Question

A series $ RLC $ circuit, driven with $ E_{rms} = 120 \,V $ at frequency $ 50 \,Hz $ , contains an inductance with $ X_{L} =100 \, \Omega $ , a capacitance with $ X_{c} = 110 \, \Omega $ and an unknown resistance $ R $ . For what value of $ R $ , the power factor is $ 0.9 $ ?

• A. $ 20 \, \Omega $
• B. $ 42 \, \Omega $
• C. $ 59 \, \Omega $
• D. $ 110 \, \Omega $

Answer 1

Answer: (A)

Given, $E_{rms}=120\,V $
$f=50\,Hz $
$X_{L}=100\,\Omega$
$X_{c}=110\, \Omega$
Power factor $= 0.9 $
To find $R$ = ?
As power factor in $LCR$ circuit
$\cos\,\phi=\frac{R}{\sqrt{R^{2}+\left(\omega L-\frac{1}{\omega C}\right)^{2}}}$
$0.9=\frac{R}{\sqrt{R^{2}+\left(100-110\right)^{2}}}$
$0.9=\frac{R}{\sqrt{R^{2}+100}}$
$\Rightarrow 0.81=\frac{R^{2}}{R^{2}+100}$
$\Rightarrow R=20\,\Omega$