Question

The phase difference between voltage and current in an AC circuit containing a resistor and an inductor in series is $\phi_{1}$ . When the inductor is replaced by a capacitor, the phase difference is changed to $\phi_{2}$ . The phase difference when all the three elements are connected in series with the same AC source will be

• A. $\left(\text{tan}\right)^{- 1} \left(\text{tan} \left(\phi\right)_{1} - \text{tan} \left(\phi\right)_{2}\right)$
• B. $\left(\text{tan}\right)^{- 1} \left(\text{tan} \left(\phi\right)_{2} - \text{tan} \left(\phi\right)_{1}\right)$
• C. $\left(\text{cos}\right)^{- 1} \left(\text{cos} \left(\phi\right)_{1} - \text{cos} \left(\phi\right)_{2}\right)$
• D. $\left(\text{cos}\right)^{- 1} \left(\text{cos} \left(\phi\right)_{2} - \text{cos} \left(\phi\right)_{1}\right)$

Answer 1

Answer: (A)

$\text{tan } \phi_{1} = \frac{\text{X}_{\text{L}}}{\text{R}} \text{, } \text{tan } \phi_{2} = \frac{\text{X}_{\text{C}}}{\text{R}}$
$\text{tan } \phi = \frac{\text{X}_{\text{L}} - \text{X}_{\text{C}}}{\text{R}} = \text{tan } \phi_{1} - \text{tan } \phi_{2}$
$\phi = \left(\text{tan}\right)^{- 1} \left(\text{tan} \left(\phi\right)_{1} - \text{tan} \left(\phi\right)_{2}\right)$