Understanding RL and RLC Circuits
Learn about inductor behavior, time constants, and resonance in electrical circuits
RL Time Constant
The RL time constant τ_L = L/R determines how quickly current changes in a resistor-inductor circuit. Unlike RC circuits where voltage changes gradually, RL circuits have current that changes gradually.
τ_L = L / R
- • L = inductance (henries)
- • R = resistance (ohms)
- • τ_L = time constant (seconds)
Step Response
When voltage is suddenly applied to an RL circuit, current rises exponentially toward its final value. The inductor initially acts like an open circuit, then gradually allows more current.
i(t) = (V_s/R) × (1 - e^(-t/τ_L))
Starting from i = 0, current exponentially approaches V_s/R
RLC Natural Frequency
In a series RLC circuit, energy oscillates between the inductor (magnetic) and capacitor (electric). The natural frequency determines how fast this oscillation occurs without damping.
ω_0 = 1 / √(LC)
- • ω_0 = natural frequency (rad/s)
- • f_0 = ω_0 / (2π) = resonant frequency (Hz)
Damping Ratio ζ
The damping ratio compares actual damping to critical damping. It determines whether the circuit oscillates (underdamped), returns fastest without overshoot (critical), or decays slowly (overdamped).
ζ = R / 2 × √(C/L)
- • ζ < 1: Underdamped (oscillates)
- • ζ = 1: Critically damped
- • ζ > 1: Overdamped
Quality Factor Q
The Q factor measures how "sharp" or selective a resonant circuit is. Higher Q means narrower bandwidth and less energy loss per cycle. At resonance, voltages across L and C can be Q times the source voltage!
Q = 1 / (2ζ) = ω_0L / R
- • High Q (>10): Sharp resonance, ringing
- • Low Q (<1): Heavily damped
Inductor Behavior
Inductors store energy in their magnetic field and oppose changes in current. When current is interrupted suddenly, they generate voltage spikes (back-EMF) that can damage components. Real circuits need protection.
V_L = L × (di/dt)
Rapid current change (high di/dt) produces high voltage!
Key Time Milestones in RL Circuits
1τ_L
↑ 63.2%
↓ 36.8%
2τ_L
↑ 86.5%
↓ 13.5%
3τ_L
↑ 95.0%
↓ 5.0%
4τ_L
↑ 98.2%
↓ 1.8%
5τ_L
↑ 99.3%
↓ 0.7%
↑ Step response (% of final) | ↓ Decay (% remaining)
Educational Tool Only
This calculator uses idealized models for learning purposes. Real circuits have parasitic resistance, non-ideal components, and require proper simulation tools for actual design. Always consult qualified electrical engineers for safety-critical applications.
Frequently Asked Questions
Common questions about RL and RLC circuit analysis
The time constant τ_L = L/R determines how quickly current changes in an RL circuit. After one time constant, the current reaches about 63.2% of its final value during a step response. After 5 time constants, the circuit is approximately 99.3% complete. A larger inductance means a slower response, while a larger resistance speeds up the response.
Related Calculators
Explore more physics and electrical engineering tools
RC Circuit Time Constant
Analyze capacitor charging and discharging behavior
Ohm's Law Calculator
Calculate voltage, current, resistance, and power
Waves Calculator
Frequency, wavelength, and wave speed relationships
Simple Harmonic Motion
Analyze oscillating systems and periodic motion
Power & Efficiency
Calculate electrical power and system efficiency