First-Order System Chapter 5 The standard form for a first-order TF is: where: Chapter 5 Consider the response of this system to a step of magnitude, M: Substitute into (5-16) and rearrange,

Chapter 5 Take L-1 (cf. Table 3.1), Let steady-state value of y(t). From (5-18), t ___ 0 0 0.632 0.865 0.950 0.982 0.993 Chapter 5 Note: Large means a slow response.

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 5

Chapter 5 For a sine input (1st order process) output is... By partial fraction decomposition,

Chapter 5 Inverting, note: f is not a function of t but of t and w. this term dies out for large t Chapter 5 note: f is not a function of t but of t and w. For large t, y(t) is also sinusoidal, output sine is attenuated by… (fast vs. slow w)

Integrating Process Chapter 5 Not all processes have a steady-state gain. For example, an “integrating process” or “integrator” has the transfer function: Chapter 5 Consider a step change of magnitude M. Then U(s) = M/s and, L-1 Thus, y(t) is unbounded and a new steady-state value does not exist.

Chapter 5 Common Physical Example: Consider a liquid storage tank with a pump on the exit line: Assume: Constant cross-sectional area, A. Mass balance: Eq. (1) – Eq. (2), take L, assume steady state initially, For (constant q), Chapter 5