A system whose transfer function is G(s) = Kωn²/(s² + 2ζωn s + ωn²), where ωn is the natural angular frequency and ζ the damping ratio. For 0 < ζ < 1 the step response approaches the final value while oscillating, for ζ = 1 it settles fastest without overshoot and for ζ > 1 it approaches slowly. Mechanical spring–mass–damper systems, RLC circuits and position control loops take this form. Higher-order systems are also often approximated by their two dominant poles.
Japanese name: 二次遅れ系
Second-order system appears in 9 practice questions across 2 topics, including Control Engineering Basics, Circuit Analysis: AC, Linear Circuits and Transients.
Series RLC circuit switched onto a DC voltage · View in diagrams
The overshoot is exp(−πζ/√(1 − ζ²)) and depends on ζ alone: about 16% at ζ = 0.5 and about 4.3% at ζ ≈ 0.707 (1/√2). The overshoot peaks at π/ωd, where ωd = ωn√(1 − ζ²) is the angular frequency of the decaying oscillation. The smaller ζ is, the faster the response rises, but the larger the overshoot and oscillation.
Relation to pole positions
For 0 < ζ < 1 the poles are at s = −ζωn ± jωd, at a distance ωn from the origin, and the angle θ they make with the negative real axis satisfies cos θ = ζ. The real part ζωn sets the rate of decay, and the ±2% settling time is estimated at about 4/(ζωn). This formula is an approximation derived from the envelope of the oscillation, and the true value is often a little shorter.
Seen in the frequency response
On a Bode plot, with ωn as the corner, the gain falls at −40 dB/dec at high frequencies and the phase changes from 0° to −180°. When ζ is below about 0.707, a resonant peak appears in the gain near ωn. In a feedback system, the smaller the phase margin, the smaller the closed-loop ζ; as a rule of thumb, ζ is about one hundredth of the phase margin in degrees.
Method 1 (compare with the standard form): the characteristic equation is s² + (R/L)s + 1/(LC) = 0, and comparing it with the standard form s² + 2ζωn s + ωn² gives ωn = 1/√(LC) = 1×10⁵ rad/s and 2ζωn = R/L. From these, ζ = (R/2)√(C/L) = 25 × √(0.1×10⁻⁶/1×10⁻³) = 25 × 0.01 = 0.25. Method 2 (ratio to the critical resistance): the resistance for critical damping is 2√(L/C) = 200 Ω, so ζ = R/(2√(L/C)) = 50/200 = 0.25 as well. Since ζ < 1, the step response overshoots the final value and settles while oscillating: a damped oscillation.
For 0 < ζ < 1 the poles are a complex-conjugate pair and the response approaches the final value while oscillating (underdamped). For ζ = 1 the poles are a repeated negative real root, and the response settles fastest among responses without overshoot (critically damped). For ζ > 1 the poles are two negative real numbers, and the response approaches slowly without overshoot (overdamped). For ζ = 0 the oscillation continues without decaying, and for ζ < 0 the system is unstable.
FAQ
What damping ratio should a design aim for?
For a fast response with a little overshoot, ζ = 0.5 to 0.8 is often chosen. ζ ≈ 0.707 gives an overshoot of about 4.3% and is also the boundary below which a resonant peak appears in the frequency response. Where overshoot is not allowed, the design aims for ζ ≥ 1.
Can the second-order formulas be used for third- and higher-order systems?
If the two poles closest to the imaginary axis (the dominant poles) are much slower than the others, the response can be estimated by approximating with those two poles. When other poles or zeros are close by, the error grows, so checking with a numerical calculation is the safe approach.
Second-order system in the explanations
The overshoot of a standard second-order system is exp(−πζ/√(1 − ζ²)) and does not depend on the natural angular frequency ωn.
If θ is the angle between the negative real axis and the line from the origin to the pole in the s-plane, then ζ = cos θ, and the larger θ is, the smaller the damping ratio.
Method 2 (damping ratio): comparing with the standard form s² + 2ζωn s + ωn² gives ζ = (R/2)√(C/L), and critical damping ζ = 1 gives the same R = 2√(L/C) = 200 Ω.