Step 1: Understanding the Concept:
In a series LCR (Inductor, Capacitor, Resistor) circuit, the alternating current (AC) experiences opposition to its flow, known as impedance (\(Z\)).
The voltage across the inductor (\(V_L\)) leads the current by \(90^\circ\), the voltage across the capacitor (\(V_C\)) lags the current by \(90^\circ\), and the voltage across the resistor (\(V_R\)) is in phase with the current.
Step 2: Key Formula or Approach:
Using a phasor diagram, the net reactive voltage is \(V_L - V_C\) (assuming \(X_L>X_C\)), which lies on the y-axis, and the resistive voltage \(V_R\) lies on the x-axis.
The phase angle \(\phi\) is the angle between the net applied emf and the current.
From the impedance triangle, we know:
\[ \tan \phi = \frac{\text{Net Reactance}}{\text{Resistance}} = \frac{X_L - X_C}{R} \]
Step 3: Detailed Explanation:
Looking at the trigonometric relations derived from the impedance triangle:
- \(\tan \phi = \frac{X_L - X_C}{R}\)
- \(\cos \phi = \frac{R}{Z}\) (Power Factor)
- \(\sin \phi = \frac{X_L - X_C}{Z}\)
Option (A) perfectly matches the standard relation for the tangent of the phase angle.
Step 4: Final Answer:
The correct formula is \(\tan \phi = \frac{X_L - X_C}{R}\).