AC Circuit Cheat Sheet (R, L, C)
1. Basic Quantities
- Angular frequency: ω = 2πf
- Inductive reactance: XL = ωL
- Capacitive reactance: XC = 1 / (ωC)
2. Pure Components
Pure Resistor (R)
- Impedance: Z = R
- Phase angle: φ = 0°
- Voltage and current are in phase
Pure Inductor (L)
- Impedance: Z = jωL
- Phase angle: φ = +90°
- Current lags voltage
Pure Capacitor (C)
- Impedance: Z = 1 / (jωC)
- Phase angle: φ = −90°
- Current leads voltage
3. RL Circuit (Series)
- Impedance: Z = √(R² + (ωL)²)
- Phase angle: φ = tan⁻¹(ωL / R)
- Current lags voltage
Special Cases
- ωL >> R → behaves like inductor (φ ≈ 90°)
- R >> ωL → behaves like resistor (φ ≈ 0°)
4. RC Circuit (Series)
- Impedance: Z = √(R² + (1/ωC)²)
- Phase angle: φ = tan⁻¹(−1 / (ωCR))
- Current leads voltage
Special Cases
- 1/ωC >> R → behaves like capacitor (φ ≈ −90°)
- R >> 1/ωC → behaves like resistor (φ ≈ 0°)
5. RLC Circuit (Series)
- Impedance: Z = √[R² + (XL − XC)²]
- Phase angle: φ = tan⁻¹((XL − XC)/R)
6. Key Conditions in RLC
Inductive Case
- XL > XC
- φ > 0
- Current lags
Capacitive Case
- XC > XL
- φ < 0
- Current leads
Resonance Condition
- XL = XC → ωL = 1/ωC
- Frequency: f₀ = 1 / (2Ï€√(LC))
- Z = R (minimum)
- φ = 0°
- Current is maximum
7. Power Factor
Power factor = cosφ
| Condition | Power Factor |
|---|---|
| Pure Resistor | 1 |
| Inductive | Lagging |
| Capacitive | Leading |
| Resonance | 1 |
8. Current Amplitude
I₀ = V₀ / Z
Summary
- L → Lag (Inductor)
- C → Lead (Capacitor)
- Resonance → Maximum current
- High impedance → Low current
| Property | Ideal Resistor (R) | Ideal Capacitor (C) | Ideal Inductor (L) |
|---|---|---|---|
| Primary Function | Dissipates electrical energy as heat. | Stores energy in an electric field. | Stores energy in a magnetic field. |
| Energy Storage | None | \(U=\frac{1}{2}CV^2\) | \(U=\frac{1}{2}LI^2\) |
| Current Type | Conduction current only. | Conduction current in wires; displacement current through the dielectric. | Conduction current only. |
| Charge Carriers Inside Component | Electrons move through the resistor. | No charge carriers cross the dielectric. | Electrons move through the wire coil. |
| Dominant Field | Weak electric field drives current. | Electric field between the plates. | Magnetic field around the coil. |
| Electric Field | Present | Strong | Present (usually small) |
| Magnetic Field | Weak (due to current) | Weak (due to current in leads) | Strong |
| Voltage–Current Relationship | \(V=IR\) | \(I=C\frac{dV}{dt}\) | \(V=L\frac{dI}{dt}\) |
| Impedance | \(R\) | \(\frac{1}{j\omega C}\) | \(j\omega L\) |
| Reactance | 0 | \(X_C=\frac{1}{\omega C}\) | \(X_L=\omega L\) |
| Phase Difference (Voltage vs Current) | 0° (in phase) | Current leads voltage by 90° | Current lags voltage by 90° |
| Power Consumption | Consumes real power. | Average real power = 0 (ideal). | Average real power = 0 (ideal). |
| Energy Transfer | Electrical → Heat | Electrical ↔ Electric field | Electrical ↔ Magnetic field |
| DC Steady-State Behavior | Current flows continuously. | Acts as an open circuit. | Acts as a short circuit. |
| High-Frequency Behavior | Unchanged | Behaves like a short circuit. | Behaves like an open circuit. |
| Low-Frequency Behavior | Unchanged | Behaves like an open circuit. | Behaves like a short circuit. |
| Opposes Changes In | Neither | Voltage | Current |
| Continuity Rule | Current is continuous through the resistor. | Conduction current equals displacement current. | Current is continuous through the inductor. |
| Displacement Current | Not significant. | Present between capacitor plates. | Not applicable. |
| Magnetic Flux | Very small | Very small | Primary stored quantity |
| Electric Flux | Small | Primary stored quantity | Small |
| Physical Mechanism | Collisions of charge carriers convert energy into heat. | Charge separation creates a changing electric field. | Current creates a changing magnetic field. |
| Typical Symbol | R | C | L |