Question 269: AC fundamentals

Question

The power factor of a CR circuit is $\frac{1}{\sqrt{2}}$, If the frequency of a.c. signal is halved, then the power factor of the circuit will become

Accepted Answer

For capacitive reactance,
$$ \mathrm{X}_{\mathrm{C}} \propto \frac{1}{\omega} $$
When frequency is halved,
$$ X_C^{\prime}=2 X_C $$
Now,
$$ \begin{aligned} \cos \phi= & \frac{\mathrm{R}}{\sqrt{\mathrm{R}^2+\mathrm{X}_{\mathrm{c}}^2}} \ 	herefore \quad \frac{(\cos \phi)^{\prime}}{(\cos \phi)} & =\frac{\mathrm{R}}{\sqrt{\mathrm{R}^2+\left(\mathrm{X}_{\mathrm{c}}^{\prime}ight)^2}} 	imes \frac{\sqrt{\mathrm{R}^2+\mathrm{X}_{\mathrm{c}}^2}}{\mathrm{R}} \ & =\sqrt{\frac{\mathrm{R}^2+\mathrm{X}_{\mathrm{c}}^2}{\mathrm{R}^2+4 \mathrm{X}_{\mathrm{c}}^2}} \end{aligned} $$
$$ \begin{array}{ll}  & 	ext { But, } 	an \phi=\frac{\mathrm{X}_{\mathrm{C}}}{\mathrm{R}} \ 	herefore \quad & (\cos \phi)^{\prime}=(\cos \phi) 	imes \sqrt{\frac{\mathrm{R}^2+\mathrm{R}^2 	an ^2 \phi}{\mathrm{R}^2+4 \mathrm{R}^2 	an ^2 \phi}} \ & 	ext { Given: } \cos \phi \frac{1}{\sqrt{2}} \Rightarrow 	an \phi=1 \ 	herefore \quad & (\cos \phi)^{\prime}=\frac{1}{\sqrt{2}} 	imes \sqrt{\frac{\mathrm{R}^2+\mathrm{R}^2}{\mathrm{R}^2+4 \mathrm{R}^2}}=\frac{1}{\sqrt{2}} 	imes \frac{\sqrt{2}}{\sqrt{5}}=\frac{1}{\sqrt{5}} \end{array} $$

Answer

$\frac{1}{\sqrt{3}}$

Answer

$\frac{1}{\sqrt{5}}$

Answer

$\frac{1}{\sqrt{7}}$

Answer

$\frac{1}{\sqrt{11}}$

AC fundamentals

The power factor of a CR circuit is \frac{1}{\sqrt{2}}, If the frequency of a.c. signal is halved, then the power factor of the circuit will become

Choose the correct answer: