Impedance Formula
Calculate total impedance in AC circuits from resistance, capacitive reactance, and inductive reactance.
Find phase angle for series and parallel RLC circuits.
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The Formula
Z = √(R² + (X_L - X_C)²)
Impedance is the total opposition to alternating current (AC) in a circuit. It combines resistance (R) with reactance from inductors (X_L) and capacitors (X_C).
Variables
| Symbol | Meaning |
|---|---|
| Z | Impedance (Ohms, Ω) |
| R | Resistance (Ohms) |
| X_L | Inductive reactance = 2πfL (Ohms) |
| X_C | Capacitive reactance = 1/(2πfC) (Ohms) |
Example 1
R = 30 Ω, X_L = 50 Ω, X_C = 10 Ω
Z = √(30² + (50 - 10)²)
Z = √(900 + 1600)
Z = √2500 = 50 Ω
Example 2
R = 100 Ω, X_L = 80 Ω, X_C = 80 Ω (resonance)
Z = √(100² + (80 - 80)²) = √(10000 + 0)
Z = 100 Ω (at resonance, impedance equals pure resistance)
When to Use It
Use the impedance formula when:
- Analyzing AC circuits with resistors, capacitors, and inductors
- Calculating current flow in AC power systems
- Designing audio equipment and radio frequency circuits
- Matching impedance between source and load for maximum power transfer
Key Notes
- At resonance (X_L = X_C), the reactive terms cancel and impedance equals pure resistance R — this is the minimum impedance point in a series RLC circuit
- Impedance is a complex number: Z = R + j(X_L − X_C); the formula gives its magnitude; the phase angle is φ = arctan((X_L − X_C) / R)
- This formula is for series RLC circuits; in a parallel circuit, impedances combine as reciprocals (1/Z = 1/Z₁ + 1/Z₂ + ...) and the math is different
Key Notes
- Complex impedance: Z = R + jX: R is resistance (real part, always ≥ 0); X is reactance (imaginary part, can be positive or negative). Inductive reactance X_L = 2πfL is positive; capacitive reactance X_C = −1/(2πfC) is negative.
- Magnitude and phase: |Z| = √(R² + X²); φ = arctan(X/R): |Z| is the ratio of peak voltage to peak current (like resistance in DC). φ is the phase angle — positive means voltage leads current (inductive); negative means current leads voltage (capacitive).
- Resonance: X_L + X_C = 0: At the resonant frequency f₀ = 1/(2π√(LC)), the inductive and capacitive reactances cancel. The circuit becomes purely resistive (Z = R), and current is maximized (series) or minimized (parallel).
- Impedance matching: Maximum power transfer occurs when the load impedance equals the source impedance (conjugate match: Z_load = Z_source*). Mismatched impedances cause reflections in transmission lines and reduce power efficiency.
- Applications: Impedance calculations are fundamental in AC circuit analysis, filter design (low-pass, high-pass, band-pass), radio frequency (RF) engineering, audio speaker crossover networks, and biomedical devices (EEG and ECG electrode impedance).