Transfer Functions and Block Diagrams

A transfer function $G(s) = \dfrac{Y(s)}{U(s)}$ (with zero initial conditions) captures a linear system's dynamics. Examples:

• First-order: $G(s) = \dfrac{K}{\tau s + 1}$ (thermal system, RC circuit).
• Second-order: $G(s) = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$ (mass–spring–damper, RLC).
• DC motor speed: approximately second-order from voltage to speed.

Block diagram rules: series blocks multiply; parallel blocks add; a feedback loop reduces to $G/(1 + GH)$. Signal-flow graphs with Mason's gain formula handle more complex systems.