Step-by-Step Reasoning for Any LCR Problem

Step 1: Identify given quantities List R ($\Omega$), L (H), C (F), V_rms (V), f (Hz). Convert all to SI units first.

Step 2: Compute reactances $X_L = 2\pi f L \qquad X_C = \frac{1}{2\pi f C}$ Check units: (Hz)(H) → ($s^{-1}$)(H) → ($s^{-1}$)(V·s/A) → V/A = $\Omega$ ✓

Step 3: Determine circuit nature

• If X_L > X_C: circuit is inductive → current lags voltage
• If X_C > X_L: circuit is capacitive → current leads voltage
• If X_L = X_C: resonance → Z = R, φ = 0

Step 4: Compute impedance $Z = \sqrt{R^2 + (X_L - X_C)^2}$

Step 5: Compute current $I_\text{rms} = \frac{V_\text{rms}}{Z}$

Step 6: Compute phase angle and power factor $\tan\phi = \frac{X_L - X_C}{R} \qquad \cos\phi = \frac{R}{Z}$

Step 7: Compute power $P = I_\text{rms}^2 \times R = V_\text{rms} \times I_\text{rms} \times \cos\phi$ (Both methods must give the same answer — use as a cross-check.)

Step 8: Compute individual voltages if needed $V_R = I_\text{rms} R \qquad V_L = I_\text{rms} X_L \qquad V_C = I_\text{rms} X_C$ Phasor sum check: $V = \sqrt{V_R^2 + (V_L - V_C)^2} = V_\text{supply}$ ✓

Common pitfalls in this chain:

• Forgetting to convert C from μF to F (×10^{-6})
• Adding V_R + V_L + V_C algebraically (must use phasors)
• Using peak values instead of RMS in power formula
• Concluding Z = 0 at resonance (Z = R)