Second-Order Step Response: Overdamped, Critically, Underdamped

What defines the step response of a second-order system?

A second-order system's step response is fully determined by two numbers: natural frequency ωₙ and damping ratio ζ. Every mass-spring-damper, RLC filter, and PID-controlled servo loop that can be reduced to a second-order transfer function inherits its entire transient personality — fast or slow, smooth or ringing — from these two parameters alone.

Standard form: natural frequency ωₙ and damping ratio ζ

The canonical second-order transfer function is:

G(s) = ωₙ² / (s² + 2ζωₙs + ωₙ²)

ωₙ, in rad/s, sets the time scale of the response — roughly how fast the system would oscillate if there were no damping at all. ζ, dimensionless, sets the shape — whether the system oscillates, settles smoothly, or crawls. Everything downstream (overshoot, settling time, ringing frequency) is algebra on these two values.

Pole location in the s-plane as a function of ζ

The denominator's roots — the system's poles — sit at:

s = -ζωₙ ± ωₙ√(ζ² - 1)

When ζ² > 1 the square root is real, giving two real poles on the negative real axis. When ζ² < 1 it's imaginary, giving a complex-conjugate pair. When ζ = 1, the two poles collide at s = -ωₙ. The step response is just the inverse Laplace transform of G(s)/s, so the pole locations aren't an abstraction — they are literally the exponents in the time-domain solution.

Why step input reveals system dynamic behavior

A step input excites every mode of the system simultaneously and holds the excitation constant afterward, which makes it the cleanest way to see how a system moves from one equilibrium to another. Because the input itself has no dynamics of its own (it's a constant for t > 0), anything you see in the output — overshoot, ringing, lag — comes from the plant's poles, not from the test signal. That's why step response, rather than ramp or impulse response, is the standard bench test for tuning loops.

How does damping ratio ζ determine system behavior?

Damping ratio ζ determines step response shape because it directly controls whether the system's poles are real (no oscillation possible) or complex (oscillation guaranteed). There are three regimes, split exactly at ζ = 1.

Overdamped case: ζ > 1 (two real negative poles)

Two distinct real poles, s₁ and s₂, both negative. The response is a sum of two decaying exponentials with no sinusoidal term, so it cannot overshoot the final value — it approaches asymptotically, slowed by whichever pole is closer to the origin.

Critically damped case: ζ = 1 (repeated real pole)

A single repeated pole at s = -ωₙ. The response is (1 + ωₙt)e^(-ωₙt) scaled to the step size — still no oscillation, but it reaches steady state faster than any overdamped case with the same ωₙ.

Underdamped case: ζ < 1 (complex conjugate poles)

Complex poles at s = -ζωₙ ± jωₙ√(1-ζ²) produce a damped sinusoid riding on an exponential envelope. The imaginary part is the oscillation frequency (the damped natural frequency ωd); the real part sets how fast that oscillation decays. This is why the underdamped case exhibits overshoot and oscillation: the complex pole pair mathematically forces a sine/cosine term into the solution, and that term crosses the final value before settling.

Relationship between ζ and pole locations

Geometrically, all poles for a fixed ωₙ lie on a circle of radius ωₙ in the s-plane. ζ = cos(θ), where θ is the angle between the pole vector and the negative real axis. θ = 0° is critically/overdamped (poles on the real axis), θ approaching 90° is a nearly undamped oscillator. This geometric view is the fastest way to reason about root-locus tuning without running a simulation.

What is the time-domain response for each damping mode?

Overdamped: slow, monotonic rise without overshoot

c(t) = 1 - (s₂e^(s₁t) - s₁e^(s₂t))/(s₂ - s₁), normalized to a unit step. No sign reversal in the derivative, so the curve is monotonic — useful whenever overshoot would be physically dangerous (overcurrent in a motor, over-pressure in a valve).

Critically damped: fastest approach to steady-state without overshoot

Among all non-oscillating responses, ζ = 1 reaches 2% of final value in the least time. Push ζ below 1 and you trade that margin for speed at the cost of overshoot; push ζ above 1 and you only get slower.

Underdamped: overshoot followed by oscillation and decay

The envelope e^(-ζωₙt) decays geometrically while the system rings at ωd = ωₙ√(1-ζ²). Lower ζ means a higher, longer-lived overshoot; ζ in the 0.4–0.7 range is the usual sweet spot for control loops that need speed but can't tolerate wild ringing.

Settling time and rise time trade-offs across modes

How do you measure key performance metrics from step response?

Peak overshoot percentage (%OS) in underdamped case

%OS = 100 · e^(-ζπ/√(1-ζ²)). For ζ = 0.5, that's about 16.3%; for ζ = 0.707, about 4.3%. This formula is the one you invert when a spec says "overshoot must be under 10%" — solve for the minimum ζ first, then pick ωₙ for speed.

