RLC Impedance Calculator

Calculate impedance, reactance, phase angle, and power factor for series and parallel RLC circuits.

Impedance in AC Circuits

Impedance (Z) is the total opposition to alternating current in a circuit. Unlike resistance, which is purely real and frequency-independent, impedance is a complex quantity that includes both resistive and reactive components. The resistive component dissipates energy as heat; the reactive components (inductors and capacitors) store energy in magnetic and electric fields respectively and return it to the circuit each cycle.

At DC (f = 0 Hz), an inductor is a short circuit and a capacitor is an open circuit. As frequency increases, inductive reactance rises linearly while capacitive reactance falls. This frequency dependence is the basis for filters, oscillators, tuned circuits, and impedance matching networks throughout electronics and power engineering.

Series RLC Formulas

Inductive reactance: XL = 2πfL (Ω), L in Henries
Capacitive reactance: XC = 1 / (2πfC) (Ω), C in Farads
Net reactance: X = XL − XC

Impedance magnitude: |Z| = √(R² + X²)
Phase angle: θ = arctan(X / R)
Power factor: PF = cos(θ) = R / |Z|

Phasor form: Z = R + jX = |Z|∠θ

In a series circuit, current is the same through all components but voltage drops differ. Voltages add as phasors: V_R is in phase with current, V_L leads by 90°, V_C lags by 90°. At resonance (XL = XC), the circuit appears purely resistive and current is maximised.

Parallel RLC Formulas

Conductance: G = 1/R
Inductive susceptance: BL = −1/XL (negative, lagging)
Capacitive susceptance:BC = +1/XC (positive, leading)
Total admittance: Y = G + j(BC + BL)

Impedance: |Z| = 1/|Y| = 1/√(G² + B²)
Phase angle: θ = −arctan(B/G)

Resonant frequency: f₀ = 1 / (2π√(LC))

In a parallel circuit, voltage is the same across all components but branch currents differ. At parallel resonance, the inductive and capacitive branch currents cancel, leaving only the resistive current. The circuit impedance is at its maximum (a "tank circuit" that looks like a high resistance to the source).

Resonance and Quality Factor

Resonant frequency: f₀ = 1 / (2π√(LC)) [Hz]
Angular frequency: ω₀ = 1 / √(LC) [rad/s]

Quality factor (series): Q = ω₀L/R = 1/(ω₀CR)
Quality factor (parallel): Q = R/(ω₀L) = Rω₀C
Bandwidth (−3 dB): BW = f₀/Q = R/(2πL)

Q factor quantifies the sharpness of resonance. A high-Q circuit (Q > 10) has a narrow bandwidth — useful in radio tuners and bandpass filters. A low-Q circuit (<1) is overdamped and used in wideband applications. At resonance in a series circuit, voltage across L or C can be Q times the source voltage — a common source of overvoltage stress in high-frequency designs.

Worked Example — 50 Hz Mains Filter

Series RLC: R = 10 Ω, L = 100 mH (0.1 H), C = 100 µF (100×10⁻⁶ F), f = 50 Hz

ω = 2π × 50 = 314.16 rad/s
XL = 314.16 × 0.1 = 31.42 Ω
XC = 1 / (314.16 × 100×10⁻⁶) = 31.83 Ω
X = XL − XC = 31.42 − 31.83 = −0.41 Ω (slightly capacitive)

|Z| = √(10² + 0.41²) = √100.17 = 10.01 Ω
θ = arctan(−0.41/10) = −2.3° (lagging — capacitive)
PF = cos(−2.3°) = 0.9992

Resonant frequency: f₀ = 1/(2π√(0.1 × 100×10⁻⁶)) = 50.33 Hz
→ Circuit is very nearly at resonance at 50 Hz, hence Z ≈ R = 10 Ω

Practical Applications

Power factor correction: Industrial loads are typically inductive (motors, transformers). Adding capacitor banks in parallel raises the power factor toward unity, reducing reactive current and thus line losses and utility penalties. The capacitor susceptance partially cancels the inductive susceptance of the load.

RF tuned circuits: AM radio receivers use parallel LC tanks (resonant circuits) to select a carrier frequency. The tank presents high impedance at its resonant frequency, passing that signal while attenuating adjacent frequencies. Q factors of 50–200 are typical.

EMI filters: Switch-mode power supplies generate high-frequency noise. LC low-pass filters on the input and output attenuate switching harmonics by presenting high XL to high-frequency currents. Combined with Y-capacitors to ground, this meets conducted emission limits (EN 55032, CISPR 32).

Impedance matching: Maximum power transfer occurs when source and load impedances are complex conjugates. In RF systems (antennas, amplifiers), L-networks, pi-networks, and T-networks transform impedances to achieve matching across a desired bandwidth.

Disclaimer

This calculator assumes ideal components (no parasitic resistance in inductors, no equivalent series resistance in capacitors, no lead inductance). Real components deviate significantly at high frequencies. Always verify with simulation (SPICE) and measurement for production designs. For power electronics, consult a qualified electrical engineer.

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