Draw the phasor diagram of the voltages in a series LCR circuit and obtain its impedance and phase angle. What is resonance? Obtain the resonant frequency and show how the resistance affects the current near it.

1. 1 · Set up. The same current flows through R, L and C, so take the current phasor I₀ as the reference.
2. 2 · Draw the phasor diagram. [I₀ and V_R along the x-axis; V_L up and V_C down; the resultant (V_L − V_C); E₀ as the diagonal at angle φ to I₀]
3. 3 · Add the voltages. E₀² = V_R² + (V_L − V_C)² = I₀²[R² + (X_L − X_C)²].
4. 4 · Write the impedance. Z = E₀/I₀ = √(R² + (X_L − X_C)²); tan φ = (X_L − X_C)/R.
5. 5 · Find resonance. The current is largest when X_L = X_C: 2πνL = 1/2πνC, so ν_r = 1/(2π√(LC)) and Z = R.
6. 6 · Sketch the curve. [Peak current against frequency: a peak at ν_r, taller and sharper for a smaller R]
7. 7 · Link to life. A radio's tuning circuit uses resonance to pick out one station.
8. 8 · Close in one line. So, at resonance the circuit acts as a pure resistance, and the current is in phase with the voltage.

Scoring tip: Draw V_L and V_C on one vertical line in opposite directions. Then (V_L − V_C) and the right triangle follow at once.