Calculate impedance (Z), reactance, phase angle, resonant frequency and Q factor for series and parallel RLC circuits.
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RLC Circuit Formulas
Key equations for series and parallel RLC circuits:
Series RLC
Inductive Reactance
XL = 2π × f × L
Capacitive Reactance
XC = 1 / (2π × f × C)
Impedance
Z = √(R² + (XL−XC)²)
Phase Angle
φ = arctan((XL−XC)/R)
Resonant Frequency
f₀ = 1 / (2π√(LC))
Q Factor
Q = (1/R) × √(L/C)
Parallel RLC
Admittance
Y = √((1/R)² + (1/XC−1/XL)²)
Impedance
Z = 1 / Y
Resonant Frequency
f₀ = 1 / (2π√(LC))
Q Factor
Q = R × √(C/L)
Worked Examples
1Series RLC at 1 kHz
R = 10 Ω, L = 10 mH, C = 100 µF
XL = 2π×1k×0.0162.83 Ω
XC = 1/(2π×1k×100µ)1.59 Ω
Z = √(10²+61.24²)≈ 61.9 Ω
2Resonant frequency
L = 10 mH, C = 100 µF
f₀ = 1/(2π√(LC))≈ 159 Hz
At f₀: XL = XCZ = R = 10 Ω
Q = (1/10)×√(0.01/100µ)≈ 1.0
3Inductive circuit
R = 50 Ω, L = 100 mH, C = 0
f = 500 Hz
XL = 2π×500×0.1314.2 Ω
Z = √(50²+314.2²)≈ 318 Ω
4Capacitive circuit
R = 100 Ω, L = 0, C = 10 µF
f = 1 kHz
XC = 1/(2π×1k×10µ)15.92 Ω
Z = √(100²+15.92²)≈ 101.3 Ω
Frequently Asked Questions
How do I calculate RLC impedance? ▾
For series RLC: Z = √(R² + (XL − XC)²), where XL = 2πfL and XC = 1/(2πfC). For parallel: Z = 1 / √((1/R)² + (1/XC − 1/XL)²). The phase angle is φ = arctan((XL−XC)/R).
What is the resonant frequency of an RLC circuit? ▾
f₀ = 1 / (2π√(LC)). At resonance, XL = XC and net reactance is zero. Series RLC has minimum impedance (= R); parallel RLC has maximum impedance.
What does a positive or negative phase angle mean? ▾
Positive φ means the circuit is inductive (XL > XC) — voltage leads current. Negative φ means capacitive (XC > XL) — current leads voltage. At resonance φ = 0° (purely resistive).
What is Q factor and why does it matter? ▾
Q (quality factor) indicates how selective a resonant circuit is. High Q = narrow bandwidth = sharper resonance peak. Bandwidth BW = f₀ / Q. For series RLC: Q = (1/R)√(L/C).
What is the difference between impedance and resistance? ▾
Resistance (R) opposes current in DC and AC circuits and dissipates energy as heat. Impedance (Z) is the total opposition to AC current, combining resistance and reactance. Z = √(R² + X²). Unlike R, impedance depends on frequency.