R, C and L Circuits
Phase relationships of voltage and current in R, C and L circuits, in parallel, series and series-parallel; power dissipation; impedance, phase angle, power factor and current calculations; true, apparent and reactive power.
R, C and L circuits, summary notes
- In a purely resistive circuit, voltage and current are in phase.
- In a purely inductive circuit, current LAGS voltage by 90°. Inductive reactance X_L = 2πfL rises with frequency.
- In a purely capacitive circuit, current LEADS voltage by 90°. Capacitive reactance X_C = 1/(2πfC) falls with frequency.
- Impedance combines resistance and reactance: Z = √(R² + (X_L − X_C)²), measured in ohms.
- Phase angle: tan φ = (X_L − X_C) / R. Power factor = cos φ.
- True power (watts) = VI·cos φ; apparent power (volt-amperes) = VI; reactive power is in VAR.
- At resonance X_L = X_C, so impedance is purely resistive, minimum in a series circuit, maximum in a parallel circuit.
- Mnemonic CIVIL: in C, I leads V; in L, V leads I.
- ⚠ Exam trap: X_L rises with frequency while X_C falls. Reversing these two is the classic error.
- Inductive reactance
- X_L = 2πfL (ohms)
- Capacitive reactance
- X_C = 1 / (2πfC) (ohms)
- Impedance
- Z = √(R² + (X_L − X_C)²)
- Phase angle
- tan φ = (X_L − X_C) / R
- Power factor
- PF = cos φ = R / Z
- Resonant frequency
- f_r = 1 / (2π√(LC))
Find the reactance of a 0.1 H inductor at 400 Hz.
X_L = 2πfL = 2π × 400 × 0.1 = 251 Ω.
A circuit has R = 30 Ω, X_L = 60 Ω and X_C = 20 Ω. Find the impedance.
Net reactance = 60 − 20 = 40 Ω. Z = √(30² + 40²) = √(900 + 1600) = √2500 = 50 Ω.
Reactance vs frequency
Series LC resonance
AC power & power factor
R, C, L concept map
R, C and L Circuits
R, C and L circuits quiz
R, C and L Circuits, quiz
1. The impedance triangle shown has a resistance of 3 Ω and a reactance of 4 Ω. The impedance Z is:
5 Ω7 Ω12 Ω2. The reactance of a 100 µF capacitor at 50 Hz is approximately:
31.8 Ω3.18 Ω314 Ω3. The reactance of a 0.1 H inductor at 50 Hz is approximately:
31.4 Ω3.14 Ω0.03 Ω4. A circuit has a resistance of 8 Ω and a net reactance of 6 Ω. Its impedance is:
10 Ω14 Ω2 Ω5. As frequency increases, the reactance of a capacitor:
DecreasesIncreasesStays constant6. In a purely inductive circuit, the current:
Lags the voltage by 90°Leads the voltage by 90°Is in phase with the voltage7. A circuit has R = 30 Ω, X_L = 60 Ω and X_C = 20 Ω. The impedance is:
50 Ω110 Ω70 Ω8. The reactance of a 0.1 H inductor at 400 Hz is approximately:
251 Ω40 Ω628 Ω9. In a purely capacitive circuit, the current:
Leads the voltage by 90°Lags the voltage by 90°Is in phase with the voltage10. Power factor is defined as:
cos φsin φtan φ