Page 277 - Dynamics and Control of Nuclear Reactors

APPENDIX D Laplace transforms and transfer functions 279

Exercises

D.1. Determine the inverse Laplace transforms of the following:
s +1
ð
ð s +2Þ s +3Þ
s +1
2
ð s +2Þ
s +1
s +4
2
D.2. The Laplace transform of a time function x(t) is given by
6
XsðÞ ¼
ð
ss +1Þ s +3Þ
ð
(a) Determine the time function x(t).
(b) Calculate the value of x(t) for t¼0.
(c) Calculate the value of x(t) as t goes to infinity (same as the steady-state value).
D.3. A time domain function f(t) is given by

ftðÞ ¼ sin tðÞ + cos tðÞ + e t

Determine the Laplace transform F(s) of f(t), and simplify your answer
in the form of a ratio of two polynomials in s.

D.4. Consider the following transfer function of a second order system:

YsðÞ 2
GsðÞ ¼ ¼
2
XsðÞ s + s +2
(a) Calculate the roots (poles) of the denominator polynomial.
(b) If the input x(t) is a unit step function, determine the response y(t) to this input.
You may use the method of residues or partial fraction.

(c) Make a plot of this step response. You may use the MATLAB command step
(sys) where ‘sys’ is defined by the transfer function G(s). Comment on the
characteristics of this second order system response.
(d) Is this system stable or unstable? Explain.
D.5. A system has a transfer function given by
1
GsðÞ ¼
s +1