. It is quite obvious that if, in no commutation period, the instantaneous value of the inverter current becomes negative, then operation of the inverter is possible without installing a compensating capacitor.

The presence of controlled zero valves leads to the fact that, in the inverter under consideration, complete compensation of reactive energy by instantaneous values occurs when the load power factor is equal to

\[ \cos \varphi_{\text{н}} \geq \cos \omega_{\text{м}} T_{\text{к}} . \tag{1} \]

This occurs because the reactive (lagging) current in the load phases is closed through the zero valves, bypassing the power source.

It follows from expression (1) that, in order to increase the zone of the complete internal-compensation mode, it is necessary to reduce the ratio \(f_{\text{к}}/f_{\text{м}}\). Thus, at \(f_{\text{к}}=24 f_{\text{м}}\), the inverter will be fully compensated at \(\varphi_{\text{н}} \leq 15^\circ\), and at \(f_{\text{к}}=12 f_{\text{м}}\), at \(\varphi_{\text{н}} \leq 30^\circ\).

If the value of the load angle \(\varphi_{\mathrm{n}}\) lies within the limits

\[ \omega_{\mathrm{m}} T_{\mathrm{k}} \leqslant \varphi_{\mathrm{n}} \leqslant \frac{\pi}{3}, \]

then intervals of time appear during which the load current of each phase is closed through the power source, i.e., the instantaneous value of the source current becomes negative.

The value of the compensating capacitance is determined according to the expression

\[ C_d = \frac{\displaystyle \int_t^{t+\Delta t} i_d \cdot dt}{\Delta U_c}. \tag{2} \]

where \(\Delta t\) is the time interval during which the current is directed from the inverter to the source;

\(\Delta U_c\) is the permissible value of the change in voltage across the capacitance.

Since the average value of the current consumed by the inverter is positive for any period of the carrier frequency, the compensating capacitor does not accumulate energy from one carrier-frequency period to the next; rather, charging during the interval when the inverter current is negative, it discharges to its former voltage during the time when the current is positive.

Therefore, to determine the value of the capacitance of the compensating capacitor it is sufficient to find one interval for which the amount of electricity going to charge the capacitor is maximal. Obviously, the amount of electricity delivered to the capacitor will be maximal precisely in the first switching period after the no-current mode. Then, if \(I_m\) denotes the amplitude of the phase current at \(\lambda = 1\), and \(\lambda\) denotes the modulation depth, then for \(\varphi_{\mathrm{n}} \geq \dfrac{2\pi}{n}\) the required value of the compensating capacitance is

\[ C_d = \frac{\lambda \cdot I_m}{f_{\mathrm{k}} \cdot \Delta U_c} \sin \left( \varphi - \frac{3}{2}\frac{\pi}{n} \right). \tag{3} \]

If

\[ \frac{\pi}{n} \leqslant \varphi_{\mathrm{n}} \leqslant \frac{2\pi}{n}, \]

then the value of the compensating capacitance is determined from the expression

\[ C_d = \frac{\lambda \cdot I_m}{f_{\mathrm{k}} \cdot \Delta U_c} \left\{ \sin \left( \varphi - \frac{3}{2}\frac{\pi}{n} \right) + \left[ 1 - \cos \left( \frac{2\pi}{n} - \varphi \right) \right] \right\}. \tag{4} \]

Expressions (3) and (4) show that the value of the capacitance of the compensating capacitor does not depend on the output frequency and is determined by the value of the load power factor and by the switching frequency. Moreover, as the switching frequency increases, the value of the capacitance \(C_d\) decreases.

REFERENCES

1. M. A. Zhitkov. Bridge inverter with pole commutation in sinusoidal PWM mode. Izvestiya TPI, vol. 262, 1972.

2. A. S. Sandler, Yu. M. Gusyatskii. Thyristor Inverters with Pulse-Width Modulation. Moscow, “Energiya,” 1968.