Page 75 - Biosystems Engineering

56 Chapter Two

Quadratic Lag (Second-Order System with a Pair of Complex
Conjugate Poles)

ω 2 ω 2
Gs() = n ; Gj ) = n (2.72)
ω
(
s + ζω s + ω 2 ω 2 − ω 2 + ζω j ω
2
2
2
n n n n
The magnitude for this transfer function is given by the following
expression:
|(Gjω )|= 1
⎛ ω 2 ⎞ 2 ⎛ ω ⎞ 2
⎜ ⎜1 − 2 ⎟ ⎟ +⎜2 ζ ⎟
⎝ ω n⎠ ⎝ ω n ⎠
or (2.73)

⎛ ω 2 ⎞ 2 ⎛ ω ⎞ 2
|(Gjω )|=−20 log ⎜1 − ⎟ +⎜2 ζ ⎟ dB
10 ⎜ ω 2 ⎟ ω
⎝ n⎠ ⎝ n ⎠
This frequency dependency of the magnitude can again be simplified by
a linear asymptotic approximation, using the following asymptotes:

ω
G jω ≈ −20log (
for << 1:| ( )| 1) = 0 dB
ω 10
n
(2.74)
ω ⎛ ω ⎞ ⎛ ω ⎞ ⎞
2
for >> 1: ||(Gjω )|≈−20 log ⎜ ⎟ =−40 log ⎜ ⎟ dB
ω 10 ⎝ ω n ⎠ 10 ⎝ ω n ⎠
2
n
The straight-line approximation for magnitude is thus a line at 0 dB
for low frequencies, changing to a line of slope –40 dB/decade at
the break point where the frequency ω equals the undamped natu-
ral frequency ω . The shape of the true curve depends on the value
n
of the damping factor ζ. For ω = ω and ζ = 0, | (Gjω= ∞ dB . The
)|
n
slope of –40 dB/decade for frequencies higher than the undamped
natural frequency is also called a roll-off rate of 40 dB/decade.
The phase shift for this quadratic lag system is given by the fol-
lowing expression:
⎡ ω ⎤
⎢ ζ 2 ω ⎥
ω =
ω
Gj ) arg[ Gj )] = − arctan ⎢ n ⎥ (2.75)
(
(
⎢ ω ⎥
2
⎢ 1 − 2 ⎥
⎣ ω n ⎦ ⎦