SC · Laplace

First- and Second-Order Responses

A single pole location fixes the shape of the response. A first-order system is a simple exponential set by a time constant τ; a second-order system has a damping ratio ζ and natural frequency ωn that split it into over-, critically, and underdamped. Move the damping ratio and watch the response turn from ringing into a smooth settle.

Tune the damping for the fastest settle

Above is the s-plane, with × marking the poles of a second-order system. Underdamped, the poles are a complex pair off the real axis and the response overshoots on its way; overdamped, two poles sit far apart on the real axis and it crawls. Below is its step response. Move the damping ratio ζ toward 1 to find critical damping, where the two poles meet on the real axis, the fastest reach to the target with no overshoot. ωn raises the overall speed.

s-plane (× = pole)

unit step response

damping ratio ζ0.30

natural frequency ωn1.40

Pole position and behavior

ζ = 0.30 · ωn = 1.40

s = -0.42 ± j1.34

The damping ratio ζ is below 1, so the poles are a complex pair. The step response overshoots the target and rings before settling. The smaller ζ, the larger the overshoot and the longer the ringing.

Far from critical

First order: a single time constant

A first-order system H(s)=1/(τs+1) has a single pole on the real axis at s=−1/τ. Its step response is 1−e^(−t/τ), an exponential that rises smoothly toward the target with no oscillation. The time constant τ is the time to reach about 63% of the final value, and by roughly 4τ it has essentially settled. The closer the pole to the origin (larger τ) the slower it is; the farther, the faster. One pole position sets the speed.

Observefirst order y(t) = 1 − e^−t/τ

First order is a simple exponential with time constant τ.

Choose2nd-order roots s = ?

The second-order roots come from the characteristic equation.

Fill inunderdamped ζ ? ⟹ oscillates

ζ<1 gives complex poles, so it oscillates.

On your owncritical (fastest, no overshoot) ζ = ?

ζ=1 is critical, fastest with no overshoot.

Second order: a transient set by two poles

The standard second-order system H(s)=ωn²/(s²+2ζωn s+ωn²) has poles that are the two roots of the characteristic equation s²+2ζωn s+ωn²=0. Solving, s=−ζωn±ωn√(ζ²−1). When the damping ratio ζ is below 1 the term under the root is negative, so the poles become a complex pair −ζωn±jωd, where ωd=ωn√(1−ζ²) is the frequency it actually oscillates at. The pole’s real part −ζωn sets how fast the amplitude decays, the imaginary part ωd how fast it oscillates. The natural frequency ωn is the distance from the origin to the pole, a measure of overall speed, while ζ is the angle of the pole off the real axis, which sets the shape.

Over, critical, and underdamped behavior

A single damping ratio ζ decides the kind of behavior. ζ>1, overdamped, has two real roots and is slow with no overshoot; ζ=1, critically damped, is a double root and the fastest with no overshoot; ζ<1, underdamped, has complex poles and overshoots before settling. At ζ=0 the poles sit on the imaginary axis and it oscillates forever. The size of the underdamped overshoot is set by ζ alone and grows as ζ shrinks. The settling time is about 4/(ζωn), so the larger the pole’s real part, the faster it settles. So a design tunes overshoot with ζ and speed with ωn, separately.

Back to the first screen

With the damping ratio ζ small, the poles were a complex pair tilted far off the real axis, so the response overshot heavily and rang for a long time; raising ζ to 1 brought the two poles together on the real axis for the fastest settle with no overshoot. Pushing ζ higher, overdamped, split the poles and made it sluggish. Raising ωn moved the poles away from the origin and sped the same shape up wholesale. The angle of the poles off the real axis set the overshoot, the distance from the origin set the speed. The shape and speed of the response were written right into the location of the poles on the s-plane.

A first-order system H(s)=1/(τs+1) has one pole at s=−1/τ, so its step response 1−e^−t/τ is a simple exponential with time constant τ. A second-order system H(s)=ωn²/(s²+2ζωn s+ωn²) has poles s=−ζωn±ωn√(ζ²−1): ζ>1 overdamped (real roots, slow), ζ=1 critically damped (double root, fastest with no overshoot), ζ<1 underdamped (complex poles −ζωn±jωd, oscillation). Overshoot is set by ζ, settling time 4/(ζωn) by ζωn. The shape and speed of the response are exactly the pole locations.

On to the next unit

That completes the story of continuous-time signals and systems. The Laplace transform turned differential equations into algebra in s, and pole positions told us the response. But computers and digital devices cannot handle signals continuously; they grab a value at regular intervals. The first unit of the next chapter, the sampling theorem, asks when a continuous signal sampled at discrete instants can still be rebuilt exactly. The Nyquist criterion, that you must sample faster than twice the highest frequency in the signal, appears together with the aliasing that arises when you sample too slowly.