Page 295 - Dynamics and Control of Nuclear Reactors

APPENDIX E Frequency response analysis of linear systems 297

Exercises

E.1 Construct asymptotic Bode plots for the open-loop transfer functions given
below. Verify your plots by comparison with the Bode plots generated by
using MATLAB function “bode”.
ð
40 s + 2Þ
(a) GsðÞ ¼ 2
s s +8ð Þ s +10Þ
ð
9
(b) GsðÞ ¼ 2
ð
ð s+1Þ s+3Þ
(c) GsðÞ ¼ 10 e s
s +10
E.2 Consider the following frequency response function
1
ð
GjωÞ ¼
2
1+ jω=4 ω=4Þ
ð
Construct a log-log plot of jG(f)j vs. f , for a frequency range of 0.01–10Hz. (Sub-
stitute ω¼2πf, and obtain an expression for jG(f)j; compute and plot jG(f)j as a func-
tion of f).
Estimate the frequency at which a peak is seen in the jG(f)j graph.
E.3 Consider the following transfer function of a linear system.
1
GsðÞ ¼ , where s is the Laplace variable; the unit of time is ‘second’.
ð s +0:2Þ
p ffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
Hint: If z¼x+jy, is a complex number, the magnitude of z is z jj ¼ x + y , and

the phase angle is φ ¼ tan 1 y .
x
(a) State the frequency response function of G(s).
(b) Compute the magnitude of the frequency response function G(jω) for
ω¼0rad/s.
(c) Compute the phase angle (degrees) of G(jω) for ω¼0rad/s.
(d) Compute the magnitude of G(jω) for ω¼0.2rad/s. Simplify your answer.
(e) Compute the phase angle (degrees) of G(jω) for ω¼0.2rad/s. Simplify
your answer.
(f) Make a Bode plot of G(jω) using the MATLAB function “bode”.
(g) Determine the break frequency (rad/s) of the device defined by the above
transfer function G(s).
(h) If this device is a resistance temperature detector (RTD), determine its time
constant (sec).
(i) What is the meaning of the magnitude of the frequency response function,
G(jω), of a linear system at any frequency, ω?

E.4 The neutron power to reactivity transfer function model of a BWR has the fol-
p ffiffiffiffiffiffiffi
lowing zeros and poles. [j ¼ 1].
Zeros: 0.03, 0.18+j0.27, 0.18 – j0.27.
Poles: 0.25, 21.7, 0.045+j0.32, 0.045 – j0.32.