1. Input Displacement Factor (DSF):

$$DSF =\cos \alpha$$

2. Input Power Factor (PF):

$$P F=\left(\frac{I_{1}}{I_{\mathrm{rms}}}\right) \cos \phi_{1}$$

$$=\left(\frac{2 \sqrt{2} I_{d}}{\frac{\pi}{I_{d}}}\right) \cos \alpha P F=\frac{2 \sqrt{2}}{\pi} \cos \alpha$$

3. D.C. Voltage Ratio:

$$r=\frac{\frac{1}{\pi} \int_{\alpha}^{(\pi+\alpha)} E_{m} \sin \omega t \cdot \mathrm{d}(\omega t)}{\frac{1}{\pi} \int_{0}^{\pi} E_{m} \sin \omega t \cdot \mathrm{d}(\omega t)}, r=\cos \alpha$$

4. Input Current Distortion Factor:

$$=\frac{I_{1}}{I_{\mathrm{rms}}}=\frac{2 \sqrt{2} I_{d}}{\pi \cdot I_{d}}=\frac{2 \sqrt{2}}{\pi}$$

5. Input Harmonic Factor $\left(\boldsymbol{I}_{H}\right)$:

We know that

$$ I_{H}=\frac{\left(I_{\text { rms }}^{2}-I_{1}^{2}\right)^{1 / 2}}{I_{1}} $$

Substituting the values of $I_{\mathrm{rms}}$ and $I_{1}$

$$I_{H}=\frac{\left[I_{d^{2}}-\left(\frac{2 \sqrt{2} I_{d}}{\pi}\right)^{2}\right]^{1 / 2}}{2 \sqrt{2} I_{d} / \pi}=\frac{\left[I_{d^{2}}-\frac{8 I_{d^{2}}}{\pi^{2}}\right]^{1 / 2}}{2 \sqrt{2} I_{d} / \pi}$$

$$=\frac{\left[\frac{\pi^{2} I_{d^{2}}-8 I_{d^{2}}}{\pi^{2}}\right]^{1 / 2}}{2 \sqrt{2} \mathrm{Id} / \pi}$$

$$=\left[\frac{\pi^{2}-8}{8}\right]^{1 / 2}$$

$$ I_{H}=\left[\frac{\pi^{2}}{8}-1\right]^{1 / 2}$$

6. Voltage Ripple Factor $\left(k_{\mathrm{V}}\right)$:

We have the equation for $k_{V}$ as

$$k_{V}=\frac{\sqrt{E_{\mathrm{dc} \operatorname{ms}}^{2}-E_{\mathrm{dc}}^{2}}}{E_{\mathrm{dc}}}=\frac{\left[\frac{E_{m}^{2}}{2}-\left(\frac{2 E_{m}}{\pi} \cos \alpha\right)^{2}\right]^{1 / 2}}{\frac{2 E_{m}}{\pi} \cos \alpha}$$

$$=\frac{E_{m}\left[\frac{1}{2}-\frac{4}{\pi} \cos ^{2} \alpha\right]^{1 / 2}}{2 E_{m} \cos \alpha / \pi}=\frac{\pi\left[\frac{1}{2}-\frac{4}{\pi^{2}} \cos ^{2} \alpha\right]^{1 / 2}}{2 \cos \alpha}$$

$$=\frac{\pi\left[\frac{\pi^{2}-8 \cos ^{2} \alpha}{2 \pi^{2}}\right]^{1 / 2}}{2 \cos \alpha}=\left[\frac{\pi^{2}-8 \cos ^{2} \alpha}{2 \cos ^{2} \alpha}\right]^{1 / 2}$$

$$k_{V}=\left(\frac{\pi^{2}}{8 \cos ^{2} \alpha}-1\right)^{1 / 2}$$

7. Active Power Input $(P_i)$: Only the fundamental component contributes to the mean a.c. input power and the mean power as the harmonic components of current is zero.

Therefore, the mean a.c. input power in a given line is given by:

$$ \begin{aligned} P_{i}=& \text { RMS line voltage } \times \text { RMS fundamental component of current } \\ & \times \text { displacement factor. } \end{aligned} $$

$$=E_{\mathrm{rms}} \times I_{1} \times \cos \alpha$$ $$=E_{\mathrm{rms}} \times \frac{2 \sqrt{2} I_{d}}{\pi} \cos \alpha=\frac{2 E_{m}}{\pi} I_{d} \cos \alpha$$

8. Reactive Power Input $(Q_i)$: Reactive power input is given by $$Q_{i}=E_{\mathrm{rms}} \cdot I_{1} \sin \alpha=\frac{2 E_{m}}{\pi} I_{d} \sin \alpha$$