Settling time Ts: when response stays within ±2% band

Ts ≈ 4/(ζωₙ) for the common ±2% criterion (use 3/(ζωₙ) for ±5%). Because Ts depends on the product ζωₙ — the real part of the pole — two very different (ζ, ωₙ) pairs can give identical settling times with very different overshoot and ringing counts.

Rise time Tr: 10% to 90% crossing

For underdamped systems a common approximation is Tr ≈ 1.8/ωₙ (near ζ ≈ 0.5–0.8); for overdamped and critically damped systems there's no closed form and you read it off the simulated or measured curve directly.

Peak time Tp and natural frequency relation

Tp = π/ωd = π/(ωₙ√(1-ζ²)) — the time at which the derivative of the step response first hits zero. Measuring Tp and %OS off an oscilloscope trace lets you back-calculate both ζ and ωₙ from a single step test, without ever writing the transfer function down first. Full derivations of these formulas, plus Ess (steady-state error), are covered in Performance Specs: %OS, Ts, Tr, Tp, Ess (Steady-State Error).

Why choose one damping mode over another in design?

The right damping mode depends on which failure mode costs you more: a slow response or an overshooting one.

Overdamped: robustness to noise and overshoot rejection

Useful in thermal loops, battery charge controllers, or anywhere an overshoot (overheating a sample, overcharging a cell) is a hard limit that must never be crossed, even under a fast load step or sensor glitch.

Critically damped: optimal speed without oscillation

The default choice for positioning systems (camera gimbals, linear stages) where you want the fastest possible move that still arrives without hunting back and forth.

Underdamped: faster transient response at cost of ringing

Chosen when raw speed matters more than overshoot — e.g., a current loop nested inside a slower outer voltage loop, where a few percent of ringing settles out before the outer loop even notices.

Real-world trade-offs: stability margin vs. closed-loop bandwidth

Pushing ζ down increases closed-loop bandwidth (faster Tr) but erodes phase margin in the open-loop sense and raises the resonant peak — the system becomes more sensitive to gain variation and more prone to actual instability if the real plant has unmodeled poles your second-order approximation ignored.

How do poles and zeros shape the frequency response?

Underdamped resonance peak near ωₙ

The closed-loop magnitude response of an underdamped system peaks near ω ≈ ωₙ√(1-2ζ²) (valid for ζ < 0.707), with peak magnitude Mp = 1/(2ζ√(1-ζ²)). Below ζ ≈ 0.707 this peak is visible; above it, the response rolls off monotonically with no bump.

Quality factor Q and its link to damping ratio

Q = 1/(2ζ) is the same number electrical engineers know from RLC filters — high Q means a sharp resonance and a ringy step response, low Q means a broad, damped roll-off.

Overdamped roll-off without peaking

Overdamped and critically damped systems have Q ≤ 0.5 and show pure low-pass roll-off with no magnitude peak, which is why they're preferred ahead of anything sensitive to resonant excitation.

Connection to steady-state error and disturbance rejection

A higher resonant peak amplifies disturbances and sensor noise that happen to sit near ωₙ — directly relevant to how much high-frequency noise makes it into your plant output, a topic covered in Noise Amplification: T(s) at High Frequency. Steady-state error, by contrast, is set by the system's DC gain and type, not by ζ — a system can be heavily underdamped and still have zero Ess, or overdamped with a large one.

What happens in practical circuits and control loops?

RLC circuit example: L, R, C map to ωₙ and ζ

For a series RLC circuit with the capacitor voltage as output, ωₙ = 1/√(LC) and ζ = (R/2)√(C/L). A circuit with L = 10 mH, C = 100 nF gives ωₙ = 1/√(10e-3 × 100e-9) ≈ 31,623 rad/s (about 5.03 kHz). To hit ζ = 0.707 you'd need R = 2ζ√(L/C) ≈ 2(0.707)√(10e-3/100e-9) ≈ 447 Ω — pull the actual inductor's ESR and capacitor tolerance from their datasheets before trusting this number on a board.

Feedback control: how proportional and derivative gain tune damping

In a PID loop, proportional gain Kp raises ωₙ (pushes the closed-loop pole frequency up) while derivative gain Kd raises ζ directly by adding damping torque proportional to the error's rate of change. Tuning by hand often means raising Kp for speed, then raising Kd until overshoot drops to the target %OS — a direct, physical application of the formulas above.

Measurement noise vs. damping mode selection

Derivative terms amplify high-frequency sensor noise, so an underdamped design that relies heavily on Kd for its ζ can trade overshoot rejection for a noisier control signal — a well-known conflict between transient shaping and noise immunity in the same loop.

Saturation and nonlinearity effects on transient response

Actuator saturation (a motor driver hitting its current limit, a PWM duty cycle clamping at 100%) breaks the linear second-order model during the fastest part of the transient, usually making the real overshoot larger and the real settling time longer than the ζ/ωₙ formulas predict — always verify with a bench step test, not just the linear model.

Related reading

• First-Order Step Response: Rise Time, Settling Time — comparing dominant-pole behavior.