Question

The phase difference between voltage and current in an AC circuit containing a resistor and an inductor in series is ${\varphi }_1$ . When the inductor is replaced by a capacitor, the phase difference is changed to ${\varphi }_2$ . The phase difference when all the three elements are connected in series with the same AC source will be

• A. ${\left(\text{tan}\right)}^{-1}\left(\text{tan}{\left(\varphi \right)}_1-\text{tan}{\left(\varphi \right)}_2\right)$
• B. ${\left(\text{tan}\right)}^{-1}\left(\text{tan}{\left(\varphi \right)}_2-\text{tan}{\left(\varphi \right)}_1\right)$
• C. ${\left(\text{cos}\right)}^{-1}\left(\text{cos}{\left(\varphi \right)}_1-\text{cos}{\left(\varphi \right)}_2\right)$
• D. ${\left(\text{cos}\right)}^{-1}\left(\text{cos}{\left(\varphi \right)}_2-\text{cos}{\left(\varphi \right)}_1\right)$

Answer 1

Answer: (A)

$\text{tan }{\varphi }_1=\frac{{\text{X}}_{\text{L}}}{\text{R}}\text{, }\text{tan }{\varphi }_2=\frac{{\text{X}}_{\text{C}}}{\text{R}}$
$\text{tan }\varphi =\frac{{\text{X}}_{\text{L}}-{\text{X}}_{\text{C}}}{\text{R}}=\text{tan }{\varphi }_1-\text{tan }{\varphi }_2$
$\varphi ={\left(\text{tan}\right)}^{-1}\left(\text{tan}{\left(\varphi \right)}_1-\text{tan}{\left(\varphi \right)}_2\right)$