Question

For a series RLC circuit which of the following statement is not correct?

• A. \( \omega L>\frac{1}{\omega C} \), \( \phi \) is positive. In this case the voltage leads the current by an angle \( \phi \)
• B. \( \omega L<\frac{1}{\omega C} \), \( \phi \) is negative. In this case the current lags the voltage by an angle \( \phi \)
• C. \( \omega L = \frac{1}{\omega C} \), \( \phi = 0^\circ \). In this case the voltage and current are in phase
• D. The impedance is purely resistive and minimum when \( \phi = 0 \), Z=R

Hint:

\(\phi\) is positive for inductive circuits (V leads I).
\(\phi\) is negative for capacitive circuits (I leads V).
\(\phi = 0\) for resonant circuits (V and I in phase).
Awkward phrasing like "lagging by a negative angle" usually implies leading by the positive magnitude.

Answer 1

The Correct Option is B

Let \(\phi\) be the phase angle by which voltage V leads current I in a series RLC circuit (\(\phi = \phi_V - \phi_I\)). Then \(\tan\phi = (X_L - X_C)/R\). (a) If \(\omega L>1/(\omega C)\) (i.e., \(X_L>X_C\)), the circuit is inductive. \(X_L - X_C>0\), so \(\phi\) is positive. Voltage leads current by \(\phi\). This is CORRECT. (c) If \(\omega L = 1/(\omega C)\) (i.e., \(X_L = X_C\)), the circuit is at resonance. \(X_L - X_C = 0\), so \(\phi = 0^\circ\). Voltage and current are in phase. This is CORRECT. (d) When \(\phi = 0\) (resonance), \(X_L = X_C\). Impedance \(Z = \sqrt{R^2 + (X_L-X_C)^2} = R\). This is purely resistive and is the minimum impedance. This is CORRECT. (b) If \(\omega L<1/(\omega C)\) (i.e., \(X_L<X_C\)), the circuit is capacitive. \(X_L - X_C<0\), so \(\phi\) is negative. Let \(\phi = -\alpha\) where \(\alpha>0\). This means voltage leads current by \(-\alpha\), which implies voltage lags current by \(\alpha\), or current leads voltage by \(\alpha\). The statement says "current lags the voltage by an angle \(\phi\)". If "current lags by \(\theta\)" means \(\phi_V - \phi_I = \theta\) (where \(\theta\) is the amount of lag for current, so \(\phi_I = \phi_V - \theta\)). The statement is: current lag amount = \(\phi\). Since \(\phi\) is negative (e.g., \(-30^\circ\)), it means current lags by \(-30^\circ\), which implies current leads by \(30^\circ\). This physical outcome (current leading) is correct for a capacitive circuit. However, the phrasing "lags by an angle \(\phi\)" where \(\phi\) itself is defined as the angle by which V leads I and is negative, is unconventional. A "lag" angle is typically positive. If it states current lags by \(\phi\) (a negative value), it means current leads by \(|\phi|\). While the physical meaning ends up correct, the terminology "lags by a negative angle" is what makes this statement "not correct" in terms of standard phrasing. It should say "current leads voltage by \(|\phi|\)" or "voltage lags current by \(|\phi|\)". Therefore, statement (b) is considered "not correct" due to this terminological issue. \[ \boxed{\parbox{0.9\textwidth}{\centering \( \omega L<\frac{1}{\omega C} \), \( \phi \) is negative. In this case the current lags the voltage by an angle \( \phi \) }} \]