First Order Systems

Transfer function

H (s) = \frac{b_{0}}{s + a_{0}}, a_{0} \neq 0

Frequency response:

\begin{aligned} Y (s) & = H (s) U (s) \\ = \frac{b_{0}}{s + a_{0}} \times \frac{1}{s} \\ = \frac{b_{0}}{a_{0}} (\frac{1}{s} - \frac{1}{s + a_{0}}) \end{aligned}

Taking the Laplace transform:

y (t) = \frac{b_{0}}{a_{0}} (1 - e^{- a_{0} t}), t \geq 0

Rise time is the time required to go from 10% to 90% of the final value.

t_{r} = \frac{l n (9)}{a_{0}}

Settling time is the time required to get within 2% of the final value and stay there.

t_{s} \approx \frac{3.91}{a_{0}}

Time constant $\frac{b_0}{a_0}(1-e^{-1})$ , i.e. roughly 63% of the final value.

τ = \frac{1}{a_{0}}

Bandwidth is the frequency where,

| H (j ω_{B W}) | = \frac{1}{\sqrt{2}} H (0)

For a first order system,

ω_{B W} = a_{0}