Page 73 - Biosystems Engineering

54 Chapter Two

The phase shift is –90°, which means that the output lags behind on
the input with 90°, regardless of the frequency. The Bode plot of an
integral term transfer function is illustrated in Fig. 2.11.

Derivative Term (Zero at the Origin)
ω
Gs() = sG j ) = jω (2.63)
;
(
The magnitude can then be calculated as
ω
|(Gjω )| ω or |(Gjω )|= 20 log ( ) dB (2.64)
=
10
This means that the magnitude increases by 20 dB for a tenfold
increase in frequency. With the frequency ω plotted on a logarithmic
scale, the magnitude is represented by a straight line of slope 20 dB per
decade of frequency, which is passing through 0 dB for ω = 1 rad/s.
The phase shift for this derivative term is
⎡ ω⎤
ω
=
=
ω
Gj ) arg[ Gj )] arctan ⎢ ⎥ =+90° (2.65)
(
(
⎣ ⎦
0
The phase shift is 90°, which means that the output would lead the
input by 90° regardless of the frequency. The Bode plot of a derivative
term transfer function is illustrated in Fig. 2.11.
Simple Lag (Real Pole First-Order System)
ω
Gs() = 1 ; Gj ) = 1 (2.66)
(
1 + s τ 1 + jωτ
The magnitude can then be calculated as
1
|(Gjω )|= or |(Gjω )|= 20 log 1 + ω τ dB (2.67)
22
22
1 + ωτ 10
This frequency dependency of the magnitude is often simplified by a
linear asymptotic approximation, using the following asymptotes:

)
for ωτ << 1:| G jω ≈ 20log ( 1) = 0 dB
|
(
10
(2.68)
G(
for ωτ >> 1:| jω)|≈−20 log ( ωτ) dB
j
10
The latter is a straight line of slope –20 dB per decade, which inter-
sects the 0 dB line when ωτ = 1 (i.e., at ω =1/τ). This frequency is
termed the break-point, or corner frequency.
The phase shift for this simple lag is
ωτ⎤
τ
=
ω
ω
Gj ) arg[ Gj )] = − arctan ⎡ ⎢ ⎣ 1 ⎥ ⎦ =− arctan(ωτ) (2.69)
(
(