Key ideas
In alternating current the voltage and current oscillate sinusoidally. Inductors and capacitors oppose the current by an amount that depends on frequency, and the total opposition is the impedance.
- The effective (rms) value of a sinusoid is its peak divided by √2. Mains voltages and meter readings are rms values.
- Inductive reactance X_L = 2πfL grows with frequency. Capacitive reactance X_C = 1 ÷ (2πfC) falls with frequency.
- In a series RLC circuit the impedance is Z = √(R² + (X_L − X_C)²), and the rms current is V_rms ÷ Z. The reactances oppose each other and do not add arithmetically with R.
- At resonance, f₀ = 1 ÷ (2π√(LC)), X_L equals X_C, the impedance is just R, and the current is largest. An ideal transformer scales voltage by the turns ratio.
Key equations
V_rms = V_peak ÷ √2X_L = 2πfLX_C = 1 ÷ (2πfC)Z = √(R² + (X_L − X_C)²)f₀ = 1 ÷ (2π√(LC))V_s ÷ V_p = N_s ÷ N_p
Worked example
A series circuit has R = 30 Ω, X_L = 80 Ω and X_C = 40 Ω on a 120 V rms supply. The net reactance is 80 − 40 = 40 Ω, so Z = √(30² + 40²) = 50 Ω. The current is 120 ÷ 50 = 2.4 A rms, and the phase angle is arctan(40 ÷ 30) = 53°.
Common mistakes
- Adding resistance and reactance directly instead of in quadrature.
- Mixing peak and rms values in the same calculation.
- Forgetting that X_C goes down as frequency goes up, while X_L goes up.
Practice
Test yourself with practice questions on this topic: AC circuits