082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

672 Chapter 21 The Laplace transform
We note that ( ( )
) jω
jω −z 1 can be written as −z1 +1 . Applying this process
−z 1
toeach bracket inthe numerator and denominator yields
( )( )···( )
−z1 −z2 −zm
G (jω) =K ( )( ) )
−p1 −p2 ···(
−pn
( ) ( ) ( )
jω jω jω
+1 +1 ... +1
−z
× 1
−z 2
−z
( ) ( ) ( m
)
jω jω jω
jω +1 +1 ... +1
−p 1
−p 2
−p n
The Bode plot isalogarithmic plot of20 log 10
|G(jω)|.So we wishtoplot
( ) ( ) )
−z1 −z2 ···(
−zm
∣ ∣∣∣∣
20log 10
|G (jω)| = 20log 10
K ( )( ) )
−p1 −p2 ···(
−pn
( ) ( ) ( )
jω jω jω
+1 +1 ... +1
−z
· 1
−z 2
−z
( m ) ( ) ( )
jω jω jω
jω +1 +1 ... +1
∣
−p 1
−p 2
−p n
Using the laws of logarithms (page 86) this can be rewritten as
20log 10
|G (jω)|
∣ ∣∣∣∣
( ) ( ) )
−z1 −z2 ···(
−zm = 20log 10
K ( )( ) )
−p1 −p2 ···(
−pn ∣ +20log jω
10 ∣ +1
−z ∣
1
∣ ∣∣∣ jω
+20log 10
+1
−z ∣ +···+20log jω
10∣
+1
2
−z ∣
m
∣ ∣∣∣ jω
−20log 10
|jω| −20log 10
+1
−p ∣
1
∣ ∣∣∣ jω
−20log 10
+1
−p ∣ −···−20log jω
10∣
+1
2
−p ∣
n
Inthisform the contribution ∣ of individual poles and zeros can bestudied.
∣∣∣∣
( ) ( ) )
−z1 −z2 ···(
−zm Theterm20log 10
K ( )( ) )
−p1 −p2 ···(
−pn ∣ isconstant.Ithasnodependenceon
frequency, ω.
Theterm20log 10
|jω|hasavalueof0at ω = 1.At ω = 10ithasavalueof20,at
ω = 100 it has a value of 40, and so on. At ω = 0.1 it has a value of −20, at 0.01, a
value of −40, and so on. If we plot this term by itself using a logarithmic frequency
scaleitwouldthereforebeastraightlinepassingthroughthepointwhere ω = 1with
a gradient of 20 dBfor ∣ each decade, or factor of 10, increase infrequency.
∣∣∣ jω
Theterm20log 10
+1
−z ∣ canbeexaminedbyconsideringtheinfluenceatdif-
1
∣ ∣ ferent frequencies. If
ω ∣∣∣ ∣∣∣ jω
∣ ≪ 1, then 20log
z 10
+1
1
−z ∣ ≈ 20log 10
|1| = 0. So the
1
termhas very littleinfluence.