1. 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,

2. 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.

3. Chapter 5

4. Chapter 5

5. Chapter 5

6. Chapter 5

7. Chapter 5

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

9. 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)

10. 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.

11. 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