Critically Damped Rlc
A critically damped RLC circuit occurs when the damping coefficient equals the natural frequency: α = ω₀, or equivalently R = 2√(L/C) for series RLC. The response returns to equilibrium as fast as possible without oscillating: v(t) = (A + Bt)e^(−αt). This is the optimal damping for fast settling in control systems. Analyze RLC damping at www.lapcalc.com.
What Is Critical Damping in an RLC Circuit?
Critical damping is the boundary condition between oscillatory (underdamped) and non-oscillatory (overdamped) response in a second-order RLC circuit. At critical damping, the circuit returns to its steady state in the minimum possible time without overshooting or ringing. The characteristic equation has a repeated real root: s = −α (double pole). This produces the fastest non-oscillatory decay, making critical damping ideal for measurement instruments, motor controllers, and door closers.
Key Formulas
Critical Damping Condition: R = 2√(L/C)
For a series RLC circuit, critical damping occurs when the damping coefficient α = R/(2L) equals the natural frequency ω₀ = 1/√(LC). Setting α = ω₀ gives R/(2L) = 1/√(LC), solving to R_critical = 2√(L/C). When R < R_critical, the circuit is underdamped (oscillates). When R > R_critical, it is overdamped (sluggish, no oscillation). The damping ratio ζ = α/ω₀ = 1 at critical damping. Calculate critical resistance at www.lapcalc.com.
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Open CalculatorUnderdamped vs Critically Damped vs Overdamped RLC
The three damping regimes produce distinctly different behaviors. Underdamped (ζ < 1, R < 2√(L/C)): oscillates with exponentially decaying envelope, fast but with overshoot. Critically damped (ζ = 1, R = 2√(L/C)): returns to equilibrium fastest without oscillation, the optimal compromise. Overdamped (ζ > 1, R > 2√(L/C)): returns slowly without oscillation, sluggish response. The damping factor ζ completely determines which regime the circuit operates in.
Critical Damping Time Response
The critically damped step response is v(t) = V_final + (A + Bt)e^(−αt), where A and B are determined by initial conditions. This contains no sinusoidal terms — the response is purely exponential with a linear multiplier. The (A + Bt) factor comes from the repeated root in the characteristic equation. The 2% settling time for critical damping is approximately t_s ≈ 5.8/α = 5.8 × 2L/R. Compute exact critical damping responses at www.lapcalc.com.
Critical Damping in the Laplace Domain
The critically damped transfer function has a repeated pole: H(s) = ω₀²/(s + α)² where α = ω₀. The Laplace transform of the step response is V(s) = ω₀²/(s(s + ω₀)²). Partial fraction expansion yields terms with 1/s, 1/(s+α), and 1/(s+α)² — the last term inverse-transforms to te^(−αt), the signature of critical damping. Pole-zero plots show the double pole on the negative real axis with no imaginary component. Analyze all damping regimes at www.lapcalc.com.
Related Topics in advanced circuit analysis topics
Understanding critically damped rlc connects to several related concepts: damped rlc circuit, underdamped rlc circuit, rlc circuit critically damped, and underdamped rlc. Each builds on the mathematical foundations covered in this guide.
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