Impedance Formula (AC Circuits)
Impedance combines resistance and reactance in AC circuits.
Learn Z = √(R² + X²) and complex impedance with worked examples.
The Formula
Z = R + jX (complex form)
Impedance (Z) is the total opposition to current flow in an alternating current (AC) circuit. It extends the concept of resistance to include the effects of capacitors and inductors, which store and release energy rather than dissipating it.
In DC circuits, only resistance matters. In AC circuits, capacitors and inductors create reactance (X), which depends on the frequency of the signal. Capacitive reactance: X_C = 1/(2πfC). Inductive reactance: X_L = 2πfL. The net reactance X = X_L - X_C.
The phase angle θ = arctan(X/R) tells you how much the current leads or lags the voltage. In a purely resistive circuit, θ = 0° (voltage and current are in phase). In a purely inductive circuit, θ = 90° (current lags voltage). In a purely capacitive circuit, θ = -90° (current leads voltage).
Variables
| Symbol | Meaning |
|---|---|
| Z | Impedance (in ohms, Ω) |
| R | Resistance (in ohms, Ω) |
| X | Net reactance = X_L - X_C (in ohms, Ω) |
| X_L | Inductive reactance = 2πfL (in ohms) |
| X_C | Capacitive reactance = 1/(2πfC) (in ohms) |
| f | Frequency of the AC signal (in hertz, Hz) |
| θ | Phase angle (in degrees or radians) |
Example 1
A series circuit has R = 30 Ω and X_L = 40 Ω (no capacitor). What is the impedance?
X = X_L = 40 Ω (no capacitive reactance)
Z = √(R² + X²) = √(30² + 40²) = √(900 + 1600) = √2500
Phase angle: θ = arctan(40/30) = arctan(1.333) ≈ 53.1°
Z = 50 Ω at a phase angle of 53.1°
Example 2
An RLC circuit has R = 100 Ω, L = 0.2 H, and C = 10 μF at f = 60 Hz. Find the impedance.
X_L = 2π × 60 × 0.2 = 75.4 Ω
X_C = 1/(2π × 60 × 10×10⁻⁶) = 1/0.003770 = 265.3 Ω
X = X_L - X_C = 75.4 - 265.3 = -189.9 Ω (capacitive)
Z = √(100² + 189.9²) = √(10,000 + 36,062) = √46,062
Z ≈ 214.6 Ω
When to Use It
The impedance formula is essential for analyzing any AC circuit.
- Designing audio equipment and speaker systems
- Calculating current in power distribution systems
- Matching impedance for maximum power transfer in radio frequency circuits
- Filter design for electronics and signal processing