Question

In an AC series circuit supply voltage \(V_{\text{rms}} = 100\) volts; \(R = 80\,\Omega\), \(X_L = 80\,\Omega\) and source frequency is \(f = 50\) Hz. Find the power factor.

• A. \( \dfrac{1}{\sqrt{2}} \)
• B. \( \dfrac{1}{2} \)
• C. \( \dfrac{3}{4} \)
• D. \( \dfrac{\sqrt{3}}{2} \)

Answer 1

The Correct Option is A

Concept: In an AC series circuit containing resistance and inductive reactance, the total impedance \(Z\) is given by \[ Z = \sqrt{R^2 + X_L^2} \] The power factor of the circuit is defined as \[ \text{Power Factor} = \cos\phi = \frac{R}{Z} \] where

• \(R\) = resistance
• \(X_L\) = inductive reactance
• \(Z\) = total impedance

For an \(RL\) circuit, the current lags the voltage, so the power factor is called a lagging power factor. Step 1: Find the impedance of the circuit.} Given \[ R = 80\,\Omega, \qquad X_L = 80\,\Omega \] \[ Z = \sqrt{R^2 + X_L^2} \] \[ Z = \sqrt{80^2 + 80^2} \] \[ Z = \sqrt{6400 + 6400} \] \[ Z = \sqrt{12800} \] \[ Z = 80\sqrt{2} \] Step 2: Calculate the power factor.} \[ \cos\phi = \frac{R}{Z} \] \[ \cos\phi = \frac{80}{80\sqrt{2}} \] \[ \cos\phi = \frac{1}{\sqrt{2}} \] Thus, the power factor of the circuit is \[ \boxed{\dfrac{1}{\sqrt{2}}} \]