Analyze RLC series and parallel circuits. Calculate impedance, phase, resonant frequency, Q factor, damping ratio, and frequency response with voltage distribution tables.
The RLC circuit — containing a resistor (R), inductor (L), and capacitor (C) — is one of the most important building blocks in electrical engineering. These three passive components create a second-order system capable of resonance, filtering, and oscillation. Understanding RLC behavior is essential for designing audio crossovers, radio receivers, power supply filters, and control systems.
In a series RLC circuit, all three components share the same current, and their voltages add as phasors. In a parallel RLC circuit, all three share the same voltage, and their currents add as phasors. The resonant frequency — where inductive and capacitive reactances cancel — is identical for both configurations, but the impedance behavior at resonance is opposite: minimum for series, maximum for parallel.
This calculator provides complete RLC analysis including impedance magnitude and phase, resonant frequency, quality factor, damping ratio classification, voltage distribution across components, and a detailed frequency response table showing how the circuit behaves across a range of frequencies.
RLC circuit analysis involves complex numbers, trigonometric functions, and cascading formulas that are tedious to compute by hand. This calculator instantly determines impedance, phase, resonance, Q factor, and damping characteristics for both series and parallel configurations. The frequency response table and damping visualization help you intuitively understand circuit behavior without plotting Bode diagrams manually.
Series RLC Impedance: Z = R + j(X_L − X_C) |Z| = √(R² + (X_L − X_C)²) φ = arctan((X_L − X_C)/R) Parallel RLC Admittance: Y = 1/R + j(1/X_C − 1/X_L) |Z| = 1/|Y| Resonant Frequency: f₀ = 1/(2π√(LC)) Q (series) = (1/R)√(L/C) Q (parallel) = R√(C/L) Damping Ratio (series): ζ = R/(2√(L/C))
Result: |Z| = 1541 Ω at −88.1°
At 1 kHz, the inductive reactance X_L = 62.8 Ω and capacitive reactance X_C = 1592 Ω. The net reactance is −1529 Ω (capacitive), giving |Z| = √(50² + 1529²) ≈ 1541 Ω at a phase angle of −88.1°. The resonant frequency is about 5.03 kHz where X_L = X_C.
Impedance is the AC generalization of resistance. In an RLC circuit, impedance is a complex quantity with a real part (resistance) and an imaginary part (reactance). The magnitude |Z| determines how much the circuit opposes current flow, while the phase angle determines the timing relationship between voltage and current. Positive phase means voltage leads current (inductive), negative means voltage lags current (capacitive), and zero phase means they are in sync (resistive).
When an RLC circuit is subjected to a sudden change (step input), its transient response depends on the damping ratio. Underdamped circuits ring at the natural frequency with exponentially decaying amplitude — this is useful in oscillators but problematic in power supplies. Critically damped circuits settle fastest without overshooting, making them ideal for control systems. Overdamped circuits are sluggish but stable.
Series RLC circuits are natural bandpass filters when the output is taken across R, and notch filters when taken across L+C. By cascading multiple RLC stages with staggered resonant frequencies, you can create wideband filters with steep roll-off. Active filter designs using op-amps can simulate RLC behavior without bulky inductors, which is why active filters dominate at audio frequencies.
In a series RLC, components share the same current and impedance is minimum at resonance. In a parallel RLC, components share the same voltage and impedance is maximum at resonance. The resonant frequency is the same for both.
The damping ratio ζ describes how oscillations decay in an RLC circuit. When ζ < 1 (underdamped), the circuit oscillates with decaying amplitude. At ζ = 1 (critically damped), it returns to equilibrium fastest without oscillating. When ζ > 1 (overdamped), it returns slowly without oscillation.
Bandwidth = f₀/Q. A high Q means narrow bandwidth (selective filtering), while a low Q means wide bandwidth. For example, Q = 10 at f₀ = 1 MHz gives a 100 kHz bandwidth.
Passive RLC circuits cannot amplify power, but at resonance, the voltage across L or C in a series circuit can exceed the source voltage by a factor of Q. This is called voltage magnification and is useful in antenna matching.
Power factor = cos(φ) = R/|Z|. It ranges from 0 (purely reactive, no real power) to 1 (purely resistive, all real power). At resonance, the power factor equals 1 because the reactive components cancel.
First set f₀ = 1/(2π√(LC)) to fix the product LC. Then for series RLC, set R = (1/Q)√(L/C) to achieve the desired Q. You have one degree of freedom — choose L (or C) based on practical component availability, then calculate the other.