A series LCR circuit is connected to an AC source v = vₘ sin ωt. Using a phasor diagram, derive the expression for its impedance and the phase angle. When is the current in phase with the voltage?

1. 1 · State the set-up. R, L and C in series; the same current i = iₘ sin(ωt + φ) flows through all three.
2. 2 · Write the voltages. V_R = iₘR in phase with I; V_L = iₘX_L leading by π/2; V_C = iₘX_C lagging by π/2.
3. 3 · Draw the phasor diagram. [I and V_R along the horizontal; V_L up; V_C down; resultant vₘ at angle φ; impedance triangle beside it]
4. 4 · Add the phasors. vₘ² = V_R² + (V_C − V_L)² = iₘ²[R² + (X_C − X_L)²].
5. 5 · Write Z and φ. Z = √(R² + (X_C − X_L)²); tan φ = (X_C − X_L)/R.
6. 6 · Discuss the cases.
   • X_C > X_L: capacitive, the current leads.
   • X_L > X_C: inductive, the current lags.
   • X_L = X_C: φ = 0, the current is in phase (resonance).
7. 7 · Real life. A fan's capacitor shifts the phase of the current in one winding so the motor starts by itself.
8. 8 · Close in one line. So, iₘ = vₘ/Z, and the nature of the circuit depends on whether X_L or X_C is larger.

Scoring tip: Draw the phasor diagram neatly with arrowheads and labels; it can earn a full mark on its own.