When a sinusoidal voltage is applied across RL parallel circuit so that R = XL the phase angle will

Question

When a sinusoidal voltage is applied across RL parallel circuit so that R = XL the phase angle will be

Examoogle · Accepted Answer

$45^\circ$ lagging

Explanation: In a parallel RL circuit, the total current $I$ lags the applied voltage $V$ by an angle $\phi$. Since the resistor and inductor are connected in parallel, they share the same voltage $V$, and the circuit impedance involves the ratio of the resistive current $I_R = \frac{V}{R}$ and inductive current $I_L = \frac{V}{X_L}$.

Working Principle: The total current in a parallel RL circuit is given by $I = I_R - jI_L$. The phase angle $\phi$ is defined as $\phi = -\arctan(\frac{I_L}{I_R})$. Given $R = X_L$, then $I_L = \frac{V}{X_L} = \frac{V}{R} = I_R$. Substituting these into the formula: $\phi = -\arctan(\frac{V/R}{V/R}) = -\arctan(1) = -45^\circ$. The negative sign indicates a lagging power factor.

Key Formulas: $\phi = -\arctan(\frac{R}{X_L})$ — Phase angle of parallel RL circuit; $I = \sqrt{I_R^2 + I_L^2}$ — Total current magnitude

Key Points: In parallel circuits, voltage is the common reference.; Inductors cause the current to lag behind the voltage.; When resistance equals inductive reactance, current magnitudes are equal, leading to equal real and reactive components of the total current.; A phase angle of $45^\circ$ indicates a power factor of $\cos(45^\circ) = 0.707$ lagging.

Advantages: Easy to determine phase using current components; Voltage remains constant across parallel branches

Disadvantages: Total current increases as frequency decreases; Does not reach resonance like RLC circuits

Applications: Induction motor modeling; Transformer equivalent circuits

Additional Info: The phase angle $\phi$ is the angle between the total current and the reference voltage.. Option B (45° leading) is incorrect because inductors never lead voltage.. Options C and D are incorrect as 90° only occurs in purely inductive or purely capacitive circuits.

Answer: A is correct — The phase angle in a parallel RL circuit with $R = X_L$ is $45^\circ$ lagging because the resistive and inductive current components are equal in magnitude.

Exam Tip: Always remember: in Series RL, $	an(\phi) = \frac{X_L}{R}$, whereas in Parallel RL, $	an(\phi) = \frac{R}{X_L}$ (with a negative sign for lagging current).

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