Page 280 - Dynamics and Control of Nuclear Reactors

282 APPENDIX E Frequency response analysis of linear systems

Consider an input δx(t) of the form
δxtðÞ ¼ Αsin ωtðÞ (E.3)
As shown in Table D.1 in Appendix D, the Laplace transform of δx(t)is
Αω
δXsðÞ ¼ (E.4)
s + ω 2
2
A¼amplitude of the sinusoidal function.
ω ¼ frequency rad=secondÞof sin ωtÞ:
ð
ð
Substituting for δX(s), Eq. (E.2) becomes.
Αω
δYsðÞ ¼ GsðÞ (E.5)
s + ω 2
2
If G(s) is stable and has n distinct poles, p i , one may write the inverse Laplace trans-
form, δy(t), as follows:
Aω
δytðÞ ¼ k 1 e p 1 t + k 2 e p 2 t + … + k n e p n t + inverse Laplace transform of (E.6)
2
s + ω 2
The time-domain terms corresponding to the poles of G(s) go to zero as t!∞,
because all the poles have negative real parts in a stable system. Thus, for the
1
steady-state response, it is sufficient to consider the poles of s + ω 2 . Since these are
2
given by s¼ jω, δY(s) is expressed as (we are not interested in the terms containing
the poles of G(s)).
GsðÞΑω
δYsðÞ ¼ (E.7)
ð s + jωÞ s jωÞ
ð
The inverse Laplace transform of Eq. (E.7) is as follows:
G jωÞΑωe jωt GjωÞΑωe jωt
ð
ð
δytðÞ ¼ + (E.8)
ð 2jωÞ 2jω
G(jω) is a complex number and may be expressed using its magnitude and phase
angle:
ð
ð
j
GjωÞ ¼ Gjωjexp jϕωðÞÞ (E.9)
ð
where
n o 1=2
2 2
½
ð
ð
j GjωÞj ¼ ½ ReGjωފ +ImGjωފ (E.10a)
ð
ð
ð
tan φωðÞ ¼ ImGjωÞ=ReGjωÞ (E.10b)
This complex plane representation is illustrated in Fig. E.2.
The time dependent response then becomes
jφ jωt
Α GjωÞje e Α GjωÞje jφ jωt
e
ð
j
ð
j
δytðÞ ¼ (E.11)
2j 2j
Or