Page 69 - Biosystems Engineering

50 Chapter Two

Sinusoidal steady-state transfer relation G(jω) is a complex function of ω
that can be expressed in terms of a real and an imaginary part; that is,
Gj ) = Re[( ω jIm[( ω (2.44)
ω
Gj )] +
(
Gj )]
Because a complex function can be represented in polar form, Eq. (2.44)
can be rewritten as
ω
=
ω
ω
(
(
Gj ) | Gj )| e jG j ) (2.45)
(
Using the expression in Eq. (2.45), Eq. (2.43) becomes
1 jG j )
ω
A = |( ω e (2.46)
(
Gj )|
1 j 2
Similarly, to obtain A , both sides of Eq. (2.40) are multiplied by (s + jω),
2
and the Laplace variable s is replaced by –jω:
A = ⎡( s + j ) (ω G s)ω ⎤ = ⎡ Gs ()ω⎤ ⎤
2 ⎢ s + ω 2 ⎥ ⎢ s − jω ⎥
2
⎣ s ⎦ =− jω ⎣ ⎦ =−s jω
G( −jω) 1
= =− |( e − jG jω) (2.47)
(
Gjω)|
− j 2 j 2
Using Eqs. (2.46) and (2.47), Eq. (2.6) is rewritten as
|(jω
+
j ω
+
j
⎣
yt
[( )] t→∞ = AG j 2 )| { e j ⎡ ω t G ( jω ) ⎤ ⎦ − e − ⎡ ⎣ t G ( jω) ⎤ ⎦ }
= AG jω)|sin( ω t + G jω) ) (2.48)
(
|(
Because G(jω) is a complex function with a numerator and denomina-
tor, Eq. (2.44) is also rewritten as
ω
Nj )] +
Nj ) Re[( ω jIm[( ω
(
Nj )]
Gj ) = = (2.49)
ω
(
ω
ω
D
(
(
Dj ) Re[ Djω)] + j Im[ (j ω)]
ω
Magnitude |(Gjω and phase Gj ) then become
(
)|
Nj )]} +{
Nj )]}
{ Re[( ω 2 Im[( ω 2 2
ω
=
ω
mag Gj )] | Gj )|= (2.50)
[(
(
{ Re[(Djω } +{ Im[(Djω } 2
2
)]
)]
( ω
( ω =
phase Gj )] = Gj ) arg[ Gj )]
( ω
[
⎛ Im[ Nj ( ω))]⎞ ⎛ ⎛ Im[ (Djω )]⎞
= arctan ⎜ ⎝ Re[ ( )]⎠ ⎟ − arctan ⎜ ⎝ Re[ (Djω )]⎠ ⎟ (2.51)
Njω