Physics · Alternating Current · NEET
By convention φ is the angle by which the applied VOLTAGE leads the CURRENT. We take the current as the reference (drawn along the horizontal axis of the phasor diagram) because in a series circuit the same current flows through R, L and C. So a positive φ means voltage is ahead of current, i.e. current lags voltage. This happens when the circuit is net inductive (X_L > X_C).
Compare X_L and X_C. If X_L > X_C the net reactance is inductive, φ is positive, and current LAGS voltage. If X_C > X_L the net reactance is capacitive, φ is negative, and current LEADS voltage. If X_L = X_C (resonance) the reactances cancel, φ = 0, and current is in phase with voltage. Simple rule: whichever of L or C has the bigger reactance decides the behaviour.
In the phasor diagram the resistor voltage V_R = IR lies along the current (horizontal), while (V_L − V_C) = I(X_L − X_C) is vertical (90 degrees ahead). The angle of the resultant source voltage with the horizontal has tan φ = vertical/horizontal = (X_L − X_C)/R. The impedance Z is the hypotenuse, so cos φ = R/Z and sin φ = (X_L − X_C)/Z. tan uses the two perpendicular sides, which are R and (X_L − X_C), not Z.
At resonance X_L = X_C, so the numerator (X_L − X_C) = 0 and tan φ = 0, giving φ = 0. The circuit behaves as a pure resistor: impedance is minimum (Z = R), current is maximum, current and voltage are in phase, and power factor cos φ = 1. This is the only condition where a series LCR circuit is purely resistive.
No, but they are directly linked. The phase angle φ is the angle between voltage and current. The power factor is cos φ. So once you find φ from tan φ = (X_L − X_C)/R, the power factor is just cos φ = R/Z. NEET often asks for one after giving data for the other, so learn both together.
To an ac power supply of 220 V at 50 Hz, a resistor of 20 Ω, a capacitor of reactance 25 Ω and an inductor of reactance 45 Ω are connected in series. The current in the circuit and the phase angle between the current and the voltage are, respectively:
An ac source is connected in the given series circuit. For V = 220 sin(100πt + φ) volt, the value of the phase angle φ will be:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
tan φ = (X_L − X_C)/R, where X_L = ωL is the inductive reactance, X_C = 1/(ωC) is the capacitive reactance, and R is the resistance. Equivalently cos φ = R/Z with Z = sqrt(R^2 + (X_L − X_C)^2).
In a net inductive circuit (X_L > X_C), the current lags the applied voltage. In a net capacitive circuit (X_C > X_L), the current leads. At X_L = X_C (resonance) they are in phase.
The phase angle can approach but never reach 90°. φ = 90° would need R = 0 (a pure L or pure C). With any real resistance present, |φ| stays below 90°, and it becomes exactly 0° at resonance.
The power factor is cos φ. Since cos φ = R/Z, a small phase angle means a high power factor (more real power delivered), and a 90° phase angle gives cos φ = 0 (wattless current, no average power).
NEET almost every year gives R, X_L and X_C (or V_L, V_C, V_R) and asks for φ, the power factor, or the current. One formula, tan φ = (X_L − X_C)/R together with Z = sqrt(R^2 + (X_L − X_C)^2), solves nearly all of these questions quickly.
