Series RLC Circuit Impedance

Impedance of a Series RLC Circuit

1. Series RLC Circuit Components

• R = Resistance (Ω)
• L = Inductance (H)
• C = Capacitance (F)

All components are in series, so the same current flows through each.

2. Impedance Definition

The total impedance Z is the opposition to AC current, combining resistance and reactance:

Z = R + j(XL - XC)

• j = √-1 (imaginary unit)
• XL = 2πfL (inductive reactance in Ω)
• XC = 1/(2πfC) (capacitive reactance in Ω)
• f = AC frequency in Hz

3. Magnitude of Impedance

|Z| = √(R² + (XL - XC)²)

4. Phase Angle

φ = tan⁻¹((XL - XC)/R)

φ > 0 → circuit is inductive
φ < 0 → circuit is capacitive
φ = 0 → resonance (XL = XC)

5. Step-by-Step Calculation Example

Given:

• R = 10 Ω
• L = 0.1 H
• C = 100 μF = 100 × 10⁻⁶ F
• f = 50 Hz

Step 1: Reactances

XL = 2πfL ≈ 31.42 Ω

XC = 1/(2πfC) ≈ 31.83 Ω

Step 2: Net Reactance

X = XL - XC = 31.42 - 31.83 = -0.41 Ω

Step 3: Impedance Magnitude

|Z| = √(R² + X²) = √(10² + (-0.41)²) ≈ 10.008 Ω

Step 4: Phase Angle

φ = tan⁻¹(X/R) = tan⁻¹(-0.41/10) ≈ -2.35°

Slightly capacitive because X < 0.

6. Summary Formulas

• Inductive Reactance: XL = 2πfL
• Capacitive Reactance: XC = 1/(2πfC)
• Impedance: Z = R + j(XL - XC)
• Magnitude: |Z| = √(R² + (XL - XC)²)
• Phase Angle: φ = tan⁻¹((XL - XC)/R)