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Physics · Glossary

What is First-order linear system?

Definition 7.1 University Physics — Year 1 · Chapter 7 — Transient Regimes: First and Second Order

A quantity x(t)x(t) obeys a first-order linear equation with constant coefficients when

τ  ⁣dx ⁣dt+x=x∞,\tau\,\frac{\dd x}{\dd t} + x = x_\infty ,

with τ>0\tau > 0 the time constant and x∞x_\infty a constant (the value imposed by the source). Its steady state is x=x∞x = x_\infty; the transient regime is the approach to it from the initial value x(0)=x0x(0) = x_0.

The first-order step response from x_0 = 0: the initial tangent meets the asymptote at t =; 63\% of the way at , 95\% at 3. Every first-order transient is this curve, stretched by and shifted by x_0 and x_∈fty.
The first-order step response from x0=0x_0 = 0: the initial tangent meets the asymptote at t=τt = \tau; 63%63\% of the way at τ\tau, 95%95\% at 3τ3\tau. Every first-order transient is this curve, stretched by τ\tau and shifted by x0x_0 and x∞x_\infty.
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