Figure 1 shows the full-wave centre tap phase-controlled thyristor-circuit with inductive load and freewheeling diode. The load voltage and current-waveforms are also shown in Fig.2.

As shown in Fig.2, the thyristors are triggered at angle $\alpha .$ The variable d.c. voltage at the load is obtained by varying this firing angle $\alpha .$ From the same figure, it is also clear that as the supply voltage goes through zero at $180^{\circ}$ , the load voltage cannot be negative since the freewheeling diode, $D_{f}$ , starts conducting and clamps the load voltage to zero volts. A constant load current is maintained by freewheeling current through the diode. The conduction period of thyristors and diode is also shown in Fig. 2. The stored-energy in the inductive load circulates current through the feedback-diode in the direction shown in Fig.1. The rate of decay of this current depends upon the time-constant of the load.

The average d.c. output voltage can be calculated as,

$$E_{\mathrm{dc}}=\frac{1}{\pi} \int_{\alpha}^{\pi} E_{m} \cdot \sin \omega t \cdot \mathrm{d}(\omega t)=\frac{E_{m}}{\pi}[1+\cos \alpha]$$

The d.c. load-current is given by

$$ I_{\mathrm{dc}}=\frac{E_{m}}{\pi R}[1+\cos \alpha] $$

It is also observed from Fig. 2, that the freewheeling diode, $D_{F}$ , carries the load-current during the firing-angle $\alpha$ when the thyristors are not conducting.

Hence, the current through the diode $D_{F}$ is given by

$$I D_{f}=I_{\mathrm{dc}} \frac{\alpha}{\pi}=\frac{E_{m}}{\pi R}(1+\cos \alpha) \frac{\alpha}{\pi}$$

$$I D_{f}=\frac{E_{m}}{\pi^{2} R}\left(\alpha+\alpha^{\prime} \cos \alpha\right)$$