RC/RL/RLC Circuit Analyzer

Analyze transient and AC steady-state behavior of RC, RL, and RLC circuits. Four modes cover charging and discharging capacitors, inductive current rise, damped oscillations with underdamped, critically damped, and overdamped classification, plus impedance resonance with quality factor and bandwidth. Interactive JSXGraph plots and SVG circuit diagrams update in real time.

Circuit Diagram

Controls

Circuit Mode

Presets

Operation

Source Voltage (V₀)10 V

Resistance (R)1.0k Ω

Capacitance (C)1.0 μF

Results

Time Constant (τ)

1.00 ms

V at 5τ

9.933 V

Step-by-Step

\tau = RC = (1000)(0.000001) = 0.001 \text{ s}

V_C(t) = V_0\left(1 - e^{-t/\tau}\right) = 10\left(1 - e^{-t/0.001}\right)

\text{At } t = 5\tau = 0.005 \text{ s: } V_C \approx 0.993 \times V_0

Voltage & Current vs Time

Voltage Current

Reference Guide

RC Circuits

An RC circuit contains a resistor and capacitor in series. The time constant $\tau = RC$ sets how fast the capacitor charges or discharges.

V_C(t) = V_0\left(1 - e^{-t/\tau}\right) \quad \text{(charging)}

V_C(t) = V_0\,e^{-t/\tau} \quad \text{(discharging)}

After $5\tau$ , the capacitor reaches approximately 99.3% of its final value. The current always starts at $V_0/R$ and decays exponentially.

RL Circuits

An RL circuit contains a resistor and inductor in series. The time constant $\tau = L/R$ governs how quickly current rises to its steady-state value.

I(t) = \frac{V_0}{R}\left(1 - e^{-t/\tau}\right)

The inductor opposes changes in current. At $t = 0$ , the inductor acts like an open circuit (no current). At steady state, it acts like a wire with current $V_0/R$ .

RLC Resonance

An RLC series circuit can oscillate, with the damping ratio $\zeta$ determining the behavior.

\zeta = \frac{R}{2}\sqrt{\frac{C}{L}}, \quad \omega_0 = \frac{1}{\sqrt{LC}}

Underdamped (ζ < 1) oscillates with decaying amplitude. Critically damped (ζ = 1) returns to equilibrium fastest without oscillating. Overdamped (ζ > 1) returns slowly with no oscillation.

Impedance and Phase

In AC analysis, each component has a frequency-dependent impedance. The total series impedance combines resistance and reactance.

|Z| = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}

At resonance $\omega_0 = 1/\sqrt{LC}$ , the reactance cancels and $|Z| = R$ (minimum impedance, maximum current). The quality factor $Q = \omega_0 L/R$ measures the sharpness of the resonance peak.