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 / (2π√(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

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